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30600 04 繖 7 縺6ｧ80ｷ3 ｯ0ｷ3 ｪ ｭ ｮ07 縺00 06 ｭ0503 ｭ ｯ 70ｷ300 7 76ｧ805 ｮ0705 ｮ ｯ0ｷ3 x. 070303 罍陦ｧ0 0503 0503 09 ｧ 0 04 縺0700308 ｧ 0 040003 ﾀ0903 ﾀ05 ｧ. 7 ﾀ ﾀ0503070 ｧ06 00607 閨06 閨0406007 0503 09 0 ｧ. 7 ﾀ ﾀ0503070 ｧ06 d 00607 閨06 0503 09 0 ｧ A(x ) ｧ B(x ) 0 03 ﾀ ｧ: d = jx, x j = (x, x ) : () : 6 50600 ｧ ｧ0 AB ( 00050603 ｧ 06 ﾀ09 07) 0 04 04000600030503 0305 縺06 09 0 03 ﾀ ｧ: AB = x, x : () 3. 7 ﾀ ﾀ0503070 ｧ06 d 00607 閨060503 09 0 ｧ A(x ; y ) ｧ B(x ; y ) 0 040003 ﾀ- 0903 ﾀ05 ｧ: d = (x, x ) +(y, y ) : (3) 4. 7 03 06 09 09 ｧ ｧ 0 03 ﾀ ｧ 090303 罍陦ｧ0 05 0 04 04000600030503 0305 縺06 09, ｧ00 ｧ 040609-0503,! AB 0 040003 ﾀ0903 ﾀ05 ｧ ﾀ0 00030 A(x ; y ) ｧ 0903009030 B(x ; y ): 04 x,! AB = X = x, x ; 04 y,! AB = Y = y, y : (4). 703 ﾀ05 縺03 ｧ0504 0 ｧ ﾀ0003040308 03 ﾀ ｧ 0503 09 ｧ A(,5), B(+4) ｧ C(,) ｧ 0 0805 ｧ 040600 ｧ ｧ003 AB, BC ｧ AC 0305 縺06 090304 0 03 ﾀ ｧ. 7 縺030406 罍ｧ0504, 0503 AB + BC = AC.. 6 5030403000 ｧ0504 04 縺06 閨03 閨0600606 0604 070060 ｧ06 閨0007 0503 0609 A(+), B(,4) ｧ C(+5). 3. 703 ﾀ05 縺03 ｧ0504 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(,4; ), B(0;,) ｧ C(3; 3) ｧ 0304 縺06 閨0600 ｧ0504 060503 0406 罍ｧ00605 ｧ 06050003. 4. 6 70309 0504, 0503 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(,3;,), B(0;,) ｧ C(,; 5) 04 縺07003060503000400308. 5. 703 ﾀ05 縺03 ｧ0504 0503 09 ｧ A(,4; 0), B(,; 4) ｧ0503 09 ｧ A, B, ﾀ ｧ0-00605 罍ｧ 00306 00030 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oy. 6 503 ｧ ﾀ00 ｧ0504 0406 罍ｧ00605 05 040609 ｧ ｧ ABB A. 6. ｯ03 09 B ﾀ ｧ000605 罍ｧ 0 A(4;,) 0305003 ﾀ ｧ05060004003 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ03 0406 縺04030503 090303 罍陦ｧ0 050030503 060500. 7 0805 ｧ 閨00 ｧ006 AB.

30 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 7. 7 0805 ｧ 0503 0906, 06 0006000606 0 5 06 陦ｧ0 ｧ09 09 09 0305 0503 09 ｧ A(; ), 05 09 ｧ 0305 03 ﾀ ｧOy. 8. 7 03 ﾀ ｧ 03 罍陦ｧ0 05 0 0805 ｧ0503 0906, 06 000600060603050503 09 ｧ A(4;,) 0 5 06 陦ｧ0 ｧ09. 70307 ﾀ0 ｧ0504 0403 ﾀ05 縺03060 ｧ060, 0403 06006 04030006 0605 ﾀ07 閨04 縺0600060 ｧ07. 9. 7 03 ﾀ ｧ 03 ﾀ09 ｧ ﾀ ﾀ 0 0805 ｧ 0503 0906, 06 0006000606 03050503 09 ｧ A(a; b) 0 c 06 陦ｧ0 ｧ09. 0ｷ3 ﾀ ﾀ0006 閨0304 0504 縺0600060 ｧ06 04 罍ｧ c>jbj, c = jbj ｧ c<jbj. 0. 7 03 ﾀ ｧ Ox 0 0805 ｧ 0503 0906, 03 陦ｧ0 09030403 06 0006000606 03050 00 090303 罍陦ｧ0 05 ｧ 0305 0503 09 ｧ A(8; 4).. 7 0805 ｧ 0906005 ｧ 陦ｧ06 ﾀ 09 縺0605, 0304 ｧ ﾀ 00030503 0309030003 05 縺060605030004-0 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(4; 3), B(,3; ) ｧ C(;,6).. 6 7 003 0503 09 ｧ A(; 6) ｧ B(0; ); 0403 ﾀ05 縺03 ｧ0504 0406090503,! AB, 060503,!,! 09030040300600503 0 03 ﾀ0708 ｧ 0403 ｧ ﾀ00 ｧ0504 04 x AB, 04 繦 AB ｧ 閨00 ｧ006 AB. 3. 6 5 0503 0906 A(; 5) 04 罍ｧ000307060 ﾀ ｧ00, 04 縺03060909 ｧ ｧ 09030503 縺0308 0 03 ﾀ ｧ 090303 罍陦ｧ0 05 04003: X =3 ｧ Y =3. 0504 縺06 閨0600 ｧ0504 090300609 0406090503,! AB, ｧ 0303 07 060060503 ﾀ ｧ0006, ｧ 040600 ｧ ｧ006 ﾀ ｧ0003. 4. 6 5 0503 0906 A(,3;,) 04 罍ｧ000307060 ﾀ ｧ00, 04 縺03060909 ｧ07 09030503 縺0308 Y = =,, 04 縺03060909 ｧ07 X 0403000307 ｧ050600040. 0504 縺06 閨0600 ｧ0504 090300609 0406090503,! AB, ｧ 0303 07 060060503 ﾀ ｧ0006, 06 ﾀ00 ｧ 0606 040600 ｧ ｧ0 040 5. 5 ). 7 ｧ ﾀ0003040308 03 ﾀ ｧ 0403 ﾀ05 縺03 ｧ0504 0503 09 ｧ A(), B(,3) ｧ C(,) ｧ 0 0805 ｧ 040600 ｧ ｧ003 AB, BC ｧ CA 0305 縺06 090304 0 03 ﾀ ｧ. 7 縺030406 罍ｧ0504, 0503 AB + BC + CA =0. 6. 7 040003 ﾀ0903 ﾀ05 ｧ 0403 ﾀ05 縺03 ｧ0504 0503 09 ｧ A(,7; 0) ｧ B(0; ) ｧ0503 09 ｧ A ｧ B, ﾀ ｧ000605 罍ｧ 00306 0503 09 0 A ｧ B 0305003 ﾀ ｧ05060004003 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ03 0406 縺04030503 ｧ 05 縺060504060503 090303 罍陦ｧ0 0500308 0605000304. 6 503 ｧ ﾀ00 ｧ0504 0406 罍ｧ00605 05-040609 ｧ ｧ ABB A. 7. 7 03 ﾀ ｧ 03 罍陦ｧ0 05 0 0805 ｧ 0503 0906, 03 陦ｧ0 09030403 06 0006000606 0305 0-00 090303 罍陦ｧ0 05 ｧ 0305 0503 09 ｧ A(,; 5). 8. 7 03 ﾀ ｧ 03 ﾀ09 ｧ ﾀ ﾀ 0 0805 ｧ 0503 0906, 06 0006000606 03050503 09 ｧ A(,; 3) 0 3 5 06 陦ｧ0 ｧ09. 9. 0504 縺06 閨0600 ｧ0504 0906005 ｧ 陦ｧ06 ﾀ 09 縺0605, 0304 ｧ ﾀ 00030503 0309030003 05 縺06-06050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(,3;,), B(5; 3) ｧ C(6;,4). 0. 6 7 003 0503 09 ｧ A(x ; y ) ｧ B(x ; y ).6 50 0006 090303 罍陦ｧ0 05 04 罍ｧ- 00030706003 ﾀ ｧ0003, ｧ 0303 07 0600306 0406090503 0 ｧ,! OA ｧ,! OB. 703 ﾀ05 縺03 ｧ0504 ｧ08 04003 閨0608 ﾀ0504060600606,! OC ｧ 閨0309 0504, 0503 04 縺03060909 ｧ07 04003 閨0608 ﾀ05040606-00608 0 090303 罍陦ｧ0 0500606 03 ﾀ04 040 ﾀ060006 04 縺03060909 ｧ08 ﾀ03 ﾀ05 040007060 ｧ08 0 0506 0706 03 ﾀ04.. 6 7 003 0503 09 ｧ A(; ), B(3; 5), C(5; ) ｧ D(;,). 6 5 0503 0906 A 04 罍ｧ00030706003 ﾀ ｧ0003,! AB,,! AC ｧ,! AD. 7 0805 ｧ 04 縺03060909 ｧ ｧ 0 03 ﾀ ｧ 090303 - 陦ｧ0 05 04003 閨0608 ﾀ0504060600608 ﾀ ｧ0003 ｧ 0606 040600 ｧ ｧ006. )6 509 07 閨030 04 05 0706 0403 ﾀ0006 06 縺0503 04 罍ｧ0406 閨06003 ｧ, 09030503 縺0306 縺0609030060 閨06-0605 ﾀ07 閨0007 0 ｧ07 0 閨030 ｧ00 ｧ 閨0007 0403040503 縺060 ｧ08.

3x. 6 70600060 ｧ06 0305 縺06 09 04 00030 030500300060 ｧ ｧ x. 6 70600060 ｧ06 0305 縺06 09 04 00030 030500300060 ｧ ｧ. 700030 閨04 05 縺0606050300040 ｧ09 ｧ 0003050306050300040 ｧ09. 6 70600060 ｧ06 0305 縺06 09 04 00030 030500300060 ｧ ｧ. 6 7 003 0503 09 ｧ A(x ; y ) ｧ B(x ; y ). 070303 罍陦ｧ0 0503 0503 09 ｧ M (x; y), 閨06000700608 0305 縺06 0309 AB 04030500300060 ｧ ｧ AM : MB =, 0304 縺06 閨0600070605 ﾀ07 0403 0703 縺00600 0: x = x + x + ; y = y + y + : () 6 5 ﾀ05003 ﾀ05 ｧ, 04 罍ｧ 閨0600060 ｧ ｧ 0403040300 0, 05. 06. 04 030500300060 ｧ ｧ =:=, x = x + x ; y = y + y : (). 700030 閨04 0003050306050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(x ; y ), B(x ; y ), C(x 3 ; y 3 ), :::, F (x n ; y n ) 040 S = h i x y x x y + y x x 3 y + :::+ n y n 3 x y : (3) 6 503 07060 ｧ06 04 ｧ x y x y 04003 x y, x y ｧ0 0304 0605 ﾀ07 0304 縺06 閨0600 ｧ- 050600060 040503 縺030503 0403 縺07 閨09 ).. 703 ﾀ05 縺03 ｧ05040503 09 ｧ A(,; ) ｧ B(3; 6) ｧ 0 0805 ｧ0503 0906 M (x; y), 閨06000700606 AB 04030500300060 ｧ ｧ AM : MB =3:. 3. 6 7 003 0503 09 ｧ A(,; ) ｧ B(3; 6). 7 閨0600 ｧ0504 0305 縺06 0309 AB 04 030500300060 ｧ ｧ AM : MB =,3 :. 4. 6 50503 09 08 A(x ) ｧ B(x ) 03 ﾀ ｧ Ox 0403006006003 0 ﾀ ﾀ03 m ｧ m. 7 0805 ｧ 0906005 0 ﾀ ﾀ 05050308 ﾀ ｧ ﾀ0506003. 5. 6 50503 09 08 A(x ), B(x ) ｧ C(x 3 ) 03 ﾀ ｧ Ox 0403006006003 ﾀ030305-040605 ﾀ0504060003 0 ﾀ ﾀ03 m, m ｧ m 3. 70309 0504, 0503 0906005 0 ﾀ ﾀ 05050308 ﾀ ｧ ﾀ0506003 0306 閨0605 040503 0906 x = m x + m x + m 3 x 3 m + m + m 3. 6. 7 090300903 03 閨003 縺03 閨0030503 ﾀ0506 縺07007 閨00 ｧ00308 40 ﾀ0 ｧ 0 ﾀ ﾀ0308 500 05 0 ﾀ 0706003 00 縺03 0 ﾀ ﾀ0308 00 05 ｧ 400 05. 0504 縺06 閨0600 ｧ0504 0906005 0 ﾀ ﾀ 05050308 ﾀ ｧ ﾀ0506003. 7. 6 50503 09 08 A(,; 4), B(3;,) ｧ C(; 3) 0403006006003 ﾀ030305040605- ﾀ0504060003 0 ﾀ ﾀ03 60 05, 40 05 ｧ 00 05. 0504 縺06 閨0600 ｧ0504 0906005 0 ﾀ ﾀ 05050308 ﾀ ｧ- ﾀ0506003. 8. 0504 縺06 閨0600 ｧ0504 ﾀ06 縺06 陦ｧ003 ﾀ0503 縺030 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(;,), B(4; 3) ｧ C(,; ). 9. 6 5 05 縺0606050300040 ｧ0906 ﾀ 0406 縺00 ｧ0 0 ｧ O(0; 0), A(8; 0) ｧ B(0; 6) 0304 縺06 閨0600 ｧ0504 閨00 ｧ006 006 陦ｧ 003 OC ｧ 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ03 OD. ) 0503 0304 縺06 閨0600 ｧ0506000708 0403 陦縺0303003 ｧ 00030706003 04 0500. 4, x.

3 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 30. 7 0805 ｧ 0906005 0 ﾀ ﾀ 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(;,), B(6; 4) ｧ C(; 6). ｰ09 0 ｧ06. ｨC06005 0 ﾀ ﾀ 05 縺0606050300040 ｧ09 0 0803 陦ｧ05 ﾀ07 04 0503 0906 0406 縺06 ﾀ06 06-0 ｧ07 060503 006 陦ｧ 0. 3. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(; 0), B(5; 3) ｧ C(; 6). 3. 70309 0504, 0503 0503 09 ｧ A(; ), B(,; 7) ｧ C(0; 4) 000607 05 0 03 閨00308 04 縺0700308. 33. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04 060503 縺060806050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(3; ), B(4; 6), C(6; 3) ｧ D(5;,). 34. 6 50503 09 08 A(,3;,) ｧ B(4; 6) 04 罍ｧ00030706003 04 000006000400306 ﾀ ｧ0003, ﾀ030305040605 ﾀ0504060003 0400306 30 7 ｧ 40 7. 7 0305 縺06 0906 AB 0 0805 ｧ 0503 0906 04 罍ｧ000307060 ｧ07 04003 閨0608 ﾀ0504060600608. 35. 6 50503 09 08 O(0; 0), A(;,5) ｧ B(4; ) 0403006006003 ﾀ030305040605 ﾀ05-04060003 0 ﾀ ﾀ03 500 05, 00 05 ｧ 00 05. 0504 縺06 閨0600 ｧ0504 0906005 0 ﾀ ﾀ 05050308 ﾀ ｧ- ﾀ0506003. 36. 6 505 縺0606050300040 ｧ0906 ﾀ 0406 縺00 ｧ0 0 ｧ A(,; 0), B(6; 6) ｧ C(;,4) 0304 縺06 閨0600 ｧ0504 閨00 ｧ006 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ03 AE. 37. 7 0805 ｧ 0906005 0 ﾀ ﾀ 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(x ; y ), B(x ; y ) ｧ C(x 3 ; y 3 ). 38. 7 0805 ｧ 0906005 0 ﾀ ﾀ 060503 縺0608060503000400308 03 閨003 縺03 閨00308 閨03 ﾀ09 ｧ ﾀ 0406-00 ｧ0 0 ｧ A(,; ), B(3; 6), C(5; ) ｧ D(0;,6). ｰ09 0 ｧ06. 703 0703 縺00600 0, 04030006 0600030 04 06 37, 0 0805 ｧ 0906005 縺03 0 ﾀ ﾀ 05 縺0606050300040 ｧ090304 ABC ｧ ADC ｧ 閨0600 ｧ0504 ﾀ ﾀ0503070 ｧ06 00607 閨06 0 ｧ0 ｧ 04 030500300060 ｧ ｧ, 0303 050030 030500300060 ｧ06 0400030 閨0608 05 縺0606050300040 ｧ090304. 39. 6 7 003 0503 09 ｧ A(; ) ｧ B(4; 4). 7 03 ﾀ ｧOx 0304 縺06 閨0600 ｧ0504 0503 0906 C 05 09, 05030303 0400030 閨04 4ABC 030300 040 5, ｧ 0403 ﾀ05 縺03 ｧ0504 4ABC. 40. 6 505 縺0606050300040 ｧ0906 ﾀ 0406 縺00 ｧ0 0 ｧ A(,; ), B(;,4) ｧ C(4; 5) 09 07 07 ﾀ0503 縺030 04 縺03 閨030007060 04 0 04 0400060 ｧ ｧ 03030803 0406 罍ｧ00605 04 縺03-05 ｧ04 ﾀ03040308 ﾀ05 縺060009 ｧ 0 03 閨006 05 縺060504 ﾀ04030608 閨00 ｧ003. 0504 縺06 閨0600 ｧ0504 090300903 M, N ｧ P 04 縺03 閨030007060 ｧ08 ﾀ0503 縺030 ｧ 0 0805 ｧ 030500300060 ｧ06 k 040003-0 陦ｧ 4MNP 09 0400030 陦ｧ 4ABC. x 3. ｰ 040060 ｧ06 00 ｧ0 ｧ ｧ 09 09 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 ｰ 040060 ｧ060 00 ｧ0 ｧ ｧ 0 0304 0605 ﾀ07 06 040060 ｧ06 ﾀ 0406 縺0600600030 ｧ x ｧ y; 09030503 縺03006 06 閨03040006050403 縺070605 090303 罍陦ｧ0 0503 0006030308 0503 09 ｧ 05050308 00 ｧ0 ｧ ｧ ｧ 050300040903 030 ｧ. 6 50803 閨070 ｧ06 04 06 040060 ｧ06 00 ｧ0 ｧ ｧ 0406 縺06006000306 x ｧ y 0 0304 0605 ﾀ07 05060906-0 ｧ0 ｧ 090303 罍陦ｧ0 05 0 ｧ, 0306090406000306 0403 ﾀ050307000306 04 00605 0 ｧ. 7-04 罍ｧ006, 04 06 040060 ｧ ｧ 0309 縺0607003 ﾀ05 ｧ ( 4) x + y = R 0406 縺06006000306 x ｧ y 050609060 ｧ06 090303 罍陦ｧ0 0503, 0403 ﾀ05030700 07 R 04 00605. ｪ05030303 ﾀ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 00 ｧ0 ｧ ｧ 09 09 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 030300 060 ｧ08 03 陦ｧ0 090304030 ﾀ040308 ﾀ0504030, 00607003:

3x 3. ｰ 040060 ｧ06 00 ｧ0 ｧ ｧ 09 09 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 3 ) 04 070504 04 縺03 ｧ 0403000400606 (0506090600606) 0503 0906 M (x; y) 00 ｧ0 ｧ ｧ; ) 04 ｧ ﾀ 0504 04060 ﾀ0504030 030300606 ﾀ040308 ﾀ050403 04 ﾀ0608 0503 0609 M 00 ｧ0 ｧ ｧ; 3) 040803 閨070 ｧ06 04 050503 04060 ﾀ050403 0305 縺06 09 ｧ ( ｧ 06050003) 0403 ｧ0504 06 縺06 05060906-0 ｧ06 090303 罍陦ｧ0 0503 0503 09 ｧ M (x; y) ｧ 06 縺06 000306 04 06. 4. 70309 0504, 0503 06 040060 ｧ060 0309 縺0607003 ﾀ05 ｧ ﾀ 陦ｧ06 ﾀ030 R ｧ ﾀ 0906005 縺030 04 0 0006 090303 罍陦ｧ0 05 0306 閨0605 x + y = R. 4. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ ﾀ 0906005 縺030 C(3; 4) ｧ 陦ｧ06- ﾀ030 R =5.6ｧ80607 05 00 ｧ 0 05050308 0309 縺0607003 ﾀ05 ｧ 0503 09 ｧ A(,; ), B(; 3), O(0; 0) ｧ D(4; )? 43. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 00 ｧ0 ｧ ｧ, 0403 09030503 縺0308 閨04 ｧ070605 ﾀ07 0503 09 M (x; y), 0400306 000600 07 0305 0503 0609 A(0; ) ｧ B(4;,). 6ｧ80607 05 00 ｧ 0 05050308 00 ｧ0 ｧ ｧ 0503 09 ｧ C(,; ), D(;,), E(0;,) ｧ F (; )? 44. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 06090503 罍ｧ ｧ 0503 09 ｧ M (x; y), 09030503 07 04 罍ｧ ﾀ0403060 閨04 ｧ07060 ｧ ｧ 03 ﾀ05 0605 ﾀ07 0405 縺0306 00040006 0305 0503 09 ｧ A(0; 9), 060 0305 0503 09 ｧ B(0; ). 45. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 06090503 罍ｧ ｧ 0503 09 ｧ M (x; y), 09030503 07 04 罍ｧ ﾀ0403060 閨04 ｧ07060 ｧ ｧ 03 ﾀ05 0605 ﾀ07 04 閨040306 0300 ｧ0706 09 0503 0906 A(,; ), 060 090503 0906 B(,4; 4). 46. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 090303 罍陦ｧ0 0500308 0605000304. 47. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, ﾀ0600 ﾀ ﾀ0503070 ｧ08 0305 09 07 閨0308 ｧ 09030503 縺0308 閨03 0503 0609 F (; 0) ｧ F (,; 0) 040 5. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ06 0403 0606 06 040060 ｧ06. 48. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 04003-06 0006000308 0305 0503 09 ｧ F (; ) ｧ030503 ﾀ ｧOx. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ06 0403 0606 06 040060 ｧ06. 49. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 00 ｧ0 ｧ ｧ, 0403 09030503 縺0308 閨04 ｧ070605 ﾀ07 0503 09 M (x; y), 03 ﾀ05 04 07 ﾀ04 04 閨040306 00040006 0305 03 ﾀ ｧOx, 060 0305 03 ﾀ ｧOy. 50. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ: ) y =x +5;)y =7, x; 3)y =x; 4) y =4;5)y =4, x. 5. 0504 縺06 閨0600 ｧ0504 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 00 ｧ0 ｧ ｧ y = x, 4x +3 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 ｧ 0403 ﾀ05 縺03 ｧ0504 0606. 5. 0504 縺06 閨0600 ｧ0504 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 00 ｧ0 ｧ08: ) 3x, y =;)y = x +4x; 3)y =x +4. 703 ﾀ05 縺03 ｧ0504 0505 ｧ 00 ｧ0 ｧ ｧ. 53. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 04003-06 0006000308 0305 03 ﾀ ｧOy ｧ03050503 09 ｧ F (4; 0), ｧ 0403 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ06 0403 0606 06 040060 ｧ06. 54. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 00 ｧ0 ｧ ｧ, 0403 09030503 縺0308 閨04 ｧ070605 ﾀ07 0503 09 M (x; y), 0400306 000600 07 03050 00 090303 罍陦ｧ0 05 ｧ 03050503 09 ｧ A(,4; ). 6ｧ80607 05 00 ｧ 0 05050308 00 ｧ0 ｧ ｧ 0503 09 ｧ B(,; ), C(; 3), D(; 7)?

34 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 55. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 06090503 罍ｧ ｧ 0503 09 ｧ M (x; y), 09030503 07 04 罍ｧ ﾀ0403060 閨04 ｧ07060 ｧ ｧ 03 ﾀ05 0605 ﾀ07 04 閨040306 0300 ｧ0706 09 0503 0906 A(0;,), 060 090503 0906 B(0; 4). 703 ﾀ05 縺03 ｧ0504 05 06090503 罍ｧ06 閨04 ｧ07060 ｧ07. 56. 0504 縺06 閨0600 ｧ0504 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 00 ｧ0 ｧ08: ) x +5y +0=0;)y =3, x, x ;3)y =4, x. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ. 57. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 04003-06 0006000308 0305 03 ﾀ ｧOx ｧ03050503 09 ｧ F (0; ), ｧ 0403 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ06 0403 0606 06 040060 ｧ06. 58. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 003 ﾀ0504 ﾀ ﾀ0503070 ｧ08 0305 09 07 閨0308 ｧ 09030503 縺0308 閨03 0503 0609 F (,;,) ｧ F (; ) 040 4. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ06 0403 0606 06 040060 ｧ06. x 4. ｰ 040060 ｧ06 04 縺0700308: ) ﾀ 0605000304030 0903050707 ｧ09 ｧ06005030, ) 030300606, 3) 04 0305 縺06 09 08 0 03 ﾀ0708. ｰ 040060 ｧ06 04 縺0700308 ﾀ 0605000304030 0903050707 ｧ09 ｧ06005030 y = kx + b: () 7 00605 k 04060 05 005060 ﾀ06 060500 0 0900030 04 縺0700308 09 03 ﾀ ｧ Ox (k = = tg ) ｧ0 0304 0605 ﾀ07 0605000304030 0903050707 ｧ09 ｧ06005030, ｧ00 ｧ ｧ00305 0 0900030030 04 縺0700308. 7 00605 b 040600 ｧ ｧ0 0305 縺06 09 0 03 ﾀ ｧOy, ｧ00 ｧ 0 00040 07 03 罍陦ｧ0 05.. 050300606 06 040060 ｧ06 04 縺0700308 Ax + By + C =0: () 05 ﾀ03030306 ﾀ0006 ｧ: ) 04 罍ｧ C =0 y =, A x 04 縺070 07 04 縺030803 陦ｧ05 06 縺06 0 0003 090303 罍陦ｧ0 05; B 03) 04 罍ｧ B =0 x =, C A = a 04 縺070 07 04 00000600040 03 ﾀ ｧ Oy; 04) 04 罍ｧ A =0 y =, C = b 04 縺070 07 04 00000600040 03 ﾀ ｧ Ox; B 05) 04 罍ｧ B = C =0 Ax =0, x =0 03 ﾀ04Oy; ) 04 罍ｧ A = C =0 By =0, y =0 03 ﾀ04Ox. 3. ｰ 040060 ｧ06 04 縺0700308 04 0305 縺06 09 08 0 03 ﾀ0708 x a + y =; (3) b 05 閨06 a ｧ b 040600 ｧ ｧ003 0305 縺06 090304, 0305 ﾀ0609 0600308 04 縺0700308 0 03 ﾀ0708 090303 罍陦ｧ0 05. 59. 703 ﾀ05 縺03 ｧ0504 04 縺0700606, 0305 ﾀ0609 0600606 0 03 ﾀ ｧOy 0305 縺06 0309 b = =3 ｧ ﾀ03 ﾀ05 0400070600606 ﾀ03 ﾀ0406 Ox 06050300: ) 45 ;)35. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 0505 ｧ08 04 縺0700308. 60. 703 ﾀ05 縺03 ｧ0504 04 縺0700606, 0305 ﾀ0609 0600606 0 03 ﾀ ｧOy 0305 縺06 0309 b = =, 3 ｧ ﾀ03 ﾀ05 0400070600606 ﾀ03 ﾀ0406 Ox 06050300: ) 60 ;)0. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 0505 ｧ08 04 縺0700308.

3x 4. ｰ 040060 ｧ06 04 縺0700308 5 6. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0 0003 0903-03 罍陦ｧ0 05 ｧ ﾀ03 ﾀ05 0400070600608 ﾀ 03 ﾀ0406 Ox 06050300: ) 45 ;)60 ;3)90 ; 4) 0 ;5)35. 6. 703 ﾀ05 縺03 ｧ0504 04 縺0700606, 04 縺030803 閨0700606 06 縺06 0 0003 090303 罍陦ｧ0 05 ｧ 06 縺06 0503 0906 (,; 3), ｧ 0 04 ｧ ﾀ 0504 060606 040060 ｧ06. 63. 0504 縺06 閨0600 ｧ0504 04 00605 縺03 k ｧ b 閨0007 09 07 閨0308 ｧ 04 縺0700308: ) x, 3y =6; ) x +3y =0; 3) y =,3; 4) x 4 + y 3 =. 64. 703 ﾀ05 縺03 ｧ0504 04 縺0700306: ) 3x +4y =; ) 3x, 4y =0; 3) x, 5=0; 4) y +5=0. 65. 0504 縺06 閨0600 ｧ0504 04 00605 縺03 k ｧ b 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 A(; 3) ｧ ﾀ03 ﾀ05 0400070600608 ﾀ Ox 06050300 45. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05050308 04 縺0700308. 66. ｰ 040060 ｧ07 04 縺0700308: ) x, 3y = 6;) 3x, y +4 = 0 04 罍ｧ0406 ﾀ05 ｧ 09 04 ｧ 閨06 04 0305 縺06 09 08 0 03 ﾀ0708. 67. 6 7 003 0503 09 ｧ O(0; 0) ｧ A(,3; 0). 7 0305 縺06 0906 OA 0403 ﾀ05 縺03060 04 000006000305 00, 陦ｧ 05030 00 ｧ 09030503 縺030503 0406 縺06 ﾀ0609 0605 ﾀ07 04 0503 0906 B(0; ). 7 04 ｧ ﾀ 0504 06 040060 ｧ07 ﾀ0503 縺030 ｧ 陦ｧ 05030 000608 04 000006000305 00. 68. 7 04 ｧ ﾀ 050406 040060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 A(4; 3) ｧ 0305 ﾀ0609 0600608 0305 090303 罍陦ｧ0 050030503 060500 05 縺0606050300040 ｧ09 0400030 閨0406, 04-00308 3. 69. 7 縺0700306 y =, ｧ y =40406 縺06 ﾀ0609 0605 04 縺0700606 3x, 4y, 5=0 ﾀ030305040605 ﾀ0504060003 04 0503 09 08 A ｧ B. 703 ﾀ05 縺03 ｧ0504 0406090503,! AB, 0304 縺06 閨0600 ｧ0504 060503 閨00 ｧ006 ｧ 060503 04 縺03060909 ｧ ｧ 0 03 ﾀ ｧ 090303 罍陦ｧ0 05. 70. 6ｧ80607 05 00 ｧ 0503 09 ｧ A(3; 5), B(; 7), C(,;,3) ｧ D(,;,6) 0 04 縺0700308 y =x, ｧ00 ｧ 0706 030 ｧ 080403000609 ｧ00 ｧ 080 ｧ070609 05050308 04 縺0700308? 7. 07 090304 05060300605 罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀ00 006 04060 ﾀ0504: ) y>3x +;)y<3x +;3)x + y, 4 > 0; 4)x + y, 4 < 0? 7. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ05 ｧ ), 090303 罍陦ｧ0 0503 0503 0609 09030503 縺0308 06 閨03040006-050403 縺070605 006 04060 ﾀ0504 0: ) y<, x; x>,; y>,; ) y>, x; x<4; y<0; 3) x 4 + y 6 ; y > x +; x >,4: 73. ｯ03 09 M (x; y) 閨04 ｧ070605 ﾀ07 05 09, 0503 003 ﾀ0504 0904 陦 050304 ﾀ- ﾀ0503070 ｧ08 0305 00606 閨03 0503 0609 A(,a; a) ｧ B(a;,a) 03 ﾀ05 0605 ﾀ07 0400308 4a. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0606 05 06090503 罍ｧ ｧ. ) ｮ00030403 08030300 ﾀ050409 閨06 ﾀ04 03 0 0605 ﾀ0504 040003 ﾀ0903 ﾀ05 ｧ xoy, 090303 罍陦ｧ0 0503 09 07 閨0308 0503 09 ｧ 09030503 縺0308 06 閨03040006050403 縺070605 00609030503 縺030 06 ﾀ000304 ｧ070 (0 04 罍ｧ006, 006 04060 ﾀ0504 0). 050300 ﾀ0504 0 0304 0605 ﾀ07 009006050308, 06 ﾀ00 ｧ 04 00606 04090006 06003 0503 09 ｧ, 000607 0 ｧ06 0 05-0 ｧ0906 030300 ﾀ05 ｧ. 6 5 04 縺0305 ｧ040030 ﾀ0006 06 030300 ﾀ0504 0 0304 0605 ﾀ07 030509 縺03050308.

36 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 74. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 06090503 罍ｧ ｧ 0503 09 ｧ M (x; y), 04 縺03060909 ｧ07 09030503 縺0308 0 03 ﾀ04 Ox 閨04 ｧ070605 ﾀ07 ﾀ03 ﾀ0903 縺03 ﾀ050406 m 06 / ﾀ, 0 03 ﾀ04 Oy ﾀ03 ﾀ0903 縺03 ﾀ050406 n 06 / ﾀ. 7 000400306 0403000307060 ｧ06 0503 09 ｧ M 0 (a; b). 75. 703 ﾀ05 縺03 ｧ0504 04 縺0700306, 000306 04 00605 0 ｧ: ) b =,, ' = =60 ｧ)b =,, ' = 0, ｧ 0 04 ｧ ﾀ 0504 ｧ0806 040060 ｧ07. 76. 0504 縺06 閨0600 ｧ0504 04 00605 縺03 k ｧ b 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 (,; 3) ｧ ﾀ03 ﾀ05 0400070600608 ﾀ Ox 06050300 45. 703 ﾀ05 縺03 ｧ0504 04 縺0700606 ｧ 0 04 ｧ ﾀ 0504 060606 040060 ｧ06. 77. 7 040030306 陦縺0600 07 05 040609 ｧ07 ﾀ 03 ﾀ00304 0 ｧ070 ｧ 8 ﾀ0 ｧ ﾀ0 ｧ0060605 03 ﾀ05 縺0308 06050300 45. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 ﾀ0503 縺030 05 040609 ｧ ｧ, 04 罍ｧ00704 03 ﾀ04 Ox 03030004000606 03 ﾀ00304 0 ｧ06 ｧ 03 ﾀ04 Oy 03 ﾀ04 ﾀ ｧ000605 罍ｧ ｧ 05 040609 ｧ ｧ. 78. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 ﾀ0503 縺030 縺03003 ﾀ 陦ｧ 05030 00070 ｧ 0 ﾀ0 ｧ 6 ﾀ0, 04 罍ｧ00704 03030004000606 陦ｧ 05030 0004 03 ﾀ04 Ox ｧ 006004000606 03 ﾀ04 Oy. 79. 7 04 ｧ ﾀ 050406 040060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 (,4; 6) ｧ0305 ﾀ0609 0600608 0305 03 ﾀ0608 090303 罍陦ｧ0 05 05 縺0606050300040 ｧ09 0400030 閨0406 6. 80. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 00 ｧ0 ｧ ｧ, 0403 09030503 縺0308 閨04 ｧ070605 ﾀ07 0503 09 M (x; y),03 ﾀ05 04 07 ﾀ04 04 閨040306 00040006 0305 03 ﾀ ｧOx, 060 0305 04 縺0700308 x =,3. 8. 7 縺0700306 x =, ｧ x =30406 縺06 ﾀ0609 0605 04 縺0700606 y =x +04 0503 09 08 A ｧ B. 0504 縺06 閨0600 ｧ0504 閨00 ｧ006 0406090503,! AB ｧ 060503 04 縺03060909 ｧ ｧ 0 03 ﾀ ｧ 090303 罍陦ｧ0 05. x 5. ｰ050300 00607 閨06 04 縺070030 ｧ. ｰ 040060 ｧ06 0406 09 04 縺0700308, 04 縺030803 閨070 ｧ08 06 縺06 000606 0503 0906. ｰ 040060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 閨0406 000306 0503 09 ｧ. ｯ03 09 0406 縺06 ﾀ06 060 ｧ07 閨040608 04 縺0700308. ｰ050300 ', 0305 ﾀ ｧ05 000308 04 縺0305 ｧ04 ﾀ03040308 ﾀ05 縺060009 ｧ 0305 04 縺0700308 y = = k x + b 閨03 04 縺0700308 y = k x + b, 0304 縺06 閨0600070605 ﾀ07 0703 縺006000308 6 70007 04 縺0700308, 000308 06 040060 ｧ070 ｧ 0703 縺00600 () 04 罍ｧ00605 04 ｧ tg ' = k, k +k k : () A x + B y + C =0 ｧ A x + B y + C =0; tg ' = A B, A B A A + B B : ｰ ﾀ000304 ｧ06 04 0000060004003 ﾀ05 ｧ: k = k ｧ00 ｧ A A = B B. ｰ ﾀ000304 ｧ06 0406 縺04060 陦ｧ09060007 縺003 ﾀ05 ｧ: k =, k ｧ00 ｧ A A + B B =0.

3x 5. ｰ050300 00607 閨06 04 縺070030 ｧ. ｰ 040060 ｧ06 0406 09 04 縺0700308 7. ｰ 040060 ｧ06 0406 09 04 縺0700308,04 縺030803 閨070 ｧ08 06 縺06 0006060503-0906 A(x ; y ): y, y = k(x, x ): () 3. ｰ 040060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 閨0406 000306 0503 09 ｧ A(x ; y ) ｧ B(x ; y ): y, y y, y = x, x x, x : (3) 4. ｪ05030303 0 0805 ｧ 0503 0906 0406 縺06 ﾀ06 060 ｧ07 00604 000006000400308 04 縺0700308 A x + B y + C =0 ｧ A x + B y + C =0,00607003 縺0600 ｧ0504 ﾀ0304006 ﾀ05003 ｧ08 06 040060 ｧ07. 7030006 ｧ0:,C B,C B A,C A x = A,C ; y = B A B A : B A B 8. 0504 縺06 閨0600 ｧ0504 06050300 00607 閨06 04 縺070030 ｧ: ) y =x, 3, y = x +; ) 5x, y +7=0, x, 3y +=0; 3) x + y =0, y =3x, 4; 4) 3x +y =0, 6x +4y +9=0; 5) 3x, 4y =6, 8x +6y =; 6) x a + y b =, x b + y a =. 83. ｮ 縺06 陦ｧ 04 縺0700308 3x,y+7 = 0, 6x,4y,9 = 0, 6x+4y,5 = 0, x +3y, 6=00609 0504 04 000006000400306 ｧ 0406 縺04060 陦ｧ09060007 縺00306. 84. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0406 09 04 縺0700308, 04 縺030803 閨070 ｧ08 06 縺06 0503-0906 A(; 3). 6 50303 0504 ｧ 0505030503 0406 09 04 縺0700306, ﾀ03 ﾀ05 040007060 ｧ06 ﾀ 03 ﾀ0406 Ox 06050003: ) 45 ;)60 ;3)35 ;4)0, ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ08. 85. 703 ﾀ05 縺03 ｧ0504 0503 0906 A(,; 5) ｧ 04 縺0700606 x, y = 0. 7 04 ｧ- ﾀ 0504 06 040060 ｧ06 0406 09 04 縺0700308, 04 縺030803 閨070 ｧ08 06 縺06 A, ｧ 040303 0504 ｧ 0406 09 : ) 04 縺0700606, 04 000006000400606 000308; ) 04 縺0700606, 0406 縺04060 陦ｧ0906-0007 縺00606 09 000308. 86. 6 50503 09 08 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 x, 5y, 0 = 0 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 0403 ﾀ ﾀ05 040006003 0406 縺04060 陦ｧ09060007 縺03 09 05050308 04 縺0700308. 7 04 ｧ ﾀ 0504 ｧ08 06 040060 ｧ07. 87. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 09 ｧ A(,; 3) ｧ B(4;,).

38 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 88. 6 505 縺0606050300040 ｧ0906 ﾀ 0406 縺00 ｧ0 0 ｧ A(,; 0), B(; 6) ｧ C(4; ) 04 縺030406 閨06003 0403 ﾀ0305 BD ｧ 006 陦ｧ 0 BE. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 ﾀ0503- 縺03003 AC, 006 陦ｧ 003 BE ｧ 0403 ﾀ030503 BD. 89. 7 0805 ｧ 0400605 縺0600 ｧ06 06050003 05 縺0606050300040 ｧ09, ﾀ0503 縺03003 09030503 縺030503 003 06 040060 ｧ070 ｧ x +y =0, x +4y, 6=0, x, 4y, 6=0. ｰ09 0 ｧ06. ｪ05030303 0 0805 ｧ 0400605 縺0600 ｧ06 06050003 05 縺0606050300040 ｧ09, 00607003 06050003040306 0903050707 ｧ09 ｧ0600503 ﾀ0503 縺030 040304 ｧ ﾀ 0504 04 0403 縺07 閨0906 06030304 0 ｧ07: k >k > k, k 3 +k k 3, >k 3, 05060 0403 ｧ ﾀ00070504 05 005060 ﾀ03 0605000304 0403 0703 縺00600 0 k, k, +k k k 3, k. ｰ0306 陦ｧ0504 ﾀ07040505030 ｧ 06 縺050607, 0403006 ﾀ05 ｧ04 03 閨006 ｧ 0406 縺00 ｧ0 04 +k k 3 0 0006 090303 罍陦ｧ0 05. 90. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700308, 04 縺030803 閨070 ｧ08 06 縺06 0 0003 0903-03 罍陦ｧ0 05 0403 060500030 45 09 04 縺0700308 y =4, x. 9. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700308, 04 縺030803 閨070 ｧ08 06 縺06 0503 0906 A(,; ) 0403 060500030 45 09 04 縺0700308 x +3y =6. 9. 0ｷ3 0503 09 ｧ A(5; 4) 04030803 陦ｧ05 0006 ﾀ040605 0403 060500030 ' = arctg 0903 ﾀ ｧOx ｧ0305006060305 07 0605 ﾀ07. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 060060503 ｧ 0305 070600030503 0006 0608. 93. 0504 縺06 閨0600 ｧ0504 0406 縺00 ｧ003 ｧ 06050003 05 縺0606050300040 ｧ09, ﾀ0503 縺03003 09030503- 縺030503 003 06 040060 ｧ070 ｧ x +3y =0, x =3, x, y +3=0. 94. 0505 縺06 0309 04 縺0700308 3x +y = 6, 0305 ﾀ06 06000308 03 ﾀ070 ｧ 090303 - 陦ｧ0 05, ﾀ000607 ｧ05 05 ｧ04030506006 0308 040030306 陦縺0600030503 04 縺0700306050300040030503 05 縺06-06050300040 ｧ09. 7 0805 ｧ 0406 縺00 ｧ006 04 縺070030503 060500, 06 ﾀ00 ｧ ｧ 0406 ﾀ05003, 0503 030 000607 ｧ05 080403000609 000308 04 縺0700308. 95. 6 7 0 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(,; 0), B(; 4) ｧ C(4; 0). 7 04 ｧ ﾀ 0504 06 040060 ｧ07 ﾀ0503 縺030 05 縺0606050300040 ｧ09, 006 陦ｧ 003 AE, 0403 ﾀ030503 AD ｧ 0 0805 ｧ 閨00 ｧ006 006 陦ｧ 003 AE. 96. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 ﾀ0503 縺030 ｧ 0 0805 ｧ 06050003 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(0; 7), B(6;,) ｧ C(; ). 97. 7 縺070 07 x, y +8 = 0 0406 縺06 ﾀ0609 0605 03 ﾀ ｧOx ｧ Oy 040503 09 08 A ｧ B. ｯ03 09 M 閨0600 ｧ05 AB 04030500300060 ｧ ｧ AM : MB =3:. 7-04 ｧ ﾀ 0504 06 040060 ｧ06 0406 縺04060 陦ｧ09060007, 0403 ﾀ ﾀ05 04000600030503 04 0503 0906 M 09 04 縺0700308 AB. 98. 703 ﾀ05 縺03 ｧ0504 05 縺0606050300040 ｧ09, ﾀ0503 縺03003 09030503 縺030503 003 06 04006-0 ｧ070 ｧ x+y =4, 3x,y =0, x,3y,8 =0; 0 0805 ｧ 06050003 ｧ 0400030 閨04 05 縺0606050300040 ｧ09. 99. 7 0805 ｧ 0503 0906 0406 縺06 ﾀ06 060 ｧ07 006 陦ｧ 0 ｧ 0503 0906 0406 縺06 ﾀ06 060 ｧ07 0403- ﾀ0305 05 縺0606050300040 ｧ09, 0406 縺00 ｧ003 09030503 縺030503 A(,4; ), B(;,5) ｧ C(5; 0). 00. 0ｷ3 0503 09 ｧ A(,5; 6) 04030803 陦ｧ05 0006 ﾀ040605 0403 060500030 ' = = arctg (,) 0903 ﾀ ｧOx ｧ0305 07 0605 ﾀ07 0305 03 ﾀ ｧOx, 05060 0305 03 ﾀ ｧ Oy. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 ﾀ0608 05 縺06080006 0608.

3x 6. 703 縺0 000400306 06 040060 ｧ06 04 縺0700308 9 x 6. 703 縺0 000400306 06 040060 ｧ06 04 縺0700308. 7 ﾀ ﾀ0503070 ｧ06 0305 0503 09 ｧ 閨03 04 縺0700308. ｰ 040060 ｧ07 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ. ｰ 040060 ｧ06 0406 09 04 縺0700308, 04 縺030803 閨070 ｧ08 06 縺06 0503 0906 0406 縺06 ﾀ06 060 ｧ07 閨040608 000308 04 縺0700308. 703 縺0 000400306 06 040060 ｧ06 04 縺0700308 x cos + y sin, =0; () 05 閨06 閨00 ｧ0 0406 縺04060 陦ｧ09060007 (003 縺0 00 ｧ), 03040600600030503 ｧ 0 00 090303 - 陦ｧ0 05 0 04 縺0700606, 06050300 0 0900030 0505030503 0406 縺04060 陦ｧ09060007 09 03 ﾀ ｧ Ox. ｪ05030303 04 罍ｧ0406 ﾀ05 ｧ 030300606 06 040060 ｧ06 04 縺0700308 Ax + By + C =009 003 縺0 0004-003006 04 ｧ 閨06, 00607003 04 ﾀ06 0006003 060503 06000307 ｧ0504 0 003 縺0 ｧ 縺06060 ｧ08 000307 ｧ05060004 M =, 04 07050308 ﾀ03 0 09030, 04 縺0305 ｧ040304030003070030 0 0906 ﾀ04030303 - A + B 0030503 00060 C.. 7 ﾀ ﾀ0503070 ｧ06 d 0305 0503 09 ｧ (x 0 ; y 0 ) 閨03 04 縺0700308 0 08 閨060; 06 ﾀ00 ｧ 04 0006040606 ﾀ0504 003 縺0 00040030503 06 040060 ｧ07 04 縺0700308 0 006 ﾀ0503 05060906-0 ｧ08 090303 罍陦ｧ0 05 0403 陦ﾀ05 04 ｧ0 090303 罍陦ｧ0 0503 (x 0 ; y 0 ) ｧ 04030006 06000306 ｧ ﾀ0003 0403 040060 0403 03 ﾀ0300060500308 040600 ｧ ｧ006: d = jx 0 cos + y 0 sin, j; () ｧ00 ｧ d = jax 0 + By 0 + Cj A + B : ( 0 ) 3. ｰ 040060 ｧ07 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 0605000304 00607 閨06 04 縺070030 ｧ Ax + By + + C =0 ｧ A x + B y + C =0: Ax + By + C = A x + B y + C : (3) A + B A + B 4. ｰ 040060 ｧ06 0406 09 04 縺0700308,04 縺030803 閨070 ｧ08 06 縺06 0503 0906 0406 縺06- ﾀ06 060 ｧ07 閨040608 000308 04 縺0700308: (Ax + By + C) +(A x + B y + C )=0: (4) 030307003 0403000307 ｧ0504 =, ｧ ﾀ090006 ｧ04 0505 ｧ0 ｧ 0406 09 (4) 040503 縺0606 ｧ 0-00308 04 縺0700308. 0. 7 罍ｧ0406 ﾀ05 ｧ 09 003 縺0 0004003006 04 ｧ 閨06 06 040060 ｧ07 04 縺0700308: ) 3x, 4y, 0 = 0; ) x + y +3=0; 3) y = kx + b. 0. 703 ﾀ05 縺03 ｧ0504 04 縺0700606, 06 ﾀ00 ｧ 閨00 ｧ0 003 縺0 00 ｧ =, 06050300 0 0900030 0606 09 03 ﾀ ｧ Ox 04060: ) 45 ;) 35 ;3) 5 ;4) 35. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 0505 ｧ08 04 縺0700308. 03. 7 0805 ｧ ﾀ ﾀ0503070 ｧ07 0305 0503 0609 A(4; 3), B(; ) ｧ C(; 0) 閨03 04 縺0700308 3x +4y, 0=0. 703 ﾀ05 縺03 ｧ0504 0503 09 ｧ ｧ 04 縺0700606.

30 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 04. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0305 0 00 090303 罍陦ｧ0 05 閨03 04 縺0700308 x,, 5y +39=0. 05. 70309 0504, 0503 04 縺0700306 x, 3y =6 ｧ 4x, 6y =504 00-00060004003, ｧ 0 0805 ｧ ﾀ ﾀ0503070 ｧ06 00607 閨06 0 ｧ0 ｧ. ｰ 09 0 ｧ 06. 7 03 閨00308 ｧ 04 縺0700308 04 07050404 縺03 ｧ 04030004006060503 0906 ｧ 0 0805 ｧ ﾀ ﾀ0503070 ｧ06 0305 00606 閨03 陦縺06050308 04 縺0700308. 06. 7 0805 ｧ k ｧ 06 ﾀ000304 ｧ07, 0503 04 縺070 07 y = kx +5 06 00060 0305 0 00 090303 罍陦ｧ0 05 0 ﾀ ﾀ0503070 ｧ06 d = 5. 07. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 06 00060-00308 0305 04 縺0700308 4x, 3y =00 ﾀ ﾀ0503070 ｧ06 d =4. 08. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 04 縺0700308, 06 0006000308 0305 0503 09 ｧ A(4;,) 0 ﾀ ﾀ0503070 ｧ06 d =4 ｧ 04 000006000400308 04 縺0700308 8x, 5y =0. 09. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 0605000304 00607 閨06 04 縺070030 ｧ x +3y =0 ｧ 3x +y =0. 0. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 0605000304 00607 閨06 04 縺070030 ｧ 3x +4y = ｧ y =0.. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 06090503 罍ｧ ｧ 0503 09 ｧ M (x; y), 09030503 07 04 罍ｧ ﾀ0403060 閨04 ｧ07060 ｧ ｧ 03 ﾀ05 0605 ﾀ07 0405 縺0306 00040006 0305 04 縺0700308 y =x, 4, 060 0305 04 縺0700308 y =4, x.. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 M 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 x + y +6= 0 ｧ 3x +5y, 5 = 0 ｧ 06 縺06 0503 0906 N (;,) (006 0 0803 閨07 0503 09 ｧ M). 3. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 M 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 5x, y +0=0 ｧ 8x +4y +9=0 ｧ 04 00-0006000400308 04 縺0700308 x +3y =0(006 0 0803 閨07 0503 09 ｧ M). 4. 7 0805 ｧ 閨00 ｧ006 0403 ﾀ030503 BD 0405 縺0606050300040 ｧ0906 ﾀ 0406 縺00 ｧ0 0 ｧ A(,3; 0), B(; 5) ｧ C(3; ). 5. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 A(; 4) ｧ06 0006000308 0305 0 00 090303 罍陦ｧ0 05 0 ﾀ ﾀ0503070 ｧ06 d =. 6. 7 縺030406 罍ｧ0504, 0503 0503 09 ｧ A(,4;,3), B(,5; 0), C(5; 6) ｧ D(; 0) ﾀ000607 05 0406 縺00 ｧ0 0 ｧ 05 040609 ｧ ｧ, ｧ 0 0805 ｧ 0606 0403 ﾀ030506. 7. ｪ06 縺06 0 0003 090303 罍陦ｧ0 05 04 縺030406 閨060 04 縺070 07 0 03 陦ｧ0 090304030 ﾀ ﾀ0503070 ｧ ｧ 0305 0503 0609 A(; ) ｧ B(4; 0). 7 0805 ｧ 050503 ﾀ ﾀ0503070 ｧ06. 8. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 06 00060-00308 0305 04 縺0700308 x +y, 5=00 ﾀ ﾀ0503070 ｧ06, 0400306 5. 9. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 06090503 罍ｧ ｧ 0503 09 ｧ M (x; y), 09030503 07 04 罍ｧ ﾀ0403060 閨04 ｧ07060 ｧ ｧ 03 ﾀ05 0605 ﾀ07 04 閨040306 00040006 0305 04 縺0700308 y = x, 060 0305 04 縺0700308 y =,x. 0. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 M 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 x, 3y +5=0 ｧ 3x + y, 7=0 ｧ 0406 縺04060 陦ｧ- 09060007 縺00308 09 04 縺0700308 y =x (006 0 0803 閨07 0503 09 ｧ M).

3x 7. ｮ00600 000306 ｧ 0 04 縺0700606 x 7. ｮ00600 000306 ｧ 0 04 縺0700606. ｪ06 縺06 0 0003 090303 罍陦ｧ0 05 04 縺030406 ﾀ05 ｧ 04 縺0700606, 0303 060600606 ﾀ 04 縺070030 ｧ x + y = a ｧ x =005 縺0606050300040 ｧ09 0400030 閨0406 a.. 6 7 003 0503 09 ｧ A(,4; 0) ｧ B(0; 6). ｪ06 縺06 ﾀ06 縺06 陦ｧ006 0305 縺06 09 AB 04 縺030406 ﾀ05 ｧ 04 縺0700606, 0305 ﾀ0609 0600606 0 03 ﾀ ｧOx 0305 縺06 0309, 04 閨040306 03030004-00 ｧ08, 060 0 03 ﾀ ｧ Oy. 3. 6 7 003 0503 09 ｧ A(,; 0) ｧ B(;,). 7 0305 縺06 0906 OA 0403 ﾀ05 縺03060 04 000006000305 00 OACD, 陦ｧ 05030 00 ｧ 09030503 縺030503 0406 縺06 ﾀ0609 0605 ﾀ07 04 0503 0906 B. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 ﾀ0503 縺030, 陦ｧ 05030 000608 04 000006000305 00 ｧ 0 0805 ｧ 06050300 CAD. 4. 7 0805 ｧ 06050003 ｧ 0400030 閨04 05 縺0606050300040 ｧ09, 0303 0304 00030503 04 縺07-0030 ｧ y =x, y =,x ｧ y = x + b. 5. 0ｷ3 0 00 090303 罍陦ｧ0 05 04 縺030406 閨06003 閨0406 04 ｧ0003 0406 縺04060 陦ｧ0906-0007 縺00306 04 縺0700306, 0303 06060 ｧ06 ﾀ 04 縺0700308 x +y = a 040030306 陦縺06000308 05 縺0606050300040 ｧ09. 7 0805 ｧ 0400030 閨04 0505030503 05 縺0606050300040 ｧ09. 6. 7 0805 ｧ 0400605 縺0600 ｧ06 06050003 05 縺0606050300040 ｧ09, 06 ﾀ00 ｧ 003 06 04006-0 ｧ07 060503 ﾀ0503 縺030: (AB) x, 3y +3 = 0 ｧ (AC) x +3y +3 = 0 ｧ 03 ﾀ00304 0 ｧ06 D(,; 3) 0403 ﾀ030503 AD. 7. 6 7 003 06 040060 ｧ07 03030903040308 ﾀ0503 縺030 040030306 陦縺0600030503 05 縺06-06050300040 ｧ09 3x + y =0 ｧ x, 3y =0 ｧ0503 09 (5; 0) 0 060503 03 ﾀ00304 0 ｧ ｧ. 7 0805 ｧ 0406 罍ｧ00605 ｧ 0400030 閨04 05 縺0606050300040 ｧ09. 8. 6 5 05 縺0606050300040 ｧ0906 ABC 003: ) 06 040060 ｧ06 ﾀ0503 縺03003 (AB) 3x +y =;)06 040060 ｧ06 0403 ﾀ030503 (BM) x +y =4;3)06 040060 ｧ06 0403 ﾀ030503 (AM ) 4x + y = 6, 05 閨06 M 0503 09 0406 縺06 ﾀ06 060 ｧ07 0403 ﾀ0305. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 ﾀ0503 縺030 AC, BC ｧ 0403 ﾀ030503 CM. 9. 6 70406 ﾀ0503 縺03003 04 000006000305 00 003 06 040060 ｧ070 ｧ y = = x, ｧ 5y = x +6. 6 7 ｧ 05030 00 ｧ 060503 0406 縺06 ﾀ0609 0605 ﾀ07040 0006 090303 - 陦ｧ0 05. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 閨040608 陦縺0605 ｧ08 ﾀ0503 縺030 04 000006000305 00 ｧ 060503 陦ｧ 05030 000608. 30. 6 7 0 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(0;,4), B(3; 0) ｧ C(0; 6). 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0406 縺00 ｧ003 C 0305 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ03 060500 A. 3. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 06090503 罍ｧ ｧ 0503 09 ｧ M (x; y), 閨04 ｧ0706-00608 ﾀ07 05 09, 0503 ﾀ0600 ﾀ ﾀ0503070 ｧ08 0305 00606 閨03 04 縺0700308 y = x ｧ y =,x= 03 ﾀ05 0605 ﾀ07 0403 ﾀ050307000308 ｧ 0400308 5. 3. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ05 ｧ, 090303 罍陦ｧ0 0503 0503 0609 09030503 縺0308 06 閨03040006050403- 縺070605 006 04060 ﾀ0504 0: ) x, <y<0 ｧ x>0; ), 6 y 6 x 6 ; 3) < x + y<8, x>0 ｧ y>0. 33. ｮ0503 縺03003 AB ｧ BC 04 000006000305 00 003 06 040060 ｧ070 ｧ x,y +5=0 ｧ x,y +4 = 0, 陦ｧ 05030 00 ｧ 060503 0406 縺06 ﾀ0609 0605 ﾀ07 04 0503 0906 M (; 4). 7 0805 ｧ 閨00 ｧ003 060503 0403 ﾀ0305. 34. 7 0805 ｧ 0406 縺00 ｧ003 04 縺0700306050300040030503 040030306 陦縺0600030503 05 縺06-06050300040 ｧ09, 06 ﾀ00 ｧ 0 0406 縺00 ｧ0 04 縺070030503 060500 C(3;,) ｧ 06 04006-0 ｧ06 05 ｧ04030506006 03 3x, y +=0.

3 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 35. 6 7 003 閨0406 0406 縺00 ｧ003 05 縺0606050300040 ｧ09 A(,4; 3) ｧ B(4;,) ｧ 0503 09 0406 縺06 ﾀ06 060 ｧ07 0403 ﾀ0305 M (3; 3). 7 0805 ｧ 05 縺06050406 0406 縺00 ｧ006 C. 36. 6 503 ｧ ﾀ00 ｧ0504 090303 罍陦ｧ0 0503 0406 縺00 ｧ003 縺03003, 06 ﾀ00 ｧ ｧ 0406 ﾀ05003 06 040060 ｧ07 閨040608 060503 ﾀ0503 縺030: x +y =4 ｧ x +y =0, ｧ06 040060 ｧ06 03 閨00308 ｧ 060503 陦ｧ 05030 000608: y = x +. 37. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ07 ﾀ0503 縺030 05 縺0606050300040 ｧ09, 0 07 03 閨006 060503 0406 縺00 ｧ006 A(0; ) ｧ06 040060 ｧ07 0403 ﾀ0305: (BM) x + y = 4 ｧ (CM) y =x, 05 閨06 M 0503 09 0406 縺06 ﾀ06 060 ｧ07 0403 ﾀ0305. 38. 6 7 003 04 縺070 07 x +y, 4 = 0 ｧ0503 09 A(5; 7). 7 0805 ｧ: ) 04 縺03060909 ｧ06 B 0503 09 ｧ A 0 000606 04 縺0700606; ) 0305 07060 ｧ06 C 0503 09 ｧ A 04 000308 04 縺0700308. ｰ 09 0 ｧ 06. 7 04 ｧ ﾀ 04 06 040060 ｧ06 0406 縺04060 陦ｧ09060007 AB ｧ 縺0600 ｧ04 060503 ﾀ0304006 ﾀ05003 ﾀ 06 040060 ｧ060 000308 04 縺0700308, 0 08 閨060 0503 0906 B, 09030503 07 06 ﾀ0504 ﾀ06 縺06 陦ｧ0 AC. 39. 6 7 0 04 縺070 07 x + y, 6=0 ｧ 0 00608 閨0406 0503 09 ｧ A ｧ B ﾀ 03 罍陦ｧ0 05 0 ｧ y A =6 ｧ y B =,. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0403 ﾀ030503 AD 05 縺0606050300040 ｧ09 AOB, 0 0805 ｧ 0606 閨00 ｧ006 ｧ DAB. x 8. 0509 縺0607003 ﾀ0504 ｰ 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ ﾀ0906005 縺030 04 0503 0906 C(a; b) ｧ 陦ｧ06 ﾀ030, 040030 R: (x, a) +(y, b) = R : () ｭ ﾀ00 ｧ 04 06 040060 ｧ ｧ () ﾀ09 縺030504 ﾀ09030309 ｧ, 0503 03003 04 罍ｧ00605 04 ｧ x + y + mx + ny + =0: () ｪ05030303 0305 06 040060 ｧ07 () 0304070504 0406 縺060805 ｧ 09 06 040060 ｧ06 04 ｧ (), 00607003 04 0006040308 ﾀ05 ｧ 06 040060 ｧ07 () 0403 閨0600 ｧ0504 04030000306 0904 陦 0503: x + m + y + n = m 4 + n, : (3) 4 40. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ ﾀ 0906005 縺030 C(,4; 3), - 陦ｧ06 ﾀ030 R =5 ｧ 0403 ﾀ05 縺03 ｧ0504 0606. 6ｧ80607 05 00 ｧ 0 05050308 0309 縺0607003 ﾀ05 ｧ 0503 09 ｧ A(,;,), B(3; ), O(0; 0)? 4. 6 7 0 0503 09 (,4; 6). 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, 陦ｧ - 00605 縺030 09030503 縺0308 ﾀ000607 ｧ05 0305 縺06 0309 OA. 4. 703 ﾀ05 縺03 ｧ0504 0309 縺0607003 ﾀ05 ｧ: ) x + y, 4x +6y, 3 = 0; ) x + y, 8x =0; 3) x + y +4y =0. 43. 703 ﾀ05 縺03 ｧ0504 0309 縺0607003 ﾀ0504 x + y +5x =0, 04 縺0700606 x + y =0 ｧ 0 0805 ｧ 0503 09 ｧ ｧ08 0406 縺06 ﾀ06 060 ｧ07. 44. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, 09 ﾀ 0600608 ﾀ07 03 ﾀ0608 090303 - 陦ｧ0 05 ｧ 04 縺030803 閨0700608 06 縺06 0503 0906 A(; ).

3x 8. 0509 縺0607003 ﾀ0504 3 45. 7 0805 ｧ 06050300 00607 閨06 陦ｧ06 ﾀ 0 ｧ 0309 縺0607003 ﾀ05 ｧ x + y +4x,, 6y =0, 04 縺030406 閨0600030 ｧ 04 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 03 ﾀ0406 Oy. 46. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 09 ｧ A(,; 3), B(0; ) ｧ C(;,). ｰ 09 0 ｧ 06. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 ｧ ﾀ090300308 0309 縺0607003 ﾀ05 ｧ 04 04 ｧ 閨06 x + + y + mx + ny + =0, 0403 陦ﾀ05 04 ｧ0504 04 0060503 090303 罍陦ｧ0 0503 09 07 閨0308 0503 09 ｧ ｧ 05060 0 0805 ｧ m, n ｧ. 47. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 0309 縺0607003 ﾀ05 ｧ x + y +4x, 4y =0 ﾀ 04 縺0700308 y =,x ｧ 06 縺06 0503 0906 A(4; 4). 48. 0504 縺06 閨0600 ｧ0504 030300 ﾀ0504 ﾀ0403000307060 ｧ07 09 罍ｧ040308 y =,,x, 4x. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040606. 49. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 0309 縺0607003 ﾀ05 ｧ x + y,, 8x, 4y +6=0, 04 縺030406 閨06000308 ｧ 0 00 090303 罍陦ｧ0 05. 50. 6 7 0 0503 09 A(a; 0). ｯ03 09 M 閨04 ｧ070605 ﾀ07 05 09, 0503 04 4OMA 06050300 OMA 03 ﾀ05 0605 ﾀ07 04 縺070030. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 閨04 ｧ07060 ｧ07 0503 09 ｧ M. 5. 6 7 003 0503 09 ｧ A(,6; 0) ｧ B(; 0). 7 0805 ｧ 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609, ｧ 09030503 縺0308 0305 縺06 09 ｧ OA ｧ OB 04 ｧ 閨003 0403 040030 ｧ 060500 0 ｧ. 5. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 0503 09 ｧ M (x; y), 閨04 ｧ070600608 ﾀ07 05 09, 0503 ﾀ0600 0904 陦 050304 ﾀ ﾀ0503070 ｧ08 0305 00606 閨03 0503 0609 A(,a; 0), B(0; a) ｧ C(a; 0)03 ﾀ05 0605 ﾀ07 0400308 3a. 53. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 0503 09 ｧ M (x; y), 閨04 ｧ070600608 ﾀ07 05 09, 0503 ﾀ0600 0904 陦 050304 ﾀ ﾀ0503070 ｧ08 0305 00606 閨03 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 090303 罍陦ｧ0 05-00308 0605000304 03 ﾀ05 0605 ﾀ07 0400308 a. 54. 6 7 0 0309 縺0607003 ﾀ0504 x + y = a. 0ｷ3 06060503 09 ｧ A(a; 0)04 縺03-0406 閨06003 04 ﾀ060403 0030700306 0803 罍閨03. 0504 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 ﾀ06 縺06 陦ｧ0 0505 ｧ08 0803 罍. 55. 6 7 003 0503 09 ｧ A(,3; 0) ｧ B(3; 6). 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309- 縺0607003 ﾀ05 ｧ, 陦ｧ 00605 縺030 09030503 縺0308 ﾀ000607 ｧ05 0305 縺06 0309 AB. 56. 7 0805 ｧ 0906005 縺03 ｧ 陦ｧ06 ﾀ03 0309 縺0607003 ﾀ050608: ) x + y, 6x + +4y, 3 = 0; )x + y +5x, 7y +; 5=0;3)x + y +7y =0. 703 ﾀ05 縺03 ｧ0504 0309 縺0607003 ﾀ05 ｧ. 57. 0509 縺0607003 ﾀ0504 09 ﾀ 0605 ﾀ07 03 ﾀ ｧ Ox 040 0006 090303 罍陦ｧ0 05 ｧ 04 縺03-0803 陦ｧ05 06 縺06 0503 0906 A(0;,4). 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ ｧ 0 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 0 ｧ 090303 罍陦ｧ0 0500308 0605000304. 58. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0-0003 090303 罍陦ｧ0 05 ｧ 06 縺06 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 x + y + a =0 ﾀ 0309 縺0607003 ﾀ050406 x + y = a.

34 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 59. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308, 04 縺030406 閨06000308 ｧ 0-00 090303 罍陦ｧ0 05 09 0309 縺0607003 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 09 ｧ A(;,), B(0;,) ｧ C(,3; 0). 60. 7 0805 ｧ 06050300 00607 閨06 陦ｧ06 ﾀ 0 ｧ 0309 縺0607003 ﾀ05 ｧ x + y, 4x + +6y, 5=0, 04 縺030406 閨0600030 ｧ 04 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 03 ﾀ0406 Ox. 6. 70309 0504, 0503 0503 09 A(3; 0) 000607 ｧ05 0400605 罍ｧ 0309 縺0607003 ﾀ05 ｧ x + + y, 4x +y +=0, ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ06 0803 罍閨03, 閨06000700608 ﾀ07 04 0503 0906 A 0403040300 0. ｰ09 0 ｧ06. 0ｷ3 ﾀ09030 07 0803 罍 0406 縺04060 陦ｧ09060007 縺0 09 CA,05 閨06 C 0906005 0309 縺0607003 ﾀ05 ｧ. 6. ｯ03 09 M (x; y) 閨04 ｧ070605 ﾀ07 05 09, 0503 ﾀ0600 0904 陦 050304 ﾀ ﾀ0503-070 ｧ08 0305 00606 閨03 0 00 090303 罍陦ｧ0 05 ｧ 閨03 0503 09 ｧ A(,a; 0) 03 ﾀ05 0605 ﾀ07 0400308 a. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 閨04 ｧ07060 ｧ07 0503 09 ｧ M. 63. 6 7 0 0309 縺0607003 ﾀ0504 x + y =4.0ｷ3 0503 09 ｧ 0606 A(,; 0) 04 縺03-0406 閨060 0803 罍 AB ｧ 04 縺03 閨030007060 0 ﾀ ﾀ0503070 ｧ06 BM = AB. 0504 縺06- 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 M. 64. 0505 縺06 0309 AM = a 0406 縺060060 0605 ﾀ07 0403 040003 ﾀ0903 ﾀ05 ｧ xoy, 03 ﾀ05-04 07 ﾀ04 04 00000600040030 Ox, 05 09, 0503 0006040308 090300609 060503 A ﾀ09030004 ｧ05 0403 0309 縺0607003 ﾀ05 ｧ x + y = a. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 閨04 ｧ07060 ｧ07 0503 09 ｧ M. x 9. 70000 ｧ04 ﾀ 70000 ｧ04 ﾀ030 0 0304 0605 ﾀ07 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609; ﾀ0600 ﾀ- ﾀ0503070 ｧ08 0305 09 07 閨0308 ｧ 09030503 縺0308 閨03 閨040608 000308 0503 0609 F ｧ F (07030906- ﾀ0304) 06 ﾀ0504 0403 ﾀ05030700 07 040600 ｧ ｧ0 a; 0303000400 07 F F. 07 0030 ｧ 06 ﾀ090306 (04 縺03 ﾀ050608000606) 06 040060 ｧ06 050000 ｧ04 ﾀ x a + y =: () b 70000 ｧ04 ﾀ, 000308 06 040060 ｧ060 (), ﾀ ｧ000605 罍ｧ 060 0305003 ﾀ ｧ05060004003 03 ﾀ0608 090303 罍陦ｧ0 05 ( 罍ｧ ﾀ. ). 7 00605 縺03 a ｧ b 0 0304 0605 ﾀ07 0403000603 ﾀ070 ｧ 050000 ｧ04 ﾀ. 7 ｧ ﾀ. 706 ﾀ0504 a>b, 050305 07030906 ﾀ03 F ｧ F 0 0803 閨0705 ﾀ07 0 03 ﾀ ｧ Ox 0 ﾀ ﾀ0503070 ｧ ｧ c = a, b 0305 0906005. 050500300060 ｧ06 c = e< 0 0304 0605 ﾀ07 0509 ﾀ0906005 罍ｧ-

3x 9. 70000 ｧ04 ﾀ 5 ﾀ ｧ050605030 050000 ｧ04 ﾀ. 7 ﾀ ﾀ0503070 ｧ07 0305 0503 09 ｧ M (x; y) 050000 ｧ04 ﾀ 閨03 060503 07030906 ﾀ0304 (070309 000400306 陦ｧ06 ﾀ-0406090503 縺03) 0304 縺06 閨0600070605 ﾀ07 0703 縺00600 0 ｧ r = a, ex; r = a + ex: () ｭ ﾀ00 ｧ 0706 a<b,050307030906 ﾀ03 0 0803 閨0705 ﾀ07 0 03 ﾀ ｧ Oy, c = b, a, e = c b, r = b ey. 65. 703 ﾀ05 縺03 ｧ0504 050000 ｧ04 ﾀ x +4y =6, 0 0805 ｧ 060503 07030906 ﾀ03 ｧ 0509 ﾀ- 0906005 罍ｧ ﾀ ｧ050605. 66. 7 04 ｧ ﾀ 0504 09 0030 ｧ 06 ﾀ090306 06 040060 ｧ06 050000 ｧ04 ﾀ, 0 07, 0503: ) ﾀ ﾀ0503070 ｧ06 00607 閨06 07030906 ﾀ 0 ｧ 04003 8, 0 00 07 0403000603 ﾀ04 b = 3; ) 0303000400 07 0403000603 ﾀ04 a =6, 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 e =0; 5. 67. 7 0805 ｧ 0 000606 0403000603 ﾀ04 b ｧ 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 e 050000 ｧ04 ﾀ, ｧ006-060060503 03030004000606 0403000603 ﾀ04 a =5 ｧ 04 00605 c, 0400308: ) 4,8; ) 4; 3) 3; 4),4; 5) 0. 703 ﾀ05 縺03 ｧ0504 09 07 閨0308 ｧ 050000 ｧ04 ﾀ0304. 68. 6ｦ00600007 閨04 ｧ070605 ﾀ07 0403 050000 ｧ04 ﾀ06, 04 03 閨0030 ｧ 07030906 ﾀ0304 09030503 縺030503 0 0803 陦ｧ05 ﾀ07 ｮ030000906. 7 ｧ006004000606 ﾀ ﾀ0503070 ｧ06 0305 6ｦ006000 ｧ 閨03 ｮ0300009 04003 04 罍ｧ0300 ｧ ｧ05060004003 47,5 0000 090, 0 ｧ03030004000606 5,5 0000 090. 7 0805 ｧ 03030004000606 0403000603 ﾀ04 ｧ 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 03 縺03 ｧ0503 6ｦ006000 ｧ. 69. 70000 ｧ04 ﾀ, ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004003 03 ﾀ0608 090303 罍陦ｧ0 05, 04 縺03-0803 陦ｧ05 06 縺06 0503 09 ｧ M (; 3) ｧ B(0; ). 7 04 ｧ ﾀ 0504 060503 06 040060 ｧ06 ｧ 0 0805 ｧ ﾀ ﾀ0503070 ｧ07 0305 0503 09 ｧ M 閨03 07030906 ﾀ0304. 70. 70000 ｧ04 ﾀ, ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004003 03 ﾀ0608 090303 罍陦ｧ0 05, 07030906 ﾀ03 09030503 縺030503 0 0803 閨0705 ﾀ07 0 03 ﾀ ｧ Ox, 04 縺030803 陦ｧ05 06 縺06 0503 0906 M (,4; ) ｧ ｧ0060605 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 e = 3. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 4 050000 ｧ04 ﾀ ｧ 0 0805 ｧ 070309 000400306 陦ｧ06 ﾀ-0406090503 縺03 0503 09 ｧ M. 7. 7 0805 ｧ 閨00 ｧ006 0803 罍閨03 050000 ｧ04 ﾀ x +y =8, 閨06000700608 06050300 00607 閨06 03 ﾀ070 ｧ 0403040300 0. 7. 7 0805 ｧ 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 050000 ｧ04 ﾀ, 06 ﾀ00 ｧ ﾀ ﾀ0503070 ｧ06 00607 閨06 07030906 ﾀ 0 ｧ 04003 ﾀ ﾀ0503070 ｧ06 00607 閨06 0903009 0 ｧ 03030004000308 ｧ 0 000308 0403-000603 ﾀ0608. 73. 6 5 050000 ｧ04 ﾀ x +4y =40404 ｧ ﾀ 0 04 04 ｧ000400308 05 縺0606050300040 ｧ09, 03 閨0 ｧ 0406 縺00 ｧ0 09030503 縺030503 ﾀ030404 0605 ﾀ 0903009030 03030004000308 0403000603 ﾀ ｧ. 0504 縺06 閨0600 ｧ0504 090303 罍陦ｧ0 0503 閨040608 陦縺0605 ｧ08 0406 縺00 ｧ0 05 縺0606050300040 ｧ09. ｰ 09 0 ｧ 06. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 03 閨00308 ｧ ﾀ0503 縺030, ｧ0060600608 0 090003k = tg 30, ｧ 0 0805 ｧ 0503 09 ｧ 0606 0406 縺06 ﾀ06 060 ｧ07 ﾀ 050000 ｧ04 ﾀ030. 74. 7 050000 ｧ04 ﾀ06 9x +5y = 5 0 0805 ｧ 0503 0906, ﾀ ﾀ0503070 ｧ06 0305 09030503 縺0308 閨03 04 04030503 07030906 ﾀ 04 060503 縺06 030300040006 ﾀ ﾀ0503070 ｧ07 0305 00606 閨03 000604030503 07030906 ﾀ. 75. 05 罍陦ｧ0 0503 04 ﾀ0608 0503 0609 0309 縺0607003 ﾀ05 ｧ x + y =36 ﾀ0309 006003 0405 縺0306. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04030006 06000308 003040308 09 罍ｧ040308. 76. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 0503 09 ｧ M, 09030503 07 04 罍ｧ ﾀ0403060 閨04 ｧ- 07060 ｧ ｧ 03 ﾀ05 0605 ﾀ07 04 閨040306 0300 ｧ0706 09 0503 0906 F (,; 0), 060 09 04 縺0700308 x =,4.

36 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 77. 0505 縺06 0309 AB 0403 ﾀ050307000308 閨00 ｧ003 a + b 閨04 ｧ070605 ﾀ07 05 09, 0503 060503 090300609 A ﾀ09030004 ｧ05 0403 03 ﾀ ｧ Ox, 090300609 B 040303 ﾀ ｧOy. 0504 縺06- 閨0600 ｧ0504 05 06090503 罍ｧ06 閨04 ｧ07060 ｧ07 0503 09 ｧ M 0305 縺06 09, 閨06000700608 060503 0 ﾀ05 ｧ BM = a ｧ MA = b (050000 ｧ0405 ｧ 06 ﾀ09 ｧ08 09 ｧ 縺09060004 6ｧ806030 罍閨03 6 5 ｧ0 ｧ). 78. 6 7 003 0309 縺0607003 ﾀ05 ｧ x + y = b ｧ x + y = a (b<a). 7 縺03 ｧ 0403000400308 0006 OBA 0406 縺06 ﾀ0609 0605 ｧ08 ﾀ030305040605 ﾀ0504060003 04 0503 09 08 B ｧ A, ｧ 09030503 縺0308 04 縺030406 閨06003 04 縺0700306, 04 000006000400306 03 ﾀ070 090303 罍陦ｧ- 0 05, 閨03 0406 縺06 ﾀ06 060 ｧ07 ｧ08 04 0503 0906 M. 0504 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 M. 79. 7 04 ｧ ﾀ 0504 04 縺03 ﾀ050608000606 06 040060 ｧ06 050000 ｧ04 ﾀ, 06 09030503 縺030503 ﾀ- ﾀ0503070 ｧ07 0305 03 閨0030503 ｧ 07030906 ﾀ0304 閨03 09030090304 03030004000308 03 ﾀ ｧ 04003 5 ｧ. 80. 70000 ｧ04 ﾀ, ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004003 03 ﾀ0608 090303 罍陦ｧ0 05, 04 縺03-0803 陦ｧ05 06 縺06 0503 09 ｧ M ( 3; 6) ｧ A(6; 0). 7 04 ｧ ﾀ 0504 060503 06 040060 ｧ06, 0 0805 ｧ 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 ｧ ﾀ ﾀ0503070 ｧ07 0305 0503 09 ｧ M 閨03 07030906 ﾀ0304. 8. 7 0805 ｧ 閨00 ｧ006 0803 罍閨03 050000 ｧ04 ﾀ x a + y =, 0 04 040006000308 b 0403 陦ｧ 05030 00 ｧ 04 縺0700306050300040 ｧ09, 0403 ﾀ05 縺030600030503 0 03 ﾀ0708 050000 ｧ04 ﾀ. 8. 7 0805 ｧ 03030 ｧ06 0503 09 ｧ 050000 ｧ04 ﾀ x +4y =4 ｧ0309 縺0607003 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 07030906 ﾀ03 050000 ｧ04 ﾀ ｧ ｧ0060600608 0906005 04 060503 080406 縺08-0060809 0406 縺00 ｧ006. 83. 7 04 縺0700308 x =,5 0 0805 ｧ 0503 0906, 03 陦ｧ0 09030403 06 0006000606 0305 0800060403050309 07030906 ﾀ ｧ0305080406 縺080060809 0406 縺00 ｧ003 050000 ｧ04 ﾀ x +5y =0. 84. 7 050000 ｧ04 ﾀ06 x +5y = 00 0805 ｧ 0503 0906, 陦ｧ06 ﾀ-0406090503 縺03 09030503 縺0308 0406 縺04060 陦ｧ09060007 縺003. ｰ09 0 ｧ06. 0ｷ3 ﾀ090300306 0503 09 ｧ ﾀ060504 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 050000 ｧ04 ﾀ030 0309 縺0607003 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 07030906 ﾀ03 050000 ｧ04 ﾀ ｧ ｧ0060600608 0906005 04 0-0006 090303 罍陦ｧ0 05. 85. 縺03 ﾀ09 ｧ ﾀ ﾀ03 0503 0609 0309 縺0607003 ﾀ05 ｧ x + y =406040600 ｧ 06003 04 閨040306. 0504 縺06 閨0600 ｧ0504 04030006 06000606 09 罍ｧ040606. 86. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 0503 09 ｧ M, 09030503 07 04 罍ｧ ﾀ0403060 閨04 ｧ- 07060 ｧ ｧ 03 ﾀ05 0605 ﾀ07 0405 縺0306 0300 ｧ0706 09 0503 0906 A(; 0), 060 09 04 縺0700308 x =9. x 0. 06 ｧ0406 縺030300 06 ｧ0406 縺0303000308 0 0304 0605 ﾀ07 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609; 003 ﾀ0504 ﾀ ﾀ0503070 ｧ08 0305 09 07 閨0308 ｧ 09030503 縺0308 閨03 閨040608 000308 0503 0609 F ｧ F (0703-0906 ﾀ0304) 06 ﾀ0504 0403 ﾀ05030700 07 040600 ｧ ｧ0 a (0 < a <F F ). 07 0030 ｧ 06 ﾀ090306 (04 縺03 ﾀ050608000606) 06 040060 ｧ06 05 ｧ0406 縺03030003 x a, y =: () b

3x 0. 06 ｧ0406 縺030300 7 06 ｧ0406 縺030300, 00 07 06 040060 ｧ060 (), ﾀ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 03 ﾀ0608 090303 罍陦ｧ0 05 ( 罍ｧ ﾀ. ). 050 0406 縺06 ﾀ0609 0605 03 ﾀ04Ox 040503 09 08 A(a; 0) ｧ A (,a; 0) 0406 縺00 ｧ0 08 05 ｧ0406 縺03030003 ｧ 006 0406 縺06 ﾀ0609 0605 03 ﾀ04Oy. 7 00605 a 0 0304 0605 ﾀ07 0406-006 ﾀ050406000308 0403000603 ﾀ0406, b 00 ｧ00308 0403000603 ﾀ0406. 7 00605 c = a + b 7 ｧ ﾀ. 06 ﾀ0504 ﾀ ﾀ0503070 ｧ06 0305 07030906 ﾀ 閨03 0906005. 050500300060 ｧ06 c a = e> 0 0304 0605 ﾀ07 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605030 05 ｧ0406 縺03030003. 7 縺0700306 y = b x 0 0304 0605 ﾀ07 ﾀ ｧ004a 050305 0 ｧ 05 ｧ0406 縺03030003. 7 ﾀ ﾀ0503070 ｧ07 0305 0503 09 ｧ M (x; y) 05 ｧ0406 縺03030003 閨03 0606 0703-0906 ﾀ0304 (070309 000400306 陦ｧ06 ﾀ-0406090503 縺03) 0304 縺06 閨0600070605 ﾀ07 0703 縺00600 0 ｧ r = jex, aj; r = jex + aj: () 06 ｧ0406 縺030300, 06 09030503 縺0308 a = b, 0 0304 0605 ﾀ07 04003 ﾀ0503 縺03000608, 060606 04006-0 ｧ06 x,y = a, 06 040060 ｧ07 ﾀ ｧ004050305 y = x. 06 ｧ0406 縺03030003 x a, y b = ｧ y b, x =0 0304 0605 ﾀ07 ﾀ0304 縺07070600030 ｧ. a 87. 703 ﾀ05 縺03 ｧ0504 05 ｧ0406 縺03030006 x,4y =6 ｧ 0606 ﾀ ｧ00405030503. 7 0805 ｧ 07030906 ﾀ03, 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 ｧ 06050300 00607 閨06 ﾀ ｧ004050305 0 ｧ. 88. 7 05 ｧ0406 縺03030006 x, 4y =604 0705 0503 09 M ﾀ 03 罍陦ｧ0 050308, 0400308. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0305 00606 閨03 07030906 ﾀ0304. 89. 7 04 ｧ ﾀ 0504 09 0030 ｧ 06 ﾀ090306 06 040060 ｧ06 05 ｧ0406 縺03030003, 0 07, 0503: ) ﾀ ﾀ0503070 ｧ06 00607 閨06 07030906 ﾀ 0 ｧ c =0, 00607 閨06 0406 縺00 ｧ0 0 ｧ a = = 8; ) 0406006 ﾀ05040600 07 0403000603 ﾀ04 a = 5, 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 e = = ;. 90. 06 ｧ0406 縺030300 ﾀ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 03 ﾀ0608 090303 罍陦ｧ0 05, 04 縺03-0803 陦ｧ05 06 縺06 0503 0906 M (6;, ) ｧ ｧ0060605 00 ｧ00606 0403000603 ﾀ04 b =. 7 04 ｧ ﾀ 0504 060606 040060 ｧ06 ｧ 0 0805 ｧ ﾀ ﾀ0503070 ｧ07 0305 0503 09 ｧ M 閨03 07030906- ﾀ0304.

38 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 9. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 ｧ0406 縺03030003, ｧ0060600608 0406 縺00 ｧ003 04 0703-0906 ﾀ 08, 07030906 ﾀ03 04 0406 縺00 ｧ0 08 050000 ｧ04 ﾀ x 5 + y =. 9 9. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 ｧ0406 縺03030003, ｧ0060600608 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 e =,04 縺030803 閨0700608 06 縺06 0503 0906 (a; a 3) ｧ ﾀ ｧ000605 罍ｧ 00308 0305003- ﾀ ｧ05060004003 03 ﾀ0608 090303 罍陦ｧ0 05. 93. 703 ﾀ05 縺03 ｧ0504 05 ｧ0406 縺03030006 y = a + x, 0 0805 ｧ 090303 罍陦ｧ0 0503 0606 07030906 ﾀ0304 ｧ 06050300 00607 閨06 ﾀ ｧ004050305 0 ｧ. 94. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 09 05 ｧ0406 縺03030006 x, 4y = =6, 04 縺030406 閨06000308 ｧ 0503 09 ｧ A(0;,). 95. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0305 07030906 ﾀ 05 ｧ0406 縺03030003 x a, y = 閨03 0606 b ﾀ ｧ004050305 ｧ 06050300 00607 閨06 ﾀ ｧ004050305 0 ｧ., y 96. 7 0805 ｧ ﾀ0503 縺03006 0904 陦 05, 0404 ｧ ﾀ 00030503 04 05 ｧ0406 縺03030006 x a, =, ｧ ｧ ﾀ ﾀ0006 閨0304 0504, 04 09 09 ｧ06 05 ｧ0406 縺03030003 00307003 0404 ｧ ﾀ 0504 0904 陦 05. b 97. 7 0805 ｧ 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 05 ｧ0406 縺03030003, ﾀ ｧ004050305 09030503 縺0308 ﾀ03- ﾀ05 0400070605 ﾀ 0406006 ﾀ050406000308 03 ﾀ0406 06050300: ) 60 ;). 98. 0504 縺06 閨0600 ｧ0504 030300 ﾀ0504 ﾀ040300030706-0 ｧ07 09 罍ｧ040308 y =, 9+x. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040606. 99. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 0503 09 ｧ M (x; y), 09030503 07 04 罍ｧ ﾀ0403060 閨04 ｧ07060 ｧ ｧ 03 ﾀ05 0605 ﾀ07 04 閨040306 0300 ｧ0706 09 04 縺0700308 x =, 060 09 0503 0906 F (4; 0). 00. 6 7 003 0503 09 ｧ A(,; 0) ｧ B(; 0). ｯ03 09 M 閨04 ｧ070605 ﾀ07 05 09, 0503 04 4AMB 06050300 B 03 ﾀ05 0605 ﾀ07 04 閨040306 030300040006 060500 A. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 閨04 ｧ07060 ｧ07. 0. 6 7 0 0503 09 A(a; 0). 70303 ﾀ ｧOy 閨04 ｧ070605 ﾀ07 0503 09 B. 7 04 縺0700308 BE, 04-000006000400308 Ox, 03050900 閨0304 0605 ﾀ07 0305 縺06 09 ｧ 7 ｧ ﾀ. 3 BM ｧ BM, 0400306 AB. 0504 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 M ｧ M. 0. 6 7 003 04 縺0700306 x = b ｧ x = a (b <a). 7 縺03 ｧ 0403000400308 0006 OA ( 罍ｧ ﾀ. 3) 0406 縺06 ﾀ0609 0605 04 縺0700606 x = b ( ｧ00 ｧ x =,b) 040503 0906 B ｧ 04 縺0700606 x = a ( ｧ00 ｧ x =,a) 040503 0906 A. 7 陦ｧ06 ﾀ030 OA 0304 ｧ ﾀ 0 閨0605, 0406 縺06 ﾀ0609 060 07 Ox 040503 0906 C. 0ｷ3 0503 0609 B ｧ C 04 縺030406 閨06003 04 縺0700306, 04 000006000400306 ﾀ030305040605 ﾀ0504060003 Ox ｧ Oy, 閨03 0406 縺06 ﾀ06 060 ｧ07 04 0503 0906 M. 0504 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 M. 03. 7 04 ｧ ﾀ 0504 09 0030 ｧ 06 ﾀ090306 06 040060 ｧ06 05 ｧ0406 縺03030003, 0 07, 0503 ﾀ ﾀ0503070 ｧ07 0305 03 閨00308 ｧ 0606 0406 縺00 ｧ0 閨03 07030906 ﾀ0304 04003 9 ｧ.

3x. 7 030300 9 04. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ ｧ004050305 05 ｧ0406 縺03030003 x, 3y = = ﾀ 0309 縺0607003 ﾀ050406, ｧ0060600608 0906005 04 04 04030 07030906 ﾀ06 05 ｧ0406 縺03030003 ｧ04 縺030803 閨0700608 06 縺06 0 0003 090303 罍陦ｧ0 05. 05. 06 ｧ0406 縺030300 04 縺030803 陦ｧ05 06 縺06 0503 0906 M (6; 3 5=), ﾀ ｧ000605 罍ｧ - 0 0305003 ﾀ ｧ05060004003 03 ﾀ0608 090303 罍陦ｧ0 05 ｧ ｧ0060605 0406006 ﾀ050406000606 0403000603 ﾀ04 a =4. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 0406 縺04060 陦ｧ09060007 縺0304, 030406006000308 ｧ 0006-04030503 07030906 ﾀ 05 ｧ0406 縺03030003 0 0606 ﾀ ｧ00405030503. 06. 7 05 ｧ0406 縺03030006 9x, 6y = 44 0 0805 ｧ 0503 0906, ﾀ ﾀ0503070 ｧ06 0305 09030503 縺0308 閨03 000604030503 07030906 ﾀ 04 閨040306 0060040006, 060 閨03 04 04030503. 07. 7 05 ｧ0406 縺03030006 x, y =40 0805 ｧ 0503 0906, 070309 000400306 陦ｧ06 ﾀ- 0406090503 縺03 09030503 縺0308 0406 縺04060 陦ｧ09060007 縺003 ( ﾀ0. 0609 0 ｧ06 09 06 84). 08. ｯ03 09 M 閨0600 ｧ05 ﾀ ﾀ0503070 ｧ06 00607 閨06 07030906 ﾀ 0 ｧ 05 ｧ0406 縺03030003 9x, 6y = 44 04030500300060 ｧ ｧ F M : MF =:3,05 閨06 F 0006040308 07030906 ﾀ05 ｧ0406 縺03030003. ｪ06 縺06 0503 0906 M 04 縺030406 閨060 04 縺070 07 0403 060500030 35 0903 ﾀ ｧOx. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 05050308 04 縺0700308 ﾀ ﾀ ｧ004050305 0 ｧ 05 ｧ0406 縺03030003. 09. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 0503 09 ｧ M, 09030503 07 閨04 ｧ070605 ﾀ07 05 09, 0503 03 ﾀ05 0605 ﾀ07 04 閨040306 00040006 0305 0503 09 ｧ F (,8; 0), 060 0305 04 縺0700308 x =,. 0. 6 7 003 0503 09 ｧ A(,a; 0) ｧ B(a; 0). ｯ03 09 M 閨04 ｧ070605 ﾀ07 05 09, 0503 06050300 MAB 03 ﾀ05 0605 ﾀ07 0405 縺0306 0060040006 04006000060503 060500 AMC 05 縺06-06050300040 ｧ09 AM B. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 閨04 ｧ07060 ｧ07 0503 09 ｧ M. x. 7 030300 7 0303000308 0 0304 0605 ﾀ07 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609; 03 陦ｧ0 09030403 06 0006000308 0305 000308 0503 09 ｧ (07030906 ﾀ ) ｧ 000308 04 縺0700308 ( 陦ｧ 縺060905 罍ｧ ﾀ03). 07 0030 ｧ 06 ﾀ090306 06 040060 ｧ06 04 03030003 ｧ0060605 閨04 04 ｧ : ) y =x 04 030300 ﾀ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Ox ( 罍ｧ ﾀ. 4); ) x =y 04 030300 ﾀ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oy ( 罍ｧ ﾀ. 5). 7 ｧ ﾀ. 4 7 ｧ ﾀ. 5 6 5 030303 ｧ08 ﾀ0006 0708 0406 縺00 ｧ0 04 03030003, 05. 06. 0503 09, 000607 0 07 0 03 ﾀ ｧ ﾀ ｧ000605 罍ｧ ｧ, 0 0803 陦ｧ05 ﾀ07 04 0 0006 090303 罍陦ｧ0 05.

330 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 7 030300 ｧ0060605 07030906 ﾀ F ;0 0503 09 ｧ M (x; y) 0 00608 r = x +. y =x ｧ 陦ｧ 縺060905 罍ｧ ﾀ06 x =, ;070309000400308 陦ｧ06 ﾀ-0406090503 7 030300 x =y ｧ0060605 07030906 ﾀ F 0; ｧ 陦ｧ 縺060905 罍ｧ ﾀ06 y =, ;070309000400308 陦ｧ06 ﾀ-0406090503 0503 09 ｧ M (x; y) 0 00608 r = y +.. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 03 陦ｧ0-09030403 06 0006000308 0305 0503 09 ｧ F (0; ) ｧ0305 04 縺0700308 y =4. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 05050308 09 罍ｧ040308 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 ｧ 0403 ﾀ05 縺03 ｧ0504 0606.. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 03 陦ｧ0-09030403 06 0006000308 0305 0 00 090303 罍陦ｧ0 05 ｧ 0305 04 縺0700308 x =,4. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 05050308 09 罍ｧ040308 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 ｧ 0403 ﾀ05 縺03 ｧ0504 0606. 3. 703 ﾀ05 縺03 ｧ0504 04 03030003, 000306 06 040060 ｧ070 ｧ: ) y =4x; ) y =,4x; 3) x = 4y; 4) x =,4y, 05 090706 ｧ08 07030906 ﾀ03 ｧ 陦ｧ 縺060905 罍ｧ ﾀ03 ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ07 陦ｧ 縺060905 罍ｧ ﾀ. 4. 7 04 ｧ ﾀ 050406 040060 ｧ06 04 03030003: ) 04 縺030803 閨0700608 06 縺06 0503 09 ｧ (0; 0) ｧ (;,3) ｧ ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Ox; )04 縺030803- 閨0700608 06 縺06 0503 09 ｧ (0; 0) ｧ (;,4) ｧ ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oy. 5. 07 0 05 0403 閨0406 ﾀ0030503 003 ﾀ05 ｧ0060605 0703 縺006 04 03030003 ( 罍ｧ ﾀ. 6). 7 04 ｧ ﾀ 0504 060606 040060 ｧ06 0305003 ﾀ ｧ05060004003 0609 000308 0 06 縺05060706 03 ﾀ0608, 06 ﾀ00 ｧ 04 縺0305 ｧ03 09 0 05 OA = a, 閨00 ｧ0 04 縺03000605 BC = b. 6. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, ｧ0060600608 0906005 04 07030906 ﾀ06 04 03030003 y = x ｧ09 ﾀ 0600608 ﾀ07 0606 陦ｧ 縺060905 罍ｧ ﾀ03. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 04 03030003 ｧ 0309 縺0607003 ﾀ05 ｧ. 7. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04 03030003 ｧ 0606 陦ｧ 縺060905 罍ｧ ﾀ03, 06 ﾀ00 ｧ 04-030300 04 縺030803 陦ｧ05 06 縺06 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 x + y = 0 ｧ 7 ｧ ﾀ. 6 0309 縺0607003 ﾀ05 ｧ x + y +4y =0 ｧ ﾀ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oy. 703 ﾀ05 縺03 ｧ0504 0309 縺0607003 ﾀ0504, 04 縺0700606 ｧ 04 03030006.

3x. 7 030300 3 8. 7 04 03030006 y = 6x 0 0805 ｧ 0503 0906, 070309 000400308 陦ｧ06 ﾀ- 0406090503 09030503 縺0308 04060 4,5. 9. 6ｦ006 縺09 00040 07 04030406 縺08003 ﾀ0504 04 縺030706090503 0303 0304 0 04 006-0 ｧ060 04 03030003 040309 縺0605 0606 03 ﾀ ｧ ﾀ ｧ000605 罍ｧ ｧ. 6 7 ｧ 00605 06 縺09 00 80 ﾀ0, 05000603 ｧ0 060503 0 ﾀ0. 7 09 09030 ﾀ ﾀ0503070 ｧ ｧ 0305 0406 縺00 ｧ003 04 03030003 00607003 0403006 ﾀ05 ｧ0504 ｧ ﾀ0503 0 ｧ09 ﾀ040605, 06 ﾀ00 ｧ 閨0007 0305 07060 ｧ07 0006 0608 04-00000600040030 0406 09030 030 閨030007060 03030504 0407030906 ﾀ06 04 03030003? 0. 0504 縺06 閨0600 ｧ0504 030300 ﾀ0504 ﾀ04030003-07060 ｧ07 09 罍ｧ040308 y =,,x. 703 ﾀ05 縺03- ｧ0504 09 罍ｧ040606.. 0ｷ3 0406 縺00 ｧ003 04 03030003 y = =x 04 縺030406 閨06003 04 ﾀ060403 0030700306 0803 - 閨03. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06- ﾀ09030503 006 ﾀ05 ﾀ06 縺06 陦ｧ0 0505 ｧ08 0803 罍.. 0504 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0906005 縺0304 0309 縺0607003 ﾀ050608, 09 ﾀ 06-0 ｧ08 ﾀ07 0309 縺0607003 ﾀ05 ｧ x + y =ax ｧ 03 ﾀ ｧ Oy. 3. 6 7 003 0503 09 ｧ A(0; a) ｧ B(a; a). 0505 縺06 09 ｧ OA ｧ AB 閨060006003 0 n 7 ｧ ﾀ. 7 0400308 ﾀ050608 0503 09 0 ｧ A, A, A 3, ::: ｧ B, B, B 3, ::: ( 罍ｧ ﾀ. 7). 706 ﾀ0504 M k 0503 09 0406 縺06 ﾀ06 060 ｧ07 0006 OB k ﾀ 04 縺0700308 A k M k kox. 703-09 0504, 0503 05 09 ｧ06 0503 09 ｧ M k 000607 05 0 04 03030006 y = ax. 703 ﾀ05 縺03 ｧ0504 0505 ｧ0 04 罍ｧ060030 04 03030003 y =4x, y =5x, y =3x. 4. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 03 陦ｧ0-09030403 06 0006000308 0305 0 00 090303 罍陦ｧ0 05 ｧ 0305 04 縺0700308 x =4. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 05050308 09 罍ｧ040308 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 ｧ 0403 ﾀ05 縺03 ｧ0504 0606. 5. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 03 陦ｧ- 0 09030403 06 0006000308 0305 0503 09 ｧ F (; 0) ｧ0305 04 縺0700308 y =. 7 0805 ｧ 0406 縺00 ｧ006 04 03030003, 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ Ox ｧ 0403 ﾀ05 縺03 ｧ0504 0606. 6. 7 04 ｧ ﾀ 050406 040060 ｧ06 04 03030003: ) 04 縺030803 閨0700608 06 縺06 0503 09 ｧ (0; 0) ｧ (,; ) ｧ ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Ox; )04 縺030803- 閨0700608 06 縺06 0503 09 ｧ (0; 0) ｧ (; 4) ｧ ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oy. 7. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04 03030003 ｧ 0606 陦ｧ 縺060905 罍ｧ ﾀ03, 06 ﾀ00 ｧ 04-030300 04 縺030803 陦ｧ05 06 縺06 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 y = x ｧ 0309 縺0607-003 ﾀ05 ｧ x + y +6x =0 ｧ ﾀ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Ox. 703- ﾀ05 縺03 ｧ0504 04 縺0700606, 0309 縺0607003 ﾀ0504 ｧ 04 03030006. 8. 6 5 04 03030006 y = x 0404 ｧ ﾀ 0 04 04 ｧ000400308 05 縺0606050300040 ｧ09. 0504 縺06 閨0600 ｧ0504 060503 0406 縺00 ｧ003 ( ﾀ0. 0609 0 ｧ06 09 06 73). 9. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 09 04 03030006 y = 8x, 04 縺030406 閨06000308 ｧ 0503 09 ｧ A(0;,).

33 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 30. ｪ06 縺06 07030906 ﾀ 04 03030003 y =,4x 04 縺030406 閨060 04 縺070 07 0403 060500030 0 0903 ﾀ ｧOx. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04 縺0700308 ｧ 0 0805 ｧ 閨00 ｧ006 0303 0304 04000608 ﾀ07 0803 罍閨03. x. 6 7 ｧ 縺060905 罍ｧ ﾀ03, 陦ｧ 00605 縺03 ｧ 09 ﾀ 0506000400306 09 09 罍ｧ04030 040503 縺030503 0403 縺07 閨09.6 7 ｧ 縺060905 罍ｧ ﾀ 0 ｧ 050000 ｧ04 ﾀ x a + y =(04 罍ｧ a>b) ｧ05 ｧ0406 縺03030003 b x a, y =0 0304 0605 ﾀ07 04 縺0700306, 04 000006000400306 03 ﾀ ｧ Oy ｧ0305 ﾀ0503070 ｧ06 0305 b 00606 0 ﾀ ﾀ0503070 ｧ06 a,05 閨06 e 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 09 罍ｧ040308. e ｰ 040060 ｧ07 陦ｧ 縺060905 罍ｧ ﾀ: x = a e : () ｮ040308 ﾀ050403 陦ｧ 縺060905 罍ｧ ﾀ: 030500300060 ｧ06 ﾀ ﾀ0503070 ｧ08 0305 0503 09 ｧ 09 罍ｧ- 040308 閨03 07030906 ﾀ ｧ 閨03 ﾀ030305040605 ﾀ0504060600608 陦ｧ 縺060905 罍ｧ ﾀ03 04003 0509 ﾀ0906005 罍ｧ- ﾀ ｧ05060506 09 罍ｧ040308 r = e: () d. 6 7 ｧ 00605 縺030 09 罍ｧ040308 040503 縺030503 0403 縺07 閨09 0 0304 0605 ﾀ07 050603-00605 罍ｧ 06 ﾀ090306 006 ﾀ0503 ﾀ06 縺06 陦ｧ0 04 000006000400308 0803 罍. 6 7 ｧ 00605 0 ｧ 050000 ｧ04 ﾀ ｧ 05 ｧ0406 縺03030003 0309 0304 0605 ﾀ07 0305 縺06 09 ｧ ｧ 0006 ｧ 04 縺0700308, 04 縺030803 閨070 ｧ08 06 縺06 0906005, 陦ｧ 00605 0 ｧ 04 03030003 0006 ｧ, 04 000006000400306 0606 03 ﾀ ｧ. ｰ 040060 ｧ06 ｧ 0 06 05, 閨0600070060503 0403040300 0 0803 罍閨03 ﾀ 0 0900030030 tg = k, 0306 閨0605 閨0007 09 罍ｧ040308 x a y b =: y = b a x; (3) k 閨0007 04 03030003 y =x: y = k : (4) 6 704 陦ｧ 00605 050000 ｧ04 ﾀ ｧ 05 ｧ0406 縺03030003, ｧ 09030503 縺0308 09 07 閨0308 閨0600 ｧ05 0403-040300 0 0803 罍閨03, 04 000006000400306 陦縺060503006, 0 0304 0605 ﾀ07 04 ｧ0003 ﾀ0304 縺0707060-0030 ｧ. 0ｷ308 06050003040306 0903050707 ｧ09 ｧ0600503 k ｧ k ﾀ0407 003 04 ｧ ﾀ ｧ003 ﾀ050406 kk = =, b a (06 050000 ｧ04 ﾀ ) ｧ kk = b (06 05 ｧ0406 縺03030003). a 3. ｰ 040060 ｧ07 09 ﾀ 0506000400308: x 09 050000 ｧ04 ﾀ06 a + y xx0 yy b = a + 0 b =; x 0905 ｧ0406 縺03030006 a, y xx0 b = a, yy 0 b =; 09 04 03030006 (y =x) yy 0 = (x +x 0 ),05 閨06 (x 0 ; y 0 ) 0503 09 09 ﾀ 0 ｧ07.

3x. 6 7 ｧ 縺060905 罍ｧ ﾀ03, 陦ｧ 00605 縺03 ｧ 09 ﾀ 0506000400306 09 09 罍ｧ04030-0503 0403 縺07 閨09 33 3. 703 ﾀ05 縺03 ｧ0504 050000 ｧ04 ﾀ x 5 + y =, 060503 陦ｧ 縺060905 罍ｧ ﾀ03 ｧ 0 0805 ｧ 9 ﾀ ﾀ0503070 ｧ07 0305 0503 09 ｧ 050000 ｧ04 ﾀ ﾀ 03 ﾀ09 ｧ ﾀ ﾀ0308 x =,3 閨03 04 04030503 07030906 ﾀ ｧ 04 040308 陦ｧ 縺060905 罍ｧ ﾀ03. 3. 703 ﾀ05 縺03 ｧ0504 05 ｧ0406 縺03030006 x 6, y =, 0606 陦ｧ 縺060905 罍ｧ ﾀ03 ｧ 0 0805 ｧ 9 ﾀ ﾀ0503070 ｧ07 0305 0503 09 ｧ 05 ｧ0406 縺03030003 ﾀ 03 ﾀ09 ｧ ﾀ ﾀ0308 x =5 閨03 000604030503 07030906 ﾀ ｧ 0006040308 陦ｧ 縺060905 罍ｧ ﾀ03. 33. 7 04 ｧ ﾀ 0504 09 0030 ｧ 06 ﾀ090306 06 040060 ｧ06 050000 ｧ04 ﾀ, 陦ｧ 縺060905 罍ｧ ﾀ 0 ｧ 09030503 縺030503 ﾀ000607 05 04 縺0700306 x = 4 3 ｧ 0303000400 07 0403000603 ﾀ04 09030503 縺030503 040. 34. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 ｧ0406 縺03030003, ﾀ ｧ00405030503 09030503 縺0308 y = = x, 陦ｧ 縺060905 罍ｧ ﾀ03 x = 6. 35. 703 ﾀ05 縺03 ｧ0504 050000 ｧ04 ﾀ x +4y =6, 陦ｧ 00605 y = x ｧ ﾀ0304 縺07-0706000308 06006 陦ｧ 00605 ｧ 0 0805 ｧ 閨00 ｧ003 a ｧ b 0403 ﾀ05 縺0306000308 04030006- 陦ｧ 00605 縺0304. 36. 703 ﾀ05 縺03 ｧ0504 05 ｧ0406 縺03030006 x, 4y = 4, 陦ｧ 00605 y =,x ｧ ﾀ0304 縺070706000308 06006 陦ｧ 00605 ｧ 0 0805 ｧ 06050300 00607 閨06 陦ｧ 00605 0 ｧ. 37. 7 0805 ｧ 閨00 ｧ006 05030503 陦ｧ 00605 050000 ｧ04 ﾀ x a + y b =,09030503 縺0308 04060 ﾀ040306006 ﾀ0304 縺0707060003006 陦ｧ 00605 縺06. 38. 罍ﾀ ｧ004050305 05 ｧ0406 縺03030003 x a, y = ﾀ03 ﾀ05 0400070605 ﾀ03 ﾀ0406 Ox b 06050300 60. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 陦ｧ 00605, ﾀ0304 縺07070600030503 ﾀ 陦ｧ 00605- 縺030 y = x. 6 50303 04 04 縺03 ｧ 04030004003 0305 縺06 0309 a, 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606, 陦ｧ 00605 縺03 ｧ 0803 罍閨03, 04 000006000400306 0003006 陦ｧ 00605 縺06. 39. 0504 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 ﾀ06 縺06 陦ｧ0 0803 罍 04 03030003 y =4x, ﾀ03 ﾀ05 040007060 ｧ08 ﾀ Ox 06050300 45. 40. 6 7 0 050000 ｧ04 ﾀ x 9 + y =. ｪ06 縺06 0503 0906 (,; ) 04 縺030406 ﾀ05 ｧ 4 0803 罍閨06, 閨06000700606 ﾀ07 04 05050308 0503 0906 0403040300 0. 4. 6 7 0 04 030300 y =,4x. ｪ06 縺06 0503 0906 (,;,) 04 縺030406 ﾀ05 ｧ 0803 罍閨06, 閨06000700606 ﾀ07 04 05050308 0503 0906 0403040300 0. 4. 7 04 罍ｧ006 縺06 ｧ 35 04 縺030406 罍ｧ0504 050603 縺06006 縺04030000030 ｧ07: a + b = a + b ｧ a b sin ' = ab, 05 閨06 a ｧ b 閨00 ｧ003 ﾀ03-04 縺070706000308 04030006 陦ｧ 00605 縺0304, a ｧ b 0403000603 ﾀ ｧ 050000 ｧ04 ﾀ, ' 06050300 00607 閨06 ﾀ0304 縺07070600030 ｧ 陦ｧ 00605 0 ｧ. 43. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 09 09 罍ｧ04030: ) x +4y =6;)3x,y =3;3)y =x 040503 0906 ﾀ 03 ﾀ09 ｧ ﾀ ﾀ0308 x 0 =.

334 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 44. 70309 0504, 0503 06 ﾀ00 ｧ 04 縺070 07 Ax + By + C =006 ﾀ0504 09 ﾀ - 050600040 07 09 050000 ｧ04 ﾀ06 x a + y b =,0503A a + B b = C. ｰ09 0 ｧ06. 0ｷ3 04 縺030403 縺09 ｧ030 0004003 ﾀ05 ｧ 0903050707 ｧ09 ｧ060050304 06 040060 ｧ08 xx 0 a + yy 0 b = ｧ Ax + By + C = 00304 縺06 閨0600 ｧ0504 x 0 ｧ y 0 ｧ 0403 陦ﾀ05 04 ｧ0504 ｧ080406 040060 ｧ06 x a + y b =. 45. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 09 050000 ｧ04 ﾀ06 x +4y =0, 04 000006000400308 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ06 0406 縺04030503 090303 罍陦ｧ0 050030503 060500. 46. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 09 050000 ｧ04 ﾀ06 x +y =8, 04 縺030406 閨06000308 ｧ 0503 09 ｧ (0; 6). 47. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 050000 ｧ04 ﾀ06 x a + y b =, 0305 ﾀ0609 0600608 0 03 ﾀ0708 090303 罍陦ｧ0 05 0400306 0403000307 ｧ0506000400306 0305 縺06 09 ｧ. 48. 70309 0504, 0503 06 ﾀ00 ｧ 04 縺070 07 Ax+By+C =006 ﾀ050409 ﾀ 05060004-0 07 09 05 ｧ0406 縺03030006 x a, y b =,0503A a, B b = C ( ﾀ0. 0609 0 ｧ06 09 06 44). 49. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 05060004003080905 ｧ0406 縺03030006 4x, 9y = =36, 0406 縺04060 陦ｧ09060007 縺00308 09 04 縺0700308 x +y =0. 50. 6 70309 0504, 0503 003 縺0 0004 09 050000 ｧ04 ﾀ06 06 ﾀ0504 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 060500 00607 閨06 陦ｧ06 ﾀ-0406090503 0 ｧ ﾀ030305040605 ﾀ0504060600608 0503 09 ｧ 050000 ｧ04 ﾀ. 5. 6 70309 0504, 0503 09 ﾀ 050600040 07 09 05 ｧ0406 縺03030006 06 ﾀ0504 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 060500 00607 閨06 陦ｧ06 ﾀ-0406090503 0 ｧ 0503 09 ｧ 09 ﾀ 0 ｧ07. 5. 6 70309 0504, 0503 0006 ｧ, 04030803 閨070 ｧ06 ｧ 07030906 ﾀ 04 03030003, 0305-07 0605 ﾀ07 0305 04 03030003 0403 04 縺070030, 04 00000600040030 0606 03 ﾀ ｧ. ｰ09 0 ｧ06. 70607003 0 04 ｧ ﾀ 0504 06 040060 ｧ06 003 縺0 00 ｧ MN, 0 0805 ｧ 0503 0906 N 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 03 ﾀ0406 04 03030003 ｧ 閨0309 0504, 0503 FM = FN,05 閨06 F 07030906 ﾀ 04 03030003. 53. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ ｧ004050305 05 ｧ0406 縺03030003 x 6, y 9 = ﾀ 0606 陦ｧ 縺060905 罍ｧ ﾀ 0 ｧ. 54. 703 ﾀ05 縺03 ｧ0504 050000 ｧ04 ﾀ x +4y = 6, 060503 陦ｧ 00605 y = x ｧ ﾀ0304 縺070706000308 06006 陦ｧ 00605 ｧ 0 0805 ｧ 06050300 00607 閨06 0505 ｧ0 ｧ 陦ｧ 00605 0 ｧ. 55. 0504 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 ﾀ06 縺06 陦ｧ0 0803 罍 05 ｧ0406 縺03030003 x, 4y =6, ﾀ03 ﾀ05 040007060 ｧ08 06050300 45 ﾀ03 ﾀ0406 Ox. 56. 6 7 0 05 ｧ0406 縺030300 4x, y =4. ｪ06 縺06 0503 0906 (; ) 04 縺030406 ﾀ05 ｧ 0803 罍閨06, 閨06000700606 ﾀ07 04 05050308 0503 0906 0403040300 0. 57. 7 050000 ｧ04 ﾀ06 x +y = 604 0705 0503 09 M ﾀ 03 罍陦ｧ0 050308 ｧ0305 罍ｧ09 0506000400308 03 ﾀ09 ｧ ﾀ ﾀ0308. 7 0805 ｧ 06050300 09 ﾀ 0506000400308 09 050000 ｧ04 ﾀ06 04 0503 0906 M ﾀ 04 縺0700308 OM.

3x 3. 7 縺060303 0304 0 ｧ06 閨0609 縺0503040308 090303 罍陦ｧ0 05 35 58. 70309 0504, 0503 06 ﾀ00 ｧ 04 縺070 07 Ax + By + C =006 ﾀ0504 09 ﾀ - 050600040 07 09 04 03030006 y = x, 0503B = AC ( ﾀ0. 0609 0 ｧ06 09 06 44). 59. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 04 03030006 y =8x, 04-000006000400308 04 縺0700308 x + y =0. x 3. 7 縺060303 0304 0 ｧ06 閨0609 縺0503040308 090303 罍陦ｧ0 05. 7 03030003 = + + ｧ = + +. 06 ｧ0406 縺030300 =. 070303 罍陦ｧ0 0503 (x; y) 04 000308 ﾀ ｧ ﾀ0506006 04 縺060303 060605 ﾀ07 09 090303 罍陦ｧ0-05 0 (X; Y ) 04 003040308 ﾀ ｧ ﾀ0506006 0403 0703 縺00600 0: ) 04 罍ｧ 04 00000600040030 ﾀ 閨04 ｧ0506 03 ﾀ0608 ｧ 0406 縺06006 ﾀ060 ｧ ｧ 0 00 090303 罍陦ｧ0 05 040503 0906 O (; ) x = X + ; y = Y + ; () ) 04 罍ｧ 04030403 縺030506 03 ﾀ0608 0 06050300 ' x = X cos ', Y sin '; y = X sin ' + Y cos ': (). ｰ 040060 ｧ06 y = a(x, ) + 0406 縺06003 ﾀ030 0 00 090303 罍陦ｧ0 05 04 0503 0906 O (; ) 04 罍ｧ0403 陦ｧ05 ﾀ07 09 04 ｧ 閨06 Y = ax ｧ, ﾀ0006 閨0304 05060004003, 0304 縺06 閨06-00070605 04 03030006 ﾀ 0406 縺00 ｧ00308 O (; ) ｧ03 ﾀ0406 ﾀ ｧ000605 罍ｧ ｧ, 04 000006000400308 Oy ( 罍ｧ ﾀ. 8). ｰ 040060 ｧ06 y = ax + bx + c 0403 閨0600060 ｧ060 04 04 040308 ﾀ05 ｧ 0403000030503 0904 陦 05 04 罍ｧ0403 陦ｧ05 ﾀ07 09 04 縺06 閨03 閨06006006 ｧ 0403050503006 05030706 0304 縺06 閨0600070605 04 03030006. 7 罍ｧ a > 0 04 030300 0305 0406 縺00 ｧ003 0 04 0400060 08040406 縺0809, 04 罍ｧ a<0 08040 ｧ 09. 3. ｰ 040060 ｧ06 xy = k 04 罍ｧ 04030403 縺030506 03 ﾀ0608 090303 罍陦ｧ0 05 0 06050300 ' =45 04 罍ｧ0403 陦ｧ05 ﾀ07 09 04 ｧ 閨06 X, Y =k ｧ, ﾀ0006 閨0304 05060004003, 0304 縺06 閨0600070605 7 ｧ ﾀ. 8 7 ｧ ﾀ. 9 04003 ﾀ0503 縺03000606 05 ｧ0406 縺03030006, ﾀ ｧ004050305 0 ｧ 09030503 縺0308 ﾀ000607 05 03 ﾀ ｧ 090303 罍陦ｧ- 0 05 ( 罍ｧ ﾀ. 9). ｰ 040060 ｧ06 (x, )(y, ) =k 0406 縺06003 ﾀ030 0 00 090303 罍陦ｧ0 05 040503 0906 O (; ) 04 罍ｧ0403 陦ｧ05 ﾀ07 09 04 ｧ 閨06 XY = k ｧ 0403050503006 05030706 0304 縺06 閨0600070605 04003 ﾀ0503 縺03000606 05 ｧ0406 縺03030006.

336 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 60. ) ｯ03 09 A(3; ) 04 罍ｧ 04 00000600040030 ﾀ 閨04 ｧ0506 03 ﾀ0608 090303 罍陦ｧ0 05 04030006 ｧ00 003040306 090303 罍陦ｧ0 0503 (;,). 703 ﾀ05 縺03 ｧ0504 000306 ｧ ﾀ0060060-00306 03 ﾀ ｧ 090303 罍陦ｧ0 05 ｧ 0503 0906 A. ) 7 0805 ｧ 03 ﾀ05 縺0308 06050300 04030403 縺0305 03 ﾀ0608 090303 罍陦ｧ0 05, 04 罍ｧ 09030503 縺030 0503 09 A(; 4) 04030006 ｧ05 003040606 03 ﾀ09 ｧ ﾀ ﾀ06 4. 703 ﾀ05 縺03 ｧ0504 030306 ﾀ ｧ ﾀ0506003 090303 罍陦ｧ0 05 ｧ 0503 0906 A. 6. 706 縺06006 ﾀ060 ｧ060 0 00 090303 罍陦ｧ0 05 0604 縺03 ﾀ05 ｧ0504 06 040060 ｧ07: (x, ) (x ) +(y +) +3) (y, ) =; ), =; 4 9 4 3) (y +) =4(x, 3); 4) y =,(x +) ; 5) x +4y, 6x +8y =3; 6) y, 8y =4x; 7) x, 4y +8x, 4y = 4; 8) x +6x +5=y: 703 ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧ 003040306 03 ﾀ ｧ 090303 罍陦ｧ0 05 ｧ 09 罍ｧ040306. 6. 7030403 縺0305030 03 ﾀ0608 090303 罍陦ｧ0 05 0 45 0604 縺03 ﾀ05 ｧ0504 06 040060 ｧ07: ) 5x, 6xy +5y =3; ) 3x, 0xy +3y +3=0. 703 ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧ 003040306 03 ﾀ ｧ 090303 罍陦ｧ0 05 ｧ 09 罍ｧ040306. 63. 703 ﾀ05 縺03 ｧ0504 04030503 09 0 09 罍ｧ040606 xy =,4 ｧ 04030403 縺0305030 03 ﾀ0608 0 06050300 ' =,45 04 縺060303 0304 0504 06 040060 ｧ06. 64. 706 縺06003 ﾀ030 0 00 090303 罍陦ｧ0 05 04 罍ｧ0406 ﾀ05 ｧ 09 04 ｧ 閨06 xy = k 06 04-0060 ｧ07 09 罍ｧ040308: ) xy, x =6; ) xy, x, y +8=0; 3) xy, x +y =6; 4) xy +x =3y: ｰ09 0 ｧ06. ｰ 040060 ｧ06 xy + Ax + By + C =000307003 0 04 ｧ ﾀ 0504 04 04 ｧ 閨06 (x + B)(y + A) =AB, C. 65. 703 ﾀ05 縺03 ｧ0504 04 03030003: ) y =(x, ) ; ) y =(x, ) +3; 3) y =(x +) ; 4) y =(x +), 3: 66. 703 ﾀ05 縺03 ｧ0504 04 03030003: ) y = x, 4x +5; ) y = x +x +3; 3) y =,x +x,, 0403 閨0600 ｧ04 04 04 040308 ﾀ050708 06 040060 ｧ08 04030000306 0904 陦 0503. 67. 703 ﾀ05 縺03 ｧ0504 04 03030003: ) y =4x, x ｧ)y =3+x, x, 0 08 閨07 ｧ08 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ0406 Ox. 68. ｮ05 縺0607 0403 閨03 0703005 0 閨03 ﾀ05 ｧ05 0605 0 ｧ03030004000608 0403 ﾀ030503 400 ﾀ ﾀ0503070 ｧ ｧ 0,5 0 0305 0406 縺05 ｧ09 00 ｧ, 04 縺030803 閨0700608 06 縺06 0503 0906 O 04030803 ﾀ05 縺06 ｧ. 7 0805 ｧ 0403 ﾀ030506 ﾀ05 縺06 ｧ 0 0503 罍ｧ 03005 000406 Ox 0 ﾀ ﾀ0503070 ｧ ｧ 0,75 0 0305 0503 09 ｧ O.

3x 3. 7 縺060303 0304 0 ｧ06 閨0609 縺0503040308 090303 罍陦ｧ0 05 37 69. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 04 03030003, ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004-003 03 ﾀ ｧ Oy ｧ 0305 ﾀ0609 0600608 0 00608 0305 縺06 0309 b, 0 03 ﾀ ｧ Ox 0305 縺06 09 ｧ a ｧ,a. ｰ09 0 ｧ06. 6 5 06 040060 ｧ ｧ 04 03030003 04 ｧ y = Ax + Bx + C 0403 - ﾀ05 04 ｧ0504 090303 罍陦ｧ0 0503 000308 0 04 03030006 0503 0609 (,a; 0), (a; 0) ｧ (0; b) ｧ 05060 0 0805 ｧ A, B ｧ C. 70. 7 030300 y = ax + bx + c 04 縺030803 陦ｧ05 06 縺06 0503 09 ｧ O(0; 0), A(,;,3) ｧ B(,;,4). 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, 陦ｧ 00605- 縺030 09030503 縺0308 ﾀ000607 ｧ05 0305 縺06 0309 03 ﾀ ｧ Ox, 0305 ﾀ06 06000308 04 0303000308. 7. 7 09 090308 06050300 00607003 04030406 縺0060504 03 ﾀ ｧ 090303 罍陦ｧ0 05, 05030303 ｧ ﾀ- 06 00060, ﾀ03 閨06 縺07 0 ｧ08 xy, 04 06 040060 ｧ0708: ) x, xy + y, 3=0; ) 5x, 4xy +y, 4 = 0? 703 ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧ 003040306 03 ﾀ ｧ 090303 罍陦ｧ0 05 ｧ 09 罍ｧ040306. 7. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 閨04 ｧ07060 ｧ07 040600 ｧ, 03 縺030006000308 0403 060500030 ' 09 0503 罍ｧ 0300506 ﾀ0 000400308 ﾀ0903 縺03 ﾀ050406 v 0. 0504 縺06 閨0600 ｧ0504 05 090706 0004003 ﾀ0504 0403000605 040600 ｧ ｧ 0 ｧ0403 ﾀ000606 0503 0906 05 06090503 罍ｧ ｧ ( ﾀ0304 縺0305 ｧ- 0400060 ｧ060 0403 閨0608 04 縺0600603 縺06 04). 73. 7 04 ｧ ﾀ 050406 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M (x; y), 030500300060 ｧ06 ﾀ ﾀ0503070 ｧ08 0305 09030503 縺0308 閨03 0503 09 ｧ F (4; 0) 09 ﾀ ﾀ0503070 ｧ070 閨03 04 縺0700308 x =, 04003. 74. 70309 0504, 0503 0406 縺06003 ﾀ030 0 00 090303 罍陦ｧ0 05 04 0006040606 0406-00 ｧ006 050000 ｧ04 ﾀ x a + y b = ｧ00 ｧ 04 04 040606 0406 縺00 ｧ006 05 ｧ0406 縺03030003 x a, y b =0303 06 040060 ｧ07 04 罍ｧ0403 閨0705 ﾀ07 09 03 陦ｧ0 09030403006 04 ｧ 閨06: y = =x + qx,05 閨06 = b a, q = e,. 75. 703 縺06 06000405 05 0 ｧ 74 0304 縺06 閨0600 ｧ0504 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 ｧ 05 ｧ04 09 罍ｧ040308: ) y = x, 4 x ;)y = x+ 4 x ;3)y = x. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040306, 0 08 閨07 閨0007 0406 縺040308 閨040608 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ｧ08 ﾀ 03 ﾀ0406 Ox ｧ 04 00605 縺03 a ｧ b. 76. 6 503 閨0600060 ｧ060 04030000308 0904 陦 050304 ｧ 0406 縺06003 ﾀ030 0 00 090303 - 陦ｧ0 05 0604 縺03 ﾀ05 ｧ0504 06 040060 ｧ07 00 ｧ0 ｧ08: ) x +5y, x +0y +3=0; ) x, y +6x +4y, 4=0; 3) y +4y =x; 4) x, 0x =4y, 3. 703 ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧ 003040306 03 ﾀ ｧ ｧ 09 罍ｧ040306.

338 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 77. 7030403 縺0305030 03 ﾀ0608 090303 罍陦ｧ0 05 0 45 0604 縺03 ﾀ05 ｧ0504 06 040060 ｧ06 3x, xy +3y, 8=0. 0504 縺06 閨0600 ｧ0504 090303 罍陦ｧ0 0503 07030906 ﾀ030404 ﾀ05 縺0308 ﾀ ｧ ﾀ0506006 090303 罍陦ｧ0 05. 78. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, 陦ｧ 00605 縺030 09030503 縺0308 ﾀ0006-07 ｧ05 0305 縺06 0309, 0305 ﾀ0609 0600308 0 03 ﾀ ｧ Ox 04 0303000308 y =3, x, x. 703 ﾀ05 縺03 ｧ0504 03030609 罍ｧ040306. 79. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, 陦ｧ 00605 縺030 09030503 縺0308 ﾀ0006-07 ｧ05 0305 縺06 0309 04 縺0700308 x + y = 6,0305 ﾀ06 06000308 05 ｧ0406 縺0303000308 xy = 8. 703 ﾀ05 縺03 ｧ0504 04 ﾀ06 05 罍ｧ 00 ｧ0 ｧ ｧ. 80. ｯ03 09 A 0406 縺00 ｧ0 04 03030003 y = x +6x +5, B 0503 09 0406 縺06 ﾀ06 060 ｧ07 04 03030003 ﾀ 03 ﾀ0406 Oy. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0406 縺04060 陦ｧ- 09060007, 0403 ﾀ ﾀ05 04000600030503 ｧ ﾀ06 縺06 陦ｧ003 0305 縺06 09 AB. 8. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 04 03030003, ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004-003 03 ﾀ ｧ Ox ｧ 0305 ﾀ0609 0600608 0 00608 0305 縺06 0309,4, 0 03 ﾀ ｧ Oy 0305 縺06 09 ｧ 4 ｧ,4. ｰ09 0 ｧ06. ｰ 040060 ｧ06 04 03030003 閨030007003 ｧ0060504 04 ｧ x = ay + c (0403 06006?). 8. 703 ﾀ05 縺03 ｧ0504 04030503 09 0 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 04-03030003: ) 3y =9, x ; ) y =9, 3x; 3) y =4+x; 4) x =4+y: 83. 7 04 ｧ ﾀ 050406 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M (x; y), 030500300060 ｧ06 ﾀ ﾀ0503070 ｧ08 0305 09030503 縺0308 閨03 0503 09 ｧ F (4; 0) 09 ﾀ ﾀ0503070 ｧ070 閨03 04 縺0700308 x =0 04003 =. x 4. ｮ00600 000306 ｧ 0 09 罍ｧ040306 040503 縺030503 0403 縺07 閨09 84. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, 陦ｧ 00605 縺030 09030503 縺0308 ﾀ0006-07 ｧ05 0305 縺06 0309 04 縺0700308 x a + y =,0305 ﾀ06 06000308 03 ﾀ070 ｧ 090303 罍陦ｧ0 05. b 85. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0305 0906005 0309 縺0607003 ﾀ05 ｧ x + y + ay =0 閨03 04 縺0700308 y =(a, x). 86. ｪ06 縺06 0906005 0309 縺0607003 ﾀ05 ｧ x + y =ax 04 縺030406 閨060 04 縺070 07, 04 00000600040 07 04 縺0700308 x +y = 0 ｧ 0406 縺06 ﾀ0609 060 07 0309 縺0607003 ﾀ0504 04 0503 09 08 A ｧ B. 7 0805 ｧ 0400030 閨04 4AOB. 87. 70309 0504, 0503 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 M,09030503 縺0306 06-0006003 04 m 00040006 0305 000308 0503 09 ｧ A, 060 0305 陦縺06050308 000308 0503 09 ｧ B, 06 ﾀ0504 04 縺070 07 04 罍ｧ m = ｧ0309 縺0607003 ﾀ0504 04 罍ｧ m 6=. 88. 0505 縺06 0309 AB 閨0600060 0 ﾀ05 ｧ AO = a ｧ OB = b. 70309-0504, 0503 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609, ｧ 09030503 縺0308 0305 縺06 09 ｧ AO ｧ OB 04 ｧ 閨003 0403 040030 ｧ 060500 0 ｧ, 06 ﾀ0504 04 縺070 07 04 罍ｧ a = b ｧ0309 縺0607-003 ﾀ0504 04 罍ｧ a 6= b ( 04030000030 ｧ0604 0309 縺0607003 ﾀ0504).

3x 4. ｮ00600 000306 ｧ 0 09 罍ｧ040306 040503 縺030503 0403 縺07 閨09 39 89. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 0503 09 ｧ M (x; y), 閨04 ｧ070600608 ﾀ07 05 09, 0503 ﾀ0600 0904 陦 050304 ﾀ ﾀ0503070 ｧ08 0305 00606 閨03 04 縺0700308 y = kx ｧ y = =, kx 03 ﾀ05 0605 ﾀ07 0403 ﾀ050307000308 ｧ 0400308 a. 90. 70000 ｧ04 ﾀ, ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Ox ｧ 04 縺0700308 x =,5, 04 縺030803 陦ｧ05 06 縺06 0503 09 ｧ (,; ; 8) ｧ (,5; 3). 7 04 ｧ ﾀ 0504 06 04-0060 ｧ06 050000 ｧ04 ﾀ ｧ 0403 ﾀ05 縺03 ｧ0504 060503. 9. 7 0805 ｧ 0400030 閨04 04003 ﾀ0503 縺0300060503 05 縺0606050300040 ｧ09, 0404 ｧ ﾀ 0-0030503 04 05 ｧ0406 縺03030006 x, y = a. 9. 7 0805 ｧ06050300 00607 閨06 陦ｧ 05030 00070 ｧ 04 縺0700306050300040 ｧ09, 0406 縺00 ｧ003 09030503 縺030503 0 0803 閨0705 ﾀ07 04 0503 09 08 0406 縺06 ﾀ06 060 ｧ07 050000 ｧ04 ﾀ x +3y =l ｧ05 ｧ0406 縺03030003 x, 3y =6l. 93. 0509 縺0607003 ﾀ0504 ﾀ 0906005 縺030 04 0 0006 090303 罍陦ｧ0 05 04 縺030803 陦ｧ05 06- 縺06 07030906 ﾀ03 05 ｧ0406 縺03030003 x, y = a. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 0309 縺0607003 ﾀ05 ｧ ﾀ ﾀ ｧ004050305 0 ｧ 05 ｧ0406 縺03030003. 94. 703 ﾀ05 縺03 ｧ0504 05 ｧ0406 縺03030003 xy =,4 ｧ x, y = 6 ｧ 0 0805 ｧ 0400030 閨04 4ABC, 05 閨06 A ｧ B 0406 縺00 ｧ003 閨040608 0406 縺06 ﾀ0609 060 ｧ08 ﾀ07 040605040608 05 ｧ0406 縺030300, C 0503 09 0406 縺06 ﾀ06 060 ｧ07 閨040608 陦縺0605 ｧ08 040605040608 05 ｧ0406 縺030300. 95. 6 70309 0504, 0503 04 縺03 ｧ 0406 閨060 ｧ06 ﾀ ﾀ0503070 ｧ08 0006030308 0503 09 ｧ 05 ｧ- 0406 縺03030003 0305 0606 ﾀ ｧ004050305 06 ﾀ0504 040600 ｧ ｧ0 0403 ﾀ05030700 07, 040 07 a b c. 96. 7 0805 ｧ 閨00 ｧ006 ｧ06 040060 ｧ06 0406 縺04060 陦ｧ09060007, 03040600600030503 ｧ 07030906 ﾀ 04 03030003 y =, x 0 04 縺0700606, 0305 ﾀ0609 0600606 0 03 ﾀ0708 8 090303 罍陦ｧ0 05 0305 縺06 09 ｧ a = b =. 97. 703 ﾀ05 縺03 ｧ0504 050000 ｧ04 ﾀ x +4y = 4 ｧ 04 03030006 x = 6y ｧ 0 0805 ｧ 0400030 閨04 05 040609 ｧ ｧ, 03 ﾀ00304 0 ｧ070 ｧ 09030503 縺0308 ﾀ000607 05 0303000400 07 03 ﾀ04 050000 ｧ04 ﾀ ｧ 03030 07 0803 罍 050000 ｧ04 ﾀ ｧ 04 03030003. 98. 0ｷ3 07030906 ﾀ 04 03030003 y = x, 09 09 ｧ 0906005, 0304 ｧ ﾀ 0 0309 縺0607003 ﾀ0504 05 09, 0503 03030 07 0803 罍 09 罍ｧ040308 03 陦ｧ0 09030403 06 00060 0305 0406 縺00 ｧ003 ｧ 0305 07030906 ﾀ 04 03030003. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ. 99. 7 0805 ｧ 閨00 ｧ006 ｧ06 040060 ｧ06 0406 縺04060 陦ｧ09060007, 03040600600030503 ｧ 0406 縺00 ｧ003 04 03030003 by = x +ax+a +b 0 04 縺0700606, 0305 ﾀ0609 0600606 0 03 ﾀ0708 090303 罍陦ｧ0 05 0305 縺06 09 ｧ a ｧ b. 300. 703 ﾀ05 縺03 ｧ0504 04030503 09 0 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 04-03030003 4y =, x ｧ 4x =, y ｧ 0 0805 ｧ 閨00 ｧ006 ｧ08 030300608 0803 罍閨03. 30. 7 0805 ｧ 0400030 閨04 060503 縺060806050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ 04 0503 09 08 0406 縺06 ﾀ06 060 ｧ07 04 03030003 y =4, x ﾀ03 ﾀ0406 Ox ｧ ﾀ 04 縺0700308 y =3x. 30. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0-0003 090303 罍陦ｧ0 05 ｧ 06 縺06 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 04 03030003 y = x a,, x + a ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05.

340 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 303. 6 7 0 050000 ｧ04 ﾀ x +4y =6. 0ｷ3 060503 0406 縺00 ｧ003 A(4; 0) 04 縺03-0406 閨06003 04 ﾀ060403 0030700306 0803 罍閨03. 0504 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 ﾀ06 縺06 陦ｧ0 0505 ｧ08 0803 罍 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306. 304. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06 0503 09 ｧ M (x; y), 閨04 ｧ070600608 ﾀ07 05 09, 0503 003 ﾀ0504 0904 陦 050304 ﾀ ﾀ0503070 ｧ08 0305 00606 閨03 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 090303 罍陦ｧ- 0 0500308 0605000304 03 ﾀ05 0605 ﾀ07 0400308 8. 305. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0906005 縺0304 0309- 縺0607003 ﾀ050608, 04 縺030803 閨070 ｧ08 06 縺06 0503 0906 A(3; 4) ｧ09 ﾀ 060 ｧ08 ﾀ07 03 ﾀ ｧ Ox. 306. 6 503 閨0600060 ｧ060 04030000308 0904 陦 050304 ｧ 0406 縺06003 ﾀ030 0 00 0604 縺03- ﾀ05 ｧ0504 06 040060 ｧ06 00 ｧ0 ｧ ｧ x,y,4x,6y,9 =0. 703 ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧ 003040306 03 ﾀ ｧ 090303 罍陦ｧ0 05 ｧ 09 罍ｧ040606. 307. 7 0805 ｧ 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 ﾀ06 縺06 陦ｧ0 070309 000400308 陦ｧ06 ﾀ- 0406090503 縺0304, 04 縺030406 閨06000308 ｧ 04 04030503 07030906 ﾀ 0903 04 ﾀ060 0503 09 0 05 ｧ0406-03030003 x 9, y 6 =. 308. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 050000 ｧ04 ﾀ, 04 縺030803 閨070060503 06 縺06 0503 0906 A(a;,a), 06 ﾀ00 ｧ 07030906 ﾀ03 060503 0 0803 閨0705 ﾀ07 04 0503 09 08 F (a; a) ｧ F (,a;,a). ｰ04 縺03 ﾀ05 ｧ0504 06 040060 ｧ06 04030403 縺0305030 03 ﾀ0608 090303 罍陦ｧ0 05 0 45. 309. 7030403 縺0305030 03 ﾀ0608 090303 罍陦ｧ0 05 0 06050300 ' = arctg 0604 縺03 ﾀ05 ｧ0504 06 040060 ｧ06 00 ｧ0 ｧ ｧ 3x +8xy, 3y =0. 703 ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧ 003040306 03 ﾀ ｧ 090303 罍陦ｧ0 05 ｧ 09 罍ｧ040606. 30. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, - 003 ﾀ0504 0904 陦 050304 ﾀ ﾀ0503070 ｧ08 0305 09030503 縺0308 閨03 04 縺0700308 3x +4y = 0 ｧ 閨0303 ﾀ ｧOx 03 ﾀ05 0605 ﾀ07 0403 ﾀ050307000308 ｧ 0400308 ; 4. 3. 7 04 ｧ ﾀ 050406 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M (x; y), 030500300060 ｧ06 ﾀ ﾀ0503070 ｧ08 0305 09030503 縺0308 閨03 0503 09 ｧ F e + ;0 09 ﾀ- ﾀ0503070 ｧ070 閨03 04 縺0700308 x =, 04003 e. e(e +) 3. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ05 ｧ, 090303 罍陦ｧ0 0503 0503 0609 09030503 縺0308 06 閨03040006050403- 縺070605 006 04060 ﾀ0504 0: ) R <x + y < 4R ｧ x >R =4; ) x, y >a ｧ x < 4a ; 3) xy > a ｧ jx + yj < 4a; 4) x <y +4y ｧ x + y +4x +4y<0. x 5. 050300606 06 040060 ｧ06 00 ｧ0 ｧ ｧ 040503 縺030503 0403 縺07 閨09.6ｧ8 ｧ0 ｧ0608 040503 縺030503 0403 縺07 閨09 0 0304 0605 ﾀ07 00 ｧ0 ｧ07, 0304 縺06 閨060007060 07 06 040060 ｧ060-08 ﾀ050604060 ｧ, 09030503 縺0306 04 03030060 04 ｧ 閨06 00307003 0 04 ｧ ﾀ 0504 05 09: Ax +Bxy + Cy +Dx +Ey + F =0: ()

3x 5. 050300606 06 040060 ｧ06 00 ｧ0 ｧ ｧ 040503 縺030503 0403 縺07 閨09 4 ｮ03 ﾀ05 04 ｧ0 ｧ 0903050707 ｧ09 ｧ060050304 06 040060 ｧ07 () 閨04 0304 縺06 閨0600 ｧ05060007: = B A B A B D C ; = B C E D E F : 0504 縺06 閨0600 ｧ05060004 0 0304 0605 ﾀ07 陦ｧ ﾀ09 罍ｧ0 ｧ0 005030 06 040060 ｧ07 (), 陦ｧ ﾀ09 罍ｧ0 ｧ0 005030 ﾀ05 縺00 ｧ08 060503 000600304. 6 5 04 ｧ ﾀ ｧ003 ﾀ05 ｧ 0305 0 060 ｧ08 ｧ 06 040060 ｧ06 () 0304 縺06 閨0600070605 ﾀ0006 閨06060 ｧ08 05060300605 罍ｧ 06 ﾀ09 ｧ08 0303 : 6= 0 =0 >0 70000 ｧ04 ﾀ ( 閨0608 ﾀ0504 ｧ0506000400308 ｧ00 ｧ 00 ｧ00308) ｯ03 09 <0 06 ｧ0406 縺030300 7 0406 縺06 ﾀ0609 060 ｧ08 ﾀ07 04 縺07-00308 =0 7 030300 7 04 000006000400308 04 縺0700308 ( 閨0608 ﾀ0504 ｧ0506000400308 ｧ00 ｧ 00 ｧ- 00308). 7 縺060303 0304 0 ｧ06 06 040060 ｧ07 () 09 0906005 縺06. ｭ ﾀ00 ｧ = = B A B C 6= 0, 0503 00 ｧ0 ｧ07 ｧ0060605 0906005, 090303 罍陦ｧ0 0503 09030503 縺030503 0 0803 閨0705 ﾀ07 ｧ 06 040060 ｧ08 0 (x; y) =0; (x; y) =0; () x 0 y 05 閨06 (x; y) 000604 07 ﾀ0504 06 040060 ｧ07 (). 706 縺06006 ﾀ07 0 0003 04 0906005 O (x 0 ; y 0 ) ( 罍ｧ ﾀ. 0), 04 罍ｧ0406 閨060 06 040060 ｧ06 () 09 04 ｧ 閨06 05 閨06 Ax +Bx y + Cy + F =0; (3) F = Dx 0 + Ey 0 + F = : (4) 3. 7 縺060303 0304 0 ｧ06 06 040060 ｧ07 (3) 09 03 ﾀ070 ﾀ ｧ000605 罍ｧ ｧ. 7030403 縺0305030 03 ﾀ0608 O x ｧ O y 0 00609030503 縺0308 06050300 ' ( 罍ｧ ﾀ. 0) 06 040060 ｧ06 (3) 04 罍ｧ0403 陦ｧ05 ﾀ07 09 09 0030 ｧ 06 ﾀ0903006 04 ｧ 閨06: A X + C Y + F =0: (5) 0703050707 ｧ09 ｧ0600503 A ｧ C 070400070605 ﾀ07 0903 縺0070 ｧ 06 040060 ｧ07 ｰ050300 04030403 縺0305 ' 0 0803 陦ｧ05 ﾀ07 0403 0703 縺0060006, (A + C) + =0: (6) tg ' = B A, C : (7)

34 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 4. 7 縺060303 0304 0 ｧ06 06 040060 ｧ07 00 ｧ0 ｧ ｧ 040503 縺030503 0403 縺07-09, 006 ｧ0060600608 0906005. ｭ ﾀ00 ｧ =0, 0503 00 ｧ0 ｧ07 006 ｧ0060605 0906005 ｧ00 ｧ 006 ｧ0060605 0304 縺06 閨06000600030503 0906005. ｭ06 06 040060 ｧ06 00307003 050305 04 ｧ ﾀ 0504 04 04 ｧ 閨06 (x + y) +Dx +Ey + F =0: (8) ｮ0006 08. D ｧ E 04 縺030403 縺09 ｧ030 0004003 ｧ : D = m, E = m. ｰ 040060 ｧ06 () 04 罍ｧ00605 04 ｧ (x + y) +m(x + y) +F =0,03050906 x + y =,m m, F 04 04 縺0700308: ｮ0006 08.D ｧ E 006 04 縺030403 縺09 ｧ030 0004003 ｧ. ｰ 040060 ｧ06 (8) 00307003 0406 縺0604 ｧ ﾀ 0504 04 04 ｧ 閨06 (x + y + n) +m(x, y + q) =0: (9) 7 00605 縺03 m, n ｧ q 0 08 閨0605 ﾀ07 ﾀ 040060 ｧ060 0903050707 ｧ09 ｧ060050304 04 06 04006-0 ｧ0708 (8) ｧ (9). 6 7 000606, 04 罍ｧ00704 03 ﾀ04 O X 04 縺0700606 x + y + n =0, x + y + n 03 ﾀ04 O Y 04 縺0700606 x, y + q =0( 罍ｧ ﾀ. ), 0 08 閨060: Y = +, x, y + q X = +. 703 ﾀ0006 0505030503 06 040060 ｧ06 (9) 04 罍ｧ00605 04 ｧ Y =X, 05 閨06 = 7 ｧ ﾀ. 0 7 ｧ ﾀ. jmj +. 05 ﾀ04 O X 0 04 0400070605 ﾀ07 04 0506 04030006040003 ﾀ0903 ﾀ0504, 04 09030503 縺0308 x, y + q ｧ0060605 0 09, 04 縺0305 ｧ0403040300030700308 0 0906 m, 09 09 050503 ﾀ0006 閨060605 ｧ 06 040060 ｧ07 (9). 33. 6 50307 ﾀ0 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀ00 06 040060 ｧ08: ) 4x, y =0; ) 4x + y =0; 3) x + y +x +=0; 4) x + y, 6x, 8y +5=0; 5) x + xy =0; 6) y, 6 = 0; 7) x, 3xy +y =0.

3x 5. 050300606 06 040060 ｧ06 00 ｧ0 ｧ ｧ 040503 縺030503 0403 縺07 閨09 43 34. 7 0805 ｧ 0906005 縺03 ｧ 04 縺060303 0304 0504 09 0906005 縺0606 040060 ｧ07 00 ｧ0 ｧ08: ) x +3y, 4x +6y, 7=0; ) x, y, 4x +y, 4=0; 3) x +5xy +y, 6x, 3y, 8=0. 35. 7030403 縺0305030 03 ﾀ0608 090303 罍陦ｧ0 05 04 縺060303 0304 0504 06 040060 ｧ07 09 09-0030 ｧ 06 ﾀ0903006 04 ｧ 閨06 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: ) 5x, 4xy +y = 4; ) x +4xy, y =. 36. 7 縺060303 0304 0504 0909 0030 ｧ 06 ﾀ0903006 04 ｧ 閨06 06 040060 ｧ07 ｧ 0403 ﾀ05 縺03- ｧ0504 09 罍ｧ040306: ) 3x, xy +3y, 4x, 4y, =0; ) x, 6xy + y, 4x, 4y +=0. 37. 7 縺060303 0304 0504 0909 0030 ｧ 06 ﾀ0903006 04 ｧ 閨06 06 040060 ｧ07 00 ｧ0 ｧ08: ) x +4xy +4y, 0x +0y, 50 = 0; ) x, 4xy +4y, 6x +y +8=0 ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ08. 38. 703 陦ｧ ﾀ09 罍ｧ0 ｧ0 005 0 ｧ 0304 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀ00 06 040060 ｧ08: ) x, 4xy +3y, 8x +4y +5=0; ) x +xy +4y, x +4y +4=0; 3) x +4xy +4y +3x +6y +=0. 70600 ｧ04 0406 縺040306 ｧ 05 縺06050406 06 040060 ｧ07 0305003 ﾀ ｧ05060004003 y, 0403 ﾀ05 縺03 ｧ0504 00 ｧ- 0 ｧ ｧ, 0304 縺06 閨0600070600306 0505 ｧ0 ｧ 06 040060 ｧ070 ｧ. 39. 7 罍ｧ0406 ﾀ05 ｧ 0909 0030 ｧ 06 ﾀ0903006 04 ｧ 閨06 06 040060 ｧ06 09 罍ｧ040308 y = = 3x, x +4 ｧ 0403 ﾀ05 縺03 ｧ0504 0606. 4x, 8 30. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 罍ｧ040308 040503 縺030503 0403 縺07 閨09, ｧ0060600608 0906005 縺030 0503 0906 O (; ) ｧ04 縺030803 閨0700608 06 縺06 0 0003 090303 罍陦ｧ0 05 ｧ 06 縺06 0503 09 ｧ (0; 4) ｧ (;,). 3. 70309 0504, 0503 06 040060 ｧ06 x + y = a 0304 縺06 閨0600070605 閨060506 04 03030003, 0403 ﾀ05 縺03 ｧ0504 04 03030006 ｧ 0 0805 ｧ 0606 0406 縺00 ｧ006. ｰ09 0 ｧ06. 7030406 縺0060504 03 ﾀ ｧ 090303 罍陦ｧ0 05 0 06050300 ' =,45. 3. 7 04 ｧ ﾀ 050406 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M (x; y), 030500300060 ｧ06 ﾀ ﾀ0503070 ｧ07 0305 09 07 閨0308 ｧ 09030503 縺0308 閨03 0503 09 ｧ F (m; n) 09 ﾀ ﾀ0503070 ｧ06 0305 00606 閨03 04 縺0700308 x cos + y sin, q = 0 04003 e. 050303 0 ｧ04 0903050707 ｧ09 ｧ0600503 04030006 0600030503 06 040060 ｧ07 06 縺06 A, B, C, :::, 0304 縺06 閨0600 ｧ0504 ｧ004 罍ｧ 00503 A + C ｧ A B B C.

344 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 33. 6 50307 ﾀ0 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀ00 06 040060 ｧ08: ) x, 4y =0; ) x +y +4x, 8y +=0; 3) x +5xy, 6y =0. 34. 7 縺060303 0304 0504 0909 0030 ｧ 06 ﾀ0903006 04 ｧ 閨06 06 040060 ｧ07 ｧ 0403 ﾀ05 縺03- ｧ0504 09 罍ｧ040306: ) x, xy + y, x, y, =0; ) 3x +0xy +3y, x, y +4=0. 35. 7 縺060303 0304 0504 0909 0030 ｧ 06 ﾀ0903006 04 ｧ 閨06 06 040060 ｧ07: ) x, xy + y, 0x, 6y +5=0; ) x +xy + y, 4x, 4y +3=0 ｧ 0403 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ, ｧ 0303 07 0600306 ｧ0 ｧ. 36. 703 陦ｧ ﾀ09 罍ｧ0 ｧ0 005 0 ｧ 0304 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀ00 06 040060 ｧ08: ) x, xy + y, 4x +4y +3=0; ) x, xy, 3y +6x +0y, 7=0. 70600 ｧ04 09 07 閨0306 06 040060 ｧ06 0305003 ﾀ ｧ05060004003 y, 0403 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ06, 0304 縺06- 閨0600070600606 ｧ0. 37. 7 04 ｧ ﾀ 050406 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M (x; y), 030500300060 ｧ06 ﾀ ﾀ0503070 ｧ08 0305 09030503 縺0308 閨03 0503 09 ｧ F (3; 3) 09 ﾀ ﾀ0503070 ｧ070 閨03 04 縺0700308 x + y =0 04003: ) e = ;)e =. 38. 7 04 ｧ ﾀ 050406 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M (x; y), 03 陦ｧ0 09030403 06 0006000308 0305 0503 09 ｧ F (a=; a=) ｧ0305 04 縺0700308 x + y =0, ｧ 04 罍ｧ0406 ﾀ05 ｧ 060503 09 09 0030 ｧ 06 ﾀ0903006 04 ｧ 閨06. 39. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, - 003 ﾀ0504 0904 陦 050304 ﾀ ﾀ0503070 ｧ08 0305 09030503 縺0308 閨03 04 縺0700308 x, y = ｧ 閨0303 ﾀ ｧOx 03 ﾀ05 0605 ﾀ07 0403 ﾀ050307000308 ｧ 0400308 3,. 7 縺060303 0304 0504 060503 0909 0030 ｧ 06 ﾀ0903006 04 ｧ 閨06 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606. x 6. 7030007 縺00306 090303 罍陦ｧ0 0503 706 ﾀ0504 0 040003 ﾀ0903 ﾀ05 ｧ 0 0503 09 O 04030006 ﾀ ｧ 0006 OP 04030007 縺0 07 03 ﾀ04 ( 罍ｧ ﾀ. ). ｯ0305 0403000307060 ｧ06 0503 09 ｧ M 0 040003 ﾀ0903 ﾀ05 ｧ 0304 縺06 閨0600 ｧ05 ﾀ07: ) 04030007 縺0030 060500030 ' = MOP; ) 閨00 ｧ00308 r 陦ｧ06 ﾀ-0406090503,,! OM: r = j,,! OMj. 7 罍ｧ ｧ 06 060 ｧ ｧ 06 040060 ｧ08, ﾀ0407 0304 060 ｧ08 r ｧ ', 030304 0605 04030006 003 ﾀ- ﾀ0 05 罍ｧ04 0504 04030007 縺00306 090303 罍陦ｧ0 0503 ' ｧ r 04 罍ｧ0 ｧ0 060 ｧ0 ｧ 09 09 ｧ06 060503 閨003 0403000307 ｧ0506000400306 ｧ 0305 罍ｧ09 0506000400306 0 060 ｧ07. 7 罍ｧ 0505030 0305 罍ｧ09 0506000400306 06050003 ' 0305 ﾀ ｧ050304 0605 ﾀ07 0403 ﾀ03040308 ﾀ05 縺06000906, 0305 罍ｧ09 0506000400306 r 03050900 閨03-04 0605 ﾀ07 006 0403 0006 06, 0403 060503 04 縺03 閨030007060 ｧ06 04030006 ﾀ.

3x 6. 7030007 縺00306 090303 罍陦ｧ0 0503 45 ｭ ﾀ00 ｧ 04 罍ｧ0070504 04030006 ﾀ 0 0003 閨0609 縺0503040308 04 縺07003060503000400308 090303 罍陦ｧ- 0 05, 04030007 縺00606 03 ﾀ04OP 03 ﾀ04Ox, 0503 閨0609 縺05030403 090303 罍陦ｧ0 0503 (x; y) 0503 09 ｧ M ｧ 0606 04030007 縺00306 090303 罍陦ｧ0 0503 ('; r) 0306- 閨0605 ﾀ0407 003 04 ｧ ﾀ ｧ003 ﾀ050406 x = r cos '; y = r sin '; () r = x + y ; tg ' = y x : () ｭ ﾀ00 ｧ 04 罍ｧ0070504 07030906 ﾀ 050000 ｧ04 ﾀ, 05 ｧ0406 縺03030003 ｧ 04 03030003 04030006 ﾀ, 070309 000400606 03 ﾀ04 ﾀ ｧ0006-05 罍ｧ ｧ 04030007 縺00606 03 ﾀ04, 0 04 040006000606 04 ﾀ0503-7 ｧ ﾀ. 縺03006, 04 縺0305 ｧ0403040300030700606 0300 ｧ07 08000608 0406 縺00 ｧ006, 0503 06 040060 ｧ06 04 ﾀ0608 05 縺0608 09 罍ｧ040308 04 04030007 縺00308 090303 罍陦ｧ0 05 08 0306 閨0605 03 陦ｧ0 090304030: r =, e cos ' ; (3) 05 閨06 e 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605, 04 00605. 6 70007 050000 ｧ04 ﾀ ｧ 05 ｧ0406 縺03030003 = b a. 330. 6 5 04030007 縺00308 ﾀ ｧ ﾀ0506006 090303 罍陦ｧ0 05 ('; r) 0403 ﾀ05 縺03 ｧ0504 0503 09 ｧ A(0; 3), B(=4; ), C(=; 3), D(; ), E(3=; 3). 33. 703 ﾀ05 縺03 ｧ0504 0503 09 ｧ A(=;,), B(,=; 3), C(,=4;,4), D(=3;,3). 33. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ06 r =+cos'. ｰ09 0 ｧ06. ｮ03 ﾀ05 04 ｧ0504 05 0300 ｧ0906 0 060 ｧ08 r 閨0007 ' =0; =3; =; =3;. 333. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ ( ﾀ0. ﾀ. 334 ｧ 335, 罍ｧ ﾀ. 80, 8 ｧ 86): ) r = a' ( 縺08 ｧ006 閨0304 ﾀ04 ｧ 0004); ) r = a(, cos ') (09 罍陦ｧ03 ｧ ); 3) r = a cos ' (000600 ｧ ﾀ09 05 ); 4) r = a=' (05 ｧ0406 縺030300 ｧ 06 ﾀ09 07 ﾀ04 ｧ 0004); 5) r = a( + cos ') (0600 ｧ0509 7 ﾀ09 0007): 334. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ: ) r = a; )' = 4 ;3)r = b sin '. 335. 7 04 ｧ ﾀ 0504 04 04030007 縺00308 090303 罍陦ｧ0 05 08 06 040060 ｧ07: ) 04 縺0700308, 0305 ﾀ0609 0600608 0305 04030007 縺00308 03 ﾀ ｧ 0305 縺06 0309 a ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 00608; ) 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 A(; a) ｧ 04 000006000400308 0403-0007 縺00308 03 ﾀ ｧ. 336. 7 04 ｧ ﾀ 0504 04 04030007 縺00308 090303 罍陦ｧ0 05 08 06 040060 ｧ06 04 縺0700308, 04 縺03-0803 閨0700608 06 縺06 0503 0906 A(; a) ｧ ﾀ03 ﾀ05 0400070600608 ﾀ 04030007 縺00308 03 ﾀ0406 06050300. 337. 7 04 ｧ ﾀ 0504 04 04030007 縺00308 090303 罍陦ｧ0 05 08 06 040060 ｧ06 0309 縺0607003 ﾀ05 ｧ ﾀ 0906005 縺030 04 0503 0906 C(0; a) ｧ 陦ｧ06 ﾀ030, 040030 a.

346 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 338. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: ) r =3, sin '; ) r = + cos 3'; 3) r =, sin 3'. ｰ 09 0 ｧ 06. 0504 縺06 閨0600 ｧ0504 06050003, 04 罍ｧ 09030503 縺0308 ｧ006060 r max ｧ r min. 339. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ ( ﾀ0. ﾀ. 334, 罍ｧ ﾀ. 8 ｧ 83): ) r = a sin 3' (05 縺060800060406 ﾀ05090304 07 縺03 ); ) r = a sin ' ( 060503 縺060800060406 ﾀ05090304 07 縺03 ). 340. 7 縺060303 0304 0504 09 04030007 縺0030 090303 罍陦ｧ0 05 0 06 040060 ｧ07 00 ｧ- 0 ｧ08: ) x, y = a ; ) x + y = a ; 3) x cos + y sin, =0; 4) y = x; 5) x + y = ax; 6) (x + y ) = a (x, y ). 34. 7 縺060303 0304 0504 09 閨0609 縺050304030 090303 罍陦ｧ0 05 0 06 040060 ｧ07 00 ｧ- 0 ｧ08 ｧ 0403 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ: ) r cos ' = a; ) r =a sin '; 3) r sin ' =a ; 4) r sin ' + = a ; 4 5) r = a( + cos '). 34. 7 04 ｧ ﾀ 0504 09 0030 ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 09 罍ｧ040308 040503 縺030503 0403- 縺07 閨09 : ) r = 3) r = 9 5, 4 cos ' ; ) r = 9 4, 5 cos ' ; 3, cos '. 343. 070300803 ｧ. ｪ06 縺06 0503 0906 A(=; a) 04 縺030406 閨060 04 縺070 07, 04 00000600040 07 04030007 縺00308 03 ﾀ ｧ. 7 縺03 ｧ 0403000400308 0006 OB 0406 縺06 ﾀ0609 0605 050506 04 縺0700606 040503 0906 B. 7 0006 06 0305 0503 09 ｧ B 0403 030306 0606 ﾀ0503 縺03003 030500030706003 0305 縺06 09 ｧ BM = BM = b. 0504 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 M ｧ M 04 04030007 縺00308 090303 罍陦ｧ0 05 08 ｧ 0403 ﾀ05 縺03 ｧ050409 罍ｧ040606. 344. ｮ 05 03 07 03 ｧ. 7 縺070 07 x = a 0406 縺06 ﾀ0609 0605 03 ﾀ04Ox 040503 0906 A ｧ04 縺03 ｧ 0403000400308 0006 OB 040503 0906 B. 7 0006 06 0305 0503 09 ｧ B 0403 030306 0606 ﾀ0503 縺03003 030500030706003 0305 縺06 09 ｧ BM ｧ BM, 0400306 AB. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M ｧ M 04 04030007 縺00308 ｧ 閨0609 縺0503040308 090303 罍陦ｧ0 05 08 ( 罍ｧ ﾀ. 84, ﾀ. 335).

3x 6. 7030007 縺00306 090303 罍陦ｧ0 0503 47 345. 0504 00 07 ﾀ ﾀ ｧ0 ｧ. ｯ03 09 M ('; r) 閨04 ｧ070605 ﾀ07 05 09, 0503 04 縺03- ｧ 0406 閨060 ｧ06 ﾀ ﾀ0503070 ｧ08 0305 00606 閨03 0503 0609 F (0; a) ｧ F (; a) 03 ﾀ05 0605 ﾀ07 040030 b. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 06090503 罍ｧ ｧ 閨04 ｧ07060 ｧ07 0503 09 ｧ M 04 04030007 縺00308 090303 罍陦ｧ0 05 08. 346. 07 罍陦ｧ03 ｧ. 7 04 縺03 ｧ 040300040030 0006 06 OA 0305 0503 09 ｧ A 0406- 縺06 ﾀ06 060 ｧ07 060503 ﾀ 0309 縺0607003 ﾀ050406 r = a cos ' 03050900 閨0304 0605 ﾀ07 0403 030306 ﾀ0503 縺03003 0305 縺06 0309 AM = AM = a. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 05060300605 罍ｧ- 06 ﾀ09030503 006 ﾀ05 0503 0609 M ｧ M 04 04030007 縺00308 ｧ 閨0609 縺0503040308 090303 罍陦ｧ0-05 08. 347. 07 罍陦ｧ03 ｧ (0504 ｧ09 ｧ090003 ｧ ). 07 縺0605 陦ｧ 00605 a 09-05 ｧ05 ﾀ07 0306 ﾀ0903000407060 ｧ07 0403 09 縺060506 05 09030503 0706 陦ｧ 00605 ﾀ0 縺0607 ｧ 060503. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 罍ｧ040308, 0304 ｧ ﾀ 000308 0503 090308 M 09 050700608 ﾀ07 0309- 縺0607003 ﾀ05 ｧ, 06 ﾀ00 ｧ 04030006 ﾀ ｧ 0 000400306 0403000307060 ｧ06 0503 09 ｧ M 04 罍ｧ0070504 0503 0906 09 ﾀ 0 ｧ07 09 縺06050304, 04030007 縺00606 03 ﾀ04 04 縺030406 ﾀ05 ｧ 06 縺06 0906005 縺03 09 縺06-050304 (04 0 00040030 0403000307060 ｧ ｧ). 348. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: ) r = 3 + cos '; )r =3, sin 3'; 3) r = a cos ' ( ﾀ0. 0609 0 ｧ06 09 06 338). 349. 703 ﾀ05 縺03 ｧ0504: ) r = 4( + cos '); )r =, sin '. 350. 7 04 ｧ ﾀ 0504 04 04030007 縺00308 090303 罍陦ｧ0 05 08 06 040060 ｧ06 04 縺0700308, 04 縺03-0803 閨0700608 06 縺06 000306 0503 09 ｧ A(; a) ｧ B(; b). ｰ09 0 ｧ06. 7 ﾀ ﾀ00305 縺060504 04 ｧ ﾀ ｧ003 ﾀ0504 00607 閨06 0400030 閨070 ｧ 05 縺060605030004-0 ｧ090304 AOM, BOM ｧ AOB, 05 閨06 M ('; r) 04 縺03 ｧ 040300040 07 0503 09 04 縺0700308. 35. 7 04 ｧ ﾀ 0504 09 0030 ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 09 罍ｧ040308 040503 縺030503 0403- 縺07 閨09 : ) r =, 3 cos ' ; ) r = 3) r =, 5 cos ' ;, cos '. 35. 6ｧ80600 ｧ ﾀ09 05 706 縺0060000 ｧ. ｯ03 09 M ('; r) 閨04 ｧ070605 ﾀ07 05 09, 0503 04 縺03 ｧ 0406 閨060 ｧ06 0606 ﾀ ﾀ0503070 ｧ08 0305 0503 0609 F (0; c) ｧ F (; c) 03 ﾀ05 0605 ﾀ07 040030 c. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05 06090503 罍ｧ ｧ 閨04 ｧ07060 ｧ07 04 04030007 縺00308 ｧ 閨0609 縺0503040308 090303 罍陦ｧ0 05 08. ｰ09 0 ｧ06. 703 050603 縺06006 0903 ﾀ ｧ006 ﾀ0304 FM = r + c, rc cos ' ｧ F M = r + c +rc cos ', 04 罍ｧ 060 0403 06 ﾀ000304 ｧ06 FM F M = c 4.

348 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 353. ｰ00 ｧ0509 7 ﾀ09 0007. 7 04 縺03 ｧ 040300040030 0006 06 OA 0305 0503 09 ｧ A 0406 縺06 ﾀ06 060 ｧ07 060503 ﾀ 0309 縺0607003 ﾀ050406 r = a cos ' 0403 030306 ﾀ0503 縺03003 03050003-0706003 0305 縺06 09 ｧ AM = AM = b. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 05060300605 罍ｧ 06- ﾀ09030503 006 ﾀ05 0503 0609 M 04 04030007 縺00308 090303 罍陦ｧ0 05 08. 354. ｪ060503 縺060800060406 ﾀ05090304 07 03. 070300903 0305 縺06 09 AB = = a ﾀ09030004 0705 0403 03 ﾀ070 閨0609 縺0503040308 090303 罍陦ｧ0 05. 0ｷ3 0 00 090303 - 陦ｧ0 05 0304060060 0 AB 0406 縺04060 陦ｧ09060007 OM. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M (x; y) 04 罍ｧ 04 ﾀ060403 0030700308 040300030706-0 ｧ0708 0305 縺06 09 AB. x 7. 縺00050603 ｧ 06 ﾀ09 ｧ06 09 罍ｧ040306 05 縺060504060503 ｧ 0403 ﾀ00 ｧ08 0403 縺07 閨090304 355. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040306 ( ﾀ0. ﾀ. 33, 罍ｧ ﾀ. 66{69): ) y = x 3 =3 (090603 ｧ 06 ﾀ09 07 04 030300 ); ) y = x 3 (04030006090603 ｧ 06 ﾀ09 07 04 030300 ); 3) y 3 = x 4) y = x(x, 4) (040605000604 07 04 030300 ): 356. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: ) x =3 + y =3 = a =3 ( ﾀ05 縺03 ｧ 04003 ﾀ0503 縺03000707); x =3 y =3 ) + =, b 6= a ( ﾀ05 縺03 ｧ 006 04003 ﾀ0503 縺03000707). a b ｰ 09 0 ｧ 06. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 09 罍ｧ040308 ﾀ 03 ﾀ070 ｧ Ox ｧ Oy ｧ 0406 縺040308 09 罍ｧ040308 ﾀ 04 縺070030 ｧ y = x, 040503 縺0308 ﾀ 04 縺070030 ｧ y = b a x ( 罍ｧ ﾀ. 78 0 ﾀ. 334). 357. 703 ﾀ05 縺03 ｧ0504 0 0305 縺06 0906 [,; ] 09 罍ｧ040306: ) y = x n+ ;)y = = x n ; 3) x n + y n = 04 罍ｧ n = ; ; 4. 0709 09 ｧ0 00030 0030 04 罍ｧ0300 ｧ07 0605 ﾀ07 0505 ｧ 09 罍ｧ040306, 090305 n!? ｰ 09 0 ｧ 06. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 0406 縺040308 09 罍ｧ040308 ﾀ 04 縺0700308 y = = x,040503 縺0308 09 罍ｧ040308 ﾀ 04 縺0700308 y = ｧ05 縺0605040608 09 罍ｧ040308 ﾀ 04 縺0700308 y = x. n n 6ｦ0 06 陦ｧ0 ｧ0906 0 ﾀ0005 03 04 罍ｧ0070504 0 090006050309 09000605 050308 03060 05 ｧ. 358. 罍ﾀ05 縺03 ｧ. 070300903 0305 縺06 09 AB = a ﾀ09030004 0705 0403 03 ﾀ070 閨0609 縺0503040308 090303 罍陦ｧ0 05. 7 縺0700306 AC ｧ BC, 04 000006000400306 03 ﾀ070 090303 罍陦ｧ0 05, 0406 縺06 ﾀ0609 0605 ﾀ07 04 0503 0906 C. 0ｷ3 C 0304060060 0 AB 0406 縺04060- 陦ｧ09060007 CM. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M (x; y) 04 罍ｧ 04 ﾀ060403 0030700308 0403000307060 ｧ0708 0305 縺06 09 AB.

3359. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: x 8. ｯ 0 ﾀ09060 閨0600500306 09 罍ｧ040306 49 ) y = x3 a, x (09 ｧ ﾀ ﾀ03 ｧ, 罍ｧ ﾀ. 85, ﾀ. 335); ) y = 8a3 x +4a (000309030, 罍ｧ ﾀ. 76, ﾀ. 333): 360. 07 07 07 0503 09 P (x 0 ; y 0 ) 04 03030003 y =x ﾀ0060060 04-0000060004003 03 ﾀ ｧ Ox 0 ﾀ ﾀ0503070 ｧ06 PM = OP. 7 0805 ｧ 05060300605 罍ｧ- 06 ﾀ090306 006 ﾀ0503 0503 0609 M. 36. ｮ0506 縺0706004 OA = a 04 0 0605 ﾀ07 040309 縺0605 0 00 090303 罍陦ｧ0 05 O. 6 50503 0906 A 09 006006 04 罍ｧ09 縺060400060 00 縺0 ｧ 縺030 ﾀ0506 縺0706004 AB =a, 090300609 09030503 縺030503 ﾀ09030004 ｧ05 0403 Ox. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 00 ｧ0 ｧ ｧ, 09030503 縺0606 0306 閨0605 0304 ｧ ﾀ0304 0504 04 罍ｧ 0505030 ﾀ06 縺06 陦ｧ0 M 0305 縺06 09 AB. 36. ｨC ｧ ﾀ ﾀ03 ｧ. 7 縺03 ｧ 0403000400308 0006 OA ( 罍ｧ ﾀ. 85, ﾀ. 335) 0406- 縺06 ﾀ0609 0605 0309 縺0607003 ﾀ0504 x + y = ax 040503 0906 A ｧ 04 縺0700606 x = a 04 0503 0906 B. 7 0006 06 03050900 閨0304 0605 ﾀ07 0305 縺06 0309 OM = AB. ｮ03 ﾀ05 04 ｧ0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M. 363. 7 縺03 ｧ 0403000400308 0006 OB ( 罍ｧ ﾀ. 85) 0406 縺06 ﾀ0609 0605 04 縺0700606 x = a 040503 0906 B. C 04 縺03060909 ｧ07 0503 09 ｧ B 0 03 ﾀ04 Oy ｧ M 04 縺03060909 ｧ07 0503 09 ｧ C 0 04 縺0700606 OB. 70309 0504, 0503 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 M 06 ﾀ0504 09 ｧ ﾀ ﾀ03 ｧ. 364. ｭ ﾀ00 ｧ ｧ 0406 縺00 ｧ003 04 03030003 y =,4ax 030406 ﾀ09 0504 0406 縺04060 陦ｧ- 09060007 縺03 0 09 ﾀ 0506000400306 09 05050308 09 罍ｧ040308, 0503 05060300605 罍ｧ 06 ﾀ09 ｧ0 006 ﾀ05030 03 ﾀ00304 0 ｧ08 0406 縺04060 陦ｧ09060007 縺0304 0306 閨0605 09 ｧ ﾀ ﾀ03 ｧ. 6 70309 0504. 365. 6ｧ80309030. 7 縺03 ｧ 0403000400308 0006 OA 0406 縺06 ﾀ0609 0605 0309 縺0607003 ﾀ0504 x + y = ay ｧ 04 縺0700606 y = a 040503 09 08 A ｧ B, ｧ 09030503 縺0308 04 縺030406 閨06003 04 縺0700306, 04 000006000400306 ﾀ030305040605 ﾀ0504060003 03 ﾀ ｧ Ox ｧ 03 ﾀ ｧ Oy 閨03 0406 縺06 ﾀ06 060 ｧ07 04 0503 0906 M. 0504 縺06 閨0600 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 M. 366. 6 70609 縺050304 00 ｧ ﾀ05 x 3 + y 3, 3axy =0. 70309 0504, 0503 050503 06 040060 ｧ06 04030403 縺0305030 03 ﾀ0608 090303 罍陦ｧ0 05 0 45 04 罍ｧ0403 陦ｧ05 ﾀ07 09 04 ｧ 閨06 Y X 3 (3b, X) = 3(b + X),05 閨06 b a =. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040606, 0304 縺06 閨0600 ｧ04 04 003040308 ﾀ ｧ ﾀ0506006 090303 罍陦ｧ0 05 030300 ﾀ0504 ﾀ0403000307060 ｧ07 09 罍ｧ040308 ｧ 0606 ﾀ ｧ0-00605 罍ｧ06, 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 04 縺0700308 y = x (05. 06. ﾀ 003040308 03 ﾀ0406 OX) ｧ ﾀ ｧ00405030506. 70309 0504, 0503 06 040060 ｧ06 ﾀ ｧ00405030503 04 003040308 ﾀ ｧ ﾀ0506006 090303 罍陦ｧ0 05 0306 閨0605 X =,b, 04 ﾀ05 縺0308 x + y + a =0( ﾀ0. 罍ｧ ﾀ. 79, ﾀ. 334). x 8. ｯ 0 ﾀ09060 閨0600500306 09 罍ｧ040306 367. ｨC ｧ090003 ｧ. 07 縺0605 陦ｧ06 ﾀ a 09 05 ｧ05 ﾀ07 0403 04 縺0700308 OX 0306 ﾀ0903000407060 ｧ07. ｮ03 ﾀ05 04 ｧ0504 04 00605 罍ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 09 罍ｧ040308, 0304 ｧ- ﾀ 000308 0503 090308 M 0309 縺0607003 ﾀ05 ｧ, 04 罍ｧ00704 04 00605 t 06050300 04030403 縺0305

350 0600.. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ 09 05070060503 ﾀ07 09 縺0605 ｧ 0403000307 ｧ04, 0503 04 罍ｧ t =00503 09 M 0 0803 陦ｧ05 ﾀ07 04 0 0006 090303 罍陦ｧ0 05. 368. 7 0406 縺0509 09 06 05. 7 ｧ0504, 0 00305 00 07 0 0309 縺0607003 ﾀ0504 x + y = a, 0 050304 0605 ﾀ07, 03 ﾀ05 04 07 ﾀ04 0 0507006050308. ｮ03 ﾀ05 04 ｧ0504 04-00605 罍ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 09 罍ｧ040308, 0304 ｧ ﾀ 000308 0903009030 0 ｧ05 ｧ, 06 ﾀ00 ｧ 040 0006 090300609 0 ｧ05 ｧ 0 0803 陦ｧ05 ﾀ07 04 0503 0906 (a; 0). 6ｦ0 04 00605 t 04 罍ｧ- 0070504 閨00 ｧ006 ﾀ00305 000308 閨0605 ｧ (04 陦ｧ06 ﾀ 08). 368. 0704 陦 05 罍ｧ ﾀ. 7 縺03 ｧ 0403000400308 0006 OM, ﾀ03 ﾀ05 040007060 ｧ08 ﾀ03 ﾀ0406 Oy 06050300 t (04 陦ｧ 0 08), 0406 縺06 ﾀ0609 0605 04 縺0700606 x = at 040503 0906 M. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609 M. 370. 704 ｧ09 ｧ090003 ｧ. 07 縺0605 陦ｧ06 ﾀ r 09 05 ｧ05 ﾀ07 0306 ﾀ0903000407060 ｧ07 0403 09 縺060506 陦ｧ06 ﾀ R ﾀ0 縺0607 ｧ 060503. ｮ03 ﾀ05 04 ｧ0504 04 00605 罍ｧ 06 ﾀ09 ｧ06 06 04-0060 ｧ07 09 罍ｧ040308, 0304 ｧ ﾀ 000308 0503 090308 M 09 050700608 ﾀ07 0309 縺0607003 ﾀ05 ｧ. ( 7 罍ｧ r = R 0504 ｧ09 ｧ090003 ｧ 0303 0 0605 ﾀ07 04 09 ｧ 03 ｧ 06. ｮ0. 06 347.) 37. 06 ｧ040309 ｧ090003 ｧ. 07 縺0605 陦ｧ06 ﾀ r 09 05 ｧ05 ﾀ07 0306 ﾀ090300040706-0 ｧ07 0403 09 縺060506 陦ｧ06 ﾀ R>r0400605 罍ｧ 0060503. ｮ03 ﾀ05 04 ｧ0504 04 00605 罍ｧ 06- ﾀ09 ｧ06 06 040060 ｧ07 09 罍ｧ040308, 0304 ｧ ﾀ 000308 0503 090308 M 09 050700608 ﾀ07 0309 縺0607003- ﾀ05 ｧ. ( 7 罍ｧ r = R=4 05 ｧ040309 ｧ090003 ｧ 0303 0 0605 ﾀ0704 ﾀ05 縺03 ｧ 閨06 x =3 + + y =3 = a =3.)

30600 04 6 5 ｭ07 ｯ05 7 7 縺00 縺6ｧ806 ｭ 7 7 x. ｮ000307060 ｧ06 0406090503 縺0304. ｰ000307060 ｧ06 0406090503 0 ﾀ09 0007.0504 縺06 閨0600060 ｧ07. 6 506090503 縺030 0 0304 0605 ﾀ07 0 04 040006000308 0305 縺06 0309,! AB ( 罍ｧ ﾀ. 3), 04 09030503 縺030 0503 09 A ﾀ ﾀ0 05 罍ｧ04 0605 ﾀ07 09 09 0 0003, 0503 09 B 09 09 090300609. 6 506090503 030303 0 0605 ﾀ07 ｧ00 ｧ 0609 0 ｧ060 060503 0 00 ｧ 0903009,! AB ﾀ03 ﾀ05 縺0600090308 0 0406 縺0806, ｧ00 ｧ 03 閨00308 09 090308-0 ｧ0306 閨04 030609040308, 0403 閨060006000308 0403-000607 ｧ 縺0030 00 罍ｧ0705030, 0 04 罍ｧ006 a. 0303 閨060004 ( 閨00 ｧ0 ) 0406090503 030303 0 0605 ﾀ07 j,! ABj, ｧ00 ｧ jaj, ｧ00 ｧ AB, ｧ00 ｧ a. 6 506090503 縺03, 04 000006000400306 03 閨00308 04 縺0700308, 0 0304 0605 ﾀ07 09030000 ｧ006 縺0030 ｧ. 6 506090503 縺03, 04 000006000400306 03 閨00308 040003 ﾀ0903 ﾀ05 ｧ, 0 0304 0605 ﾀ07 090300400 0 縺0030 ｧ. 6 704 0406090503 a ｧ b ( 罍ｧ ﾀ. 3) 0 0304 0605 ﾀ07 040030 ｧ, 06 ﾀ00 ｧ 030 ｧ: ) ｧ0060605 0400306 003 閨0600 ｧ; ) 09030000 ｧ006 縺003; 3) 0-04 040006003 04 03 閨006 ﾀ0503 縺03006.. ｰ000307060 ｧ06 0406090503 0 ﾀ09 0007. 7 縺03 ｧ 0406 閨060 ｧ060 040609-0503 a 0 ｧ ﾀ0003 ( ﾀ09 0007 ) m 0 0304 0605 ﾀ07 003040308 0406090503, ｧ006060 ｧ08 閨00 ｧ006 ajmj ｧ 0 04 040006000308 03 陦ｧ0 09030403 ﾀ a (04 罍ｧ m>0) ｧ00 ｧ 04 縺0305 ｧ04030403000307003 a (04 罍ｧ m<0). 3. ｮ000307060 ｧ06 0406090503 縺0304. ｮ06000308 0406090503 縺0304 a+b+c 0 0304 0605 ﾀ07 0406090503 R =,! OC ( 罍ｧ ﾀ. 4), 00309 060 ｧ08 00030 00606 OABC, 0403 ﾀ05 縺0306000606 7 ｧ ﾀ. 3 7 ｧ ﾀ. 4 ｧ 000308 0406090503 縺0304. 6 5 ﾀ05003 ﾀ05 ｧ, 04 04 000006000305 0006, 0403 ﾀ05 縺030600030 0 000308 0406090503 08,! OA = a ｧ,! OB = b, 03 閨0 0406090503 - 陦ｧ 05030 0004,! OC 06 ﾀ0504 ﾀ0600 a + b, 陦縺0605 07,! BA 06 ﾀ0504 003 ﾀ0504 a, b 000308 0406090503 縺0304. 4. 7 縺03060909 ｧ07 0406090503 0 03 ﾀ04. 706 ﾀ0504 0406090503 a ﾀ03 ﾀ05 0400070605 06050300 ' ﾀ 03 ﾀ0406 Ox. ｯ0305 04 縺03060909 ｧ07 0406090503 0 050506 03 ﾀ04 0304 縺06 閨0600070605 ﾀ07 0703 縺006-000308 04 x a = jaj cos ' = a cos ( a;ox):

35 0600.. 6 506090503 縺0 07 00050603 7 縺03060909 ｧ07 ﾀ060003 0406090503 縺0304 0 03 ﾀ04 040 ﾀ060006 04 縺03060909 ｧ08 ﾀ03 ﾀ05 040007-060 ｧ08 0406090503 縺0304 0 0506 0706 03 ﾀ04: 04 x (a + b) =04 x a + 04 x b: 37. 703 ﾀ0503 縺030 0 OA ｧ OB 04 縺0700306050300040 ｧ09 OACB 030500030706003 06 陦ｧ0 ｧ 00306 0406090503 縺03 i ｧ j ( 罍ｧ ﾀ. 5). 6 503 ｧ0504 06 縺06 i ｧ j 0406090503 縺03,! OA,,! AC,,! CB,,! BO,,! OC ｧ,! BA, 06 ﾀ00 ｧ OA =3 ｧ OB =4. 373. 706 ﾀ0504 0 罍ｧ ﾀ. 5 M ﾀ06 縺06 陦ｧ0 BC ｧ N ﾀ06 縺06 陦ｧ0 AC. 0504 縺06 閨0600 ｧ0504 0406090503 縺03,,! OM, ON,,! ｧ MN,,! 04 罍ｧ OA =3 ｧ OB =4. 374. 7 040003 ﾀ0903 ﾀ05 ｧ 003 0503 09 ｧ A(0;,), B(4; ) ｧ C(4;,). 6 50 0006 090303 罍陦ｧ0 05 04 罍ｧ- 00030706003 ﾀ ｧ0003,! OA,,! OB ｧ,! OC. 703 ﾀ05 縺03 ｧ0504 ｧ08 04003 閨0608 ﾀ0504060600606,,! OM, 0 0805 ｧ 0606 04 縺03060909 ｧ ｧ 0 03 ﾀ ｧ 090303 罍陦ｧ0 05 ｧ 040600 ｧ ｧ006. 6 503 ｧ0504 ﾀ ｧ- 7 ｧ ﾀ. 5 0003,! OA,,! OB,,! OC ｧ,,! OM 06 縺06 06 陦ｧ0 ｧ 00306 040609-0503 縺03 i ｧ j 090303 罍陦ｧ0 0500308 03 ﾀ0608. 375. 6 7 003 05 罍ｧ 090300400 0 縺00308 06 陦ｧ0 ｧ 00308 0406090503 m, n ｧ, 04 罍ｧ 060 ( [m; n) =30 ｧ ( dn; ) =60. 703 ﾀ05 縺03 ｧ0504 0406090503 u = m + +n, 3 ｧ 0403 ｧ ﾀ00 ｧ0504 060503 003 閨060004. ｰ09 0 ｧ06. 6 5 00030 00308, 0403 ﾀ05 縺0306000308 ｧ 0406090503 縺0304 m, n ｧ,3, 04 縺03 閨030007 ｧ0504 0406 縺040306 0406003 閨03 0406 縺06 ﾀ06 060 ｧ07 ﾀ 05 縺060504 ｧ0. 376. 7 縺030406 罍ｧ0504 0 00 ｧ05 ｧ 06 ﾀ09 ｧ ｧ 05060300605 罍ｧ 06 ﾀ09 ｧ 0406090503 縺00306 050307- 閨06 ﾀ0504 : ) a b, a a + = + b ; ) a, a + b a, b =. 377. 7 05 縺0608 006090300400 0 縺00308 0406090503 08,! OA = a,,! OB = b ｧ,! OC = c 0403 ﾀ05 縺03060 04 000006000604 ｧ0406. ｰ09 0504 0506 060503 0406090503 - 陦ｧ 05030 00 ｧ, 09030503 縺0306 ﾀ030305040605 ﾀ0504060003 04003 a + b, c, a, b + c, a, b, c ｧ b, a, c. 378. ｮ 040300300406 06 縺050607 ｧ 377 04 縺030406 罍ｧ0504 0406 縺06006 ﾀ05 ｧ05060004-00306 ﾀ040308 ﾀ050403 0406090503 縺00308 ﾀ060003 a + b, c = a, c + b = b + a, c = b, c + a: 379. 6 7 003 0406090503 縺03,! OA = a ｧ,! OB = b. 6 506090503,! OC = c 006 陦ｧ- 0 4OAB. 7 000307 ｧ0504 0 00 ｧ05 ｧ 06 ﾀ09 ｧ ｧ 05060300605 罍ｧ 06 ﾀ09 ｧ: ) 0406090503 c 0403 0406090503 0 a ｧ b; )0406090503 a 0403 0406090503 0 b ｧ c. 380. 6 5 04 縺0700306050300040 ｧ0906 OACB ( 罍ｧ ﾀ. 5) M ｧ N ﾀ06 縺06 陦ｧ003 ﾀ0503 縺030 BC =3 ｧ AC =4. 7 000307 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ ｧ 0 00 ｧ05 ｧ- 06 ﾀ09 ｧ 0406090503,! OC = c 0403 0406090503 0,,! OM = a ｧ ON,,! = b. ｰ09 0 ｧ06. 6 5 06 ﾀ000304 ｧ06 c = ma + nb 0403 陦ﾀ05 04 ｧ0504 0403 07060 ｧ07 a, b ｧ c 06 縺06 i ｧ j ｧ ﾀ 040 ｧ0504 0903050707 ｧ09 ｧ0600503 ﾀ000604 ｧ ﾀ04 04 04 罍ｧ i ｧ j.

3x. 7 縺07003060503000400306 090303 罍陦ｧ0 0503 0503 09 ｧ ｧ 0406090503 04 04 縺03 ﾀ05 0 ﾀ050406 53 38. 6 7 0 04 04 ｧ000400308 0006 ﾀ05 ｧ06050300040 ｧ09 OABCDE ﾀ03 ﾀ0503 縺0300308 OA = 3. 050303 0 ｧ04 06 陦ｧ0 ｧ 00306 0406090503 縺03 0 04 0400060 ｧ08,! OA,,! AB,,! BC 06 縺06 m, n ｧ, 06 ﾀ05 00304 ｧ0504 04 ｧ ﾀ ｧ003 ﾀ0504 00607 閨06 0 ｧ0 ｧ (0 04 罍ｧ- 006, ﾀ ﾀ00305 縺060 ｧ060 05 040609 ｧ ｧ OABC). 6 503 ｧ0504 05060 06 縺06 m ｧ n 0406090503 縺03,! OB,,! BC,,! EO,,! OD ｧ,! DA. 38. 6 5 040030306 陦縺06000308 05 040609 ｧ ｧ OACB ( 罍ｧ ﾀ. 6) 06050300 BOA = = 60, OB = BC = CA =, M ｧ N ﾀ06 縺06 陦ｧ003 ﾀ0503 縺030 BC ｧ AC. 6 503 ｧ0504 0406090503 縺03,! AC,,,! OM,,,! ON ｧ,,! MN 06 縺06 m ｧ n 06 陦ｧ0 ｧ - 00306 0406090503 縺03 0 04 0400060 ｧ08,! OA ｧ,! OB. 383. 6 7 003 0406090503 縺03 a ｧ b, 06050300 00607 閨0609030503 縺030 ｧ 0. 703 ﾀ05 縺03 ｧ0504 040609-0503 c = a, ; 5b ｧ 0304 縺06 閨0600 ｧ0504 060503 003 閨060004, 06 ﾀ00 ｧ a =3 ｧ b =4. 7 ｧ ﾀ. 6 384. 7 040003 ﾀ0903 ﾀ05 ｧ 003 0503 09 ｧ A(3; 3), B(,3; 3) ｧ C(,3; 0). 6 5 0 0006 090303 罍陦ｧ0 05 04 罍ｧ00030706003 ﾀ ｧ0003,! OA,,! OB ｧ,! OC. 703 ﾀ05 縺03 ｧ0504 04003 閨0608 ﾀ0504060600606,,! OM, 0 0805 ｧ 0606 04 縺03060909 ｧ ｧ 0 03 ﾀ ｧ 090303 罍陦ｧ0 05 ｧ 040600 ｧ ｧ006. 6 503 ｧ0504 ﾀ ｧ0003,! OA,,! OB,,! OC ｧ,,! OM 06 縺06 06 陦ｧ0 ｧ 00306 0406090503 縺03 i ｧ j 090303 罍陦ｧ0 0500308 03 ﾀ0608. 385. ) 6 5 05 040609 ｧ ｧ OACB ｧ006060 BC = OA=3 ｧ BCkOA. 7-000307 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ ｧ 0 00 ｧ05 ｧ 06 ﾀ09 ｧ 0406090503,! OA = a 0403 0406090503 0,! OC = c ｧ,! OB = b. ｰ09 0 ｧ06. 0ｷ3 4OBC 00307003 c 0403 ｧ0504 06 縺06 b ｧ a ｧ 05060 縺0600 ｧ0504 04030006 06000306 06 040060 ｧ06 0305003 ﾀ ｧ05060004003 a. ) ｯ03 09 B 閨0600 ｧ05 閨060506 0309 縺0607003 ﾀ05 ｧ ^ AC= 90 04030500300060 ｧ ｧ :. O 0906005 0309 縺0607003 ﾀ05 ｧ. 7 000307 ｧ0504 0406090503,! OC = c 0403 0406090503 0,! OA = a ｧ,! OB = b. x. 7 縺07003060503000400306 090303 罍陦ｧ0 0503 0503 09 ｧ ｧ 0406090503 04 04 縺03 ﾀ05 0 ﾀ050406.0504 縺06 閨0600060 ｧ06. 706 ﾀ0504 003 05 罍ｧ 04 ｧ0003 0406 縺04060 陦ｧ09060007 縺00306 0903-03 罍陦ｧ0 0500306 03 ﾀ ｧ ﾀ 03030 ｧ0,,! 0 00030 O ｧ 0 0503 09 M ( 罍ｧ ﾀ. 7). 7 縺03060909 ｧ ｧ 0606 陦ｧ06 ﾀ-0406090503 OM = r 0 03 ﾀ ｧ 090303 罍陦ｧ0 05 OM = x, OM = y ｧ OM 3 = z 0 0304 0605 ﾀ07 04 縺0700306050300040030 ｧ 090303 罍陦ｧ0 05 0 ｧ 0503 09 ｧ M ｧ00 ｧ 0406090503 r,,! = OM.. 7 陦ｧ06 ﾀ-0406090503 0503 09 ｧ 04 04 縺03 ﾀ05 0 ﾀ050406. 0303 閨060004 ｧ00 ｧ 閨00 ｧ0,,! 陦ｧ06 ﾀ-0406090503 OM = r: r = x + y + z : ()

354 0600.. 6 506090503 縺0 07 00050603 ｭ 陦ｧ0 ｧ 00306 0406090503 縺03 090303 罍陦ｧ0 0500308 03 ﾀ0608 i, j ｧ k 0 0304 0605 ﾀ07 03 縺05 0 ｧ. 7 陦ｧ06 ﾀ-0406090503 0403 07 0605 ﾀ07 06 縺06 03 縺0503: 7 ｧ ﾀ. 7 r = xi + yj + zk: () 3. 6 506090503, 000308 090303 - 陦ｧ0 05 0 ｧ 0 00 ｧ 0903009. 706 ﾀ0504 003 0503 09 ｧ A(x ; y ; z ) ｧ B(x ; y ; z ). 7 縺03060909 ｧ ｧ 0406090503 u,! = AB 0 03 ﾀ ｧ 090303 - 陦ｧ0 05 0306 閨0605:,! 04 x AB = X = x, x ;,! 04 y AB = Y = y, y ; (3),! 04 z AB = = z, z : 030307003 0 04 ｧ ﾀ 0504 0703 縺0060003, 0 000305 ｧ 00306 0703 縺00600 0 (), (): u = X + Y + = (x, x ) +(y, y ) +(z, z ) ; (4) u =,! AB = Xi + Y j + k: (5) ｭ ﾀ00 ｧ, ｧ 06050003 0406090503 u =,! AB ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05, 0503 cos = X u ; cos = Y u ; cos = u ; (6) 04 罍ｧ 060 cos + cos + cos =; (7) 05. 06. ﾀ0600 0904 陦 050304 0 04 040007060 ｧ08 0903 ﾀ ｧ006 ﾀ0304 0406090503 040. 0ｷ3 0703 縺00600 (4){(6) ﾀ0006 閨060605, 0503 0406090503 u 04040300006 0304 縺06 閨0600070605 ﾀ07 05 縺06007 ｧ ﾀ00 0 ｧ: X, Y ｧ 060503 04 縺03060909 ｧ070 ｧ ｧ00 ｧ 060503 090303 罍陦ｧ0 05 0 ｧ. 703050503006 ｧ00305 04 ｧ000605 ｧ00 ｧ 05030403 縺0705: 0 0406090503 ufx; Y ; g. 386. 703 ﾀ05 縺03 ｧ0504 0503 0906 M (5;,3; 4) ｧ 0304 縺06 閨0600 ｧ0504 閨00 ｧ006 ｧ 0-04 0400060 ｧ06 0606 陦ｧ06 ﾀ-0406090503. 387. 703 ﾀ05 縺03 ｧ0504 0406090503 r,,! = OM = i +3j +6k ｧ 0304 縺06 閨0600 ｧ0504 060503 閨00 ｧ006 ｧ 0 04 0400060 ｧ06 (04 縺030406 罍ｧ0504 0403 0703 縺0060006 cos + cos + + cos =). 388. 6 506090503 ﾀ03 ﾀ05 0400070605 ﾀ 03 ﾀ070 ｧ Ox ｧ Oz 06050003 40 ｧ 80. 7 0805 ｧ 060503 06050300 ﾀ 03 ﾀ0406 Oy. 389. 7 陦ｧ06 ﾀ-0406090503 0503 09 ｧ M ﾀ03 ﾀ05 0400070605 ﾀ03 ﾀ0406 Ox 06050300 45 ｧ ﾀ03 ﾀ0406 Oy 06050300 60. 6 700 ｧ0 060503 r = 6. 0504 縺06 閨0600 ｧ0504 090303 罍陦ｧ0 0503 0503 09 ｧ M, 06 ﾀ00 ｧ 0606 090303 罍陦ｧ0 05 z 0305 罍ｧ09 050600040, ｧ 0403 ｧ0504 0406090503,,! OM = r 06 縺06 03 縺0503 i, j, k. 390. 6 7 003 0503 09 ｧ A(; ; 3) ｧ B(3;,4; 6). 703 ﾀ05 縺03 ｧ0504 0406090503,! AB = u, 060503 04 縺03060909 ｧ ｧ 0 03 ﾀ ｧ 090303 罍陦ｧ0 05 ｧ 0304 縺06 閨0600 ｧ0504 閨00 ｧ006 ｧ 0-04 0400060 ｧ06 0406090503. 703 ﾀ05 縺03 ｧ0504 06050003 0406090503 u ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05.

3x 3. ｮ09 0007 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 閨040608 0406090503 縺0304 55 39. 703 ﾀ05 縺03 ｧ0504 04 000006000305 00 0 0406090503 08,! OA = i+j ｧ,! OB = = k, 3j ｧ 0304 縺06 閨0600 ｧ0504 060503 陦ｧ 05030 00 ｧ. 39. 6 5 0503 0906 A(; ;,) 04 罍ｧ000307060 ﾀ ｧ00 R = 7. 6ｦ00 07 閨0406 090303 罍陦ｧ0 0503 05050308 ﾀ ｧ0003 X = ｧ Y =,3, 0304 縺06 閨0600 ｧ0504 0 04 0400060 ｧ06 ｧ 090300609 0406090503, ｧ 0303 07 060060503 ﾀ ｧ0006. 393. 7 040003 ﾀ0903 ﾀ05 ｧ xoy 003 0503 09 ｧ A(4; ), B(; 3) ｧ C(0; 5) ｧ 0403 ﾀ05 縺0306003 0406090503 縺03,! OA = a,,! OB = b ｧ,! OC = c. 7 000307 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ ｧ 0 00 ｧ05 ｧ 06 ﾀ09 ｧ 0406090503 a 0403 0406090503 0 b ｧ c. 394. 6 7 003 0503 09 ｧ A(; ; 0) ｧ B(0;,; 5). 703 ﾀ05 縺03 ｧ0504 0406090503,! AB = u ｧ 0304 縺06 閨0600 ｧ0504 060503 閨00 ｧ006 ｧ 0 04 0400060 ｧ06. 395. 6 506090503,,! OM = r ﾀ03 ﾀ05 0400070605 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 0400306 03 ﾀ05 縺0306 06050003. 0504 縺06 閨0600 ｧ0504 0505 ｧ 06050003 ｧ 0403 ﾀ05 縺03 ｧ0504 0406090503 r, 06 ﾀ00 ｧ 060503 閨00 ｧ0 040 3. 396. 6 506090503 ﾀ03 ﾀ05 0400070605 ﾀ 03 ﾀ070 ｧ Oy ｧ Oz 06050003 60 ｧ 0. 07 090308 06050300 030 ﾀ03 ﾀ05 0400070605 ﾀ03 ﾀ0406 Ox? 397. 6 7 003 05 罍ｧ 0403 ﾀ0006 閨0304 0506000400306 0406 縺00 ｧ003 04 000006000305 00 A(;,; 3), B(3; ; ) ｧ C(6; 4; 4). 7 0805 ｧ 060503 06050406 縺050606 0406-00 ｧ006 D. ｰ09 0 ｧ06. 0ｷ3 04060 ﾀ0504,! AD,! = BC ﾀ0006 閨060605, 0503 04003 ｧ ｧ08 090303 - 陦ｧ0 0503: x, =6, 3 ｧ05.. 398. 7 040003 ﾀ0903 ﾀ05 ｧ xoy 0403 ﾀ05 縺03 ｧ0504 0406090503 縺03,! OA = a =i,,! OB = = b = 3i +3j ｧ,! OC = c = i +6j. 7 000307 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ ｧ 0 00 ｧ05 ｧ 06 ﾀ09 ｧ 0406090503 c 0403 0406090503 0 a ｧ b. x 3. ｮ09 0007 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 閨040608 0406090503 縺0304. 0504 縺06 閨0600060 ｧ06. ｮ09 0007 縺0030 04 縺03 ｧ 0406 閨060 ｧ060 閨040608 0406090503 縺0304 0-0304 0605 ﾀ07 04 縺03 ｧ 0406 閨060 ｧ06 ｧ08 003 閨06000608; 06000307060-00306 0 0903 ﾀ ｧ006 ﾀ 060500 00607 閨06 0 ｧ0 ｧ. ｮ09 0007 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 0406090503 a 0 040609-0503 b 030303 0 0605 ﾀ07 a b. 0ｷ305 09, a b = ab cos ': () 0ｷ3 罍ｧ ﾀ. 8 04 ｧ 閨003, 0503 b cos ' = 04 a b. 703050503006 a b = ab cos ' = a 04 a b = b 04 b a: (). ｮ040308 ﾀ0504 ﾀ09 0007 縺0030503 04 縺03 ｧ 0406- 閨060 ｧ07: 7 ｧ ﾀ. 8 I. a b = b a 0406 縺06006 ﾀ05 ｧ0506000400308 09030. II. a (b + c) =ab + a c ﾀ04 縺06 閨0600 ｧ0506000400308 09030. III. ｭ ﾀ00 ｧ akb, 0503ab = ab. 6 5 ﾀ05003 ﾀ05 ｧ, a = aa = aa cos 0 = a ; 03 ﾀ0506 a = a : (3)

356 0600.. 6 506090503 縺0 07 00050603 IV. ｭ ﾀ00 ｧ a? b, 0503a b = ab cos 90 =0. V. ｮ09 0007 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 03 縺050304: i j =0; j k =0; i k =0; i i =; j j =; k k =: VI. ｭ ﾀ00 ｧ 0406090503 縺03 a ｧ b 003 090303 罍陦ｧ0 05 0 ｧ afa x ;a y ;a z g ｧ bfb x ;b y ;b z g,0503 3. ｰ050300 00607 閨06 0406090503 0 ｧ: cos ' = a b ab a b = a x b x + a y b y + a z b z : (4) = a x b x + a y b y + a z b q q z : (5) a + a + a b + b + b x y z x y z ｰ ﾀ000304 ｧ06 04 0000060004003 ﾀ05 ｧ: b = ma ｧ00 ｧ bx b b y z = = = m. a x a y a z ｰ ﾀ000304 ｧ06 0406 縺04060 陦ｧ09060007 縺003 ﾀ05 ｧ: a b =0 ｧ00 ｧ a x b x + a y b y + a z b z =0. 399. 0504 縺06 閨0600 ｧ0504 06050300 00607 閨06 0406090503 0 ｧ a =,i + j ｧ b = i,, j +k. 400. 0504 縺06 閨0600 ｧ050406050003 4ABC ﾀ 0406 縺00 ｧ0 0 ｧ A(;,; 3), B(; ; ) ｧ C(0; 0; 5). 40. 6 7 003 0503 09 ｧ A(a; 0;0), B(0; 0; a) ｧ C(a; 0;a). 703 ﾀ05 縺03 ｧ0504 0406090503 縺03,! OC ｧ,! AB ｧ 0 0805 ｧ 06050300 00607 閨06 0 ｧ0 ｧ. 40. 7 040003 ﾀ0903 ﾀ05 ｧ 0 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ O(0; 0), A(a; 0) ｧ B(a;,a). 7 0805 ｧ 06050300, 0303 0304 000308 ﾀ0503 縺0300308 OB ｧ 006 陦ｧ 00308 OM 0505030503 05 縺0606050300040 ｧ09. 403. 7 0805 ｧ 06050300 00607 閨06 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 0 ｧ 0605000304 xoy ｧ yoz. 404. 0ｷ3 0406 縺00 ｧ003 0904 陦 05 04 縺030406 閨06003 04 縺0700306, 閨0600070 ｧ06 04 縺0305 ｧ- 0403040300030700306 ﾀ0503 縺03003 0403040300 0. 7 0805 ｧ 06050300 00607 閨060505 ｧ0 ｧ 04 縺070030 ｧ. 405. 7 0805 ｧ 06050300 00607 閨06 陦ｧ 05030 00070 ｧ 04 000006000305 00, 0403 ﾀ05 縺03-0600030503 0 0406090503 08 a =i + j ｧ b =,j + k. 406. 6 7 003 0406090503 縺03 a = i + j +k ｧ b = i, j +4k. 0504 縺06 閨0600 ｧ0504 04 b a ｧ04 a b. 407. 7 ﾀ09 縺030504 ﾀ09030309 ｧ 04 0403 07060 ｧ ｧ (i, j) j +(j, k) k +(i, k) : 408. 6 503 ｧ ﾀ00 ｧ0504: ) (m+n), 06 ﾀ00 ｧ m ｧ n 06 陦ｧ0 ｧ 00306 0406090503 縺03 ﾀ060500030 00607 閨06 0 ｧ0 ｧ 30 ; ) (a, b), 06 ﾀ00 ｧ a =, b = 4 ｧ (d a; b) = 35. 409. 7 ﾀ09 縺030504 ﾀ09030309 ｧ 04 0403 07060 ｧ0708: ) (a + b) ; ) (a + b) +(a, b) ｧ 040307 ﾀ0 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀ00 04030006 06000308 0703 縺00600. 40. 6 7 003 090300400 0 縺00306 0406090503 縺03 a, b ｧ c, 04 罍ｧ 060 a =3, b =, c =5, (d a; b) =60 ｧ (d b; c) =60. 703 ﾀ05 縺03 ｧ0504 0406090503 u = a + b, c

3x 3. ｮ09 0007 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 閨040608 0406090503 縺0304 57 ｧ 0403 ｧ ﾀ00 ｧ0504 060503 003 閨060004 0403 0703 縺0060006 u = (a + b, c) : 4. 7 0805 ｧ 040600 ｧ ｧ006 04003 閨0608 ﾀ0504060600608 060503 縺0608 090300400 0-00308 ﾀ ｧ00, 04 罍ｧ00030706000308 09 0503 0906 O, 06 ﾀ00 ｧ 040600 ｧ ｧ0 09 07 閨0308 ﾀ ｧ0003 040 0 7, 06050300 00607 閨06 閨0406007 0403 ﾀ0006 閨0304 050600040030 ｧ ﾀ ｧ00 0 ｧ - 04060 45. 4. 0504 縺06 閨0600 ｧ0504 閨00 ｧ003 陦ｧ 05030 000608 04 000006000305 00, 0403 ﾀ05 縺03-0600030503 0 0406090503 08 a = m + n ｧ b = m, n, 05 閨06 m ｧ n 06 陦ｧ0 ｧ 00306 0406090503 縺03, 06050300 00607 閨06 09030503 縺030 ｧ 60. 43. 6 7 0 0406090503 a =m, n, 05 閨06 m ｧ n 06 陦ｧ0 ｧ 00306 0406090503 縺03 ﾀ060500030 0 00607 閨06 0 ｧ0 ｧ. 7 0805 ｧ cos ( [a; m) ｧ cos (da; n). 44. 0504 縺06 閨0600 ｧ0504 06050300 00607 閨06 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 0 ｧ 閨040608 040003 ﾀ09 ｧ08 0605000304 04 04 ｧ00040030503 050605 05 陦, 04 縺030406 閨0600030 ｧ ｧ 03 閨00308 060503 0406 縺00 ｧ003. ｰ09 0 ｧ06. ｭ ﾀ00 ｧ m, n ｧ 06 陦ｧ0 ｧ 00306 0406090503 縺03 縺060306, 0503 m + n ｧ m + 0406090503 縺03, 0 04 040006000306 0403 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 0. 45. 7 03 ﾀ0708 Ox, Oy ｧ Oz 0305000307 ｧ0504 0400306 0305 縺06 09 ｧ a =4 ｧ 0 0 ｧ08 0403 ﾀ05 縺03 ｧ0504 090603. 706 ﾀ0504 M 0906005 0406 縺0800608 05 0 ｧ, N 0906005 04 040308 03030903040308 05 0 ｧ 090603. 0504 縺06 閨0600 ｧ0504 0406090503 縺03,,! OM ｧ,,! ON ｧ 06050300 00607 閨06 0 ｧ0 ｧ. 46. 6 7 003 0406090503 縺03,! OA = a ｧ,! OB = b, 04 罍ｧ 060 a =, b =4, (d a; b) =60. 0504 縺06 閨0600 ｧ0504 06050300 00607 閨06 006 陦ｧ 00308,,! OM 05 縺0606050300040 ｧ09 AOB ｧ ﾀ0503 縺0300308,! OA. 47. 0ｷ3 0406 縺00 ｧ003 04 縺0700306050300040 ｧ09 ﾀ03 ﾀ0503 縺030 0 ｧ 6 ﾀ0 ｧ 4 ﾀ0 04 縺030406 閨06003 04 縺0700306, 閨0600070 ｧ06 04 縺0305 ｧ0403040300030700306 ﾀ0503 縺03003 0403040300 0. 7 0805 ｧ 06050300 ' 00607 閨06 0 ｧ0 ｧ. 48. 6 7 003 05 罍ｧ 0403 ﾀ0006 閨0304 0506000400306 0406 縺00 ｧ003 04 000006000305 00 : A(,3;,; 0), B(3;,3; ) ｧ C(5; 0; ). 7 0805 ｧ 060503 06050406 縺050606 0406-00 ｧ006 D ｧ06050300 00607 閨06 0406090503 0 ｧ,! AC ｧ,! BD. 49. 6 7 003 0503 09 ｧ A(3; 3;,), B(0;,3; 4), C(0;,3; 0) ｧ D(0; ;,4). 703 ﾀ05 縺03 ｧ0504 0406090503 縺03,! AB = a ｧ,! CD = b ｧ 0 0805 ｧ 04 a b. 40. 6 5 040030306 陦縺06000308 05 040609 ｧ ｧ OACB ( ﾀ0. 罍ｧ ﾀ. 6) M ｧ N ﾀ06 縺06 陦ｧ003 ﾀ0503 縺030 BC = ｧ AC =. 05 ﾀ05 縺0308 06050300 05 040609 ｧ ｧ 60. 0504 縺06 閨0600 ｧ0504 06050300 00607 閨06 0406090503 0 ｧ,,! OM ｧ ON.,,! 4. 7 0805 ｧ 06050300 00607 閨06 0406090503 0 ｧ a =m +4n ｧ b = m, n, 05 閨06 m ｧ n 06 陦ｧ0 ｧ 00306 0406090503 縺03, 0303 06060 ｧ06 06050300 0. 4. 70309 0504, 0503 06050300 00607 閨06 陦ｧ 05030 00070 ｧ 04 縺0700306050300040 ｧ09, 0403 ﾀ05 縺030600030503 0 0406090503 08 a ｧ b (a? b), 0304 縺06 閨0600070605 ﾀ07 0703 縺006000308 cos ' = a, b a + b.

358 0600.. 6 506090503 縺0 07 00050603 43. 7 縺03060909 ｧ ｧ 0406 縺060060060 ｧ07 s 閨04 ｧ070600608 ﾀ07 0503 09 ｧ 0 03 ﾀ ｧ 090303 - 陦ｧ0 05 s x = 0, s y = 0, s z =, 0. 7 縺03060909 ｧ ｧ 閨0608 ﾀ0504060600608 ﾀ ｧ0003 F 0 03 ﾀ ｧ 090303 罍陦ｧ0 05 04003 F x =5 7, F y =4 7 ｧF z =3 7. 6 503 ｧ ﾀ00 ｧ0504 03030506 A ﾀ ｧ0003 F (A = F s) ｧ06050300 00607 閨06 ﾀ ｧ000308 F ｧ 0406 縺060060060 ｧ060 s. 44. 07 0406 縺00 ｧ006 04 04 ｧ00040030503 050605 05 陦 ﾀ 縺0603 縺030 a 04 罍ｧ00030706003 05 罍ｧ ﾀ ｧ0003, ｧ 0303 07 0600306 060503 0406090503 - 縺0603 0 ｧ. 0504 縺06 閨0600 ｧ0504 040600 ｧ- ｧ006 04003 閨0608 ﾀ0504060600608. ｰ09 0 ｧ06. 0ｷ3 ﾀ09030 07 040600 ｧ ｧ0 040 a (m + n + ),05 閨06 m, n ｧ 06 陦ｧ0 ｧ 00306 0406090503 縺03 000308 ﾀ ｧ00. 45. 0704 陦 05 閨0600060 0 05 罍ｧ 04030003 ﾀ03 03 陦ｧ0 0903040308 00 ｧ 罍ｧ003 ｧ - 05060 ﾀ0406 縺00605 04 04 04 ｧ00040060605 縺06060503000400606 04 罍ｧ 006. 7 0805 ｧ 06050300 00607 閨06 閨0406007 ﾀ006070030 ｧ 0406004070 ｧ 00030 00308, 0303 0304 000308 04 罍ｧ 0505030 陦ｧ - 05030 000406 0904 陦 05. x 4. 6 506090503 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 閨040608 0406090503 縺0304. 0504 縺06 閨0600060 ｧ06. 6 506090503 縺0030 04 縺03 ｧ 0406 閨060 ｧ060 0406090503 a 0 040609-0503 b 0 0304 0605 ﾀ07 05 090308 05 縺0605 ｧ08 0406090503 c ( 罍ｧ ﾀ. 9), 09030503 縺0308: ) ｧ0060605 003 閨060004, ｧ ﾀ00060003 0400308 0400030 陦ｧ 04 000006000305 00, 0403- ﾀ05 縺030600030503 0 0406090503 08 a ｧ b; ) 0406 縺04060 陦ｧ09060007 縺060 09 040003 ﾀ0903 ﾀ05 ｧ 04 000006000305 00 ; 3) 0 04 0400060 04 05 090606 ﾀ0503 縺03006, ﾀ09030503 縺0308 09 05 08000606 04 0060 ｧ06 0305 a 09 b ﾀ ﾀ0 05 罍ｧ04 0605 ﾀ07 ﾀ030406 縺00 060 ｧ0 ﾀ07 04 縺0305 ｧ04 ﾀ03040308 ﾀ05 縺060009 ｧ. ｯ 090306 ﾀ0403000307060 ｧ06 0406090503 縺0304 a, b ｧ c 0 0304 0605 ﾀ07 04 040308 ﾀ0407 090308. 6 506090503 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 030303 0 0605 ﾀ07 a b. 0ｷ305 09, a b = c; 06 ﾀ00 ｧ: ) c = ja bj = ab sin ', 7 ｧ ﾀ. 9 ) c? a ｧ c? b, 3) a, b, c ﾀ03 ﾀ05 0400070605 04 040606 ﾀ0407 0906.. ｮ040308 ﾀ0504 0406090503 縺0030503 04 縺03 ｧ 0406 閨060 ｧ07: I. a b =,b a. II. a (b + c) =ab + a c ﾀ04 縺06 閨0600 ｧ0506000400308 09030. III. ｭ ﾀ00 ｧ akb, 0503a b =0; 04 ﾀ05003 ﾀ05 ｧ, a a =0. 3. 6 506090503 縺00306 04 縺03 ｧ 0406 閨060 ｧ07 03 縺050304: i j = k; j k = i; k i = j: () 6 5030303006 04 縺03 ｧ 0406 閨060 ｧ06 0006030308 閨040608 ﾀ0060700308 0406090503 縺0304 04 0403 ﾀ0006 閨0304-05060004003 ﾀ05 ｧ 0605 ﾀ0006 閨06060 ｧ08 0406090503 ﾀ03 0 09030 +, 04 0303 0500308 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ ﾀ03 0 09030,.

3x 4. 6 506090503 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 閨040608 0406090503 縺0304 59 4. 6 503 07060 ｧ06 0406090503 縺0030503 04 縺03 ｧ 0406 閨060 ｧ07 06 縺06 090303 - 陦ｧ0 0503 ﾀ03000307 ｧ0506000608 afa x ;a y ;a z g ｧ bfb x ;b y ;b z g: a b = i j k a x a y a z b x b y b z : () 5. 700030 閨04 04 000006000305 00, 0403 ﾀ05 縺030600030503 0 0406090503 08 a ｧ b: S = ja bj; (3) 0400030 閨04 05 縺0606050300040 ｧ09, 0403 ﾀ05 縺030600030503 0 0406090503 08 a ｧ b: S4 = ja bj: (4) 46. 0504 縺06 閨0600 ｧ0504 ｧ 0403 ﾀ05 縺03 ｧ0504 0406090503 c = a b, 06 ﾀ00 ｧ: ) a =3i, b = k; )a = i + j, b = i, j; 3) a = i +3j, b = 3j +k. 7 0805 ｧ 0409 07 閨030 ﾀ0006 06 0400030 閨04 04 000006000305 00, 0403 ﾀ05 縺030600030503 0 0406090503 08 a ｧ b. 47. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(7; 3; 4), B(; 0; 6) ｧ C(4; 5;,). 48. 703 ﾀ05 縺03 ｧ0504 04 000006000305 00 0 0406090503 08 a =j + k ｧ b = = i +k ｧ 0403 ｧ ﾀ00 ｧ0504 060503 0400030 閨04 ｧ 0403 ﾀ030506. 49. 7 ﾀ09 縺030504 ﾀ09030309 ｧ ｧ 0604 縺03 ﾀ05 ｧ0504 0403 07060 ｧ07: ) i (j + k), j (i + k) +k (i + j + k); ) (a + b + c) c +(a + b + c) b +(b, c) a; 3) (a + b) (c, a) +(b + c) (a + b); 4) i (j k) +3j (i k) +4k (i j). 430. 6 70309 0504, 0503 (a, b) (a + b) = a b, ｧ 040307 ﾀ0 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 0 060 ｧ06 0505030503 050307 閨06 ﾀ0504. 43. 6 506090503 縺03 a ｧ b ﾀ03 ﾀ05 0400070605 06050300 45. 7 0805 ｧ 0400030 閨04 05 縺0606050300040 ｧ09, 0403 ﾀ05 縺030600030503 0 0406090503 08 a, b ｧ 3a +b, 06 ﾀ00 ｧ jaj = jbj =5. 43. 7 0805 ｧ 0400030 閨04 04 000006000305 00, 陦ｧ 05030 00070 ｧ 09030503 縺030503 ﾀ000607 05 0406090503 縺03 m, n ｧ 4m, 5n, 05 閨06 m ｧ n 06 陦ｧ0 ｧ 00306 0406090503 縺03, 0303 06060 ｧ06 06050300 45. ｰ09 0 ｧ06. 0ｷ3006060 a + b =m, n ｧ a, b =4m, 5n, 05 閨06 a ｧ b 0406090503 縺03- ﾀ0503 縺03003 04 000006000305 00. 706 縺06000307 ｧ04, 0 08 閨060 0406090503 b a, 003 閨060004 09030503 縺030503 ｧ 04060 06 閨040306000308 ｧ ﾀ090300308 0400030 陦ｧ. 433. 703 ﾀ05 縺03 ｧ05040406090503 縺03 a =3k,j, b =3i,j ｧ c = ab. 6 503- ｧ ﾀ00 ｧ0504 003 閨060004 0406090503 c ｧ 0400030 閨04 05 縺0606050300040 ｧ09, 0403 ﾀ05 縺030600030503 0 0406090503 08 a ｧ b. 434. 703 ﾀ05 縺03 ｧ050405 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(;,; 8), B(0; 0; 4) ｧ C(6; ; 0). 6 503 ｧ ﾀ00 ｧ0504 060503 0400030 閨04 ｧ 0403 ﾀ030506 BD.

360 0600.. 6 506090503 縺0 07 00050603 435. 6 503 ｧ ﾀ00 ｧ0504 陦ｧ 05030 00 ｧ ｧ 0400030 閨04 04 000006000305 00, 0403- ﾀ05 縺030600030503 0 0406090503 08 a = k, j ｧ b = i + j + k. 436. 6 70309 0504, 0503 (a + b) (a +b) =3a b. 437. 7 0805 ｧ 0400030 閨04 04 000006000305 00, 0403 ﾀ05 縺030600030503 0 0406090503-08 a = m +n ｧ b =m + n, 05 閨06 m ｧ n 06 陦ｧ0 ｧ 00306 0406090503 縺03, 0303 06060 ｧ06 06050300 30. x 5. ｮ00600 000306 04 縺03 ｧ 0406 閨060 ｧ06 05 縺0608 0406090503 縺0304. 0504 縺06 閨0600060 ｧ06. ｮ00600 00030 04 縺03 ｧ 0406 閨060 ｧ060 0406090503 縺0304 a, b ｧ c 0 0304 0605 ﾀ07 0403 07060 ｧ06 04 ｧ (a b) c. ｭ ﾀ00 ｧ 0406090503 縺03 a, b ｧ c 003 ﾀ0403 ｧ0 ｧ 090303 罍陦ｧ0 05 0 ｧ, 0503 (a b) c = a x a y a z b x b y b z c x c y c z : (). ｮ040308 ﾀ0504 ﾀ00600 00030503 04 縺03 ｧ 0406 閨060 ｧ07. I. 0505 0406 縺06 ﾀ05 0030409 ｧ 閨040608 0006030308 ﾀ03000307 ｧ0506000608 ﾀ00600 000306 04 縺03 ｧ 0406- 閨060 ｧ06 0060070605 0 09: (a b) c =,(a c) b =,(c b) a: () II. ｭ ﾀ00 ｧ 閨04 ｧ 05 縺0608 000308 0406090503 縺0304 04003 ｧ00 ｧ 04 0000060004003, 0503 ｧ08 ﾀ00600 000306 04 縺03 ｧ 0406 閨060 ｧ06 04003 0. III. 6ｦ00 09 ｧ 030406 09 ｧ08 080503 09 09 ｧ0809 縺06 ﾀ0509 00307003 04030060070504 006 ﾀ05 0 ｧ, (a b) c = a (b c); 0403050503006 ﾀ00600 000306 04 縺03 ｧ 0406 閨060 ｧ06 04 罍ｧ0070503 04 ｧ ﾀ0304 0504 04 04 ｧ 閨06 abc, 05.06.0306 0 090304 閨0608 ﾀ0504 ｧ08 ｧ 0306 ﾀ0903030309. 3.05030060 04 000006000604 ｧ0406,0403 ﾀ05 縺030600030503 0 0406090503 08 a, b ｧ c. V = abc (+ 04 罍ｧ 04 040308 ﾀ0407 0906,, 04 罍ｧ 0006040308 ﾀ0407 0906). 05030060 04 ｧ 0 ｧ 閨03, 0403 ﾀ05 縺0306000308 0 0406090503 08 a, b, c: V04 ｧ = 6 abc: 4. ｰ ﾀ000304 ｧ06 090300400 0 縺003 ﾀ05 ｧ. ｭ ﾀ00 ｧ a, b ｧ c 090300400 0 縺003, 0503 abc =0, ｧ 0303 05003. 7 罍ｧ 0505030 00607 閨06 a, b ｧ c ﾀ06006 ﾀ0504060605 00 ｧ006080 07 04 ｧ ﾀ ｧ003 ﾀ0504 04 ｧ c = ma + nb. 438. 703 ﾀ05 縺03 ｧ0504 04 000006000604 ｧ0406 0 0406090503 08 a = 3i +4j, b = =, 3j + k, c =j +5k ｧ 0403 ｧ ﾀ00 ｧ0504 060503 03030060. 7 040308 ｧ00 ｧ 0006040308 0306 閨0605 ﾀ0407 09 0406090503 縺0304 (a; b; c)? 439. 703 ﾀ05 縺03 ｧ0504 04 ｧ 0 ｧ 閨06 ﾀ 0406 縺00 ｧ0 0 ｧ O(0; 0; 0), A(5; ; 0), B(; 5; 0) ｧ C(; ; 4) ｧ 0403 ｧ ﾀ00 ｧ0504 060603030060, 0400030 閨04 05 0 ｧ ABC ｧ 0403 ﾀ030506 04 ｧ 0 ｧ 閨03, 030406006000606 0 050506 05 004. 440. 70309 0504, 0503 0503 09 ｧ A(;,;,), B(; ; ), C(; 3; 0) ｧ D(5; 0;,6) 000607 05 04 03 閨00308 040003 ﾀ0903 ﾀ05 ｧ. 44. 70309 0504, 0503 0406090503 縺03 a =,i +3j +k, b =i, 3j, 4k, c =,3i+j+6k 090300400 0 縺003, ｧ 000307 ｧ0504 0406090503 c 0403 0406090503 0 a ｧ b.

3x 5. ｮ00600 000306 04 縺03 ｧ 0406 閨060 ｧ06 05 縺0608 0406090503 縺0304 6 44. 70309 0504, 0503: ) (a + b) (a + c) b =,abc; ) (a +b, c) (a, b) (a, b, c) =3abc. 443. 7 0805 ｧ 03030060 050605 05 陦, 0403 ﾀ05 縺030600030503 0 0406090503 08,! OA,,! OB ｧ,! OC, 06 ﾀ00 ｧ 0505 ｧ 0406090503 縺03 0 04 040006003 0403 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ 0 090303 罍陦ｧ- 0 0500308 0605000304 ｧ 閨00 ｧ0 09 07 閨030503 0406090503 040. 444. 703 ﾀ05 縺03 ｧ0504 04 ｧ 0 ｧ 閨06 ﾀ 0406 縺00 ｧ0 0 ｧ A(; 0; 0), B(0; 3; 0), C(0; 0; 6) ｧ D(; 3; 8), 0403 ｧ ﾀ00 ｧ0504 060603030060 ｧ 0403 ﾀ030506, 030406006000606 0 05 004 ABC. 445. 703 ﾀ05 縺03 ｧ05040406090503 縺03 a = i+j+4k, b = i,j ｧ c =3i,3j+4k, 040309 0504, 0503 030 ｧ 090300400 0 縺003, ｧ 0 0805 ｧ 00 ｧ0060800606 04 ｧ ﾀ ｧ003 ﾀ0504 00607 閨06 0 ｧ0 ｧ. 446. 70309 0504, 0503 03030060 04 000006000604 ｧ0406, 0403 ﾀ05 縺030600030503 0 陦ｧ 05030 000708 05 00608 00030503 04 000006000604 ｧ0406, 04060 06 閨0403060003006 0303006006 00030503 04 000006000604 ｧ0406. 447. 6 7 003 06 陦ｧ0 ｧ 00306 0406090503 縺03 m, n ｧ. ｰ050300 ( [m; n) = X X ; (m n) sin. = =. 6 70309 0504, 0503 050305 (m n) = 448. 7 罍ｧ 0006030308 0406090503 08 a, b ｧ c 0406090503 縺03 a, b, b, c ｧ c, a 090300400 0 縺003. 6 70309 0504 050503 0 00 ｧ05 ｧ 06 ﾀ09 ｧ ｧ 05060300605 罍ｧ 06 ﾀ09 ｧ ( ﾀ- ﾀ00305 縺060 ｧ060 04 000006000604 ｧ0406, 0403 ﾀ05 縺030600030503 0 0406090503 08 a, b ｧ c). 449. 6 503 ｧ ﾀ00 ｧ0504 03030060 04 000006000604 ｧ0406 OABCO A B C, 04 09030503 縺030 003 05 罍ｧ 0406 縺00 ｧ003 0 ｧ070060503 03 ﾀ00304 0 ｧ07 O(0; 0; 0), A(;,3; 0) ｧ C(3; ; 0) ｧ 0406 縺00 ｧ0 0406 縺080060503 03 ﾀ00304 0 ｧ07 B (3; 0; 4), 000607 0 07 0 0303090304030 縺0603 縺06 BB,04 縺0305 ｧ040304030003070030 縺0603 縺06 OO.

30600 04 3 繖 7 縺6ｧ80ｷ3 ｯ0ｷ3 ｪ ｭ ｮ07 縺00 06 ｭ0503 ｭ ｯ 70ｷ300 6 5 7 705 ｮ ｯ 7 繖 7 ｮ ｯ6 5 ｭ x. ｰ 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ. ｰ 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ,04 縺030803 閨0700608 06 縺06 0503 0906 M (x ; y ; z ) ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 0406090503 縺06 NfA; B; Cg. 706 ﾀ0504 M (x; y; z) 04 縺03 ｧ 040300040 07 0503 09 040003 ﾀ0903 ﾀ05 ｧ ( 罍ｧ ﾀ. 0). ｯ0305,,,! M M?? N ｧ0403 06 ﾀ000304 ｧ06 0406 縺04060 陦ｧ09060007 縺003 ﾀ05 ｧ 0406090503 縺0304 A(x, x )+B(y, y )+C(z, z )=0: (). 050300606 06 040060 ｧ06 040003 ﾀ- 0903 ﾀ05 ｧ: Ax + By + Cz + D =0: () 6 506090503 NfA; B; Cg 0 0304 0605 ﾀ07 003-0 00040030 0406090503 縺030 09 040003 ﾀ0903 ﾀ05 ｧ () 7 ｧ ﾀ. 0 ｧ00 ｧ (). 3.05 ﾀ03030306 ﾀ0006 ｧ 06 040060 ｧ07 Ax + By + Cz + D =0: I. D = 0, Ax + By + Cz = 0 040003 ﾀ0903 ﾀ0504 04 縺030803 陦ｧ05 06 縺06 0 0003 090303 罍陦ｧ0 05. II. C =0, Ax + By + D =0 040003 ﾀ0903 ﾀ0504 04 00000600040 03 ﾀ ｧ Oz. III. C = D =0, Ax + By =0 040003 ﾀ0903 ﾀ0504 04 縺030803 陦ｧ05 06 縺06 03 ﾀ04Oz. IV. B = C = 0, Ax + D = 0 040003 ﾀ0903 ﾀ0504 04 00000600040 040003 ﾀ0903 ﾀ05 ｧ yoz. V. ｰ 040060 ｧ07 090303 罍陦ｧ0 0500308 040003 ﾀ0903 ﾀ050608: x =0, y =0, z =0. 4. ｰ 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ 04 0305 縺06 09 08 0 03 ﾀ0708: x a + y b + z c =: (3) 450. 703 ﾀ05 縺03 ｧ0504 040003 ﾀ0903 ﾀ05 ｧ: ) 5x, y +3z, 0 = 0; )3x + +y, z =0;3)3x +z =6;4)z, 7=0. 45. 703 ﾀ05 縺03 ｧ0504 040003 ﾀ0903 ﾀ0504 x +3y +6z, = 0 ｧ 0 0805 ｧ 06050003 003 縺0 00 ｧ 09 040003 ﾀ0903 ﾀ05 ｧ ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05. 45. 6 7 003 0503 09 ｧ M (0;,; 3) ｧ M (; 3; 5). 7 04 ｧ ﾀ 0504 06 04006-0 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 0906 M ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 0406090503 縺06 N,,,,! = M M.

3x. 05 ﾀ0030400306 ｧ 0 040003 ﾀ0903 ﾀ0504 63 453. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 0906 M (a; a; 0) ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 0406090503 縺06,,! OM. 703 ﾀ05 縺03 ｧ0504 040003 ﾀ- 0903 ﾀ0504. 454. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 04003-06 0006000308 0305 0503 0609 A(a;,a=; a) ｧ B(0; a=; 0). 455. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 000006000400308 03 ﾀ ｧ Ox ｧ 04 縺030803 閨0700608 06 縺06 0503 09 ｧ M (0; ; 3) ｧ M (; 4; 5) ｧ 0403 ﾀ05 縺03 ｧ0504 0606. 456. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 03 ﾀ04Ox ｧ0503 0906 M (0;,; 3). 703 ﾀ05 縺03 ｧ0504 040003 ﾀ0903 ﾀ0504. 457. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 03 ﾀ04Oz ｧ0503 0906 M (;,4; 3). 703 ﾀ05 縺03 ｧ0504 040003 ﾀ0903 ﾀ0504. 458. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 000006000400308 03 ﾀ ｧ Oy ｧ 0305 ﾀ0609 0600608 0 03 ﾀ0708 Ox ｧ Oz 0305 縺06 09 ｧ a ｧ c. 703 ﾀ05 縺03 ｧ0504 0606. 459. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 0906 M (;,; 3) ｧ0305 ﾀ0609 0600608 0 03 ﾀ0708 090303 罍陦ｧ0 05 0400306 0305 縺06 09 ｧ. 460. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 0906 M (,4; 0; 4) ｧ0305 ﾀ0609 0600608 0 03 ﾀ0708 Ox ｧ Oy 0305 縺06 09 ｧ a = 4 ｧ b =3. 46. 703 ﾀ05 縺03 ｧ0504 040003 ﾀ0903 ﾀ05 ｧ: ) x+y,z +6 = 0; )x,y,z =0; 3) y, z +8=0;4) x, 5=0;5)x + z =;6)y + z =0. 46. 703 ﾀ05 縺03 ｧ0504 040003 ﾀ0903 ﾀ0504 x, y + z, 6=0 ｧ 0 0805 ｧ 06050003 0606 003 縺0 00 ｧ ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05. 463. ｪ06 縺06 0503 0906 M (,; ; 3) 04 縺030406 閨060 040003 ﾀ0903 ﾀ0504, 0406 縺04060 陦ｧ- 09060007 縺0 07 09 OM. 7 04 ｧ ﾀ 0504 060606 040060 ｧ06. 464. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 03 ﾀ04Oy ｧ 06 縺06 0503 0906 (4; 0; 3). 703 ﾀ05 縺03 ｧ0504 040003 ﾀ0903 ﾀ0504. 465. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 000006000400308 03 ﾀ ｧ Oz ｧ 04 縺030803 閨0700608 06 縺06 0503 09 ｧ M (; ; 0) ｧ M (4; 0; 0). 703 ﾀ05 縺03 ｧ0504 040003 ﾀ- 0903 ﾀ0504. 466. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 0906 M (;,3; 5) ｧ0305 ﾀ0609 0600608 0 03 ﾀ0708 Oy ｧ Oz 04 閨040306 0303000400 ｧ06 0305- 縺06 09 ｧ, 060 0 03 ﾀ ｧ Ox. x. 05 ﾀ0030400306 ｧ 0 040003 ﾀ0903 ﾀ0504. ｰ050300, 0303 0304 000308 閨0406007 040003 ﾀ0903 ﾀ05070 ｧ: cos ' = N N NN = AA + BB + CC NN ; () 05 閨06 N ｧ N 003 縺0 000400306 0406090503 縺03 09 040003 ﾀ0903 ﾀ05070 Ax + By + Cz + D =0 ｧ A x + B y + C z + D =0. ｰ ﾀ000304 ｧ06 04 0000060004003 ﾀ05 ｧ: A A = B B = C C : ()

364 0600. 3. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 04 04 縺03 ﾀ05 0 ﾀ050406 ｰ ﾀ000304 ｧ06 0406 縺04060 陦ｧ09060007 縺003 ﾀ05 ｧ: AA + BB + CC =0: (3). 7 ﾀ ﾀ0503070 ｧ06 0305 0503 09 ｧ M 0 (x 0 ; y 0 ; z 0 ) 閨03 040003 ﾀ0903 ﾀ05 ｧ Ax + + By + Cz + D =0: d jax 0 + By 0 + Cz 0 + Dj = : (4) N 3. ｰ 040060 ｧ06 0406 09 04 ﾀ0608 040003 ﾀ0903 ﾀ050608, 04 縺030803 閨070 ｧ08 06 縺06 00 ｧ0 ｧ06 0406 縺06 ﾀ06 060 ｧ07 閨040608 000308 040003 ﾀ0903 ﾀ050608: (Ax + By + Cz + D) +(A x + B y + C z + D )=0: (5) 030307003 0403000307 ｧ0504 =, ｧ ﾀ090006 ｧ04 0505 ｧ0 ｧ 0406 09 (5) 040503 縺0606 ｧ 000308 040003 ﾀ0903 ﾀ050608. 467. 7 0805 ｧ 06050300 00607 閨06 040003 ﾀ0903 ﾀ05070 ｧ: ) x, y +z, 8=0 ｧ x + z, 6=0; ) x +z, 6=0 ｧ x +y, 4=0: 468. 7 0805 ｧ 040003 ﾀ0903 ﾀ0504, 04 縺030803 閨0700606 06 縺06 0503 0906 (; ;,) ｧ 04 000006000400606 040003 ﾀ0903 ﾀ05 ｧ x, y, 3z =0. 469. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 0906 (,;,; ) ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 040003 ﾀ0903 ﾀ05070 x, y + z, 4=0 ｧ x +y, z +4=0. 470. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 0906 (0; 0; a) ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 040003 ﾀ0903 ﾀ05070 x, y, z =0 ｧ y = x. 47. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 09 ｧ M (,;,; 0) ｧ M (; ; ) ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 040003 ﾀ0903 ﾀ05 ｧ x + +y +z, 4=0. 47. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 09 ｧ M (;,; ), M (; ; ) ｧ M 3 (; ; 4). 473. ｪ06 縺06 03 ﾀ04 Oz 04 縺030406 ﾀ05 ｧ 040003 ﾀ0903 ﾀ0504, ﾀ03 ﾀ05 0400070600606 ﾀ 040003 ﾀ- 0903 ﾀ050406 x + y, 5z =006050300 60. 474. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0305 0503 09 ｧ (5; ;,) 閨03 040003 ﾀ0903 ﾀ05 ｧ x,, y, z +4=0. 475. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0305 0503 09 ｧ (4; 3; 0) 閨03 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803- 閨0700608 06 縺06 0503 09 ｧ M (; 3; 0), M (4;,; ) ｧ M 3 (3; 0; ). 476. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 00607 閨06 04 00000600040030 ｧ 040003 ﾀ0903 ﾀ05070 ｧ 4x +3y, 5z, 8=0 ｧ 4x +3y, 5z +=0: ｰ09 0 ｧ06. 6 5 070504 0 0406 縺040308 040003 ﾀ0903 ﾀ05 ｧ 0006030606 0503 0906, 0 04 罍ｧ006 (; 0; 0), ｧ 0 0805 ｧ 0606 ﾀ ﾀ0503070 ｧ06 0305 陦縺06050308 040003 ﾀ0903 ﾀ05 ｧ. 477. ) 7 04 ｧ ﾀ 0504 06 040060 ｧ07 040003 ﾀ0903 ﾀ050608, 04 000006000400308 040003 ﾀ- 0903 ﾀ05 ｧ x, y +z, 5 = 0 ｧ06 0006000308 0305 00606 0 ﾀ ﾀ0503070 ｧ06, 0400306.

3x 3. ｰ 040060 ｧ07 04 縺0700308 65 ) 7 04 ｧ ﾀ 0504 06 040060 ｧ07 040003 ﾀ0903 ﾀ050608, 閨0600070 ｧ08 0403040300 0 閨040605 0-00308 06050300, 0303 0304 000308 040003 ﾀ0903 ﾀ05070 ｧ x +y = z ｧ z = 0, ｧ 0403 ﾀ05 縺03 ｧ0504 000306 ｧ ｧ ﾀ090300306 040003 ﾀ0903 ﾀ05 ｧ. 478. ) 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 00 ｧ- 0 ｧ06 0406 縺06 ﾀ06 060 ｧ07 040003 ﾀ0903 ﾀ050608 x, y +3z, 6=0, x +y, z +3=0 ｧ 06 縺06 0503 0906 (; ; 4). ) 7 0805 ｧ 閨0406 04 ｧ0003 0406 縺04060 陦ｧ09060007 縺00306 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨07-0 ｧ06 06 縺06 04 縺0700606 0406 縺06 ﾀ06 060 ｧ07 040003 ﾀ0903 ﾀ050608 x = y ｧ z = 0, 06 ﾀ00 ｧ 03 閨0 ｧ ｧ ﾀ090300308 040003 ﾀ0903 ﾀ050608 04 縺030803 陦ｧ05 06 縺06 0503 0906 (0; 4; ). 703- ﾀ05 縺03 ｧ0504 04 縺0700606 ｧ ｧ ﾀ090300306 040003 ﾀ0903 ﾀ05 ｧ. 479. 7 0805 ｧ 0503 0906 0406 縺06 ﾀ06 060 ｧ07 040003 ﾀ0903 ﾀ050608 x, y +3z, 9=0, x +y +z, 3=0 ｧ 3x + y, 4z +6=0. 480. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 0906 (;,; ) ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 040003 ﾀ0903 ﾀ05070 3x +y, z +4=0 ｧ x + y + z, 3=0. 703 ﾀ05 縺03 ｧ0504 0606. 48. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 09 ｧ (0;,5; 0) ｧ (0; 0; ) ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 040003 ﾀ0903 ﾀ05 ｧ x +5y + +z, 0 = 0. 703 ﾀ05 縺03 ｧ0504 0606. 48. 7 0805 ｧ 06050300 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 09 ｧ O(0; 0; 0), M (a;,a; 0) ｧ M (a; a; a), ﾀ 040003 ﾀ0903 ﾀ050406 xoy. 483. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0305 0 00 090303 罍陦ｧ0 05 閨03 040003 ﾀ0903 ﾀ05 ｧ, 04 縺03-0803 閨0700608 06 縺06 0503 09 ｧ M (a; 0;0), M (0; a; 0) ｧ M 3 (a; a; a). 484. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 03 ﾀ04Ox ｧ ﾀ03 ﾀ05 0400070600608 06050300 60 ﾀ 040003 ﾀ0903 ﾀ050406 y = x. 485. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0305 0503 09 ｧ (a; b; c) 閨03 040003 ﾀ0903 ﾀ05 ｧ, 0305 ﾀ0609-0600608 0 03 ﾀ0708 090303 罍陦ｧ0 05 0305 縺06 09 ｧ a, b ｧ c. 486. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 040003 ﾀ0903 ﾀ050608, 04 000006000400308 040003 ﾀ0903 ﾀ05 ｧ x +y + z, 8=0 ｧ06 0006000308 0305 00606 0 ﾀ ﾀ0503070 ｧ06 d =4. 487. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 00 ｧ0 ｧ06 0406 縺06 ﾀ06 060 ｧ07 040003 ﾀ0903 ﾀ050608 4x, y +3z, 6=0 ｧ x +5y, z +0=0 ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 040003 ﾀ0903 ﾀ05 ｧ x, y +5z, 5=0. x 3. ｰ 040060 ｧ07 04 縺0700308. ｰ 040060 ｧ07 04 縺0700308,04 縺030803 閨0700608 06 縺06 0503 0906 A(a; b; c) ｧ 04-000006000400308 0406090503 縺06 Pfm; n; g. 706 ﾀ0504 M (x; y; z) 04 縺03 ｧ 040300040 07 0503 09 04 縺0700308 ( 罍ｧ ﾀ. ), 050305,,! AM k P ｧ0403 06 ﾀ000304 ｧ06 04 0000060004003 ﾀ05 ｧ 0406090503 縺0304 x, a m = y, b n = z, c : () ｰ 040060 ｧ07 () 0 0304 0605 ﾀ07 09 0030 ｧ 06 ﾀ09 ｧ0 ｧ 06 040060 ｧ070 ｧ 04 縺0700308. 6 50609-0503 Pfm; n; g 0 0304 0605 ﾀ07 0 04 040007060 ｧ0 0406090503 縺030 04 縺0700308.

366 0600. 3. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 04 04 縺03 ﾀ05 0 ﾀ050406. 7 00605 罍ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 04 縺0700308 04030006 ｧ0, 04 罍ｧ 04-00704 09 07 閨0306 ｧ 030500300060 ｧ08 () 04 00605 縺06 t: 7 ｧ ﾀ. x = mt + a; y = nt + b; z = t + c: () 3. ｰ 040060 ｧ07 04 縺0700308, 04 縺030803- 閨0700608 06 縺06 閨0406 0503 09 ｧ: x, x x, x = y, y y, y = z, z z, z : (3) 4. 05030 ｧ06 06 040060 ｧ07 04 縺0700308: Ax + By + Cz + D =0; A x + B y + C z + D =0: 5. ｰ 040060 ｧ07 04 縺0700308 04 04 縺03060909 ｧ0708 04030006 ｧ0, ｧ ﾀ090006 ｧ04 ｧ 03030 ｧ08 06 040060 ｧ08 (4) 03 陦ｧ0 y, 陦縺06050308 x: (4) x = mz + a; y = nz + b: (5) ｰ 040060 ｧ07 (5) 00307003 04 ｧ ﾀ 0504 04 09 0030 ｧ 06 ﾀ090308 0703 縺006: x, a m 488. 7 0805 ｧ ﾀ0006 閨03 04 縺0700308: = y, b n = z, 0 : ) x = z +5, y =4, z ｧ ) x, 3 = y, = z, 3 0 040003 ﾀ0903 ﾀ050708 xoy ｧ xoz ｧ 0403 ﾀ05 縺03 ｧ0504 04 縺0700306. ｰ09 0 ｧ06. 703000307 ｧ0504 0406 040060 ｧ0708 04 縺0700308: ) z =0;) y =0. 489. ｰ 040060 ｧ07 04 縺0700308 x +y +3z, 3 = 0, 3x + y +4z, 4 = =00 04 ｧ ﾀ 0504: ) 04 04 縺03060909 ｧ0708; ) 04 09 0030 ｧ 06 ﾀ090308 0703 縺006. 7 0805 ｧ ﾀ0006 閨03 04 縺0700308 0 090303 罍陦ｧ0 0500308 040003 ﾀ0903 ﾀ050708, 0403 ﾀ05 縺03 ｧ0504 04 縺0700606 ｧ 0606 04 縺03060909 ｧ ｧ. 490. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 A(4; 3; 0) ｧ 04 000006000400308 0406090503 縺06 Pf,; ; g. 7 0805 ｧ ﾀ0006 04 縺07-00308 0 040003 ﾀ0903 ﾀ05 ｧ yoz ｧ 0403 ﾀ05 縺03 ｧ0504 04 縺0700606. 49. 703 ﾀ05 縺03 ｧ0504 04 縺0700606 x =4, y =3 ｧ 0 0805 ｧ 0606 0 04 040007060 ｧ08 0406090503. 49. 703 ﾀ05 縺03 ｧ0504 04 縺0700306: ) y =3, z =; ) y =, z = x +; 3) x =4, z = y ｧ 0304 縺06 閨0600 ｧ0504 ｧ08 0 04 040007060 ｧ06 0406090503 縺03. 493. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 09 ｧ A(,; ; 3) ｧ B(; 6;,), ｧ 0 0805 ｧ 0606 0 04 040007060 ｧ06 0903 ﾀ ｧ006 ﾀ03. 494. 703 ﾀ05 縺03 ｧ0504 04 縺0700606, 04 縺030803 閨0700606 06 縺06 0503 09 ｧ A(;,; 3) ｧ B(; 3; 3), ｧ 0 04 ｧ ﾀ 0504 060606 040060 ｧ07.

3x 3. ｰ 040060 ｧ07 04 縺0700308 67 495. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 05 06090503 罍ｧ ｧ 0503 09 ｧ M (x; y; z), 09030503-07, 040308 閨07 ｧ 0503 09 ｧ A(4;,3; ), 閨04 ｧ070605 ﾀ07 ﾀ03 ﾀ0903 縺03 ﾀ050406 vf; 3; g. 496. 7 04 ｧ ﾀ 0504 04 00605 罍ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 04 縺0700308: ) 04 縺030803 閨0700608 06 縺06 0503 0906 (,; ;,) ｧ 04 000006000400308 0406090503 縺06 Pf;,; 3g; ) 04 縺030803 閨0700608 06 縺06 0503 09 ｧ A(3;,; 4) ｧ B(; ; ). 497. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 (a; b; c): ) 04 0000060004003 03 ﾀ ｧ Oz; ) 0406 縺04060 陦ｧ09060007 縺003 09 03 ﾀ ｧ Oz. 498. 7 0805 ｧ 06050300 04 縺0700308 x = z,, y =,z + ﾀ 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0 0003 090303 罍陦ｧ0 05 ｧ 06 縺06 0503 0906 (;,;,). 499. 7 0805 ｧ 06050300 00607 閨06 04 縺070030 ｧ: x, y + z, 4=0, x + y,, z +5=0 ｧ x + y + z, 4=0, x +3y, z, 6=0. ｰ 09 0 ｧ 06. 7 04 040007060 ｧ08 0406090503 09 07 閨0308 ｧ 04 縺0700308 00307003 0304 縺06- 閨0600 ｧ0504 09 09 0406090503 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 003 縺0 000400308 0406090503 縺0304 040003 ﾀ0903 ﾀ050608 (P = N N ). 500. 70309 0504, 0503 04 縺070 07 x = y 3 = z 0406 縺04060 陦ｧ09060007 縺0 09 04 縺0700308 x = z +, y =, z. 50. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 (,4; 3; 0) ｧ 04 000006000400308 04 縺0700308 x, y + z =4, x + y, z =0. 50. 7 04 ｧ ﾀ 050406 040060 ｧ07 0406 縺04060 陦ｧ09060007, 03040600600030503 ｧ 0503 09 ｧ (;,3; 4) 0 03 ﾀ04 Oz. ｰ09 0 ｧ06. 0ｷ3 ﾀ09030 07 04 縺070 07 04 縺030803 陦ｧ05 06006 06 縺06 0503 0906 (0; 0; 4). 503. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 03050503 09 ｧ M (;,; 3) 閨03 04 縺0700308 x + = y 3 = + z, =. 4 5 ｰ09 0 ｧ06. ｯ03 09 A(,;,; ) 000607 ｧ05 0 04 縺0700308; Pf3; 4; 5g 0 04 040007060 ｧ08 0406090503 04 縺0700308. ｯ0305,,!,,! AMjP AMj jp AMj d = AM sin = = : P AM P 504. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 00607 閨06 04 00000600040030 ｧ 04 縺070030 ｧ x, = y + = z +3 ｧ x, = y, = z + : 505. 7 0805 ｧ ﾀ0006 閨03 04 縺0700308 x, 4 = y, = z, 0 090303 罍陦ｧ0 05-00308 040003 ﾀ0903 ﾀ050708 ｧ 0403 ﾀ05 縺03 ｧ0504 04 縺0700606. 506. ｰ 040060 ｧ07 04 縺0700308 x + y +8z, 6 = 0, x, y, z += =00 04 ｧ ﾀ 0504: ) 04 04 縺03060909 ｧ0708; ) 04 09 0030 ｧ 06 ﾀ090308 0703 縺006. 7 0805 ｧ ﾀ0006 閨03 04 縺0700308 0 090303 罍陦ｧ0 0500308 040003 ﾀ0903 ﾀ050708, 0403 ﾀ05 縺03 ｧ0504 04 縺0700606 ｧ 0606 04 縺03060909 ｧ ｧ.

368 0600. 3. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 04 04 縺03 ﾀ05 0 ﾀ050406 507. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 A(0;,4; 0) ｧ 04 000006000400308 0406090503 縺06 Pf; ; 3g, 0 0805 ｧ ﾀ0006 04 縺07-00308 0 040003 ﾀ0903 ﾀ05 ｧ xoz ｧ 0403 ﾀ05 縺03 ｧ0504 04 縺0700606. 508. 703 ﾀ05 縺03 ｧ0504 04 縺0700606 x =3, z =5 ｧ 0 0805 ｧ 0606 0 04 040007060 ｧ08 0406090503. 509. 7 0805 ｧ 0 04 040007060 ｧ08 0406090503 04 縺0700308 x + y, z =0, y = x ｧ06050003 04 縺0700308 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 ( ﾀ0. 0609 0 ｧ06 09 06 499). 50. 7 04 ｧ ﾀ 050406 040060 ｧ07 0406 縺04060 陦ｧ09060007, 03040600600030503 ｧ 0503 09 ｧ (;,3; 4) 0 03 ﾀ04 Oy. 5. 7 0805 ｧ 06050300 00607 閨06 04 縺070030 ｧ x, y, 7=0, x, z +5 = 0 ｧ 3x, y +8=0, z =3x. 5. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 0906 (,; ;,) ｧ 04 000006000400308 04 縺0700308 x, y =, y =z +. 53. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0305 0503 09 ｧ M (3; 0; 4) 閨03 04 縺0700308 y =x + +, z =x ( ﾀ0. 06 503). x 4. 7 縺070 07 ｧ 040003 ﾀ0903 ﾀ0504. ｰ050300 00607 閨06 04 縺0700308 x, a m 0903 ﾀ050406 Ax + By + Cz + D =0: = y, b n = z, c ｧ 040003 ﾀ- sin = ｰ ﾀ000304 ｧ06 ｧ08 04 0000060004003 ﾀ05 ｧ (N k P): jn Pj jam + Bn + Cj = : () NP NP ｰ ﾀ000304 ｧ06 ｧ08 0406 縺04060 陦ｧ09060007 縺003 ﾀ05 ｧ (N? P): Am + Bn + C =0: () A m = B n = C : (3). ｯ03 09 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 ｧ 040003 ﾀ0903 ﾀ05 ｧ. 7 04 ｧ ﾀ 04 04 00605 罍ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 04 縺0700308 x = mt + a, y = nt + b, z = t + c, 0403 陦ﾀ05 04 ｧ0 04 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ Ax + By + Cz + D =004006 ﾀ0503 x, y, z ｧ08 0403 07060 ｧ07 06 縺06 t. 7 08 閨060 t 0, 05060 x 0, y 0, z 0 090303 罍陦ｧ0 0503 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07. 3. ｰ ﾀ000304 ｧ06 ﾀ0403000307060 ｧ07 閨040608 04 縺0700308 04 03 閨00308 040003 ﾀ- 0903 ﾀ05 ｧ: a, a b, b c, c m n =0: (4) m n 54. 7 0805 ｧ 06050300 04 縺0700308 y =3x,, z =,3x + ﾀ 040003 ﾀ0903 ﾀ050406 x + y + z, 4=0.

3x 4. 7 縺070 07 ｧ040003 ﾀ0903 ﾀ0504 69 55. 70309 0504, 0503 04 縺070 07 x + y = + z,, = 3 04 00000600040 3 040003 ﾀ0903 ﾀ05 ｧ x + y, z =0, 04 縺070 07 x + y = + z, = +3 000607 ｧ05 3 04 05050308 040003 ﾀ0903 ﾀ05 ｧ. 56. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0503 0906 (,; ;,3) ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 04 縺0700308 x =, y, z =. 57. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 04 縺0700606 x, y, = 3 z = + 3 ｧ0503 0906 (3; 4; 0). 58. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 04 縺0700606 x, y = + ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 040003 ﾀ0903 ﾀ05 ｧ x +3y, = z +, z =4. 59. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 04 00-0006000400306 04 縺0700306 x, 3 y z, = = ｧ x + y, = z =. 50. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0 0003 090303 罍陦ｧ0 05 ｧ ﾀ03 ﾀ05 0400070600608 0400306 06050003 ﾀ 040003 ﾀ0903 ﾀ05070 ｧ 4y =3x, y =0 ｧ z =0. 7 0805 ｧ 0505 ｧ 06050003. 5. 7 0805 ｧ 0503 0906 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 x = t,, y = t +, z =, t ﾀ 040003 ﾀ0903 ﾀ050406 3x, y + z =3. 5. 7 0805 ｧ 0503 0906 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 x y, = = z + ﾀ 040003 ﾀ0903 ﾀ050406 x +y +3z, 9 = 0. 53. 7 0805 ｧ 04 縺03060909 ｧ06 0503 09 ｧ (3; ;,) 0 040003 ﾀ0903 ﾀ0504 x +y + +3z, 30 = 0. 54. 7 0805 ｧ 04 縺03060909 ｧ06 0503 09 ｧ (; 3; 4) 0 04 縺0700606 x = y = z. 55. 7 0805 ｧ 09 05 08000606 ﾀ ﾀ0503070 ｧ06 00607 閨06 00604 00000600040030 ｧ 04 縺070030 ｧ: ) x, a m ｧ x, a y, b z, c = = ; m n ) x + y z, = = ｧ x y = + z, =. 3 4 ｰ09 0 ｧ06. 7 縺06 閨040300 05 07 04 縺0700306 04 03030060 ﾀ0006 06 ﾀ09 縺060 ｧ04 060 ｧ- 0 ｧ ﾀ07, 0 罍ｧ ﾀ06060 04 000006000400306 040003 ﾀ0903 ﾀ05 ｧ, 04 09030503 縺0308 030 ｧ ﾀ040300030706003. 0ｷ3 0503 0609 A(a; b; c) ｧ A (a ; b ; c ) 04 縺030406 閨060 0406090503 縺03,! AB,,,! = A B = = Pfm; n; g ｧ,! AC,,,! = A C = P fm ; n ; g. 6 503 ﾀ0305 04 罍ｧ 003 ABCA B C ｧ 040 ｧ ﾀ0903003006 ﾀ ﾀ0503070 ｧ06. 56. 70309 0504, 0503 04 縺0700306 x = z,, y =z + ｧ x, y, = 4 = y, b n = z, c 3 = z, 0406 縺06 ﾀ0609 0605 ﾀ07, ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ,0409030503 縺0308 030 ｧ ﾀ- 040300030706003.

370 0600. 3. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 04 04 縺03 ﾀ05 0 ﾀ050406 57. 7 04 ｧ ﾀ 050406 040060 ｧ07 0406 縺04060 陦ｧ09060007, 03040600600030503 ｧ 0503 09 ｧ (; ; 0) 0 04 縺0700606 x =3z,, y =z. 58. 703 ﾀ05 縺03 ｧ0504 040003 ﾀ0903 ﾀ0504 x + y, z = 0 ｧ 04 縺0700606, 04 縺030803 閨07-00606 06 縺06 0503 09 ｧ A(0; 0; 4) ｧ B(; ; 0). 7 0805 ｧ 0503 0906 0406 縺06 ﾀ06 060 ｧ07 04 縺0700308 ﾀ 040003 ﾀ0903 ﾀ050406 ｧ 06050300 00607 閨06 0 ｧ0 ｧ. 59. 703 ﾀ05 縺03 ｧ0504 040003 ﾀ0903 ﾀ0504 y = z, 04 縺0700606 x =,z +, y = ｧ 0 0805 ｧ: ) 0503 0906 ｧ08 0406 縺06 ﾀ06 060 ｧ07; ) 06050300 00607 閨06 0 ｧ0 ｧ. 530. 7 0805 ｧ 04 縺03060909 ｧ06 0503 09 ｧ (3; ;,) 0 040003 ﾀ0903 ﾀ0504 3x + y + + z, 0=0. 53. 7 0805 ｧ 04 縺03060909 ｧ06 0503 09 ｧ (; ; 8) 0 04 縺0700606 x, y = =z., 53. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 04 00-0006000400306 04 縺0700306 x, y = + z,, = ｧ x y 3 = + z,, =. 3 533. 70309 0504, 0503 04 縺0700306 x +3 y = + z = + ｧ x =3z,4, y = z +0406 縺06 ﾀ0609 0605 ﾀ07, 0 0805 ｧ 0503 0906 ｧ08 0406 縺06 ﾀ06 060 ｧ07. 534. 7 04 ｧ ﾀ 050406 040060 ｧ07 0406 縺04060 陦ｧ09060007, 03040600600030503 ｧ 0503 09 ｧ (; 0;,) 0 04 縺0700606 x + y, = z =.,3 535. 7 0805 ｧ09 05 08000606 ﾀ ﾀ0503070 ｧ06 00607 閨06 04 縺070030 ｧ x =,y = = z ｧ x = y =. x 5. ｮ0706 罍ｧ 06 ﾀ09 ｧ06 ｧ 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ06 04030406 縺08003 ﾀ05 ｧ. ｰ 040060 ｧ06 ﾀ0706 罍ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ ﾀ 0906005 縺030 C(a; b; c) ｧ 陦ｧ06 ﾀ030 R: (x, a) +(y, b) +(z, c) = R : (). ｰ 040060 ｧ06 F (x; y) =0, 006 ﾀ03 閨06 縺07 00606 z, 0304 縺06 閨0600070605 09 ｧ00 ｧ0- 陦罍ｧ 06 ﾀ090606 04030406 縺08003 ﾀ0504 ﾀ 0303 060600608; 04 000006000400308 03 ﾀ ｧ Oz. 縺0 0003-05 ｧ 003 09 07 閨0306 ｧ 06 040060 ｧ08: ) F (y; z) = 0 ｧ)F (x; z) = 00304 縺06 閨06-00070605 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ090606 04030406 縺08003 ﾀ0504 ﾀ 0303 060600608, 04 000006000400308: ) Ox; ) Oy. 3. ｰ 040060 ｧ06 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ ﾀ 0 04-0400070600608 F (x; y) = 0, z = 0 ｧ ﾀ 0303 060600608, 04 000006000400308 0406090503 縺06 Pfm; n; g. ｰ 040060 ｧ06 04 縺03 ｧ 0403000400308 0303 060600608 0306 閨0605 x, x 0 = y, y m 0 z = = n,05 閨06 (x 0; y 0 ;0) 0503 09 0 0 04 0400070600608. 0504 縺06 閨0600 ｧ04 0305 ﾀ06 x 0 ｧ y 0 ｧ 0403 陦ﾀ05 04 ｧ04 ｧ08 04 06 040060 ｧ06 0 04 0400070600608, 04030006 ｧ0 06 040060 ｧ06 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ: x, m z; y, n z F =0: ()

3x 5. ｮ0706 罍ｧ 06 ﾀ09 ｧ06 ｧ 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ06 04030406 縺08003 ﾀ05 ｧ 7 536. 7 0805 ｧ 0906005 ｧ 陦ｧ06 ﾀ ﾀ0706 縺03: ) x + y + z, 3x +5y, 4z =0; ) x + y + z =az ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ 0303 07060 ｧ06 040503 縺0308 ﾀ0706 縺03. 537. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 ﾀ0706 罍ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ, 0404 ｧ ﾀ 000308 04 050605 05 陦, 0303 0304 000308 040003 ﾀ0903 ﾀ05070 ｧ 3x, y +6z, 8 = 0; x =0; y =0; z =0: 538. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, ﾀ0403-00030706000308 04 閨040306 0300 ｧ0706 09 0503 0906 A(; 0; 0), 060090503 0906 B(,4; 0; 0). 539. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 ﾀ0706 罍ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ, 04 縺030803 閨07-00608 06 縺06 0309 縺0607003 ﾀ0504 x + y + z = a, x + y + z = a ｧ 06 縺06 0503 0906 (a; a; a). ｰ09 0 ｧ06. 0ｷ3 ﾀ090300306 06 040060 ｧ06 閨030007003 ｧ0060504 04 ｧ x + y + z, a + (x + y + z, a) =0: 540. 703 ﾀ05 縺03 ｧ0504 04 0006040308 ﾀ ｧ ﾀ0506006 090303 罍陦ｧ0 05 04030406 縺08003 ﾀ05 ｧ: ) y + z =4; ) y = ax; 3) xz =4; 4) x + y = ax. 54. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 03 陦ｧ0-09030403 06 0006000308 0305 04 縺0700308 x = a, y =0 ｧ 040003 ﾀ0903 ﾀ05 ｧ yoz. 703 ﾀ05 縺03- ｧ0504 04030406 縺08003 ﾀ0504. 54. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 05 縺0608 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ08 04030406 縺08003 ﾀ050608, 0304 ｧ ﾀ 000308 0309030003 ﾀ0706 縺03 x + y + z, ax = 0 ﾀ 0303 06060 ｧ0 ｧ, 04 00000600040030 ｧ ﾀ030305040605 ﾀ0504060003: ) 03 ﾀ ｧ Ox; )03 ﾀ ｧOy; 3)03 ﾀ ｧOz. 543. 7 罍ｧ ﾀ0304 0504 04 0406 縺04030 030905 00506 0006040308 ﾀ ｧ ﾀ0506003 090303 罍陦ｧ0 05 09 罍ｧ040606 6 5 ｧ04 ｧ 0 ｧ: x + y + z =6; x + y =4x; 0403 ﾀ05 縺03 ｧ04 0606 0503 09 ｧ 04 罍ｧ x =0; ｧ4. 70309 0504, 0503 04 縺03060909 ｧ07 09 罍ｧ040308 0 040003 ﾀ0903 ﾀ0504 xoz 06 ﾀ0504 04 030300. 544. 7 0805 ｧ 0906005 ｧ 陦ｧ06 ﾀ 0309 縺0607003 ﾀ05 ｧ x + y + z =0y; x +y +z, 9=0: ｰ09 0 ｧ06. ｨC06005 0309 縺0607003 ﾀ05 ｧ 06 ﾀ0504 04 縺03060909 ｧ07 0906005 00 0 040003 ﾀ- 0903 ﾀ0504 ( ﾀ0. 06 530). 545. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ ﾀ 0-04 0400070600608 y =4x, z =0 ｧ ﾀ 0303 060600608, 04 000006000400308 0406090503 縺06 Pf; ; 3g. 546. 703 ﾀ05 縺03 ｧ0504 04 0406 縺04030 030905 00506 04030406 縺08003 ﾀ0504 (x+y) +az = a 0403 ﾀ06 060 ｧ070 040003 ﾀ0903 ﾀ05070 ｧ x = 0, y = 0, z = 0, z = h 6 a ｧ 040309 0504, 0503 0505 04030406 縺08003 ﾀ0504 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 07 ﾀ 0303 06060 ｧ0 ｧ, 04 00000600040030 ｧ 04 縺0700308 x + y = a, z =0.

37 0600. 3. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 04 04 縺03 ﾀ05 0 ﾀ050406 547. 0ｹ0 x + y + z =4z 03 ﾀ04060060 0006 0 ｧ, 04 00000600040030 ｧ 04 縺0700308 x =0, y = z. 7 0805 ｧ 0703 縺006 05060 ｧ 00 0 040003 ﾀ0903 ﾀ05 ｧ xoy. ｰ09 0 ｧ06. 70607003 0 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ090308 04030406 縺08003- ﾀ05 ｧ, 0303 0304 000308 0006 0 ｧ, 09 ﾀ 050600040030 ｧ 09 00 縺06. 6ｦ0 0606 0 04 0400070600606 04 罍ｧ0070504 00 ｧ0 ｧ06 ﾀ06 060 ｧ07 00 040003 ﾀ0903 ﾀ050406, 04 縺030803 閨0700608 06 縺06 0906005 00 ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 0006 0. 548. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0906005 C 04030406 縺08003 ﾀ05 ｧ x + y + z, x + y, 3z =0 ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 04 縺0700308 OC. 549. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 06 00060-00308 04 閨040306 00040006 0305 0 00 090303 罍陦ｧ0 05, 060 0305 0503 09 ｧ (0;,3; 0). 550. 7 0805 ｧ 04 縺03060909 ｧ06 0 040003 ﾀ0903 ﾀ0504 z = 0 ﾀ06 060 ｧ07 00 縺03040308 04030406 縺08003 ﾀ05 ｧ x + y + z =4(x, y, z) 040003 ﾀ0903 ﾀ050406, 04 縺030803 閨0700608 06 縺06 0906005 00 ｧ 0406 縺04060 陦ｧ09060007 縺00308 09 04 縺0700308 x =0, y + z =0. 55. 6 5 0006040308 ﾀ ｧ ﾀ0506006 090303 罍陦ｧ0 05 0403 ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ05 ｧ: ) z =4, x ;)y + z =4z; 3)y = x 3. 55. 703 ﾀ05 縺03 ｧ0504 04 0406 縺04030 030905 00506 0006040308 ﾀ ｧ ﾀ0506003 090303 罍陦ｧ0 05 09 罍ｧ- 040606 0406 縺06 ﾀ06 060 ｧ07 09 ｧ00 ｧ0 陦縺0304 x + z = a ｧ x + y = a. ｰ09 0 ｧ06. 703 ﾀ05 縺03 ｧ04 04 040003 ﾀ0903 ﾀ050708 xoz ｧ xoy 06050406 縺05 ｧ 0 04 040007-060 ｧ08 0309 縺0607003 ﾀ050608, 閨0600 ｧ0504 ｧ08 04 罍ｧ0300 ｧ07060003 0 0400306 ﾀ05 ｧ (0 04 罍ｧ- 006, 0 4) ｧ 06 縺06 0503 09 ｧ 閨0600060 ｧ07 04 縺030406 ﾀ05 ｧ 0303 06060 ｧ06 09 ｧ00 ｧ0 陦縺0304 閨03 ｧ08 0406 縺06 ﾀ06 060 ｧ07 ( ﾀ0. 罍ｧ ﾀ. 60, ﾀ. 30). 553. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ ﾀ 0303-060600608, 04 000006000400308 0406090503 縺06 Pf; ; g, ｧ ﾀ 0 04 0400070600608 x + + y =4x, z =0. 554. 703 ﾀ05 縺03 ｧ0504 05060003, 0305 0 ｧ 06000306 04030406 縺08003 ﾀ05070 ｧ y = x, z = 0, z = 4, x = 4, ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ07 陦ｧ 05030 000608 05 0 ｧ, 000607 00608 04 040003 ﾀ0903 ﾀ05 ｧ x =4. x 6. 07030 ｧ 06 ﾀ09 ｧ06 04030406 縺08003 ﾀ05 ｧ ｧ 04030406 縺08003 ﾀ05 ｧ 04 0060 ｧ07.07030 ｧ 06 ﾀ09 ｧ06 04030406 縺08003 ﾀ05 ｧ. 706 ﾀ0504 09030 ｧ 06 ﾀ09 07 04030406 縺08003 ﾀ0504 ｧ0060605 0406 縺00 ｧ006 040 0006 090303 罍陦ｧ0 05, 0 04 0400070600606 F (x; y) = 00 x 040003 ﾀ0903 ﾀ05 ｧ z = h. ｰ 040060 ｧ06 0303 060600608 0306 閨0605: y z = = x 0 y 0 h,05 閨06 (x 0 ; y 0 ; h) 0503 09 0 04 0400070600608. 0504 縺06 閨0600 ｧ04 0305 ﾀ06 x 0 ｧ y 0 ｧ 0403 陦ﾀ05-04 ｧ04 ｧ08 04 06 040060 ｧ06 F (x; y) =0, 04030006 ｧ0 06 040060 ｧ06 09030 ｧ 06 ﾀ090308 04030406 縺08-003 ﾀ05 ｧ ﾀ 0406 縺00 ｧ00308 04 0 0006 090303 罍陦ｧ0 05: xh F z ; yh =0: () z

3x 6. 07030 ｧ 06 ﾀ09 ｧ06 04030406 縺08003 ﾀ05 ｧ ｧ 04030406 縺08003 ﾀ05 ｧ 04 0060 ｧ07 73 ｭ ﾀ00 ｧ 0406 縺00 ｧ0 0903006 ﾀ 0306 閨0605 04 0503 0906 (a; b; c), 0503 06 040060 ｧ06 04 罍ｧ00605 04 ｧ (x, a)(h, c) (y, b)(h, c) F + a; + b =0: () z, c z, c ｰ 040060 ｧ06 () 03 閨003 縺03 閨003 0305003 ﾀ ｧ05060004003 x, y, z, 06 040060 ｧ06 () 03-003 縺03 閨003 0305003 ﾀ ｧ05060004003 x, a, y, b ｧ z, c. 703 03 閨003 縺03 閨003 ﾀ05 ｧ 06 040060 ｧ07 00307003 06 0 0504 06 040060 ｧ06 09030 ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ.. 7030406 縺08003 ﾀ05 ｧ 04 0060 ｧ07: ｰ 040060 ｧ07 09 罍ｧ040308 05 ﾀ04 04 0060 ｧ07 ｰ 040060 ｧ06 04030406 縺08003 ﾀ05 ｧ 04 0060 ｧ07 F (x; y) =0; Ox F (x; y + z )=0 z =0 Oy F ( x + z ;y)=0 F (x; z) =0; Ox F (x; y + z )=0 y =0 Oz F ( x + y ;z)=0 F (y; z) =0; Oy F (y; x + z )=0 x =0 Oz F ( x + y ;z)=0 555. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09030 ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ ﾀ 0406 縺00 ｧ00308 040 0006 090303 罍陦ｧ0 05 ｧ 0 04 0400070600608 x + y = a, z = c. 703 ﾀ05 縺03- ｧ0504 ｧ 0303 07060 ｧ06 04030406 縺08003 ﾀ05 ｧ. 556. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09030 ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ ﾀ 0406 縺00 ｧ00308 040503 0906 A(0;,a; 0) ｧ 0 04 0400070600608 x =y, z = h. 703 ﾀ05 縺03 ｧ0504 ｧ 0303 07060 ｧ06 04030406 縺08003 ﾀ05 ｧ. 557. 0504 縺06 閨0600 ｧ0504 0406 縺00 ｧ006 0903006 ﾀ x +(y, a), z = 0, 060503 0 04 0400070600606 04 040003 ﾀ0903 ﾀ05 ｧ z = a ｧ 0403 ﾀ05 縺03 ｧ0504 0903006 ﾀ. 558. 0504 縺06 閨0600 ｧ0504 0406 縺00 ｧ006 0903006 ﾀ x =yz, 060503 0 04 0400070600606 04 040003 ﾀ0903 ﾀ05 ｧ z = h ｧ 0403 ﾀ05 縺03 ｧ0504 0903006 ﾀ. 559. 0ｷ3 ﾀ ﾀ0006 閨0304 0504 04030406 縺08003 ﾀ0504 0903003 ｧ ) ｧ00 ｧ 0900 ｧ0 (a, x ) y = h z 0403 ﾀ06 060 ｧ070 040003 ﾀ0903 ﾀ05070 ｧ z =0, y = h, x = c (c 6 a) ｧ 0403 ﾀ05 縺03 ｧ0504 0903003 ｧ 04 030300 ﾀ05 ｧ z > 0. 560. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04030406 縺08003 ﾀ05 ｧ, 0303 0304 000308 04 006-0 ｧ060 09 罍ｧ040308 z = x, y = 0: ) 040309 縺0605 03 ﾀ ｧOz; ) 040309 縺0605 03 ﾀ ｧ Ox. 703 ﾀ05 縺03 ｧ0504 030306 04030406 縺08003 ﾀ05 ｧ. ) 0703003 ｧ 閨030 0 0304 0605 ﾀ07 04030406 縺08003 ﾀ0504, 0303 0304 00 07 閨04 ｧ07060 ｧ060 04 縺0700308, 04-000006000400308 000308 040003 ﾀ0903 ﾀ05 ｧ ｧ 0406 縺06 ﾀ0609 0600608 000606 09 罍ｧ040606 ｧ 000606 04 縺0700606.

374 0600. 3. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 04 04 縺03 ﾀ05 0 ﾀ050406 56. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04030406 縺08003 ﾀ05 ｧ, 0303 0304 000308 04 006-0 ｧ060 040309 縺0605 03 ﾀ ｧOz: )09 罍ｧ040308 z = e,x, y =0;)09 罍ｧ040308 z = 4 x, y =0. 703 ﾀ05 縺03 ｧ0504 030306 04030406 縺08003 ﾀ05 ｧ (04 0006040308 ﾀ ｧ ﾀ0506006 090303 罍陦ｧ0 05). 56. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09030 ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ ﾀ 0406 縺00 ｧ00308 O(0; 0; 0), 0 04 0400070600608 x +(y,6) +z =5, y =3 ｧ 0 罍ｧ ﾀ0304 0504 04030406 縺08003 ﾀ0504. 563. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09030 ｧ 06 ﾀ090308 04030406 縺08003 ﾀ05 ｧ ﾀ 0406 縺00 ｧ00308 C(0;,a; 0), 0 04 0400070600608 x + y + z =5, y =3 ｧ 0 罍ｧ ﾀ0304 0504 04030406 縺08003 ﾀ0504. 564. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04030406 縺08003 ﾀ05 ｧ, 0303 0304 000308 04 006-0 ｧ060 04 縺0700308 z = y, x = 0: ) 040309 縺0605 03 ﾀ ｧOy; )040309 縺0605 Oz, ｧ 0 罍ｧ ﾀ0304 0504 030306 04030406 縺08003 ﾀ05 ｧ. 565. 70309 0504, 0503 ﾀ06 060 ｧ06 0903006 ﾀ z = xy 040003 ﾀ0903 ﾀ050406 x + y = =a 06 ﾀ0504 050000 ｧ04 ﾀ, ｧ 0 0805 ｧ 060503 0403000603 ﾀ ｧ. x 7. 70000 ｧ04 ﾀ03 ｧ, 05 ｧ0406 縺03030003 ｧ 閨03 ｧ 04 03030003 ｧ 閨03. 07 0030 ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07. 07 縺03006 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ08, ﾀ06-006 ﾀ0504060605 0006 ﾀ0504 03 ﾀ0030400308 04 ｧ 閨0304 04030406 縺08003 ﾀ050608 040503 縺030503 0403 縺07 閨09, 0304 縺06- 閨0600070600308 ﾀ0006 閨06060 ｧ0 ｧ 09 0030 ｧ 06 ﾀ09 ｧ0 ｧ (04 縺03 ﾀ05060800 ｧ0 ｧ) 06 040060 ｧ070 ｧ: I. 70000 ｧ04 ﾀ03 ｧ : x II. 06 ｧ0406 縺03030003 ｧ 閨03: a + y b + z c =. 8 >< >: x a + y x a + y III. 0703006 ﾀ 040503 縺030503 0403 縺07 閨09 : b, z = 03 閨00304030003 ﾀ0500308; c b, z =, 閨040604030003 ﾀ0500308. c x 8 >< IV. 7 03030003 ｧ 閨03 (04 罍ｧ q > 0): >: a + y b, z c =0. x + y q x =z 050000 ｧ0405 ｧ 06 ﾀ09 ｧ08; =z 05 ｧ0406 縺030300 ｧ 06 ﾀ09 ｧ08., y q. 7 縺0700300 ｧ0060800306 0303 06060 ｧ06. ｪ06 縺06 09 07 閨0606 0503 0906 03-00304030003 ﾀ050030503 05 ｧ0406 縺03030003 ｧ 04 縺030803 閨0705 閨0406 060503 04 縺0700300 ｧ0060800306 0303 06-060 ｧ06: x a + z = + y x ; c b a + z c ｧ x a, z c =, y b x a, z c =, y ; b = + y b :

3x 7. 70000 ｧ04 ﾀ03 ｧ, 05 ｧ0406 縺03030003 ｧ 閨03 ｧ 04 03030003 ｧ 閨03 75 ｪ06 縺06 09 07 閨0606 0503 0906 05 ｧ0406 縺030300 ｧ 06 ﾀ09030503 04 03030003 ｧ 05030706 04 縺030803 閨0705 閨0406 060503 04 縺0700300 ｧ0060800306 0303 06060 ｧ06 (04 罍ｧ >0 ｧ q>0): x + y q x, y q =; ｧ = z x + y q x, y q = z; =: 3.07 縺060503040306 ﾀ06 060 ｧ07. 7 04 ﾀ0608 04030406 縺08003 ﾀ050708, ｧ006060 ｧ08 050000 ｧ04-05 ｧ 06 ﾀ09 ｧ06 ﾀ06 060 ｧ07, ｧ0060605 ﾀ07 05 090706 ｧ 09 縺060503040306 ﾀ06 060 ｧ07. 7 ｧ0303000400 ｧ06 09 縺060503040306 ﾀ06 060 ｧ07 050000 ｧ04 ﾀ03 ｧ x a + y b + z = (04 罍ｧ a > b > c) 0 - c 0803 閨0705 ﾀ07 0 ﾀ0706 縺06 x + y + z = b. 07 縺060503040306 ﾀ06 060 ｧ07 050000 ｧ0405 ｧ 06 ﾀ09030503 04 03030003 ｧ x + y q ﾀ0706 縺06 x + y + z =z (04 罍ｧ >q). = z, 04 縺030803 閨070 ｧ06 06 縺06 0406 縺00 ｧ006, 0 0803 閨0705 ﾀ07 0 566. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04030406 縺08003 ﾀ05 ｧ, 0303 0304 000308 04 006-0 ｧ060 050000 ｧ04 ﾀ x a + z =, y =0040309 縺0605 03 ﾀ ｧOz. c 567. 703 ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ0504 x 9 + y 4 + z = ｧ0 0805 ｧ 0400030 陦ｧ 5 0606 ﾀ06 060 ｧ08 040003 ﾀ0903 ﾀ05070 ｧ: ) z =3;)y =. 568. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04030406 縺08003 ﾀ05 ｧ, 0303 0304 000308 04 006-0 ｧ060 09 罍ｧ040308 x a, z =, y =0:)040309 縺0605 c 03 ﾀ ｧ Oz; )040309 縺0605 03 ﾀ ｧOx. 703 ﾀ05 縺03 ｧ0504 030306 04030406 縺08003 ﾀ05 ｧ (04 0006040308 ﾀ ｧ ﾀ0506006 090303 罍陦ｧ0 05). 569. 703 ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ05 ｧ: ) x + y, z =4; ) x, y + z +4=0., z 36 570. 703 ﾀ05 縺03 ｧ0504 05 ｧ0406 縺03030003 ｧ x 6 + y 4, = ｧ 0 0805 ｧ 060503 0303 06060 ｧ06, 04 縺030803- 閨070 ｧ06 06 縺06 0503 0906 (4; ;,3). 57. 7 ｧ05070 07 003 閨060004 09 ｧ00 ｧ0 陦 08 09 縺06-060 09 04030403 縺0305030 0406 縺080060503 09 縺0605 0 ( 罍ｧ ﾀ. ). 0504 縺06 閨0600 ｧ0504 06 040060 ｧ06 04030006 060-00308 0800 ｧ00608 05030809 04030406 縺08003 ﾀ05 ｧ, 06 ﾀ00 ｧ 0309-7 ｧ ﾀ.

376 0600. 3. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 04 04 縺03 ﾀ05 0 ﾀ050406 縺0607003 ﾀ05 ｧ 0606 03 ﾀ00304 0 ｧ08 000607 05 04 040003 ﾀ0903 ﾀ050708 z = c, ｧ08 0906005 縺03 0 03 ﾀ ｧ Oz, ｧ08 陦ｧ06 ﾀ03 04003 a. 7 ﾀ ﾀ00305 縺060504 ﾀ0500306 ﾀ0006 ｧ 04 罍ｧ =90, 0, 80. ｰ09 0 ｧ06. ｯ03 09 M (x; y; z) 閨0600 ｧ05 ﾀ ﾀ0503070 ｧ06 00607 閨06 0503 09 0 ｧ A(a cos t; a sin t;,c); B(a cos (t + ); a sin (t + ); c) 04030500300060 ｧ ｧ AM : MB =(c + z) :(c, z). 57. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 04030406 縺08003 ﾀ05 ｧ, 0303 0304 000308 04 006-0 ｧ060 04 03030003 az = x, y = 0 040309 縺0605 03 ﾀ ｧOz. 703 ﾀ05 縺03 ｧ0504 0403-0406 縺08003 ﾀ0504 0403 ﾀ06 060 ｧ070 040003 ﾀ0903 ﾀ05070 ｧ: z = a, x =0, y =0. 573. 703 ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ05 ｧ: ) z = x + y ; ) z = c, x a, y. b 574. 703 ﾀ05 縺03 ｧ0504 (04 0006040308 ﾀ ｧ ﾀ0506006 090303 罍陦ｧ0 05) 04030406 縺08003 ﾀ0504 x,, y =4z ｧ 0 0805 ｧ 0606 0303 06060 ｧ06, 04 縺030803 閨070 ｧ06 06 縺06 0503 0906 (3; ; ). 575. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 0305003-00060 ｧ06 ﾀ ﾀ0503070 ｧ08 0305 09 07 閨0308 ｧ 09030503 縺0308 閨03 040003 ﾀ0903 ﾀ05 ｧ x = a 09 ﾀ ﾀ0503070 ｧ070 閨03 0503 09 ｧ F (a; 0;0) 04003. 703 ﾀ05 縺03 ｧ0504 04030406 縺08-003 ﾀ0504. 576. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 0305003-00060 ｧ06 ﾀ ﾀ0503070 ｧ08 0305 09 07 閨0308 ｧ 09030503 縺0308 閨03 0503 09 ｧ F (0; 0; a) ｧ 閨03 040003 ﾀ0903 ﾀ05 ｧ z = a 04003. 703 ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ0504. 577. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 03 陦ｧ0-09030403 06 0006000308 0305 0503 09 ｧ F (,a; 0;0) ｧ0305 040003 ﾀ0903 ﾀ05 ｧ x = a. 703- ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ0504. 578. 7 0805 ｧ 0 ｧ0303000400 ｧ06 09 縺060503040306 ﾀ06 060 ｧ07 050000 ｧ04 ﾀ03 ｧ x 69 + + y 5 + z 9 =. x 579. 0504 縺06 閨0600 ｧ050409 縺060503040306 ﾀ06 060 ｧ07 050000 ｧ0405 ｧ 06 ﾀ09030503 04 03030003 ｧ 5 + y 9 = z, 04 縺030803 閨070 ｧ06 06 縺06 0 0003 090303 罍陦ｧ0 05. 580. 7 04 0504 ｧ 0403 ﾀ05 縺03 ｧ0504 09 07 閨0606 ｧ 04030406 縺08003 ﾀ050608: ) x + y + z =az; 6) x =az; ) x + y =az; 7) x =yz; 3) x + z =az; 8) z =+x + y ; 4) x, y =az; 9) (z, a) = xy; 5) x, y = z ; 0) (z, x) +4(z, x) =y :

3x 7. 70000 ｧ04 ﾀ03 ｧ, 05 ｧ0406 縺03030003 ｧ 閨03 ｧ 04 03030003 ｧ 閨03 77 58. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700300 ｧ0060800308 0303 06060 ｧ08 05 ｧ0406-03030003 ｧ x, y + z =4,04 縺030803 閨070 ｧ08 06 縺06 0503 0906 (; 4; 4). 58. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 03 陦ｧ0-09030403 06 0006000308 0305 0503 09 ｧ F (0; 0; a=) ｧ0305 040003 ﾀ0903 ﾀ05 ｧ z =,a=. 703 ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ0504. 583. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609, 03 陦ｧ0-09030403 06 0006000308 0305 0503 09 ｧ F (0; 0; a=) ｧ0305 040003 ﾀ0903 ﾀ05 ｧ z = 3a=. 703 ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ0504. 584. 7 0805 ｧ 0 ｧ00600400 ｧ06 09 縺060503040306 ﾀ06 060 ｧ07 05 ｧ0406 縺03030003 ｧ x 5 + y 9, 3z 5 =: 585. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 04 縺0700300 ｧ0060800308 0303 06060 ｧ08 05 ｧ0406-030300 ｧ 06 ﾀ09030503 04 03030003 ｧ x (4; 3; 0)., y 6 9 =z, 04 縺030803 閨070 ｧ08 06 縺06 0503 0906

30600 04 4 6 56ｧ9 ｮ0ｹ0 縺00 縺6ｧ806 ｭ 7 7 x. 0504 縺06 閨0600 ｧ050600 ｧ. 0504 縺06 閨0600 ｧ050600 ｧ. 0504 縺06 閨0600 ｧ050600060 040503 縺030503 0403 縺07 閨09 0 0304-0605 ﾀ07 ｧ ﾀ0003, 030303 0 0600306 ﾀ ｧ0040300030 a b a b ｧ 0304 縺06 閨0600070600306 04060 ﾀ0504030 a b = a b a b, a b : () 0504 縺06 閨0600 ｧ050600060 05 縺060504060503 0403 縺07 閨09 0 0304 0605 ﾀ07 ｧ ﾀ0003, 030303 0 0600306 a b c ﾀ ｧ0040300030 a b c ｧ 0304 縺06 閨0600070600306 04060 ﾀ0504030 a 3 b 3 c 3 a b c a b c a 3 b 3 c 3 = a b c a b 3 c, c a b 3 a 3 c + b c : 3 a 3 b () 3 0504 縺06 閨0600 ｧ050600 ｧ 040503 縺030503 0403 縺07 閨09, 040803 閨070 ｧ06 04 04 040606 ﾀ0504 04060 ﾀ0504 (), 04030006 0605 ﾀ07 ｧ 00030503 0304 縺06 閨0600 ｧ05060007 05 縺060504060503 0403 縺07 閨09 0403 06 縺09 ｧ04-0 ｧ060 03 閨00308 ﾀ05 縺0309 ｧ ｧ 03 閨0030503 ﾀ0503000309 ｧ 0 0304 0605 ﾀ07 060503 0 ｧ003 0 ｧ. ｱ03-00600 () 0 0304 0605 ﾀ07 0703 縺006000308 000307060 ｧ07 0304 縺06 閨0600 ｧ05060007 05 縺060504060503 0403- 縺07 閨09 0403 050006006005 0 0406 縺040308 ﾀ05 縺0309 ｧ.. ｮ040308 ﾀ0504 0304 縺06 閨0600 ｧ0506000608: I. 6 50600 ｧ ｧ0 0304 縺06 閨0600 ｧ05060007 006 ｧ 0060 ｧ05 ﾀ07 0305 006003 ﾀ05 縺0309 ﾀ0503000309 0 ｧ. II. 6 50600 ｧ ｧ0 0304 縺06 閨0600 ｧ05060007 0305 0406 縺06 ﾀ05 0030409 ｧ 閨040608 0006030308 04 0000060004-00308 060503 縺07 閨0304 0060070605 0 09 0 0303 0500308. 0ｷ3 ﾀ040308 ﾀ0504 I ｧ II ﾀ0006 閨060605, 0503 0304 縺06 閨0600 ｧ05060004 00307003 000307 ｧ0504 0403 050006006005 0 000603030503 縺07, 05 09 09 09 05050305 縺07 00307003 ﾀ 閨0600 0504 0406 縺040308 ﾀ05 縺03090308. III. 0504 縺06 閨0600 ｧ05060004 ﾀ 閨0406007 03 陦ｧ0 090304030 ｧ 04 00000600040030 ｧ 縺07 0 ｧ - 04060 0060006. IV. 05030 ｧ08 000307 ｧ05060004 0500060060050304 03 閨0030503 縺07 00307003 0403006 ﾀ05 ｧ 0 09 0304 縺06 閨0600 ｧ05060007. V. 6 50600 ｧ ｧ0 0304 縺06 閨0600 ｧ05060007 006 ｧ 0060 ｧ05 ﾀ07, 06 ﾀ00 ｧ 09 050006006005 0 03 閨0030503 縺07 04 罍ｧ03 04 ｧ0504 05000600600503 04 00000600040030503 縺07, 0600030706000306 0 04 縺03 ｧ - 0403000400306 03 陦ｧ0 0903040306 ｧ ﾀ0003. 7 04 罍ｧ006 : a b c a a b c a 3 b 3 c 3 = + mc b + nc c a + mc b + nc c a 3 + mc 3 b 3 + nc 3 c 3 :

3x. 0504 縺06 閨0600 ｧ050600 ｧ 79 ｮ 040300300406 0505030503 ﾀ040308 ﾀ0504 00307003 04 000603030 縺07 閨06 0304 縺06 閨0600 ｧ05060007 05 縺060504060503 0403 縺07 閨09 ﾀ 閨0600 0504 閨04 0060007, 060 0604 縺03 ﾀ05 ｧ05 ﾀ07 000307060 ｧ06 0304 縺06 閨0600 ｧ05060007 0403 050006006005 0 0505030503 縺07. 3. 700030 閨04 05 縺0606050300040 ｧ09 ﾀ0406 縺00 ｧ0 0 ｧ A(x ; y ), B(x ; y ), C(x 3 ; y 3 ): S = x y x y x 3 y 3 : (3) 6 503 ｧ ﾀ00 ｧ0504 0304 縺06 閨0600 ｧ050600 ｧ: 586. 3, : 587. 3 3, : 588.. 4 6 6,0,4 5 589. a, sin cos : 590. a a, cos sin. 59. sin sin cos cos. 6 503 ｧ ﾀ00 ｧ0504 0304 縺06 閨0600 ｧ050600 ｧ, 000307 ｧ04 ｧ08 0403 050006006005 0 0406 縺04030503 ﾀ0503000309 : 3 4 a a 59. 5, 3 : 593., a a, a. 6 503 ｧ ﾀ00 ｧ0504 0304 縺06 閨0600 ｧ050600 ｧ, 000307 ｧ04 ｧ08 0403 050006006005 0 05030503 縺07, 09030503 縺0308 ﾀ03 閨06 縺07 ｧ05 0 ｧ03030004000606 ｧ ﾀ0003 006000608: b,x x 594. 0 b 0 b 0,b : 595. 0,x, x,x. ｰ04 縺03 ﾀ05 ｧ0504 ｧ 0403 ｧ ﾀ00 ｧ0504 0304 縺06 閨0600 ｧ050600 ｧ: a,a a 5 596. a a a,a,a : 597. 3,4 7,3,5. 598. 600. 60. 6,4 x x 6 4 4 3 8 : 599. y y z z + cos + sin, sin + cos. cos sin cos sin. 0.

380 0600.4. 6 503 ﾀ00 07 00050603 60. 7 0805 ｧ 0400030 閨04 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(; 3); B(4;,) ｧ C(6; 5): 603. 6ｧ80607 05 00 ｧ 0 03 閨00308 04 縺0700308 0503 09 ｧ A(; 3); B(; 4) ｧ C(3; 5)? 604. 7 04 ｧ ﾀ 0504 ﾀ 040300300406 0304 縺06 閨0600 ｧ05060007 05 縺060504060503 0403 縺07 閨09 06 04-0060 ｧ06 04 縺0700308, 04 縺030803 閨0700608 06 縺06 0503 09 ｧ: ) (x ; y ) ｧ (x ; y ); ) (; 3) ｧ (,; 5). ｰ04 縺03 ﾀ05 ｧ0504 ｧ 0403 ｧ ﾀ00 ｧ0504 0304 縺06 閨0600 ｧ050600 ｧ: 605. 607.,3 6,6, ax a + x ay a + y az a + z : 606. m + a m, a a n + a n, a a a, a a. sin 3 cos 3 sin cos sin cos : 608. ｰ 09 0 ｧ 06. 6 5 06 607 0403006 ﾀ05 ｧ a 0 09 0304 縺06 閨0600 ｧ05060007, 05060 ｧ 0406 縺040308 ｧ 040503 縺0308 ﾀ05 縺0309 0403 06 ﾀ0504 05 縺06050406 ｧ 0403006 ﾀ05 ｧ (x, z) ｧ (y, z) 0 09 0304 縺06 閨0600 ｧ05060007. 609. 6 70309 0504, 0503 x + x y + y x, x y, y x y 60. 7 0805 ｧ x ｧ 06 040060 ｧ08: ) x 4 9 x 3 =0; ) x 3 x, 0 4 =0 x y : = x y ｧ 04 縺030406 罍ｧ0504 0403 陦ﾀ05 00304090308 0903 縺00608 04 0304 縺06 閨0600 ｧ05060004. x. ｮ ｧ ﾀ0506003 00 ｧ0060800308 06 040060 ｧ08. ｮ ｧ ﾀ05060 閨040608 00 ｧ0060800308 06 040060 ｧ08 ﾀ 閨0406007 006 ｧ - 0406 ﾀ050030 ｧ a x + b y = c ; a x + b y = c ().

3x. ｮ ｧ ﾀ0506003 00 ｧ0060800308 06 040060 ｧ08 8 ｧ0060605 縺0600060 ｧ06 c b c b a c a x = a c ; y = b a b a () b a b 04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 0304 縺06 閨0600 ｧ05060004 ﾀ ｧ ﾀ0506003 = a b a b 6= 0.. ｮ ｧ ﾀ05060 閨040608 03 閨003 縺03 閨00308 00 ｧ0060800308 06 040060 ｧ08 ﾀ 05 縺06007 006 ｧ 0406 ﾀ050030 ｧ a x + b y + c z =0; a x + b y + c z =0 ｧ0060605 縺0600060 ｧ07, 0304 縺06 閨0600070600306 0703 縺00600 0 ｧ x = k b c a b c ; y =,k c a a c ; z = k a b b ; (4) 05 閨06 k 04 縺03 ｧ 0403000400306 ｧ ﾀ0003. 3. ｮ ｧ ﾀ05060 05 縺0608 03 閨003 縺03 閨00308 00 ｧ0060800308 06 040060 ｧ08 ﾀ 05 縺06007 006 ｧ 0406 ﾀ050030 ｧ a x + b y + c z =0; a x + b y + c z =0; a 3 x + b 3 y + c 3 z =0 ｧ0060605 030500 ｧ 00306 0305 0 縺0600060 ｧ07, 06 ﾀ00 ｧ 0304 縺06 閨0600 ｧ05060004 ﾀ ｧ ﾀ0506003 a b c = a b c a 3 b 3 c 3 =0; ｧ 0303 05003. 4. ｮ ｧ ﾀ05060 05 縺0608 00 ｧ0060800308 06 040060 ｧ08 ﾀ 閨0406007 006 ｧ - 0406 ﾀ050030 ｧ a x + b y = c ; a x + b y = c ; (6) a 3 x + b 3 y = c 3 a b c ﾀ0304006 ﾀ050, 090305 = a b c =0 ｧ ﾀ ｧ ﾀ05060 006 ﾀ03 閨06 縺07 ｧ05 040304 縺003 a 3 b 3 c 3 04 縺0305 ｧ0403 縺06 ｧ040308 06 040060 ｧ08. 5. ｮ ｧ ﾀ05060 05 縺0608 00 ｧ0060800308 06 040060 ｧ08 ﾀ 05 縺06007 006 ｧ - 0406 ﾀ050030 ｧ (3) (5) a x + b y + c z = d ; a x + b y + c z = d ; a 3 x + b 3 y + c 3 z = d 3 (7)

38 0600.4. 6 503 ﾀ00 07 00050603 04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 0304 縺06 閨0600 ｧ05060004 ﾀ ｧ ﾀ0506003 a b c = a b c a 3 b 3 c 3 6=0; ｧ0060605 ﾀ0006 閨060600606 06 陦ｧ0 ﾀ050406000306 縺0600060 ｧ06: 05 閨06 x = x ; y = y ; z = z ; (8) d b c a x = d b c d 3 b 3 c 3 ; d c a y = a d c a 3 d 3 c 3 ; b d z = a b d a 3 b 3 d 3 : 6. 706 ﾀ0304006 ﾀ0500306 ｧ 0060304 縺06 閨060006000306 ﾀ ｧ ﾀ0506003. 050303 0 - ｧ0 0006040306 ﾀ05 ｧ 06 040060 ｧ08 (7) 06 縺06 X, X ｧ X 3. 706 ﾀ0504 0304 縺06 閨0600 ｧ05060004 ﾀ ｧ ﾀ0506003 (7) =0. 7 罍ｧ 0505030 0403 00307003 閨04 04 縺06 閨0403000307060 ｧ07. I. 7000600600503 閨040608 ﾀ05 縺0309 0304 縺06 閨0600 ｧ05060007 04 縺030403 縺09 ｧ030 0004003, 0 04 罍ｧ006 a = b = c = m. ｯ0305 X = mx ｧ: a b c ) 06 ﾀ00 ｧ d 6= md,0503 ﾀ ｧ ﾀ05060 006 ﾀ0304006 ﾀ050 (0406 縺040306 閨04 06 040060 ｧ07 04 縺0305 ｧ0403 縺06 ｧ0403); ) 06 ﾀ00 ｧ d = md,0503 ﾀ ｧ ﾀ05060 0060304 縺06 閨06000600 (06 ﾀ00 ｧ 0406 縺040306 ｧ 05 縺06050406 06 040060 ｧ07 006 04 縺0305 ｧ0403 縺06 ｧ0403). II. 6 5 0304 縺06 閨0600 ｧ05060006 00605 ﾀ05 縺0309 ﾀ 04 縺030403 縺09 ｧ030 00040030 ｧ 050006006005 0 ｧ. ｯ0305 ﾀ06006 ﾀ0504060605 030500 ｧ 00306 0305 0 ｧ ﾀ00 m ｧ n 05 09 ｧ06, 04 罍ｧ 09030503 縺0308 mx + + nx = X 3, ｧ: ) 06 ﾀ00 ｧ md + nd 6= d 3,0503 ﾀ ｧ ﾀ05060 006 ﾀ0304006 ﾀ050 ; ) 06 ﾀ00 ｧ md + nd = d 3,0503 ﾀ ｧ ﾀ05060 0060304 縺06 閨06000600. ｪ ｧ ﾀ00 m ｧ n 00307003 0403 閨0303 0504 ｧ00 ｧ 0706 0 0805 ｧ ｧ08 ｧ 06 040060 ｧ08 a m + + a n = a 3, b m + b n = b 3, c m + c n = c 3. 70600 ｧ0504 ﾀ 040300300406 0304 縺06 閨0600 ｧ0506000608 ﾀ ｧ ﾀ0506003 06 040060 ｧ08: n n 3x +y =7; ax, 3y =; 6. 6. 4x, 5y =40: ax, y =: n 5x +y =4; mx, ny =(m, n) ; 63. 64. 7x +4y =8: x, y = n (04 罍ｧ m 6= n): 70600 ｧ0504 ﾀ ｧ ﾀ0506003 06 040060 ｧ08: ( x, 3y + z, =0; 65. x +5y, 4z +5=0; 4x + y, 3z +4=0: n x, 5y +z =0; 67. x +4y, 3z =0: 66. 68. ( x, 4y +3z =; x, y +4z =3; 3x, y +5z =: ( 3x +y, z =0; x, y +3z =0; x +3y, 4z =0;

369. 6. ( 3x +y, z =0; x, y +3z =0; x + y, z =0: ( x +y +3z =4; x + y, z =3; 3x +3y +z =7: x 3. 0703004000609 ﾀ00306 ｧ ﾀ00 83 ( x +y +3z =4; 60. x +4y +6z =3; 3x + y, z =: 6. ( x +y +3z = 4; x + y, z = 3; 3x +3y +z =0: 63. 706 縺06 ﾀ0609 0605 ﾀ07 00 ｧ 04 03 閨00308 0503 0906 04 縺0700306: ) x, 3y =6, 3x + y =9, x +4y =3; ) x, 3y =6, x +y =4, x, 5y =5? 6 5030403000 ｧ0504 04 030303 ｧ08 ﾀ0006 0708 0403 ﾀ05 縺03060 ｧ06. 70600 ｧ0504 ﾀ ｧ ﾀ0506003 00 ｧ0060800308 06 040060 ｧ08: ( x, y + z = ; 64. 3x +y +z =,; 65. x, y + z = : 66. 68. n 3x +y +z =0; 5x +y +3z =0: ( x, y +3z =0; x +y, 5z =0; 3x + y, z =0: 67. 69. x 3. 0703004000609 ﾀ00306 ｧ ﾀ00 ( x +y, 3z =5; x, y, z =; x +3y +4z =6: ( 3x, y +z =0; x +3y, 5z =0; x + y + z =0: ( x, y + z = 4; x +3y, z = 3; 4x, y + z =:. 0504 縺06 閨0600060 ｧ07. 0703004000609 ﾀ0030 ｧ ﾀ00030 0 0304 0605 ﾀ07 0403 07060 ｧ06 04 ｧ x + yi, 0409030503 縺030 x ｧ y 0406006 ﾀ050406000306 ｧ ﾀ00, i 00609030503 縺0308 ﾀ ｧ0040300, 06 ﾀ00 ｧ 04 罍ｧ 0505030 04 罍ｧ0070503 06 ﾀ000304 ｧ07: ) x +0i = x, 0+yi = yi ｧ i = i, (,)i =,i; ) x + yi = x + y i 050305 ｧ 050300040903 050305, 090305 x = x ｧ y = y ; 3) (x + yi) +(x + y i)=(x + x )+(y + y )i; 4) (x + yi)(x + y i)=(xx, yy )+(xy + x y)i. 0ｷ3 06 ﾀ000304 ｧ08 ) ｧ 4) 04030006 0605 ﾀ07 ﾀ050604060 ｧ ｧ ﾀ00 i: i =,; i 3 =,i; i 4 =; i 5 = i ｧ05.. () 0703004000609 ﾀ00306 ｧ ﾀ0003 x + yi, 0409030503 縺030 y 6= 0,0 0304 0605 ﾀ07 00 ｧ0030 ｧ ﾀ00030. ｪ ｧ ﾀ0003 i 0 0304 0605 ﾀ07 00 ｧ00308 06 陦ｧ0 ｧ090608.. 6 70608 ﾀ0504 ｧ07 0 0903004000609 ﾀ0030 ｧ ｧ ﾀ00 0 ｧ. ｮ000307060 ｧ06, 0403 ｧ05 0 ｧ06, 06000307060 ｧ06 ｧ 0403 0406 閨060 ｧ06 04 ﾀ05060406004 0903004000609 ﾀ00308 ｧ ﾀ0600 00307003 04030403000070504 0403 04 04 ｧ00 0 0505 ｧ08 閨0608 ﾀ0504 ｧ08 0 00030503 00060 0 ｧ ﾀ 00600308 ﾀ0506-040600608 ｧ ﾀ00 i 0403 0703 縺00600 0 (). 6 70600060 ｧ06 0903004000609 ﾀ00308 ｧ ﾀ0600 ｧ ｧ 040006 060 ｧ06 0903 縺007 ｧ 0903004000609 ﾀ0030503 ｧ- ﾀ00 0304 縺06 閨0600070605 ﾀ07 09 09 閨0608 ﾀ0504 ｧ07 0303 0500306.

384 0600.4. 6 503 ﾀ00 07 00050603 3. ｯ 罍ｧ050300300605 罍ｧ 06 ﾀ09 07 0703 縺0 0903004000609 ﾀ0030503 ｧ ﾀ00. 0703004000609 ﾀ00306 ｧ ﾀ0003 x+yi 0304 縺06 閨0600070605 ﾀ07 04 縺0308 0406006 ﾀ050406000308 ｧ ﾀ0600 (x; y) ｧ 0403050503006 ｧ 0303 07 0605 ﾀ07 0503 090308 M (x; y) 040003 ﾀ0903 ﾀ05 ｧ ｧ00 ｧ 0606 陦ｧ06 ﾀ-0406090503- 縺030,,! r = OM ( ﾀ0. 罍ｧ ﾀ. ). 6 700 ｧ0 0505030503 0406090503 r = x + y 0 0304-0605 ﾀ07 003 閨0600060 0903004000609 ﾀ0030503 ｧ ﾀ00, 06050300 060503 ' ﾀ03 ﾀ0406 Ox 0 0304 0605 ﾀ07 縺0506006005030 0903004000609 ﾀ0030503 ｧ ﾀ00. ｯ 09 09 09 x = r cos ', y = r sin ', 0503 x + yi = r(cos ' + i sin '): () 4. 6 70608 ﾀ0504 ｧ07 0 0903004000609 ﾀ0030 ｧ ｧ ﾀ00 0 ｧ 04 05 罍ｧ0503003-00605 罍ｧ 06 ﾀ090308 0703 縺006: r(cos ' + i sin ')r (cos ' + i sin ' )= =(rr ) cos (' + ' )+i sin (' + ' ) ; (3) r(cos ' + i sin ') r (cos ' + i sin ' ) = r cos (', ' )+isin (', ' ) ; r (4) n r(cos ' + i sin ') = r n (cos n' + i sin n'); (5) n r(cos ' + i sin ') = n r cos ' +k + i sin ' +k ; (6) n n 05 閨06 k =0; ; ; :::;n,. ｱ03 縺0060003 (5) ｧ (6) 0 0304 0605 ﾀ07 0703 縺00600 0 ｧ 0306 04. 5. ｱ03 縺00600 7080006 : 6. 6ｧ80305 罍ｧ070 0903004000609 ﾀ0030503 ｧ ﾀ00 : e i' = cos ' + i sin ': (7) ln z =lnr + i' 0 +ki; (8) 05 閨06 ' 0 0 060 ｧ06 縺0506006005 ', 06 閨03040006050403 縺070600606 006 04060 ﾀ0504 0, < <'6. 6 503 07060 ｧ06 ln r + i' 0 0 0304 0605 ﾀ07 0500 040030 0 060 ｧ060 000305 - 罍ｧ070. 630. 6 5030403000 ｧ0504 閨0608 ﾀ0504 ｧ07: ) (+3i)(3, i); ) (a + bi)(a, bi); 3) (3, i) ; 4) ( + i) 3 ; 5) +i, i ; 6) i +i. 63. 70600 ｧ0504 06 040060 ｧ07: ) x +5= 0; ) x, x +5= 0; 3) x +4x +3= 0 ｧ 04 縺030406 罍ｧ0504 0403 陦ﾀ05 00304090308 0903 縺00608 04 06 040060 ｧ06. ｮ0006 閨06060 ｧ06 0903004000609 ﾀ00306 ｧ ﾀ00 ｧ 0303 ｧ0504 0406090503 0 ｧ, 0304 縺06 閨06-00 ｧ0504 ｧ08 003 閨0600 ｧ ｧ 縺050600600503 ｧ 04 ｧ ﾀ 0504 04 05 罍ｧ050300300605 罍ｧ 06 ﾀ090308 0703 縺006: 63. ) z =3; ) z =,; 3) z =3i; 4) z =,i. 633. ) z =, i; ) z =+i 3; 3) z =, 3, i.

3x 3. 0703004000609 ﾀ00306 ｧ ﾀ00 85 634. ), +i ; ) sin + i(, cos ). 635. ｪ ｧ ﾀ00, 000306 04 08 63{634, 04 ｧ ﾀ 0504 04 0703 縺006 re 'i (04 罍ｧ, <'6 ). 636. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ05 ｧ 0503 0609 z 0403 06 ﾀ000304 ｧ070: ) jzj < 3; ) jzj < ｧ = <'<; 3) < jzj < 4 ｧ, <'<,=. 637. 70309 0504, 0503 jz, z j 06 ﾀ0504 ﾀ ﾀ0503070 ｧ06 00607 閨06 0503 09 0 ｧ z ｧ z. 638. 6 7 0 0503 09 z 0 =, +3i. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ0504 0503 0609 z, 閨0007 09030503 縺0308 jz, z 0 j <. 639. ｪ ｧ ﾀ0003, ﾀ0304 縺070706000306 ﾀ z, 030303 0 0605 ﾀ07 06 縺06 z. 6 70309 0504, 0503 z z = jzj. 640. 6 503 ｧ ﾀ00 ｧ0504 04030703 縺0060006 0306 04 : ) ( + i) 0 ; ) (, i 3) 6 ; 3) (, +i) 5 ; 4) + cos 4 + i sin 4 4 ; 5), 3+i 3. 64. 6 503 ｧ0504 sin 3 ｧ cos 3 06 縺06 070600909 ｧ ｧ 060500, ｧ ﾀ04030004 0607 050307 閨06 ﾀ050403 (cos + i sin ) 3 = cos 3 + i sin 3. 64. 7 0805 ｧ 04 ﾀ06 0 060 ｧ07 z = 6 ｧ ｧ 0303 ｧ0504 ｧ08 陦ｧ06 ﾀ- 0406090503 0 ｧ, 0403 ﾀ05 縺03 ｧ04 09 縺0605 陦ｧ06 ﾀ, 040030503. 643. 7 0805 ｧ: ) 3 ; ) 3 i; 3) 6,; 4) 3,+i. 644. 7 0805 ｧ: ) i; ) 3,+i; 3) 4,8+8i 3. 645. 70600 ｧ0504 閨0406 0006000306 06 040060 ｧ07: ) x 3 +8=0;)x 4 +4 = 0. 646. 7 0805 ｧ 0500 0400306 0 060 ｧ06 000305 罍ｧ070 : ) ln (,); )ln ( + i); 3) ln i; 4) ln (x + yi); 5) ln (, i). 647. 7 0805 ｧ ﾀ060006 sin x + sin x + sin 3x + :::+ sin nx. ｰ09 0 ｧ06. 703 0703 縺0060006 7080006 0060 ｧ0504 sin x = exi, e,xi 648. 7 0805 ｧ ﾀ060006 cos x + cos x + cos 3x + :::+ cos nx. 649. 6 70309 0504 050307 閨06 ﾀ050403 x 5, =(x, )(x, x cos 7 + )(x, x cos 44 +): i ｧ05.. 650. 6 503 ｧ ﾀ00 ｧ0504: ) 4, 3i 4+3i ; ) (a + bi)3, (a, bi) 3. ｮ0006 閨06060 ｧ06 0903004000609 ﾀ00306 ｧ ﾀ00 ｧ 0303 ｧ0504 0406090503 0 ｧ, 0304 縺06 閨06-00 ｧ0504 ｧ08 003 閨0600 ｧ ｧ 縺050600600503 ｧ 04 ｧ ﾀ 0504 04 05 罍ｧ050300300605 罍ｧ 06 ﾀ090308 0703 縺006 ｧ 04 0703 縺006 re 'i (04 罍ｧ, <'6 ):

386 0600.4. 6 503 ﾀ00 07 00050603 65. ) z =4+4i; ) z =, +i 3; 3) z =, i. 65. ) z =5; ) z =,i; 3) z =,,,. 653. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ0504 0503 0609 z 0403 06 ﾀ000304 ｧ070 < jzj < 3 ｧ =4 <'<3=4: 654. 6 7 0 0503 09 z 0 = 3, 4i. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ0504 0503 0609 z, 閨0007 09030503 縺0308 jz, z 0 j < 5. 655. 6 503 ｧ ﾀ00 ｧ0504 04030703 縺0060006 0306 04 : ) (, i) 6 ; ) ( + i ) 5 ; 3) + cos 3 + i sin 3 6. 656. 6 503 ｧ0504 sin 4 ｧ cos 4 06 縺06 070600909 ｧ ｧ 060500, ｧ ﾀ04030004 0607 050307 閨06 ﾀ050403 (cos + i sin ) 4 = cos 4 + i sin 4. 657. 7 0805 ｧ 04 ﾀ06 0 060 ｧ07 0903 縺00608: ) 4,; ) 5 ｧ ｧ 0303 ｧ0504 ｧ08 陦ｧ06 ﾀ-0406090503 0 ｧ. 658. 70600 ｧ0504 06 040060 ｧ07: ) x 3, 8=0; ) x 6 +64=0; 3) x 4, 8 = 0. 659. 7 0805 ｧ ﾀ060006 ( ﾀ0. 06 647) cos x + cos 3x + cos 5x + :::+ cos (n, )x: x 4. ｰ 040060 ｧ07 0403 ﾀ00 ｧ08 ﾀ0506040600608 ｧ 04 罍ｧ0300 ｧ0706000306 縺0600060 ｧ06 06 040060 ｧ08. 070603 ｧ 06 ﾀ090306 06 040060 ｧ06: x 3 + ax + bx + c =0: () ｭ ﾀ00 ｧ x, x, x 3 0903 縺0 ｧ 06 040060 ｧ07 (), 0503 06 040060 ｧ06 00307003 04 ｧ ﾀ 0504 04 04 ｧ 閨06 (x, x )(x, x )(x, x 3 ) = 0. 0505 ﾀ06 a =,(x + x + x 3 ), b = x x + x x 3 + x x 3, c =,x x x 3. ｰ 040060 ｧ06 x 3 + ax + bx + c =004 罍ｧ0403 陦ｧ05 ﾀ07 09 04 ｧ 閨06 z 3 + z + q =0 0403 陦ﾀ05 00304090308 x = z, a 3. ｰ 040060 ｧ06 z3 + z + q =0 縺0600 0605 ﾀ07 0403 0703 縺0060006 07 罍 003: s r z = 3, q + q 4 + 3 7 + 3 s, q, r q 4 + 3 7 = u + v: I. ｭ ﾀ00 ｧ = q 4 + 3 7 > 0, 0503 z = u + v, z ; 3 =, u + v i 3,05 閨06 u ｧ v 0406006 ﾀ050406000306 0 060 ｧ07 0903 縺00608 u ｧ v. u, v II. ｭ ﾀ00 ｧ = q 4 + 3 7 =0,0503z = 3q, z = z 3 =, 3t =, z.

3x 4. ｰ 040060 ｧ07 0403 ﾀ00 ｧ08 ﾀ0506040600608 87 r III. ｭ ﾀ00 ｧ = q 4 + 3 7 < 0, 0503z =, r 3 cos ', 3 z ; 3 =, 3 ' cos,05 閨06 3 0 cos ' =, q r, 3. 7.0505 閨0600060 ｧ06 0406006 ﾀ05040600030503 0903 縺007 06 040060 ｧ07 f(x) = =0. 030607 閨06 a ｧ b ﾀ03 閨06 縺07 ｧ05 ﾀ07 06 陦ｧ0 ﾀ050406000308 0903 縺06004 06 040060 ｧ07 f(x) =0, 06 ﾀ00 ｧ f(a) ｧ f(b) ｧ0060605 00306 0 09 ｧ, f(x) 00604 縺06 縺03040 0 0305 縺06 0906 [a; b] ｧ0400605 罍ｧ 0060503 ｧ0060605 04 縺03 ｧ 0403 閨00606 f 0 (x) 6= 0. 706 閨060 ﾀ ｧ05 0504 06006, 0503 0 0505030 0305 縺06 0906 ｧ f 00 (x) 6= 0. 3. ｮ0403 ﾀ0303 0803 罍 04 罍ｧ0300 ｧ070600030503 縺0600060 ｧ07 06 040060 ｧ07 f(x) =0. 706 ﾀ0504 0 050305 ｧ 09030090304 0305 縺06 09 [a; b], 0305 閨060007060060503 0903 縺06004, 7 ｧ ﾀ. 3 0 09030503 縺030 f( 0 )f 00 ( 0 ) < 0. ｯ0305 04 罍ｧ0300 ｧ07060 ｧ060 09 0903 縺006 x 0306 閨0605 0503 09 0406 縺06 ﾀ06 060 ｧ07 ﾀ Ox 0803 罍閨03 AB ( 罍ｧ ﾀ. 3): = 0, f( 0) ; k f(b), f(a) 05 閨06 k =. b, a 4. ｮ0403 ﾀ0303 09 ﾀ 0506000400308 ( ﾀ0403 ﾀ0303 7040605030 ). 706 ﾀ0504 0 050305 ｧ 09030090304 0305 縺06 09 [a; b], 0 09030503 縺030 f( 0 )f 00 ( 0 ) > 0. ｯ0305 04 罍ｧ0300 ｧ07060 ｧ060 09 0903 縺006 x 0306 閨0605 0503 09 0406 縺06 ﾀ06 060 ｧ07 ﾀ Ox 09 ﾀ 0506000400308 0909 罍ｧ040308 y = f(x) 040503 0906 [ 0 ;f( 0 )] ( 罍ｧ ﾀ. 3): = 0, f( 0) k ; 05 閨06 k = f 0 ( 0 ). 7 罍ｧ00600707 0403040503 縺003 ﾀ0403 ﾀ0303 0803 罍陦ｧ09 ﾀ 0506000400308, 00307003 ﾀ03 ﾀ05 04 ｧ0504 05 03-00 ｧ0906 j j f() j f() j k j k j j j :: j :: j :: j :: j :: j :: j :: j :: j ; ()

388 0600.4. 6 503 ﾀ00 07 00050603 05 閨06 k ｧ k 0 090003003 0803 罍 ｧ 09 ﾀ 0506000400308, =, f() ｧ =, f() : k k 703 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ 04030006 0600308 04 05 0300 ｧ0906 () 0 060 ｧ08 ｧ ﾀ0803- 閨0705 ﾀ07 09 ｧ ﾀ0903003006 0903 縺006. 5. ｮ0403 ﾀ0303 ｧ0506 09 ｧ08. ｭ ﾀ00 ｧ 06 040060 ｧ06 f(x) =000307003 04 罍ｧ0406 ﾀ05 ｧ 09 04 ｧ 閨06 x = '(x), 04 罍ｧ 060 04 00609030503 縺0308 0309 縺06 ﾀ05003 ﾀ05 ｧ 0903 縺007 j' 0 (x)j < 0 6 ｧ x 0 0006030306 ｧ ﾀ0003 04 05050308 0309 縺06 ﾀ05003 ﾀ05 ｧ, 0503 ﾀ0803 閨070 07 ﾀ07 0403 ﾀ0006 閨0304 05060004003 ﾀ0504 04 罍ｧ0300 ｧ0706000308 縺0600060 ｧ08 0306 閨0605 x = '(x 0 ); x = '(x ); x 3 = '(x );::: 6 5 06 040060 ｧ0708 660, 66 ﾀ 縺06 陦ｧ 0906000308 000307 ｧ0506000608 ﾀ04030303-0030503 00060 0403 閨0303 0504 03 陦ｧ0 0903 縺06004, 閨0600 ｧ0504 0006040606 ﾀ0504 0 x,x ｧ 05060 0 0805 ｧ 03 ﾀ05 000400306 0903 縺0 ｧ: 660. ) x 3, 4x + x +6= 0; ) x 3, 4x, 4x, 5=0. 70600060 ｧ06 04 縺030406 罍ｧ0504 ﾀ03 ﾀ05 0400060 ｧ060 0403 07060 ｧ08 x + x + x 3 ; x x + x x 3 + x x 3 ; x x x 3 : 66. ) x 3, 5x, x +4=0; ) x 4 + x 3 +x, 4=0; 3) 9x 3 +8x, x, =0; 4) 4x 3, 4x + x, =0. 70600 ｧ0504 0403 0703 縺0060006 07 罍 003 ﾀ0006 閨06060 ｧ06 06 040060 ｧ07: 66. ) z 3, 6z, 9=0; ) z 3, z, 6 = 0. 663. ) z 3, z, 8=0; ) z 3 +6z, 7=0. 664. x 3 +9x +8x +9=0. 665. 6 7 003 06 040060 ｧ06 f (x) =x 4, x, 0 = 0. ｮ03 ﾀ05 04 ｧ04 05 0300 ｧ0906 0 090304 f (x) 04 罍ｧ x =0; ; ; :::, 0304 縺06 閨0600 ｧ0504 05 0 ｧ0903 0403000307 ｧ05060004-0030503 0903 縺007 ｧ 0403 ｧ ﾀ00 ｧ0504 060503 ﾀ 0503 003 ﾀ050406 閨03 0,0 0403 ﾀ0403 ﾀ030306 0803 罍 ｧ 09 ﾀ 0506000400308. 666. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ08 y = x3, 0304 縺06 閨0600 ｧ0504 05 07 ｧ 06-3 ﾀ09 ｧ 05 0 ｧ0903 0903 縺00608 06 040060 ｧ07 x 3, 6x +3 = 0 ｧ 0403 ｧ ﾀ00 ｧ0504 0903 縺0 ｧ ﾀ 0503 003 ﾀ050406 閨03 06 陦ｧ0 ｧ0903 05 縺060504060503 0 09. 667. 703 ﾀ0403 ﾀ030306 ｧ0506 09 ｧ08 (0403 ﾀ0006 閨0304 0506000400308 04 罍ｧ0300 ｧ07060 ｧ08) 0 0805 ｧ 0406006 ﾀ050406000306 0903 縺0 ｧ 06 040060 ｧ08: ) x 3 +60x, 80 = 0; ) x =4x; 3) x 3 + l x + l 3 =0; 4) x 4, x, =0. 668. 703 閨0303 縺030 03 閨0030503 0903 縺007 ﾀ 縺06 陦ｧ 0906000308 000307 ｧ0506000608 ﾀ04030303-0030503 00060 縺0600 ｧ0504 06 040060 ｧ07: ) x 3 +8x +5x +8=0; ) x 3, 3x +4=0. 6 70007 04 縺030406 縺09 ｧ ﾀ03 ﾀ05 04 ｧ0504 0403 07060 ｧ07 P x i, P x i x j ｧ x x x 3.

3x 4. ｰ 040060 ｧ07 0403 ﾀ00 ｧ08 ﾀ0506040600608 89 669. 703 0703 縺0060006 07 罍 003 縺0600 ｧ0504 06 040060 ｧ07: ) z 3 +8z, 9=0; ) z 3, 6z, 4=0; 3) z 3, 3z +=0; 4) x 3 +6x +9x +4=0: 670. 703 ﾀ05 縺03 ｧ04 05 07 ｧ09 070600909 ｧ ｧ y = x4, 0304 縺06 閨0600 ｧ0504 05 0 ｧ0903 5 0903 縺00608 06 040060 ｧ07 x 4 +3x,5 = 0 ｧ 0403 ｧ ﾀ00 ｧ0504 0903 縺0 ｧ ﾀ 0503 003 ﾀ050406 閨03 0,0. 67. 7 0805 ｧ ﾀ0503 003 ﾀ050406 閨03 0,0 0403000307 ｧ0506000400306 0903 縺0 ｧ 06 04006-0 ｧ08: ) x 3 +50x, 60 = 0; ) x 3 + x, 3 = 0. 67. 703 ﾀ0403 ﾀ030306 ｧ0506 09 ｧ08 0 0805 ｧ 0406006 ﾀ050406000308 0903 縺06004 06 04006-0 ｧ07 x 3 +x, 8=0, 0403 ｧ ﾀ000707 0403 ﾀ0006 閨0304 0506000400306 04 罍ｧ0300 ｧ07060 ｧ07 0403 0703 縺0060006 x = 3 8, x.

30600 04 5 6 56 5 ｭ6 7 ｭ 70ｷ3 ｭ 6 5 繖 7 縺6ｧ80ｷ36ｦ0 x. 706 縺06006000306 040600 ｧ ｧ003 ｧ 070600909 ｧ ｧ. 0505 縺06 09 ｧ ｧ ｧ00506 縺04 0003. 030030706 ﾀ050403 ｧ ﾀ0600 x, 06 閨03040006050403 縺0706-0 ｧ08 006 04060 ﾀ0504 0 a<x<b,0 0304 0605 ﾀ07 ｧ00506 縺04 00030 ｧ030303 0 0605 ﾀ07 (a; b). 030030706 ﾀ050403 ｧ ﾀ0600 x, 06 閨03040006050403 縺07060 ｧ08 006 04060 ﾀ0504 0 a 6 x 6 b, 0 0304 0605 ﾀ07 0305 縺06 09030 ｧ030303 0 0605 ﾀ07 [a; b]. 70904 ｧ04 000600500306 006 04060 ﾀ0504 (04 罍ｧ a>0) x <a ; ｧ00 ｧ jxj <a; ｧ00 ｧ, a<x<a 0304 縺06 閨0600070605 ｧ00506 縺04 00, ﾀ ｧ000605 罍ｧ 00308 0305003 ﾀ ｧ05060004003 0060007.. 706 縺06006000306 040600 ｧ ｧ003 ｧ 070600909 ｧ ｧ. ｭ ﾀ00 ｧ 09 07 閨03006 0-060 ｧ06 0406 縺06006000308 x 0403 ﾀ05 040006003 04 ﾀ030305040605 ﾀ0504 ｧ06 03 閨003 ｧ ﾀ0003, 0503 0406 縺060060-0 07 y, 0304 縺06 閨060007060 07 ﾀ030403090604003 ﾀ050406 0505 ｧ08 ｧ ﾀ0600, 0 0304 0605 ﾀ07 03 閨003 0 00308 070600909 ｧ0608 x. 706 縺0600600 07 x 0 0304 0605 ﾀ07 04 罍ｧ 0505030 縺0506006005030, 00 07 ﾀ030403090604003 ﾀ0504 0 060 ｧ08 縺0506006005 030300 ﾀ050406 0304 縺06 閨0600060 ｧ07 070600909 ｧ ｧ. ｯ03, 0503 y 06 ﾀ0504 070600909 ｧ07 x, ﾀ ｧ0040300 ｧ 06 ﾀ09 ｧ 04 ｧ ﾀ0304 0605 04 04 ｧ 閨06 y = = f(x), ｧ00 ｧ y = F (x), ｧ00 ｧ y = '(x) ｧ05.04. ｮ ｧ0040300 f(x) ｧ00 ｧ F (x) ｧ 05. 04. 030303 0 0605 09030 ﾀ030305040605 ﾀ0504 ｧ07 0406 縺06006000308 x ｧ y, 04 ﾀ05003 ﾀ05 ｧ, 030 003070605 03 0 0504 ﾀ030403090604003 ﾀ0504 閨0608 ﾀ0504 ｧ08 ｧ00 ｧ 030406 09 ｧ08, 09030503 縺0306 00607003 04030403000 ｧ0504 0 x, 05030303 04030006 ｧ0504 ﾀ030305040605 ﾀ0504060600606 0 060 ｧ06 y. 673. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ05 ｧ ｧ 0060060 ｧ07 0406 縺06006000308 x, 06 閨03040006050403 縺07-0600608 006 04060 ﾀ0504 0: ) jxj < 4; ) x 6 9; 3) jx, 4j < ; 4), <x, 3 6 ; 5) x > 9; 6) (x, ) 6 4. 674. 6ｦ0 04 ｧ ﾀ 0504 006 04060 ﾀ0504 0 ｧ ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ00506 縺04 0003 ｧ 006006-0 ｧ07 0406 縺06006000308: [,; 3]; (0; 4); [,; ]. 675. 0504 縺06 閨0600 ｧ0504 030300 ﾀ0504 ｧ 0060060 ｧ07 0406 縺06006000308 x =, t,05 閨06 t 04 罍ｧ0 ｧ0 0605 0006030306 0 060 ｧ06 6. 6 5 08 676 { 678 0403 ﾀ05 縺03 ｧ0504 04030503 09 0 0 0305 縺06 0906 jxj 6 3 05 07 ｧ09 ｧ 0609 000308 070600909 ｧ08: 676. ) y =x; ) y =x +; 3) y =x, : 677. ) y = x ; ) y = x +; 3) y = x, : 678. ) y = x3 x3 x3 ; ) y = +; 3) y = 3 3 3, :

3x. 706 縺06006000306 040600 ｧ ｧ003 ｧ 070600909 ｧ ｧ 9 679. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 ｧ 070600909 ｧ08: ) y = 6 x ;)y =x ;3)y = = log x. 07 090606 03 ﾀ0303060003 ﾀ0504 04 ﾀ0403000307060 ｧ ｧ 0505 ｧ08 09 罍ｧ040308 0305003- ﾀ ｧ05060004003 03 ﾀ0608 090303 罍陦ｧ0 05 00307003 00605 ｧ0504? 680. 703 ﾀ05 縺03 ｧ0504 0 03 閨0030 06 縺05060706 05 07 ｧ09 ｧ 070600909 ｧ08: ) y = = sin x; )y = cos x 0403 0503 09 0, 04 09030503 縺0308 y ｧ0060605 0 ｧ03030004000606, 0 ｧ006004000606 ｧ0060006040306 0 060 ｧ07. ｮ000307060 ｧ060 03 罍陦ｧ0 05 0505 ｧ08 09 罍ｧ040308 0403 ﾀ05 縺03 ｧ0504 0 05030 0706 06 縺05060706 05 07 ｧ09 070600909 ｧ ｧ y = sin x + cos x. 68. 7 0805 ｧ 0903 縺0 ｧ x ｧ x 070600909 ｧ ｧ y =4x, x ｧ 0403 ﾀ05 縺03 ｧ0504 0606 05 07 ｧ09 0 0305 縺06 0906 [x, ; x +]. 68. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 ｧ 070600909 ｧ08: ) y = jxj; ) y =,jx, j; 3) y = jxj,x. 6 5 08 683 { 686 0 0805 ｧ 030300 ﾀ05 ｧ 0304 縺06 閨0600060 ｧ07 0406006 ﾀ050406000308 0 060 ｧ08 070600909 ｧ08 ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ08 05 07 ｧ09 ｧ. 683. ) y = x +; ) y = 9, x ; 3) y = 4x, x. 684. ) y =,x + 4+x; ) y = arcsin x, : 685. ) y = x( x) ; ) y = x 4, x: 4 686. ) y =, sin x; ) y =, x 6, x : 687. ) f (x) =x, x +; 0403 ｧ ﾀ00 ｧ0504 f (0), f (), f (,), f (), f (a +); ) '(x) = x, 3 3 ; 0403 ｧ ﾀ00 ｧ0504 '(0), '(,), ', x + ', x. '(x) 688. F (x) =x ; 0403 ｧ ﾀ00 ｧ0504: F (b), F (a) a + h ) ; ) F, F b, a a, h f 689. f (x) =x, '(x) =x 3 (b), f (a) ; 0403 ｧ ﾀ00 ｧ0504 '(b),. '(a) 690. F (x; y) =x 3, 3xy, y ; 0403 ｧ ﾀ00 ｧ0504 F (4; 3) ｧ F (3; 4). 69. ｱ0600909 ｧ07 f (x) 0 0304 0605 ﾀ07 060500308, 06 ﾀ00 ｧ f (,x) =f (x); 006 060500308, 06 ﾀ00 ｧ f (,x) =,f (x). ｰ09 0504, 09 09 ｧ06 ｧ ﾀ0006 閨06060 ｧ08 070600909 ｧ08 060500306 ｧ09 09 ｧ06 006 060500306: ) f (x) = sin x x ; ) '(x) = ax, a ; 3) F x + (x) =ax + a ; x 4) (x) =a x, x ; 5) (x) =x sin x, x 3 ; 6) f (x) =x + x..

39 0600. 5. 6 50406 閨060 ｧ06 04 0 00 ｧ 69. ｮ06 縺06 陦ｧ0 0006030308 0803 罍閨03 05 07 ｧ09 00609030503 縺0308 070600909 ｧ ｧ f (x) 000607 ｧ05 04030006 05 07 ｧ09 05050308 070600909 ｧ ｧ. 6ｦ0 04 ｧ ﾀ 0504 050503 ﾀ040308 ﾀ050403 0706009-09 ｧ ｧ 006 04060 ﾀ0504030. 7 縺030406 罍ｧ0504, 0503 0505 ｧ0 ﾀ040308 ﾀ0504030 030300 0605 0706009-09 ｧ07 f (x) =x. 693. 07 09 07 ｧ 050006006005 縺00308 070600909 ｧ08 030300 0605 ﾀ040308 ﾀ0504 0 ｧ f () = 0, f (a) =, f (xy) =f (x)+f (y)? 694. 07 09 07 ｧ 050006006005 縺00308 070600909 ｧ08 030300 0605 ﾀ040308 ﾀ0504 0 ｧ f (0) =, f () = a, f (x + y) =f (x)f (y)? 695. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ05 ｧ ｧ 0060060 ｧ07 0406 縺06006000308 x, 06 閨03040006050403 縺07-0600608 006 04060 ﾀ0504 0: ) jxj < 3; ) x 6 4; 3) jx, j < ; 4) (x, ) 6 4. 696. 0504 縺06 閨0600 ｧ0504 030300 ﾀ0504 ｧ 0060060 ｧ07 0406 縺06006000308 x =+ t,05 閨06 t 04 罍ｧ0 ｧ0 0605 0006030306 0 060 ｧ06 6. 697. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 ｧ 070600909 ｧ08: ) y =4, x3 0 0305 縺06 0906 jxj 6 ; ) y =3; 5+3x, x 00607 閨060503 09 0 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ0406 03 ﾀ09 ｧ ﾀ ﾀ. 698. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 ｧ 070600909 ｧ08: ) y = x, 4+jx, j 0 0305 縺06 0906 [,; 5]; ) y =, cos x 0 0305 縺06 0906 jxj 6. 699. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 ｧ 070600909 ｧ08: ) y =, 4 x ; ) y =,x. 700. 7 0805 ｧ 030300 ﾀ05 ｧ 0304 縺06 閨0600060 ｧ07 0406006 ﾀ050406000308 0 060 ｧ08 0706009-09 ｧ08: ) y = 4, x ; ) y = x +, 3, x; 3) y =, cos x; 4) y = + ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ08 05 07 ｧ09 ｧ. 70. ) 6 70007 070600909 ｧ ｧ f (x) = 4 x, 4 x + x + 0403 ｧ ﾀ00 ｧ0504 f (0), f (,), f (,=), f (x, ), f (=); '(x ) 閨0007 070600909 ｧ ｧ '(x) =x 3 + h), '(x, h) 0403 ｧ ﾀ00 ｧ0504 ; h 3) 閨0007 070600909 ｧ ｧ f (x) =4x, x 0403 ｧ ﾀ00 ｧ0504 f (a +), f (a, ).

3x. 7 縺06 閨060003 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ ｧ 070600909 ｧ ｧ 93 x. 7 縺06 閨060003 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ ｧ 070600909 ｧ ｧ. 706 ﾀ0903006 003 0 000306 ｧ 0306 ﾀ0903006 003 0303000400 ｧ06. ｪ ｧ ﾀ000304 07 0403 ﾀ0006 閨0304 05060004003 ﾀ0504. 706 ﾀ0504 09 07 閨03006 0 0506-0004003006 ｧ ﾀ0006 n =; ; 3;::: 0403 00609030503 縺03006 0903006 0403 ﾀ05 040006003 04 ﾀ03-0305040605 ﾀ0504 ｧ06 ｧ ﾀ0003 x n. ｯ0305 05030403 縺0705, 0503 0505 ｧ0 0304 縺06 閨0600060 0403 ﾀ0006 閨0304-05060004003 ﾀ0504 ｧ ﾀ0600 x, x, x 3, ::: ｧ00 ｧ, 0903 縺03 06, 0403 ﾀ0006 閨0304 05060004003 ﾀ0504 fx n g = = fx ;x ;x 3 ; :::g. 0505 閨06000400306 ｧ ﾀ00 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ fx n g 0 0304-0605 ﾀ07 0606 050006006005 0 ｧ. 06030403 縺0705 06006, 0503 0406 縺0600600 07 x n 04 縺03030605 0605 0 06-0 ｧ06 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ fx n g.. 7 縺06 閨0600 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ (04 縺06 閨0600 0406 縺06006000308). ｪ ｧ ﾀ0003 a 0 0304 0605 ﾀ07 04 縺06 閨0600030 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ fx n g, ｧ00 ｧ 04 縺06 閨0600030 0406 縺06006000308 x n (030303 0 0605 ﾀ07 x n! a), 06 ﾀ00 ｧ 閨0007 04 ﾀ0709030503 ">0 0 08 閨0605 ﾀ07 04 ｧ ﾀ0700606 0305 " ｧ ﾀ0003 n 0 05 090306, 0503 jx n, aj <" 閨0007 04 ﾀ0608 0 0506 000400308 n>n 0. 0ｷ300506 縺04 00 (a, "; a + ") 0 0304 0605 ﾀ07 "-0309 縺06 ﾀ05003 ﾀ050406 ｧ ﾀ00 a ( ｧ00 ｧ 0503 09 ｧ a). ｯ 09 ｧ0 0303 030, x n! a 030303 0 0605, 0503 閨0007 04 ﾀ0709030503 ">0 0 08 閨0605 ﾀ07 05 090306 ｧ ﾀ0003 n 0, 0503 閨0007 04 ﾀ0608 n>n 0 ｧ ﾀ00 x n 0306 閨0605 0 0803 陦ｧ0504 ﾀ07 04 "-0309 縺06 ﾀ05003 ﾀ05 ｧ ｧ ﾀ00 a. 3. 7 縺06 閨0600 070600909 ｧ ｧ. 706 ﾀ0504 070600909 ｧ07 f(x) 0304 縺06 閨0600060 04 00609030503- 縺0308 "-0309 縺06 ﾀ05003 ﾀ05 ｧ 0503 09 ｧ a, ｧ ﾀ090006 060 ｧ060 03030504 003070605 ﾀ 00308 0503 09 ｧ a. 06030403 縺0705, 0503 ｧ ﾀ0003 b 070400070605 ﾀ07 04 縺06 閨0600030 070600909 ｧ ｧ f(x) 04 罍ｧ x! a (04 ｧ000605 f(x)! b 04 罍ｧ x! a ｧ00 ｧ lim f(x) =b), 06 ﾀ00 ｧ 閨0007 000603030503 ">0 ﾀ06006 ﾀ0504060605 x!a 04 ｧ ﾀ0700606 0305 " ｧ ﾀ0003 >0 05 090306, 0503 jf(x), bj <"04 罍ｧ 0 < jx, aj <. 縺0 000305 ｧ 003, lim f(x) = b, 06 ﾀ00 ｧ 閨0007 04 ﾀ0709030503 ">0 ﾀ06006 ﾀ0504060605 04 ｧ ﾀ07- x!a 00606 0305 " ｧ ﾀ0003 N 05 090306, 0503 jf(x), bj <"04 罍ｧ jxj >N. ｰ040305 縺060300070605 ﾀ07 05 090706 04 ｧ ﾀ04 lim x!a f(x) =, 09030503 07 030303 0 0605, 0503 閨0007 04 ﾀ0709030503 ｧ ﾀ00 A> 0 ﾀ06006 ﾀ0504060605 04 ｧ ﾀ0700606 0305 A ｧ ﾀ0003 05 090306, 0503 jf(x)j >A04 罍ｧ 0 < jx, aj <. ｭ ﾀ00 ｧ x! a ｧ 04 罍ｧ 0505030 x<a, 0503 04 ｧ000605 x! a, 0; 0 000305 ｧ 003, 06 ﾀ00 ｧ x! a ｧ 04 罍ｧ 0505030 x>a, 0503 04 ｧ000605 x! a +0. ｪ ｧ ﾀ00 f(a, 0) = = lim x!a,0 f(x) ｧ f(a +0)= lim x!a+0 f(x) 0 0304 0605 ﾀ07 ﾀ030305040605 ﾀ0504060003 04 縺06- 閨0600030 ﾀ000604 070600909 ｧ ｧ f(x) 04 0503 0906 a ｧ 04 縺06 閨0600030 ﾀ04 04 070600909 ｧ ｧ f(x) 04 0503 0906 a. 6 70007 ﾀ06006 ﾀ05040304 0 ｧ07 04 縺06 閨0600 070600909 ｧ ｧ f(x) 04 罍ｧ x! a 00603030803- 陦ｧ003 ｧ 閨03 ﾀ05 0503 003, 05030303 03030003 f(a, 0) = f(a +0). 6 5006 ﾀ0503 x! 0, 0 ｧ x! 0+004 ｧ000605 x!,0 ｧ x! +0 ﾀ030305040605 ﾀ0504060003. 4. 706 ﾀ0903006 003 0 000306. ｭ ﾀ00 ｧ lim (x) =0, 05. 06. 06 ﾀ00 ｧ j(x)j <" x!a 04 罍ｧ 0 < jx, aj <("), 0503070600909 ｧ07 (x) 0 0304 0605 ﾀ07 0306 ﾀ0903006 003 0 000308 04 罍ｧ x! a. 縺0 000305 ｧ 003 0304 縺06 閨0600070605 ﾀ07 0306 ﾀ0903006 003 0 00 07 (x) 04 罍ｧ x!. 5. 706 ﾀ0903006 003 0303000400 ｧ06. ｭ ﾀ00 ｧ 閨0007 000603030503 ﾀ09030004 060503 閨003 03030004-00030503 ｧ ﾀ00 N ﾀ06006 ﾀ0504060605 05 090306 (N ), 0503 04 罍ｧ 0 < jx, aj <(N ) 0403-040300006003 04060 ﾀ050403 jf(x)j >N, 0503 070600909 ｧ07 f(x) 0 0304 0605 ﾀ07 0306 ﾀ0903006 003 03030004000308 04 罍ｧ x! a. 縺0 000305 ｧ 003 0304 縺06 閨0600070605 ﾀ07 0306 ﾀ0903006 003 0303000400 07 f(x) 04 罍ｧ x!.

394 0600. 5. 6 50406 閨060 ｧ06 04 0 00 ｧ 70. 70300 05 07 n =0; ; ; 3; :::, 0 04 ｧ ﾀ 0504 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ 0 060 ｧ08 0406 縺06006000308: = ; =, n ; =, n : n 7 ｧ0 07 ﾀ 09 09030503 n 003 閨060004 09 07 閨0308 ｧ 0406 縺06006000308 ﾀ 閨0600 0605 ﾀ07 ｧ 0306- 閨0605 03 ﾀ05 04 0504 ﾀ07 0060040006 0,00, 0060040006 00030503 0403000307 ｧ050600040030503 "? 703. 7 04 ｧ ﾀ 0504 0403 ﾀ0006 閨0304 05060004003 ﾀ0504 0 060 ｧ08 0406 縺06006000308 x = =+ (,)n. 7 ｧ0 07 ﾀ 09 09030503 n 003 閨060004 003 ﾀ05 ｧ x, ﾀ 閨0600 0605 ﾀ07 ｧ n + 0306 閨0605 03 ﾀ05 04 0504 ﾀ07 0060040006 0,0, 0060040006 00030503 0403000307 ｧ050600040030503 "? 704. 7 罍ｧ03 04000707 09 3 ( ｧ00 ｧ 0403 ｧ05 07 ｧ 3) ﾀ0 00, 05060 0,, 040305030 0,0 ｧ 05.., 04 ｧ ﾀ 0504 08 閨06 ﾀ0705 ｧ 0030 ｧ09 0403 ﾀ0006 閨0304 05060004003 ﾀ05070 ｧ 04 罍ｧ0300 ｧ07060 ｧ07 0406 縺06006000308 09 04 縺06 閨060006: x n! 3+0, x n! 3, 0. 705. 6ｦ0 04 ｧ ﾀ 0504 08 閨06 ﾀ0705 ｧ 0030 ｧ09 0403 ﾀ0006 閨0304 05060004003 ﾀ05070 ｧ 04 罍ｧ0300 ｧ- 07060 ｧ07 0406 縺06006000308 09 04 縺06 閨0600 0: x n! 5+0, x n! 5, 0, x n!!, +0, x n!,, 0, x n! +0, x n!, 0, x n! ; +0, x n! ;, 0. 706. 6 70309 0504, 0503 lim x =4. 70307 ﾀ0 ｧ0504 05 0300 ｧ09 0 ｧ 0 060 ｧ08 x! x ｧ x. 707. 6 70309 0504, 0503 lim (x, )=5. 703 0003006 ｧ ﾀ0006 ">0 x!3 0 0805 ｧ 0 ｧ03030004000606 ｧ ﾀ0003 >0 05 090306, 05030303 04 罍ｧ 000603030 x ｧ -0309 縺06 ﾀ05003 ﾀ05 ｧ ｧ ﾀ00 3 0 060 ｧ06 070600909 ｧ ｧ y =x, 0309 0003 ﾀ04 04 "-0309 縺06 ﾀ05003 ﾀ05 ｧ ｧ ﾀ00 5. 70307 ﾀ0 ｧ0504 05 07 ｧ 06 ﾀ09 ｧ. 708. 6 70309 0504, 0503 lim x!, 000608-0309 縺06 ﾀ05003 ﾀ05 ｧ ｧ ﾀ00, 00607003 04 070504 0 060 ｧ06 x, 05030303 0-060 ｧ06 070600909 ｧ ｧ y =3, x, x 030500 ｧ 0003 ﾀ04 0305 0606 04 縺06 閨0600 0060040006 060 0 " =0; 000? 709. 6 70309 0504, 0503 sin 06 ﾀ0504 0306 ﾀ0903006 003 0 00 07 04 罍ｧ! 0. ｰ 09 0 ｧ 06. ｮ 閨0600 0504 06 縺050607 ｧ 040309 0504, 0503 j sin j < jj. 70. 6 70309 0504, 0503 lim sin x = sin a. x!a (3, x, x )=4. 0ｷ3 09 090308 0 ｧ03030004- ｰ09 0 ｧ06. 703000307 ｧ04 x = a +, ﾀ03 ﾀ05 04 ｧ0504 003 ﾀ0504 sin x, sin a ｧ 05060 0403000307 ｧ0504! 0. 7. 6 70309 0504, 0503 lim 060 ｧ08 x ｧ 3x +4 x 3x +4 =3. 70307 ﾀ0 ｧ0504 05 0300 ｧ09 0 ｧ 0 - x! x 04 罍ｧ x =; 0; 00; 000; ::: 4x, 3 7. 6 70309 0504, 0503 lim x! x + =. 7 罍ｧ 09 09 ｧ08 x 0 060 ｧ07 070600909 ｧ ｧ 0306 閨0605 030500 ｧ 0504 ﾀ07 0305 ﾀ0403060503 04 縺06 閨0600 0060040006 060 0 0,00?

3x. 7 縺06 閨060003 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ ｧ 070600909 ｧ ｧ 95, x 73. 6 70309 0504, 0503 lim =,0;5. 7 罍ｧ 09 09 ｧ08 x 0 060 ｧ07 x! +4x 070600909 ｧ ｧ 0306 閨0605 030500 ｧ 0504 ﾀ07 0305 ﾀ0403060503 04 縺06 閨0600 0060040006 060 0 0,0? 74. 6 70309 0504, 0503 lim 0; 333:::3 n! {z } =, ﾀ03 ﾀ05 04 ｧ04 003 ﾀ05 ｧ 3 3, n 0 090304, 0;3; 3, 0;33; 3, 0;333; :::;, 0; 333:::3 3 {z }. n 0 090304 75. 7 04 ｧ ﾀ 0504 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ: ) x n = n n + ; ) x n =, n n + ; 3) x n = (,) n n n + ; 8 cos n(=) n +(,)n 4) x n = ; 5) x n = ; n +4 n 6) x n =,n a cos n. ｮ06006 ﾀ0504060605 00 ｧ lim x n 0409 07 閨030 04 罍ｧ006 縺06 ｧ 06006 030 04060? n!+ 76. 7 0805 ｧ lim x!+0 3 x, ｧ lim x!,0 3 x, ｧ 040307 ﾀ0 ｧ0504 05 0300 ｧ09 0 ｧ. 77. 7 0805 ｧ lim =x ｧ lim =x ｧ 040307 ﾀ0 ｧ0504 05 0300 ｧ09 0 ｧ. x!0+0 x!0,0 78. 6 50307 ﾀ0 ｧ0504 0503 00308 ﾀ003 ﾀ00 0806 ﾀ0003040030809 04 ｧ ﾀ0608: ) =0; ) 0 = ; 3) 3 = ; 4) 3, =0; 5) lg 0 =,; 6) tg90 =. 79. 70309 0504, 0503 lim x! 04 05060004003 ﾀ05 ｧ 0 060 ｧ08 sin x: sin x 006 ﾀ06006 ﾀ0504060605, ﾀ03 ﾀ05 04 ｧ04 0403 ﾀ0006 閨03- ) 04 罍ｧ x = n; ) 04 罍ｧ x = +n; 3) 04 罍ｧ x =, +n (n =0; ; ; 3; 4; :::). 70. 70309 0504, 0503 lim sin x!0 x 006 ﾀ06006 ﾀ0504060605. 7. 70309 0504, 0503 lim x sin =004 罍ｧ 000603030 ﾀ0403 ﾀ030306 04 罍ｧ0300 ｧx!0 x 07060 ｧ07 x 090. 7. 6 509 縺0605 陦ｧ06 ﾀ R 0404 ｧ ﾀ 0 04 04 ｧ000400308 0003050306050300040 ｧ09 ﾀ ｧ ﾀ00030 ﾀ0503 縺030 n ｧ ﾀ0503 縺0300308 a n. 0504 ｧ ﾀ 04 0309030003 09 縺0605 0904 陦 05, 0403-09 0504, 0503 a n <",09 09 050300040903 n> 8R=", 05.06. a n! 0, 090305 n!. 73. 706 ﾀ0504 r n 040307060 04 04 ｧ00040030503, 0404 ｧ ﾀ 00030503 04 09 縺0605 n- 06050300040 ｧ09. 6 70309 0504, 0503 lim r n = R, 05 閨06 R 陦ｧ06 ﾀ 09 縺0605. n!

396 0600. 5. 6 50406 閨060 ｧ06 04 0 00 ｧ 74. 6 506 縺00 ｧ0 B 05 縺0606050300040 ｧ09 ABC 0406 縺060060 0605 ﾀ07 0403 04 縺0700308 BE k AC, 0060305 0 ｧ 060003 06 000707 ﾀ04 0404 0403. 07 09 0306 閨0605 04 罍ｧ 0505030 ｧ 0060070504 ﾀ07 ﾀ0503 縺03003 05 縺0606050300040 ｧ09, 060503 0400030 閨04, 0400605 縺0600 ｧ06 06050003 ｧ 04006000 ｧ08 06050300 BCD? 75. 7 04 ｧ ﾀ 0504 08 閨06 ﾀ0705 ｧ 0030609 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ 04 罍ｧ0300 ｧ07060 ｧ08 0406 縺06006000308 09 04 縺06 閨0600 0: x n! 4+0; x n! 4, 0; x n!,;5 +0; x n!,;5, 0. 76. 6 70309 0504, 0503: ) lim x 3 =7; x!3 77. 6 70309 0504, 0503 lim ) lim (x +x) =3. x! 5x + = ;5, 040309 04, 0503 003 ﾀ0504 x! x 5x +, ;5 06 ﾀ0504 0306 ﾀ0903006 003 0 00 07 04 罍ｧ x 0306 ﾀ0903006 003 0303000400030. x 70307 ﾀ0 ｧ0504 05 0300 ｧ090608, 040300 05 07 x =; 0; 00; 000; ::: 78. 6 70309 0504, 0503 lim cos x = cos a ( ﾀ0. 06 709). x!a 79. 7 04 ｧ ﾀ 0504 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ 0 060 ｧ08 0406 縺06006000308: ) x n =+, n ; ) x n =(,) n + n ; 3) x n =(,) n (n + ); 4) x n = n sin (n=) : n + 07 09 07 ｧ 0403 ﾀ0006 閨0304 05060004003 ﾀ050608 ｧ0060605 04 縺06 閨0600 04 罍ｧ n! +? 730. 7 0805 ｧ: ) lim =(x,) ; ) lim =(x,) ; x!,0 x!+0 3) lim 3 tg x ; 4) lim 3 tg x ; 5) lim x!=4,0 x!=4+0 x!=+0 6) lim x!=,0 ; 7) lim +tg x x!+ 73. 6 70309 0504, 0503 lim n!, 0; 6; 3, 0; 66; :::;, 0; 666:::6 3 {z }. n 0 090304 a +a x. + tg x ; 0; 666:::6 {z } =, ﾀ03 ﾀ05 04 ｧ04 003 ﾀ05 ｧ, 3 3 n 0 090304 73. 706 ﾀ0504 n 0400605 縺0600 ｧ08 06050300 04 04 ｧ00040030503 n-06050300040 ｧ09. 6 70309 0504, 0503 lim n =. n! 733. 7 04 縺03 閨030007060 ｧ ｧ 0305 縺06 09 AB = a ﾀ04 04 04 0705 0503 09 M 0 ﾀ ﾀ0503070 ｧ ｧ BM = x. 7 0805 ｧ lim x! AM.

3x 3. ｮ040308 ﾀ0504 04 縺06 閨06000304 97 x 3. ｮ040308 ﾀ0504 04 縺06 閨06000304. 7 ﾀ09 縺0305 ｧ06 0060304 縺06 閨0600060003 ﾀ050608 04 ｧ 0 0 ｧ. 7 縺06 閨0600 0403 ﾀ050307000308 04060 ﾀ 00308 0403 ﾀ050307000308. : lim(u + v) = limu + limv; 3 06 ﾀ00 ｧ lim u ｧ limv ﾀ06006 ﾀ0504060605: : lim(uv) = limu limv; 4. lim u v = limu, 06 ﾀ00 ｧ limu ｧ limv ﾀ06006 ﾀ0504060605 ｧlimv 6= 0. limv 5. ｭ ﾀ00 ｧ 閨0007 04 ﾀ0608 0 060 ｧ08 x 04 00609030503 縺0308 0309 縺06 ﾀ05003 ﾀ05 ｧ 0503 09 ｧ a, 09 縺03006, 03030504 003070605, x = a, 070600909 ｧ ｧ f(x) ｧ '(x) 04003 ｧ 03 閨0 ｧ 0 ｧ08 ｧ0060605 04 縺06 閨0600 04 罍ｧ x! a, 0503 ｧ040503 07 ｧ0060605 050305 0706 04 縺06 閨0600. 70503 ﾀ040308 ﾀ050403 04 罍ｧ0060070605 ﾀ07 04 罍ｧ ﾀ09 縺0305 ｧ ｧ 0060304 縺06 閨0600060003 ﾀ050608 04 ｧ 0 0 ｧ. 7 04 罍ｧ006, x, a = x + a 04 罍ｧ 0006030308 x, 09 縺03006 x = a. 703 ﾀ040308x, a ﾀ050406 5 lim x!a x, a x, a = lim (x + a) =a. x!a 7 0805 ｧ 04 縺06 閨060003: x, 4x + + sin x 734. ) lim ; ) lim x! x + x!=4, cos 4x. x, 4 735. lim x! x, (040307 ﾀ0 ｧ0504 05 0300 ｧ090608). x, 736. lim x! x, 3x. x, 9 737. lim + x!3 x, x,. 3 ｰ09 0 ｧ06. 70600 ｧ0504 06 736 閨0406007 ﾀ0403 ﾀ0303 0 ｧ: ) 040300 05 07 x = =+; ) 00 05 07 0 0060 05060004 0 000307 ｧ050600 ｧ. tg x 738. lim : 739. lim x! sin x x 740. lim : x!0 +3x, 3 x, 74. lim : x! x, x!=4 74. lim x!a 743. lim x!0 sin x, cos x : cos x ax, x x, a : 3 +mx, : x ｰ 09 0 ｧ 06. 6 5 06 74 0403000307 ｧ0504 x = t 6, 04 06 743 + + mx = t 3. +x,, x 744. lim. x!0 x, tg x, +tg x. 745. lim x!

398 0600. 5. 6 50406 閨060 ｧ06 04 0 00 ｧ x, 746. ) lim x! 3x, 4x ; 5x 3, 7x ) lim x!, x. 3 ｰ09 0 ｧ06. 030307003 縺0600 ｧ0504 閨0406007 ﾀ0403 ﾀ0303 0 ｧ: ) 閨0600 ｧ04 ｧ ﾀ00 ｧ- 05060004 ｧ 0 0060 05060004 0 x 04 0403 ﾀ000608 ﾀ050604060 ｧ; ) 0403000307 ｧ04 x ==. 747. lim x! 749. lim x! 75. lim n! 3x, : 748. lim x + x, 6x x! : 750. lim n! 3x + n + : 75. lim n, n! x 3, x + : 3n, n : ++3+:::+ n : 9n 4 + 7 0805 ｧ 04 縺06 閨060003: 3x +6 753. lim x!, x 3 +8 : 755. lim x!, 9, x 754. lim : x!3 3x, 3 x, x, : 756. lim x 3 + x!+0 + cos x : sin x 5x, 3x + 3n + 757. lim : 758. lim x! x +4x + n! 3n : + 5x 759. lim x!, x +=x. 760. lim n! +3+5+:::+(n, ). ++3+:::+ n, x, 3 76. lim x!7 x, 49 : 76. lim x!=4 sin x, cos x,. cos x, sin x x 4. 7 縺06 閨0600 030500300060 ｧ07 sin 04 罍ｧ! 0 ｭ ﾀ00 ｧ 06050300 0403 07060 04 陦ｧ 0 08, 0503 sin lim =; lim!0!0 sin =:

3x 5. 7060304 縺06 閨0600060003 ﾀ05 ｧ 04 ｧ, ｧ 0 99 7 0805 ｧ 04 縺06 閨060003: sin 4x sin (x=3) 763. lim : 764. lim. x!0 x x!0 x ｰ 09 0 ｧ 06. 6 5 06 763 06000307 ｧ0504 ｧ ﾀ00 ｧ05060004 ｧ 0 0060 05060004 0 4 ( ｧ00 ｧ 0403000307 ｧ0504 4x = ). tg x 765. lim x!0 x : 768. lim x!0 770. ) lim x!0 sin (x=), cos x 766. lim : 767. lim x!0 x x!0 x sin x. sin 3x : 769. lim x +, h!0 arctg x ; ) lim x x!= sin (x + h), sin (x, h). h arcsin (, x). 4x, ｰ09 0 ｧ06. 703000307 ｧ0504 04 04 罍ｧ006 縺06 ) arctg x =, 04 04 罍ｧ006 縺06 ) arcsin (, x) =., cos x tg x, sin x 77. lim : 77. lim. x!0 x x!0 x 3 7 0805 ｧ 04 縺06 閨060003: x 773. lim x!0 sin 3x :, cos x 775. lim x!,0 x sin 4x 774. lim : x!0 x +, : 776. lim x!0 x sin x sec x, :, cos mx, cos x + tg x 777. lim : 778. lim : x!0 x x!0 x sin x sin (x, ) 779. lim +,=(x,) (0403000307 ｧ0504 x =+). x! x, 4 780. ) lim h!0 cos (x + h), cos (x, h) h sin x 78. lim. x!0 +xsin x, cos x ; ) lim x!, arcsin (x +). x +x x 5. 7060304 縺06 閨0600060003 ﾀ05 ｧ 04 ｧ, ｧ 0 7 0805 ｧ 04 縺06 閨060003: 78. lim x!+ ( x +3x, x). 783. lim x! x,,. x,

300 0600. 5. 6 50406 閨060 ｧ06 04 0 00 ｧ 784. lim x!+ ( x + x +, x, x). 785. lim x! x,,. x 3, 8 786. lim x!0 sin x,. 4 sin (x=) +3+:::+(n, ) 787. lim, n. n! n +3 788. lim (, x)tg x (0403000307 ｧ0504 x =, ). x! 789. lim ( x +, x, 4x). x!, 790. lim x!, x + + 4. x, 4 7 0805 ｧ 04 縺06 閨060003: 79. lim (x, x, x +). 79. lim (x, x, a ). x!+ x!+ sin x 793. lim x!= cos x, tg x. ++3+:::+ n 794. lim, n. n! n + 795. lim, tg x (0403000307 ｧ0504 x = x!= + ). x 6. ｮ00600 000306 04 罍ｧ006 縺03 0 0403 ｧ ﾀ00060 ｧ06 04 縺06 閨06000304 7 0805 ｧ 04 縺06 閨060003: x +4, 796. ) lim x!0 sin 5x 797. ) lim x! 4 x, 3 x, ; ; ) lim x!0 ) lim x!0 798. lim x!, ( x + ax, x, ax). +xsin x, cos x : sin x x 4 +x, :

3x 7. ｮ 040060 ｧ06 0306 ﾀ0903006 003 0 000308 0, x x, sin x 799. ) lim x! 3 ; +8x 3 +,x ) lim x!, 5x : 800. ) lim x!, 80. ) lim x!0 x 3 + sin (x +), cos x x( +x, ) x + x, ; ) lim x!, x +x : cos (x=) ; ) lim x! x, : sin (, x) 80. ) lim ; ) lim x! x, 3x 4 803. ) lim x! 804. ) lim x!=+0 n!+ ; ) lim n!,, 0 n +0 n+ : 3, 0 n +0 n+ :, x, 4 =x + cos x (x +) ; ) lim cos, x x!, 3 : x + x 7. ｮ 040060 ｧ06 0306 ﾀ0903006 003 0 000308.0504 縺06 閨0600060 ｧ07. 706 ﾀ0504 04 罍ｧ x! a 070600909 ｧ ｧ (x) ｧ (x) 070400070605 ﾀ07 0306 ﾀ0903006 003 0 00030 ｧ. ｯ0305 : I. ｭ ﾀ00 ｧ lim = 0, 0503 0 0304 0605 ﾀ07 0306 ﾀ0903006 003 0 000308 0403 ﾀ00060503 x!a 0403 縺07 閨09 0305003 ﾀ ｧ05060004003. II. ｭ ﾀ00 ｧ lim = A (0903006 060 ｧ030500 ｧ 060 0305 0), 0503 0 0304 0605 ﾀ07 x!a n 0306 ﾀ0903006 003 0 000308 n-0503 0403 縺07 閨09 0305003 ﾀ ｧ05060004003. III. ｭ ﾀ00 ｧ lim =,0503 ｧ 0 0304 0605 ﾀ07 050904 ｧ04 00060050030 ｧ 0306 ﾀ0903x!a 006 003 0 00030 ｧ. 70904 ｧ04 0006005003 ﾀ0504 04 ｧ ﾀ0304 0605 ﾀ07 05 09:.. ｮ040308 ﾀ0504 050904 ｧ04 000600500308 0306 ﾀ0903006 003 0 000308: ) 7 003 ﾀ0504 閨040608 050904 ｧ04 000600500308 0306 ﾀ0903006 003 0 000308 06 ﾀ0504 0306 ﾀ0903006 003 0 00 07 0403 ﾀ00060503 0403 縺07 閨09 0305003 ﾀ ｧ05060004003 09 07 閨0308 ｧ 0 ｧ08. ) ｭ ﾀ00 ｧ ｧ ﾀ060003 006 ﾀ0903000409 ｧ08 0306 ﾀ0903006 003 0 000308 00308 0403 縺07 閨090304 030503 縺03 ﾀ ｧ0504 0306 ﾀ0903006 003 0 000306 0403 ﾀ00 ｧ08 0403 縺07 閨090304, 0503 03 ﾀ05 0400 07 ﾀ07 ﾀ0504, 0-0304 060 07 0500 0400308, 050904 ｧ04 00060050 04 ﾀ0608 ﾀ060006. 0ｷ3 0406 縺04030503 ﾀ040308 ﾀ0504 ﾀ0006 閨060605, 0503 050904 ｧ04 000600500306 0306 ﾀ0903006 003 0 000306 003050605 ﾀ 閨0600 0504 ﾀ07 04 罍ｧ0300 ｧ07060003 040030 ｧ ﾀ03 ﾀ09030004 060503 閨003 0 000308 0305003 ﾀ ｧ- 0506000400308 040305 縺0600003 ﾀ050406. 703050503006 0 09 003 04 罍ｧ006007060 09 09 閨0007 030303 0-060 ｧ07 050904 ｧ04 0006005003 ﾀ05 ｧ 0306 ﾀ0903006 003 0 000308, 05 09 ｧ 閨0007 04 ｧ ﾀ ｧ 04 罍ｧ0300 ｧ07060-0030503 04060 ﾀ0504 ｧ08 閨03 ﾀ05 0503 003 0 000308 0 060 ｧ08. 805. 0504 縺06 閨0600 ｧ0504 0403 縺07 閨09 ｧ 0306 ﾀ0903006 003 0 000308: ), cos x; ) tg x, sin x 0305003 ﾀ ｧ05060004003 0306 ﾀ0903006 003 0 000308 x. 70309 0504 0 06-05060706, 0503 04 罍ｧ 0600600400060 ｧ ｧ 060500 x 04 閨040306 040600 ｧ ｧ0, cos x 06006004-00 0605 ﾀ07 04 罍ｧ0300 ｧ ｧ0506000400304 060503 縺06, 040600 ｧ ｧ0 tg x, sin x 04 罍ｧ0300 ｧ ｧ05060004003 04 0403 ﾀ06004.

30 0600. 5. 6 50406 閨060 ｧ06 04 0 00 ｧ 806. 0504 縺06 閨0600 ｧ0504 0403 縺07 閨09 ｧ 0306 ﾀ0903006 003 0 000308: ) sin 4 x, x 5 ; ) sin x + x 4 ; 3) +x 3, 0305003 ﾀ ｧ05060004003 0306 ﾀ0903006 003 0 000308 x. 807. 0504 縺06 閨0600 ｧ0504 0403 縺07 閨0309 0 0003 ﾀ05 ｧ 08 ﾀ05 縺06000309 09 縺06050304030503 ﾀ0605006005 0305003 ﾀ ｧ05060004003 0306 ﾀ0903006 003 0 000308 閨0605 ｧ ﾀ0605006005. 808. 6 70309 0504, 0503 04 罍ｧ x! 0: ) sin mx mx; ) tgmx mx; 3) 3 +x, 3 x. 809. 0703050707 ｧ09 ｧ06005 030300600030503 ﾀ00 ｧ 縺060 ｧ07 050600 04 罍ｧ0 ｧ0 0605 ﾀ07 04 罍ｧ0300 ｧ07060003 040030 0605 縺03060003006 0903050707 ｧ09 ｧ0600506 00 ｧ006080030503 ﾀ- 00 ｧ 縺060 ｧ07. 7 050904 ｧ04 0006005003 ﾀ05 ｧ 09 09 ｧ08 0306 ﾀ0903006 003 0 000308 050503 03 ﾀ003-04 003? 80. 703 050603 縺06006 lim = lim, 06 ﾀ00 ｧ, ｧ 03 陦ｧ0 ｧ 04 縺06 閨06000304 ﾀ06006 ﾀ0504060605, 0 0805 ｧ 04 縺06 閨060003: sin 5x ) lim ; ) lim x!0 sin x x!0 sin ax + x tg bx 3x + sin x ; 3) lim x!0 sin x, x ; 3 8. 07 040007 0403 閨03 ｧ ﾀ04 縺070605 ﾀ07 05 09, 0503 0606 陦ｧ06 ﾀ ﾀ05 縺060 ｧ05 ﾀ07 09 0060006. 0504 縺06 閨0600 ｧ0504 0403 縺07 閨09 ｧ 0306 ﾀ0903006 003 0 000308 04030406 縺08003 ﾀ05 ｧ ｧ 03030-060 09 0400 ｧ 0305003 ﾀ ｧ05060004003 0606 陦ｧ06 ﾀ. 8. 0504 縺06 閨0600 ｧ0504 0403 縺07 閨09 ｧ 0306 ﾀ0903006 003 0 000308: ) +x, ; ) sin x, sin x; 3), cos 0305003 ﾀ ｧ05060004003 0306 ﾀ0903006 003 0 000308 x. 83. 6 70309 0504, 0503 04 罍ｧ x! 0: ) arctg mx mx; ) +x, x; 3), cos3 x ; 5 sin x. x + 3 x 8. 70604 縺06 縺0304003 ﾀ0504 070600909 ｧ ｧ. 0504 縺06 閨0600060 ｧ06. ｱ0600909 ｧ07 f(x) 0 0304 0605 ﾀ07 00604 縺06 縺030400308 04 罍ｧ x = a, 06 ﾀ00 ｧ 030 0304 縺06 閨0600060 04 00609030503 縺0308 0309 縺06 ﾀ05003 ﾀ05 ｧ a ｧ lim f(x) =f(a): x!a 70503 0304 縺06 閨0600060 ｧ06 ﾀ03 閨06 縺07 ｧ05 05 09 ｧ06 060503 縺06 06 ﾀ000304 ｧ07 00604 縺06 縺0304003 ﾀ05 ｧ: ) f(x) 閨0300070 03030504 0304 縺06 閨0600060 04 00609030503 縺0308 0309 縺06 ﾀ05003 ﾀ05 ｧ a; ) 閨030007003 ﾀ06006 ﾀ05040304 0504 0903006 00306 04 縺06 閨060003 lim f(x) ｧ lim x!a,0 3) 0505 ｧ 04 縺06 閨060003 ( ﾀ000604 ｧ ﾀ04 04 ) 閨030007003 03030504 03 陦ｧ0 090304030 ｧ; 4) 0505 ｧ 04 縺06 閨060003 閨030007003 03030504 04003 f(a). x!a+0 f(x);

3x 8. 70604 縺06 縺0304003 ﾀ0504 070600909 ｧ ｧ 03 ｱ0600909 ｧ07 0 0304 0605 ﾀ07 00604 縺06 縺030400308 0 0305 縺06 0906 [x ;x ], 06 ﾀ00 ｧ 030 006-04 縺06 縺03040 04 09 07 閨0308 0503 0906 0400605 罍ｧ 0305 縺06 09, 0 060503 05 0 ｧ09 08 lim f(x) = x!x+0 = f(x ) ｧ lim f(x) =f(x ). x!x,0 70006006005 縺00306 070600909 ｧ ｧ: ﾀ0506040600 07 x n, 040309 050600040 07 a x, 000305 罍ｧ07-0 ｧ 06 ﾀ09 07, 05 罍ｧ050300300605 罍ｧ 06 ﾀ09 ｧ06 ｧ ｧ0 0303 0500306, 05 090706 ｧ08 ﾀ0600, 04 縺03- ｧ 0406 閨060 ｧ06, ﾀ0500306 00604 縺06 縺0304003 04 罍ｧ 04 ﾀ0709030 x, 04 罍ｧ 09030503 縺030 030 ｧ ｧ0060605 0304 縺06 閨060006000306 0 060 ｧ06.. 7 縺030403 070600909 ｧ ｧ. ｱ0600909 ｧ07 ｧ0060605 縺0304 04 罍ｧ x = a, 06 ﾀ00 ｧ 030 0304 縺06 閨0600060 ﾀ000604 ｧ ﾀ04 04 0305 a, 003 04 0503 0906 a 006 ﾀ03030006 閨06003 08030507 0303 03 閨003 ｧ 06 ﾀ000304 ｧ08 00604 縺06 縺0304003 ﾀ05 ｧ. 7 00 ｧ 0605 閨04 03 ﾀ0030400308 04 ｧ 縺0304. ) 7 縺0304 I 縺03 090305 ﾀ06006 ﾀ0504060605 0903006 00306 04 縺06 閨060003 lim f(x) x!a,0 ｧ lim x!a+0 f(x), 05.06.090305 0403040300006003 040503 縺0306 06 ﾀ000304 ｧ06 00604 縺06 縺0304003 ﾀ05 ｧ ｧ 006 0403040300006003 03 ﾀ05 000400306 ( ｧ00 ｧ 08030507 0303 03 閨003 ｧ 0 ｧ08). 7 04 罍ｧ006, 070600909 ｧ07 y = x, a, 040 07, 04 罍ｧ x<a ｧ + 04 罍ｧ jx, aj x>a, ｧ0060605 04 罍ｧ x = a 縺0304 I 縺03 ( 罍ｧ ﾀ. 4), 05 09 09 09 ﾀ06006 ﾀ0504060605 04 縺06 閨060003 y =, ｧ y =+,0030505 ｧ 04 縺06 閨060003 006 04003. lim x!a,0 lim x!a+0 ) 7 縺0304 II 縺03 090305 lim f(x) ﾀ000604 ｧ00 ｧ ﾀ04 04 04060. x!a 7 04 罍ｧ006, 070600909 ｧ07 y = f(x) = a ( 罍ｧ ﾀ. 5) ｧ0060605 04 罍ｧ x = a x, a 縺0304 II 縺03. 6 5 ﾀ06 陦縺030300306 070600909 ｧ ｧ, 0 0060 05060004 09030503 縺0308 04 罍ｧ x = a 7 ｧ ﾀ. 4 7 ｧ ﾀ. 5 04060 0, ｧ ﾀ00 ｧ05060004 006 04060 0, ｧ0060605 04 罍ｧ x = a 縺0304 II 縺03. ｱ0600909 ｧ07 f(x) = =x ( 89, 罍ｧ ﾀ. 38 0 ﾀ. 74) 05 090706 ｧ0060605 04 罍ｧ x =0 縺0304 II 縺03, 05 09 09 09 lim f(x) =0,003 lim f(x) =. x!,0 x!+0 84. ｰ09 0504 0503 0906 縺0304 070600909 ｧ ｧ y = 4 x,, 0 0805 ｧ y ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 04030503 09 0 lim y, lim x!+0 x! x =,; 0; ; 3; 4 ｧ 6: lim y, x!,0

304 0600. 5. 6 50406 閨060 ｧ06 04 0 00 ｧ 85. 7 0805 ｧ 0503 09 ｧ 縺0304 ｧ 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 ｧ 070600909 ｧ08: ) y =, 6 4 ; ) y = tg x; 3) y = x 4, x. 86. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ ( x= 04 罍ｧ x 6= ; y = 0 04 罍ｧ x = ｧ0609 0504 0503 0906 0606 縺0304. 07 09 ｧ06 ｧ 060503 縺0608 06 ﾀ000304 ｧ08 00604 縺06 縺0304-003 ﾀ05 ｧ 04 05050308 0503 0906 0403040300006003 ｧ 09 09 ｧ06 006 0403040300006003? 87. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 ｧ 070600909 ｧ08: ) y = x + jx +j ｧ)y = = x x + +. 07 09 ｧ06 ｧ 06 ﾀ000304 ｧ08 00604 縺06 縺0304003 ﾀ05 ｧ 040503 09 08 縺0304 jx +j 0505 ｧ08 070600909 ｧ08 0403040300006003 ｧ 09 09 ｧ06 006 0403040300006003? 88. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ 8 < sin x 04 罍ｧ x 6= 0; y = f (x) = x : 04 罍ｧ x =0 ｧ0609 0504 0503 0906 0606 縺0304. 07 09 ｧ06 ｧ 06 ﾀ000304 ｧ08 00604 縺06 縺0304003 ﾀ05 ｧ 04 00608 0403040300006003 ｧ 09 09 ｧ06 00605? 89. ｰ09 0504 0503 0906 縺0304 070600909 ｧ ｧ y = =x, 0 0805 ｧ lim y, x!,0 lim y, lim y ｧ 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ. 07 09 ｧ06 06 ﾀ000304 ｧ07 006x!+0 x! 04 縺06 縺0304003 ﾀ05 ｧ 04 0503 0906 縺0304 006 0403040300006003? 80. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ 8 >< 0; 5x 04 罍ｧ jxj < ; y = f (x) = ; 5 04 罍ｧ jxj =; >: 3 04 罍ｧ jxj > ｧ0609 0504 0503 09 ｧ 0606 縺0304. 8. 7 0805 ｧ 0503 09 ｧ 縺0304 ｧ 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 ｧ 070600909 ｧ08 ) y = ; ) y + = arctg a =x x, a ; 3) y = x3, x jx, j. 8. ｮ090300040903 03 閨003 0 00308 070600909 ｧ08 003 06 040060 ｧ060 x,, y =0? 0504 縺06 閨0600 ｧ0504 ｧ 0 ｧ08: ) 060500606 070600909 ｧ06; ) 006 060500606 070600909 ｧ06 05 09, 05030303 030 ｧ ｧ00600 ｧ 0903006 00306 縺030403 (I 縺03 ) 04 罍ｧ x = ; ; 3; :::, ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ08 05 07 ｧ09 ｧ.

3x 9. 罍ﾀ ｧ00405030503 05 83. ｰ09 05040503 0906 縺0304 070600909 ｧ ｧ y = x x, 0 0805 ｧ + lim y, x!,,0 y ｧ 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 0403 0503 09 0 x =,6;,4;,3, lim y, lim x!,+0 x!, ; 0;. 84. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 8 070600909 ｧ ｧ < 04 罍ｧ x =0 ｧ x = ; y = f (x) = 4, x 04 罍ｧ 0 < jxj < ; : 4 04 罍ｧ jxj > ｧ0609 0504 0503 09 ｧ 縺0304. 07 09 ｧ06 06 ﾀ000304 ｧ07 00604 縺06 縺0304003 ﾀ05 ｧ 0403040300-006003 04 0503 09 08 縺0304 ｧ 09 09 ｧ06 00605? 85. 7 0805 ｧ 0503 09 ｧ 縺0304 ｧ 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 ｧ 070600909 ｧ08: ) y =, jxj x ; ) y ==(x,) ; 3) y =, =x ; 4) y = x3 + x ; 5) y = 4, x jxj j4x, x 3 j : 86. ｮ090300040903 03 閨003 0 00308 070600909 ｧ08 003 06 040060 ｧ060 x + + y = 4? 0504 縺06 閨0600 ｧ0504 ｧ 0 ｧ08: ) 閨0406 00604 縺06 縺030400306 0 0305 縺06 0906 jxj 6 ; ) 0506 ｧ 0 ｧ08, 09030503 07 0305 罍ｧ09 050600040 0 0305 縺06 0906 jxj 6 6 ｧ 0403000307 ｧ050600040 閨0007 04 ﾀ0608 03 ﾀ05 000400308 閨030406 ﾀ05 ｧ00308 0 060 ｧ08 x. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 ｧ0609 0504 縺030403 0403 ﾀ0006 閨00608 070600909 ｧ ｧ. x 9. 罍ﾀ ｧ00405030503 罍ﾀ ｧ0040503050308 09 罍ｧ040308 0 0304 0605 ﾀ07 04 縺070 07, 09 09030503 縺0308 0060305 0 ｧ- 060003 04 罍ｧ0300 ｧ07 0605 ﾀ07 0503 09 09 罍ｧ040308 04 罍ｧ 0606 06 00060 ｧ ｧ 0403 09 罍ｧ040308 04 0306 ﾀ0903-006 003 ﾀ0504. I. ｭ ﾀ00 ｧ lim f(x) =, 0503 04 縺070 07 x = a 06 ﾀ0504 ﾀ ｧ004050305 09 罍ｧ040308 x!a y = f(x). 7 04 罍ｧ006, 09 罍ｧ04 07 y = a ｧ0060605 ﾀ ｧ00405030506 x = a ( 罍ｧ ﾀ. 5). x, a II. ｭ ﾀ00 ｧ 04 04 040308 ﾀ05 ｧ 06 040060 ｧ07 09 罍ｧ040308 y = f(x) 00307003 0403 閨0600 ｧ0504 00 ｧ0060800606 ﾀ0504 y = f(x) =kx + b + (x) 05 09, 0503 03 ﾀ05 0400 07 ﾀ07 ﾀ0504 (x)! 0, 090305 x!, 0503 04 縺070 07 y = kx + b 06 ﾀ0504 ﾀ ｧ004050305 09 罍ｧ040308. 7 罍ｧ006 縺03: ) 09 罍ｧ04 07 y = x3 + x + = x ++ ｧ0060605 ﾀ ｧ00405030506 x x y = x +( ｧ ﾀ ｧ00405030506 x =0); ) 09 罍ｧ04 07 y = a x, a =0+ a x, a ｧ0060605 ﾀ ｧ00405030506 y =0( 罍ｧ ﾀ. 5). f(x) III. ｭ ﾀ00 ｧ ﾀ06006 ﾀ0504060605 0903006 00306 04 縺06 閨060003 lim = k ｧ lim x!+ ｧ00 ｧ, x!+ ｧ00 ｧ, x [f(x), kx] =k, 0503 04 縺070 07 y = kx + b 06 ﾀ0504 ﾀ ｧ004050305. 87. 0504 縺06 閨0600 ｧ0504 ﾀ ｧ00405030503 09 罍ｧ040308 y =, 4 x ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 04030503 09 0 x = ; ; 4.

306 0600. 5. 6 50406 閨060 ｧ06 04 0 00 ｧ 6 5 08 88{830 0 0805 ｧ ﾀ ｧ00405030503 09 罍ｧ040308, 0403 閨0600 ｧ04 ｧ 陦縺0303 ｧ 00 ｧ0060800606 0906000606 ﾀ0504; 0403 ﾀ05 縺03 ｧ0504 ﾀ ｧ00405030503 ｧ 09 罍ｧ040306: 88. ) y = x + ; ) y = x x ; 3) y = x x + x : + 89. ) y = jxj, ; ) y = x, x, ; 3) y = ax + b x mx + n : 830. ) y =, 4x x3 ; ) y = +x x + ; 7 0805 ｧ ﾀ ｧ00405030503 09 罍ｧ040308 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: 83. ) x, y = a ; ) x 3 + y 3 =3axy; 3) y = x, arctg x; x 4) y = arctg a, x : 83. ) y = x +, x, ; ) y = x ++ x, ; 3) y = x,. x 4x, x3 3) y = x : +4 833. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: ) y = x4 + 3x ; ) y = x3 + x, ｧ x + 04 03030003, 09 09030503 縺030 0505 ｧ 09 罍ｧ040306 ﾀ ｧ004050305 ｧ 06 ﾀ09 ｧ 04 罍ｧ0300 ｧ07 0605 ﾀ07. 834. 7 0805 ｧ ﾀ ｧ00405030503 09 罍ｧ040308: ) y =, x ;)y =,x + + x ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306 0403 0503 09 0 x = ; ;. 835. 7 0805 ｧ ﾀ ｧ00405030503 09 罍ｧ040308 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: ) y = x, 4 x x x3 ; ) y = ; 3) y = ; 4) y = x +4, x x, 4, x. x 0. ｪ ｧ ﾀ0003 ｪ ｧ ﾀ00030 e 0 0304 0605 ﾀ07 04 縺06 閨0600 lim n! + n n = lim n!, + n n = lim ( + ) = = e:!0 70503 ｧ ﾀ0003 ｧ 罍 09 ｧ030 000400306 ｧ 04 罍ｧ0300 ｧ07060003 04003 e = ;788::: 6ｧ803-05 罍ｧ07003 ﾀ 03 ﾀ00304 0 ｧ060 e 0 0304 0605 ﾀ07 0 0506 00040030 ｧ ｧ030303 0 0605 ﾀ07 log e x =lnx. 6 706 ﾀ0705 ｧ 00308 000305 罍ｧ070: lg x = M ln x, 05 閨06 M =0;4349:::

3x 0. ｪ ｧ ﾀ0003 e 07 7 0805 ｧ 04 縺06 閨060003: 836. lim, 5 n (0403000307 ｧ0504, 5 n! n n = ). 837. ) lim, n ; ) lim + 4 n+3 : n! 3n n!, n 838. ) lim ( + x) =x ; ) lim x!0 x!0 839. ) lim n! n n + n ; ) lim x! (, 4x) (,x)=x : x, x : x + 840. ) lim n[ln (n +3), ln n]; ) lim ( + 3 tg x) ctg x. n! x!0 84. lim (cos x) ctg x (0403000307 ｧ0504 sin x = ). x!0 ln ( + ) e,x, 84. ) lim ; ) lim!0 x!0 x ｰ 09 0 ｧ 06. 6 5 04 罍ｧ006 縺06 ) 0403000307 ｧ0504 e,x, =. ; 3) lim x!0 a x,. x 843. 7 0805 ｧ 0403 ﾀ0006 閨0304 0506000400306 0906000306 ｧ ﾀ00, 00607 閨06 09030503 縺030 ｧ ﾀ03 閨06 縺07 ｧ05 ﾀ07 0403 07060 ｧ06 6(, ; 0,00 ). 7 0805 ｧ 04 縺06 閨060003: 844. ) lim + 3n n, n= 3 ; ) lim : n! n n! n 3x, x e,3x, 845. ) lim ; ) lim : x! 3x + x!0 x 846. lim (sin x) tg x (0403000307 ｧ0504 cos x = ). x!=4 t 847. ) lim t!0 ln ( + xt) ; ) lim n! n ln n, ln (n +).

30600 04 6 7 7050ｷ36ｦ06 5056 7 7 縺00 0ｷ3 6 70ｷ3 ｱ ｱ ｭ 7 ｭ 7ｨC0ｷ3 縺6ｧ8 x. 7 縺03 ｧ 0403 閨00306 00050603 ｧ 06 ﾀ09 ｧ08 ｧ 05 罍ｧ050300300605 罍ｧ 06 ﾀ09 ｧ08 070600909 ｧ08. 0504 縺06 閨0600060 ｧ07. 7 縺03 ｧ 0403 閨00308 070600909 ｧ ｧ y = f(x) 040503 0906 x 0-0304 0605 ﾀ07 04 縺06 閨0600 f(x +x), f(x) y lim = lim x!0 x x!0 x : () ｭ ﾀ00 ｧ 05050305 04 縺06 閨0600 0903006 00308, 0503 070600909 ｧ07 f(x) 0 0304 0605 ﾀ07 陦ｧ070706 縺060-09 ｧ 縺060600308 040503 0906 x; 04 罍ｧ 0505030 030 0309 0304 0605 ﾀ07 030307 05060004003 ｧ 00604 縺06 縺0304-00308 04 05050308 0503 0906. ｭ ﾀ00 ｧ 0706 04 縺06 閨0600 () 04060 + ( ｧ00 ｧ,), 0503 0306 閨060 05030403 罍ｧ0504, 0503 070600909 ｧ07 f(x) ｧ0060605 04 0503 0906 x 0306 ﾀ0903006 00606 04 縺03 ｧ 0403 閨00606, 03 閨0 0903 04 罍ｧ 閨030403000 ｧ050600040030 06 ﾀ000304 ｧ ｧ, 0503 070600909 ｧ07 04 05050308 0503 0906 00604 縺06 縺03040. 7 縺03 ｧ 0403 閨0 07 030303 0 0605 ﾀ07 y 0 ｧ00 ｧ f 0 (x), ｧ00 ｧ dy, ｧ00 ｧ df(x). 7 0803-07 閨060 ｧ06 04 縺03 ｧ 0403 閨00308 0 0304 0605 ﾀ07 陦ｧ070706 縺06009 ｧ 縺0304 0 ｧ060 070600909 ｧ ｧ.. 05 ﾀ0030400306 0703 縺0060003 陦ｧ070706 縺06009 ｧ 縺0304 0 ｧ07: ) (c) 0 =0; ) (x n ) 0 = nx n, ; 3) (cu) 0 = cu 0 ; 4) (u + v) 0 = u 0 + v 0 ; 5) (uv) 0 = u 0 v + uv 0 ; u v 0 = u0 v, v 0 u 6) v ; 7) ( x) 0 = x ; 8) (sin x) 0 = cos x; 9) (cos x) 0 =, sin x; 0) (tg x) 0 = cos x ; ) (ctg x)0 =, sin x. y 848. 6 503 ｧ ﾀ00060 ｧ060 lim x!0 x 0 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: ) y = x 3 ; ) y = x 4 ; 3) y = x; 4) y = sin x; 5) y = x ; 6) y = ; 7) y = ; 8) y = tg x; x x 9) y = x 3 ; 0) y = +x; ) y = 3x + ; ) y = +x.

3x. 7 縺03 ｧ 0403 閨00306 00050603 ｧ 06 ﾀ09 ｧ08 ｧ 05 罍ｧ050300300605 罍ｧ 06 ﾀ09 ｧ08 070600909 ｧ08 09 7 0805 ｧ 04030703 縺00600 0 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 849. ) y = x3 3, x +4x, 5; ) y = bx + c : a 850. ) y = x5 5, x3 3 + x; ) y =, x : 85. ) y = x + x; ) y =, a, x : 85. ) y = 0 x 3 ; ) y = x + x + x 3 : 853. ) y = x + x, 5x 5 ; ) y =3x, 6 x: 854. ) y =6 3 x, 4 4 x; ) y =, 3 x : 855. ) y = x, 3x 3 ; ) y = 4 8, 6 x 3 : x 856. ) y = x, sin x; ) y = x, tg x: 857. ) y = x cos x; ) y = x ctg x: 858. ) y = cos x x ; ) y = x x + : 859. ) y = x tg x ; ) y = :, 4x x 860. ) f (x) = cos x, sin x ; ) '(x) = x x + : 86. ) s = gt ; ) x = a(t, sin t): 86. f (x) = x3 3, x + x; 0403 ｧ ﾀ00 ｧ0504 f 0 (0), f 0 (), f 0 (,). 863. f (x) =x, x ; 0403 ｧ ﾀ00 ｧ0504 f 0 (), f 0 (,). 864. f (x) = ( x, ) ; 0403 ｧ ﾀ00 ｧ0504 0; 0 f 0 (0; 0). x 7 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 865. ) y =(a, bx ) 3 ; ) y =(+ 3 x) : 866. ) y = 0x 5, 4x 4 ; ) y = 3 3 x, x :

30 0600. 6. 7 縺03 ｧ 0403 閨0 07 ｧ 陦ｧ070706 縺06009 ｧ 00 867. ) y = x + sin x; ) y = x + ctg x: 868. ) y = x sin x; ) y = x tg x: 869. ) y = x cos x; ) s = t, t : 870. ) y = x, x, 3x 3 ; ) y = x, x + : 87. ) y = + 3 x 3 ; ) y = cos x +sinx : 87. f (x) = 3 x ; 0 0805 ｧ f 0 (,8). 873. f (x) = x x, ; 0 0805 ｧ f 0 (0), f 0 () ｧ f 0 (,). x. 7 縺03 ｧ 0403 閨0 07 ﾀ00030700308 070600909 ｧ ｧ ｭ ﾀ00 ｧ y = f(u), u = '(x), 0503y 0 0304 0605 ﾀ07 070600909 ｧ0608 0305 070600909 ｧ ｧ ｧ00 ｧ ﾀ00030700308 070600909 ｧ0608 0305 x. ｯ0305 dy = dy du du ｧ00 ｧ y0 = f 0 (u) u 0 : () ｱ03 縺0060003 04 縺06 閨03 閨060060503 04 05 07 04 罍ｧ00605 05060406 縺04 03030 ｧ08 04 ｧ : ) (u n ) 0 = nu n, u 0 ; ) (sin u) 0 = cos u u 0 ; 3) (cos u) 0 =, sin u u 0 ; 4) ( u) 0 = u0 u ; 5) (tg u) 0 = u0 cos u ; 6) (ctg u)0 =, u0 sin u : 7 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 874. ) y = sin 6x; ) y = cos (a, bx): 875. ) y = sin x + cos x ; ) y = 6 cos x 3 : 876. ) y =(, 5x) 4 ; ) y = 3 (4 + 3x) : 877. ) y = (, x ) 5 ; ) y =, x ; 3) y = cos 4x. 878. y = x, sin x: 879. y = sin 4 x = (sin x) 4. 880. ) y = sin x; ) y = cos x; 3) y = sec x. 88. y = sin 3 x + cos 3 x: 88. y = tg 3 x, 3 tg x +3x: 883. y = 4 + cos x: 884. y = sin x:

3x 3. 07 ﾀ 050600040 07 ｧ 003 縺0 0004 09 040003 ﾀ090308 09 罍ｧ040308 885. y = + sin x,, sin x. 886. y = ( + cos 4x) : 5 887. y = x ctg3 3 : 888. y = sin x x cos x : 889. y = x, : x, 890. y = : 89. s = a cos t x a : 89. ) r = a cos '; ) r = r' + cos ' + 4 893. f (t) = a + b, ab cos t; 0403 ｧ ﾀ00 ｧ0504 f 0 (=); f 0 (); f 0 3 894. f (x) = x + x; 0 0805 ｧ f 0 (). :. 7 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 895. y = 4x + sin 4x: 896. y = x, x : 897. y = sin 4 x + cos 4 x: 898. y = 3 + cos 6x: 899. ) y = tg x + 3 tg3 x + 5 tg5 x; ) y = sin x 3. r + sin x 900. y =, sin x : 90. s = t, sin t : 90. r = cos 4, ' 4x + : 903. y = x : 904. f (t) = + cos t ; 0 0805 ｧ f 0. x 3. 07 ﾀ 050600040 07 ｧ 003 縺0 0004 09 040003 ﾀ090308 09 罍ｧ040308 ｰ050003040308 0903050707 ｧ09 ｧ06005 09 ﾀ 0506000400308 0909 罍ｧ040308 y = f(x) 040503 0906 09 罍ｧ040308 (x 0 ; y 0 ) 04060 0 060 ｧ06 04 縺03 ｧ 0403 閨00308 070600909 ｧ ｧ f(x) 04 0503 0906 x 0 : k = tg ' = f 0 (x 0 )=y 0 j x=x0 : () ｪ ｧ ﾀ0003 k 0 0304 0605 ｧ00305 0 0900030030 09 罍ｧ040308 04 0503 0906 (x 0 ; y 0 ). ｰ 040060 ｧ06 09 ﾀ 0506000400308 040503 0906 M (x 0 ; y 0 ) 0 09 罍ｧ040308 ( 罍ｧ ﾀ. 6): y, y 0 = k(x, x 0 ): ()

3 0600. 6. 7 縺03 ｧ 0403 閨0 07 ｧ 陦ｧ070706 縺06009 ｧ 00 ｰ 040060 ｧ06 003 縺0 00 ｧ: y, y 0 =, k (x, x 0); (3) 05 閨06 k 0304 縺06 閨0600070605 ﾀ07 0703 縺006000308 (). 0505 縺06 09 ｧ TA = y 0 ctg ', AN = y 0 tg ' ( 罍ｧ ﾀ. 6) 0 0304 0605 ﾀ07 ﾀ030305-040605 ﾀ0504060003 0403 閨09 ﾀ 0506000400308 ｧ 0403 閨003 縺0 000406, 閨00 ｧ003 0305 縺06 090304 MT ｧ MN 閨00 ｧ0 0 ｧ 09 ﾀ 0506000400308 ｧ 003-0 00 ｧ. 7 ｧ ﾀ. 6 905. 7 0805 ｧ 0 090003003 04 03030003 y = x 040503 09 08 x =. 906. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ - 0506000400308 ｧ 003 縺0 00 ｧ 09 04 03030006 y = =4, x 040503 0906 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 03 ﾀ0406 Ox (04 罍ｧ x>0) ｧ 0403 ﾀ05 縺03 ｧ0504 04 03030006, 09 ﾀ 0506000400606 ｧ 003 縺0 0004. 6 5 08 907{90 0 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 05060004003080909 罍ｧ04030 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306 ｧ 09 ﾀ 05060004-00306: 907. 0709 罍ｧ040308 y = x3 3 040503 0906 x =,. 908. 0709 罍ｧ040308 y = x 3 040503 09 08 x =0 ｧ x =. 909. 07 00030903006 y = 8 040503 0906 x =. 4+x 90. 07 ﾀ ｧ006 ﾀ03 ｧ 閨06 y = sin x 040503 0906 x =. 9. 703 09 09 ｧ0 060500030 09 罍ｧ04 07 y = sin x 0406 縺06 ﾀ0609 0605 03 ﾀ04Ox? 9. 703 09 09 ｧ0 060500030 0406 縺06 ﾀ0609 0605 ﾀ07 09 罍ｧ040306 y = x ｧ y =8, x? 93. 7 0805 ｧ 閨00 ｧ006 0403 閨09 ﾀ 0506000400308, 0403 閨003 縺0 00 ｧ, 09 ﾀ 0506000400308 ｧ 003 縺0 00 ｧ 09 罍ｧ040308: ) y = x ; ) y = x 3 040503 0906 x =. 94. 6 70309 0504, 0503 0403 閨09 ﾀ 050600040 07 04 03030003 y = x 040 06 閨040306000308 03 ﾀ09 ｧ ﾀ ﾀ06 0503 09 ｧ 09 ﾀ 0 ｧ07, 0403 閨003 縺0 0004 040. 95. 6 506 040060 ｧ ｧ 04 03030003 y = x + bx + c 0304 縺06 閨0600 ｧ0504 b ｧ c, 06 ﾀ00 ｧ 04 030300 09 ﾀ 0605 ﾀ07 04 縺0700308 y = x 040503 0906 x =. 96. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 05060004003080905 ｧ0406 縺03030006 xy = 404 0503 09 08 x = ｧ x =,4 ｧ 0 0805 ｧ 06050300 00607 閨06 09 ﾀ 050600040030 ｧ. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ09 ﾀ 0506000400306.

3x 4. ｮ0006 ｧ 006 陦ｧ070706 縺06009 ｧ 縺0606003 ﾀ05 ｧ 00604 縺06 縺030400308 070600909 ｧ ｧ 3 6 5 08 97{99 0 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 05060004003080909 罍ｧ04030 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306 ｧ 09 ﾀ 0506000400306 09 0 ｧ0: 97. y =4x, x 040503 09 08 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ0406 Ox. 98. y =4, x 040503 09 08 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ0406 Oy. 99. y =(4+x) 3 040503 09 08 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ Ox ｧ Oy. 90. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0406 縺00 ｧ003 04 03030003 y = x, 4x +50305 09 ﾀ 0506000400308 09 00608 04 0503 0906 0406 縺06 ﾀ06 060 ｧ07 04 03030003 ﾀ 03 ﾀ0406 Oy. 9. 703 09 09 ｧ0 060500030 04 縺070 07 y = 0;5 0406 縺06 ﾀ0609 0605 09 罍ｧ040606 y = cos x? 9. 6 509 090308 0503 0906 09 ﾀ 050600040 07 09 04 03030006 y = x +4x 04 00-000600040 03 ﾀ ｧ Ox? 93. 6 509 090308 0503 0906 04 03030003 y = x, x +5 00607003 04 縺030406- ﾀ05 ｧ 09 ﾀ 0506000400606, 05030303 030 030300 0406 縺04060 陦ｧ09060007 縺0 09 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ06 0406 縺04030503 090303 罍陦ｧ0 050030503 060500? 94. 7 0805 ｧ 閨00 ｧ006 0403 閨09 ﾀ 0506000400308, 0403 閨003 縺0 00 ｧ, 09 ﾀ 0506000400308 ｧ 003 縺0 00 ｧ 09 罍ｧ040308 y = 040503 0906 x =. +x 95. 07 09 ｧ06 06050003 0303 060605 04 030300 y = x ﾀ09 ｧ ﾀ ﾀ03 09030090304 09030503 縺0308 04003 ｧ 4? x 4. ｮ0006 ｧ 006 陦ｧ070706 縺06009 ｧ 縺0606003 ﾀ05 ｧ 00604 縺06 縺030400308 070600909 ｧ ｧ 4 ﾀ06060803 罍閨0308, 03-. ｰ05000304 07 0503 09. ｯ03 09 A(x ; y ) 09 罍ｧ040308 y = f(x) ( 罍ｧ ﾀ. 7) 0 0304 0605 ﾀ07 06050003040308, 06 ﾀ00 ｧ 04 05050308 0503 0906 04 縺03 ｧ 0403 閨0 07 y 0 006 ﾀ06006 ﾀ0504060605, 003 ﾀ06006 ﾀ0504060605 000604 07 ｧ 04 04 07 00 ｧ - 00306 04 縺03 ｧ 0403 閨00306: lim x!,0 y x = k y lim x!+0 x = k. 0ｷ3 06050003040308 0503 09 ｧ 0403-0803 閨0705 閨04 09 ﾀ 0506000400308 0006 ﾀ 0 0900030 0 ｧ k ｧ k.. ｯ03 09 0403 04 05 ﾀ 0406 縺05 ｧ- 09 000400308 09 ﾀ 0506000400308. ｯ03 09 B(x ; y ) ( 罍ｧ ﾀ. 7) 0 0304 0605 ﾀ07 0503 090308 0403 04 05 ﾀ 0406 縺05 ｧ09 000400308 09 ﾀ 0506000400308, 06 ﾀ00 ｧ 04 05050308 0503 0906 04 縺03 ｧ 0403 閨0 07 y 0 006 ﾀ06-006 ﾀ0504060605, 003 ﾀ06006 ﾀ0504060605 000604 07 ｧ 04-04 07 0306 ﾀ0903006 00306 04 縺03 ｧ 0403 閨00306 0030503 ｧ 7 ｧ ﾀ. 7 0 09 (+ ｧ,). ｯ 09 07 0503 09 070400070605 ﾀ07 ﾀ050030 ﾀ0006 060 06050003040308. 0ｷ3 00606 04030803 陦ｧ05 03 陦ｧ0 0406 縺05 ｧ09 000400308 09 ﾀ 0506000400308 0006 ｧ00 ｧ, 00307003 ﾀ ｧ05 0504, 0503 ｧ 00606 04030803 閨0705 閨04 ﾀ00 ｧ0400 ｧ08 ﾀ07 09 ﾀ 0506000400308 0006. 3. ｯ03 09 0406 縺0605 ｧ03 ﾀ 0406 縺05 ｧ09 000400308 09 ﾀ 0506000400308. ｯ03-09 C(x 3 ; y 3 ) ( 罍ｧ ﾀ. 7) 0 0304 0605 ﾀ07 0503 090308 0406 縺0605 ｧ03 ﾀ 0406 縺05 ｧ09 000400308 09 ﾀ 0506000400308, 06 ﾀ00 ｧ 04 00608 ﾀ06006 ﾀ0504060605 0306 ﾀ0903006 0 07 04 縺03 ｧ 0403 閨0 07

34 0600. 6. 7 縺03 ｧ 0403 閨0 07 ｧ 陦ｧ070706 縺06009 ｧ 00 y lim x!,0 lim y x!+0 x y 0 = x = =+ ( ｧ00 ｧ,). 6 5 05 090308 0503 0906 ﾀ06006 ﾀ05-04060605 0406 縺05 ｧ09 00040 07 09 ﾀ 050600040 07. 6 50503 09 08 A ｧ B 070600909 ｧ07 y = f(x) 006 ｧ0060605 04 縺03 ｧ 0403 閨00308; 04 0503 0906 C 030 ｧ0060605 0306 ﾀ0903006 00606 04 縺03 ｧ 0403 閨00606. 6 503 04 ﾀ0608 05 縺0608 0503 09 08 070600909 ｧ07 00604 縺06 縺03040, 003 006 陦ｧ070706 縺06009 ｧ 縺06060. 96. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ y = x ( ｧ00 ｧ y = jxj) ｧ 0 0805 ｧ 0006040606 y, 0 ｧ04 040606 y 0 + 04 縺03 ｧ 0403 閨00306 04 06050003040308 0503 0906 05 07 ｧ09. 97. 7 0305 縺06 0906 [0; 4] 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ y = 0; 5 (x, ) ｧ 0 0805 ｧ 0006040606 y, 0 ｧ04 040606 y0 + 04 縺03 ｧ 0403 閨00306 04 06050003040308 0503 0906 05 07 ｧ09 070600909 ｧ ｧ. sin x 98. 7 0305 縺06 0906 [,; ] 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ y = ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 05060004003080406050003040308 0503 0906 09 罍ｧ040308. 99. 7 0305 縺06 0906 [0; ]0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ y = +cos x, 0 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 05060004003080406050003040308 0503 0906 09 罍ｧ040308 ｧ 0 0805 ｧ 06050300 00607 閨06 0 ｧ0 ｧ. 930. 7 0305 縺06 0906 [,; ] 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ y = 3 x ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308040503 0906 x =0. 93. 7 0305 縺06 0906 [0; 4] 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ y =,, 3 (x, ) ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 00608 04 0503 0906 x =. 93. 7 0305 縺06 0906 [,; ] 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 y 3 =4x ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 00608 04 0503 0906 x =0. 933. 7 0305 縺06 0906 [0; 4] 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 y 3 = 4(, x) ｧ 0 04 ｧ- ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 00608 04 0503 0906 x =. 934. 7 0305 縺06 0906 [0; ] 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ y =,, cos x ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 09 09 罍ｧ040308 04 06050003040308 0503 0906. 935. 7 0305 縺06 0906 [,; 0] 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ y = = 3 (x +), ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 09 罍ｧ040308 04 0503 0906 x =,. 936. 7 0305 縺06 0906 [,; 5] 0403 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ y = j4x,x j ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 04 06050003040308 0503 0906 x =0 ｧ 0 0805 ｧ 06050300 00607 閨06 0 ｧ0 ｧ. x 5. 7 縺03 ｧ 0403 閨00306 000305 罍ｧ070 ｧ 06 ﾀ09 ｧ08 ｧ 040309 0506000400308 070600909 ｧ08 05 ﾀ0030400306 0703 縺0060003: (ln u) 0 = u0 u ; (eu ) 0 = e u u 0 ; (a u ) 0 = a u ln a u 0 :

3x 5. 7 縺03 ｧ 0403 閨00306 000305 罍ｧ070 ｧ 06 ﾀ09 ｧ08 ｧ 040309 0506000400308 070600909 ｧ08 5 7 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 937. ) y = x ln x; ) y = +lnx ; 3) y =lg(5x). x 938. ) y =lnx, x, x ; ) y =ln(x +x): 839. ) y =ln(+cosx); ) y = ln sin x, sin x: 940. y =ln, x + x +. 94. y =ln a + x a, x : 94. y =ln x, x : r 943. y =lntg 4 + x +x : 944 y =ln, x : 945. y =ln(x + a + x ). 946. y = x, 4ln(+ x). 947. ) y = cos x sin x +lntg x ; ) y =ln x,. ax 4 948. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 09 罍ｧ040308 y =lnx 040503 0906 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 03 ﾀ0406 Ox. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ09 ﾀ 0506000400606. 949. 70309 0504, 0503 04 030300 y = x 09 ﾀ 0605 ﾀ07 09 罍ｧ040308 y =lnx, e ｧ 0 0805 ｧ 0503 0906 09 ﾀ 0 ｧ07. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040306. 7 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 950. ) y = x +3 x ; ) y = x x ; 3) y = x e x : 95. ) y = a sin x ; ) y = e,x ; 3) y = x e,x : 95. y =(e x=, e,x= ): 953. y = xe x : 954. y = +ex, e x : 955. y = ex=a cos x a : 956. ) y = e,x (sin x + cos x); ) y =ln(e,x + xe,x ). e x 957. y =ln x + : 958. y =(eax, e,ax ). 959. f (t) =ln(+a,t ); 0 0805 ｧ f 0 (0). 960. 703 09 09 ｧ0 060500030 09 罍ｧ04 07 y = e x 0406 縺06 ﾀ0609 0605 03 ﾀ04Oy? 96. 6 70309 0504, 0503 閨00 ｧ0 0403 閨09 ﾀ 0506000400308 04 0006030308 0503 0906 09 罍ｧ040308 y = e x=a 040 a.

36 0600. 6. 7 縺03 ｧ 0403 閨0 07 ｧ 陦ｧ070706 縺06009 ｧ 00 96. 7 縺06 閨04 罍ｧ050600040030 000305 罍ｧ070 ｧ 縺0304 0 ｧ060 0 0805 ｧ 04 縺03 ｧ 0403-00306 070600909 ｧ08: ) y = x x ;)y = x sin x. 7 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 963. y = ln cos x, cos x., 964. y =ln x, x, : 965. y =ln + x +. x 966. y = ln (sin x + + sin x). 967. y =ln 969. y =ln x, x : 968. y = ln tg x + ln cos x: r sin x : 970. y = ln ( + sec x):, sin x 97. y = a ln ( x + a + x), x + ax. 97. y = ae,x=a + xe,x=a : 973. y = a (ex=a + e,x=a ): 974. y = ex + e,x e x, e,x : 975. y =ln(ex + e 4x +): 976. y =ln s e 4x e 4x + : 977. y = x=x : 978. f (t) =ln +tg t, tg t ; 0 0805 ｧ f 0 (=3). 979. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 09 罍ｧ040308 y =, e x= 04 0503 0906 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 03 ﾀ0406 Oy. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040606, 09 ﾀ 0506000400606 ｧ ﾀ ｧ00405030506 09 罍ｧ040308. x 6. 7 縺03 ｧ 0403 閨00306 0303 0500308 05 罍ｧ050300300605 罍ｧ 06 ﾀ09 ｧ08 070600909 ｧ08 (arcsin u) 0 = (arctg u) 0 = 7 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 980. y =, x + arcsin x. u 0, u ; (arccos u)0 =, u0, u ; u0 +u ; (arcctg u)0 =, u0 +u :

3x 7. 7 縺03 ｧ 0403 閨00306 05 ｧ0406 縺030300 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08 7 98. y = x, arctg x: 98. y = arcsin, 4x: 983. y = arcsin x a : 984. y = arctg x a : 985. y = arccos (, x): 986. y = arcctg +x, x : 987. ) y = x, x + arcsin x; ) y = arcsin (e 3x ). r +x 988. y = arctg x +ln : 989. y = arccos., x x 990. y = x arctg x a, a ln (x + a ). 7 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 99. y = arcsin x: 99. y = arctg 6x,. 993. ) y = arccos (, x ); ) y = arcctg x, x. 994. y = e x, e x + arcsin e x. 995. y = x arccos x,, x. r 996. y = arctg e x e x + +ln e x,. t 997. s = 4t, t + 4 arcsin. 998. y = arccos, x + x, 4x. 999. f (z) =(z +)arctg e,z ; 0 0805 ｧ f 0 (0). x 7. 7 縺03 ｧ 0403 閨00306 05 ｧ0406 縺030300 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08,. 0504 縺06 閨0600060 ｧ07. 6 503 07060 ｧ07 ex e,x, ex + e,x ｧ ｧ0803050030006-0 ｧ07 0 0304 0605 ﾀ07 ﾀ030305040605 ﾀ0504060003 05 ｧ0406 縺030300 ｧ 06 ﾀ09 ｧ0 ｧ ﾀ ｧ006 ﾀ030; 0903 ﾀ ｧ006 ﾀ030; 05 005060 ﾀ030 ｧ 090305 005060 ﾀ030 ｧ 030303 0 0605 ﾀ07 sh x = ex, e,x ; ch x = ex + e,x ; th x = sh x ch x ; ch x cth x = sh x :. ｮ040308 ﾀ0504 05 ｧ0406 縺030300 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08: ) ch x, sh x =; 4) sh 0 = 0; ch0= ; ) ch x +sh x =chx; 5) (sh x) 0 =ch x; (ch x) 0 =shx; 3) sh x =shx ch x; 6) (th x) 0 = ch x ; (cth x)0 =, sh x :

38 0600. 6. 7 縺03 ｧ 0403 閨0 07 ｧ 陦ｧ070706 縺06009 ｧ 00 7 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 000. ) y =sh x; ) y = x, th x; 3) y = ch x,. 00. f (x) =sh x +ch x ; 0 0805 ｧ f 0 (0) + f (0). 00. ) y =ln[ch x]; ) y =thx + cth x: 003. ) y = x, cth x; ) y = ln [th x]: 004. ) y = arcsin [th x]; ) y = +sh 4x: 005. 6ｧ8 ｧ0 ｧ07 y = a (ex=a + e,x=a )=ach x 0 0304 0605 ﾀ07 09060400308. a 7 04 ｧ ﾀ 0504 06 040060 ｧ06 003 縺0 00 ｧ 09 05050308 00 ｧ0 ｧ ｧ 04 0503 0906 x = a. 703 ﾀ05 縺03- ｧ0504 09 罍ｧ040606 ｧ 003 縺0 0004. 006. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 05060004003080909 罍ｧ040308 y = shx 04 0503 0906 x =,. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ09 ﾀ 0506000400606 09 00608. 007. 6 70309 0504, 0503 04 縺03060909 ｧ07 03 罍陦ｧ0 0503 0006030308 0503 09 ｧ 09060400308 00 ｧ0 ｧ ｧ y = a ch x 0 0606 003 縺0 0004 06 ﾀ0504 040600 ｧ ｧ0 0403 ﾀ05030700 07, 04- a 0 07 a. x 8. ｮ00600 000306 04 罍ｧ006 縺03 ｧ ｧ 0 陦ｧ070706 縺06009 ｧ 縺0304 0 ｧ06 7 0805 ｧ 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: x, 008. ) y = + arcsin x x ; ) y = tg x + ln cos x. 009. y = 4x, +arcctg 4x,. 00. x =ln(e t +), arctg (e t ). 0. y = 4 ln ( x, 4) + x, 4x. 0. s = 4 tg4 t, tg t, ln (cos t). 03. f (x) =(x + a ) arctg x a, ax; 0 0805 ｧ f 0 (a). 04. ) y =ln x, a 05. f (x) = arcsin x, x ; 0 0805 ｧ f 0 (5). x ; ) y = x(cos ln x + sin ln x). 06. '(u) =e,u=a cos u a ; 040309 0504, 0503 '(0) + a'0 (0) = 0.

3x 9. 7 縺03 ｧ 0403 閨00306 0403 ﾀ00 ｧ08 0403 縺07 閨090304 9 07. f (y) =arctg y a, ln 4 y 4, a 4 ; 0 0805 ｧ f 0 (a). 08. F (z) = cos z + sin z ; 040309 0504, 0503 F 4, 3F 0 4 =3. 09. 70309 0504, 0503 070600909 ｧ07 s = 06 閨03040006050403 縺070605 陦ｧ070706t ln ct 縺06009 ｧ 0004003006 06 040060 ｧ06 t ds dt + s =,ts. 00. 70309 0504, 0503 070600909 ｧ07 x = t, e,t 06 閨03040006050403 縺070605 陦ｧ07- t 0706 縺06009 ｧ 0004003006 06 040060 ｧ06 t dt +x = e,t. x 9. 7 縺03 ｧ 0403 閨00306 0403 ﾀ00 ｧ08 0403 縺07 閨090304 706 ﾀ0504 003 0 0000 ｧ 閨0007 070600909 ｧ ｧ y = f(x) 0606 04 縺03 ｧ 0403 閨00606 y 0 = f 0 (x). 7 縺03 ｧ 0403 閨0 07 0305 05050308 04 縺03 ｧ 0403 閨00308 0 0304 0605 ﾀ07 04 縺03 ｧ 0403 閨00308 040503 縺030503 0403- 縺07 閨09 070600909 ｧ ｧ f(x) ｧ030303 0 0605 ﾀ07 y 00 ｧ00 ｧ f 00 (x) ｧ00 ｧ d y. 縺0 000305 ｧ 003 0304 縺06 閨0600070605 ﾀ07 ｧ 030303 0 0605 ﾀ07 04 縺03 ｧ 0403 閨0 07 05 縺060504060503 0403 縺07 閨09 y 000 = f 000 (x) = d3 y 3, 04 縺03 ｧ 0403 閨0 07 06050406 縺05030503 0403 縺07 閨09 y IV = f IV (x) = d4 y 4, ｧ 04030303006 04 縺03 ｧ 0403 閨0 07 n-0503 0403 縺07 閨09 y (n) = f (n) (x) = dn y n. 0. 7 0805 ｧ 04 縺03 ｧ 0403 閨00606 040503 縺030503 0403 縺07 閨09 070600909 ｧ ｧ: ) y = sin x; ) y = tg x; 3) y = +x. 0. 7 0805 ｧ 04 縺03 ｧ 0403 閨00606 05 縺060504060503 0403 縺07 閨09 070600909 ｧ ｧ: ) y = cos x; ) y = ; 3) y = x sin x. x 03. 7 0805 ｧ 04 縺03 ｧ 0403 閨00606 05 縺060504060503 0403 縺07 閨09 070600909 ｧ ｧ: ) y = x ln x; ) s = te,t ; 3) y = arctg x a. 04. s = t, t + arcsin t ; 0 0805 ｧ d3 s dt. 3 7 0805 ｧ 04 縺03 ｧ 0403 閨00606 n-0503 0403 縺07 閨09 070600909 ｧ ｧ: 05. ) e,x=a ; ) ln x; 3) x: 06. ) x n ; ) sin x; 3) cos x:

30 0600. 6. 7 縺03 ｧ 0403 閨0 07 ｧ 陦ｧ070706 縺06009 ｧ 00 07. 703 ﾀ0006 閨0304 050600040030 陦ｧ070706 縺06009 ｧ 縺0304 0 ｧ060 04030406 ﾀ05 ｧ 0703-0060003 6ｧ80608030 ｧ09 : (uv) 00 = u 00 v +u 0 v 0 + uv 00 ; (uv) 000 = u 000 v +3u 00 v 0 +3u 0 v 00 + uv 000 ; (uv) IV = u IV v +4u 000 v 0 +6u 00 v 00 +4u 0 v 000 + uv IV ｧ05.. 08. 703 0703 縺0060006 6ｧ80608030 ｧ09 0 0805 ｧ 04 縺03 ｧ 0403 閨00606 040503 縺030503 0403- 縺07 閨09 070600909 ｧ ｧ: ) y = e x cos x; ) y = a x x 3 ; 3) y = x sin x. 09. 703 0703 縺0060006 6ｧ80608030 ｧ09 0 0805 ｧ 04 縺03 ｧ 0403 閨00606 05 縺060504060503 0403- 縺07 閨09 070600909 ｧ ｧ: ) y = e,x sin x; ) y = x ln x; 3) y = x cos x. 030. f (x) =xe x=a ; 0 0805 ｧ f 000 (x), f (n) (x), f (n) (0). 03. f (x) = ( + x) m ; 0 0805 ｧ f (0), f 0 (0), f 00 (0), f 000 (0), ::: :::;f (n) (0). x 03. f (x) = ; 040309 0504, 0503 04 罍ｧ n > +x f (n) n, 3 5 ::: (n, 3) (0) = (,) n, n: 033. f (x) = ; 040309 0504, 0503, x n! 04 罍ｧ n =m; f (n) (0) = 0 04 罍ｧ n =m, : ｰ 09 0 ｧ 06. 7 罍ｧ0060 ｧ0504 050307 閨06 ﾀ050403, x = +x + :, x 034. 7 縺03 陦ｧ070706 縺06009 ｧ 縺0304 04 050307 閨06 ﾀ050403 (x, )(x + x 3 + ::: :::+ x n )=x n+, x 05 罍ｧ 0403 x ｧ 0403000307 ｧ04 05060 x =, 0 0805 ｧ P ﾀ060006 n (n +)n(n, ) k(k, ) = ｧ 05060 ﾀ060006 0904 陦 050304 k= 3 ｧ ﾀ0600 0 0506 00040030503 縺07 nx k= k = + + :::+ n n(n + )(n +) = : 6 035. 7 0805 ｧ 04 縺03 ｧ 0403 閨00606 040503 縺030503 0403 縺07 閨09 070600909 ｧ ｧ: ) y = e,x ; ) y = ctg x; 3) y = arcsin x.

3x 0. 7 縺03 ｧ 0403 閨0 07 006070400308 070600909 ｧ ｧ 036. 7 0805 ｧ 04 縺03 ｧ 0403 閨00606 n-0503 0403 縺07 閨09 070600909 ｧ ｧ: ) y = a x ; ) y = +x ; 3) y = sin x. 037. f (x) = arcsin x ; 0 0805 ｧ f (), f 0 () ｧ f 00 (). 038. 703 0703 縺0060006 6ｧ80608030 ｧ09 0 0805 ｧ 04 縺03 ｧ 0403 閨00606 05 縺060504060503 0403- 縺07 閨09 070600909 ｧ ｧ: ) y = x 3 e x ; ) y = x sin x a ; 3) y = xf 0 (a, x)+3f (a, x). 039. 70309 0504, 0503 070600909 ｧ07 y = e x cos x 06 閨03040006050403 縺070605 陦ｧ07-0706 縺06009 ｧ 0004003006 06 040060 ｧ06 y IV +4y =0. 040. 70309 0504, 0503 070600909 ｧ07 y = xe,=x 06 閨03040006050403 縺070605 06 04006-0 ｧ06 x 3 y 00, xy 0 + y =0. 04. f (x) =x e,x=a ; 040309 0504, 0503 f (n) n(n, )(,)n (0) =. a n, 04. f (x) =e,x ; 040309 0504, 0503 f (n) (0) =,(n, )f (n,) (0); f (m,) (0) = 0; f m (0) = (,) m (m, )(m, 3) ::: 5 3 : 043. f (x) =x n ; 040309 0504, 0503 f () + f 0 () + f 00 ()! + :::+ f (n) () n! = n : x 0. 7 縺03 ｧ 0403 閨0 07 006070400308 070600909 ｧ ｧ ｭ ﾀ00 ｧ 06 040060 ｧ06 F (x; y) =0, 006 縺060006000306 0305003 ﾀ ｧ05060004003 y, 0304 縺06 閨06-00070605 y 09 09 03 閨003 0 00606 070600909 ｧ06 x, 0503y 0 0304 0605 ﾀ07 006070400308 070600909 ｧ0608 x. ｪ05030303 0 0805 ｧ 04 縺03 ｧ 0403 閨00606 y 0 05050308 006070400308 070600909 ｧ ｧ, 00607003 030306 ﾀ05 ｧ 06 040060 ｧ07 F (x; y) =004 縺03 陦ｧ070706 縺06009 ｧ 縺0304 0504 0403x, ﾀ ﾀ0 05 罍ｧ04 07 y 09 09 070600909 ｧ06 0305 x. 0ｷ3 04030006 0600030503 06 040060 ｧ07 0 08 閨060 ｧ ﾀ090300606 04 縺03 ｧ 0403 閨00606 y 0. ｪ05030303 0 0805 ｧ y 00,00607003 06 040060 ｧ06 F (x; y) =0 閨04 07 閨03 04 縺03 陦ｧ070706- 縺06009 ｧ 縺0304 0504 0403x ｧ05.. 7 0805 ｧ y 0 ｧ 06 040060 ｧ08: 044. ) x + y = a ; ) y =x; 3) x a, y b =. 045. ) x + xy + y =6; ) x + y, xy =0: 046. ) x =3 + y =3 = a =3 ; ) e y, e,x + xy =0: 047. e x sin y, e,y cos x =0.

3 0600. 6. 7 縺03 ｧ 0403 閨0 07 ｧ 陦ｧ070706 縺06009 ｧ 00 048. x = y + arcctg y. 049. e xy, x + y 3 =0; 0 0805 ｧ dy 04 罍ｧ x =0. 050. 7 0805 ｧ y 00 ｧ 06 040060 ｧ08: ) x + y = a ; ) ax + by, xy = c; 3) x m y n =. 05. x a + y b =; 0 0805 ｧ y00 040503 0906 (0; b). 05. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 05060004003080909 罍ｧ040308 x +y +4x,, y, 3=0040503 09 08 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 03 ﾀ0406 Oy. 053. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 003 縺0 00 ｧ 05 ｧ0406 縺03030003 x,y =9, 04 縺030406 閨06000308 ｧ 0503 09 ｧ (5; 4), ﾀ ﾀ ｧ004050305 0 ｧ. 054. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 04 0503 0906 (x 0 ; y 0 ) 0909 罍ｧ- 040308: ) x a + y b =; ) y =x. 055. 7 04 ｧ ﾀ 050406 040060 ｧ07 09 ﾀ 0506000400308 09 ﾀ05 縺03 ｧ 閨06 x =3 +y =3 = = a =3 040503 09 08 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 04 縺0700308 y = x. 056. 703 09 09 ｧ0 060500030 0406 縺06 ﾀ0609 0605 ﾀ07 09 罍ｧ040306 x + y =5 ｧ y =4x? 057. 7 0805 ｧ y 0 ｧ 06 040060 ｧ08: ) x a + y b =; ) x3 + y 3, 3axy =0. 058. 7 0805 ｧ y 00 ｧ 06 040060 ｧ08: ) x, y = a ; ) (x, ) +(y, ) = R ; 3) arctg y = x + y; 4) x + xy + y = a : 059. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308090309 縺0607003 ﾀ05 ｧ x +y + +4x, 4y +3=0040503 09 08 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 03 ﾀ0406 Ox. 703 ﾀ05 縺03 ｧ0504 0309 縺0607003 ﾀ0504 ｧ09 ﾀ 0506000400306. 060. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 050000 ｧ04 ﾀ06 x +4y =6 040503 0906, 04 09030503 縺0308 閨0600 ｧ05 ﾀ07 0403040300 0 0305 縺06 0309 09 ﾀ 0506000400308, 0305 ﾀ06 060-00308 03 ﾀ070 ｧ 090303 罍陦ｧ0 05, ｧ 09030503 07 000607 ｧ05 04 0406 縺040308 06050406 縺05 ｧ. 06. te,s= + se,t= =; 0 0805 ｧ ds 04 罍ｧ t =0. dt 06. t ln x, x ln t =; 0 0805 ｧ 04 罍ｧ t =. dt 063. x sin y, cos y + cos y =0; 0 0805 ｧ y 0 04 罍ｧ y = =.

3x. 6 7 ｧ070706 縺06009 ｧ 00 070600909 ｧ ｧ 3 x. 6 7 ｧ070706 縺06009 ｧ 00 070600909 ｧ ｧ ｭ ﾀ00 ｧ 070600909 ｧ07 y = f(x) 陦ｧ070706 縺06009 ｧ 縺06060 04 0503 0906 x, 05. 06. ｧ0060605 04 05050308 0503 0906 0903006 00606 04 縺03 ｧ 0403 閨00606 y 0,0503 y x = y0 +, 05 閨06! 0 04 罍ｧ x! 0; 0305 ﾀ06 y = y 0 x + x: () 0600 040 07 ﾀ0504 y 0 x 04 罍ｧ 0060 ｧ07 y 070600909 ｧ ｧ, 00 ｧ006080 07 0305003 ﾀ ｧ- 05060004003 x, 0 0304 0605 ﾀ07 陦ｧ070706 縺06009 ｧ 00030 070600909 ｧ ｧ ｧ 030303 0 0605 ﾀ07 dy: dy = y 0 x: () 703000307 ｧ04 04 0703 縺0060006 () y = x, 04030006 ｧ0 = x 0 x = x =x, ｧ 0403050503006 dy = y 0 : (3) ｱ03 縺00600 (3) 0406 縺0 ｧ 04 05030 ﾀ0006 06, 06 ﾀ00 ｧ x 06 ﾀ0504 070600909 ｧ07 003040308 0406 縺06-006000308 t. 0ｷ3 () ﾀ0006 閨060605, 0503 y dy, 05. 06. 04 罍ｧ 閨03 ﾀ05 0503 003 0 00030 =x 04 罍ｧ 0060 ｧ06 070600909 ｧ ｧ 04 罍ｧ0300 ｧ07060003 04003 0606 陦ｧ070706 縺06009 ｧ 0006. 6 5 ﾀ05003 ﾀ05 ｧ, 閨0007 00 ｧ0060800308 070600909 ｧ ｧ y = ax + b ｧ006060: y = dy. 7 0805 ｧ 陦ｧ070706 縺06009 ｧ 0003 070600909 ｧ08: 064. ) y = x n ; ) y = x 3, 3x +3x: 065. ) y = +x ; ) s = gt : 066. ) r =', sin '; ) x = t : 067. ) d(sin t); ) d(, cos u): a 068. ) d x + arctg x ; ) d( +ln); a 3) d cos ' ; 4) d arcsin : x 069. 7 080307 閨060 ｧ060 陦ｧ070706 縺06009 ｧ 00 09 07 閨030503 00060 06 040060 ｧ07 0 0805 ｧ dy ｧ 06 040060 ｧ08: ) x + y = a ; ) xy = a ; 3) x, xy, y =0. 070. ) y = x ; 0 0805 ｧ 04 罍ｧ0300 ｧ07060003 ｧ 0060060 ｧ06 y (y dy), 090305 x ｧ 0060070605 ﾀ07 0305 閨03,0; ) y = x; 0 0805 ｧ 04 罍ｧ0300 ｧ07060003 ｧ 0060060 ｧ06 y, 090305 x ｧ 0060070605 ﾀ07 0305 00 閨03 0. 07. ) ｮ0503 縺030 090603 x = 5 0 0; 0 0. 0504 縺06 閨0600 ｧ0504 03 ﾀ03000605-00606 ｧ 0305003 ﾀ ｧ0506000400606 040305 縺0600003 ﾀ0504 04 罍ｧ 0403 ｧ ﾀ00060 ｧ ｧ 03030060 090603.

34 0600. 6. 7 縺03 ｧ 0403 閨0 07 ｧ 陦ｧ070706 縺06009 ｧ 00 ) 6 700 ｧ0 0506000605 070030503 04 縺030403 s = b + f 3b,05 閨06 b ﾀ ﾀ0503070 ｧ06 00607 閨06 0503 09 0 ｧ 0403 閨0406 ﾀ, f 0 ｧ0303000400 ｧ08 04 縺0305 ｧ03. 7 ﾀ090300040903 06040600 ｧ ｧ05 ﾀ07 04 縺0305 ｧ03 f, 090305 04 縺030403 0305 0 05 縺0604 0 ｧ07 06 閨00 ｧ- 0 ｧ05 ﾀ07 0 ds? 07. ) ｮ09 090308 0503 003 ﾀ050406 00607003 ｧ 006 罍ｧ0504 03 ﾀ09 ｧ ﾀ ﾀ06 09 罍ｧ040308 y = x x 04 罍ｧ x 6, 05030303 04 罍ｧ 0403 ｧ ﾀ00060 ｧ ｧ 0606 03 罍陦ｧ0 0503 閨030406- ﾀ05 ｧ0504 040305 縺0600003 ﾀ0504 006 0303000606 0,? ) ｮ09 090308 0305003 ﾀ ｧ0506000400308 0503 003 ﾀ050406 00607003 ｧ 006 罍ｧ0504 陦ｧ06 ﾀ 00, 05030303 04 罍ｧ 0403 ｧ ﾀ00060 ｧ ｧ 03030060 00 閨030406 ﾀ05 ｧ0504 040305 縺0600003 ﾀ0504 006 0303000606 %? 073. 0504 縺06 閨0600 ｧ0504 04 罍ｧ0300 ｧ07060003: ) 0400030 閨04 09 縺06050304030503 0903000409 ; ) 03030060 ﾀ0706 罍ｧ 06 ﾀ090308 0303030003 09 ｧ. ｮ 040 ｧ0504 ﾀ ｧ08 0503 0030 ｧ 0 060 ｧ- 070 ｧ. 7 0805 ｧ 陦ｧ070706 縺06009 ｧ 0003 070600909 ｧ08: 074. ) y = x, x ; ) r = cos (a, b'); 3) s =, t : 075. ) y = ln cos x; ) z = arctg 4u, ; 3) s = e,t : 076. ) d( x + ); ) d(tg, ); 3) d(bt, e,bt ): 077. ) y = x 3 ; 0304 縺06 閨0600 ｧ0504 y ｧ dy ｧ 0403 ｧ ﾀ00 ｧ0504 ｧ08 04 罍ｧ ｧ 0060060 ｧ ｧ x 0305 閨03,98. ) 706 罍ｧ03 0903000603 0 ｧ07 0 07050 ｧ09 T = l=980 ﾀ, 05 閨06 l 閨00 ｧ0 0 07050 ｧ09 04 ﾀ 005 ｧ00605 08. 07 09 00607003 ｧ 0060 ｧ0504 閨00 ｧ006 0 07050 ｧ09 l =0 ﾀ0, 05030303 0406 罍ｧ03 0903000603 0 ｧ07 0600600400 ｧ00 ﾀ07 0 0, ﾀ? 3) ｮ09 090308 0503 003 ﾀ050406 00607003 ｧ 006 罍ｧ0504 03 ﾀ09 ｧ ﾀ ﾀ06 09 罍ｧ040308 xy = =404 罍ｧ x > 0; 5, 05030303 04 罍ｧ 0403 ｧ ﾀ00060 ｧ ｧ 0606 03 罍陦ｧ0 0503 閨030406 ﾀ05 ｧ0504 040305 縺0600003 ﾀ0504 006 0303000606 0,? x. 7 00605 罍ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 09 罍ｧ040308 706 ﾀ0504 09 罍ｧ04 07 0 04 00605 罍ｧ 06 ﾀ09 ｧ0 ｧ 06 040060 ｧ070 ｧ x = f(t) ｧ y = '(t). 050303 0 07 0503 09 0 ｧ 04 縺03 ｧ 0403 閨00306 0403 04 00605 縺06, 0 08 閨060: dy = _y _x ; d y d(_y= _x) = = y _x, x _y _x 3 : 078. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040306 0403 04 00605 罍ｧ 06 ﾀ09 ｧ0 06 040060 ｧ070: ) x = t ; y = t3 ; ) x = t ; y = t3 3, t. 0ｷ3 ﾀ090006 ｧ04 ｧ 06 040060 ｧ08 t, 0 04 ｧ ﾀ 0504 06 040060 ｧ06 09 07 閨0308 09 罍ｧ040308 04 030303 0030 04 ｧ 閨06: F (x; y) =0.

3x. 7 00605 罍ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 09 罍ｧ040308 5 7 罍ｧ0406 ﾀ05 ｧ 09 04 ｧ 閨06 F (x; y) =0( ｧ00 ｧ y = f (x)) 06 040060 ｧ07 09 罍ｧ- 040308, 000308 04 00605 罍ｧ 06 ﾀ09 ｧ: 079. ) x = a cos t; y = b sin t; ) x = a cos 3 t; y = a sin 3 t. 080. ) x = et + e,t ; y = et, e,t ; ) x = tg t; y = cos t. 08. 703 ﾀ05 縺03 ｧ0504 08 0406 縺05090609, ｧ00 ｧ 08050403000404060050609, 09 縺0605 ( ﾀ0. - 06 368) x = a(cos t + t sin t); y = a(sin t, t cos t); 04 07 t 0 060 ｧ07: 0, =,, 3=,. 08. 703000307 ｧ04 y = xt, 04030006 ｧ0504 04 00605 罍ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 08 閨0609 縺050304 00 ｧ ﾀ05 09 x 3 +y 3,3axy =0( ﾀ0. 06 366) ｧ ｧ ﾀ ﾀ0006 閨0304 0504 閨04 ｧ07060 ｧ06 0503 09 ｧ 0403 09 罍ｧ040308 04 罍ｧ 003003050300030 ｧ 0060060 ｧ ｧ t: ) 0305 0 閨03 +; )0305 0 閨03,; 3)0305, 閨03,. 083. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 09 ｧ090003 ｧ 閨06 ( ﾀ0. 06 367) x = a(t,sin t), y = a(,cos t) 040503 0906, 05 閨06 t = =. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ09 ﾀ 0506000400606. 084. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 05060004003080905 ｧ040309 ｧ090003 ｧ 閨06 ( ﾀ05 縺03- ｧ 閨06) x = a cos 3 t, y = a sin 3 t 040503 0906 t = =4. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ 09 ﾀ 0506000400606. ｰ09 0 ｧ06. 6 70007 0403 ﾀ05 縺03060 ｧ07 09 罍ｧ040308 ﾀ03 ﾀ05 04 ｧ0504 05 0300 ｧ0906 0 060 ｧ08 x ｧ y 04 罍ｧ t =0;=4; =; 3=4 ｧ05.. 085. 7 0805 ｧ d y ｧ 06 040060 ｧ08: ) x = a cos t; y = a sin t; ) x = t ; y = t3 3, t; 3) x = a(t, sin t); y = a(, cos t). 086. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040306, 000306 04 00605 罍ｧ 06 ﾀ09 ｧ0 ｧ 06 04006-0 ｧ070 ｧ: ) x =t, ; y =, 4t ; ) x = t 3 ; y = t,, 0 08 閨07 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ｧ08 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 ｧ 00605 ｧ04, 0503 閨0007 040503 縺0308 09 罍ｧ040308 dy = 04 罍ｧ t =0. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 罍ｧ040308 04 04 ｧ 閨06 F (x; y) =0.

36 0600. 6. 7 縺03 ｧ 0403 閨0 07 ｧ 陦ｧ070706 縺06009 ｧ 00 087. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 09 ｧ090003 ｧ 閨06 x = a(t, sin t); y = a(, cos t) 040503 0906 t =3=. 703 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ09 ﾀ 0506000400606. 088. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 09 0406 縺050906 09 縺0605 040503 0906 t = =4. x = a(cos t + t sin t); y = a(sin t, t cos t) 089. 7 0805 ｧ d y ｧ 06 040060 ｧ08: ) x = cos t; y = sin t; ) x = t ; y = t + t 3 ; 3) x = e t ; y = e 3t.

30600 04 7 7 70ｷ36ｧ8056 9 ｭ 70ｷ300 7 7050ｷ36ｦ06 5056 7 705 x. ｮ0903 縺03 ﾀ0504 ｧ06 ﾀ0903 縺060 ｧ06 706 ﾀ0504 0503 09 閨04 ｧ070605 ﾀ07 0403 03 ﾀ ｧ Ox ｧ 04 003006005 t ｧ0060605 090303 罍陦ｧ0 0506 x = f(t). ｯ0305 04 003006005 t ﾀ0903 縺03 ﾀ0504 v = 06 ﾀ0903 縺060 ｧ06 w = x lim t!0 t =, dt v lim t!0 t = dv dt = d x. dt 090. 6ｦ0060 ｧ0500308 ﾀ0 縺07 040303 縺0300060 0406 縺05 ｧ09 0004003 040406 縺08 ﾀ0 0004-00308 ﾀ0903 縺03 ﾀ050406 a 0/ ﾀ. 7 09 090308 0403 ﾀ030506 x 030 0306 閨0605 06 縺06 t ﾀ06-09060? 0504 縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀ0504 ｧ 06 ﾀ0903 縺060 ｧ06 閨04 ｧ07060 ｧ07 ﾀ0 縺07. ｪ06 縺06 ﾀ090300040903 ﾀ0609060 ﾀ0 縺07 閨03 ﾀ05 ｧ0500605 0 ｧ0403 ﾀ000608 0503 09 ｧ ｧ 0 09 09030 ﾀ- ﾀ0503070 ｧ ｧ 0305 06000 ｧ? 09. ｯ060003 閨04 ｧ070605 ﾀ07 0403 04 縺0700308 Ox 0403 0903006 x = t3 3, t +3t. 0504 縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀ0504 ｧ06 ﾀ0903 縺060 ｧ06 閨04 ｧ07060 ｧ07. 6 509 09 ｧ06 00300600503 05060003 0060070605 0 04 0400060 ｧ06 閨04 ｧ07060 ｧ07? 09. 0703000603 0506000400306 閨04 ｧ07060 ｧ06 0 0506 罍ｧ 000400308 0503 09 ｧ ﾀ030406 縺00-0605 ﾀ07 0403 0903006 x = a cos!t. 0504 縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀ0504 ｧ 06 ﾀ0903 縺060 ｧ06 閨04 ｧ07060 ｧ07 04 0503 09 08 x = a ｧ x = 0. 70309 0504, 0503 06 ﾀ0903 縺060 ｧ06 d x ｧ03050900030060 ｧ06 x ﾀ0407 003 08 陦ｧ070706 縺06009 ｧ 0004003009 06 040060 ｧ060 dt d x dt =,! x. 093. 6 5 0 0600606 ﾀ07 0 0803040306 09030006 ﾀ03, 閨06 縺07 ｧ04 0600306 0503 縺003 030, t ﾀ0609060 04030403 ｧ04 0605 ﾀ07 0 06050300 ' = a + bt, ct,05 閨06 a, b ｧ c 0403000307 ｧ0506000400306 0403 ﾀ050307000306. 0504 縺06 閨0600 ｧ0504 06050003040606 ﾀ0903 縺03 ﾀ0504 ｧ 06 ﾀ0903 縺060 ｧ06 04 0060 ｧ07. 070305 09030006 ﾀ03 03 ﾀ05 00304 ｧ05 ﾀ07? 094. 07030006 ﾀ03 陦ｧ06 ﾀ a 09 05 ｧ05 ﾀ07 0403 04 縺0700308. ｰ050300 ' 04030403 縺0305 09030006 ﾀ t ﾀ0609060 0304 縺06 閨0600070605 ﾀ07 06 040060 ｧ060 ' = t+ t. 0504 縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀ0504 ｧ 06 ﾀ0903 縺060 ｧ06 閨04 ｧ07060 ｧ07 0906005 09030006 ﾀ. 095. 706 ﾀ0504 v ﾀ0903 縺03 ﾀ0504 ｧw 06 ﾀ0903 縺060 ｧ06 0503 09 ｧ, 閨04 ｧ070600608 ﾀ07 0403 03 ﾀ ｧ Ox. 70309 0504, 0503 w= vdv.

38 0600.7. 7 罍ｧ000307060 ｧ07 04 縺03 ｧ 0403 閨00308 096. ｯ03 09 閨04 ｧ070605 ﾀ07 04 縺0700300 ｧ00608003 05 09, 0503 v =ax,05 閨06 v ﾀ0903 縺03 ﾀ0504, x 04 縺0308 閨06000308 04060504 ｧ a 0403 ﾀ05030700 07. 0504 縺06 閨0600 ｧ0504 06 ﾀ0903 縺060 ｧ06 閨04 ｧ07060 ｧ07. 097. ｯ060003 ﾀ 0403 ﾀ030503 0 0 03 縺030006003 0406 縺05 ｧ09 0004003 040406 縺08 ﾀ0 0004-00308 ﾀ0903 縺03 ﾀ050406 0 0/ ﾀ. 7 09 090308 0403 ﾀ030506 x 03003 0306 閨0605 06 縺06 t ﾀ0609060? 0504 縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀ0504 ｧ06 ﾀ0903 縺060 ｧ06 閨04 ｧ07060 ｧ07. ｪ06 縺06 ﾀ090300040903 ﾀ0609060 05060003 閨03 ﾀ05 ｧ0500605 0 ｧ0403 ﾀ000608 0503 09 ｧ ｧ0 09 090308 0403 ﾀ030506? 098. ｮ03 ﾀ06 04 0703 縺006 0403000600 陦ｧ06 ﾀ R ﾀ0 0 0403000070605 ﾀ07 0403 閨0308 ﾀ 0403 ﾀ050307000308 ﾀ0903 縺03 ﾀ050406 a 00/ ﾀ. 0504 縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀ0504 0403040300060 ｧ07 06 縺0304007 0 0403 ﾀ030506 06 縺0304007 h ﾀ0 ｧ 040309 0504, 0503 030 0303 05003 04 縺030403-09 ｧ030 00040 0400030 陦ｧ ﾀ04030303 閨00308 04030406 縺08003 ﾀ05 ｧ 07 ｧ 閨0903 ﾀ05 ｧ. ｰ09 0 ｧ06. 05030060 00 縺0304030503 ﾀ0605006005 V = h. 050306 R, h 3 ﾀ05 ｧ 0505030503 04060 ﾀ0504 00607003 04 縺03 陦ｧ070706 縺06009 ｧ 縺0304 0504 0403t, 04 罍ｧ 060 dv = dt a (0403 06 ﾀ000304 ｧ06). 099. 6ｦ0 04 ｧ ﾀ ｧ003 ﾀ0504 00607 閨06 090300 ｧ 06 ﾀ0504030 x 0406006 ﾀ0504, 04030006 06-0030503 04 00609030503 縺0308 08 ｧ0 ｧ 06 ﾀ090308 縺06 0909 ｧ ｧ, ｧ 04 縺060060060 t 0403 07 0605 ﾀ07 06 040060 ｧ060 x = A(, e,kt ). 0504 縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀ0504 縺06 0909 ｧ ｧ. 00. 706 ﾀ0504 0605000304 07 ﾀ0903 縺03 ﾀ0504 d' d! =!, 06050003040306 06 ﾀ0903 縺060 ｧ06 dt dt = = ". 70309 0504, 0503 d(! ) d' =". x. ｯ0603 縺06003 03 ﾀ 縺06 閨0060. ｯ0603 縺060 703000007. ｭ ﾀ00 ｧ f(x): ) 00604 縺06 縺03040 0 0305 縺06 0906 [a; b], ) ｧ0060605 04 縺03 ｧ 0403 閨00606 0400605 罍ｧ 0060503, 3) f(a) =f(b), 050300607 閨06 a ｧ b 0 08- 閨0605 ﾀ07 05 090306 x = c, 04 罍ｧ 09030503 縺030 f 0 (c) =0: (). ｯ0603 縺060 6ｧ8 05 007. ｭ ﾀ00 ｧ f(x): ) 00604 縺06 縺03040 0 0305 縺06 0906 [a; b], ) ｧ0060605 04 縺03 ｧ 0403 閨00606 0400605 罍ｧ 0060503, 0503 00607 閨06 a ｧ b 0 08 閨0605 ﾀ07 05 090306 x = c, 04 罍ｧ 09030503 縺030 f(b), f(a) =(b, a)f 0 (c): () 3. ｯ0603 縺060 070300 ｧ. ｭ ﾀ00 ｧ f(x) ｧ '(x): ) 00604 縺06 縺0304003 0 0305 縺06 0906 [a; b], ) ｧ0060605 04 縺03 ｧ 0403 閨00306 0400605 罍ｧ 0060503, 04 罍ｧ 060 ' 0 (x) 6= 0,050300607 閨06 a ｧ b 0 08 閨0605 ﾀ07 05 090306 x = c, 04 罍ｧ 09030503 縺030 f(b), f(a) '(b), '(a) = f 0 (c) ' 0 (c) : (3)

3x. ｯ0603 縺06003 03 ﾀ 縺06 閨0060 9 705 ｧ 050603 縺06003 003 ﾀ0705 0 04 0 ｧ06 050603 縺060 03 ﾀ 縺06 閨0060 04030503006, 0503 04 0 ｧ08 05030403 罍ｧ05 ﾀ07 03 00609030503 縺030 0 060 ｧ ｧ x = c, ﾀ 縺06 閨0060 00607 閨06 a ｧ b. 06060300605 罍ｧ 06 ﾀ09 ｧ 050603 縺06003 703000007 ｧ 6ｧ8 05 007 06050406 縺07 0605, 0503 0 閨060506 AB 00604 縺06 縺030400308 09 罍ｧ040308 y = f(x), ｧ0060600608 04 09 07 閨0308 0503 0906 0304 縺06 閨0600060-00606 09 ﾀ 0506000400606 ｧ 006 ｧ0060600608 0503 0609 0403 04 05, 0 08 閨0605 ﾀ07 0400605 縺06000707 0503 09, 09 ﾀ 050600040 07 04 09030503 縺0308 04 00000600040 0803 罍閨06 AB. 7 閨0605 08, ﾀ03 閨06 縺07 0 ｧ08 06050003040306 0503 09 ｧ ｧ00 ｧ 0503 09 ｧ 0403 04 05, 06 ﾀ000304 ｧ07 050603 縺060 03 ﾀ 縺06 閨0060, 03 0604 ｧ 閨003, 006 0403040300006003. ｯ0603 縺06006 703000007 04 ﾀ050030 ﾀ0006 06 04 罍ｧ f(b) =f(a) =00703 縺00600 ｧ 縺060605 05 09: 00607 閨06 閨0406007 0903 縺0070 ｧ a ｧ b 070600909 ｧ ｧ f(x) 0 08 閨0605 ﾀ07 0403 09 0800608 006 縺06 03 陦ｧ0 0903 縺06004 0606 04 縺03 ｧ 0403 閨00308 f 0 (x), 06 ﾀ00 ｧ f(x) 00604 縺06 縺03040 0 0305 縺06 0906 [a; b] ｧ ｧ0060605 04 縺03 ｧ 0403 閨00606 0400605 罍ｧ 0060503. 0. 7 縺030406 罍ｧ0504, 0503 00607 閨06 0903 縺0070 ｧ 070600909 ｧ ｧ f (x) =x,4x + +30 0803 陦ｧ05 ﾀ07 0903 縺06004 0606 04 縺03 ｧ 0403 閨00308. 70307 ﾀ0 ｧ0504 05 07 ｧ 06 ﾀ09 ｧ. 0. 7 罍ｧ0060 ｧ0 00 ｧ 050603 縺060 703000007 09 070600909 ｧ ｧ f (x) =, 3 x 0 0305 縺06 0906 [,; ]? 70307 ﾀ0 ｧ0504 05 07 ｧ 06 ﾀ09 ｧ. 03. 703 ﾀ05 縺03 ｧ0504 閨060506 AB ^ 09 罍ｧ040308 y = j sin xj 0 0305 縺06 0906 [,=; =]. 703 06006 0 閨060506 00605 09 ﾀ 0506000400308, 04 000006000400308 0803 罍閨06 AB? 07 090306 ｧ 06 ﾀ000304 ｧ08 050603 縺06003 703000007 閨06 ﾀ04 006 0403040300006003? 04. 6 5 09 090308 0503 0906 09 ﾀ 050600040 07 09 04 03030006 y = x 04 00-000600040 0803 罍閨06, ﾀ050705 ｧ04 0600608 0503 09 ｧ A(,; ) ｧ B(3; 9)? 70307 ﾀ0 ｧ0504 05 07 ｧ 06 ﾀ09 ｧ. 05. 7 04 ｧ ﾀ 0504 0703 縺0060006 6ｧ8 05 007 閨0007 070600909 ｧ ｧ f (x) =x 0 0305 縺06 0906 [a; b] ｧ 0 0805 ｧ c. 70307 ﾀ0 ｧ0504 05 07 ｧ 06 ﾀ09 ｧ. 06. 7 04 ｧ ﾀ 0504 0703 縺0060006 6ｧ8 05 007 閨0007 070600909 ｧ ｧ f (x) = x 0 0305 縺06 0906 [; 4] ｧ 0 0805 ｧ c. 07. 70309 0504, 0503 0 0305 縺06 0906 [,; ] 050603 縺060 6ｧ8 05 007 006-04 罍ｧ0060 ｧ0 09 070600909 ｧ070 4 x ｧ, 3 x. 70307 ﾀ0 ｧ0504 05 07 ｧ 06 ﾀ09 ｧ. 08. 703 ﾀ05 縺03 ｧ0504 AB ^ 09 罍ｧ040308 y = j cos xj 0 0305 縺06 0906 [0; =3]. 703 06006 0 閨060506 00605 09 ﾀ 0506000400308, 04 000006000400308 0803 罍閨06 AB? 07 090306 ｧ 06 ﾀ000304 ｧ08 050603 縺06003 6ｧ8 05 007 閨06 ﾀ04 006 0403040300006003? 09. 706 ﾀ0504 f (x) = x 04 罍ｧ jxj < ; 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 05050308 04 罍ｧ jxj > : 070600909 ｧ ｧ ｧ, 04 0704 0 0060 0503 09 ｧ O(0; 0) ｧ B(; ), 040309 0504, 0503 00607 閨06 O ｧ B 0 0505030 05 07 ｧ0906 00605 0503 09 ｧ, 09 ﾀ 050600040 07 04 09030503 縺0308 030300 0303 04 00000600040 OB. 07 09 ｧ06 06 ﾀ000304 ｧ07 050603 縺06003 6ｧ8 05 007 閨0007 05050308 070600909 ｧ ｧ 0 0305 縺06 0906 [0; ] 0403040300006003 ｧ 09 09 ｧ06 00605? 0. 70306 04 縺03000600 ﾀ ﾀ0503070 ｧ06 00607 閨06 ﾀ05 009 ｧ070 ｧ ﾀ03 ﾀ 縺06 閨00608 ﾀ0903 縺03 ﾀ050406 v 0 =40090/. ｯ0603 縺060 6ｧ8 05 007 06050406 縺07 0605, 0503 030300 003006005 閨04 ｧ07060 ｧ07, 04 09030503 縺0308 ｧ ﾀ05 ｧ00 07 ( 006 ﾀ 縺06 閨00707) ﾀ0903 縺03 ﾀ0504 閨04 ｧ07060 ｧ07 ds 030300 040 40 090/. 70309 0504 050503. dt

330 0600.7. 7 罍ｧ000307060 ｧ07 04 縺03 ｧ 0403 閨00308. 6 7 003, 0503 f (x) 00604 縺06 縺03040 0 0305 縺06 0906 [a; b] ｧ ｧ0060605 04 縺03- ｧ 0403 閨00606 0409 07 閨0308 0503 0906 0400605 罍ｧ 0060503. 7 罍ｧ0060 ｧ04 050603 縺06006 703000007 09070600909 ｧ ｧ x f (x) (x) = b f (b) a f (a) ; 04030006 ｧ0504 050603 縺06006 6ｧ8 05 007. 6 50307 ﾀ0 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 0 060 ｧ06 070600909 ｧ ｧ (x). f (b), f (a). 7 04 ｧ ﾀ 0504 0703 縺0060006 070300 ｧ '(b), '(a) = f 0 (c) ' 0 閨0007 0706009- (c) 09 ｧ08 f (x) =x 3 ｧ '(x) =x ｧ 0 0805 ｧ c. 3. 06060300605 罍ｧ 06 ﾀ09 ｧ 050603 縺060 070300 ｧ 06050406 縺07 0605, 0503 0 閨060506 09 罍ｧ040308 x = '(t), y = f (t) 閨0007 0 060 ｧ08 t 0 0305 縺06 0906 a 6 t 6 b 0 08 閨0605 ﾀ07 0400605 縺06000707 0503 09, 04 09030503 縺0308 09 ﾀ 050600040 07 04 00000600040 0803 罍閨06, 06 ﾀ00 ｧ 070600909 ｧ ｧ '(t) ｧ f (t) 0 0305 縺06 0906 [a; b] 06 閨03040006050403 縺070605 06 ﾀ000304 ｧ070 050603 縺06003 070300 ｧ. 6 70309 0504 050503. 4. 7 04 ｧ ﾀ 0504 0703 縺0060006 6ｧ8 05 007 04 04 ｧ 閨06 f (x +x), f (x) = = xf 0 (x + x), 05 閨06 0 <<, 閨0007 070600909 ｧ08: ) f (x) =x ; ) f (x) =x 3, ｧ 040309 0504, 0503 閨0007 0406 縺040308 070600909 ｧ ｧ 006 04 ｧ ﾀ ｧ05 0305 x, 閨0007 040503 縺0308 04 ｧ ﾀ ｧ05 0305 x ｧ x. 5. 70309 0504, 0503 0 = 0 + 0; 05. 00 + 6. ｮ 040300300406 0703 縺0060003 070300 ｧ 閨0309 0504, 0503 06 ﾀ00 ｧ 0503 05 閨06 0 <<: f (0) = f 0 (0) = f 00 (0) = :::= f (n,) (0) = 0; f (x) x n = f (n) (x) ; n! 7. 7 04 ｧ ﾀ 0504 0703 縺0060006 6ｧ8 05 007 f (b), f (a) =(b, a)f 0 (c) 閨0007 070600909 ｧ ｧ f (x) =x 3 ｧ 0 0805 ｧ c. 8. 7 04 ｧ ﾀ 0504 0703 縺0060006 6ｧ8 05 007 ｧ 0 0805 ｧ c 閨0007 070600909 ｧ08: ) f (x) =arctg x 0 0305 縺06 0906 [0; ]; ) f (x) = arcsin x 0 0305 縺06 0906 [0; ]; 3) f (x) =lnx 0 0305 縺06 0906 [; ]:

3x 3. 7 ﾀ09 縺0305 ｧ06 0060304 縺06 閨0600060003 ﾀ050608. 7 04 ｧ0003 6ｧ80304 ｧ05 0007 3 9. 7 04 ｧ ﾀ 0504 0703 縺0060006 070300 ｧ ｧ 0 0805 ｧ c 閨0007 070600909 ｧ08: ) sin x ｧ cos x 0 0305 縺06 0906 [0; =]; ) x ｧ x 0 0305 縺06 0906 [; 4]: 0. 703 ﾀ05 縺03 ｧ0504 05 07 ｧ09 070600909 ｧ ｧ y = jx, j 0 0305 縺06 0906 [0; 3]. 703 06006 閨06 ﾀ04 0060004 07 04 縺030406 ﾀ05 ｧ 09 ﾀ 0506000400606, 04 000006000400606 0803 罍閨06? 07 090306 ｧ 06 ﾀ000304 ｧ08 050603 縺06003 6ｧ8 05 007 閨06 ﾀ04 006 0403040300006003?. 6 509 090308 0503 0906 09 ﾀ 050600040 07 09 09 罍ｧ040308 y =4, x 04 00-000600040 0803 罍閨06, ﾀ050705 ｧ04 0600608 0503 09 ｧ A(,; 0) ｧ B(; 3)? 70307 ﾀ0 ｧ0504 05 07 ｧ 06 ﾀ09 ｧ. x 3. 7 ﾀ09 縺0305 ｧ06 0060304 縺06 閨0600060003 ﾀ050608. 7 04 ｧ0003 6ｧ80304 ｧ05 0007. 7060304 縺06 閨0600060003 ﾀ0504 ｭ ﾀ00 ｧ lim x!a 0 ｧ08 ﾀ06006 ﾀ0504060605. f(x) = lim '(x) =0,0503 lim x!a x!a 0. 706 縺040306 04 04 ｧ0003 6ｧ80304 ｧ05 0007. 0 f(x) '(x) = lim f 0 (x) x!a ' 0, 090305 0403 ﾀ0006 - (x). 7060304 縺06 閨0600060003 ﾀ0504. 6 50503 縺0306 04 04 ｧ0003 6ｧ80304 ｧ05 0007. f 0 (x) f(x) ｭ ﾀ00 ｧ lim f(x) = lim '(x) =, 0503 lim x!a x!a x!a '(x) = lim x!a ' 0, 090305 0403 ﾀ0006 - (x) 0 ｧ08 ﾀ06006 ﾀ0504060605. 3. 7060304 縺06 閨0600060003 ﾀ05 ｧ 0,,, ｧ 0 0 ﾀ0403 閨0705 ﾀ07 09 006-0304 縺06 閨0600060003 ﾀ05070 0 0 ｧ 040605060 00050603 ｧ 06 ﾀ09 ｧ08 04 縺060303 0304 0 ｧ08. 7 0805 ｧ 04 縺06 閨060003: sin 3x e x,. lim : 3. lim x!0 x x!0 sin x : x, a 4. lim x!a x n, a n :, cos ax 6. lim x!0, cos bx x, 5. lim x! ln x : : 7. lim x!0, cos x x : 8. lim x!0 x, sin x x 3 : 9. lim x!0 tg x, sin x x, sin x : 30. ) lim x!+ e x ; x3 ) lim x!, e x : 3. lim x3 x! ln x x : ln x tg x 3. lim : 33. lim x!0 ctg x x!= tg 3x :

33 0600.7. 7 罍ｧ000307060 ｧ07 04 縺03 ｧ 0403 閨00308 34. lim x! (, x)tg x : 36. lim x!+ xn e,x : 35. lim x ln x: x!0 37. lim x x : x!0 38. lim x!0 (sin x) tg x : 39. lim x! + 3 x x : 40. 0504 縺06 閨0600 ｧ0504 0403 縺07 閨0309 0306 ﾀ0903006 003 0 000308 xe x, sin x 0305003- ﾀ ｧ05060004003 x! 0. 4. 6 70309 0504, 0503 04 罍ｧ x! 0: ) x, arctg x x3 3 ; ) ax, b x x ln a b ; 3) e x,, x x ; 4) x, ln ( + x) x : 4. 6 70309 0504, 0503 (04 罍ｧ x! 0) x,sin x x3 ｧ0305 ﾀ06 sin x 6 x ﾀ 040305 縺0600003 ﾀ050406, 04 罍ｧ0300 ｧ07060003 0400308 x 3 =6. 6 503 ｧ ﾀ00 ｧ0504 sin ｧ sin 6 ｧ 0309060 ｧ0504 040305 縺0600003 ﾀ0504. 43. 6 70309 0504, 0503 (04 罍ｧ! 0) 3 +,, 3, 9 ｧ0305 ﾀ06 3 + + ﾀ 040305 縺0600003 ﾀ050406 3 9. 6 503 ｧ ﾀ00 ｧ0504 3 ; 006, 3 0; 99, 3 65, 3 0 ｧ 0309060 ｧ0504 040305 縺0600003 ﾀ0504. 7 0805 ｧ 04 縺06 閨060003: e ax, e bx x, arctg x 44. lim : 45. lim : x!0 sin x x!0 x 3 46. lim x!=a 48. lim x!=6, sin ax (ax, ) : 50. lim x!0 e x, ln ( + x) : 47. lim x!0, sin x : 49. lim cos 3x x!=4 a x, l x tg x :, tg x cos x : ln x 5. lim x!, x : 3 5. lim x!0 (, e x )ctg x: 53. lim x! x =(,x) : 54. lim x!0 x sin x, : 55. lim (e x + x) =x : x x!0 56. 6 70309 0504, 0503 04 罍ｧ x! 0 arcsin x, x x3 6.

3x 4. 6 503 ﾀ05 0 ｧ06 ｧ 06030304 0 ｧ06 070600909 ｧ ｧ 33 57. 6 70309 0504, 0503 (04 罍ｧ! 0) +,,, 8 ｧ 0305 ﾀ06 + + ﾀ 040305 縺0600003 ﾀ050406, 04 罍ｧ0300 ｧ07060003 0400308 8. 6 503 ｧ ﾀ00 ｧ0504 ; 006, ; 004, 0; 998, 0; 994, 65, 85 ｧ 0309060 ｧ0504 040305 縺0600003 ﾀ0504. x 4. 6 503 ﾀ05 0 ｧ06 ｧ 06030304 0 ｧ06 070600909 ｧ ｧ. 03 09 ﾀ ｧ0060 ｧ 0 ｧ0 ｧ0060. 0504 縺06 閨0600060 ｧ07: I. ｱ0600909 ｧ07 f(x) 0 0304 0605 ﾀ07 0403 ﾀ05 0600608 04 0503 0906 x 0, 06 ﾀ00 ｧ 04 006-09030503 縺0308 "-0309 縺06 ﾀ05003 ﾀ05 ｧ 05050308 0503 09 ｧ f(x 0, h) <f(x 0 ) <f(x 0 + h) 04 罍ｧ 000603030 0403000307 ｧ050600040030 h<". II. ｱ0600909 ｧ07 f(x) 0 0304 0605 ﾀ07 0403 ﾀ05 0600608 0 0305 縺06 0906 [a; b], 06 ﾀ00 ｧ 閨0007 0006030308 x ｧ x 0 0505030 0305 縺06 0906 f(x ) <f(x ), 090305 x <x. 縺0 000305 ｧ 003 0304 縺06 閨0600070605 ﾀ07 06030304 0 ｧ06 070600909 ｧ ｧ 04 0503 0906 ｧ 0 0305 縺06 0906. III. ｱ0600909 ｧ07 f(x) 0 0304 0605 ﾀ07 ｧ0060600608 0509 ﾀ05 縺060060 (0 09 ﾀ ｧ0060 ｧ00 ｧ 0 ｧ0 ｧ0060)040503 0906 x 0, 06 ﾀ00 ｧ f(x 0 ) 070400070605 ﾀ07 0 ｧ0303000400 ｧ0 ｧ00 ｧ 0 ｧ00600400 ｧ0 0 060 ｧ060 070600909 ｧ ｧ 04 00609030503 縺0308 閨0406 ﾀ0503 縺03000608 0309 縺06 ﾀ05003 ﾀ05 ｧ 05050308 0503 09 ｧ.. 6 703 ﾀ05 0503 00306 04 罍ｧ 0 09 ｧ 0403 ﾀ05 0 ｧ07 ｧ 06030304 0 ｧ07 070600909 ｧ ｧ y = f(x) (04 0503 0906 ｧ 0 0305 縺06 0906): 06 ﾀ00 ｧ y 0 > 0, 0503070600909 ｧ07 0403 ﾀ05 0605; 06 ﾀ00 ｧ y 0 < 0, 0503070600909 ｧ07 06030304 0605. 3. 70603030803 陦ｧ00306 06 ﾀ000304 ｧ06 0509 ﾀ05 縺060060. ｱ0600909 ｧ07 y = f(x) 003070605 ｧ0060504 0509 ﾀ05 縺060060 050300040903 04 0503 09 08, 05 閨06 y 0 =0 ｧ00 ｧ 006 ﾀ06006 ﾀ0504060605. ｯ 09 ｧ06 0503 09 ｧ 0 0304 0605 ﾀ07 09 罍ｧ05 ｧ 06 ﾀ09 ｧ0 ｧ. 6 5 0 ｧ08 09 ﾀ 050600040 07 ｧ00 ｧ 0503 罍ｧ- 03005 00040 (y 0 =0), ｧ00 ｧ 0406 縺05 ｧ09 00040 (04 0503 0906 0403 04 05 ), ｧ00 ｧ 00605 0304 縺06- 閨060006000308 09 ﾀ 0506000400308 (0 04 罍ｧ006, 04 06050003040308 0503 0906). 6 5 閨040608 0403 ﾀ0006 閨0 ｧ08 ﾀ0006 0708 y 0 006 ﾀ06006 ﾀ0504060605. 4.6 703 ﾀ05 0503 00306 06 ﾀ000304 ｧ07 0509 ﾀ05 縺060060. ｭ ﾀ00 ｧ070600909 ｧ07 f(x) 00604 縺06 縺03040 04 0503 0906 x 0 ｧ ｧ0060605 04 00609030503 縺0308 0309 縺06 ﾀ05003 ﾀ05 ｧ x 0,09 縺03006, 03030504 003070605, 0503 09 ｧ x 0, 0903006 00606 04 縺03 ｧ 0403 閨00606 ｧ 06 ﾀ00 ｧ 04 罍ｧ 0406 縺060803 閨06 x 06 縺06 x 0 : y 0 0060070605 0 09 ﾀ + 0,, 0503f(x 0 )=y max, y 0 0060070605 0 09 ﾀ, 0 +, 0503f(x 0 )=y min, y 0 006 0060070605 0 09, 0503 0509 ﾀ05 縺060060 00605. ｯ 縺0605 ｧ08 ﾀ0006 08 ｧ0060605 006 ﾀ0503 04 030303090030406000308 0503 0906 (04 罍ｧ y 0 > 0 ｧ00 ｧ y 0 < 0), 05 090706 04 0503 0906 0406 縺0605 ｧ03 ｧ0406050003040308 0503 0906. 0ｷ305 09, 05030303 0 0805 ｧ 0509 ﾀ05 縺060060 070600909 ｧ ｧ, 00607003: ) 7 0805 ｧ y 0 ｧ09 罍ｧ05 ｧ 06 ﾀ09 ｧ06 0503 09 ｧ, 04 09030503 縺0308 y 0 =0 ｧ00 ｧ 006 ﾀ06006- ﾀ0504060605. ) 0504 縺06 閨0600 ｧ0504 0 09 y 0 ﾀ000604 ｧ ﾀ04 04 0305 09 07 閨0308 09 罍ｧ05 ｧ 06 ﾀ090308 0503 09 ｧ, ﾀ03 ﾀ05 04 ｧ04 05 0300 ｧ0906, 0 04 罍ｧ006, 04 ｧ

334 0600.7. 7 罍ｧ000307060 ｧ07 04 縺03 ｧ 0403 閨00308 6 7 000606 00307003 0 0805 ｧ y max ｧ y min ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606. 7 罍ｧ ﾀ. 8 0403 ﾀ05 縺03060 09 罍ｧ04 07, ﾀ030305040605 ﾀ050406060 07 04 罍ｧ0406 閨06000308 04030006 05 0300 ｧ0906. 5. 6 703 ﾀ05 0503 00306 06 ﾀ000304 ｧ07 0509 ﾀ05 縺060060 (040503 縺0308 ﾀ0403 ﾀ0303 ｧ ﾀ ﾀ0006- 閨0304 0 ｧ07). ｭ ﾀ00 ｧ 04 00609030503 縺0308 0503 0906 x = x 0 : ) y 0 = 0, y 00 < 0, 0503 f(x 0 )= = y max ; ) y 0 = 0, y 00 > 0, 0503 f(x 0 )= = y min ; 3) y 0 = 0 ｧ y 00 = 0, 0503 040304 縺03 ﾀ 03 ﾀ05 0605 ﾀ07 006 縺06000600030 ｧ 00607003 0303-7 ｧ ﾀ. 8 05 ｧ0504 ﾀ07 09 0406 縺0403006 ﾀ0403 ﾀ030306 ｧ ﾀ ﾀ0006 閨0304-0 ｧ07. 0ｷ3 ﾀ ﾀ0006 閨0304 0504 0403 ﾀ05 0 ｧ06 ｧ 06030304 0 ｧ06 070600909 ｧ08: 58. ) y = x ; ) y = x 3 ; 3) y = ; 4) y =lnx: x 59. ) y = tg x; ) y = e x ; 3) y =4x, x : 7 0805 ｧ 0509 ﾀ05 縺060060 070600909 ｧ ｧ ｧ 0403 ﾀ05 縺03 ｧ0504 0606 05 07 ｧ09 ): 60. y = x +4x +5: 6. y =4x, x3 3 : 6. y = x3 64. y = x4 3, x, 3x: 63. y =+x, x4 4 : 4, x3 : 65. y = x + x : 66. y = 3 x, : 67. y = +x : 68. y = x, 6x +3 : 69. y = x (, x): x, 3 70. y =, 3 (x, 4) : 7. y = e,x : ) 6 5 08 65, 68, 73 ｧ 00609030503 縺0308 陦縺0605 ｧ08 閨0007 0403 ﾀ05 縺03060 ｧ07 09 罍ｧ040308 00607003 0 0805 ｧ 0606 ﾀ ｧ00405030503 ( ﾀ0. 0500. 5, x 9).

3x 4. 6 503 ﾀ05 0 ｧ06 ｧ 06030304 0 ｧ06 070600909 ｧ ｧ 35 7. y = x + cos x 04 ｧ00506 縺04 0006 (0; ). 73. x =4x, tg x 04 ｧ00506 縺04 0006, ;. 74. y = +lnx : 75. y = x, arctg x: x 76. ) y = xe,x= ; ) y = x ln x: 77. ) y = sin x ; ) y = e x, : 78. y = sin 4 x + cos 4 x: 79. y = x, x: 80. y = 4 x x + : 8. y = x (x, )(x, 4) : 8. y = x + x : 83. y = x=3 +(x, ) =3 : 84. y = x5 5, x4 + x 3 : 85. y = x 3 (x +) : 86. y = x, : 87. y = x3 x x, : 3 88. y =tg x, tg x: 89. y = x + ln (cos x): 90. ) y =ln +x, arctg x; ) y = jxj(x +). 9. y = x e,x. 9. y =3 3 (x +), x. 7 0805 ｧ 0509 ﾀ05 縺060060 070600909 ｧ ｧ ｧ 0403 ﾀ05 縺03 ｧ0504 0606 05 07 ｧ09: 93. y =4x, x : 94. y = x +x, 3: 95. y = x3 97. y = x x, : 99. y = x4 0. y = 3 + x : 96. y = x 3 +6x +9x: 98. y = x3 + x4 4 : 4, x : 00. y =x, 3 3 x : (x, ) x + : 0. y = xe,x = : 03. y = x, lnx: 04. y = x =3 (x, 5):

336 0600.7. 7 罍ｧ000307060 ｧ07 04 縺03 ｧ 0403 閨00308 05. y = sin x, x 04 ｧ00506 縺04 0006 (,=; =). 06. y =x + ctg x 04 ｧ00506 縺04 0006 (0; ). 07. y = x + arcctg x. 08. y =+ 3 (x, ). 09. y = sin x + cos x 04 ｧ00506 縺04 0006 (0; ). 0. y =3x 4, 8x 3 +6x :. y = ln x x :. y = 3, x x + : 3. y = x + x : 4. ) y = ae,x cos x (04 罍ｧ x>0); ) y =3x 5, 5x 3. (4, x)3 3(x +) 5. y = : 6. y = : 9(, x) x +8 7. y = x, x 4 : 8. y =(, x )(, x 3 ): 9. y =. ) y = +x + x, x + x : 0. y = x +,x: (x +3)3 (x +) ; ) y =, cos x: x 5. 6ｦ0 ｧ 03 0 ｧ0303000400 ｧ08 ｧ 0 ｧ00600400 ｧ08 0 060 ｧ0708 040600 ｧ ｧ0. 706000605090308 閨00 ｧ00308 0 0 00607003 030503 縺03 陦ｧ0504 04 罍ｧ000605 0600606 09 閨03006 04 縺07003060503000400606 0400030 閨0906 0 ｧ03030004000608 0400030 陦ｧ. 0504 縺06 閨0600 ｧ0504 006 縺03 04 縺07003060503000400308 0400030 閨09 ｧ. 3. 7 000307 ｧ0504 ｧ ﾀ0003 0 0 閨04 ﾀ00 05 0600308 05 09, 05030303 04 縺03 ｧ - 0406 閨060 ｧ06 ｧ08 03030003 0 ｧ0303000400 ｧ0. 4. 6 505 縺0606050300040 ｧ09 ﾀ 03 ﾀ00304 0 ｧ060 a ｧ 0403 ﾀ03050308 h 0404 ｧ ﾀ 0 04 縺07003-06050300040 ｧ09 0 ｧ03030004000608 0400030 陦ｧ. 0504 縺06 閨0600 ｧ0504 0400030 閨04 04 縺070030605030004-0 ｧ09. 5. 0ｷ3 0904 陦 050030503 00 ｧ ﾀ05 09 縺05030 ﾀ03 ﾀ0503 縺0300308 a 0403 縺06 0605 ﾀ07 0403 060500 0 03 陦ｧ0 0903040306 0904 陦 0503 ｧ ｧ 03 ﾀ05 04000608 ﾀ07 ﾀ05 ｧ ﾀ090006 ｧ04 0605 ﾀ07 04 縺0700306050300040 07 0903 縺030309. 07 090304 閨0300070 03030504 ﾀ0503 縺030 0403 縺06 060030503 0904 陦 05, 05030303 03030060 0903 縺030309 ｧ 030300 0 ｧ0303000400 ｧ0? 6. 0504 縺06 閨0600 ｧ0504 006 縺03 030509 縺0305030503 03 ﾀ ﾀ06080 ﾀ 0904 陦 050030 閨0030 03030060030 3 0 3 05 09, 05030303 0 030300 ｧ0903040906 060503 ﾀ05060 ｧ 閨0 0403000003 0 ｧ006004000606 090300 ｧ 06 ﾀ050403 0 0506 罍ｧ 00. 7. 7030903040306 ﾀ0503 縺03003 ｧ 006004000606 03 ﾀ00304 0 ｧ06 05 040609 ｧ ｧ 04003 0 ﾀ0. 0504 縺06 閨0600 ｧ0504 0606 03030004000606 03 ﾀ00304 0 ｧ06 05 09, 05030303 0400030 閨04 05-040609 ｧ ｧ 030300 0 ｧ03030004000608.

3x 5. 6ｦ0 ｧ 03 0 ｧ0303000400 ｧ08 ｧ 0 ｧ00600400 ｧ08 0 060 ｧ0708 040600 ｧ ｧ0 37 8. 6 5 0403000609 縺0605 0404 ｧ ﾀ 0 05 040609 ｧ07, 03 ﾀ00304 0 ｧ06 09030503 縺0308 06 ﾀ0504 陦ｧ 00605 0403000609 縺0605. 0504 縺06 閨0600 ｧ0504 06050300 05 040609 ｧ ｧ 04 罍ｧ 03 ﾀ00304 0 ｧ ｧ 05 09, 05030303 0400030 閨04 05 040609 ｧ ｧ 030300 0 ｧ03030004000608. 9. ｮ06 060 ｧ06 050300060007 ｧ0060605 0703 縺006 04 縺0700306050300040 ｧ09, 0406-000600030503 0403000609 縺0605030. 706 罍ｧ00605 ﾀ06 060 ｧ07 8 0. 7 罍ｧ 09 09030 陦ｧ06 ﾀ06 0403000609 縺0605 0400030 閨04 ﾀ06 060 ｧ07 0306 閨0605 0 ｧ03030004000608? 30. 6 50300 ｧ ｧ 0403 A 04 縺030403 陦ｧ05 ﾀ07 0403 0 006 06000308 04 縺0700308 09 0503 縺03 閨06 B 07060006 0 07 閨03 縺0305. 703 09 09 ｧ0 060500030 09 04 縺03060905 ｧ 縺060600308 07060006 00308 閨03 縺030506 00607003 04 縺030406 ﾀ05 ｧ 0003 ﾀ ﾀ06 ﾀ 0403 A, 05030303 閨03 ﾀ05 0409 05 縺06 0304 ｧ A 04 B 030300 0 ｧ0303000606 閨060006040308, 06 ﾀ00 ｧ ﾀ0503 ｧ003 ﾀ0504 05030003-09 ｧ000300605 04 罍ｧ 0406 縺060403 0906 0403 0003 ﾀ ﾀ06 04 m 閨03 縺030706, 060 0403 07060006-00308 閨03 縺030506? 3. 6 704 ｧ ﾀ0503 0 ｧ09 ﾀ040605 ﾀ040300030706003 04 30 0 陦縺0605 0305 陦縺0605. 7 04 縺0700308, ﾀ0306 陦ｧ0070600608 ｧ08, 0 0805 ｧ 0 ｧ00600606 03 ﾀ0406006000606 0503 0906, 06 ﾀ00 ｧ ﾀ ｧ0003 ﾀ040605 ｧ ﾀ0503 0 ｧ090304 0305003 ﾀ0705 ﾀ07, 09 09 7 : 8. 3. 6 704 ﾀ 003000605 0006050705 04 03 閨00308 040003 ﾀ0903 ﾀ05 ｧ ｧ 04 縺0700300 ｧ00608003 0403 060500030 0 ﾀ 03 陦ｧ0 0903040308 ﾀ0903 縺03 ﾀ050406 v 090/. 6 5 00609030503 縺0308 003-006005 03 陦ｧ0 ﾀ 003000605 04 罍ｧ0006050600 04 0503 0906 0406 縺06 ﾀ06 060 ｧ07 00 ｧ0 ｧ08 閨04 ｧ07060 ｧ07, 040503 縺0308 006 閨030006050600 閨03 00606 0 a 090. ｪ06 縺06 09 090306 04 縺06007 ﾀ ﾀ0503070 ｧ06 00607 閨06 ﾀ 003000605 0 ｧ 0306 閨0605 0 ｧ00600400 ｧ0 ｧ 06006 04003 050503 ﾀ ﾀ050307-0 ｧ06? 33. 7 0009 04 縺0700306050300040030503 ﾀ06 060 ｧ07 ﾀ03 ﾀ04030303 閨003 030406 縺05030 ｧ 09030-09 0 ｧ 04003006 縺003 0 05 縺0607060 0403 04 ﾀ0608 閨00 ｧ006. ｮ05 縺0600 0606 04 縺0305 ｧ03 0303 05003 04 縺030403 縺09 ｧ030 00040 00300600506 ｧ006 縺09 ｧ ｧ ﾀ06 060 ｧ07 03 0009 ｧ I = = xy3,05 閨06 x ｧ y 006 縺03 03 0009 ｧ. 0504 縺06 閨0600 ｧ0504 006 縺03 03 0009 ｧ 04 罍ｧ 0 ｧ006004000608 ﾀ05 縺060006 04 縺0305 ｧ03, 06 ﾀ00 ｧ 03 0009 0403 縺06 0 ｧ 09 縺060500030503 03 縺06040 ﾀ 陦ｧ 00605 縺030 D. 34. 6 503 ﾀ090300040903 03030060 00 030300040006 03030060 0 ｧ0303000400060503 09 ｧ00 ｧ0 陦, 0404 ｧ ﾀ 00030503 04 05050305 00? 35. 6 704 0903 罍ｧ 閨03 00 ｧ 罍ｧ00308,4 0 ｧ,6 0 0406 縺06 ﾀ0609 0605 ﾀ07 0403 04 縺070030 060500030. 0504 縺06 閨0600 ｧ0504 0 ｧ03030004000606 閨00 ｧ006 0006 ﾀ050 ｧ0903, 09030503- 縺0606 00307003 0406 縺06006 ﾀ05 ｧ (0503 罍ｧ 03005 0004003) ｧ 03 閨0030503 0903 罍ｧ 閨03 04 陦縺06-050308. 36. 6 5 0903006 ﾀ ﾀ 陦ｧ06 ﾀ030 4 閨0 ｧ 0403 ﾀ03050308 6 閨0 0404 ｧ ﾀ 0 09 ｧ00 ｧ0 陦 0 ｧ0303000400060503 03030060. 7 0805 ｧ 05050305 03030060. 37. 6 5 0403000609 縺0605 陦ｧ06 ﾀ R 0404 ｧ ﾀ 0 04 縺0700306050300040 ｧ09 0 ｧ03030004000608 0400030 陦ｧ. 0504 縺06 閨0600 ｧ0504 060503 006 縺03. 38. 7 04 03030006 y = x 0 0805 ｧ 0503 0906, 0 ｧ00600606 06 0006000606 0305 04 縺0700308 y =x, 4. 39. 07 縺05 ｧ0 0403040600060 0 ﾀ0506006. 7 ｧ070 ｧ08 0606 090300609 0 b ﾀ0, 0406 縺080 ｧ08 0 a ﾀ0 04030006 0500 0 030006 05060007. 7 09 09030 ﾀ ﾀ0503070 ｧ ｧ 0305 ﾀ0506003 閨030007060 04 ﾀ05 0504 0 030006 05060004, 05030303 ﾀ ﾀ00305 縺060504 09 縺05 ｧ006 0403 0 ｧ0303000400 ｧ0 060500030?

338 0600.7. 7 罍ｧ000307060 ｧ07 04 縺03 ｧ 0403 閨00308 40. 05030 07 閨00 ｧ0 ﾀ05060 ｧ 0303 070600030503 0 0400 006 閨030 ( 罍ｧ ﾀ. 9) 閨0300070 03030504 040 90 0. 7 罍ｧ 09 090308 00 ｧ 罍ｧ006 x 0903 罍ｧ 閨03 0400030 閨04 05 縺0608 03 ﾀ05 000400308 090300 05 0306 閨0605 0 ｧ03030004000608? 4. 6 5 04 縺07003060503000400308 05 縺0606050300040 ｧ09 ﾀ 05 ｧ04030506006 0308 8 ﾀ0 ｧ 060500030 60 0404 ｧ ﾀ 0 04 縺0700306050300040 ｧ09, 03 ﾀ00304 0 ｧ06 09030503 縺030503 ﾀ040300030706003 0 05 ｧ- 04030506006 06. 07 09030403 閨030007003 03030504 006 縺03 04 縺0700306050300040 ｧ09, 05030303 060503 0400030 閨04 030300 0 ｧ03030004000608? 4. 6 7 003 0503 09 ｧ A(0; 3) ｧ B(4; 5). 7 03 ﾀ ｧ Ox 0 0805 ｧ 0503 0906 M 05 09, 05030303 ﾀ ﾀ0503070 ｧ06 S = AM + MB 03030003 0 ｧ00600400 ｧ0. 43. ｮ0304 縺0305 ｧ0400060 ｧ06 03 0009 ｧ 04 縺03 閨030004003006 ﾀ07 05 ｧ06 04 縺030403 縺09 ｧ03-0 0004003 0400030 陦ｧ 04030406 縺06 0030503 ﾀ06 060 ｧ07. 0504 縺06 閨0600 ｧ0504 006 縺03 03 0009 ｧ, 0403 縺06 000308 ｧ 09 縺060500030503 03 縺06040 陦ｧ 00605 縺030 D, 05 09, 05030303 ﾀ0304 縺0305 ｧ- 0400060 ｧ06 0606 ﾀ07 05 ｧ06 03030003 0 ｧ0303000400 ｧ0. 44. 0ｷ3 09 縺0605 0403 縺06 0605 ﾀ07 ﾀ06090503, ﾀ03 閨06 縺07 0 ｧ08 06050300, 05060 ﾀ06090503 ﾀ0406 縺050304 0605 ﾀ07 04 0903006 ﾀ. 7 罍ｧ 09 09030 0 060 ｧ ｧ 060500 03030060 0903006 ﾀ 0306 閨0605 0 ｧ0303000400 ｧ0? 45. 06 縺06 0406 ﾀ030 P,000607 0 ｧ08 0 0503 罍ｧ 03005 000400308 040003 ﾀ0903 ﾀ05 ｧ, 00607003 ﾀ 閨04 ｧ0060504 04 罍ｧ00030706000308 09 006006 ﾀ ｧ000308 F ( 罍ｧ ﾀ. 30). 703 09 09 ｧ0 7 ｧ ﾀ. 9 7 ｧ ﾀ. 30 060500030 09 0503 罍ｧ 0300506 00607003 0 04 04 ｧ0504 ﾀ ｧ0006 F, 05030303 030 030300 0 ｧ006004000608. 0703050707 ｧ09 ｧ06005 05 縺060 ｧ07 =0; 5. x 6. 7 04 0400060 ｧ06 04030406090003 ﾀ05 ｧ ｧ0503 09 ｧ 0406 縺0605 ｧ03 09 罍ｧ040308. 703 ﾀ05 縺03060 ｧ06 09 罍ｧ040308.6 5030406090003 ﾀ0504. 07 罍ｧ04 07 0 0304 0605 ﾀ07 0403040609000308 08040406 縺0809 (08040 ｧ 09) 04 0503 0906 x = x 0, 06 ﾀ00 ｧ 04 00609030503 縺0308 0309 縺06 ﾀ05003 ﾀ05 ｧ 05050308 0503 09 ｧ ( ﾀ000604 ｧ ﾀ04 04 ) 09 罍ｧ04 07 ﾀ0403000307060 080 ｧ070609 (080403000609) 09 ﾀ 0506000400308 04 05050308 0503 0906. ｭ ﾀ00 ｧ 04 0503 0906 x = x 0 : ) y 00 > 0, 050309 罍ｧ04 07 040304060900 08040 ｧ 09; ) y 00 < 0, 050309 罍ｧ04 07 040304060900 08040406 縺0809.. ｯ03 090308 0406 縺0605 ｧ03 0 0304 0605 ﾀ07 0503 09, 04 09030503 縺0308 09 罍ｧ04 07 0406 縺06-0803 陦ｧ05 ﾀ 03 閨00308 ﾀ0503 縺03003 09 ﾀ 0506000400308 0 陦縺06050606 ( ｧ, ﾀ0006 閨0304 05060004003, 0060070605 0 04 0400060 ｧ06 04030406090003 ﾀ05 ｧ). 70603030803 陦ｧ0030 06 ﾀ000304 ｧ060 0503 09 ｧ 0406 縺0605 ｧ03 07040007-0605 ﾀ07 0503, 0503 04 00608 y 00 =0 ｧ00 ｧ 006 ﾀ06006 ﾀ0504060605, 閨03 ﾀ05 0503 0030 0503, 0503 y 00 04 罍ｧ 0505030 0060070605 0 09.

3x 6. 7 04 0400060 ｧ06 04030406090003 ﾀ05 ｧ ｧ 0503 09 ｧ 0406 縺0605 ｧ03 09 罍ｧ040308 39 3.6 70007 0403 ﾀ05 縺03060 ｧ07 09 罍ｧ040308 縺0609030060 閨060605 ﾀ07 0304 縺06 閨0600 ｧ0504: ) ﾀ ｧ0-00605 罍ｧ06; ) 030300 ﾀ0504 ﾀ0403000307060 ｧ07; 3) 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ Ox ｧ Oy; 4)0503 09 ｧ 縺0304 070600909 ｧ ｧ y = '(x) ｧ00 ｧ x = f(y) ｧ ﾀ ｧ00405030503; 5) 0403 ﾀ05 0 ｧ06 ｧ00 ｧ 06030304 0 ｧ06 y ｧ00 ｧ x ｧ 0509 ﾀ05 縺060 000400306 0503 09 ｧ; 6) 0 04-0400060 ｧ06 04030406090003 ﾀ05 ｧ ｧ 0503 09 ｧ 0406 縺0605 ｧ03. 46. 0ｷ3 ﾀ ﾀ0006 閨0304 0504 0 04 0400060 ｧ06 04030406090003 ﾀ05 ｧ ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ- 040306: ) y = x ; ) y = x 3 ; 3) y = e x ; 4) y =lnx; 5) y = x 5=3 : 47. 0504 縺06 閨0600 ｧ0504 0509 ﾀ05 縺060 000400306 0503 09 ｧ ｧ 0503 09 ｧ 0406 縺0605 ｧ03 09 罍ｧ- 040308 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: ) y = x3 6, x ; ) y = e,x ; 3) y = x +x ; 4) y ==x. 7 罍ｧ00600707 00609030503 縺0306 ｧ 04 04 ｧ00 04. 3, 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306, - 000306 04 08 48{6 06 040060 ｧ070 ｧ: 48. y =x +9: 49. y =,x, 4x. ｰ 09 0 ｧ 06. 6 5 06 48 0304 縺06 閨0600 ｧ0504 ﾀ ｧ000605 罍ｧ06, 030300 ﾀ0504 ﾀ04030003-07060 ｧ07 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ, 04 06 49 0503 0906 0509 ﾀ05 縺060060 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ Ox. 50. y = sin x; y = cos x: 5. y =shx; y =chx. ｰ 09 0 ｧ 06. 6 5 08 50, 5 0304 縺06 閨0600 ｧ0504 0503 09 ｧ 0509 ﾀ05 縺060060 ｧ 0406 縺0605 ｧ03. 5. y =ln(x +): 53. y = e,x. ｰ 09 0 ｧ 06. 6 5 08 5, 53 0304 縺06 閨0600 ｧ0504 030300 ﾀ0504 ﾀ0403000307060 ｧ07 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ, ﾀ ｧ00405030506 ｧ 0 04 0400060 ｧ06 04030406090003 ﾀ05 ｧ. 54. ) y = x 3 ; ) y =(x +3) 3 : 55. ) y =+ x, 4 ; ) y = 3 x, x : 3 56. ) y = e ln x x ; ) y = exe,x : 57. ) y = x + 4 x + ; ) y = x 4, x : 58. ) y = x, ln x; ) y = a (ex=a + e,x=a ): 59. ) y = x4 x 3, ; ) y = 4 x + x : 4 60. ) y =x, x 4 ; ) x(y, x) =4: 6. y =(x +) =3, (x, ) =3. 6. y = xe,x.

30600 04 8 7 ｭ05 7 7 ｭ6 7 ｭ6ｧ8 ｭ 7 76ｧ9 0ｷ3 7 ｯ ｭ06 7 縺6ｧ8 x. 7060304 縺06 閨060006000308 ｧ0050605 00. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 000307060 ｧ060. 7060304 縺06 閨06000600030 ｧ0050605 00030 R f (x) 0 0304 0605 ﾀ07 0706009-09 ｧ07 F (x) +C, ﾀ03 閨06 縺07 0 07 04 縺03 ｧ 0403000400306 0403 ﾀ050307000306 C, 陦ｧ070706 縺06009 ｧ 00 09030503 縺0308 04060 0403 閨030050605 0004003006 0403 07060 ｧ06 f(x), 05.06. f(x) = F (x) +C; 06 ﾀ00 ｧ d[f (x) +C] =f (x) :. ｯ 0300 ｧ09 03 ﾀ0030400308 ｧ0050605 000304: : x n = xn+ n + + C 6: sin x =, cos x + C: (n 6=,): : =lnjxj + C: 7: x 3: a x = ax + C: 8: 4: 5: ln a e x = e x + C: 9: cos x = sin x + C: 0: cos = tg x + C: x sin x +x =, x = =,ctg x + C: 8 < arctg x + C ｧ00 ｧ :,arcctg x + C : 8 < arcsin x + C ｧ00 ｧ :, arccos x + C: 3. ｮ040308 ﾀ0504 0060304 縺06 閨06000600030503 ｧ0050605 00 : I: d u= u : II: du = u + C: III: Au = A u : IV: (u + v) = u+ v :

3x. 7060304 縺06 閨060006000308 ｧ0050605 00. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 000307060 ｧ060 4 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 000307060 ｧ060 06 ﾀ0504 04 罍ｧ0406 閨060 ｧ06 00030503 ｧ0050605 00 (0403 ﾀ040308 ﾀ050406 IV)09 ﾀ060006 0303000606 04 縺03 ﾀ050308 ｧ0050605 000304. 63. 6 5 ﾀ0006 閨06060 ｧ08 04060 ﾀ0504 08 0403000 ｧ0504 04 縺030406006000306 006 ﾀ05 : ) d( ) = x; ) d( ) = x 3 ; 3) d( ) = cos x; 4) d( ) = x ; 5) d( ) = ; 6) d( ) = cos x +x : 7 0805 ｧ 05060 ｧ0050605 0003 x, x 3 ｧ05.. 7 0805 ｧ ｧ0050605 0003: 64. ) x +x + ; ) x 0x 8 +3 : x 4 65. ) 66. ) x, ; ) x 3 ( x + 3 x) ; ) (x +) : x 3 x, : 4 x 3 67. ) 68. ) 69. ) 70. ) 7. ) 7. ) ( x, ) 3 x, ; ) x e x, e,x ; ) a x x : 3 x : cos x ; ) ctg x : cos x sin x 3, sin x cos x ; ) ctg x : cos x sin x ; ) cos x : +x, 3 ; ), x + a,x x 3 x 4 +x :

34 0600. 8. 7060304 縺06 閨060006000308 ｧ0050605 00 7 0805 ｧ ｧ0050605 0003: (x, ) 73. ) x 3 ; ), 3 x x : x 74. ) x, x ; ) 3 75. ) x + x + ; ) x 3 76. ) e x + e,x ; ) a x cos x, sin3 x 77. : 78. tg x. sin x ( x +) : x sin x, cos x : + a,x x 5 : x. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 0403 陦ﾀ05 00304090308 ｧ 0060403 ﾀ 縺06 陦ﾀ050406000306 703000307 ｧ04 x = '(u), = ' 0 (u) du, 04030006 ｧ0 f(x) = f '(u) ' 0 (u) du: () ｯ 090306 04 縺060303 0304 0 ｧ06 ｧ0050605 00 0 0304 0605 ﾀ07 ｧ0050605 罍ｧ 縺0304 0 ｧ060 0403 - ﾀ05 00304090308. 6 5 04 縺03 ﾀ050308 ﾀ0006 0708 040406 閨060 ｧ06 003040308 0406 縺06006000308 u 縺0609030060 閨060605 ﾀ07 0403-0403000070504 04 06006, 04 罍ｧ00600707 ﾀ0006 閨06060 ｧ06 04 縺060303 0304 0 ｧ07 陦ｧ070706 縺06009 ｧ 00 : = a d(ax + b); x= d(x ); cos x= d(sin x); x = d(ln x) ｧ05. 04.; ｧ030303 0 07 003 ﾀ00060003 0403 07060 ｧ06 04 ﾀ09030309 08 06 縺06 u. ｯ 090308 04 罍ｧ060 ｧ00506-05 罍ｧ 縺0304 0 ｧ07 0 0304 0605 0060403 ﾀ 縺06 陦ﾀ05040600030. 7 0805 ｧ ｧ0050605 0003: 79. cos 3x : 80. sin x. ｰ 09 0 ｧ 06. 6ｦ0 06 79 00307003 縺0600 ｧ0504 閨0406007 ﾀ0403 ﾀ0303 0 ｧ: ) 0403-000307 ｧ04 3x = u, x = u=3, = du=3; ) 04 罍ｧ0406 閨07 ｧ0050605 00 09 04 ｧ 閨06 R 3 cos 3xd(3x).

3x. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 0403 陦ﾀ05 00304090308 ｧ 0060403 ﾀ 縺06 陦ﾀ050406000306 43 8. e,3x : 8. cos 5x : 83. e x= + e,x= : 84. 4x, : 85. 87. 89. (3, x) 4 : 86. 3, x : 88. x, 5 : 90. x, 5x +7 3 5, 6x : sin (a, bx) : x x + : ｰ 09 0 ｧ 06. 6ｦ0 ｧ 89{98 縺0600 0605 ﾀ07 0403 0703 縺0060006 u 0 u = du =lnjuj + C; u 05. 06. 06 ﾀ00 ｧ ｧ ﾀ00 ｧ05060004 0403 閨030050605 000400308 陦縺0303 ｧ 06 ﾀ0504 04 縺03 ｧ 0403 閨0 07 0305 0 0060 05060007; 0503 ｧ0050605 00 04060 000305 罍ｧ07006 0 0060 05060007. 9. 93. 95. 97. 99., 0x : 9. ctg x : 94. cos x : 96. sin x cos x cos x : 98. + sin x sin x cos x : 300. e x, 3e x : tg x : sin x + 3 cos x : x(+lnx) : cos 3 x sin x : ｰ 09 0 ｧ 06. 6ｦ0 06 99 00307003 縺0600 ｧ0504 0403 陦ﾀ05 00304090308 sin x = u ｧ00 ｧ 0060403 ﾀ 縺06 陦ﾀ0504060003, 0060 ｧ04 cos x 06 縺06 d(sin x). 30. 303. cos x sin 4 x : 30., cos x : 304. sin x sin x cos 3 x : sin x cos x :

344 0600. 8. 7060304 縺06 閨060006000308 ｧ0050605 00 305. e cos x sin x : 306. e x3 x : ｰ 09 0 ｧ 06. 6ｦ0 06 306 00307003 縺0600 ｧ0504 0403 陦ﾀ05 00304090308 x 3 = u ｧ00 ｧ 0060403 ﾀ 縺06 陦ﾀ0504060003, 0060 ｧ04 x 06 縺06 3 d(x3 ). 307. 309. e,x x : 308. x +x : 30. e x x : 3 x 3, 8x : ｰ 09 0 ｧ 06. 6ｦ0 06 309 00307003 縺0600 ｧ0504 0403 陦ﾀ05 00304090308 x + =u ｧ00 ｧ 0060403 ﾀ 縺06 陦ﾀ0504060003, 04 ｧ ﾀ 04 ｧ0050605 00 04 04 ｧ 閨06 R (x +) = d(x +). x x 3. : 3. : +x 3, x 33. 35. 3 sin x + cos x : 34. +lnx : x 3 + 4 sin x cos x : 36., 6x 5 x 4 : 7 0805 ｧ ｧ0050605 0003: 37. 39. 3. (e x + e,x ) : 38., 4x : 30. 3 +3x: 3. sin 3 x cos x : cos (a, bx) : 6, x 3 x : x, sin x 33. : 34. : +x cos x + sin x 35. : 36. e sin x cos x : sin x x 37., x : 38. 3 (a, : bx) 3

3x 3. 0ｷ30050605 0003 ﾀ040609 ｧ 00040030503 04 ｧ 45 x 3. 0ｷ30050605 0003 04 ｧ,,, + ｧ 09 0 ｧ0 04 罍ｧ0403 閨070 ｧ06 ﾀ07 39. 70309 0504, 0503: ) a + x = a arctg x + C, 0403000307 ｧ04 x = a tg t; a ) x = arcsin + C, 0403000307 ｧ04 x = a sin t; a, x a 3) x, a = a ln x, a x + a + C, 000307 ｧ04 x, a = a + x + a, x = a x, a a x, a, ; x + a 4) 330. ) 33. ) 33. ) 333. ) 334. ) 335. ) x + k =lnjx + x + kj + C, 0403000307 ｧ04 x + k = t,x. x, 5 ; ) 4, x ; ) x, 4 ; ) 5, x ; ) x 3, x 4 ; ) 3, 4x ; ) x +9 : x +5: x +3 : x 4+x 6 : b x, a : x 3 x8, : 336. ) 337. ) 5x, 3x, 4 ; x +4 ) x, : 4 x + ; ) x + x +, x :

346 0600. 8. 7060304 縺06 閨060006000308 ｧ0050605 00 x 338. x : + 339. x4 x, : 3 ｰ09 0 ｧ06. 6 5 08 338, 339 00607003 ｧ 0403 閨030050605 000400308 00604 04 ｧ000400308 陦縺0303 ｧ ｧ ﾀ090006 ｧ0504 0906000306 0403 07060 ｧ06. 340. x +4x +5 : 34. x +6x +3 : ｰ 09 0 ｧ 06. 6 5 08 340{347 00607003 ｧ 0904 陦 050030503 05 縺0608 00060 0403 閨0600 ｧ0504 04030000308 0904 陦 05. 34. 344. 346. x +x +3: 343. 4x, x : 345. +3x, x : 347., x, x : x +3x +3 : : 3x, x, 7 0805 ｧ ｧ0050605 0003: 3 348. x +3 + 6. x, 3 349. +., x +x 4x, 5 x 350. : 35. x +5 x, : x4 35. x : + 353. e x :, e x 354. 356. 358. x x 4 +0; 5 : 355. x, x +5 : 357. x x + x + : 359. x +4x +9 : 5, 4x, x : : 4x +4x +3

3x 4. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 0403 ﾀ05070 47 x 4. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 0403 ﾀ05070 0ｷ3 0703 縺0060003 陦ｧ070706 縺06009 ｧ 00 04 縺03 ｧ 0406 閨060 ｧ07 d(uv) = udv+ vdu04030006-0605 ﾀ07 0703 縺00600 ｧ0050605 罍ｧ 縺0304 0 ｧ07 0403 ﾀ05070 udv = uv, v du: 705 0703 縺00600 006 04 ﾀ060503 04 罍ｧ0060070605 ﾀ07 050305, 090305 0403 ｧ0050605 00030 ｧ0060605 ﾀ07 04 縺03 ｧ 0406 閨060 ｧ06 00050603 ｧ 06 ﾀ090308 ｧ 05 0 ﾀ09060 閨0600500308 070600909 ｧ08, 0-04 罍ｧ006 R x e x ｧ00 ｧ R x ln x. 7 罍ｧ 0505030 u 04 罍ｧ0 ｧ0 0605 ﾀ07 070600909 ｧ07, 09030503 07 陦ｧ070706 縺06009 ｧ 縺0304 0 ｧ060 0604 縺030 0605 ﾀ07, dv 05 ﾀ0504 0403 閨030-050605 00040030503 0403 07060 ｧ07, ﾀ03 閨06 縺07 0 07, ｧ0050605 00 0305 09030503 縺0308 ｧ 0406 ﾀ05060 ｧ00 ｧ 003070605 03030504 0 08 閨060. 0ｷ3 05 0 ﾀ09060 閨0600500308 070600909 ｧ08 u 030303 003 04 罍ｧ0 ｧ0 0605 ﾀ07 ln x, arctg x ｧ arcsin x. 7 04 罍ｧ006, 04 ｧ0050605 0006 R x ln x u 00607003 04 罍ｧ0070504 ln x ( 006 x ), 04 ｧ0050605 0006 R x e x u 00607003 04 罍ｧ0070504 x ( 006 e x ). 7 0805 ｧ ｧ0050605 0003: 360. 36. 364. ln x : 36. xe x : 363. x cos x : 365. x ln (x, ) : x arctg x : e x sin x : 366. 70309 0504, 0503 x + k = x x + k + k ln (x + (x + k)) + C: 7 0805 ｧ ｧ0050605 0003: 367. (ln x) : 368. ln x 369. : 370. x 37. arcsin x : 37. 373. ln (x +): 374. x sin x : arcsin x +x : x 3 e,x : cos (ln x) :

348 0600. 8. 7060304 縺06 閨060006000308 ｧ0050605 00 7 0805 ｧ ｧ0050605 0003: x 375. ln x : 376. x e,x= : x 377. arctg x : 378. cos x : arcsin (x=) 379. e x cos x : 380. :, x x cos x 38. sin 3 x : 38. arctg x, : x 5. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 05 罍ｧ050300300605 罍ｧ 06 ﾀ09 ｧ08 070600909 ｧ08. 0ｷ30050605 0003 0305 0904 陦 050304 ｧ 陦縺0605 ｧ08 060500308 ﾀ0506040600608 ﾀ ｧ- 006 ﾀ ｧ 0903 ﾀ ｧ006 ﾀ 0 0803 閨0705, 04 罍ｧ00600707 ﾀ0006 閨06060 ｧ06 0703 縺0060003 04030 ｧ0706-0 ｧ07 ﾀ050604060 ｧ: sin, cos x x = ; cos + cos x sin x x = ; sin x cos x = :.0ｷ30050605 0003 0305 0906030304 ｧ 陦縺0605 ｧ08 006 060500308 ﾀ0506040600608 ﾀ ｧ 006 ﾀ ｧ 0903 ﾀ ｧ006 ﾀ 0 0803 閨0705, 0305 閨06000707 0305 006 060500308 ﾀ050604060 ｧ 03 陦ｧ0 000307 ｧ05060004 ｧ 040300 05 07 0903070600909 ｧ06 0400308 003040308 0406 縺06006000308 u. 0ｷ30050605 00 R cos m x sin n x0 0803 陦ｧ05 ﾀ07 0403 04 04 ｧ0006, 06 ﾀ00 ｧ m ｧ n 0303 060500306, ｧ 0403 04 04 ｧ0006, 06 ﾀ00 ｧ m ｧ00 ｧ n 006 0605003. 7 0805 ｧ ｧ0050605 0003: 383. 385. 387. 389. 39. 393. sin 3x : 384. (, sin x) : 386. sin x cos x : 388. sin x cos 4 x : 390. sin x cos 3 x : 39. cos 7 x : 394. ( + cos x) : cos 4 x : sin 4 x cos 4 x : sin 5 x : sin 3 x cos 3 x : ( + cos x) 3 : 395. cos3 x sin x : 396. sin3 x cos x :

3x 5. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 05 罍ｧ050300300605 罍ｧ 06 ﾀ09 ｧ08 070600909 ｧ08 49 397. sin x = sin x + cos x =? sin x cos x 398. ) sin x ; ) cos x : cos x + sin x 399. : 400. sin x sin x, cos x : 40. tg 3 x : 40. ctg 3 x : ｰ 09 0 ｧ 06. 6 5 06 40 0403000307 ｧ0504 tgx = t, x = arctg t. 403. sin 3x cos x : 404. cos mx cos nx. ｰ 09 0 ｧ 06. 6 5 08 403{406 04 罍ｧ0060 ｧ0504 0703 縺0060003 sin cos = sin ( + ) + sin (, ) ; cos cos = cos ( + ) + cos (, ) ; sin sin = cos (, ), cos ( + ) : 405. ) 406. sin 3x sin 5x ; ) sin 5x, cos x +. 4 4 sin mx sin nx. 407. 0ｷ30050605 罍ｧ 縺0607 0403 ﾀ05070, 04030406 ﾀ05 ｧ 0703 縺0060003 0804030 ｧ07060 ｧ07 ﾀ050604060 ｧ09: ) sin n x=, n cos x sinn, x + n, sin n, x; n ) cos n x= n sin x x n cosn, +, cos n, x n ｧ0403 0505 ｧ0 0703 縺00600 0 0 0805 ｧ: ) sin 6 x; ) cos 6 x. 408. 7 0805 ｧ ｧ0050605 0003: ) sin 3 x ; ) cos 3 x. ｰ 09 0 ｧ 06. 7 罍ｧ0060 ｧ0504 0703 縺0060003 ｧ 407 09 ｧ0050605 00 0 ｧ cos x. sin x

350 0600. 8. 7060304 縺06 閨060006000308 ｧ0050605 00 7 0805 ｧ ｧ0050605 0003: 409. ( + 3 cos x) : 40. 4. sin 4 x cos x : 4. 43. sin 3 x cos x : 44. 45. (sin x, cos x) : sin x sin3 x + 46. 47. cos x : 48. sin 4 x : cos 5 x : ( + sin x) 3 : sin 3x sin x : sin x + cos x : 6 x 6. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 09 ｧ030 000400308 00050603 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08. ｭ ﾀ00 ｧ 0403 閨030050605 00040 07 陦縺030304 00604 04 ｧ00040 07, 050300607003 ｧ ﾀ090006 ｧ0504 ｧ 00606 0906000306 0403 07060 ｧ06.. 6ｦ00 0060 05060004 04 04 ｧ000400308 陦縺0303 ｧ 00 05 0605 ﾀ07 0 000307 ｧ050600 ｧ 04 ｧ (x, a) ｧ (x + x + q), 04 04 ｧ00040 07 陦縺030304 00 05 0605 ﾀ07 0 ﾀ060006 050006006005 縺00308 陦縺03030608 ﾀ0006 閨06060 ｧ0 0303 030: P (x) (x, a) (x + x + q) ::: = A x, a + A (x, a) + :::+ A (x, a) + + M x + N x + x + q + M x + N (x + x + q) + :::+ M x + N (x + x + q) + :::; 05 閨06 P (x) 040300 ｧ0030 ﾀ050604060 ｧ 0 ｧ0706 ﾀ050604060 ｧ 0 0060 05060007. 7 0805 ｧ ｧ0050605 0003: 49. ) 40. x 3 x 4 ; ) ; 3) x, x + a x, 4 x +7 : 4. (x, )(x, 3) x + x, : x 5 x 3, a 3. 4. 44. 3x +x, 3 : 43. x 3, x x + : 45. x 3, x (x +) 3 x, x : 3x, a x 4, ax 3 :

3x 6. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 09 ｧ030 000400308 00050603 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08 5 x, 5x + 5x, 46. : 47. x 3, x + x x 3, 3x, : 5x + 4x, ; 4 48. : 49. x +x +0 x, 0; x +0; : 7 ｰ 09 0 ｧ 06. 6 5 06 48 0403 閨0600 ｧ0504 04 0 0060 05060006 04030000308 0904 陦 05 ｧ 05060 0403000307 ｧ0504 x +=t. 430. 43. 434. ) x + x +4 : 43. x 3 + x +4x +4 x : 3 +8 433. (x + b ) ; ) 7x, 5 x 3, x +5x : 3x +x + (x +) (x +) : (x + b ) 3 : ｰ09 0 ｧ06. 703000307 ｧ0504 x = b tg t ｧ 05060 (0403 040503 縺030 04 罍ｧ006 縺06) ｧ ﾀ- 04030004 0304 0504 0703 縺0060006 ) ｧ 407. 435. ) 436. (x +) (x +x +5) ; ) 4x ( + x)( + x ) : 437. (x, 6x : + 0) 3 x + x 4 +4x : +4 7 0805 ｧ ｧ0050605 0003, 006 04 罍ｧ00600707 03030060503 0060503 0060304 縺06 閨060006000308 0903050707 ｧ09 ｧ060050304: 438. x(x : 439. + a) (x + a)(x + b). ｰ09 0 ｧ06 09 0 438{44. 6 5 ｧ ﾀ00 ｧ05060006 0403 閨030050605 000400308 陦縺0303 ｧ 0 04 ｧ ﾀ 0504 003 ﾀ0504 000307 ｧ0506000608 0 0060 05060007, 閨0600 ｧ04 ｧ0050605 00 0 ﾀ030305040605 ﾀ0504060600606 ｧ ﾀ0003. 440. 44. x, x : 44. x 4, x : 443. (x, 3)(x +) : x 3 +4x :

35 0600. 8. 7060304 縺06 閨060006000308 ｧ0050605 00 7 0805 ｧ ｧ0050605 0003: x, 444. : 445. (x, )(x, ) 5x, 4 446. : 447. x 3, x, 4x +4 5x, 8 448. : 449. x 3, 4x +4x x, a 450. : 45. x 3 + a x 45. x 3, : 8 453. 3x + x + x, 3 : x +6 (x, )(x +) : x + x 3, x +x : x 3 + x +x + : x (x +x +) : 6 5 08 454{457 04030403000 ｧ0504 ｧ0050605 罍ｧ 縺0304 0 ｧ06, 006 04 罍ｧ030605 07 09 0060503 閨06 0060304 縺06 閨060006000308 0903050707 ｧ09 ｧ060050304: 454. x +5x : 455. x 4 +3x : 456. x 4, : 457. x 4, x, : x 7. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 00609030503 縺0308 ｧ 罍 09 ｧ030 000400308 00050603 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08 n. 0ｷ30050605 00 R x; ax + b, 05 閨06 R(x; y) 09 ｧ030 00040 07 070600909 ｧ07, 0 0803 陦ｧ05 ﾀ07 0403 陦ﾀ05 00304090308 ax + b = t n, ｧ0050605 00 0303000606 03030060503 04 ｧ R x m n ; ax m + b x m, 0403 陦ﾀ05 00304090308 ax m + b = t n. Mx + N.0ｷ30050605 00 0 0803 陦ｧ05 ﾀ07 0403 陦ﾀ05 00304090308 (x, ) ax + bx + c x, = t. 3. ｯ 罍ｧ050300300605 罍ｧ 06 ﾀ09 ｧ06 0403 陦ﾀ05 0030409 ｧ. 07 09 ｧ030 0004003006 05 罍ｧ050300300605 罍ｧ 06 ﾀ0903006 04 ｧ 閨06 04 罍ｧ0403 閨0705 ﾀ07 ｧ0050605 0003 R x; a, x 0403 陦ﾀ05 00304090308 x = a sin t; R x; a + x 0403 陦ﾀ05 00304090308 x = a tg t:

3x 7. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 ｧ 罍 09 ｧ030 000400308 00050603 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08 53 4 a0 x m + a x m, + :::+ a m. 0ｷ3 ｧ0050605 00 00307003 0403 閨06ax + bx + c 00 ｧ0504 00050603 ｧ 06 ﾀ090606 ﾀ0504 0403 0703 縺0060006 a0 x m + :::+ a m =(A 0 x m, + :::+ A m, )W + A m W W ; 05 閨06 W = ax + bx + c. 0703050707 ｧ09 ｧ0600503 A 0 0803 閨0705 ﾀ07 0403 ﾀ0006 陦ｧ070706 縺060-09 ｧ 縺0304 0 ｧ07 04060 ﾀ0504 ｧ 03 ﾀ0403030307 閨060 ｧ07 060503 0305 0 0060 05060007 ﾀ 040060 ｧ060 0903-050707 ｧ09 ｧ060050304 ﾀ000604 ｧ ﾀ04 04 04 罍ｧ 03 陦ｧ0 0903040308 ﾀ0506040600708 x. 5. 0ｷ30050605 00 0305 陦ｧ070706 縺06009 ｧ 00040030503 03 ｧ0030 x m (a + bx n ) 0306 縺0605 ﾀ07 04 0903006 0030 04 ｧ 閨06 04 05 縺0608 ﾀ0006 0708: ) 090305 0906000306 ｧ ﾀ0003, 000307060 ｧ060; ) 090305 m + 0906000306 ｧ ﾀ0003, 0403 陦ﾀ05 - n 00304090308 a + bx n = t s ;3)090305 m + + 0906000306 ｧ ﾀ0003, 0403 陦ﾀ05 00304090308 n ax,n + b = t s,05 閨06 s 0 0060 05060004 陦縺0303 ｧ. 0ｷ3 ﾀ04030004 0607 0403 陦ﾀ05 0030409 ｧ 04., 0 0805 ｧ ｧ0050605 0003: 458. 460. 46. x + : 459. 3 3x + : 46. x + x 3 x 3 + 3 : 463. x 4 + x x ++ : x a, x : x 3 x +: 0ｷ3 ﾀ04030004 0607 0403 陦ﾀ05 003040906 04., 0 0805 ｧ ｧ0050605 0003: 464. x : 465. x, x : x +x + 466. x : 467. : ax, x (x +) x +x + 7 0805 ｧ ｧ0050605 0003, ｧ ﾀ04030004 0607 0403 陦ﾀ05 0030409 ｧ 04. 3 : 468. 470. 47. a, x : 469. x 4, x : 47. 3+x, x : 473. (4 + x ) 3 : x (a + x ) 5 : x (, x ) 3 :

354 0600. 8. 7060304 縺06 閨060006000308 ｧ0050605 00 7 0805 ｧ ｧ0050605 0003, 04 罍ｧ00600707 04 04 ｧ0003 04. 4 : 474. x +4x x +x +: 475. x 3, x, x : 476. x + k : 477. ax, x : 7 0805 ｧ ｧ0050605 0003 0305 陦ｧ070706 縺06009 ｧ 000400308 03 ｧ00300304: 478. x 4 : 479. +x 3 x 3 : 3, x 3 x 480. 3 : 48. x ( + x ) 3 (a, bx : ) 3= 7 0805 ｧ ｧ0050605 0003: x, 48. : 483. x, x 484. : 485. x + : 3x +, 3 x : a, x x + 486. x x 3 : 487. : 3 x, x +, x x 3 488. : 489. : x ++ +x + 4, x 490. x x +x : 49. (x, ) x, x : r x x 49. : 493. 4, x, x : ｰ 09 0 ｧ 06. 6 5 06 493 0403000307 ｧ0504 x = sin t. x 494. 4x + x : 495. : 5 +4x, x 496. 498. x 3 +x : 497. x, x 3 : 499. 3 x +x : x : 3x, x,

3x 8. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 00609030503 縺0308 05 0 ﾀ09060 閨0600500308 070600909 ｧ08 55 x 8. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 00609030503 縺0308 05 0 ﾀ09060 閨0600500308 070600909 ｧ08 07 09 ｧ030 0004003006 00050603 ｧ 06 ﾀ0903006 04 ｧ 閨06 04 罍ｧ0403 閨0705 ﾀ07 ｧ0050605 0003: R(e x ) 0403 陦ﾀ05 00304090308 e x = t, x =lnt, dt = t ; R(tg x) 0403 陦ﾀ05 00304090308 tg x = t, x = arctg t, dt = +t ; R(sin x; cos x) 0403 陦ﾀ05 00304090308 tg x = t, sin x t = =, t dt, = +t +t. +t, cos x = 7 0805 ｧ ｧ0050605 0003: e x, e x 500. e x + : 50. e 3x 50. e : x + 503. 504. 5 + 3 cos x : 505. 506. sin 4 x : 507. tg 4 x : sin x : 3 sin x + 4 cos x : + 3 cos x : ｰ 09 0 ｧ 06. 6 5 08 506, 507, 5, 53, 05 閨06 0403 ｧ0050605 00030 sin x ｧ cos x ﾀ03 閨06 縺07 05 ﾀ07 050300040903 04 060500308 ﾀ050604060 ｧ, 0006 0006 04 罍ｧ0060070504 0403 - ﾀ05 0030409 ｧ tg x = t; sin x = t +t ; cos x = dt ; = +t +t : 7 0805 ｧ ｧ0050605 0003: e x 508. e x, : 509. 50. 5. e 3x e x, : 5. cos 4 x : 53. tg 5 x : 3 + cos x : + 3 sin x :

356 0600. 8. 7060304 縺06 閨060006000308 ｧ0050605 00 54. sin x + sin x : + cos x 55. sin 3 x : e x + +tg x 56. : 57. sin x : e x, x 9. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 05 ｧ0406 縺030300 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08. 06 ｧ0406 縺030300 ｧ 06 ﾀ09 ｧ06 0403 陦ﾀ05 0030409 ｧ : ch x =shx + C: : sh x=chx + C: 3: =thx + C: 4: =, cth x + C: ch x sh x 0ｷ30050605 0003 0305 0904 陦 050304 ｧ 陦縺0605 ｧ08 060500308 ﾀ0506040600608 ch x ｧ sh x 0 0803 閨0705 ﾀ07 04 罍ｧ0060060 ｧ060 0703 縺00600: ch x = chx + ; sh x chx =, ; sh x ch x = sh x : 0ｷ30050605 0003 0305 006 060500308 ﾀ0506040600608 sh x ｧ ch x 0 0803 閨0705 ﾀ07 05060 0706 ﾀ0403- ﾀ0303030, 0503 ｧ ｧ0050605 0003 0305 006 060500308 ﾀ0506040600608 sin x ｧ cos x. 06 ｧ0406 縺030300 ｧ 06 ﾀ09 ｧ06 0403 陦ﾀ05 0030409 ｧ ｧ00305 04 罍ｧ0060070605 ﾀ07 04 罍ｧ 0 080307 閨060 ｧ ｧ ｧ0050605 000304 04 ｧ R x; x, a 0403 陦ﾀ05 00304090308 x = a ch t; R x; x + a 0403 陦ﾀ05 00304090308 x = a sh t: x + x 7 罍ｧ 0505030: 06 ﾀ00 ｧ x = a ch t, 0503t =ln, a a, 06 ﾀ00 ｧ x = a sh t, 0503t x =ln + x + a. a 7 0805 ｧ ｧ0050605 0003: 58. ) 59. 5. sh 3x; ) ch 3 x : 50. ch x + : 5. (+shx) : th x : th x, :

3x 0. ｮ00600 000306 04 罍ｧ006 縺03 0 ｧ0050605 罍ｧ 縺0304 0 ｧ06 57 53. 55. x + a : 54. (x +4) 3 : 56. x, a : (x, 5) 3 : 7 0805 ｧ ｧ0050605 0003: 57. sh 3 3x : 58. 59. sh 4 x ch x : 530. sh x ch x : cth x : 53. 533. ch x +: 53. x x, 3: 534. +shx ch x : x +3 : x x 0. ｮ00600 000306 04 罍ｧ006 縺03 0 ｧ0050605 罍ｧ 縺0304 0 ｧ06 7 0805 ｧ ｧ0050605 0003: +x arctg x 535. : 536. : x +x 537. x 3 + ax : 538. + sin x : 539. : 540. x(, x) sin x=a + cos x=b : 54. x cos x : 54. e x + e x : r, x cos x 543. : 544. +x sin 4 x : 545. x tg cos x x : 546. sin x : 547. 549. sin x b + cos x : 548. ax, b : 550. (ax + b) 4 x + x : 3 x 4 + x :

358 0600. 8. 7060304 縺06 閨060006000308 ｧ0050605 00 55. (sin x : + cos x) 55. x a + b ln x : x 553. (a, bx 3 ) n : 554., x, x : arctg x 555. : ( + x) 556. : 3 x e x, 557. : 558. 559. 56. ) 56. ) e x +4 ctg 4 x : 560. cos x ; ) cos 3x x + a + x ; ) (x + )( + 4, x x sin x sin 3x : : x +, x : : x +) 563. 565. 567. 569. 57. 573. 575. 577. 579. 58. x 4 + x : +x 564. : x 3, x x 3 x : 566. x 3, +tg x : arcsin x sin x : 568. x cos 4 x : cos x ln (cos x) : 570. : sin 4 x sin x e 3x, e x : sin 3 57. x cos 5 x : ln (x +) : 574. x, sin x : + sin x : 576. x x 4, x, : arctg x e,x : 578. : x tg x sin x : ln (x +) 580. : x 3 a x a x :, sin x + 58. : x

3x 0. ｮ00600 000306 04 罍ｧ006 縺03 0 ｧ0050605 罍ｧ 縺0304 0 ｧ06 59 583. 585. 587. 589. s (x +) 3 : 584. (x, ) x : 586. x, x, a : 588. ax + x cos3 x + : 590. sin x x arcsin x, x : x (x +) 4 : 4x + x 3 + x, x : x 4 +4 :

30600 04 9 05 7 7 ｭ6 7 ｭ6ｧ8 ｭ 7 76ｧ9 0ｷ3 7 ｯ ｭ06 7 縺6ｧ8 x. 6 503 ｧ ﾀ00060 ｧ06 0304 縺06 閨06000600030503 ｧ0050605 00 706 ﾀ0504 0 0305 縺06 0906 [a; b] 0304 縺06 閨0600060 070600909 ｧ07 f(x). 7 030304060 0305 縺06 0309 [a; b] 0 n ﾀ050608 0503 09 0 ｧ a = x 0 <x <x <:::<x n = b. 0ｷ3 09 07 閨030503 ｧ00506 縺04 00 (x i, ;x i ) 0403 040060 04 縺03 ｧ 0403000400606 0503 0906 i ｧ ﾀ03 ﾀ05 04 ｧ0 ﾀ060006 np P f( i )x i,05 閨06 x i = x i, x i,. ｮ0600 04 ｧ n f( i )x i 0 0304 0605 ﾀ07 i= ｧ0050605 000400308 ﾀ06000308, 0606 04 縺06 閨0600 04 罍ｧ max x i! 0, 06 ﾀ00 ｧ 030 ﾀ06006 ﾀ0504060605 ｧ 0903006 060, 0 0304 0605 ﾀ07 0304 縺06 閨06000600030 ｧ0050605 00030 0305 070600909 ｧ ｧ f(x) 04 04 縺06 閨0600 08 0305 a 閨03 b ｧ030303 0 0605 ﾀ07 b a f(x) = lim nx max x i!0 i= i= f( i )x i : () ｱ0600909 ｧ07 f(x) 04 0505030 ﾀ0006 06 0 0304 0605 ﾀ07 ｧ0050605 罍ｧ 縺060600308 0 0305 縺06 0906 [a; b]. 6 70007 ｧ0050605 罍ｧ 縺0606003 ﾀ05 ｧ 閨03 ﾀ05 0503 003, 05030303 0 0305 縺06 0906 [a; b] 0706009-09 ｧ07 030300 00604 縺06 縺03040 ｧ00 ｧ 0706 ｧ00600 0903006 00306 ｧ ﾀ0003 0903006 00308 縺03-040304. 706 ﾀ0504 f(x) 00604 縺06 縺03040 0 [a; b]. ｯ0305 0 0505030 0305 縺06 0906 ﾀ06006 ﾀ0504060605 0060304 縺06 閨060006000308 ｧ0050605 00 f(x) = F (x)+c () ｧ ｧ0060605 006 ﾀ0503 0703 縺00600 b a f(x) = F (b), F (a) = f(x) b a ; (3) 05. 06. 0304 縺06 閨060006000308 ｧ0050605 00 0305 00604 縺06 縺030400308 070600909 ｧ ｧ 04060 003 ﾀ05 ｧ 0 060 ｧ08 0406 縺04030303 00308 070600909 ｧ ｧ ( ｧ00 ｧ 0060304 縺06 閨06000600030503 ｧ0050605 00 ) 04 罍ｧ 0406 縺080060 ｧ 0 ｧ070060 04 縺06 閨0600 08. ｱ03 縺00600 (3) 0 0304 0605 ﾀ07 0703 縺006000308 7 04 06-05030 {6ｧ80608030 ｧ09.

3x. 6 503 ｧ ﾀ00060 ｧ06 0304 縺06 閨06000600030503 ｧ0050605 00 6 59. ｮ03 ﾀ05 0400060 ｧ060 ｧ0050605 000400308 ﾀ0600 ｧ 0406 縺060803 閨030 09 04 縺06 閨060006 0 0805 ｧ ｧ0050605 0003: a a a ) 0 x; ) 0 x ; 3) 0 e x ; 4) 0 sin x. ｰ 09 0 ｧ 06. 7 罍ｧ 縺0600060 ｧ ｧ 040503 縺030503 ｧ 06050406 縺05030503 04 罍ｧ006 縺0304 0403 ﾀ04030004-0304 0504 ﾀ07 縺06 06000405 05 0 ｧ 034 ｧ 647. 59. 6 503 ｧ ﾀ00 ｧ0504 080 ｧ070060609 ｧ 080406 縺080060609 ｧ0050605 000400306 ﾀ060003 s 5 ｧ S 5 閨0007 ｧ0050605 00, 03 ｧ04 0305 縺06 0309 [; ] 0 04070504 0400308 x ﾀ050608. ｮ 040 ｧ0504 ﾀ 0503 0030 0 060 ｧ060 ｧ0050605 00. ｰ09 0 ｧ06. s 5 = 5P i= m i x, S 5 = 5P i= M i x, 05 閨06 m i 0 ｧ006004000606, M i 0 ｧ03030004000606 0 060 ｧ06 0403 閨030050605 000400308 070600909 ｧ ｧ 04 i-0 ﾀ05 ｧ 0030 04 縺030060706050906. 6 503 ｧ ﾀ00 ｧ0504: 3 593. x 3 : 594. x + x4 : 595. 597. 599. 4 x : 596. a 3 a a + x : 598. 0 3 0 4, x : e x=3 : =4 : x + 600. sin 4x : 60. 0 9 : 60. x, 4 0 =3 =4 +tg x ( + tg x) : ｰ 09 0 ｧ 06. 6 5 06 60 00607003 04 罍ｧ0060 ｧ0504 0403 陦ﾀ05 003040906 x = t ; 04 罍ｧ 0505030 04 縺06 閨060003 ｧ0050605 00 ｧ 00600705 ﾀ07, 0503 04 ｧ ﾀ0304 0605 ﾀ07 04 04 ｧ 閨06 05 03-00 ｧ0903 x j 4 j 9. 縺0 000305 ｧ 003 04 06 60 04 罍ｧ ｧ0050605 罍ｧ 縺0304 0 ｧ ｧ 0403 陦ﾀ05 00304t j j 3 090308 tg x = t 00607003 ﾀ030305040605 ﾀ0504060003 ｧ 0060 ｧ0504 04 縺06 閨060003.

36 0600. 9. 0504 縺06 閨060006000308 ｧ0050605 00 603. 605. 607. 4 0 0 = + x + : 604. e x + : 606. sin x cos x : 608. 0 a= 0 a x 4, x : r x a, x : x a, x : 609. 0 ln (x +): 60. 0 +x : 0 0 3 3 6. ( + x : 6. ) 3 x + x : 63. 0ｷ3 0703 縺0060003 ｧ 407 04030006 ｧ0504, 0503 ｧ 0403 ｧ ﾀ00 ｧ0504: = ) = 0 sin x; ) sin n x= n, n = = 0 sin 4 x; 3) sin n, x ; = sin 6 x. 0 0 0 6 503 ｧ ﾀ00 ｧ0504: 64. 66. 68. a 0 3 0 4 (x, ax) : 65. x 4, x : 67. ( + x) : 69. 3 =6 =8 0 x : cos x : e x +e x :

3x. 6 503 ｧ ﾀ00060 ｧ06 0400030 閨0608 63 60. 5 x 4x +5 : 6., x : 6. = x cos x : 63. =4 tg 3 x : 0 64. 0ｷ3 0703 縺0060003 ｧ 407 04030006 ｧ0504, 0503 ｧ 0403 ｧ ﾀ00 ｧ0504: ) = = 0 cos x; ) 0 cos n x= n, n = = 0 cos 4 x; 3) cos n, x ; = cos 6 x. 0 0 0 x. 6 503 ｧ ﾀ00060 ｧ06 0400030 閨0608. 700030 閨04 09 罍ｧ040300 ｧ0060800308 05 040609 ｧ ｧ A ABB, 04 罍ｧ0006-07 00608 09 03 ﾀ ｧ Ox ( 罍ｧ ﾀ. 3): S = x X lim yx = x!0 y : () x 6 7 ｧ070706 縺06009 ｧ 00 0406 縺06006000308 0400030 陦ｧ A AMM 04060 ds = y. 7 ｧ ﾀ. 3 7 ｧ ﾀ. 3 ｭ ﾀ00 ｧ 09 罍ｧ04 07 0 06 040060 ｧ070 ｧ x = f(t) ｧ y = '(t), 0503 ds = = '(t) f 0 (t) dt.

364 0600. 9. 0504 縺06 閨060006000308 ｧ0050605 00. 700030 閨04 09 罍ｧ040300 ｧ0060800308 05 040609 ｧ ｧ, 04 罍ｧ000607 00608 09 03 ﾀ ｧ Oy: S = X y lim xy = y!0 xdy: () y 6 7 ｧ070706 縺06009 ｧ 00 0406 縺06006000308 0400030 陦ｧ ds = xdy. 3. 700030 閨04 ﾀ06090503 OAB ( 罍ｧ ﾀ. 3) 09 ｧ 04 03 08, 000308 04 04030007 縺00308 090303 罍陦ｧ0 05 08: S = X lim '!0 r ' = ' 6 7 ｧ070706 縺06009 ｧ 00 0406 縺06006000308 0400030 陦ｧ ds = r d'. ' r d': (3) 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04, 0305 0 ｧ 06000606 00 ｧ0 ｧ070 ｧ: 65. y =4, x ;y=0: 66. x a + y b =: 67. y =x; x = h: 68. y =3, x, x ;y =0: 69. xy =4;x=; 630. y =lnx; x = e; y =0: x =4;y=0: 63. y =x +4;x=0: 63. y = x 3 ;y=8;x=0: 633. y =(4, x) 3 ;x=0: 634. 70605000608 09 罍ｧ040308 4(y, x )+x 3 =0: 635. y = x ;y =, x : 636. y = x +4x; y = x +4: 637. a y = x 3 (a, x): 638. (y, x) = x 3 ;x =: 639. 70605000608 ﾀ05 縺030703 ｧ 閨03 y (a, x) =x(x, a). 640. ｨC060400308 00 ｧ0 ｧ0608 y = a (ex=a + e,x=a ), x = a ｧ y =0. 64. 05 閨00308 縺090308 09 ｧ090003 ｧ 閨03 x = a(t, sin t), y = a(, cos t) ｧ 03 ﾀ0406 Ox. 64. 罍ﾀ05 縺03 ｧ 閨0308 x = a cos 3 t, y = a sin 3 t. 643. 6ｧ80600 ｧ ﾀ09 050308 r = a cos '. 644. 07 罍陦ｧ03 ｧ 閨0308 r = a(, cos '). 645. r = 3 + sin ' 00607 閨06 ﾀ006070030 ｧ 0 ｧ0303000400 ｧ0 ｧ 0 ｧ006004-00 ｧ0 陦ｧ06 ﾀ-0406090503 0 ｧ. 646. r =, cos 3' 00607 閨06 ﾀ006070030 ｧ 0 ｧ0303000400 ｧ0 ｧ 0 ｧ006004-00 ｧ0 陦ｧ06 ﾀ-0406090503 0 ｧ.

3x 3. 05030060 050600 04 0060 ｧ07 65 647. r = a cos ': 648. r = a sin 3': 649. r = a(sin ' + cos '): 650. r = a ' ; 4 6 ' 6 : 65. r = a sin 3 ',000607 00608 0 ｧ0706 04030007 縺00308 03 ﾀ ｧ. 3 65. 70605000608 閨0609 縺050304 00 ｧ ﾀ05 x 3 + y 3, 3axy =0( ﾀ0. 罍ｧ ﾀ. 79 0 ﾀ. 334) (0406 縺060805 ｧ 09 04030007 縺0030 090303 罍陦ｧ0 05 0). sin ' cos 'd' ｰ 09 0 ｧ 06. 6 5 ｧ0050605 0006 0403000307 ｧ0504 tg' (sin 3 = u, ' + cos 3 ') 閨0600 ｧ04 ﾀ0 00 ｧ ﾀ00 ｧ05060004 ｧ 0 0060 05060004 0 cos 6 '. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04, 0305 0 ｧ 06000606 00 ｧ0 ｧ070 ｧ: 653. y =6x, x, y =0. 654. y = x 3, y =8, x =0. 655. y =, x ｧ x =,3. 656. y + x 4 = x. 657. y = x +4x +5, x =0, y =0 ｧ 0 ｧ0 ｧ0 000400308 03 罍陦ｧ0 050308. 658. 05 閨00308 0403000604030000308 ﾀ ｧ006 ﾀ03 ｧ 閨03 y = sin x ｧ y =0. 659. 4y = x ｧ y =4x. 660. xy =6 ｧ x + y, 7=0. 66. 70605000608 09 罍ｧ040308 x 3 + x, y =0. 66. r =3, cos ' 00607 閨06 ﾀ006070030 ｧ 0 ｧ0303000400 ｧ0 ｧ 0 ｧ006004-00 ｧ0 陦ｧ06 ﾀ-0406090503 0 ｧ. 663. r = + sin 3' 00607 閨06 ﾀ006070030 ｧ 0 ｧ0303000400 ｧ0 ｧ 0 ｧ006004-00 ｧ0 陦ｧ06 ﾀ-0406090503 0 ｧ. 664. r = a sin ': 665. r = a cos 3'. 666. r = ae ' 0305 ' =, 閨03 ' =. 667. 050300608 ﾀ05 ｧ 050000 ｧ04 ﾀ0304 x a + y x = ｧ b b + y a =(0406 縺060805 ｧ 09 04030007 縺0030 090303 罍陦ｧ0 05 0). 668. r = a( + sin ') ｧ r = a. x 3. 05030060 050600 04 0060 ｧ07.05030060 050600, 0303 0304 00030503 04 0060 ｧ060 040309 縺0605 03 ﾀ ｧOx 09 罍ｧ040300 ｧ- 0060800308 05 040609 ｧ ｧ A ABB ( 罍ｧ ﾀ. 33), 05 閨06 AB ^ 閨0605 09 罍ｧ040308 y = f(x), 0304 縺06 閨0600070605 ﾀ07 0703 縺006000308 x X V = lim y x = y : () x!0 x 6 7 ｧ070706 縺06009 ｧ 00 0406 縺0600600030503 03030060 dv = y.

366 0600. 9. 0504 縺06 閨060006000308 ｧ0050605 00.05030060 050600, 0303 0304 00030503 04 006-0 ｧ060 040309 縺0605 03 ﾀ ｧOy 09 罍ｧ040300 ｧ0060800308 05 0406-09 ｧ ｧ, 04 罍ｧ000607 00608 09 03 ﾀ ｧ Oy, 0304 縺06 閨0600070605 ﾀ07 0703 縺006000308 V = X y lim x y = y!0 x dy: () y 7 ｧ ﾀ. 33 6 7 ｧ070706 縺06009 ｧ 00 0406 縺0600600030503 03030060 dv = = x dy. 0504 縺06 閨0600 ｧ0504 03030060 050600, 0303 0304 00030503 04 0060 ｧ060 07 ｧ0506 縺03, 0305-0 ｧ 06000308 00 ｧ0 ｧ070 ｧ: 669. y =x ｧ x = h 040309 縺0605 03 ﾀ ｧOx. 670. x a, y = ｧ y = b 040309 縺0605 03 ﾀ ｧOy. b 67. xy =4, x =, x =4, y =0040309 縺0605 03 ﾀ ｧOx. 67. y =(x +4) 3 ｧ x =0040309 縺0605 03 ﾀ ｧOy. 673. x + y = a 040309 縺0605 04 縺0700308 x = b>a. ｰ09 0 ｧ06. dv = (b + x) dy, (b, x) dy =4bx dy. 674. y = a ch x, x = a, y =0040309 縺0605 03 ﾀ ｧOx. a 675. y =4, x, x =0040309 縺0605 03 ﾀ ｧOy. 676. (y, a) = ax, x =0, y =a 040309 縺0605 03 ﾀ ｧOx. 677. y = cos x ｧ y =, 040309 縺0605 04 縺0700308 y =, 04 罍ｧ, 6 x 6. 678. y = x,x, x =,4 ｧ y =0040309 縺0605 03 ﾀ ｧOy. 679. y = cos x,, x =0, y =0(04 罍ｧ x>0) 040309 縺0605 03 ﾀ ｧOx. 3 680. y = a, x a ｧ x + y = a 040309 縺0605 03 ﾀ ｧOy. 0504 縺06 閨0600 ｧ0504 03030060 050600, 0303 0304 00030503 04 0060 ｧ060 07 ｧ0506 縺03, 0305-0 ｧ 06000308 00 ｧ0 ｧ070 ｧ: 68. y = sin x (03 閨00308 0403000604030000308), y =0040309 縺0605 03 ﾀ ｧOx. 68. x, y =4, y = 040309 縺0605 03 ﾀ ｧOy. 683. y =, x =, y =0040309 縺0605 03 ﾀ ｧOx. +x

3x 4. 6 700 ｧ0 閨0605 ｧ 040003 ﾀ090308 09 罍ｧ040308 67 684. x a + y =040309 縺0605 03 ﾀ ｧOy. b 685. x =3 + y =3 = a =3 040309 縺0605 03 ﾀ ｧOx. 686. y = x 3, x =0, y =8040309 縺0605 03 ﾀ ｧOy. 687. x, y = a, x = a 040309 縺0605 03 ﾀ ｧOx. 688. y = x, y =4040309 縺0605 04 縺0700308 x =. ｰ09 0 ｧ06. dv = ( + x) dy, (, x) dy. 689. 05 閨00308 縺09 ｧ 09 ｧ090003 ｧ 閨03 x = a(t, sin t), y = a(, cos t) 040309 縺0605 03 ﾀ ｧOx. 690. (y, 3) +3x =0, x =,3 040309 縺0605 03 ﾀ ｧOx. x 4. 6 700 ｧ0 閨0605 ｧ 040003 ﾀ090308 09 罍ｧ040308. 6 700 ｧ0 閨0605 ｧ AB ^ 09 罍ｧ040308 y = f(x): x B s = +y 0 : () x A +y 0 = + dy.. 6 700 ｧ0 閨0605 ｧ AB ^ 09 罍ｧ040308 x = f(t), y = '(t): t B 6 7 ｧ070706 縺06009 ｧ 00 閨0605 ｧ ds = s = 3. 6 700 ｧ0 閨0605 ｧ AB ^ 09 罍ｧ040308 r = f('): s = t A ' B 0504 縺06 閨0600 ｧ0504 閨00 ｧ006 閨0605 ｧ 09 罍ｧ040308: _x +_y dt: () ' A r + r 0 d': (3) 69. y = x 3,0305 ﾀ06 06000308 04 縺0700308 x =4=3. 69. 6 5 ﾀ0608 09 罍ｧ040308 x + y = a. 693. 6 5 ﾀ0608 09 罍ｧ040308 x =3 + y =3 = a =3. 694. y =(x +) 3,0305 ﾀ06 06000308 04 縺0700308 x =4. 695. 05 閨00308 縺09 ｧ 09 ｧ090003 ｧ 閨03 x = a(t, sin t); y = a(, cos t): 696. x = t6 6, y =, t4 4 090303 罍陦ｧ0 05. 00607 閨06 0503 09 0 ｧ 0406 縺06 ﾀ06 060 ｧ07 03 ﾀ070 ｧ

368 0600. 9. 0504 縺06 閨060006000308 ｧ0050605 00 697. y = x,, 0305 ﾀ06 06000308 03 ﾀ0406 Ox. ｰ09 0 ｧ06. ﾀ 0504 04030703 縺0060006 ｧ 366. +x 00307003 ｧ00 ｧ 0 0805 ｧ 0403 ﾀ05070, ｧ00 ｧ 0 04 ｧ- 698. y = a (ex=a + e,x=a )=a ch x a 00607 閨06 04 縺070030 ｧ x = a. 699. y =lnx 0305 x =3=4 閨03 x ==5. +x ｰ09 0 ｧ06. 0ｷ30050605 00 0 0803 陦ｧ05 ﾀ07 0403 陦ﾀ05 00304090308 x +x = t. 700. y = ln ( cos x) 00607 閨06 ﾀ006070030 ｧ 0503 09 0 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 Oy ｧ Ox. 70. ) 9y = x(x,3) 00607 閨06 0503 09 0 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ0406 Ox. ) e y th x =0305 x = 閨03 x =. 70. ) 07 罍陦ｧ03 ｧ 閨03 r = a(, cos '). ) 706 縺04030503 04 ｧ0509 ﾀ04 ｧ 00 ｧ r = a'. 703. 6 5 ﾀ0608 09 罍ｧ040308 r = a sin 3 ' 3. 704. 06 ｧ0309 07 0 ｧ0504 0403 閨040600060 04 0503 09 08 A ｧ B, 0 0803 閨070 ｧ08 ﾀ07 0 03 閨00308 0403 ﾀ030506 0 ﾀ ﾀ0503070 ｧ ｧ AB =b, ｧ ｧ0060605 ﾀ05 縺060006 04 縺0305 ｧ03 f. ｮ ｧ05 07 0703 縺006 0 ｧ05 ｧ 04 0303000308, 040309 0504, 0503 閨00 ｧ0 0 ｧ05 ｧ s b + f 3 b 04 罍ｧ 閨03 ﾀ05 0503 003 0 00030 f b. ｰ 09 0 ｧ 06. 7 罍ｧ0060 ｧ0504 04 罍ｧ0300 ｧ0706000606 0703 縺0060006 + + ｧ 57. 0504 縺06 閨0600 ｧ0504 閨00 ｧ006 閨0605 ｧ 09 罍ｧ040308: 705. y = 4 9 (, x)3,0305 ﾀ06 06000308 04 縺0700308 x =,. 706. y = ln (sin x) 0305 x = =3 閨03 x ==3. 707. y =ln(, x ) 0305 x =,= 閨03 x ==. 708. y =x, 0305 ﾀ06 06000308 04 縺0700308 x = =. 709. x = t, y = t 3 (t, 3) 00607 閨06 0503 09 0 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ0406 Ox.

3x 5. 700030 閨04 04030406 縺08003 ﾀ05 ｧ 04 0060 ｧ07 69 x 5. 700030 閨04 04030406 縺08003 ﾀ05 ｧ 04 0060 ｧ07. 700030 閨04 04030406 縺08003 ﾀ05 ｧ,0303 0304 000308 04 0060 ｧ060 040309 縺060503 ﾀ ｧ Ox 閨0605 ｧ ^ AB 09 罍ｧ040308 y = f(x): P x = ^ AB y ds; 05 閨06 ds = + dy :. 700030 閨04 04030406 縺08003 ﾀ05 ｧ,0303 0304 000308 04 0060 ｧ060 040309 縺060503 ﾀ ｧ Oy 閨0605 ｧ ^ AB 09 罍ｧ040308 x = '(y): P y = ^ AB x ds; 05 閨06 ds = + dy : 0504 縺06 閨0600 ｧ0504 0400030 閨04 04030406 縺08003 ﾀ05 ｧ, 0303 0304 000308 04 0060 ｧ060 09 罍ｧ- 040308: 70. x + y = R 040309 縺0605 03 ﾀ ｧOx. 7. y = x =, 0305 ﾀ06 06000308 04 縺0700308 y =; 5, 040309 縺0605 03 ﾀ ｧOy. 7. y = a ch x a 00607 閨06 x = a 040309 縺0605 03 ﾀ ｧOx. 73. 4x + y =4040309 縺0605 03 ﾀ ｧOy. ｰ 09 0 ｧ 06. 7 罍ｧ00704 y 006 04 ｧ ﾀ ｧ00606 0406 縺06006000606, 04030006 ｧ0, 0503 ｧ ﾀ09030 07 0400030 閨04 P = 6, 3y dy. 6 7000606 04 罍ｧ006007060 0403 陦ﾀ05 003040906 y = 4 3 sin t. 0 74. 05 閨00308 04030006040300003 09 罍ｧ040308 y = sin x 040309 縺0605 03 ﾀ ｧOx. 75. 05 閨00308 縺09 ｧ 09 ｧ090003 ｧ 閨03 x = a(t, sin t), y = a(, cos t) 040309 縺0605 03 ﾀ ｧOx. 76. 7060500 ｧ 09 罍ｧ040308 x = t, y = t 3 (t, 3) 040309 縺0605 03 ﾀ ｧOx. 77. x + y = a 040309 縺0605 04 縺0700308 x = b>a. ｰ09 0 ｧ06. dp =(b + x) ds +(b, x) ds. 0504 縺06 閨0600 ｧ0504 0400030 閨04 04030406 縺08003 ﾀ05 ｧ, 0303 0304 000308 04 0060 ｧ060 0403-09 縺0605 Ox: 78. 6 70605 ｧ 09 罍ｧ040308 y = x3 0305 x =, 閨03 x =. 3

370 0600. 9. 0504 縺06 閨060006000308 ｧ0050605 00 79. 6 70605 ｧ 09 罍ｧ040308 y =4+x, 0305 ﾀ06 06000308 04 縺0700308 x =. 70. 6 5 ﾀ0608 09 罍ｧ040308 x = a cos 3 t, y = a sin 3 t. 7. 6 70605 ｧ 09 罍ｧ040308 x = t3 3, y =4, t 060 ｧ07 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05. 00607 閨06 0503 09 0 ｧ 0406 縺06 ﾀ06- x 6. 6ｦ0 ｧ ｧ 07 ｧ ｧ09 ｧ 7. 0504 縺06 閨0600 ｧ0504 ﾀ ｧ0006 0400060 ｧ07 0403 閨03 0 0406 縺05 ｧ09 000400308 04 縺07-003060503000400308 000006 ﾀ 03 ﾀ00304 0 ｧ060 8 0 ｧ 0403 ﾀ03050308 6 0. 0504 縺06 閨0600 ｧ0504 05 090706 ﾀ ｧ0006 0400060 ｧ07 0 0 ｧ0700606 0403000304 ｧ006 000006. 73. 0504 縺06 閨0600 ｧ0504 ﾀ ｧ0006 0400060 ｧ07 0403 閨03 0 0406 縺05 ｧ09 000400606 05 縺06-060503000400606 0400030 閨0906, 03 ﾀ00304 0 ｧ06 09030503 縺0308 a ﾀ040300030706003 0 04030406 縺08-003 ﾀ05 ｧ 0403 閨03, 0403 ﾀ0305 040 h. 74. 0504 縺06 閨0600 ｧ0504 ﾀ ｧ0006 0400060 ｧ07 0403 閨03 0 0406 縺05 ｧ09 000400308 04030006-09 縺0605, 陦ｧ 00605 09030503 縺030503 R ﾀ0403000307060 0 04030406 縺08003 ﾀ05 ｧ 0403 閨03. 75. 7000305 ｧ0 ｧ0060605 0703 縺006 05 040609 ｧ ｧ ﾀ 0406 縺080 ｧ0 03 ﾀ00304 0 ｧ060 0 0, 0 ｧ070 ｧ0 0 0 ｧ 0403 ﾀ03050308 6 0. 0504 縺06 閨0600 ｧ0504 ﾀ ｧ0006 0400060 ｧ07 0403 閨03 0 04000305 ｧ006. 76. 7 0805 ｧ 00300600503 ｧ006 縺09 ｧ ｧ 0305003 ﾀ ｧ05060004003 03 ﾀ0608 Ox ｧ Oy 0400030 陦ｧ 04 縺0700306050300040 ｧ09, 0305 0 ｧ 0600030503 00 ｧ0 ｧ070 ｧ x = 0, x = a, y =0 ｧ y = b. ｰ09 0 ｧ06. 7 03 ｧ04 04 縺0700306050300040 ｧ09 0 0503 罍ｧ 03005 000400306 0400030 閨09 ｧ, 06000307 ｧ0 09 07 閨0606 0400030 閨0906 0 0904 陦 05 0606 ﾀ ﾀ0503070 ｧ07 0305 03 ﾀ ｧOx, 05.06. 0 y. ｮ0600 ｧ 縺0607 ｧ 0406 縺0608 閨07 09 04 縺06 閨060006, 04030006 ｧ0 X b J x = lim ayy = ay dy: y!0 0 a 縺0 000305 ｧ 003 J y = bx. 0 77. 7 0805 ｧ 003006005 ｧ006 縺09 ｧ ｧ 0305003 ﾀ ｧ05060004003 03 ﾀ0608 Ox ｧ Oy 040003-0 陦ｧ 05 縺0606050300040 ｧ09, 0305 0 ｧ 0600030503 00 ｧ0 ｧ070 ｧ x = 0, y = 0 ｧ x a + y b =. 78. 7 0805 ｧ 003006005 ｧ006 縺09 ｧ ｧ 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oy 0400030 陦ｧ, 0305 0 ｧ 06000308 00 ｧ0 ｧ070 ｧ x =, y = x ｧ y =0. 79. 7 0805 ｧ ﾀ05 05 ｧ 06 ﾀ09 ｧ06 00300600503 0305003 ﾀ ｧ05060004003 Ox ｧ Oy ｧ 090303 罍陦ｧ0 0503 0906005 0 ﾀ ﾀ 05 縺0606050300040 ｧ09, 0303 0304 00030503 00 ｧ0 ｧ070 ｧ x =0, y =0 ｧ x + y = a. a a ｰ09 0 ｧ06. ｮ05 05 ｧ 06 ﾀ09 ｧ06 00300600503: M x = xy dy, M y = xy. 070303 罍陦ｧ0 0503 0906005 0 ﾀ ﾀ: x c = M y S, y c = M x,05 閨06 S 0400030 閨04 07 ｧ0506 縺03. S 0 0

3x 6. 6ｦ0 ｧ ｧ 07 ｧ ｧ09 ｧ 7 730. 7 0805 ｧ 0906005 0 ﾀ ﾀ 0400030 陦ｧ, 0305 0 ｧ 06000308 00 ｧ0 ｧ070 ｧ a y = = bx, x = a ｧ y =0. 73. 7 0805 ｧ 0906005 0 ﾀ ﾀ 0403000609 縺0605 x + y = a, 0305 ﾀ06 0600030503 03 ﾀ0406 Ox. 73. ) 6 503 ｧ ﾀ00 ｧ0504 03030506, 09030503 縺0606 00607003 05 05 ｧ0504 0 040309 - ｧ04 0 ｧ06 0403 閨03 ｧ 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09030503 03 ﾀ ﾀ06080 ﾀ 陦ｧ06 ﾀ030 03 ﾀ00304 0 ｧ07 0,5 0, 06 ﾀ00 ｧ 04 0 000400308 003006005 06 縺030406004 0403 閨03 04 03 ﾀ ﾀ0608006 04060,8 0 ｧ 0 0, 0 0 ｧ0706 04030406 ﾀ09 060060503 0403 閨06 03050406 罍ﾀ05 ｧ07 04 09 ｧ00 ｧ0 陦縺06. ) 6 503 ｧ ﾀ00 ｧ0504 03030506, 09030503 縺0606 00607003 05 05 ｧ0504 0 040309 ｧ04-0 ｧ06 0403 閨03 ｧ 0403000600 陦ｧ06 ﾀ030 R 0. 733. 0504 縺06 閨0600 ｧ0504 03030506, 09030503 縺0606 00607003 05 05 ｧ0504, 05030303 0403-0070504 0 ﾀ ﾀ06 m ﾀ 04030406 縺08003 ﾀ05 ｧ 06000 ｧ 0 0403 ﾀ030506 h. ｰ 09 0 ｧ 06. ｮ ｧ00 F 0600030503 04 罍ｧ050707060 ｧ07 0 ﾀ ﾀ0503070 ｧ ｧ x 0305 0906005 06000 ｧ 0304 縺06 閨0600070605 ﾀ07 ｧ 04 縺030403 縺09 ｧ ｧ F : mg = R : x,05 閨06 R 陦ｧ06 ﾀ 0600030503 00. 734. 0703050600 ｧ0060605 0703 縺006 04 03030003 ｧ 04 0060 ｧ07 05000603 ｧ00308 H = = 0; 5 0 ｧ 陦ｧ06 ﾀ030 03 ﾀ00304 0 ｧ07 R = 0; 4 0. 0504 縺06 閨0600 ｧ0504 03030506, 09030503 縺0606 00607003 05 05 ｧ0504 0 040309 ｧ04 0 ｧ06 0403 閨03 ｧ 05 09030503 0 040300-00600030503 09030500. 735. 6 5 09 ｧ00 ｧ0 陦縺06 0403 0403 縺000060 0 0803 陦ｧ05 ﾀ07 0403 閨0608 03030060030 V 0 = =0; 0 3 ﾀ 0400060 ｧ060 0 =; 033 0 5 7. 0504 縺06 閨0600 ｧ0504 03030506 ｧ 03-0506 縺0 ｧ 06 ﾀ09030503 ﾀ07 05 ｧ07 0403 閨0608 閨0303030060 V =0; 03 0 3. ( 703 0903006 703080007{03 罍ｧ030505 V = 0 V 0.) 736. 6 503 ｧ ﾀ00 ｧ0504 03030506 ﾀ050707060 ｧ07 0 0,00 0 006 閨00308 04 縺030403-000309 ｧ 閨00 ｧ00308 0 ﾀ 陦ｧ06 ﾀ030 ﾀ06 060 ｧ07 00. ｰ 09 0 ｧ 06. ｮ ｧ00 F 7 0 050707060 ｧ07 04 縺030403000309 ｧ 閨00 ｧ00308 l 0 ｧ 0400030 閨0406 ﾀ06 060 ｧ07 s 00 04 罍ｧ 06 閨00 ｧ0060 ｧ ｧ 0606 0 x 0 0304 縺06 閨0600070605 ﾀ07 0703 縺006000308 F = = E sx,05 閨06 E 003 閨060004 0604 縺060503 ﾀ05 ｧ. 6 70007 006 陦ｧ 00307003 04 罍ｧ0070504 E l ; 0 5 7/00. 737. 6ｦ0 09 090306 04 縺06007 0403, 0 040300007060 07 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ08 ﾀ03- ﾀ06 ﾀ 0400030 閨0406 03 ﾀ00304 0 ｧ07 S = 40 ﾀ0 ｧ 0403 ﾀ03050308 H = 40 ﾀ0, 04030506 0605 06 縺06 03050406 罍ﾀ05 ｧ06 0 閨006 0400030 閨0406 s = ﾀ0? ｰ09 0 ｧ06. ｮ0903 縺03 ﾀ0504 ｧ ﾀ0506 060 ｧ07 07 ｧ 閨0903 ﾀ05 ｧ 04 罍ｧ 06 縺0304006 0606 0 0403 ﾀ030506 x ﾀ0 0304 縺06 閨0600070605 ﾀ07 0403 0703 縺0060006 v = gx, 05 閨06 0903050707 ｧ09 ｧ06005, 04 ｧ- ﾀ070 ｧ08 0305 0407 0903 ﾀ05 ｧ 07 ｧ 閨0903 ﾀ05 ｧ, 0703 縺003 ﾀ03 ﾀ06 ｧ03050406 罍ﾀ05 ｧ07. 0303 04 罍ｧ0060 閨06 ﾀ04, 09 09 ｧ 04 06 738, =0; 6. 738. 6ｦ0 09 090306 04 縺06007 0403 04030506 0605 ｧ 09030 ｧ 06 ﾀ090308 0403 縺03009 ｧ 0403- ﾀ03050308 H =40 ﾀ0, 陦ｧ06 ﾀ030 0 ｧ070060503 03 ﾀ00304 0 ｧ07 r =0; 3 ﾀ0 ｧ 0406 縺08-0060503 R =6 ﾀ0 ( ﾀ0. 0609 0 ｧ06 09 06 737)? 739. 0504 縺06 閨0600 ｧ0504 ﾀ ｧ0006 0400060 ｧ07 0403 閨03 0 0406 縺05 ｧ09 000400606 05 縺06-060503000400606 0400030 閨0906 0403 ﾀ03050308 h, 03 ﾀ00304 0 ｧ06 09030503 縺0308 a 04 0000060004003

37 0600. 9. 0504 縺06 閨060006000308 ｧ0050605 00 04030406 縺08003 ﾀ05 ｧ 0403 閨03, 04 縺0305 ｧ040304030003070 07 0406 縺00 ｧ0 0 0803 陦ｧ05 ﾀ07 0 0403-0406 縺08003 ﾀ05 ｧ 0403 閨03. 740. 0504 縺06 閨0600 ｧ0504 ﾀ ｧ0006 0400060 ｧ07 0403 閨03 0 0406 縺05 ｧ09 000400308 04-030300 ｧ 06 ﾀ09 ｧ08 ﾀ0605006005, 03 ﾀ00304 0 ｧ06 09030503 縺030503 04003 4 0 ｧ ﾀ040300030706003 0 04030406 縺08003 ﾀ05 ｧ 0403 閨03, 0406 縺00 ｧ0 0 0803 陦ｧ05 ﾀ07 0 05000603 ｧ006 4 0. 74. 7 0805 ｧ 05000603 ｧ006 x, 0 09030503 縺0308 04 縺07003060503000400308 000006 0403- ﾀ03050308 h 閨0600 ｧ05 ﾀ07 0503 罍ｧ 03005 0004003 0 05 09 ｧ06 閨0406 ﾀ05 ｧ, 040600 ｧ ｧ0 ﾀ ｧ0003 0400060 ｧ07 0 09030503 縺0306 03 陦ｧ0 090304. 74. ｨC ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 07 09 ｧ ﾀ0506 縺0 ﾀ 0503 罍ｧ 03005 000400308 03 ﾀ0406 0 0403-000304 ｧ006 0 0403000060 0 ﾀ00030 (04000305003 ﾀ0504 0,9). 0504 縺06 閨0600 ｧ0504 ﾀ ｧ0006 040006-0 ｧ07 0 ﾀ00 0 09 07 閨0606 ｧ 040003 ﾀ09 ｧ08 ﾀ050600309 09 ｧ00 ｧ0 陦, 06 ﾀ00 ｧ 陦ｧ06 ﾀ 0606 04060 0. 743. 0504 縺06 閨0600 ｧ0504 003006005 ｧ006 縺09 ｧ ｧ 0305003 ﾀ ｧ05060004003 Ox 0400030 陦ｧ 06050406 縺05 ｧ 09 縺0605 x = a cos t, y = a sin t. 744. 7 0805 ｧ 090303 罍陦ｧ0 0503 0906005 0 ﾀ ﾀ 0400030 陦ｧ, 0305 0 ｧ 06000308 00 ｧ0 ｧ070 ｧ y =4, x ｧ y =0. 745. 6 503 ｧ ﾀ00 ｧ0504 03030506, 00603030803 陦ｧ00606 閨0007 040309 ｧ04 0 ｧ07 0403 閨03 ｧ 07003, ｧ0060600608 0703 縺006 0903006 ﾀ ( ﾀ 0406 縺00 ｧ00308 0 閨006), 0403 ﾀ0305 0903-0503 縺030503 H =0, 陦ｧ06 ﾀ 03 ﾀ00304 0 ｧ07 R =0; 3 0. 746. 0504 縺06 閨0600 ｧ0504 03030506 陦ｧ 03 05 ｧ 06 ﾀ09030503 ﾀ07 05 ｧ07 0403 閨0608 03030-060030 V 0 = 0; 0 3 ｧ ﾀ 0400060 ｧ060 0 = ; 033 0 5 7 閨03 03030060 V =0; 03 0 3. ( 罍陦ｧ 03 05 ｧ 06 ﾀ090306 ﾀ07 05 ｧ06 04 縺03 ｧ ﾀ0803 陦ｧ05 0403 0903006 706 ﾀ- ﾀ030 : V = 0 V 0,05 閨06 ; 4.) 747. 6ｦ0 09 090306 04 縺06007 0403, 0 040300007060 07 0006 0703 縺003 0403-000600 陦ｧ06 ﾀ030 40 ﾀ0, 04030506 0605 ｧ 03050406 罍ﾀ05 ｧ07 0 閨006 0400030 閨0406 ﾀ0? ( ｮ0. 0609 0 ｧ06 09 06 737; 0403000307 ｧ0 0903050707 ｧ09 ｧ06005 0407 0903- ﾀ05 ｧ =0; 8.) x 7. 706 ﾀ0303 ﾀ050406000306 ｧ0050605 0003. 0504 縺06 閨0600060 ｧ07: + b I. 0ｷ30050605 00030 f(x) 0 0304 0605 ﾀ07 lim f(x), 06 ﾀ00 ｧ 05050305 b!+ a a 04 縺06 閨0600 ﾀ06006 ﾀ0504060605 ｧ 0903006 060. 縺0 000305 ｧ 003 0304 縺06 閨0600070605 ﾀ07 ｧ0050605 0003 b + f(x) ｧ f(x).,, II. ｭ ﾀ00 ｧ f(x) 00604 縺06 縺03040 閨0007 04 ﾀ0608 0 060 ｧ08 x 0305 縺06 09 [a; b], 09 縺03006 0503 09 ｧ c, 0409030503 縺0308 f(x) ｧ0060605 縺0304 II 縺03, 0503 ｧ0050605 00030 0305 f(x) 04 04 縺06 閨0600 08 0305 a 閨03 b 0 0304 0605 ﾀ07 ﾀ0600 c," b lim f(x) + lim f(x) ; "!0!0 a c+ 06 ﾀ00 ｧ 0505 ｧ 04 縺06 閨060003 ﾀ06006 ﾀ0504060605 ｧ 0903006 003.

3x 7. 706 ﾀ0303 ﾀ050406000306 ｧ0050605 0003 73 0ｷ30050605 0003 ﾀ 0306 ﾀ0903006 0030 ｧ 04 縺06 閨0600 0 ｧ ｧ ｧ0050605 0003 0305 縺030400308 (0060305 0 ｧ 06000308) 070600909 ｧ08 0 0304 0605 ﾀ07 006 ﾀ0303 ﾀ05040600030 ｧ. ｭ ﾀ00 ｧ 04 罍ｧ0406 閨06000306 04030006 04 縺06 閨060003 0903006 003, 0503 05030403 縺0705, 0503 006 ﾀ0303 ﾀ05-0406000306 ｧ0050605 0003 ﾀ0803 閨0705 ﾀ07, 06 ﾀ00 ｧ 00605, 0503 ﾀ0803 閨0705 ﾀ07.. ｮ0803 陦ｧ003 ﾀ0504 006 ﾀ0303 ﾀ05040600030503 ｧ0050605 00 ﾀ0503 06 ﾀ05 0-0400 ｧ04 0605 ﾀ07 0060503 閨030 ﾀ 040060 ｧ07: 06 ﾀ00 ｧ 04 罍ｧ x>ajf(x)j 6 '(x) ｧ ﾀ0803 陦ｧ05 ﾀ07, 0503 ﾀ0803 陦ｧ05 ﾀ07 ｧ + a + a '(x) f(x). 縺0 000305 ｧ 00308 04 罍ｧ 0 09 ﾀ0803 陦ｧ003 ﾀ05 ｧ 00307003 0609 0504 ｧ 閨0007 ｧ0050605 00 0305 縺030400308 070600909 ｧ ｧ. 6 503 ｧ ﾀ00 ｧ0504 ｧ0050605 0003: 748. ) 6 x ; ) x ; 3) x ; 4) x n. 749. ) e,x ; ) xe,x ; 3) +x ; 0 0 4) x x, ; 5) x + x ; 6) x e,x= : 0 750. ) x x, ; ) arctg x x ; 3) (x +) : 75. ) (4, x) ; ) 8 0 (x, ) ; 3) 75. 0ｷ3 ﾀ ﾀ0006 閨0304 0504 ﾀ0803 陦ｧ003 ﾀ0504 ｧ0050605 000304: 0 (x :, ) 3 ) 4) +x ; ) 3 8 x 3, ; 3) e,x ; x 0 sin x x x ; 5) x 4 + ; 6) e,x : 0

374 0600. 9. 0504 縺06 閨060006000308 ｧ0050605 00 753. ) 0 b x n ; ) a (04 罍ｧ b>a). (b, x) n ｰ09 0 ｧ06. 7 ﾀ ﾀ00305 縺060504 05 罍ｧ ﾀ0006 07: n =, <, n = ｧ n =+>. 754. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04, 090006 06000606 00607 閨06 000309030030 y = = ｧ ﾀ ｧ0040503050308 05050308 09 罍ｧ040308. +x 755. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04, 090006 06000606 00607 閨06 09 罍ｧ040308 y = = xe,x = ｧ 0606 ﾀ ｧ0040503050308 (04 罍ｧ x>0). 756. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04, 090006 06000606 00607 閨06 09 ｧ ﾀ ﾀ03 ｧ 閨0308 y = = x3 ｧ 0606 ﾀ ｧ0040503050308. a, x ｰ09 0 ｧ06. 703000307 ｧ04 x = a sin t, 0406 縺060805 ｧ 09 04 00605 罍ｧ 06 ﾀ09 ｧ0 06 040060 ｧ070. 757. 7 0805 ｧ 03030060 050600, 0303 0304 00030503 04 0060 ｧ060 09 ｧ ﾀ ﾀ03 ｧ 閨03 x3 y = 040309 縺0605 0606 ﾀ ｧ00405030503 ( ﾀ0. 06 756). a, x 758. 0504 縺06 閨0600 ｧ0504 0400030 閨04 04030406 縺08003 ﾀ05 ｧ, 0303 0304 000308 04 006-0 ｧ060 040309 縺0605 03 ﾀ ｧOx 0306 ﾀ0903006 00308 閨0605 ｧ 09 罍ｧ040308 y = e,x 04 罍ｧ 04030003-07 ｧ0506000400308 x. 759. 7 0805 ｧ 03030060 050600, 0303 0304 00030503 04 0060 ｧ060 040309 縺0605 03 ﾀ ｧ Ox 0306 ﾀ0903006 00308 04060504 ｧ 09 罍ｧ040308 y = x, x 04 罍ｧ x >. 760. 70309 0504, 0503 04 罍ｧ m 090600030 ｧ 0403000307 ｧ050600040030 ): ) e,x x m = m!; ) e,x x m+ = m!. 0 0 76. 6 503 ｧ ﾀ00 ｧ0504 ｧ0050605 0003: ) x ; ) 0 ) ｱ0600909 ｧ07 x e,x3 ; 3) e ln x x ; 4) x ln x. e,x x t, =,(t) 0 0304 0605 ﾀ07 05 00-070600909 ｧ0608 0305 t. 7 罍ｧ 090600030 0 t>, 09 09 050503 ﾀ0006 閨060605 ｧ ｧ 760, ),,(t) =(t, )! 70300 05 07 閨06 ﾀ04 t =, 04030006 ｧ0 06 ﾀ000304003 0! =,() = 0 e,x x 0 =. 703050503006 04 罍ｧ0070503 ﾀ ｧ05 0504 0! =.

3x 8. ｮ 縺06 閨00606 0 060 ｧ06 070600909 ｧ ｧ 75 ｰ09 0 ｧ06. 6 5 04 罍ｧ006 縺06 3) 04 罍ｧ 0 080307 閨060 ｧ ｧ lim 04 04 ｧ0003 6ｧ80304 ｧ05 0007. x! ln x x 04 罍ｧ0060 ｧ0504 76. ) x +x ; ) ( + x) ; 3) 3 x + x 4. 0 763. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04, 090006 06000606 00607 閨06 09 罍ｧ040308 y = = e,x ｧ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 (04 罍ｧ x>0). 764. 7 0805 ｧ 03030060 050600, 0303 0304 00030503 04 0060 ｧ060 040309 縺0605 03 ﾀ ｧ Oy 0400030 陦ｧ 0306 ﾀ0903006 00308 閨00 ｧ003, 090006 06000308 00607 閨06 00 ｧ0 ｧ070 ｧ: xy =4; y =; x =0: 765. 0504 縺06 閨0600 ｧ0504 03030060 050600, 0303 0304 00030503 04 0060 ｧ060 09 罍ｧ- 040308 y = xe,x= (04 罍ｧ x>0) 040309 縺0605 0606 ﾀ ｧ00405030503. x 8. ｮ 縺06 閨00606 0 060 ｧ06 070600909 ｧ ｧ ｯ0603 縺060 03 ﾀ 縺06 閨0060. ｭ ﾀ00 ｧ0 0305 縺06 0906 [a; b] 070600909 ｧ07 f(x) 00604 縺06- b 縺03040, 0503 00607 閨06 04 縺06 閨0600 0 ｧ ｧ0050605 00 f(x) 0 08 閨0605 ﾀ07 05 090306 x = c, a 04 罍ｧ 09030503 縺030 b f(x) =(b, a)f(c): () a 6ｦ00 060 ｧ06 070600909 ｧ ｧ R b f(x) a y m = f(c) = () b, a 0 0304 0605 ﾀ07 ﾀ 縺06 閨0 ｧ0 0 060 ｧ060 070600909 ｧ ｧ f(x) 0 0305 縺06 0906 [a; b]. 766. 0504 縺06 閨0600 ｧ0504 ﾀ 縺06 閨00606 0 060 ｧ06 070600909 ｧ ｧ: ) y = sin x 0 0305 縺06 0906 [0; ]; ) y = tg x 0 0305 縺06 0906 [0; =3]; 3) y =lnx 0 0305 縺06 0906 [; e]; 4) y = x 0 0305 縺06 0906 [a; b]; 5) y = 0 0305 縺06 0906 [,; ]. +x ｰ09 0504 0 06 縺05060706 ﾀ 縺06 閨00606 0 060 ｧ06 070600909 ｧ ｧ 04 09 07 閨030 04 罍ｧ006 縺06.

376 0600. 9. 0504 縺06 閨060006000308 ｧ0050605 00 x 9. ｱ03 縺00600 05 040609 ｧ08 ｧ 0703 縺00600 ｮ ｧ004 ﾀ030. ｱ03 縺00600 05 040609 ｧ08: b a f(x) h " y 0 + y n X n, + i= # y i ; (I) 05 閨06 h =(b,a)=n, y 0, y, y, :::, y n 040030305 ﾀ0503070 ｧ06 03 罍陦ｧ0 050309 罍ｧ040308 y = f(x) 0 0305 縺06 0906 [a; b]. 70305 縺0600003 ﾀ0504 0703 縺0060003 (I): (b, a)h "(h) 6 jy 00 j max : (). 7 030300 ｧ 06 ﾀ09 07 0703 縺00600 ｮ ｧ004 ﾀ030 閨0007 閨040608 04030003 ﾀ: b a f(x) h 3 (y 0 +4y + y ); 05 閨06 h =(b, a)=. 3. ｱ03 縺00600 ｮ ｧ004 ﾀ030 閨0007 n 04030003 ﾀ: b a f(x) h 3 " y 0 + y n +4 nx i= 05 閨06 h =(b, a)=n. 70305 縺0600003 ﾀ0504 0703 縺00600 (II) ｧ (III): (II) # n, X y i, + y i ; (III) i= (b, a)h4 "(h) 6 jy IV j max ; () 80 05. 06. 0703 縺00600 (II) 070400070605 ﾀ07 0503 00308 閨0007 04 030300 040503 縺0308 ｧ 05 縺0605040608 ﾀ05060406-00608: y = a + bx + cx + 3. 767. 6 503 ｧ ﾀ00 ｧ0504 0403 0703 縺0060006 05 040609 ｧ08 ln = 040305 縺0600003 ﾀ0504 0403 0703 縺0060006 (). ｧ 768. 703 0703 縺0060006 ｮ ｧ004 ﾀ030 (III) 0403 ｧ ﾀ00 ｧ0504 ｧ0050605 0003 0 0 ｧ0504 ﾀ 0503 0030 ｧ 0 060 ｧ070 ｧ ｧ0050605 000304. x ｧ 0309060 ｧ0504 5 x 3 x 4, 0309060 ｧ0504 040305 縺0600003 ﾀ0504 04030703 縺0060006 () ｧ 縺06 06000405 0503 ﾀ 04-

3x 9. ｱ03 縺00600 05 040609 ｧ08 ｧ 0703 縺00600 ｮ ｧ004 ﾀ030 77 769. 703 0703 縺0060006 ｮ ｧ004 ﾀ030 (III) 0403 ｧ ﾀ00 ｧ0504 ｧ0050605 0003: ) 0 4 +x 3 (n = 4); ) = 0 3, cos x (n =6); 3) +x 4 (n =4) ｧ 0309060 ｧ0504 040305 縺0600003 ﾀ0504, 040300 05 07 04 0703-0 0060006 () 04 罍ｧ0300 ｧ07060003 h 4 jy IV j max j 4 yj max. 770. 7 0805 ｧ 0403 0703 縺0060006 ｮ ｧ004 ﾀ030 (II) 03030060 0303 09 ｧ 0403 ﾀ03050308 50 ﾀ0 ﾀ 陦ｧ 00605 縺030 09 07 閨030503 閨0 0 ﾀ0 ｧ ﾀ 陦ｧ 00605 縺030 ﾀ 縺06 閨0060503 ﾀ06 06-0 ｧ07 30 ﾀ0. 77. 6 5030406 ﾀ05 ｧ 0703 縺0060003 03030060 04 ｧ 0 ｧ 閨03 ｧ 00 ｧ 0703 縺0060003 ｮ ｧ004 ﾀ030 (II). 77. 6 503 ｧ ﾀ00 ｧ0504 ln = 0403 030300608 0703 縺0060006 ｮ ｧ004 ﾀ030 (III) x (04 罍ｧ n =0) ｧ 0309060 ｧ0504 040305 縺0600003 ﾀ0504 0403 0703 縺0060006 (). 773. 7 0805 ｧ 閨00 ｧ006 閨0605 ｧ 050000 ｧ04 ﾀ x = 5 cos t, y = 3 sin t, 04 罍ｧ- 0060 ｧ04 09 ｧ0050605 0006, 0304 縺06 閨06000706006006 0406 縺040606 06050406 縺0504 04 ﾀ0608 閨0605 ｧ, 0703 縺0060006 ｮ ｧ004 ﾀ030 (II). 774. 6 503 ｧ ﾀ00 ｧ0504 04 罍ｧ0300 ｧ07060003 =6 4, x, 04 罍ｧ0060 ｧ04 09 ｧ0050605 0006 0703 縺0060006 ｮ ｧ004 ﾀ030 (II). 0 775. 6 503 ｧ ﾀ00 ｧ0504 4 = 0403 030300608 0703 縺0060006 ｮ ｧ004 ﾀ030 +x 0 (III) (04 罍ｧ n =0) ｧ 0309060 ｧ0504 040305 縺0600003 ﾀ0504, 040300 05 07 04 0703 縺0060006 () 04 罍ｧ0300 ｧ07060003 h 4 jy IV j max j 4 yj max. 776. 7 ﾀ ﾀ0 05 罍ｧ04 07 0400030 閨04 ﾀ05 ｧ 09 縺0605, 0305 0 ｧ 0600030503 09 罍ｧ- 4 040308 x + y =3, 040309 0504, 0503 3, x =4 +8; 0 0805 ｧ, 0 0403 ｧ ﾀ000707 ｧ0050605 00 0403 0703 縺0060006 ｮ ｧ004 ﾀ030 (04 罍ｧ n =4). 777. 6 503 ｧ ﾀ00 ｧ0504 0403 0703 縺0060006 ｮ ｧ004 ﾀ030 (III) 閨00 ｧ006 閨0605 ｧ 04030006-040300003 ﾀ ｧ006 ﾀ03 ｧ 閨03 y = sin x, 03 ｧ04 0305 縺06 0309 [0; ] 0 0006 ﾀ0504 0400308 ﾀ050608.

3. 07 罍ｧ04 ｧ 0 : 0600 04 0 07 70ｷ36 50ｷ36ｦ0 7 76ｧ805 ｮ0705 0ｷ3 7 705 ｮ ｯ 7 繖 7 ｮ ｯ6 5 ｭ 7 705 07 70ｷ36 505 x. 07 罍ｧ04 ｧ 0 040003 ﾀ090308 09 罍ｧ040308. ｨC06005 ｧ 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003. 70403000605. 7 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003: k = d' ds = y 00 : () ( + y 0 ) 3= R = ( + y0 ) 3= jy 00 j 3. 070303 罍陦ｧ0 0503 0906005 09 罍ｧ04 ｧ 003: X = x, +y0 y 00 y 0 = x + _x +_y x _y, y _x _y; = (_x +_y ) 3= : () jy _x, x _yj Y = y + +y0 y 00 = y + _x +_y y _x, x _y _x: (3) 06060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0906005 縺0304 09 罍ｧ04 ｧ 003 C(X; Y ) 0 0304 0605 ﾀ07 050403-0006050308. ｰ 040060 ｧ07 (3) ｧ 0306 閨0605 04 00605 罍ｧ 06 ﾀ09 ｧ0 ｧ 06 040060 ｧ070 ｧ 050403-00060503. 4. 7 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003 09 罍ｧ040308 r = f('), 05 閨06 r ｧ ' 04030007 縺00306 090303 罍陦ｧ0 0503: R = (r + r 0 ) 3= jr +r 0, rr 00 j : (4) 0504 縺06 閨0600 ｧ0504 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ 09 縺0605 09 罍ｧ- 04 ｧ 003 09 罍ｧ040308 04 0606 0406 縺00 ｧ006: 778. y =4x, x : 779. y = e,x : 780. x +4y =4: 78. x = a(t, sin t); 78. y = xe,x : y = a(, cos t): 0504 縺06 閨0600 ｧ0504 090303 罍陦ｧ0 0503 0906005 09 罍ｧ04 ｧ 003 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ09 縺0605 09 罍ｧ04 ｧ 003 09 罍ｧ040308: 783. xy =4040503 0906 x =.

3803. xy =4: 804. x = a cos 3 t; y = a sin 3 t: x. 07 罍ｧ04 ｧ 0 040003 ﾀ090308 09 罍ｧ040308 79 784. y =lnx 040503 0906 0406 縺06 ﾀ06 060 ｧ07 ﾀ Ox. 785. y =, x3 + 3 04 0503 0906 0406 縺06 ﾀ06 060 ｧ07 ﾀ Ox. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05040300060503 09 罍ｧ040308 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ0606 05040300060506: 786. y =, x : 787. x = cos t, y = sin t. 788. x, y = a ( ｧ00 ｧ x = a ch t ｧ y = a sh t). 789. x = a(cos t + t sin t), y = a(sin t, t cos t). 790. 7 0805 ｧ 0 09 ﾀ ｧ0 000400606 09 罍ｧ04 ｧ 006 09 罍ｧ040308 y = e x. 79. 6 70309 0504, 0503 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003 09060400308 00 ｧ0 ｧ ｧ y = a ch x a 04 0006030308 0503 0906 04060 y 03 ﾀ0406 Ox. a ｧ 04060 0305 縺06 0906 003 縺0 00 ｧ 00607 閨06 09 罍ｧ040308 ｧ 79. 0504 縺06 閨0600 ｧ0504 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003 04 04 縺03 ｧ 0403000400308 0503 0906 09 罍ｧ- 040308: ) r = a(, cos '); ) r = a cos '; 3) r = a cos '. 0504 縺06 閨0600 ｧ0504 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ 09 縺0605 09 罍ｧ- 04 ｧ 003 09 罍ｧ040308 04 0606 0406 縺00 ｧ006: 793. y = +x : 794. x, y =4: 795. y = sin x: 796. y = x +4x: 0504 縺06 閨0600 ｧ0504 090303 罍陦ｧ0 0503 0906005 09 罍ｧ04 ｧ 003 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ09 縺0605 09 罍ｧ04 ｧ 003 09 罍ｧ040308: 797. y = e x 040503 0906 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ Oy. 798. y = x3 3 040503 0906 (,;,=3). 799. y = x 3 040503 0906 (; ). 800. y = cos x 040503 0906 x = 4. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 05040300060503 09 罍ｧ040308 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606 ｧ0606 05040300060506: 80. y =(x +): 80. x = t ; y = t3 3 :

380 0600. 0. 07 罍ｧ04 ｧ 0 040003 ﾀ090308 ｧ 04 縺03 ﾀ05 0 ﾀ050406000308 09 罍ｧ040308 805. 70309 0504, 0503 04 0006030308 0503 0906 ﾀ05 縺03 ｧ 閨03 x =3 + y =3 = a =3 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003 04060 3 3 ajxyj. x. 6 700 ｧ0 閨0605 ｧ 09 罍ｧ040308 04 04 縺03 ﾀ05 0 ﾀ050406 6 7 ｧ070706 縺06009 ｧ 00 閨0605 ｧ: ds = + dy + dz, ｧ00 ｧ ds = _x +_y +_z dt. t 6 700 ｧ0 閨0605 ｧ: s = _x +_y +_z dt. t 7 0805 ｧ 閨00 ｧ006 閨0605 ｧ 09 罍ｧ040308: 806. x = t, y = t, z = t3 3 0305 t =0 閨03 t =3. 807. x = 3 cos t, y = 3 sin t, z =4t 0305 t =0 閨03 04 縺03 ｧ 040300040030503 t. 808. y = x, z = x 3 6 0305 x =0 閨03 x =3. 7 0805 ｧ 閨00 ｧ006 閨0605 ｧ 09 罍ｧ040308: 809. x = t, sin t, y =, cos t, z = 4 sin t 0305 t =0 閨03 t =. 80. x = e t, y = e,t, z = t 0305 t =0 閨03 t =. 8. y = ln x, z = x 0305 x = 閨03 x =. x 3. 7 縺03 ｧ 0403 閨0 07 0406090503-070600909 ｧ ｧ 0403 ﾀ09 0007 縺06 ｧ 0606 00608 0 ｧ 06 ﾀ090306 ｧ 05060300605 罍ｧ 06 ﾀ090306 0 060 ｧ06. ｭ ﾀ0506 ﾀ050406000308 05 縺060805 00 ｧ09 09 罍ｧ040308 7 陦ｧ06 ﾀ-0406090503 r = xi + yj + zk 0503 09 ｧ 09 罍ｧ040308 x = x(t), y = y(t), z = = z(t) 06 ﾀ0504 0406090503-070600909 ｧ07 ﾀ09 0007 t. 7 縺03 ｧ 0403 閨0 07 _r = _xi+ _yj+ _zk 06 ﾀ0504 05 00506009 ｧ 000400308 0406090503 ｧ ｧ0060605 003 閨060004 j_rj = _x +_y +_z = _s = ds dt. 703050503006, 06 ﾀ00 ｧ t 04 縺06007, 09 罍ｧ04 07 05 06090503 罍ｧ07 閨04 ｧ07060 ｧ07, 0503 _r = v 06 ﾀ0504 0406090503 ﾀ0903 縺03 ﾀ05 ｧ, r = w 0406090503 06 ﾀ0903 縺060 ｧ07. ｪ06 縺06 0503 0906 M (x; y; z) 09 罍ｧ040308 ( 罍ｧ ﾀ. 34) 04 縺030406 閨060 05 罍ｧ 040003 ﾀ0903 ﾀ05 ｧ: ) 0406 縺04060 陦ｧ09060007 縺00606 09 _r; 030 0 0304 0605 ﾀ07 003 縺0 000400308; ) ﾀ03 閨06 縺07 00606 _r ｧ r; 030 0 0304 0605 ﾀ07 ﾀ0304 罍ｧ09 ﾀ 0600608 ﾀ07; 3) 0406 縺04060 陦ｧ09060007 縺00606 09 0406 縺04030 閨04060. 050 ｧ 0303 060605 06 ﾀ0506 ﾀ050406000308 05 縺060805 00 ｧ09 (05 罍ｧ05 陦) 09 罍ｧ040308.

3x 3. 7 縺03 ｧ 0403 閨0 07 0406090503-070600909 ｧ ｧ 0403 ﾀ09 0007 縺06 8 6 5 0406 縺06 ﾀ06 060 ｧ ｧ 040003 ﾀ0903 ﾀ050608 ｧ006060 05 罍ｧ 04 縺0700306: 09 ﾀ 0506000400606; 03 ｧ003-0 0004 ｧ 0500 0400606 003 縺0 0004, 0304 縺06 閨0600070600306 0406090503 0 ｧ: ) _r 05 00506009 ｧ 000400308, ) B = _r r 03 ｧ003 縺0 000400308, 3) N = B _r 0500 0400308 003 縺0 000400308. ｭ 陦ｧ0 ｧ 00306 0406090503 縺03 0505 ｧ08 0 04 0400060 ｧ08 030303 0 ｧ0,, ; 030 ｧ ﾀ0407-003 04 ｧ ﾀ ｧ003 ﾀ050406 d ds = d ds ｧ =. 706 ﾀ0504 M (X; Y ; ) 0503 09 09 ﾀ 0506000400308 ( 罍ｧ ﾀ. 34). ｯ0305,,,! MM k_r ｧ ｧ 06 ﾀ000304 ｧ07 04 0000060004003 ﾀ05 ｧ 0406090503 縺0304 04030006 ｧ0 06 040060 ｧ07 09 ﾀ 0506000400308 X, x _x = Y, y _y =, z : (I) _z 706 ﾀ0504 M (X; Y ; ) 0503 09 0 003 縺0 000400308 040003 ﾀ0903 ﾀ05 ｧ. ｯ0305,,,! MM? _r ｧ ｧ 06 ﾀ0003-04 ｧ07 0406 縺04060 陦ｧ09060007 縺003 ﾀ05 ｧ 0406090503 縺0304 04030006 ｧ0 06 040060 ｧ06 003 縺0 000400308 040003 ﾀ0903 ﾀ05 ｧ: _x(x, x) + _y(y, y) + +_z(, z) =0: (II) ｰ 040060 ｧ07 03 ｧ003 縺0 00 ｧ ｧ 0500 04-00308 003 縺0 00 ｧ 04030006 ｧ0, 0060 ｧ04 04 7 ｧ ﾀ. 34 06 040060 ｧ0708 (I) _x, _y, _z ﾀ030305040605 ﾀ05-04060003 0 B x, B y, B z ｧ00 ｧ 0 N x, N y, N z. ｰ 040060 ｧ06 ﾀ0304 罍ｧ09 ﾀ 0600608 ﾀ07 040003 ﾀ0903 ﾀ05 ｧ 04030006 ｧ0, 0060 ｧ04 04 06 040060 ｧ ｧ (II) _x, _y, _z 0 B x, B y, B z. 8. 7 陦ｧ06 ﾀ-0406090503 閨04 ｧ070600608 ﾀ07 0503 09 ｧ 04 003006005 t 0 06 04-0060 ｧ060 r =4ti, 3tj. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06, ﾀ0903 縺03 ﾀ0504 ｧ 06 ﾀ0903 縺06-0 ｧ06 閨04 ｧ07060 ｧ07. 83. ｰ 040060 ｧ06 閨04 ｧ07060 ｧ07 r =3ti +(4t, t )j. 0504 縺06 閨0600 ｧ0504 05-06090503 罍ｧ06 ｧ ﾀ0903 縺03 ﾀ0504 閨04 ｧ07060 ｧ07. 703 ﾀ05 縺03 ｧ0504 05 06090503 罍ｧ06 ｧ 0406090503 縺03 ﾀ0903 縺03 ﾀ05 ｧ 04 00300600503 t =0; ; ｧ3 ﾀ. 84. 6 5 06 83 0304 縺06 閨0600 ｧ0504 06 ﾀ0903 縺060 ｧ06 w 閨04 ｧ07060 ｧ07 ｧ 060503 05 00506009 ｧ 000400606 w = dv ｧ 003 縺0 000400606 dt w n = w, w ﾀ03 ﾀ05-040007060 ｧ06 04 0006030308 003006005 t ｧ 04 罍ｧ t =0. 85. ｰ 040060 ｧ06 閨04 ｧ07060 ｧ07 r = a cos t i + b sin t j. 0504 縺06 閨0600 ｧ0504 05 06090503 罍ｧ06, ﾀ0903 縺03 ﾀ0504 ｧ 06 ﾀ0903 縺060 ｧ06 閨04 ｧ07060 ｧ07 ｧ 0403 ﾀ05 縺03 ｧ0504 0406090503 縺03 ﾀ0903 縺03 ﾀ05 ｧ ｧ06 ﾀ0903 縺060 ｧ07 04 0503 09 08 t =0;=4; =. 6 5 08 86{88 0 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 04 縺0700308 ｧ 003 縺0 000400308 040003 ﾀ0903 ﾀ05 ｧ 09 罍ｧ040308: 86. x = t, y = t, z = t 3 04 0006030308 0503 0906 ｧ 04 罍ｧ t =.

38 0600. 0. 07 罍ｧ04 ｧ 0 040003 ﾀ090308 ｧ 04 縺03 ﾀ05 0 ﾀ050406000308 09 罍ｧ040308 87. y = x, z = x 04 0006030308 0503 0906 (x > 0) ｧ 04 罍ｧ x =4. 88. x + y =0, y + z =5040503 0906 (; 3; 4). ｰ09 0 ｧ06. 6 5 0704 陦ｧ070706 縺06009 ｧ 00 0305 0006040308 ｧ 04 040308 ﾀ050608 09 07 閨030503 06 040060 ｧ07, 0 0805 ｧ 05060 030500300060 ｧ07 : dy : dz. 89. 7 0805 ｧ 05 00506009 ｧ 000400308 _r, 03 ｧ003 縺0 000400308 B ｧ0500 0400308 003 縺0 000400308 N 0406090503 縺03 09 罍ｧ040308 x =, sin t, y = cos t, z = t 04 0503 0906 t =0. 7 0805 ｧ 05 090706, ｧ 04 050308 0706 0503 0906. 80. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 0500 0400308 003 縺0 00 ｧ, 03 ｧ003 縺0 00 ｧ ｧ ﾀ03-04 罍ｧ09 ﾀ 0600608 ﾀ07 040003 ﾀ0903 ﾀ05 ｧ 0909 罍ｧ040308 x = t, y = t, z = t 3 040503 0906 t =. 8. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 0500 0400308 003 縺0 00 ｧ ｧ 03 ｧ003 縺0 00 ｧ 09 罍ｧ- 040308 x = e t, y = e,t, z = t 040503 0906 t =0. 8. 70309 0504, 0503 06 040060 ｧ07 x = t cos t, y = t sin t, z = t 0304 縺06 閨0600070605 09030 ｧ 06 ﾀ090606 04 ｧ00503040606 00 ｧ0 ｧ06, ｧ 0 04 ｧ ﾀ 0504 06 040060 ｧ07 0500 0400308 003 縺0 00 ｧ, 03 ｧ003 縺0 00 ｧ ｧ 09 ﾀ 0506000400308 09 00608 04 0 0006 090303 - 陦ｧ0 05. 83. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 09 04 ｧ00503040308 00 ｧ0 ｧ ｧ x = = a cos t, y = a sin t, z = bt 04 0006030308 0503 0906 ｧ 04 罍ｧ t = =. 70309 0504, 0503 04 ｧ0050304 07 00 ｧ0 ｧ07 0406 縺06 ﾀ0609 0605 0303 06060 ｧ06 09 ｧ00 ｧ0 陦 x +y = a 0403 03 陦ｧ0 090304030 060500030 = arccos b a + b. 84. 7 0805 ｧ 06050003 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 05 00506009 ｧ 00040030503 0406090503 09 罍ｧ040308 x =az ｧ y =bz 040503 0906 z = ab. 85. 70003 ﾀ0903 ﾀ0504 y =0,0 09030503 縺0308 0 09 罍ｧ04 07 z = x, y =0, 0 09 縺06 ｧ04 0605 ﾀ07 0 09 ｧ00 ｧ0 陦 x + y =y. 7 04 ｧ ﾀ 0504 04 00605 罍ｧ 06- ﾀ09 ｧ06 06 040060 ｧ07 0303 0304 00030503 09 罍ｧ040308 04 ｧ005 ｧ 0304 縺06 閨0600 ｧ0504 03 ｧ003-0 000400308 0406090503 09 罍ｧ040308 04 0006030308 0503 0906 ｧ 04 0503 0906 t = =, 05 閨06 t 06050300 04030403 縺0305 040003 ﾀ0903 ﾀ05 ｧ. 86. 7 陦ｧ06 ﾀ-0406090503 閨04 ｧ070600608 ﾀ07 0503 09 ｧ 04 003006005 t 0 06 04-0060 ｧ060 r = a(t, sin t)i + a(, cos t)j. 0504 縺06 閨0600 ｧ0504 ｧ 0403 ﾀ05 縺03 ｧ0504 0406090503 縺03 ﾀ0903 縺03 ﾀ05 ｧ ｧ06 ﾀ0903 縺060 ｧ07 04 罍ｧ t = = ｧ t =. 6 5 08 87{89 0 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 09 09 罍ｧ- 040308: 87. y = x, z =x 040503 0906 x =. 88. x + y + z =4, x +y, z =040503 0906 (; ; 3) ( ﾀ0. 06 88). 89. x =t, y =lnt, z = t 040503 0906 t =.

3x 4. 07 罍ｧ04 ｧ 0 ｧ 09 縺06 060 ｧ06 04 縺03 ﾀ05 0 ﾀ050406000308 09 罍ｧ040308 83 830. r = e t i + e,t j + t k. 7 0805 ｧ 06050003 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 03 ｧ003 縺0 00040030503 0406090503 b 040503 0906 t =0. 83. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 0500 0400308 003 縺0 00 ｧ ｧ 03 ｧ003 縺0 00 ｧ 09 罍ｧ- 040308 y = x, z = y 040503 0906 x =. 83. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 0500 0400308 003 縺0 00 ｧ ｧ 03 ｧ003 縺0 00 ｧ 09 罍ｧ- 040308 x = t, sin t, y =, cos t, z = 4 sin t 040503 0906 t =. x 4. 07 罍ｧ04 ｧ 0 ｧ 09 縺06 060 ｧ06 04 縺03 ﾀ05 0 ﾀ050406000308 09 罍ｧ040308 07 罍ｧ04 ｧ 0 =R 06 ﾀ0504 04 縺06 閨0600 030500300060 ｧ07 060500 ' 04030403 縺0305 09 ﾀ 0506000400308 09 閨00 ｧ006 閨0605 ｧ s, 090305 s! 0. 07 縺06 060 ｧ06 = 06 ﾀ0504 04 縺06 閨0600 030500300060 ｧ07 060500 04030403 縺0305 03 ｧ003 縺0 00 ｧ 09 s, 090305 s! 0. ｯ 09 09 09 ' jj ｧ jj, 0503=R ｧ = ｧ ﾀ00060003 0309 0304 0605 ﾀ07 003 閨0600070 ｧ 0406090503 縺0304: d ds = R ; d ds =, : () ｭ ﾀ00 ｧ 09 罍ｧ04 07 0 06 040060 ｧ060 r = r(t), 0503 j_r rj = R j_rj ; 3 = _r r... r j_r rj : () 833. 7 縺03 陦ｧ070706 縺06009 ｧ 縺0304 04 04060 ﾀ050403 v = v 0403 t, ﾀ 040300300406 0406 縺040308 0703 縺0060003 () 04030006 ｧ0504 000307060 ｧ06 06 ﾀ0903 縺060 ｧ07 w 0 05 005060-09 ｧ 000400306 ｧ 003 縺0 000400306: w = v + v R : 834. ｯ03 09 閨04 ｧ070605 ﾀ07 0403 04 03030006 x = t, y = t, t,05 閨06 t 04 縺06007 閨04 ｧ07060 ｧ07. 0504 縺06 閨0600 ｧ0504 09 罍ｧ04 ｧ 006 =R 05 06090503 罍ｧ ｧ ｧ 05 005060-09 ｧ 000400306 ｧ 003 縺0 000400306 06 ﾀ0903 縺060 ｧ07 04 003006005 t ｧ 04 罍ｧ t =0. 835. ｯ03 09 閨04 ｧ070605 ﾀ07 0403 050000 ｧ04 ﾀ06 x = 4 cos t, y = 3 sin t, 05 閨06 t 04 縺06007 閨04 ｧ07060 ｧ07. 0504 縺06 閨0600 ｧ0504 09 罍ｧ04 ｧ 006 =R 05 06090503 罍ｧ ｧ ｧ 05 00506009 ｧ 000400306 ｧ 003 縺0 000400306 06 ﾀ0903 縺060 ｧ07 04 罍ｧ t =. 4 836. 6 70007 閨04 ｧ07060 ｧ07 ﾀ 06 040060 ｧ060 r = ti + t j + 3 t3 k 0304 縺06- 閨0600 ｧ0504 09 罍ｧ04 ｧ 006 =R 05 06090503 罍ｧ ｧ ｧ 05 00506009 ｧ 000400306 ｧ 003 縺0 000400306 06 ﾀ0903 縺060 ｧ07 04 0006030308 003006005 t ｧ 04 罍ｧ t =. 0504 縺06 閨0600 ｧ0504 09 罍ｧ04 ｧ 006 =R ｧ 09 縺06 060 ｧ06 = 09 罍ｧ040308: 837. x = t, y = t, z = t 3 04 0006030308 0503 0906 ｧ 04 罍ｧ t =0. 838. x = e t, y = e,t, z = t 04 0006030308 0503 0906 ｧ 04 罍ｧ t =0. 839. y = x, z = x3 04 0006030308 0503 0906 ｧ 04 罍ｧ x =.

384 0600. 0. 07 罍ｧ04 ｧ 0 040003 ﾀ090308 ｧ 04 縺03 ﾀ05 0 ﾀ050406000308 09 罍ｧ040308 840. 70309 0504, 0503 0 04 04030 04 ｧ00506 (x = a cos t, y = a sin t, z = bt) 09 縺06 060 ｧ06 0403000307 ｧ05060004003, 0 000604030 (x = a cos t, y = =,a sin t, z = bt) 0305 罍ｧ09 05060004003. 0504 縺06 閨0600 ｧ0504 09 罍ｧ04 ｧ 006 =R ｧ 09 縺06 060 ｧ06 = 09 罍ｧ040308: 84. x =t, y =lnt, z = t 04 0006030308 0503 0906 ｧ 04 罍ｧ t =. 84. x = y, z = x 04 0006030308 0503 0906 ｧ 04 罍ｧ y =. 843. x = e t sin t, y = e t cos t, z = e t 040503 0906 t =0.

30600 04 ｪ 罍ｮ ｯ 76ｧ9 ｭ 7 7050ｷ36ｦ06 5056 7 76ｧ9 ｭ, 7056ｧ8 76ｧ9 ｭ 6 70ｷ3 ｱ ｱ ｭ 7 ｭ 7ｨC0ｷ3 縺6ｧ86ｧ9 0ｷ3 0ｷ36ｦ 7 70ｷ36ｧ8056 9 ｭ 70ｷ300 x. ｱ0600909 ｧ ｧ 閨040608 0406 縺06006000308 ｧ ｧ08 05060300605 罍ｧ 06 ﾀ090306 ｧ 0303 07060 ｧ06. 0504 縺06 閨0600060 ｧ06. 706 縺0600600 07 z 0 0304 0605 ﾀ07 03 閨003 0 00308 0706009-09 ｧ0608 0406 縺06006000308 x ｧ y, 06 ﾀ00 ｧ 09 07 閨0308 04 縺06 0 060 ｧ08 x ｧ y 04 00609030503 縺0308 030300 ﾀ05 ｧ ｧ08 ｧ 0060060 ｧ07 0403 ﾀ05 040006003 04 ﾀ030305040605 ﾀ0504 ｧ06 03 閨003 0 060 ｧ06 z. ｱ06009-09 ｧ030 000400606 04 ｧ ﾀ ｧ003 ﾀ0504 z 0305 x ｧ y 04 ｧ ﾀ0304 0605 04 04 ｧ 閨06 z = F (x; y): (). 06060300605 罍ｧ 06 ﾀ090306 ｧ 0303 07060 ｧ06. ｰ 040060 ｧ06 () 05060300605 罍ｧ- 06 ﾀ09 ｧ 0304 縺06 閨0600070605 00609030503 縺0606 04030406 縺08003 ﾀ0504. 7 0 060 ｧ08 x ｧ y 0304 縺06 閨06-00070605 0 040003 ﾀ0903 ﾀ05 ｧ xoy 0503 0906 P (x; y), z = F (x; y) 040400 ｧ09 0506 ﾀ030305-040605 ﾀ0504060600608 0503 09 ｧ M (x; y; z) 0 04030406 縺08003 ﾀ05 ｧ. 703050503006 05030403 縺0705, 0503 z 06 ﾀ0504 070600909 ｧ07 0503 09 ｧ P (x; y), ｧ 04 ｧ000605 z = F (P ). 3. 7 縺06 閨0600 070600909 ｧ ｧ lim F (P )=A, 06 ﾀ00 ｧ 003 ﾀ0504 F (P ), A P!P0 06 ﾀ0504 0306 ﾀ0903006 003 0 00 07, 090305 = P 0 P! 0 04 罍ｧ 000603030 ﾀ0403 ﾀ030306 04 罍ｧ0300 ｧ- 07060 ｧ07 P 09 P 0 (0 04 罍ｧ006, 0403 0006030308 00 ｧ0 ｧ ｧ). 4. 70604 縺06 縺0304003 ﾀ0504 070600909 ｧ ｧ. ｱ0600909 ｧ07 F (x; y) 0 0304 0605 ﾀ07 00604 縺06 縺030400308 040503 0906 P 0, 06 ﾀ00 ｧ lim P!P0 F (P )=F (P 0 ). 0ｷ30 06 05030403 縺07, 0706009-09 ｧ07 F (x; y) 00604 縺06 縺03040 04 00609030503 縺0308 0503 0906 (x; y), 06 ﾀ00 ｧ lim x!0 y!0 F (x +x; y +y) =F (x; y): 844. ｰ09 0504 030300 ﾀ05 ｧ ｧ 0060060 ｧ08 x ｧ y, 閨0007 09030503 縺0308 ﾀ0006 閨0606-0 ｧ06 070600909 ｧ ｧ ｧ0060605 0406006 ﾀ050406000306 0 060 ｧ07: ) z = x + y ; ) az = a, x, y ; 3) z = 4) z = a, x, y ; 5) z = xy; 4 x + y ; xy 6) z = ; 7) z =, x, y y, x ; ｧ 0403 ﾀ05 縺03 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ06 ｧ 0303 07060 ｧ07 070600909 ｧ08 0403 ﾀ06 060 ｧ070 04030406 縺08003 ﾀ050608 040003 ﾀ0903 ﾀ05070 ｧ x =0, y =0, z =0 ｧ z = h.

386 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 845. 6 7 0 0406 罍ｧ00605 05 縺0606050300040 ｧ09. 0504 縺06 閨0600 ｧ0504 0400030 閨04 S 05 縺0606050300040 ｧ09 09 09 070600909 ｧ06 閨040608 060503 ﾀ0503 縺030 x ｧ y. 0504 縺06 閨0600 ｧ0504 ｧ 0403 ﾀ05 縺03 ｧ0504 030300 ﾀ0504 0403 0030700308 0 060 ｧ08 x ｧ y. 846. 6 70007 070600909 ｧ ｧ F (x; y) = x, y 0403 ｧ ﾀ00 ｧ0504 F (3; ), F (; 3), x, y F (; ), F (; ), F (a; a), F (a;,a). x 4 + y 4,xy,0503F (tx; ty) = 847. 6 70309 0504, 0503 06 ﾀ00 ｧ F (x; y)= = t F (x; y). 848. 6 70007 z = x, xy = y 0304 縺06 閨0600 ｧ0504 x z, y z ｧ z. 6 503 ｧ ﾀ00 ｧ0504 x z, y z, z, 06 ﾀ00 ｧ x ｧ 0060070605 ﾀ07 0305 閨03,, y ｧ 0060070605 ﾀ07 0305 閨03,9. 849. 70309 0504, 0503 06 040060 ｧ06 x, y, z = 0 0304 縺06 閨0600070605 z 09 09 0306 ﾀ ｧ ﾀ0006000306 00030706 ﾀ050403 03 閨003 0 00308 070600909 ｧ08 x ｧ y, ｧ 09030503 縺0308 閨0406 00604 縺06 縺0304003. ｰ09 0504 030300 ﾀ0504 0304 縺06 閨0600060 ｧ07 04 ﾀ0608 0505 ｧ08 070600909 ｧ08 ｧ 0403 ﾀ05 縺03 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 ｧ 0303 07060 ｧ06 0403000307 ｧ0506000400308 00604 縺06 縺030400308 070600909 ｧ ｧ. 7 罍ｧ0406 ﾀ05 ｧ 04 罍ｧ006 03 閨003 0 00308, 003 - 縺030400308 070600909 ｧ ｧ z = F (x; y), 0304 縺06 閨0600070600308 05060 0706 06 040060 ｧ060 x, y = z. 850. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ 06 縺030400608 (04 罍ｧ z =0; ; ｧ 05..) 0706009-09 ｧ08: ) z = r, x 4, y ; ) z = x, y; 3) z = x, y ; 4) z = xy: 85. 70309 0504, 0503 04 罍ｧ x! 0 ｧ y! 0 0403 07060 ｧ06 u = y x, y 003070605 ﾀ05 縺060 ｧ0504 ﾀ07 09 00060303006 04 縺06 閨060006. 7 罍ｧ0406 ﾀ05 ｧ 04 罍ｧ006 縺03 05 09030503 04 罍ｧ0300 ｧ07060 ｧ07 0503 09 ｧ (x; y) 090503 0906 (0; 0), 04 罍ｧ 09030503 縺030 lim u =3, lim u =, lim u =, lim u =0, lim u =,. ｰ09 0 ｧ06. 7 ﾀ ﾀ00305 縺060504 ｧ 0060060 ｧ06 x ｧ y 04 閨030004 04 縺0700308 y = kx. 85. 70309 0504, 0503: ) lim x!0 y!0, xy +4 xy =, ; ) lim 4 x!0 y!0 sin (xy) =; xy sin (xy) 3) lim =0 x!0 x y!0 04 罍ｧ 000603030 ﾀ0403 ﾀ030306 04 罍ｧ0300 ｧ07060 ｧ07 0503 09 ｧ (x; y) 090503 0906 (0; 0). ｰ09 0 ｧ06. 703000307 ｧ0504 xy =.

3x. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306 0406 縺04030503 0403 縺07 閨09 87 853. 0ｷ3 0303 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ 070600909 ｧ06 z = F (x; y) = ｧ0609 0504 00 ｧ0 ｧ ｧ 0606 縺0304. 8 < : 04 罍ｧ xy > 0; 0 04 罍ｧ xy =0;, 04 罍ｧ xy < 0 854. ｰ09 0504 030300 ﾀ05 ｧ 0304 縺06 閨0600060 ｧ07 070600909 ｧ08: r ) z = x + y; ) z = 4 x + y ; 3) z c =, x a, y b ; 4) z x =, c a, y b ; 5) z = x + x, y ; 6) z = x + y ｧ 0403 ﾀ05 縺03 ｧ0504 05060300605 罍ｧ 06 ﾀ09 ｧ06 ｧ 0303 07060 ｧ07 0505 ｧ08 070600909 ｧ08. 855. 6 70309 0504, 0503 06 ﾀ00 ｧ F (x; y)= x,0503f(a; b)+f (b; a)=. x, y 856. 70309 0504, 0503 06 040060 ｧ06 z 4 = 4, x, y 0304 縺06 閨0600070605 z 09 09 0306 ﾀ ｧ ﾀ0006000306 00030706 ﾀ050403 03 閨003 0 00308 070600909 ｧ08 x ｧ y, ｧ 0903-0503 縺0308 閨0406 00604 縺06 縺0304003. ｰ09 0504 030300 ﾀ0504 0304 縺06 閨0600060 ｧ07 04 ﾀ0608 0505 ｧ08 070600909 ｧ08 ｧ 0403 ﾀ05 縺03 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 ｧ 0303 07060 ｧ06 070600909 ｧ ｧ, 0403-000307 ｧ0506000400308 04 030300 ﾀ05 ｧ x + y 6 ｧ0305 罍ｧ09 0506000400308 04006 0606. 857. 703 ﾀ05 縺03 ｧ0504 05060300605 罍ｧ 06 ﾀ090306 ｧ 0303 07060 ｧ06 03 閨003 0 00308 070600909 ｧ ｧ z = F (x; y), 0304 縺06 閨0600070600308 06 040060 ｧ060 x + y + z = a, 0403000307 ｧ0506000400308 04 030300 ﾀ05 ｧ x + y 6 a ｰ09 0504 00 ｧ0 ｧ06 0606 縺0304. 4 ｧ0305 罍ｧ09 0506000400308 04006 0606. x. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306 0406 縺04030503 0403 縺07 閨09 7 縺03 ｧ 0403 閨0 07 070600909 ｧ ｧ z = F (x; y) 0403 x, 0 08 閨0600 07 04 04 縺06 閨04030003-07060 ｧ ｧ, 0503 y 03 ﾀ05 0605 ﾀ07 0403 ﾀ05030700030, 0 0304 0605 ﾀ07 ﾀ0500308 04 縺03 ｧ 0403 閨00308 z 0403 x ｧ030303 0 0605 ﾀ07 @z @x ｧ00 ｧ F 0 (x; y). 縺0 000305 ｧ 003 0304 縺06 閨0600070605 ﾀ07 ｧ 030303x @z 0 0605 ﾀ07 ﾀ050 07 04 縺03 ｧ 0403 閨0 07 z 0403 y: @y = F 0 (x; y). y 7 0805 ｧ ﾀ0500306 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 858. z = x 3 +3x y, y 3 : 859. z =ln(x + y ): 860. z = y x : 86. z = arctg y x : 86. z = xy x, y : 863. u =ln 3, : x 3 t

388 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 864. c = a + b, ab cos. 865. u = y x + z y, x z : 866. u = xe,yx : 867. u = x, t x +t : 868. = arcsin (t x): 869. 6 70309 0504, 0503 06 ﾀ00 ｧ, z =ln x + y,0503 x @z @x + y @z @y = : 870. 6 70309 0504, 0503 06 ﾀ00 ｧ z = x sin y x,0503 x @z @x + y @z @y = z : 87. 6 70309 0504, 0503 06 ﾀ00 ｧ u = e x=t,0503 x @u @x + t @u @t =0: 87. 6 70309 0504, 0503 06 ﾀ00 ｧ u = x y,0503 x y @u @x + @u ln x @y =u: 873. 7 ｧ0706, 04 06 898, 0306 閨0605 閨0309 0 ﾀ0006 閨06060 07 050603 縺060 7080006 : ｭ ﾀ00 ｧ z = F (x; y) 03 閨003 縺03 閨0 07 070600909 ｧ07 ﾀ050604060 ｧ n, 0503 x @z @x + y @z @y = nz. 7 縺030406 罍ｧ0504 050506 050603 縺06006 7080006 閨0007 070600909 ｧ08: ) z = x 3 + xy, y 3 ; ) z = x + xy + y ; 3) z = x 3, y 3 ; 4) z = ex=y : 7 0805 ｧ ﾀ0500306 04 縺03 ｧ 0403 閨00306 070600909 ｧ08: 874. z = cos (ax, by): 875. z = arcsin y x : 876. z = x 3y, x : 877. u = ln sin (x, t): 878. u = sin (x + y), sin x, sin y.

3x 3. 7030000308 陦ｧ070706 縺06009 ｧ 00 0406 縺04030503 0403 縺07 閨09 89 879. 6 70309 0504, 0503 06 ﾀ00 ｧ u = x + y + z,0503 @u @u @u + + =: @x @y @z 880. 6 70309 0504, 0503 06 ﾀ00 ｧ z = e x=y ln y, 0503 x @z @x + y @z @y = z ln y : r l @T 88. 6 70309 0504, 0503 06 ﾀ00 ｧ T =,0503 l g 88. 6 70309 0504, 0503 06 ﾀ00 ｧ z = e x= sin 4, y @z @x + @z = @y ex sin y : @l + g @T @g =0.,0503 883. 7 縺030406 罍ｧ0504 050603 縺06006 7080006 0303 03 閨003 縺03 閨00308 070600909 ｧ0708 ( ﾀ0. 06 873) 閨0007 070600909 ｧ08: ) z = x3 x, y ; ) z = x + y ; 3) z = arctg y x. x 3. 7030000308 陦ｧ070706 縺06009 ｧ 00 0406 縺04030503 0403 縺07 閨09 ｭ ﾀ00 ｧ 070600909 ｧ07 z = F (x; y) ｧ0060605 04 0503 0906 (x; y) 00604 縺06 縺030400306 ﾀ0500306 04 縺03 ｧ 0403 閨00306, 0503 0606 04030000306 04 罍ｧ 0060 ｧ06 003070605 03030504 04 縺06 陦ﾀ05 040006003 04 04 ｧ 閨06 z = @z @z x + y + "; () @x @y 05 閨06 "! 0 04 罍ｧ = jxj + jyj! 0. ｯ0305 0403 07060 ｧ06 @z @z x+ @x @y y 06 ﾀ0504 0500 040 07 ﾀ0504 0403000030503 04 罍ｧ 0060 ｧ07 z; 030 0 0304 0605 ﾀ07 0403000030 陦ｧ070706 縺06009 ｧ 00030 070600909 ｧ ｧ ｧ 030303 0 0605 ﾀ07 dz: dz = @z @z x + y: () @x @y 70300 05 07 04 0703 縺0060006 () z 040030: ) x; )y, 0 08 閨060: =x, dy =y. 703050503006 dz = @z @z + dy: (3) @x @y

390 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 0ｷ3 () ﾀ0006 閨060605, 0503 z dz; (4) 05. 06. 04 罍ｧ 閨03 ﾀ05 0503 003 0 000308 x ｧ y 04030000306 04 罍ｧ 0060 ｧ06 070600909 ｧ ｧ 04 罍ｧ- 0300 ｧ07060003 04003 0606 040300003006 陦ｧ070706 縺06009 ｧ 0006 (0500. 5, x 7). ｱ0600909 ｧ07 F (x; y) 0 0304 0605 ﾀ07 陦ｧ070706 縺06009 ｧ 縺060600308 04 0503 0906 (x; y), 06 ﾀ00 ｧ 030 ｧ0060605 04 05050308 0503 0906 04030000308 陦ｧ070706 縺06009 ｧ 00. 884. 7 0805 ｧ 04030000306 陦ｧ070706 縺06009 ｧ 0003 070600909 ｧ08: ) z = x y; ) z = xy x, y ; 3) u = es=t ; 4) z = x + y. 885. 7 0805 ｧ 0 060 ｧ06 0403000030503 陦ｧ070706 縺06009 ｧ 00 070600909 ｧ ｧ: ) z = y x 04 罍ｧ x =, y =, =0;, dy =0; ; ) u = e xy 04 罍ｧ x =, y =, =,0;, dy =0;. 886. 6 503 ｧ ﾀ00 ｧ0504 dz ｧ z 閨0007 070600909 ｧ ｧ z = xy 04 罍ｧ x = 5, y =4, x =0;, y =,0;. 887. 703 陦ﾀ ｧ05 0504 04 罍ｧ0300 ｧ07060003 ｧ 0060060 ｧ06 070600909 ｧ ｧ ' = arctg y x, 090305 x ｧ 0060070605 ﾀ07 0305 閨03,, y 0305 3 閨03,5. 888. 7 罍ｧ 閨060703 縺0 09 ｧ ｧ 09 ｧ00 ｧ0 陦 060503 陦ｧ06 ﾀ R 06040600 ｧ ｧ00 ﾀ07 ﾀ 0 ﾀ0 閨03 0,5 ﾀ0, 0403 ﾀ0305 H 0600600400 ｧ00 ﾀ04 ﾀ03 00 ﾀ0 閨03 98 ﾀ0. 7 0805 ｧ 04 罍ｧ0300 ｧ07060003 ｧ 0060060 ｧ06 03030060 V 0403 0703 縺0060006 V dv. 889. 07 05060503 04 縺0700306050300040030503 05 縺0606050300040 ｧ09, ｧ 006 縺06000306 ﾀ 0503-003 ﾀ050406 閨030; ﾀ0, 0309 00 ｧ ﾀ04 040030 ｧ 7;5 ﾀ0 ｧ 8 ﾀ0. 0504 縺06 閨0600 ｧ0504 03 ﾀ0300060500606 040305 縺0600003 ﾀ0504 04 罍ｧ 0403 ｧ ﾀ00060 ｧ ｧ 05 ｧ04030506006 03. 890. 7 0805 ｧ 04030000306 陦ｧ070706 縺06009 ｧ 0003 070600909 ｧ08: ) z = y x, x y ; ) s = x ln t; 3) u = x + y + z. 89. 7 0805 ｧ 0 060 ｧ06 dz ｧ z 閨0007 070600909 ｧ ｧ z =ln(x + y ), 090305 x ｧ 0060070605 ﾀ07 0305 閨03,, y 0305 閨03 0,9. 89. 703 陦ﾀ ｧ05 0504 04 罍ｧ0300 ｧ07060003 ｧ 0060060 ｧ06 070600909 ｧ ｧ z = arcsin y x, 090305 x ｧ 0060070605 ﾀ07 0305 5 閨03 4,5, y 0305 3 閨03 3,3. 893. 7 罍ｧ 閨060703 縺0 09 ｧ ｧ 0903006 ﾀ 060503 陦ｧ06 ﾀ R 06040600 ｧ ｧ00 ﾀ07 ﾀ 30 ﾀ0 閨03 30, ﾀ0, 0403 ﾀ0305 H 0600600400 ｧ00 ﾀ04 ﾀ 60 ﾀ0 閨03 59,5 ﾀ0. 7 0805 ｧ 04 罍ｧ0300 ｧ07060003 ｧ 0060060 ｧ06 03030060 0403 0703 縺0060006 V dv.

3x 4. 7 縺03 ｧ 0403 閨00306 ﾀ00030700308 070600909 ｧ08 9 x 4. 7 縺03 ｧ 0403 閨00306 ﾀ00030700308 070600909 ｧ08. ｭ ﾀ00 ｧ z = F (x; y), x = f(t), y = '(t), 0503z 0 0304 0605 ﾀ07 ﾀ00030700308 070600909 ｧ0608 0305 t. 7 罍ｧ 0505030 dz dt = @z @x dt + @z dy @y dt ; () 06 ﾀ00 ｧ 070600909 ｧ ｧ F, f ｧ ' 陦ｧ070706 縺06009 ｧ 縺0606003.. ｭ ﾀ00 ｧ z = F (x; y), 05 閨06 x = f(u; v), y = '(u; v), ｧ 06 ﾀ00 ｧ 070600909 ｧ ｧ F, f ｧ ' 陦ｧ070706 縺06009 ｧ 縺0606003, 0503 @z @u = @z @x @x @u + @z @y @y @u ; @z @v = @z @x @x @v + @z @y @y @v : () 894. 7 0805 ｧ 0403 0703 縺0060006 () dz ｧ 06 040060 ｧ08: dt ) z = x + xy + y, x = t, y = t; ) z = x + y, x = sin t, y = cos t. 7 縺030406 罍ｧ0504 04 縺06 閨04 罍ｧ0506000400308 0403 陦ﾀ05 00304090308 0 060 ｧ08 x ｧ y 04 0403-07060 ｧ06 閨0007 070600909 ｧ ｧ z. 895. 7 0805 ｧ dz dt, 06 ﾀ00 ｧ z = y x, x =, et y =, e t. 896. 7 0805 ｧ dz, 06 ﾀ00 ｧ z =,05 閨06 uv u ｧ v 070600909 ｧ ｧ 0305 x. 897. 7 0805 ｧ dz, 06 ﾀ00 ｧ z =,05 閨06 xey y 070600909 ｧ07 0305 x. 898. ｱ0600909 ｧ07 z = F (x; y) 0 0304 0605 ﾀ07 03 閨003 縺03 閨00308, 06 ﾀ00 ｧ F (xt; yt) =t n F (x; y). 6 7 ｧ070706 縺06009 ｧ 縺0607 030306 ﾀ05 ｧ 0505030503 04060- ﾀ0504 0403 t ｧ 040300 05 07 04 縺06 06000405 0506 t =, 閨0309 0504 050603 縺06006 7080006 0303 03 閨003 縺03 閨00308 070600909 ｧ0708: x @z @x + y @z @y = nz. 899. 7 0805 ｧ @z @u ｧ @z x, 06 ﾀ00 ｧ z =,05 閨06 x = u, v, y = v +u. @v y 900. 706 ﾀ0504 z = F (x; y). 6 503 ｧ0504 @z @x ｧ @z @z 06 縺06 @y @u ｧ @z @v, 06 ﾀ00 ｧ: ) u = mx + ny, v = x + qy; ) u = xy, v = y=x. 90. 706 ﾀ0504 u = F (x; y), 05 閨06 x = r cos ', y = r sin '. 6 503 ｧ0504 @u @r ｧ @u @u 06 縺06 @' @x ｧ @u ｧ 040309 0504, 0503 @y @u @u @u @u + = + : @r r @' @x @y

39 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 90. 706 ﾀ0504 z = y + F (u), 05 閨06 u = x, y. 6 70309 0504, 0503 y @z @x + x@z = x 閨0007 0006030308 陦ｧ070706 縺06009 ｧ 縺060600308 070600909 ｧ ｧ F (u). @y 903. 7 0805 ｧ dz ｧ 06 040060 ｧ08: dt ) z = Ax +Bxy + Cy, x = sin t, y = cos t; ) z = arctg y x, x = et +, y = e t,. 904. 6 70309 0504, 0503 06 ﾀ00 ｧ z = xy + xf (u), 05 閨06 u = y=x, 0503 x @z @x + y @z @y = z + xy: 905. 6 70309 0504, 0503 06 ﾀ00 ｧ z = y'(u), 05 閨06 u = x, y,0503 @z x @x + y @z @y = z y : 906. 706 ﾀ0504 z = F (x; y). 6 503 ｧ0504 @z @x ｧ @z @z 06 縺06 @y @u ｧ @z @v, 06 ﾀ00 ｧ: ) u = x +y, v = x, y; ) u = xy, v = x + y. x 5. 7 縺03 ｧ 0403 閨00306 006070400308 070600909 ｧ08. ｰ 040060 ｧ06 F (x; y) =0, ｧ0060600606 縺0600060 ｧ06 (x 0 ; y 0 ), 0304 縺06 閨06-00070605 040309 縺06 ﾀ05003 ﾀ05 ｧ x 0 0406 縺06006000606 y 09 09 00604 縺06 縺030400606 070600909 ｧ06 x 04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 04 縺03 ｧ 0403 閨0 07 @F 6=0 ｧ 00604 縺06 縺03040 04 00609030503 縺0308 0309 縺06 ﾀ05003- @y ﾀ05 ｧ 0503 09 ｧ (x 0 ; y 0 ). ｭ ﾀ00 ｧ ﾀ0406 縺08 05030503, 04 0309 縺06 ﾀ05003 ﾀ05 ｧ 0503 09 ｧ (x 0 ; y 0 ) ﾀ06006 ﾀ0504060605 ｧ 00604 縺06- 縺03040 07 04 縺03 ｧ 0403 閨0 07 @F dy,050300607040 07 070600909 ｧ07 ｧ0060605 04 縺03 ｧ 0403 閨00606, @x 0304 縺06 閨0600070600606 0703 縺006000308 dy =,@F=@x @F=@y : (). ｰ 040060 ｧ06 F (x; y; z) =004 罍ｧ 0 000305 ｧ 00308 06 ﾀ000304 ｧ0708 0304 縺06- 閨0600070605 z 09 09 006070400606 070600909 ｧ06 x ｧ y, ｧ0060600606 ﾀ0500306 04 縺03 ｧ 0403 閨00306 @z @x =,@F=@x @F=@z ; 7 0805 ｧ dy ｧ 06 040060 ｧ08: 907. x + y, 4x +6y =0. @z @y =,@F=@y @F=@z : ()

3x 5. 7 縺03 ｧ 0403 閨00306 006070400308 070600909 ｧ08 93 908. ) x =3 + y =3 = a =3 ; ) xe y, ye x =0. 909. Ax +Bxy + Cy +Dx +Ey + F =0. 7 0805 ｧ 06050003040308 0903050707 ｧ09 ｧ06005 09 ﾀ 0506000400308 09 09 罍ｧ040308: 90. x + y =0y 040503 0906 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 04 縺0700308 x =3. 9. x 3 + y 3, axy =0040503 0906 x = y = a. 9. 7 0805 ｧ 0503 09 ｧ, 04 09030503 縺0308 09 ﾀ 050600040 07 09 09 罍ｧ040308 x + y + +x, y =04 00000600040 : ) Ox; ) Oy. 7 0805 ｧ @z @x ｧ @z @y ｧ 06 040060 ｧ08: 93. x + y + z, 6x =0: 94. z = xy. 95. cos (ax + by, cz) =k(ax + by, cz). 96. 6 70309 0504, 0503 06 ﾀ00 ｧ xyz = a 3,0503 x @z @x + y @z @y =,z: 97. 70309 0504, 0503 陦ｧ070706 縺06009 ｧ 0004003006 06 040060 ｧ06 x @z @x + y @z @y = z 06 閨03040006050403 縺070605 00607040 07 070600909 ｧ07 z, 0304 縺06 閨060007060 07 06 040060 ｧ060 (0903-0 ｧ 06 ﾀ09 ｧ08 04030406 縺08003 ﾀ050608) z=x = '(y=x). 7 0805 ｧ dy ｧ 06 040060 ｧ08: 98. x, 4y =4: 99. xy +lny +lnx =0: 90. x + y = e y=x : 9. cos (x, y) =y, x: 9. 7 0805 ｧ 06050003040308 0903050707 ｧ09 ｧ06005 09 ﾀ 05060004003080909 罍ｧ040308 y,, xy =4040503 09 08 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 04 縺0700308 x =3. 93. 706 ﾀ0504 x + y + z, zx = a. 7 0805 ｧ @z @x ｧ @z @y. 94. sin (x +y, 3z) =x +y, 3z. 70309 0504, 0503 @z @x + @z @y =: 95. 70309 0504, 0503 陦ｧ070706 縺06009 ｧ 0004003006 06 040060 ｧ06 m @z @x + n@z @y = 06 閨03040006050403 縺070605 00607040 07 070600909 ｧ07 z, 0304 縺06 閨060007060 07 06 040060 ｧ060 (09 ｧ- 00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ08 04030406 縺08003 ﾀ050608): x, mz = '(y, nz).

394 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 x 6. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306 ｧ 04030000306 陦ｧ070706 縺06009 ｧ 0003 0403 ﾀ00 ｧ08 0403 縺07 閨090304 706 ﾀ0504 0 070600909 ｧ07 z = F (x; y), ｧ006060 07 ﾀ0500306 04 縺03 ｧ 0403 閨00306 @F @x ｧ @F. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306 0305 0505 ｧ08 04 縺03 ｧ 0403 閨00308 0 0304 0605 ﾀ07 ﾀ05- @y 0030 ｧ 04 縺03 ｧ 0403 閨0030 ｧ 040503 縺030503 0403 縺07 閨09. 050 ｧ 030303 0 0605 ﾀ07: @(@F=@x) @x @(@F=@y) @x = @ F @x ; = @ F @y @x ; @(@F=@x) @y @(@F=@x) @y = @ F @x@y ; = @ F @y : 縺0 000305 ｧ 003 0304 縺06 閨0600070605 ﾀ07 ｧ 030303 0 0605 ﾀ07 ﾀ0500306 04 縺03 ｧ 0403 閨00306 05 縺060504060503 0403 縺07 閨09 ｧ 陦縺0605 ｧ08 0403 ﾀ00 ｧ08 0403 縺07 閨090304. ｮ00600 000306 04 縺03 ｧ 0403 閨00306, 030500 ｧ 060 ｧ06 ﾀ07 050300040903 0403 縺07 閨09030 陦ｧ070706- 縺06009 ｧ 縺0304 0 ｧ07, 04003, 06 ﾀ00 ｧ 030 ｧ 00604 縺06 縺0304003: @ F @x@y = @ F @y @x ; @ 3 F @x @y = @3 F @x@y @x = @3 F @y @x ｧ05.. 7030006 ｧ0 ﾀ0006 閨060600606 05 0300 ｧ0906 04 縺03 ｧ 0403 閨00308 0403 ﾀ00 ｧ08 0403 縺07 閨090304: 040503 縺030503 0403 縺07 閨09 @ F @x, @ F @x@y, @ F @y ; 05 縺060504060503 0403 縺07 閨09 @3 F @x, @ 3 F 3 @x @y, @ 3 F @x@y, @ 3 F @y ｧ05.. 3 7030000306 陦ｧ070706 縺06009 ｧ 0003 0403 ﾀ00 ｧ08 0403 縺07 閨090304 0304 縺06 閨0600070605 ﾀ07 05 09: d z = @ z @x + @ z @x@y dy + @ z @y dy : ｮ ｧ0040300 ｧ 06 ﾀ09 ｧ 050503 04060 ﾀ050403 00307003 04 ｧ ﾀ 0504 05 09: @ d z = @x + @ @y dy z: 縺0 000305 ｧ 003 ｧ05.. d 3 z = @ @x + @ 3 @y dy z 96. 7 0805 ｧ ﾀ0500306 04 縺03 ｧ 0403 閨00306 05 縺060504060503 0403 縺07 閨09 070600909 ｧ ｧ z = x 3 + x y + y 3. @ z 97. 7 縺030406 罍ｧ0504, 0503 @x @y = @ z 閨0007 070600909 ｧ08: @y @x ) z = sin (ax, by); ) z = x =y ; 3) z =ln(x, y).

3x 6. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306 0403 ﾀ00 ｧ08 0403 縺07 閨090304 95 98. 7 0805 ｧ ﾀ0500306 04 縺03 ｧ 0403 閨00306 06050406 縺05030503 0403 縺07 閨09 070600909 ｧ ｧ u = x 4 +3x y, y 4. 99. 7 0805 ｧ ﾀ0500306 04 縺03 ｧ 0403 閨00306 05 縺060504060503 0403 縺07 閨09 070600909 ｧ ｧ u = y=x. 930. 706 ﾀ0504 s =ln x, ; 04 縺030406 罍ｧ0504, 0503 @ s t @x @t + @ s @x = x. 93. 7 0805 ｧ ﾀ0500306 04 縺03 ｧ 0403 閨00306 040503 縺030503 0403 縺07 閨09 070600909 ｧ ｧ z = arctg y x. x 93. 6 70309 0504, 0503 06 ﾀ00 ｧ z = sin a, y,0503 b @ z =, z: @x + @ @y a, b 933. 6 70309 0504, 0503 06 ﾀ00 ｧ u = arctg (x, t), 0503 @ u @x + @ u @x @t =0: 934. 6 70309 0504, 0503 06 ﾀ00 ｧ s = 3 ax + bt, 0503 x @ @x + t @ s =, s @t 9 : 935. 70309 0504, 0503 070600909 ｧ07 u = xe,y=x 06 閨03040006050403 縺070605 陦ｧ070706- 縺06009 ｧ 0004003006 06 040060 ｧ06 x @ u @u @x@y + @x + @u @y = y @ u @y : 936. 6 70309 0504, 0503 06 ﾀ00 ｧ z = F (x; y) 03 閨003 縺03 閨0 07 070600909 ｧ07 n-0503 ｧ 006 縺060 ｧ07, 0503 ｧ00 ｧ ﾀ ｧ0040300 ｧ 06 ﾀ09 ｧ x @ z @x +xy @ z @x @y + y @ z = n(n, )z; @y x @ @x + y @ z = n(n, )z: @y ｰ09 0 ｧ06. 7 04060 ﾀ050403 x @x @z + y @z = nz ( ﾀ0. 06 898) 04 縺03 陦ｧ07- @y 0706 縺06009 ｧ 縺0304 0504: ) 0403 x; ) 0403 y, ｧ 縺06 06000405 0503, 0600030706000306 ﾀ030305040605- ﾀ0504060003 0 x ｧ0 y, ﾀ000307 ｧ0504 0403 00060003.

396 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 937. 7 縺030406 罍ｧ0504 04060 ﾀ050403 x @ @x + y @ z = n(n, )z 閨0007 @y 03 閨003 縺03 閨00308 070600909 ｧ08: ) z = x + xy + y ; ) z = y x ; 3) z = = y x, y ; 4) z =ln x,. 938. 7 0805 ｧ d u, 06 ﾀ00 ｧ: ) u = y x ;) u = x ln y x. 939. 6 70309 0504, 0503 06 ﾀ00 ｧ z = cos (mx + ny), 0503 d z = =,z(m+ ndy). 940. 6 70309 0504, 0503 06 ﾀ00 ｧ z =ln(ax + by), 0503: ) d 3 z =dz 3 ; ) d n z =(,) n, (n, )! dz n. 94. 6 70309 0504, 0503 06 ﾀ00 ｧ z = F (u; v), 05 閨06 u = mx + ny ｧ v = = x+qy,0503 @ z @x = m @ @u + @ @ z z, @v @x @y = m @ @u + @ @v n @ @u + q @ @ z z, @v @y = n @ @u + q @ z. @v 94. 7 縺060303 0304 0504 0403 07060 ｧ06 @ z @x, 4 @ z z @x @y +3@ @y 09 00304030 0406 縺0600600030 u =3x + y ｧ v = x + y ( ﾀ0. 06 94). 943. 7 縺060303 0304 0504 0403 07060 ｧ06 @ z @x, 4 @ z z @x @y +4@ @y 09 00304030 0406 縺0600600030 u =x + y ｧ v = y ( ﾀ0. 06 94). 944. 6 70309 0504, 0503 06 ﾀ00 ｧ z = F (u; v), 05 閨06 u ｧ v 070600909 ｧ ｧ 0305 x ｧ y, 0503 @ z u @x = 0 @ x @u + @ v0 z + u 00 @z x xx @v @u + @z v00 xx @v : 0504 縺06 閨0600 ｧ0504 0 000305 ｧ 003 @ z @x@y ｧ @ z @y. 945. 7 縺060303 0304 0504 0403 07060 ｧ06 x @ z @x, @ z y 09 00304030 0406 縺06- @y 00600030 u = xy ｧ v = y=x ( ﾀ0. 06 944). 946. 7 縺060303 0304 0504 0403 07060 ｧ06 @ z @r + @ z r @' + @z r @r 09 00304030 0406 縺0600600030 x = r cos ' ｧ y = r sin ' ( ﾀ0. 06 944).

3z = x 6. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306 0403 ﾀ00 ｧ08 0403 縺07 閨090304 97 947. 7 0805 ｧ ﾀ0500306 04 縺03 ｧ 0403 閨00306 040503 縺030503 0403 縺07 閨09 070600909 ｧ ｧ x, y. 948. 7 0805 ｧ ﾀ0500306 04 縺03 ｧ 0403 閨00306 05 縺060504060503 0403 縺07 閨09 070600909 ｧ ｧ u = x 3 t. 949. 6 70309 0504, 0503 06 ﾀ00 ｧ z = xy x, y,0503 @ z @x + @ z @x @y + @ z @y = x, y : 950. 6 70309 0504, 0503 06 ﾀ00 ｧ s =ln(ax, bt), 0503 x @ @x + t @ 3 s =: @t 95. 6 70309 0504, 0503 06 ﾀ00 ｧ z = cos x, t @ z @t + @ z @x @t =0:,0503 95. 6 70309 0504, 0503 06 ﾀ00 ｧ z = e x=y,0503y @ z @x @y = @z @y, @z @x. 953. 706 ﾀ0504 u =lnx. 7 0805 ｧ d u ｧ d 3 u. 954. 7 縺060303 0304 0504 0403 07060 ｧ06 @ z @x, @ z a 09 00304030 0406 縺06- @y 00600030 u = ax + y ｧ v = ax, y ( ﾀ0. 06 94). 955. 7 縺060303 0304 0504 0403 07060 ｧ06 x @ z @x + y @ z 09 00304030 0406 縺06- @x @y 00600030 u = y ｧ v = y ( ﾀ0. 06 944). x xf (x) y 956. 70309 0504, 0503 070600909 ｧ07 u = + ' 04 罍ｧ 0006030308 y x 閨04 07 閨03 陦ｧ070706 縺06009 ｧ 縺060600308 070600909 ｧ0708 f ｧ ' 06 閨03040006050403 縺070605 陦ｧ07-0706 縺06009 ｧ 0004003006 06 040060 ｧ06 xy @ u @x@y + @ u y @y + x@u @u +y @x @y =0:

398 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 x 7. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 04030000308 陦ｧ070706 縺06009 ｧ 000304. ｪ05030303 0403 07060 ｧ06 P+ Qdy,05 閨06 P ｧ Q 陦ｧ070706 縺06009 ｧ 縺0606-00306 070600909 ｧ ｧ x ｧ y, 03030003 0403000030 陦ｧ070706 縺06009 ｧ 00030 du, 00603030803 陦ｧ003 ｧ 閨03 ﾀ05 0503 003 04030403000060 ｧ06 06 ﾀ000304 ｧ07 @P @y = @Q @x. 6 70007 0 080307 閨060 ｧ07 u ｧ 06 ﾀ000304 ｧ08 @u @x = P ｧ @u R @y = Q 04030006 ｧ0 u = = P+ ' (y), u R = Qdy+ ' (x). 6 50304 ｧ ﾀ 04 ｧ 0406 縺04030503 0403 07060 ｧ07 04 ﾀ06 ｧ 0406 ﾀ0500306 0006003, ｧ 040503 縺030503 0006003 ﾀ y, 006 閨03 ﾀ05 060 ｧ06 04 0406 縺04030, 04030006 ｧ0 070600909 ｧ06 u.. ｪ05030303 0403 07060 ｧ06 P+ Qdy + Rdz,05 閨06 P, Q ｧ R 陦ｧ070706- 縺06009 ｧ 縺060600306 070600909 ｧ ｧ 0305 x, y ｧ z, 03030003 0403000030 陦ｧ070706 縺06009 ｧ 00030 du, 00603030803 陦ｧ003 ｧ 閨03 ﾀ05 0503 003 04030403000060 ｧ06 06 ﾀ000304 ｧ08: @P @y = @Q @x ; @P @z = @R @x ; @Q @z = @R @y : 6 70007 0 080307 閨060 ｧ07 u ｧ006060: u = P+' (y; z); u = Qdy+' (x; z); u = Rdz+' 3 (x; y): 6 50304 ｧ ﾀ 04 ｧ 0406 縺04030503 0403 07060 ｧ07 04 ﾀ06 ｧ 0406 ﾀ0500306 0006003, ｧ 040503 縺030503 ｧ 05 縺060504060503 006 閨03 ﾀ05 060 ｧ06 0006003 ﾀ y ｧ z, 04030006 ｧ0 070600909 ｧ06 u. 7 080307 閨060 ｧ06 070600909 ｧ ｧ 0403 0606 040300003006 陦ｧ070706 縺06009 ｧ 0006 0 0304 0605 ﾀ07 ｧ0-050605 罍ｧ 縺0304 0 ｧ060 0403000030503 陦ｧ070706 縺06009 ｧ 00. 7 縺030406 罍ｧ0504, 0503 ﾀ0006 閨060600606 0403 07060 ｧ06 070400070605 ﾀ07 0403000030 陦ｧ07-0706 縺06009 ｧ 00030 du, ｧ 0 0805 ｧ u: 957. (x + y) +(x, y, 3) dy. 958. x sin y+ x cos ydy. 959. (x +lny) + 960. xdy, y x + y. x y + sin y dy. 96. (yz, x) +(xz + y) dy +(xy, z) dz. 96. z, x + dy y, x z + +z dz. 7 縺030406 罍ｧ0504, 0503 ﾀ0006 閨060600606 0403 07060 ｧ06 070400070605 ﾀ07 0403000030 陦ｧ07-0706 縺06009 ｧ 00030 du, ｧ 0 0805 ｧ u: 963. (y, ) +(xy +3y) dy. 964. (sin y, y tg x) +(x cos y + ln cos x +y) dy.

3x 8. 05 ﾀ03030306 0503 09 ｧ 040003 ﾀ090308 09 罍ｧ040308 99 sin y x + + dy. x 965. y, sin y x + r x + 966. t t dt + + 967. (ln y, cos z) + 968., 3 dy z + 3y, z z t +. x x y + z dy +(y +x sin z) dz. dz. x 8. 05 ﾀ03030306 0503 09 ｧ 040003 ﾀ090308 09 罍ｧ040308 ｯ03 09 09 罍ｧ040308 F (x; y) =0 0 0304 0605 ﾀ07 03 ﾀ03030308, 06 ﾀ00 ｧ 04 05050308 0503 0906 @F @F =0 ｧ @x @y =0. ｰ050003040308 0903050707 ｧ09 ｧ06005 k = y 0 09 ﾀ 05060004003080405 090308 0503 0906 0 0803 陦ｧ05 ﾀ07 ｧ 06 040060 ｧ07 A +Bk + Ck =0,05 閨06 A, B ｧ C 0 060 ｧ07 04 縺03 ｧ 0403 閨00308 @ F @x, @ F @x@y ｧ @ F 0405050308 03 ﾀ03030308 0503 0906. 7 罍ｧ 0505030 0403 00307003 05 罍ｧ ﾀ0006 07: @y ) B, AC > 0 閨040609 ﾀ 0506000400306; 0503 09 0 0304 0605 ﾀ07 06 00030; ) B, AC < 0 0060509 ﾀ 0506000400308; 0503 09 ｧ 0300 ｧ 縺0304 00 07; 3) B, AC =0 ｧ00 ｧ ｧ 0300 ｧ 縺0304 00 07 0503 09, ｧ00 ｧ 0503 09 0403 04 05, ｧ00 ｧ 0503 09 ﾀ 003 ﾀ0304 罍ｧ0903 ﾀ00304060 ｧ07; 04 0503 09 08 0403 04 05 ｧ ﾀ 003 ﾀ0304 罍ｧ0903 ﾀ003-04060 ｧ07 ﾀ06006 ﾀ0504060605 03 閨0 03030 07 09 ﾀ 050600040 0709 閨04060 04060504070 09 罍ｧ040308. ｪ05030303 04 05 縺060504060, ﾀ0300 ｧ050600040030, ﾀ0006 06 縺0600 ｧ0504 040304 縺03 ﾀ 0309030 05060004003, 00607003 06 0 0504, ｧ0060605 ﾀ07 00 ｧ 0503 09 ｧ 09 罍ｧ040308 04 ﾀ09030004 060503 閨003 0 000308 0309 縺06 ﾀ05003- ﾀ05 ｧ ｧ ﾀ ﾀ0006 閨060600308 0503 09 ｧ. 0504 縺06 閨0600 ｧ0504 030300 ﾀ05 ｧ ﾀ0403000307060 ｧ07, 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05, 03 ﾀ03030306 0503 09 ｧ 09 罍ｧ040308 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: 969. x 3 + x, y =0: 970. y =(x +) 3 : 97. x 3, x, y =0: 97. y + x 4, x =0: 973. (y, x) = x 3 : 974. y = x(x, ) : 0504 縺06 閨0600 ｧ0504 030300 ﾀ05 ｧ ﾀ0403000307060 ｧ07, 03 ﾀ03030306 0503 09 ｧ ｧ ﾀ ｧ00405030503 09 罍ｧ040308 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: 975. (x +a) 3 + xy =0: 976. x 3, y 3, 3y =0: 977. x 3 + y 3, 3axy =0: 978. y (x, a )=x 4 :

300 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 0504 縺06 閨0600 ｧ0504 030300 ﾀ05 ｧ ﾀ0403000307060 ｧ07, 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05, 03 ﾀ03030306 0503 09 ｧ 09 罍ｧ040308 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040306: 979. y + x 3, x =0: 980. a y = x (ax, x ): 98. y = x(x +) : 98. xy =(x + a) 3 : 983. 4y = x 5 +5x 4 : 984. y, x 4 + x =0: 985. 7 0805 ｧ 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05, y max,03 ﾀ03-030606 0503 0906 ｧ ﾀ ｧ00405030506 09 罍ｧ040308 4x, y + x 3, y 3 =0 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606. 0504 縺06 閨0600 ｧ0504 030300 ﾀ05 ｧ ﾀ0403000307060 ｧ07, 03 ﾀ03030306 0503 09 ｧ ｧ ﾀ ｧ00405030503 09 罍ｧ040308: 986. ) y (a, x) =x(x, a) ( ﾀ05 縺030703 ｧ ); ) a (x + y )=x y. 987. ) x(x + y )=a(x, y ); ) a(x + y )=x(x, y ). x 9. 0505 ｧ03 060 07 ﾀ0600608 ﾀ0504 040003 ﾀ09 ｧ08 09 罍ｧ040308 07 罍ｧ04 07 0 0304 0605 ﾀ07 0305 ｧ03 0600608 ﾀ0600608 ﾀ0504 09 罍ｧ040308 F (x; y; ) = 0, 06 ﾀ00 ｧ: ) 030 09 ﾀ 0605 ﾀ07 09 07 閨0308 09 罍ｧ040308 ﾀ0600608 ﾀ0504 ; ) 09 07 07 0606 0503 09 07040007-0605 ﾀ07 0503 090308 0606 09 ﾀ 0 ｧ07 ﾀ 09 罍ｧ040308 ﾀ0600608 ﾀ0504, 030500 ｧ 00308 0305 00606 ﾀ 00308. 0505 ｧ03 060 07 ﾀ0600608 ﾀ0504 09 罍ｧ040308 F (x; y; ) =0, 06 ﾀ00 ｧ 030 ﾀ06006 ﾀ0504060605, 0 0803 陦ｧ05 ﾀ07 ｧ ﾀ090006 060 ｧ060 04 00605 ｧ 06 040060 ｧ08 F (x; y; ) =0 ｧ F 0 (x; y; ) =0: 0303070605, 03 閨0 0903, ﾀ0006 ｧ0504 ﾀ07, 0503 04030006 0600 07 0505 ｧ0 ﾀ0403 ﾀ0303030 09 罍ｧ04 07 0306- 閨0605 006 0305 ｧ03 0600608, 05060300605 罍ｧ 06 ﾀ09 ｧ0 006 ﾀ05030 03 ﾀ03030308 0503 0609 09 罍ｧ040308 ﾀ06-00608 ﾀ0504 ( ﾀ0. 0305040605 09 06 990, )). 7 0805 ｧ 0305 ｧ03 0600606 ﾀ0600608 ﾀ0504 09 罍ｧ040308 ｧ 0403 ﾀ05 縺03 ｧ0504 0305 ｧ03 0600606 ｧ 09 罍ｧ040306 ﾀ0600608 ﾀ0504 : 988. ) y = ax + a ; ) y = ax + a : 989. ) (x, a) + y = R ; ) 4ay =(x, a) : 990. ) y, =(x, a) ; ) (y, ) 3 =(x, a) ; 3) (y, ) =(x, a) 3 ; 4) 9(y, a) =(x, a) 3 : 99. 0505 縺06 0309 0403 ﾀ050307000308 閨00 ｧ003 a ﾀ09030004 ｧ05 ﾀ0403 ｧ0 ｧ 0903009 0 ｧ 0403 090303 罍陦ｧ0 050030 03 ﾀ070. 7 0805 ｧ 0305 ｧ03 0600606 ﾀ0600608 ﾀ0504 05 09 ｧ08 0305- 縺06 090304. 99. 7 0805 ｧ 0305 ｧ03 0600606 ﾀ0600608 ﾀ0504 0309 縺0607003 ﾀ050608, 04 縺030803 閨070 ｧ08 06 縺06 0 0003 090303 罍陦ｧ0 05 ｧ ｧ006060 ｧ08 0906005 0 04 03030006 y =4x. 993. 7 0805 ｧ0305 ｧ03 0600606 ﾀ0600608 ﾀ0504 0309 縺0607003 ﾀ050608, ｧ006060 ｧ08 陦ｧ - 00605 0 ｧ 陦ｧ06 ﾀ-0406090503 縺03 0503 0609 05 ｧ0406 縺03030003 xy = a.

3x 0. 07 ﾀ 050600040 07 040003 ﾀ0903 ﾀ0504 ｧ 003 縺0 0004 09 04030406 縺08003 ﾀ05 ｧ 0 994. 0ｷ3 0 00 090303 罍陦ｧ0 05 04030406 ﾀ09 0605 ﾀ07 ﾀ0 縺07 ﾀ 0 000400308 ﾀ0903 縺03 ﾀ050406 b 0403 060500030 0903 ﾀ ｧOx. 7 0805 ｧ 0305 ｧ03 0600606 ﾀ0600608 ﾀ0504 05 06090503 罍ｧ08 04 罍ｧ 00 ｧ 00308. 995. 7 0805 ｧ0305 ｧ03 0600606: ) ﾀ0600608 ﾀ0504 04 縺0700308 x cos +y sin,, = 004 罍ｧ 0403 ﾀ05030700030 ; ) ﾀ0600608 ﾀ0504 04 縺0700308 y = ax + a ; 3) ﾀ0600608 ﾀ0504 090603 ｧ 06 ﾀ09 ｧ08 04 030300 y, =(x, a) 3. 996. 7 0805 ｧ 0305 ｧ03 0600606 ﾀ0600608 ﾀ0504 0309 縺0607003 ﾀ050608 ﾀ 0906005 0 ｧ 0 03 ﾀ ｧ Ox, 陦ｧ06 ﾀ 0 ｧ 09030503 縺0308 ﾀ000607 05 ﾀ030305040605 ﾀ050406060 ｧ06 03 罍陦ｧ0 0503 04 03030003 y =4x. 997. 7 0805 ｧ 0305 ｧ03 0600606 ﾀ0600608 ﾀ0504 050000 ｧ04 ﾀ0304 x a + y b =04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 ﾀ0600 0403000603 ﾀ0608 ｧ0060605 0403 ﾀ050307000606 閨00 ｧ006 l. 998. 7 0805 ｧ 0305 ｧ03 0600606 ﾀ0600608 ﾀ0504 04 030300, ｧ006060 ｧ08 03 ﾀ04 ﾀ ｧ0-00605 罍ｧ ｧ, 04 000006000400606 03 ﾀ ｧOy, ｧ04 縺030803 閨070 ｧ08 06 縺06 0503 09 ｧ (,a; 0); (3a; 0) ｧ (0; 3a ) 04 罍ｧ 00 ｧ 00308 a. x 0. 07 ﾀ 050600040 07 040003 ﾀ0903 ﾀ0504 ｧ 003 縺0 0004 09 04030406 縺08003 ﾀ05 ｧ 706 ﾀ0504 04030406 縺08003 ﾀ0504 0 06 040060 ｧ060 F (x; y; z) =0;0403 040060 0 00608 0503 0906 M (x; y; z). ｰ 040060 ｧ07 003 縺0 00 ｧ 09 04030406 縺08003 ﾀ05 ｧ 04 05050308 0503 0906 0306 閨0605 ｰ 040060 ｧ06 09 ﾀ 0506000400308 040003 ﾀ0903 ﾀ05 ｧ: X, x @F=@x = Y, y @F=@y =, z @F=@z : () @F (X, x)+@f (Y, y) +@F (, z) =0: () @x @y @z 6 506 040060 ｧ0708 () ｧ () X, Y, 050609060 ｧ06 090303 罍陦ｧ0 0503 003 縺0 00 ｧ ｧ00 ｧ 09 ﾀ 0506000400308 040003 ﾀ0903 ﾀ05 ｧ. @F 6 506090503 N @x ; @F @y ; @F 0 0304060 003 縺0 00040030 0406090503 縺030 04030406 縺08003- @z ﾀ05 ｧ. ｭ ﾀ00 ｧ 0 04030406 縺08003 ﾀ05 ｧ 06 ﾀ0504 0503 09, 04 09030503 縺0308 @F @x = 0, @F @y = 0, @F =0, 0503 030 0 0304 0605 ﾀ07 03 ﾀ03030308. 6 5 05 090308 0503 0906 00605 0 ｧ09 ﾀ 0506000400308 @z 040003 ﾀ0903 ﾀ05 ｧ, 0 ｧ 003 縺0 00 ｧ 09 04030406 縺08003 ﾀ05 ｧ. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 09 ﾀ 0506000400308 040003 ﾀ0903 ﾀ05 ｧ 09 04030406 縺08003 ﾀ05 ｧ: 999. z = x +y 040503 0906 (; ; 3).

30 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 000. xy = z 040503 0906 (x 0 ; y 0 ; z 0 ). 00. xyz = a 3 040503 0906 (x 0 ; y 0 ; z 0 ). 00. x a + y b, z c =040503 0906 (x 0; y 0 ; z 0 ) ｧ040503 0906 (a; b; c). 003. 0504 縺06 閨0600 ｧ0504 040003 ﾀ0903 ﾀ0504, 09 ﾀ 0506000400606 09 04030406 縺08003 ﾀ05 ｧ x + +4y + z =36 ｧ 04 000006000400606 040003 ﾀ0903 ﾀ05 ｧ x + y, z =0. 004. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 003 縺0 00 ｧ 04 0503 0906 (3; 4; 5) 09 04030406 縺08-003 ﾀ05 ｧ 0903006 ﾀ x + y = z.6 509 090308 0503 0906 0903006 ﾀ 003 縺0 0004 0060304 縺06- 閨06000600? 005. 7 0805 ｧ 06050003 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 003 縺0 00 ｧ 09 04030406 縺08003 ﾀ05 ｧ x + y, xz, yz =0040503 0906 (0; ; ). 006. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 003 縺0 00 ｧ 09 04030406 縺08003 ﾀ05 ｧ x z+y z =4 040503 0906 (,; 0; ). 703 ﾀ05 縺03 ｧ0504 003 縺0 0004 ｧ 04030406 縺08003 ﾀ0504. 007. 70309 0504, 0503 09 ﾀ 0506000400306 040003 ﾀ0903 ﾀ05 ｧ 09 04030406 縺08003 ﾀ05 ｧ xyz = = a 0303 060605 ﾀ 040003 ﾀ0903 ﾀ05070 ｧ 090303 罍陦ｧ0 05 04 ｧ 0 ｧ 閨03 0403 ﾀ05030700030503 03030060. 008. 70309 0504, 0503 ﾀ0600 0904 陦 050304 0305 縺06 090304, 0305 ﾀ0609 0600308 0 03 ﾀ0708 090303 罍陦ｧ0 05 040003 ﾀ0903 ﾀ050406, 09 ﾀ 0506000400308 09 04030406 縺08003 ﾀ05 ｧ x =3 + + y =3 + z =3 = a =3, 040 0403 ﾀ050307000308 040600 ｧ ｧ006 a. 009. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0 00 090303 罍陦ｧ0 05 030509 ﾀ 0506000400308 040003 ﾀ- 0903 ﾀ05 ｧ 09 050600 ｧ0903 ｧ 閨06 y = x tg z 040503 0906 (a; a; a=4). 703 ﾀ05 縺03 ｧ0504 0403a 0406 縺08003 ﾀ0504 0403 ﾀ06 060 ｧ070: z =0;a=4; a=; a. 00. 7 04 ｧ ﾀ 0504 06 040060 ｧ06 09 ﾀ 0506000400308 040003 ﾀ0903 ﾀ05 ｧ 09 04030406 縺08003- ﾀ05 ｧ az = x + y 040503 09 08 0406 縺06 ﾀ06 060 ｧ07 0606 ﾀ 04 縺0700308 x = y = z. 0. 70309 0504, 0503 09 ﾀ 050600040 07 040003 ﾀ0903 ﾀ0504 09 04030406 縺08003 ﾀ05 ｧ x a + + y b + z c =040503 0906 0 00608 (x 0; y 0 ; z 0 ) 0304 縺06 閨0600070605 ﾀ07 06 040060 ｧ060 xx 0 a + yy 0 b + zz 0 c =: 0. 7 04 ｧ ﾀ 0504 06 040060 ｧ07 003 縺0 00 ｧ 09 04030406 縺08003 ﾀ05 ｧ x + y,,(z, 5) =0040503 0906 (4; 3; 0). 703 ﾀ05 縺03 ｧ0504 04 0406 縺04030 030905 00506 0403-0406 縺08003 ﾀ0504 ｧ 003 縺0 0004. 03. 7 0805 ｧ 06050003 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 003 縺0 00 ｧ 09 04030406 縺08003 ﾀ05 ｧ z = x, y 040503 0906 (; ; 0). 04. 7 0805 ｧ ﾀ ﾀ0503070 ｧ06 0 00 090303 罍陦ｧ0 05 030509 ﾀ 0506000400308 040003 ﾀ- 0903 ﾀ05 ｧ 09 0903003 ｧ 閨06 (a, z )x, a y =0040503 0906 (a; a; a).

3x. ｮ09 0007 縺00306 04030006. 6ｧ8 ｧ0 ｧ ｧ ｧ 04030406 縺08003 ﾀ05 ｧ 06 縺030400608 03 05. 70309 0504, 0503 ﾀ0600 0305 縺06 090304, 0305 ﾀ0609 0600308 0 03 ﾀ0708 090303 - 陦ｧ0 05 040003 ﾀ0903 ﾀ050406, 09 ﾀ 0506000400308 09 04030406 縺08003 ﾀ05 ｧ x = + y = + z = = = a =, 040 0403 ﾀ050307000308 040600 ｧ ｧ006 a. 06. 6 509 090308 0503 0906 09 ﾀ 050600040 07 040003 ﾀ0903 ﾀ0504 09 04030406 縺08003 ﾀ05 ｧ z = =4, x, y 04 00000600040 : ) 040003 ﾀ0903 ﾀ05 ｧ xoy; ) 040003 ﾀ0903 ﾀ05 ｧ x + +y + z =0? 7 04 ｧ ﾀ 0504 06 040060 ｧ07 0505 ｧ08 09 ﾀ 0506000400308 040003 ﾀ0903 ﾀ050608. x. ｮ09 0007 縺00306 04030006. 6ｧ8 ｧ0 ｧ ｧ ｧ 04030406 縺08003 ﾀ05 ｧ 06 縺030400608. 7 縺03 ｧ 0403 閨0 07 04 00030 0 04 0400060 ｧ ｧ. 06 陦ｧ06005 ｰ 040060 ｧ06 u = F (x; y) 0304 縺06 閨0600070605 u 0409 07 閨0308 0503 0906 (x; y) 00609030503 縺0308 030300 ﾀ05 ｧ, 09030503 07 0 0304 0605 ﾀ07 040300060 ﾀ09 0007 u. 6 5 閨030004 09 07 閨0308 ｧ 00 ｧ0 ｧ08 F (x; y) =u, F (x; y) =u, :::,05 閨06 u, u, ::: 0403 ﾀ050307000306, ﾀ09 0007 u 03 ﾀ05 0605 ﾀ07 0403 ﾀ05030700030 ｧ 0060070605 ﾀ07 050300040903 04 罍ｧ 0406 縺060803 閨06 0503 09 ｧ (x; y) ﾀ 03-00308 00 ｧ0 ｧ ｧ 0 陦縺06050606. 705 ｧ 00 ｧ0 ｧ ｧ 0 0304 0605 ﾀ07 ｧ 0300 ｧ0 ｧ070 ｧ ( ｧ 030506 縺0 0 ｧ, ｧ 0303 0 ｧ ｧ 05. 04.) ｧ00 ｧ 00 ｧ0 ｧ070 ｧ 06 縺030400608. ｰ 040060 ｧ06 u = F (x; y; z) 0304 縺06 閨0600070605 04030006 ﾀ09 0007 u 04 00609030503 縺0308 ﾀ05 ｧ 05 縺0608006 縺0030503 04 縺03 ﾀ05 0 ﾀ0504. 0ｷ3 0304030406 縺08003 ﾀ05070 ｧ, ｧ00 ｧ 04030406 縺08003- ﾀ05070 ｧ 06 縺030400608 0306 閨0605 F (x; y; z) =u ; F (x; y; z) =u ; ::: 706 ﾀ0504 0503 09 (x; y; z) 0406 縺060060 0605 ﾀ07 0403 04 縺0700308 x = x 0 + l cos, y = = y 0 + l cos, z = z 0 + l cos ﾀ03 ﾀ0903 縺03 ﾀ050406 dl =. ｯ0305 ﾀ09 0007 dt u = F (x; y; z) 0306 閨0605 ｧ 0060070504 ﾀ07 ﾀ03 ﾀ0903 縺03 ﾀ050406 v = du dt = du dl = @F @F @F cos + cos + @x @y @z cos = N l 0; @F 05 閨06 N @x ; @F @y ; @F 003 縺0 000400308 0406090503 ｧ 0304030406 縺08003 ﾀ05 ｧ, @z l 0 fcos ; cos ; cos g 06 陦ｧ0 ｧ 00308 0406090503 0 04 0400060 ｧ07 l. 7 縺03 ｧ 0403 閨0 07 du dl = @F @F @F cos + cos + @x @y @z cos = N l 0 0 0304 0605 ﾀ07 04 縺03 ｧ 0403 閨00308 0305 070600909 ｧ ｧ u = F (x; y; z) 04 00030 0 04-0400060 ｧ ｧ l 0 fcos ; cos ; cos g. 06 陦ｧ06005030 ﾀ09 0007 u = F (x; y; z) 0 0304 0605 ﾀ07 0406090503 grad u = = @u @x i + @u @y j + @u k. 06 陦ｧ06005 06 ﾀ0504 0406090503 ﾀ0903 縺03 ﾀ05 ｧ 0 ｧ0303 ﾀ05 縺060800060503 @z ｧ 0060060 ｧ07 ﾀ09 0007 u. 07. 706 ﾀ0504 z =4, x, y. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ 06 縺030400608 ｧ grad z 040503 0906 A(; ).

304 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 08. 706 ﾀ0504 z = arctg y. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ 06 縺030400608 ｧ grad z: x ) 04 0006030308 0503 0906 04 縺0700308 y = x; ) 04 0006030308 0503 0906 04 縺0700308 y =,x, 04 ﾀ05003 ﾀ05 ｧ 040503 09 08 (=; =), (; ), ::: 09. 0603 罍ｧ 03005 00 ｧ 0403 040300060003 ﾀ05 ｧ 0304 縺06 閨0600070605 ﾀ07 06 040060 ｧ060 h =0, x 4, y. 703 ﾀ05 縺03 ｧ0504 0503 罍ｧ 03005 00 ｧ, ﾀ030305040605 ﾀ050406060 ｧ06 0305-0060509 0 h = 00, 9 0, 8 0, 6 0 ｧ 0. 7 04 0400060 ｧ06 grad h 0304 縺06 閨0600070605 閨06 ﾀ04 0 04 0400060 ｧ06 00 ｧ0 ｧ ｧ 0 ｧ0303000606 09 縺0605030503 ﾀ09 05, 0406-00 ｧ ｧ0 09 縺0605 ｧ 006 0505030503 ﾀ09 05 0403 040300060003 ﾀ05 ｧ. 703 ﾀ05 縺03 ｧ0504 grad h 040503 0906 x = ｧ y =. 00. 7 0805 ｧ 0 ｧ03030004000606 09 縺0605 ｧ 006 04030406 縺08003 ﾀ05 ｧ z = xy 04 0503 0906 (4; ). 0. 7 0805 ｧ 04 縺03 ｧ 0403 閨00606 070600909 ｧ ｧ u = ln(e x + e y ) 04 0 04-0400060 ｧ ｧ, 04 00000600040030 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ06 090303 罍陦ｧ0 050030503 060500. 0. 7 0805 ｧ 04 縺03 ｧ 0403 閨00606 070600909 ｧ ｧ u = x + y + z 040503 0906 (; ; ) 04 0 04 0400060 ｧ ｧ lfcos 45 ; cos 60 ; cos 60 g ｧ 0 0805 ｧ grad u 04 050308 0706 0503 0906 ｧ 060503 閨00 ｧ006. 703 ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ05 ｧ 06 縺030400608. 03. 703 ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ05 ｧ 06 縺030400608 ﾀ09 0007 u = x + y,, z ｧ 0 0805 ｧ ｧ 0403 ﾀ05 縺03 ｧ0504 grad u 040503 09 08 0406 縺06 ﾀ06 060 ｧ07 03 ﾀ ｧ Ox ﾀ 04030406 縺08003 ﾀ050406 u =4. 04. 7 0805 ｧ 04 縺03 ｧ 0403 閨00606 070600909 ｧ ｧ u = x a + y b + z c 040503 0906 (a; b; c) 04 0 04 0400060 ｧ ｧ 陦ｧ06 ﾀ-0406090503 05050308 0503 09 ｧ. 4 05. 706 ﾀ0504 z =. 703 ﾀ05 縺03 ｧ0504 00 ｧ0 ｧ ｧ 06 縺030400608 ｧ grad x z 04 + y 0503 0906 (,; ) ｧ 0 0805 ｧ jgrad zj. 06. 706 ﾀ0504 u = xyz. 7 0805 ｧ 04 縺03 ｧ 0403 閨00606 du 04 0 04 0400060 ｧ ｧ, dl ﾀ03 ﾀ05 040007060060 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 0400306 06050003, 04 0006030308 0503 0906 ｧ 040503 0906 (; ; ). 07. 703 ﾀ05 縺03 ｧ0504 04030406 縺08003 ﾀ05 ｧ 06 縺030400608 ﾀ09 0007 u = x +y, z, 0304 縺06 閨0600 ｧ0504 grad u 0 04030406 縺08003 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0 0003 090303 - 陦ｧ0 05, ｧ 0403 ﾀ05 縺03 ｧ0504 060503 04 050608 0503 09 08 05050308 04030406 縺08003 ﾀ05 ｧ, 04 09030503 縺0308 y =0 ｧ z =. 08. 706 ﾀ0504 u = x + y + z. 7 0805 ｧ grad u ｧ 060503 閨00 ｧ006. 09. 703 ﾀ05 縺03 ｧ0504 ｧ 0304030406 縺08003 ﾀ05 ｧ 04030007 070600909 ｧ ｧ u = z c, x a, y b ｧ 0 0805 ｧ 04 縺03 ｧ 0403 閨00606 0305 u 040503 0906 (a; b; c) 04 0 04 0400060 ｧ ｧ 陦ｧ06 ﾀ- 0406090503 05050308 0503 09 ｧ.

3x. 709 ﾀ05 縺060060 070600909 ｧ ｧ 閨040608 0406 縺06006000308 05 x. 709 ﾀ05 縺060060 070600909 ｧ ｧ 閨040608 0406 縺06006000308. 70603030803 陦ｧ00306 06 ﾀ000304 ｧ07. ｱ0600909 ｧ07 z = F (x; y) 003070605 ｧ0060504 0509 ﾀ05 縺060060 050300040903 04 0503 09 08, 04 09030503 縺0308 @F @F =0 ｧ =0. 705 ｧ0503 09 ｧ @x @y 0 0304 0605 ﾀ07 09 罍ｧ05 ｧ 06 ﾀ09 ｧ0 ｧ..6 703 ﾀ05 0503 00306 06 ﾀ000304 ｧ07. 050303 0 ｧ0 06 縺06 A, B ｧ C 0 060 ｧ07 04 縺03 ｧ 0403 閨00308 @ F @x, @ F @x@y ｧ @ F @y 0409 罍ｧ05 ｧ 06 ﾀ090308 0503 0906 (x 0; y 0 ). ｯ0305, 06 ﾀ00 ｧ: ) B A B C > 0, 0503 F (x 0 ;y 0 )=z max 04 罍ｧ A<0, F (x 0 ;y 0 )=z min 04 罍ｧ A>0; ) B A B C < 0, 0503 0509 ﾀ05 縺060060 00605; 3) B A B C =0, 0503 0509 ﾀ05 縺060060 003070605 03030504, 003070605 ｧ 006 03030504 ( ﾀ0300 ｧ- 0506000400308 ﾀ0006 08). 3. ｰ ﾀ00030400308 0509 ﾀ05 縺060060. ｪ05030303 0 0805 ｧ 0509 ﾀ05 縺060060 070600909 ｧ ｧ z = F (x; y) 04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 x ｧ y ﾀ0407 003 06 040060 ｧ060 '(x; y) = 0, ﾀ03 ﾀ05 04 ｧ0 04 ﾀ040300305 0506000400606 070600909 ｧ06 u = F (x; y) +'(x; y). 070303 罍陦ｧ0 0503 0509 ﾀ05 縺060 000400308 0503 09 ｧ (x; y) 閨030007003 06 閨03040006050403 罍ｧ0504 05 縺060 @u 06 040060 ｧ070: '(x; y) =0, @x =0, @u =0, ｧ 09030503 縺0308 ｧ 0 0803 閨0705 ﾀ07, @y x ｧ y. 7 0805 ｧ 0509 ﾀ05 縺060060 070600909 ｧ ｧ: 030. z = x, xy + y +9x, 6y +0. 03. z = y x, y, x +6y. 03. z = x 3 +8y 3, 6xy +. 033. z =xy, 4x, y: 034. z = e x= (x + y ). 035. z = sin x + sin y + sin (x + y) 04 罍ｧ 0 6 x 6 = ｧ 0 6 6 y 6 =. 036. z = x + 04 罍ｧ x + y =. y 037. z = x + y 04 罍ｧ x + y =. 038. 0504 縺06 閨0600 ｧ0504 006 縺03 04 縺0700306050300040030503 030509 縺0305030503 03 ﾀ ﾀ06080, ｧ006060060503 0 ｧ006004000606 04030406 縺08003 ﾀ0504 04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 060503 03030060 04060 V. 039. 703 ﾀ05 縺03 ｧ0504 050000 ｧ04 ﾀ x +4y =4 ｧ 04 縺0700606 x +3y, 6= =0 ｧ 0 050000 ｧ04 ﾀ06 0 0805 ｧ 0503 09 ｧ, 0 ｧ0303000606 ｧ 0 ｧ00600606 06 0006000306 0305 04 縺0700308. 040. 7 05 ｧ0406 縺03030006 x, y =40 0805 ｧ 0503 0906, 0 ｧ00600606 06 00060-00606 0305 0503 09 ｧ (0; ).

306 0600.. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 04. 0504 縺06 閨0600 ｧ0504 006 縺03 09 ｧ00 ｧ0 陦 0 ｧ0303000400060503 03030060 04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 060503 0403000 07 04030406 縺08003 ﾀ0504 040 S =6 0. 04. ) 6 5 050000 ｧ04 ﾀ x +3y =0404 ｧ ﾀ 0504 040030306 陦縺06000308 05 縺06-06050300040 ｧ09 ﾀ 03 ﾀ00304 0 ｧ060, 04 00000600040030 03030004000308 03 ﾀ ｧ, 05 09, 05030303 0400030 閨04 05 縺0606050300040 ｧ09 030300 0 ｧ03030004000608. ) 05 ﾀ04 Ox ﾀ0403000307060 0 05 0 ｧ0906 閨040608 ﾀ 縺06. 703 09 0903006 040605 ｧ 閨030007060 04 縺030805 ｧ 0006 ﾀ040605 ｧ 0503 09 ｧ A(0; a) 04 0503 0906 B(c;,b), 05030303 05 05 ｧ0504 0 04 縺03080307 閨060 ｧ06 0505030503 ﾀ ﾀ0503070 ｧ07 0 ｧ006004000606 04 縺06007 (a >0, b>0, c>0)? a ｰ09 0 ｧ06. 70607003 0 0805 ｧ 0 ｧ0 ｧ0060070600909 ｧ ｧ T = v cos + b v cos 04 罍ｧ 06 ﾀ000304 ｧ ｧ a tg +b tg = c, 05 閨06 v ｧ v ﾀ0903 縺03 ﾀ05 ｧ ﾀ040605 04 閨040608 ﾀ 縺06 08, ｧ 06050003 04 閨060 ｧ07 ｧ 04 縺060003000060 ｧ07. 7 0805 ｧ 0509 ﾀ05 縺06006003 070600909 ｧ08: 043. z =3x +6y, x, xy, y. 044. z = x + y, x, 4 xy, y +8. 045. z =x 3, xy +5x + y. 046. z =3x, x y + y, 8x +8. 047. z = xy 04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 x + y =. 048. 7 0805 ｧ 0 ｧ0303000400 ｧ08 03030060 04 縺0700306050300040030503 04 000006000604 ｧ- 0406 04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 閨00 ｧ0 060503 陦ｧ 05030 00 ｧ 040 3. 049. ) 7 04 03030006 y =4x 0 0805 ｧ 0503 0906, 0 ｧ00600606 06 0006000606 0305 04 縺0700308 x, y +4=0. ) 6 5 050000 ｧ04 ﾀ x a + y =0404 ｧ ﾀ 0 04 縺0700306050300040 ｧ09 0 ｧ03030004000608 b 0400030 陦ｧ. 7 0805 ｧ 050506 0400030 閨04. 050. 0504 縺06 閨0600 ｧ0504 006 縺03 0903006 ﾀ 0 ｧ0303000400060503 03030060 04 罍ｧ 06 ﾀ0003-04 ｧ ｧ, 0503 060503 0303090304 07 04030406 縺08003 ﾀ0504 040 S.

30600 04 6 70ｷ3 ｱ ｱ ｭ 7 ｭ 7ｨC0ｷ3 縺6ｧ804 76ｧ9 ｭ ｰ 7 縺6 5 7 ｭ 70ｷ300 x. 70300705 ｧ06 03 陦ｧ070706 縺06009 ｧ 00040030 06 040060 ｧ ｧ. 05030309003040600030 陦ｧ070706 縺06009 ｧ 00040030 06 040060 ｧ060 n -0503 0403 縺07 閨09 0 0304 0605 ﾀ07 06 040060 ｧ06 04 ｧ F (x; y; y 0 ;y 00 ; :::;y (n) )=0; () 05 閨06 y = y(x) ｧ ﾀ09030 07 070600909 ｧ07. 6ｧ80603 07 070600909 ｧ07 y = '(x), 0303 0 06-0 07 06 040060 ｧ06 () 04 050307 閨06 ﾀ050403, 0 0304 0605 ﾀ07 縺0600060 ｧ060 0505030503 06 040060 ｧ07, 05 07 ｧ09 05050308 070600909 ｧ ｧ ｧ0050605 000400308 09 罍ｧ040308. ｭ ﾀ00 ｧ 縺0600060 ｧ06-003 04 00607040030 04 ｧ 閨06 (x; y) =0, 0503 03003 030303 003 0 0304 0605 ﾀ07 ｧ0050605 00030 06 040060 ｧ07 (). ｱ0600909 ｧ07 y = '(x; C ; :::;C n ), ﾀ03 閨06 縺07 0 07 n 006 04 ｧ ﾀ ｧ00308 04 縺03 ｧ - 0403000400308 0403 ﾀ050307000308, 0 0304 0605 ﾀ07 03030 ｧ0 縺0600060 ｧ060 06 040060 ｧ07 (), 06 ﾀ00 ｧ 030 070400070605 ﾀ07 060503 縺0600060 ｧ060 04 罍ｧ 0006030308 0 060 ｧ0708 0403 ﾀ050307000308 C, ::: :::, C n. ｭ ﾀ00 ｧ 0505 070600909 ｧ07 0605 ﾀ07 04 00607040030 04 ｧ 閨06 0403 07060 ｧ060 (x; y; C ; :::;C n ) = 0, 0503 050503 0403 07060 ｧ06 0 0304 0605 ﾀ07 03030 ｧ0 ｧ00506-05 00030 06 040060 ｧ07 (). 7 罍ｧ 04 07 04 0403 07060 ｧ ｧ y = '(x; C ; :::;C n ) ｧ00 ｧ 04 0403 07060 ｧ ｧ (x; y; C ; :::;C n ) = 00304 縺06 閨060006000306 0 060 ｧ07 0403- ﾀ05030700030 C, :::, C n, 04030006 060 ﾀ0500306 縺0600060 ｧ06 ｧ00 ｧ ﾀ030305040605 ﾀ0504060003 ﾀ0500308 ｧ0050605 00 06 040060 ｧ07 (). 0503 05003, ｧ00607 ﾀ0600608 ﾀ050403 09 罍ｧ040308, 04 0600308 06 040060 ｧ060 (x; y; C ; :::;C n )=0, ｧ ｧ ﾀ090006 07 04 00605 縺03 C, :::, C n ｧ ﾀ ｧ ﾀ0506003 06 040060 ｧ08 d =0; =0; :::; dn n =0; 04030006 ｧ0, 04030303006 05030403 縺07, 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 04 ｧ (), 閨0007 0903-0503 縺030503 (x; y; C ; :::;C n )=0070400070605 ﾀ07 03030 ｧ0 ｧ0050605 00030.. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 0406 縺04030503 0403 縺07 閨09 ｧ0060605 04 ｧ F x; y; dy =0: () 70600 ｧ04 06 040060 ｧ06 () 0305003 ﾀ ｧ05060004003 dy, 06 ﾀ00 ｧ 050503 0403 00307003, 04030006 ｧ0 dy = f (x; y): (3)

308 0600.. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 ｰ 040060 ｧ06 (3) 0304 縺06 閨0600070605 0 0900030 k = tg dy = = f(x; y) ｧ00506 05 000400308 09 罍ｧ040308 04 0503 0906 (x; y), 05. 06. 0304 縺06 閨0600070605 04030006 0 04 0400060 ｧ08 ｧ0-050605 000400308 09 罍ｧ040308. ｭ ﾀ00 ｧ 04 00609030503 縺0308 030300 ﾀ05 ｧ 070600909 ｧ07 f(x; y) 00604 縺06 縺03040 ｧ ｧ0060605 0305-0 ｧ 06000606 ﾀ0500606 04 縺03 ｧ 0403 閨00606 fy 0 (x; y), 05030309 0304 0605 ﾀ07, 0503 06 縺06 09 07- 閨0606 0400605 縺06000606 0503 0906 (x 0 ; y 0 ) 05050308 030300 ﾀ05 ｧ 04 縺0308 閨0605 06 陦ｧ0 ﾀ05040600 07 ｧ0-050605 00040 07 09 罍ｧ04 07. 6 505 090308 030300 ﾀ05 ｧ 06 040060 ｧ06 (3) ｧ0060605 030300606 縺0600060 ｧ06 y = '(x; C), ｧ00 ｧ 03030 ｧ08 ｧ0050605 00 (x; y; C) =0, ｧ 09030503 縺030503 00307003 0 0805 ｧ 06 陦ｧ0 ﾀ05-0406000306 ﾀ0500306 縺0600060 ｧ06, ｧ00 ｧ 06 陦ｧ0 ﾀ050406000308 ﾀ0500308 ｧ0050605 00, 06 閨03040006-050403 縺07060 ｧ06 0 00040030 06 ﾀ000304 ｧ070: y = y 0 04 罍ｧ x = x 0. 05. 7 縺030406 罍ｧ0504 0403 陦ﾀ05 00304090308, 0503 070600909 ｧ07 y = Cx 3 070400070605 ﾀ07 縺0600060 ｧ060 陦ｧ070706 縺06009 ｧ 00040030503 06 040060 ｧ07 3y, xy 0 =0. 703 ﾀ05 縺03 ｧ0504 ｧ0050605 000400306 09 罍ｧ040306, 04 縺030803 閨070 ｧ06 06 縺06 0503 09 ｧ: ) (; =3); ) (; ); 3) (;,=3). 05. 7 縺030406 罍ｧ0504 0403 陦ﾀ05 00304090308, 0503 陦ｧ070706 縺06009 ｧ 000400306 06 04006-0 ｧ07 ) y 00 +4y =0 ｧ)y 000, 9y 0 =0 ｧ0060605 ﾀ030305040605 ﾀ0504060003 03030 ｧ06 縺0600060 ｧ07: ) y = C cos x+c sin x ｧ)y = C +C e 3x +C 3 e,3x. 053. 703 ﾀ05 縺03 ｧ0504 04 03030003 y = Cx 04 罍ｧ C = 0; ; ｧ ﾀ03- ﾀ05 04 ｧ0504 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 ﾀ0600608 ﾀ0504 05 09 ｧ08 04 030300. 054. 703 ﾀ05 縺03 ｧ0504 ｧ 0303 07060 ｧ07 ﾀ0600608 ﾀ0504 : ) 0309 縺0607003 ﾀ050608 x + + y =Cx, ) 04 030300 y = x +Cx ｧ ﾀ03 ﾀ05 04 ｧ0504 ｧ08 陦ｧ070706 縺060-09 ｧ 000400306 06 040060 ｧ07. 055. 703 ﾀ05 縺03 ｧ0504 ｧ 0303 07060 ｧ07 0403000608 0 04 0400060 ｧ08, 0304 縺06 閨06000706-00308 09 07 閨030 ｧ 06 040060 ｧ08: dy ) dy 3) = y + x. ) dy y = x ; = y, x; 056. 703 ﾀ05 縺03 ｧ0504 ｧ 0303 07060 ｧ06 04030007 0 04 0400060 ｧ08, 0304 縺06 閨06000706-0030503 06 040060 ｧ060 dy = x + y, ﾀ 040300300406 0309 縺0607003 ﾀ050608, 04 閨030004 09030503 縺0308 dy ; ; ; 3;::: 7 罍ｧ ﾀ0304 0504 04 罍ｧ0300 ｧ07060003 ｧ0050605 0004- = 00606 09 罍ｧ040606, 04 縺030803 閨0700606 06 縺06 0 0003 090303 罍陦ｧ0 05. x. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 0406 縺04030503 0403 縺07 閨09 ﾀ 閨060007060 ｧ0 ｧ ﾀ07 0406 縺0600600030 ｧ. 05 縺050305030 000400306 05 06090503 罍ｧ ｧ. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 0406 縺04030503 0403 縺07 閨09 P+ Qdy =0; () 05 閨06 P ｧ Q 070600909 ｧ ｧ x ｧ y, 0 0304 0605 ﾀ07 06 040060 ｧ060 ﾀ 閨060007060 ｧ0 ｧ ﾀ07 0406 縺0600600030 ｧ, 06 ﾀ00 ｧ 0903050707 ｧ09 ｧ0600503 P ｧ Q 04 罍ｧ 陦ｧ070706 縺06009 ｧ 00 08 00 05-0605 ﾀ07 0 000307 ｧ050600 ｧ, 04 ｧ ﾀ070 ｧ06 050300040903 0305 x ｧ00 ｧ 050300040903 0305 y, 05. 06. 06 ﾀ00 ｧ

3x. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 0406 縺04030503 0403 縺07 閨09 09 03003 ｧ0060605 04 ｧ f (x)'(y) + f (x)' (y) dy =0: () 7 閨0600 ｧ04 0303 00060 06 040060 ｧ07 () 0 '(y)f (x), 04030006 ｧ0 f(x) f (x) + ' (y) dy '(y) =0: (3) 05030 ｧ0 ｧ0050605 00030 06 040060 ｧ07 (3), ﾀ0006 閨0304 05060004003, ｧ () 0306 閨0605 f (x) + ' (y) dy = C: (4) '(y) f (x). 05 縺050305030 00040030 ｧ 05 06090503 罍ｧ070 ｧ ﾀ0600608 ﾀ0504 00 ｧ0 ｧ08 F (x; y; a) =00 0304 0605 ﾀ07 00 ｧ0 ｧ ｧ, 0406 縺06 ﾀ0609 060 ｧ06 00 ｧ0 ｧ ｧ 00030503 ﾀ0600608- ﾀ0504 0403 04 縺070030 060500030. 7 縺03 陦ｧ070706 縺06009 ｧ 縺0304 04 06 040060 ｧ06 F (x; y; a) =0 0403 x ｧ ｧ ﾀ090006 ｧ04 a ｧ 04030006 0600030503 ｧ 00030503 06 040060 ｧ08, 04030006 ｧ0 陦ｧ07-0706 縺06009 ｧ 000400306 06 040060 ｧ06 00 ｧ0 ｧ08 00030503 ﾀ0600608 ﾀ0504 y 0 = f (x; y). ｯ03-05 陦ｧ070706 縺06009 ｧ 00040030 06 040060 ｧ060 03 縺050305030 000400308 05 06090503 罍ｧ08 0306 閨0605 y 0 =, f(x; y). 6 5 ﾀ0006 閨06060 ｧ08 陦ｧ070706 縺06009 ｧ 000400308 06 040060 ｧ0708: ) 0 0805 ｧ 0303-0 ｧ08 ｧ0050605 00; ) 0403 ﾀ05 縺03 ｧ0504 006 ﾀ090300040903 ｧ0050605 000400308 09 罍ｧ040308; 3) 0 0805 ｧ ﾀ0500308 ｧ0050605 00 0403 0 00040030 06 ﾀ000304 ｧ070 y = 404 罍ｧ x =,: 057. xy 0, y =0: 058. xy 0 + y =0: 059. yy 0 + x =0: 060. y 0 = y: 7 0805 ｧ 03030 ｧ06 ｧ0050605 0003 06 040060 ｧ08: 06. x y 0 + y =0: 06. x + xy + y 0 (y + xy) =0. 063. ' dr +(r, a) d' =0. 064. st ds =(+t ) dt. 6 5 ﾀ0006 閨06060 ｧ08 06 040060 ｧ0708 0 0805 ｧ 03030 ｧ08 ｧ ﾀ0500308 ｧ0050605 0003 0403 0 00040030 06 ﾀ000304 ｧ070: 065. y 0 x = y, y =04 罍ｧ x =4. 066. y 0 =(y + ) ctg x, y == 04 罍ｧ x = =4. 067. x y 0 + y =0, y =04 罍ｧ x =,. 068. 703 ﾀ05 縺03 ｧ0504 ｧ0050605 000400306 09 罍ｧ040306 09 07 閨030503 ｧ 06 040060 ｧ08: ) y 0 (x, 4) = xy; ) y 0 + y tg x = 0,04 縺030803 閨070 ｧ06 06 縺06 0503 09 ｧ: ) (0; ); ) (0; =); 3) (0;,=); 4) (0;,). 069. 7 0805 ｧ 09 罍ｧ040606, 04 縺030803 閨0700606 06 縺06 0503 0906 (; =3), 06 ﾀ00 ｧ 06050003040308 0903050707 ｧ09 ｧ06005 09 ﾀ 0506000400308 09 00608 04 0006030308 0503 0906 09 罍ｧ040308 0405 縺0306 030300040006 0605000304030503 0903050707 ｧ09 ｧ06005 陦ｧ06 ﾀ-0406090503 0503 09 ｧ 09 ﾀ - 0 ｧ07.

30 0600.. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 070. 07 罍ｧ04 07 04 縺030803 陦ｧ05 06 縺06 0503 0906 A(0; a), MN 04 縺03 ｧ 04030004-0 07 03 罍陦ｧ0 05 09 罍ｧ040308. 0504 縺06 閨0600 ｧ050409 罍ｧ040606 ｧ 06 ﾀ000304 ｧ07, 0503 0400030 閨04 OAMN = as, 05 閨06 s 閨00 ｧ0 閨0605 ｧ AM. 07. 7 0805 ｧ 09 罍ｧ040606, 04 縺030803 閨0700606 06 縺06 0503 0906 (a; a), 06 ﾀ00 ｧ 0403-09 ﾀ 050600040 07 04 0006030308 0503 0906 0606 040 06 閨040306000308 03 ﾀ09 ｧ ﾀ ﾀ06 0503 09 ｧ 09 - ﾀ 0 ｧ07. 07. 7 0805 ｧ 09 罍ｧ040606, 04 縺030803 閨0700606 06 縺06 0503 0906 (,;,), 06 ﾀ00 ｧ 0403 閨003 縺0 0004 0606 04 09 07 閨0308 0503 0906 040. 073. 6ｦ0 09 090306 04 縺06007 05060003, 0 05 縺06050306 閨03 00 C, 030800 陦ｧ05 ﾀ07 閨03 5 C 04 090300 0506 ﾀ 050600406 0506 縺0308 0 C, 06 ﾀ00 ｧ 閨03 60 C 03003 030800 07-0605 ﾀ07 0 0 ｧ0? ( 703 0903006 7040605030 ﾀ0903 縺03 ﾀ0504 030800 07 閨060 ｧ07 04 縺030403-09 ｧ030 00040 003 ﾀ05 ｧ 050600406 0506.) 074. 7 05 縺06 09 0 09 0 05 04 ｧ ﾀ07 060503 003 ﾀ05 ( ﾀ0. 罍ｧ ﾀ. 6) 0305 09 07 閨0308 06 陦ｧ0 ｧ0903 閨00 ｧ003 0503 罍ｧ 03005 000400308 03 0009 ｧ 040. 7 縺0600603 縺0605 07 0406 ﾀ030 09 0 05, 0 0805 ｧ 060503 0703 縺006, 06 ﾀ00 ｧ 0 050707060 ｧ06 09 0 05 040 ｧ 000608 0503 0906 04 罍ｧ0070504 H. ｰ09 0 ｧ06. 6 503 040060 0 閨060506 OC ^ ( 罍ｧ ﾀ. 6) 04 縺03 ｧ 0403000400606 0503 0906 M. 7 ﾀ0504 09 0 05 OM 0306 閨0605 閨0608 ﾀ05040304 0504 05 罍ｧ ﾀ ｧ0003: 0503 罍ｧ 03005 00040 07 H (0400060403 0305 0503 09 ｧ M), 0406 縺05 ｧ09 00040 07 0406 ﾀ x ｧ05 00506009 ｧ 00040 07 ﾀ ｧ00 0-050707060 ｧ07 T (0404 0403 0305 0503 09 ｧ M). 6 70007 040030406 ﾀ ｧ07 ﾀ0600 04 縺03060909 ｧ08 ﾀ ｧ00 0 Ox ｧ Oy 閨0300070 040070504 ﾀ07 0. 075. 0504 縺06 閨0600 ｧ0504 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606, 04 縺030803 閨0700606 06 縺06 0503 0906 P (,a; a), 06 ﾀ00 ｧ 0305 縺06 0309 AB 0006030308 09 ﾀ 0506000400308 09 00608, 090006 060-00308 00607 閨06 03 ﾀ070 ｧ 090303 罍陦ｧ0 05, 閨0600 ｧ05 ﾀ07 0503 090308 09 ﾀ 0 ｧ07 M 0403040300 0. 076. 7 0805 ｧ03 縺050305030 000400306 05 06090503 罍ｧ ｧ ﾀ0600608 ﾀ0504 04 030300 ay = = x. 703 ﾀ05 縺03 ｧ0504 ｧ08. 077. 7 0805 ｧ 03 縺050305030 000400306 05 06090503 罍ｧ ｧ ﾀ0600608 ﾀ0504 05 ｧ0406 縺030300 xy = c. 078. 7 0805 ｧ 03 縺050305030 000400306 05 06090503 罍ｧ ｧ ﾀ0600608 ﾀ0504 04030006090603 ｧ 06- ﾀ09 ｧ08 04 030300 ay = x 3. 079. 7 0805 ｧ 03 縺050305030 000400306 05 06090503 罍ｧ ｧ ﾀ0600608 ﾀ0504 050000 ｧ04 ﾀ0304 x +4y = a. 70600 ｧ0504 06 040060 ｧ07: 080. y 0 x 3 =y: 08. (x + x)y 0 =y +: 08. y 0 a + x = y: 083. ( + x )y 0 ++y =0: 084. dr + r tg 'd'=0; r =04 罍ｧ ' =. 085. y 0 = y ln x; y =04 罍ｧ x = e. 086. ( + x )y 0 + y +x = xy; y =04 罍ｧ x =0. 087. 0504 縺06 閨0600 ｧ0504 09 罍ｧ040606, 04 縺030803 閨0700606 06 縺06 0503 0906 A(,; ), 06 ﾀ00 ｧ 06050003040308 0903050707 ｧ09 ｧ06005 09 ﾀ 0506000400308 04 0006030308 0503 0906 09 罍ｧ040308-04060 0904 陦 0506 03 罍陦ｧ0 0503 0503 09 ｧ 09 ﾀ 0 ｧ07.

3x 3. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0406 縺04030503 0403 縺07 閨09 088. 07 罍ｧ04 07 04 縺030803 陦ｧ05 06 縺06 0503 0906 A(0; a), MN 04 縺03 ｧ 04030004-0 07 03 罍陦ｧ0 05 09 罍ｧ040308. 0504 縺06 閨0600 ｧ050409 罍ｧ040606 ｧ 06 ﾀ000304 ｧ07, 0503 0400030 閨04 OAMN = a(mn, a). 089. 0504 縺06 閨0600 ｧ0504 ｧ 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606, 04 縺030803 閨0700606 06 縺06 0503 0906 (,;,), 閨0007 09030503 縺0308 0305 縺06 0309 OT, 0305 ﾀ0609 0600308 0 03 ﾀ ｧ Ox 09 ﾀ - 0506000400308 09 09 罍ｧ040308 04 0006030308 0606 0503 0906, 04060 0904 陦 0506 03 ﾀ09 ｧ ﾀ ﾀ03 0503 09 ｧ 09 ﾀ 0 ｧ07. 090. 7 0805 ｧ03 縺050305030 000400306 05 06090503 罍ｧ ｧ ﾀ0600608 ﾀ0504 05 ｧ0406 縺030300 x,, y = a. 09. 0504 縺06 閨0600 ｧ0504 09 罍ｧ040606, 陦ｧ06 ﾀ-0406090503 0006030308 0503 09 ｧ 09030503 縺0308 04060 0305 縺06 0906 003 縺0 00 ｧ 00607 閨06 09 罍ｧ040308 ｧ 03 ﾀ0406 Ox. 09. 0504 縺06 閨0600 ｧ0504 00 ｧ0 ｧ06, 06 ﾀ00 ｧ 0400030 閨04, 0305 0 ｧ 0600 07 03 ﾀ070 ｧ 090303 罍陦ｧ0 05, 05050308 00 ｧ0 ｧ0608 ｧ 04 縺03 ｧ 0403000400308 0606 03 罍陦ｧ0 050308, 040 =3 0400030 陦ｧ 04 縺0700306050300040 ｧ09, 0403 ﾀ05 縺030600030503 0 090303 罍陦ｧ0 05 08 0903006 00308 0503 09 ｧ 09 罍ｧ040308. x 3. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0406 縺04030503 0403 縺07 閨09 : ) 03 閨003 縺03 閨00306, ) 00 ｧ0060800306, 3) 706 縺0060000 ｧ. 05 閨003 縺03 閨00306 06 040060 ｧ06. ｰ 040060 ｧ06 P+ Qdy =00 03-04 0605 ﾀ07 03 閨003 縺03 閨0030, 06 ﾀ00 ｧ P ｧ Q 03 閨003 縺03 閨00306 070600909 ｧ ｧ 0305 x ｧ y 03 陦ｧ- 0 090304030503 ｧ 006 縺060 ｧ07. 05003 04 罍ｧ0403 陦ｧ05 ﾀ07 09 04 ｧ 閨06 dy y = ' ｧ 縺0600 0605 ﾀ07 x 0403 陦ﾀ05 00304090308 y = u ｧ00 ｧ y = ux. x. 6ｧ8 ｧ0060800306 06 040060 ｧ06. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 0 03-04 0605 ﾀ07 00 ｧ006080030, 06 ﾀ00 ｧ 03003 0406 縺040308 ﾀ050604060 ｧ 0305003 ﾀ ｧ05060004003 ｧ ﾀ090300308 0706009-09 ｧ ｧ y ｧ 04 ﾀ0608 060604 縺03 ｧ 0403 閨00308. 6ｧ8 ｧ0060800306 06 040060 ｧ06 0406 縺04030503 0403 縺07 閨09 ｧ0060605 04 ｧ y 0 + Py = Q. 05003 ﾀ0403 陦ｧ05 ﾀ07 09 閨04060 06 040060 ｧ070 ﾀ 閨060007-060 ｧ0 ｧ ﾀ07 0406 縺0600600030 ｧ 0403 陦ﾀ05 00304090308 y = uv. 6 7 縺06050308 ﾀ0403 ﾀ0303 縺0600060 ｧ07 (04 罍ｧ 09 ｧ07 04 縺03 ｧ 0403000400308 0403 ﾀ050307000308) ﾀ03 ﾀ0503 ｧ05 04 05030, 0503 ﾀ0 00 縺06-00 060 06 040060 ｧ06 y 0 + Py = 0; 04030006 060 y =,Ae R, P. 703 陦ﾀ05 040007060 050503 縺0600060 ｧ06 04 000306 06 040060 ｧ06, ﾀ ｧ05 07 A 070600909 ｧ0608 x, ｧ 05060 0 0803 陦ｧ0 A 0 ｧ A. 3. ｰ 040060 ｧ06 706 縺0060000 ｧ y 0 + Py = Qy n 縺0600 0605 ﾀ07 05 09 0706, 09 09 ｧ 00 ｧ0060800306, 0403 陦ﾀ05 00304090308 y = uv ｧ00 ｧ 04 罍ｧ 09 ｧ0608 04 縺03 ｧ 0403000400308 0403- ﾀ050307000308. ｰ 040060 ｧ06 706 縺0060000 ｧ 04 罍ｧ0403 陦ｧ05 ﾀ07 09 00 ｧ00608003006 0403 陦ﾀ05 00304090308 z = y,n. 70600 ｧ0504 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07: 093. yy 0 =y, x: 094. x + y, xyy 0 =0: 095. ds dt = s t, t s : 096. y0, 3y x = x: 097. y 0 + y x = e,x x. 098. y 0 cos x, y sin x = sin x.

3 0600.. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 099. y 0 x + y =,xy : 00. y 0, xy =,y 3 e,x. 0. xy 0 cos y x = y cos y x, x. 0. x y 0 = y + xy: 03. xy 0 + y =lnx +. 04. x y y 0 + yx 3 =. 6 5 08 05{07 0 0805 ｧ ﾀ0500306 ｧ0050605 0003 0403 00030 0-00040030 06 ﾀ000304 ｧ070: 05. y + x + y, xy 0 =0; y =004 罍ｧ x =. 06. t ds dt 07. xy 0 = y =ts, 3; s =04 罍ｧ t =,. y +ln x ; y = e 04 罍ｧ x =. 08. 7 0805 ｧ ﾀ0600608 ﾀ050403 09 罍ｧ040308, 0403 閨09 ﾀ 050600040 07 04 0006030308 0503 0906 09030503 縺0308 06 ﾀ0504 ﾀ 縺06 閨00606 罍ｧ0700605 ｧ 06 ﾀ090306 090303 罍陦ｧ0 05 0503 09 ｧ 09 ﾀ 0 ｧ07. 09. 7 0805 ｧ 03 縺050305030 000400306 05 06090503 罍ｧ ｧ ﾀ0600608 ﾀ0504 0309 縺0607003 ﾀ050608 x + y =ax. 0. ｮ ｧ00 050309 i 04 090604 ｧ ﾀ ﾀ0304 縺0305 ｧ0400060 ｧ060 R, ﾀ 003 ｧ0 閨060909 ｧ0608 L ｧ 0500060905 縺03 閨04 ｧ070600608 ﾀ ｧ000308 E 06 閨03040006050403 縺070605 陦ｧ070706 縺06009 ｧ 0004003006 06 040060 ｧ06 L di + Ri = E: dt 70600 ｧ0504 050503 06 040060 ｧ06, ﾀ ｧ05 07 R ｧ L 0403 ﾀ05030700030 ｧ, 0500060905 縺03- 閨04 ｧ070600606 ﾀ ｧ0006 E 00 ｧ00608003 0 ﾀ05 0600608: E = kt. 7 000400306 06 ﾀ000304 ｧ07: i =004 罍ｧ t =0.. 7 0805 ｧ 0703 縺006 06 縺09 00, 0305 07 060060503 04 ﾀ06 0006 ｧ, 04030803 閨07-0 ｧ06 ｧ 000308 0503 09 ｧ, 04 0000060004003 0003006 0 04 0400060 ｧ06. ｰ09 0 ｧ06. 7 ﾀ ﾀ0 05 罍ｧ04 07 040003 ﾀ090306 ﾀ06 060 ｧ06 06 縺09 00, 04 罍ｧ0060 04 0060 000606 0503 0906 0 0003 090303 罍陦ｧ0 05, 000306 0 04 0400060 ｧ06 03 ﾀ04 Oy. 07 ﾀ 050600040 07 09 ｧ ﾀ090300308 09 罍ｧ040308 04 0503 0906 M 0303 060605 0400306 06050003 ﾀ OM ｧ 03 ﾀ0406 Oy, 05.06.0305 ﾀ0609 0605 0 03 ﾀ ｧOy 0305 縺06 0309 ON = OM. 70600 ｧ0504 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07:. xy + y =(x + xy)y 0 : 3. (a + x )y 0 + xy =: 4. xy 0 + xy = y: 5. (x +)y 0 + y = x: 6. y 0, y tg x = ctg x: 7. tds, sdt= t 3 ln t dt: 8. y 0 + xy = xy 3 : 9. y 0 + y cos x = sin x:, y ; y =04 罍ｧ x =,. x x. 3y y 0 + y 3 = x +; y =, 04 罍ｧ x =. 0. y 0 = y

3x 4. ｰ 040060 ｧ07 ﾀ 陦ｧ070706 縺06009 ｧ 00 0 ｧ 04 縺03 ｧ 0406 閨060 ｧ07 ｧ ﾀ050030503 3. (, x )y 0, xy = xy ; y =0; 5 04 罍ｧ x =0. 3. 0504 縺06 閨0600 ｧ0504 09 罍ｧ040606, 04 縺030803 閨0700606 06 縺06 0503 0906 A(a; a), 06 ﾀ00 ｧ ﾀ ﾀ0503070 ｧ06 03050 00 090303 罍陦ｧ0 05 閨03 09 ﾀ 0506000400308 04 0006030308 0503 0906 09 罍ｧ040308 04003 03 ﾀ09 ｧ ﾀ ﾀ06 05050308 0503 09 ｧ. x 4. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07, ﾀ03 閨06 縺07 0 ｧ06 陦ｧ070706 縺06009 ｧ 0003 04 縺03 ｧ 0406 閨060 ｧ07 ｧ ﾀ050030503 d(xy) =xdy+ y; d y x = xdy, y x ; d x y = y, xdy y : ｯ 09 ｧ06 06 040060 ｧ07 ｧ00305 0006050903 縺0600 0605 ﾀ07, 06 ﾀ00 ｧ ﾀ030305040605 ﾀ0504060003 04030003-07 ｧ0504 xy = u, y u = x ｧ00 ｧ y = u, y = ux. x 70600 ｧ0504 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07: 4. x dy + xy = : 5. y xdy, y 3 = x dy. ｰ 09 0 ｧ 06. 6 5 04 罍ｧ006 縺06 5 06 040060 ｧ06 04 罍ｧ0403 陦ｧ05 ﾀ07 09 04 ｧ 閨06 y d y x 6. y+(x, y 3 ) dy =0. = dy ｧ00 ｧ y du = dy: 7. y, (x, y 3 ) dy =0. 8. y cos x+ sin xdy = cos x. 9. t ds dt, s = s ln t: 30. x y ++x 3 yy 0 =0: 3. t sdt+ t 3 ds = dt: 3. xdy, y= x : 33. xy 0 +tgy =x sec y: 34. y, ye,x= + = xy 0 : x 5. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0406 縺04030503 0403 縺07 閨09 04 04030000308 陦ｧ070706 縺06009 ｧ 00 08. 0ｷ30050605 罍ｧ 縺06060 ｧ08 000307 ｧ05060004. ｭ ﾀ00 ｧ 04 陦ｧ070706 縺06009 ｧ 00040030 06 040060 ｧ ｧ P+ Qdy =0 @P @Q @y =, 0503 03003 04 罍ｧ0303 縺0605 0605 04 ｧ du =0 ｧ 060503 03030 ｧ08 ｧ0050605 00 0306 閨0605 @x u = C.. ｭ ﾀ00 ｧ @P @Q 6=, 0503 04 罍ｧ 00609030503 縺0308 06 ﾀ000304 ｧ0708 ﾀ06006 ﾀ0504060605 070600909 ｧ07 @y @x (x; y) 05 09 07, 0503 P + Q dy = du. 705 070600909 ｧ07 (x; y) 0 0304 0605 ﾀ07 ｧ0050605 罍ｧ 縺06060 ｧ0 000307 ｧ050600060.

34 0600.. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0ｷ30050605 罍ｧ 縺06060 ｧ08 000307 ｧ05060004 0006050903 0 0805 ｧ 04 ﾀ0006 0708: ) 090305 @P=@y, @Q=@x Q =(x), 050305 ln =R (x) ; @Q=@x, @P=@y ) 090305 = (y), 050305 ln =R (y) dy. P 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 x 4 070400070605 ﾀ07 ﾀ050030 ｧ ﾀ0006 070 ｧ 06 04-0060 ｧ08, ﾀ ﾀ0 05 罍ｧ04 0600308 04 0 ﾀ0503070060 04 05 0706. 70600 ｧ0504 ﾀ0006 閨06060 ｧ06 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0804 04030000308 陦ｧ070706 縺06009 ｧ 00 0809: 35. 4, y y + dy =0. x x 36. 3x e y +(x 3 e y, ) dy =0. 37. e,y +(, xe,y ) dy =0. 38. x cos y+(y, x sin y) dy =0. 7 0805 ｧ ｧ0050605 罍ｧ 縺06060 ｧ06 000307 ｧ050600 ｧ ｧ 縺0600 ｧ0504 陦ｧ070706 縺06009 ｧ 0004-00306 06 040060 ｧ07: 39. (x, y) + xdy =0. 40. x tg y+(x, sin y) dy =0. 4. (e x, y ) + ydy=0. 4. (+3x sin y), x ctg ydy=0. 70309 0504, 0503 0006040306 ﾀ05 ｧ ﾀ0006 閨06060 ｧ08 陦ｧ070706 縺06009 ｧ 000400308 06 04-0060 ｧ08 ﾀ060504 04030000306 陦ｧ070706 縺06009 ｧ 0003, ｧ 縺0600 ｧ0504 06 040060 ｧ07: 43. (3x +y) +(x, 3) dy =0. 44. (3x y, 4xy ) +(x 3, 4x y +y 3 ) dy =0. 45. (x cos y +), x sin ydy =0. 7 0805 ｧ ｧ0050605 罍ｧ 縺06060 ｧ06 000307 ｧ050600 ｧ ｧ 縺0600 ｧ0504 06 040060 ｧ07: 46. y +(yx, ) dy =0. 47. (x, 3y ) +xy dy =0. 48. (sin x + e y ) + cos xdy =0. 49. (x sin y + y) +(x cos y + x ln x) dy =0.

3x 6. ｰ 040060 ｧ07 6ｧ8 05 007 ｧ 070006 縺03 5 x 6. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0406 縺04030503 0403 縺07 閨09, 006 縺060006000306 0305003 ﾀ ｧ05060004003 04 縺03 ｧ 0403 閨00308. ｰ 040060 ｧ07 6ｧ8 05 007 ｧ 070006 縺03. ｭ ﾀ00 ｧ 06 040060 ｧ06 F (x; y; y 0 )=0040503 縺0308 ﾀ050604060 ｧ 0305003 ﾀ ｧ05060004003 y 0, 0503 03003 ｧ0060605 閨04 縺0600060 ｧ07 0305003 ﾀ ｧ05060004003 y 0 : y 0 = f (x; y) ｧ y 0 = f (x; y), 00604 縺06 縺030400308 0305003 ﾀ ｧ05060004003 x ｧ y 04 00609030503 縺0308 030300 ﾀ05 ｧ, 05060300605 罍ｧ 06 ﾀ09 ｧ 0304 縺06 閨0600070605 04 0006030308 0503 0906 (x 0 ; y 0 ) 05050308 030300 ﾀ05 ｧ 閨04 0 04 0400060 ｧ07 ｧ00506-05 000400308 09 罍ｧ040308. ｯ 09 ｧ06 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 F (x; y; y 0 )=0,09 縺03006 03030060503 ｧ0050605 00 (x; y; C) =0 ｧ ﾀ0500308 ｧ0050605 000304, ｧ00305 ｧ0060605 06006 03 ﾀ03-030308 ｧ0050605 00, 006 ﾀ03 閨06 縺07 0 ｧ08 04 縺03 ｧ 0403000400308 0403 ﾀ050307000308 ｧ 04 0503 0706 04 縺06007 006 04030006 060 ｧ08 ﾀ07 ｧ 03030060503 0 ｧ 04 罍ｧ 09 09030 0 060 ｧ ｧ 0403 ﾀ050307000308. 05 ﾀ03030308 ｧ0050605 00, 06 ﾀ00 ｧ 030 ﾀ06006 ﾀ0504060605, 00307003 04030006 ｧ0504, ｧ ﾀ090006 ｧ04 = y 0 ｧ 06 040060 ｧ08 F (x; y; ) =0 ｧ F 0 (x; y; ) =0 ｧ00 ｧ 0706 ｧ ﾀ090006 ｧ04 C ｧ 03030060503 ｧ0050605 00 (x; y; C) =0 ｧ 0 C =0.06060300605 罍ｧ 06 ﾀ09 ｧ 03 ﾀ03030308 ｧ0050605 00 0304 縺06 閨0600070605 0305 ｧ03 0600606 ﾀ0600608 ﾀ0504 ｧ0050605 000400308 09 罍ｧ040308 ).. ｰ 040060 ｧ06 6ｧ8 05 007 05 閨06 = y 0, ｧ0050605 罍ｧ 縺060605 ﾀ07 ﾀ0006 閨06060 ｧ0 0303 030. 7 縺03 陦ｧ070706 縺06009 ｧ 縺0304 04 () 0403 x, 0 08 閨060: y = xf() +'(); () = f () + xf 0 () +' 0 () d : 70503 06 040060 ｧ06 00 ｧ0060800306 0305003 ﾀ ｧ05060004003 x ｧ. 70600 ｧ04 060503, 04030006 ｧ0: d x = CA() +B(): () ｰ 040060 ｧ07 () ｧ () 0306 閨0605 04 00605 罍ｧ 06 ﾀ09 ｧ 0304 縺06 閨0600070504 03030 ｧ08 ｧ00506-05 00. 0ｷ3 ﾀ090006 ｧ04 ｧ 0 ｧ08 04 00605 (06 ﾀ00 ｧ 050503 0403 00307003), 04030006 ｧ0 03030 ｧ08 ｧ0050605 00 04 0703 縺006 (x; y; C) =0. 3. ｰ 040060 ｧ06 070006 縺03 y = x + '() (3) 070400070605 ﾀ07 ﾀ050030 ﾀ0006 060 06 040060 ｧ07 6ｧ8 05 007. 05003 ｧ0060605 03030 ｧ08 ｧ0-050605 00 y = Cx+ '(C) ｧ 03 ﾀ03030308, 04030006 060 ｧ08 ﾀ07 ｧ ﾀ090006 060 ｧ060 04 00605 ｧ 06 040060 ｧ08 y = x + '() ｧ x =,' 0 (). 50. 703 ﾀ05 縺03 ｧ0504 006 ﾀ090300040903 ｧ0050605 000400308 09 罍ｧ040308 06 040060 ｧ07 y 0 = 4y. 07 09 ｧ06 閨0406 ｧ0050605 000400306 09 罍ｧ040306 04 縺030803 閨0705 06 縺06 0503 0906 M (; 4)? ) ｮ0. 0304 縺06 閨0600060 ｧ06 0305 ｧ03 0600608 0 ﾀ. 00

36 0600.. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 5. 703 ﾀ05 縺03 ｧ0504 ｧ0050605 000400306 09 罍ｧ040306 06 040060 ｧ07 y 0 + y,, = 0. 07 09 ｧ06 閨0406 ｧ0050605 000400306 09 罍ｧ040306 04 縺030803 閨0705 06 縺06 0503 0906 M (=; = )? 5. 70309 0504, 0503 ｧ0050605 000400306 09 罍ｧ040306 06 040060 ｧ07 xy 0,, yy 0 +4x =0 ﾀ03 閨06 縺07 05 ﾀ07 0400605 罍ｧ 03 ﾀ05 縺030503 060500 00607 閨06 04 縺070030 ｧ y = x. 703 ﾀ05 縺03 ｧ0504 ｧ0050605 000400306 09 罍ｧ040306, 040300 05 07 04 03030060 ｧ0-050605 0006 0403 ﾀ050307000606 C = ; ; ｧ05.. 53. 70600 ｧ0504 06 040060 ｧ07: ) yy 0 + y 0 (x, y), x =0; ) xy 0 +xy 0, y =0 ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ0050605 000400306 09 罍ｧ040306. 54. 70600 ｧ0504 06 040060 ｧ07, 006 ﾀ03 閨06 縺07 0 ｧ06 0704003 03 閨00308 ｧ 0406 縺06-006000308: ) y =+y 0 ; ) x =y 0, y. 0 ｰ09 0 ｧ06. 050303 0 ｧ04 y 0 06 縺06, 04 縺03 陦ｧ070706 縺06009 ｧ 縺0304 0504 0406 縺040306 06 040060 ｧ06 0403 x, 040503 縺0306 0403 y. 55. 7 0805 ｧ 03030 ｧ06 ｧ 03 ﾀ03030306 ｧ0050605 0003 06 040060 ｧ08 6ｧ8 05 007 : ) y = xy 0 + y 0 ; ) y =xy 0 + xy0 ; 3) y = y0 y. 0 + 56. 7 0805 ｧ 03030 ｧ08 ｧ 03 ﾀ03030308 ｧ0050605 0003 06 040060 ｧ07 070006 縺03 ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ0050605 000400306 09 罍ｧ040306: ) y = xy 0, y 0 ; ) y = xy 0, a +y 0 ; 3) y = xy 0 + y 0. 57. 703 ﾀ05 縺03 ｧ0504 ｧ0050605 000400306 09 罍ｧ040306 06 040060 ｧ07 y 0 + y =. 07 09 ｧ06 閨0406 ｧ0050605 000400306 09 罍ｧ040306 04 縺030803 閨0705 06 縺06 0503 0906 M (; 3=4)? 58. 70600 ｧ0504 06 040060 ｧ07, 006 ﾀ03 閨06 縺07 0 ｧ06 0704003 03 閨00308 ｧ 0406 縺06-006000308: ) y = y 0 + y 03 ;)x = ay0 +y. 0 59. 70600 ｧ0504 06 040060 ｧ06 y =y 0 x + x + y0. 60. 7 0805 ｧ 03030 ｧ08 ｧ 03 ﾀ03030308 ｧ0050605 0003 06 040060 ｧ07 070006 縺03 ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ0050605 000400306 09 罍ｧ040306: ) y = y 0 x + y 0 ; ) y = xy0 + y 0 + y 0. 6. 7 0805 ｧ 09 罍ｧ040606, 09 ﾀ 05060004003060909030503 縺0308 0303 060605 ﾀ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 05 縺0606050300040 ｧ09 0403 ﾀ050307000308 0400030 陦ｧ, 0400308 a. 6. 7 0805 ｧ 09 罍ｧ040606, 09 ﾀ 050600040 07 09 09030503 縺0308 0305 ﾀ0609 0605 0 03 ﾀ0708 090303 罍陦ｧ0 05 0305 縺06 09 ｧ, ﾀ0600 09030503 縺0308 040 a.

3x 7. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0403 ﾀ00 ｧ08 0403 縺07 閨090304 7 x 7. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0403 ﾀ00 ｧ08 0403 縺07 閨090304, 閨030406 ﾀ09 060 ｧ06 04030 ｧ07060 ｧ06 0403 縺07 閨09. ｰ 040060 ｧ06 04 ｧ y (n) = f(x) 縺0600 0605 ﾀ07 0403 ﾀ0006 閨0304 050600040030 n-09 050030 ｧ0050605 罍ｧ 縺0304 0 ｧ060 04 040308 ﾀ05 ｧ. 7 罍ｧ 09 07 閨030 ｧ0050605 罍ｧ 縺0304-0 ｧ ｧ 04030006 0605 ﾀ07 03 閨0 04 縺03 ｧ 040300040 07 0403 ﾀ05030700 07, 04 0309030 050600040030 縺06-06000405 0506 n 04 縺03 ｧ 0403000400308 0403 ﾀ050307000308.. ｰ 040060 ｧ06 F (x; y 0 ;y 00 )=0, 006 ﾀ03 閨06 縺07 00606 y 04 070400308 0703 縺006, 0403 陦ﾀ05 00304090308 y 0 =, y d 00 = 04 罍ｧ0403 陦ｧ05 ﾀ07 09 04 ｧ 閨06 F x; ; d =0: 3. ｰ 040060 ｧ06 F (y; y 0 ;y 00 )=0, 006 ﾀ03 閨06 縺07 00606 x 04 070400308 0703 縺006, 0403 陦ﾀ05 00304090308 y 0 =, y d 00 = = d 04 罍ｧ0403 陦ｧ05 ﾀ07 09 04 ｧ 閨06 dy F y; ; d =0: dy 70600 ｧ0504 06 040060 ｧ07: 63. ) y 000 = 6 x 3 ;0 000400306 06 ﾀ000304 ｧ07: y =, y0 =, y 00 = 04 罍ｧ x = ; ) y 00 = 4 cos x; y = 0, y 0 = 004 罍ｧ x = 0; 3) y 00 = = +x. 64. x 3 y 00 + x y 0 =: 65. yy 00 + y 0 =0: 66. y 00 + y 0 tg x = sin x: 67. y 00 +y(y 0 ) 3 =0: 68. y 00 x ln x = y 0 : 69. y 00 tg y =(y 0 ) : 70. ) xy 00, y 0 = e x x ; ) y 00 +xy 0 =0: 7. 0504 縺06 閨0600 ｧ0504 09 罍ｧ040606 ｧ 05 ｧ03 0503 罍ｧ 03005 000400308 03 0009 ｧ, 03 陦ｧ0 090300609 09030503 縺0308 0 0500060803 閨0600 0, 0 陦縺06050308 閨0608 ﾀ0504060605 ﾀ03 ﾀ 縺06 閨030503-0600 07 ﾀ ｧ00 P (0406 ﾀ030 03 0009 ｧ 04 縺0600603 縺06 04 ｧ ﾀ ｧ05 0504 ｧ 05 ｧ03 0 ﾀ050300040903 0 00030, 0503 +y 0 ). 7. 0504 縺06 閨0600 ｧ0504 09 罍ｧ040306, 06 09030503 縺0308 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003 04 閨040306 030300040006 閨00 ｧ003 003 縺0 00 ｧ. 73. 0504 縺06 閨0600 ｧ0504 09 罍ｧ040306, 06 09030503 縺0308 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003 04060 閨00 ｧ006 003 縺0 00 ｧ. 74. 7 0305 縺06 0906 [0; ] 0304 縺06 閨0600 ｧ0504 09 罍ｧ040606, 09 ﾀ 0600606 ﾀ07 03 ﾀ ｧ Ox 040 0006 090303 罍陦ｧ0 05, 06 ﾀ00 ｧ 09 罍ｧ04 ｧ 0 0606 k = x, 05. 06. 04003006 縺003 0 ﾀ05 0605 04 閨030004 03 ﾀ ｧ Ox (0406 縺060803 閨0 07 09 罍ｧ04 07). 7 罍ｧ0070504 04 罍ｧ0300 ｧ- 07060003, 0503 +y 0.

38 0600.. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 70600 ｧ0504 06 040060 ｧ07: 75. y 00 = cos x ; y = ln, y0 =04 罍ｧ x = 4. 76. ( + x )y 00 +xy 0 = x 3 : 77. y 00 y 3 =: 78. yy 00 =(y 0 ) : 79. t d s ds dt + dt + t =0: 80. yy 00 =+y 0 : 8. y 00 tg x = y 0 +: 8. 0504 縺06 閨0600 ｧ0504 09 罍ｧ040306, 06 09030503 縺0308 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003 04060 09060306 閨00 ｧ003 003 縺0 00 ｧ. 83. 6 5 ｧ00506 縺04 0006 (,=; =) 0304 縺06 閨0600 ｧ0504 09 罍ｧ040606, 09 ﾀ 06006-06 ﾀ07 03 ﾀ ｧ Ox 040 0006 090303 罍陦ｧ0 05, 06 ﾀ00 ｧ 09 罍ｧ04 ｧ 0 0606 04 0006030308 0503 0906 k = cos x. x 8. 6ｧ8 ｧ0060800306 03 閨003 縺03 閨00306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 ﾀ 0403 ﾀ05030700030 ｧ 0903050707 ｧ09 ｧ06005 0 ｧ 05 閨003 縺03 閨00306 00 ｧ0060800306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 y (n) + y (n,) + :::+ n y =0; () 05 閨06 i 070600909 ｧ ｧ x, ｧ0060605 030300606 縺0600060 ｧ06 04 ｧ y = C y + C y + :::+ C n y n ; () 05 閨06 y, y, :::, y n 00 ｧ00608003 006 04 ｧ ﾀ ｧ00306 ﾀ0500306 縺0600060 ｧ07 06 040060 ｧ07 (), C, C, :::, C n 04 縺03 ｧ 0403000400306 0403 ﾀ050307000306. ｭ ﾀ00 ｧ 0903050707 ｧ09 ｧ0600503,, :::, n 06 040060 ｧ07 () 0403 ﾀ0503070003, 0503 ﾀ05-00306 縺0600060 ｧ07 y, y, :::, y n 0 0803 閨0705 ﾀ07 ﾀ 040300300406 08 090506 罍ｧ ﾀ05 ｧ 06 ﾀ09030503 06 040060 ｧ07 r n + r n, + :::+ n =0: (3) ) 07 07 閨03006 0406006 ﾀ0504060003006 0903 縺006 r = a 06 040060 ｧ07 (3) 09 05003 ﾀ05 ｧ m ﾀ030305040605 ﾀ0504060605 m ﾀ0500308 縺0600060 ｧ08 e ax, xe ax, :::, x m, e ax. ) 07 07 閨0308 04 縺06 00 ｧ00308 0903 縺00608 r = i 09 05003 ﾀ05 ｧ m ﾀ030305040605- ﾀ0504060605 m 04 ﾀ0500308 縺0600060 ｧ08: e x cos x; xe x cos x; :::; x m, e x cos x; e x sin x; xe x sin x; :::; x m, e x sin x: 70600 ｧ0504 06 040060 ｧ07: 84. y 00, 4y 0 +3y =0: 85. y 00, 4y 0 +4y =0: 86. y 00, 4y 0 +3y =0: 87. y 00, 4y =0: 88. y 00 +4y =0: 89. y 00 +4y 0 =0: 90. d x dt +3 dt, 4x =0: 9. 4 d d' + =0:

3x 9. 6ｧ8 ｧ0060800306 00603 閨003 縺03 閨00306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 9 9. d s dt +ds dt +s =0; 04 罍ｧ t =0 s =, s0 =. 93. y 000, 5y 00 +8y 0, 4y =0. 94. y IV, 6y =0: 95. y 000, 8y =0. 96. y 000 +3ay 00 +3a y 0 + a 3 y =0. 97. y IV +4y =0: 98. 4y IV, 3y 00, y =0. 99. 0504 縺06 閨0600 ｧ0504 06 040060 ｧ06 0903000603 0 ｧ08 0 07050 ｧ09, ﾀ03 ﾀ0503070060503 ｧ 0 ﾀ ﾀ03 m, 0403 閨04060006000308 0 0 ｧ05 ｧ 閨00 ｧ00308 l ( ﾀ0304 縺0305 ｧ0400060 ｧ060 04 縺06-00603 縺06 04 ｧ 0403000307 ｧ0504, 0503 04 罍ｧ 0 00030 06050006 03050900030060 ｧ07 sin ). 0504 縺06 閨0600 ｧ0504 0406 罍ｧ03 0903000603 0 ｧ07. 00. 6 704 03 陦ｧ0 0903040308 05 縺06 0403 閨04060006003 09 090300906 04 縺0607 ｧ003. 703 閨0608 ﾀ0504 ｧ060 03 閨0030503 05 縺06 04 縺0607 ｧ0 06 閨00 ｧ0070605 ﾀ07 0 a ﾀ0. 0504 縺06 閨0600 ｧ0504 閨04 ｧ07060 ｧ06 0406 縺04030503 05 縺06, 06 ﾀ00 ｧ 040503 縺0308 030303 縺040605 ﾀ07 ( ﾀ0304 縺0305 ｧ0400060 ｧ060 04 縺0600603 縺06 04). 0504 縺06 閨0600 ｧ0504 0406 罍ｧ03 0903000603 0 ｧ07. 0. 70600 ｧ0504 06 00 ﾀ 06 0605030 ﾀ0304 縺0305 ｧ0400060 ｧ07, 04 縺030403 縺09 ｧ- 030 00040030503 ﾀ0903 縺03 ﾀ05 ｧ 閨04 ｧ07060 ｧ07. 70600 ｧ0504 06 040060 ｧ07: 0. y 00 +3y 0 +y =0: 03. y 00 +ay 0 + a y =0: 04. y 00 +y 0 +5y =0: 05. d x dt 06. d x dt +! x =0: 07. d s dt, dt, 3x =0: + ads dt =0: 08. x tt +_x t +3x =0: 09. y 000, 3y 00 +4y =0: 0. y IV, 3y 00, 4y =0:. y IV +8y 00 +6y =0:. 7 0805 ｧ ｧ0050605 000400606 09 罍ｧ040606 06 040060 ｧ07 y 00, y =0,09 - ﾀ 0600606 ﾀ07 04 0503 0906 (0; 0) 04 縺0700308 y = x. x 9. 6ｧ8 ｧ0060800306 00603 閨003 縺03 閨00306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 ﾀ 0403 ﾀ05030700030 ｧ 0903050707 ｧ09 ｧ06005 0 ｧ. 05 ﾀ0030400306 ﾀ040308 ﾀ050403. 706 ﾀ0504 003 06 040060 ｧ07: y (n) + y (n,) + :::+ n y = f (x) 00603 閨003 縺03 閨00306; () y (n) + y (n,) + :::+ n y =0 03 閨003 縺03 閨00306 () ｧ0406 ﾀ0504 u 030300606 縺0600060 ｧ06 06 040060 ｧ07 (), y ﾀ0500306 縺0600060 ｧ06 06 040060 ｧ07 (). ｯ0305 030300606 縺0600060 ｧ06 y 06 040060 ｧ07 () 0306 閨0605 y = u + y : (3)

30 0600.. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07. 03060503 0060304 縺06 閨060006000308 0903050707 ｧ09 ｧ060050304. 7 罍ｧ 0403 ﾀ0503-07000308,, :::, n ﾀ0500306 縺0600060 ｧ06 y 0 0803 陦ｧ05 ﾀ07 0060503 閨030 0060304 縺06 閨06-0006000308 0903050707 ｧ09 ｧ060050304 04 ﾀ0006 閨06060 ｧ08 ﾀ0006 0708: ) f(x) 00030503 00060; ) f(x) =e mx (a cos nx + b sin nx); 3) f(x) 06 ﾀ0504 ﾀ0600 ｧ00 ｧ 04 縺03 ｧ 0406 閨060 ｧ06 04 縺06 閨03 閨060 ｧ08 070600909 ｧ08. 6 50505 ｧ08 ﾀ0006 0708 ﾀ0500306 縺0600060 ｧ06 y ｧ0060605 050305 0706 04 ｧ, 0503 ｧ f(x), 030500 ｧ 07 ﾀ04 0305 00606 050300040903 0903050707 ｧ09 ｧ06005 0 ｧ. 0ｷ3 ﾀ090006 060 ｧ07 ﾀ03 ﾀ05 0400070605 03 ﾀ03030306 ﾀ0006 ｧ, 090305 : ) f(x) 00030503 00060, 003 r =0 0903 縺06004 08 090506 罍ｧ ﾀ05 ｧ 06 ﾀ09030503 06 04-0060 ｧ07 09 05003 ﾀ05 ｧ k; ) f(x) =e mx (a cos nx + b sin nx), 003r = m ni 06 ﾀ0504 0903 縺06004 08 09-0506 罍ｧ ﾀ05 ｧ 06 ﾀ09030503 06 040060 ｧ07 09 05003 ﾀ05 ｧ k. 6 50505 ｧ08 03 ﾀ03030308 ﾀ0006 0708 y 030500 ｧ 0605 ﾀ07 0305 f(x) 006 050300040903 0903050707 ｧ09 ｧ060-05 0 ｧ, 003 06006 ｧ 000307 ｧ050600060 x k. 3. 03060503 04 罍ｧ 09 ｧ ｧ 04 縺03 ｧ 0403000400308 0403 ﾀ050307000308. 703000606 03030 ｧ0 04 罍ｧ060030 縺0600060 ｧ07 00603 閨003 縺03 閨0030503 00 ｧ006080030503 06 040060 ｧ07 070400070605 ﾀ07 0060503 6ｧ8 05 007, ｧ00 ｧ 0060503 04 罍ｧ 09 ｧ ｧ 04 縺03 ｧ 0403000400308 0403 ﾀ050307000308. ｭ ﾀ00 ｧ y ｧ y 006 04 ｧ ﾀ ｧ00306 ﾀ0500306 縺0600060 ｧ07 06 040060 ｧ07 y 00 +y 0 + + qy =0, 0503 縺0600060 ｧ06 06 040060 ｧ07 y 00 +y 0 +qy = f(x) 0403 0060503 閨06 6ｧ8 05 007 0 0803 陦ｧ05 ﾀ07 04 04 ｧ 閨06 y = Ay + By,05 閨06 A ｧ B 070600909 ｧ ｧ x, 06 閨03040006050403 縺07-060 ｧ06 ﾀ ｧ ﾀ0506006 06 040060 ｧ08 A 0 y + B 0 y =0; A 0 y 0 + B0 y 0 = f(x): 0505 ﾀ06 A 0 =, y f(x) w ; B0 = y f(x) w ; w = y y y 0 y 0 : 70600 ｧ0504 06 040060 ｧ07: 3. y 00, y 0 + y = e x : 4. y 00, 4y =8x 3. 5. y 00 +3y 0 +y = sin x + cos x. 6. y 00 + y = x +e x : 7. y 00 +3y 0 =9x. 8. y 00 +4y 0 +5y =5x, 3x +5. 9. y 00, 3y 0 +y = e x. 0. d x dt + k x =k sin kt.. y 00, y = xe,x :. y 00, y 0 = x, x. 3. y 00 +5y 0 +6y = e,x + e,x. 4. x +k _x +k x =5k sin kt.

3x 0. 7 罍ｧ006 縺03 陦ｧ070706 縺06009 ｧ 000400308 06 040060 ｧ08 00308 05 ｧ040304 5. y 000 + y 00 =6x + e,x : 6. y IV, 8y =7e,3x : 7.... x + _x =3t : 8. y 000 +8y = e,x : 9. ) x +4_x +4x = e,t ; ) a 3 x + ax =: 30. y 00 +4y = sin x : 3. y00, 4y 0 +5y = ex cos x : 3. y 00, y 0 + y = x, e x : 33. y 00 + y =tgx: 34. ) y 00 + y 0 = +e x ; ) y00 +4y 0 +4y = e,x x : 3 35. ｭ 陦ｧ0 ｧ09 0 ﾀ ﾀ03 閨04 ｧ070605 ﾀ07 0403 03 ﾀ ｧ Ox 0403 閨0608 ﾀ0504 ｧ060 0403- ﾀ050307000308 ﾀ ｧ0003 a, 0 04 040006000308 0403 03 ﾀ ｧ, 04 罍ｧ ﾀ0304 縺0305 ｧ0400060 ｧ ｧ 閨04 ｧ- 07060 ｧ06, ｧ ﾀ00060003 040030 ﾀ0903 縺03 ﾀ05 ｧ 閨04 ｧ07060 ｧ07. 7 0805 ｧ 09030 閨04 ｧ- 07060 ｧ07, 06 ﾀ00 ｧ 04 罍ｧ t =0 ｧ006060 x =0 ｧ ﾀ0903 縺03 ﾀ0504 v =0. 70600 ｧ0504 06 040060 ｧ07: 36. y 00 + y 0, y =6x : 37. y 00, 5y 0 +6y = 3 sin 3x: 38. y 00 +y 0 + y = e x : 39. y 00 + y 0 +; 5y = 5 cos x: 40. 4y 00, y = x 3, 4x: 4. y 00, y = e,x : 4. d s dt +ds dt +s =t3,. 43. ) y 00, my 0 + m y = sin mx; ) n 3 y 00, 4ny =8. 44. y IV +5y 00 +4y = 3 sin x. 45. y 000, 3y 00 +3y 0, y = e x. ｮ0006 閨06060 ｧ06 06 040060 ｧ07 縺0600 ｧ0504 0060503 閨030 04 罍ｧ 09 ｧ ｧ 04 縺03 ｧ 04030004-00308 0403 ﾀ050307000308: 46. y 00 +4y 0 +4y = e,x ln x. 47. ) y 00 + y = cos 3 x ; ) y00 +4y = sin x. e 48. y 00, x y 0 + y = 4, x. x 0. 7 罍ｧ006 縺03 陦ｧ070706 縺06009 ｧ 000400308 06 040060 ｧ08 00308 05 ｧ040304 0504 縺06 閨0600 ｧ0504 05 ｧ04 陦ｧ070706 縺06009 ｧ 00040030503 06 040060 ｧ07 ｧ 縺0600 ｧ0504 060503: 49. y 0 + y +x = e,x : 50. y 0 + y tg x =tgx. 5. (x, x 3 )y 0 +(x, )y = x 3.

3 ｪ ﾀ0500306 縺0600060 ｧ06 03 閨003 縺03 閨0030503 06 040060 ｧ07 (04 罍ｧ f (x) =0) 00307003 0 0805 ｧ 04 04 ｧ 閨06 y = x r,05 閨06 r 0403 ﾀ050307000306 ｧ ﾀ0003. 6 70007 0 080307 閨060 ｧ07 r 00607003 0403 - ﾀ05 04 ｧ0504 y = x r 04 03 閨003 縺03 閨00306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 ｧ 縺0600 ｧ0504 04030006 06000306 08 090506 罍ｧ ﾀ05 ｧ 06 ﾀ090306 06 040060 ｧ06 0305003 ﾀ ｧ05060004003 r. 7 罍ｧ 0505030: ) 09 07 閨03006 0406006 ﾀ0504060003006 0903 縺006 r = a 09 05003 ﾀ05 ｧ m ﾀ030305040605 ﾀ0504060605 m ﾀ0500308 縺0600060 ｧ08 x a, x a ln x, x a (ln x), :::; ) 09 07 閨0308 04 縺06 00 ｧ00308 0903 縺00608 r = i 09 05003 ﾀ05 ｧ m ﾀ030305040605 ﾀ0504060605 m 04 ﾀ0500308 縺0600060 ｧ08: x cos ( ln x); x cos ( ln x)lnx; ::: ; x sin ( ln x); x sin ( ln x) ln x; ::: 0600.. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 5. ( + x )y 0 + y(x, +x )=0. 53. t ds +ts dt = e t dt: 54. xy 0 =4(y + y). 55. xyy 0 =y + y 4 + x 4. 56. xy 00 + y 0 =lnx: 57. yy 00, y 0 =0: 58. y 00, m y = e,mx : 59. y 0 x ln x + y =lnx: 60. xy 0 + y ln y x =0: 6. y0 + y = y 3 (x, ): 6. y 000, y 00 + y 0 = x : 63. y 00 = y 0 + y 0 : 64. d3 s dt 3, 3ds dt 65. ) sin tds=, s = sin t + cos t. 4t sin t + s dt; ) yy 0 x, y =: 66. ) xy 0 + y(x tg x + ) = sec x; ) y 000 + y = e,x : 67. ) y 00, 3y 0 +y = e3x +e x ; ) y000 y = y 00 y 0 : 68. ｨC ｧ00 ｧ0 陦 陦ｧ06 ﾀ a 閨0 ｧ 0 ﾀ ﾀ0308 m = a 3 0905 0400 04 0605 04 0403 閨06 04 罍ｧ 0406 縺05 ｧ09 00040030 0403000307060 ｧ ｧ 03 ﾀ ｧ. 7 0805 ｧ 0406 罍ｧ03 0903000603 0 ｧ07, 0903-0503 縺0306 04030006 ｧ05 ﾀ07, 06 ﾀ00 ｧ 09 ｧ00 ｧ0 陦 00600030503 040305 縺06 ｧ0504 04 0403 閨06 ｧ 05060 03050406 ﾀ05 ｧ0504. ｮ0304 縺0305 ｧ0400060 ｧ06 閨04 ｧ07060 ｧ06 04 罍ｧ0070504 04 罍ｧ0300 ｧ07060003 04-0030 0060006. 69. 703000308 07060006 00308 00 ﾀ 陦ｧ06 ﾀ 0 ｧ 04030406 縺08003 ﾀ050608 a ｧ a ｧ0060605 0403 ﾀ050307000606 050600406 0506 縺06 0400605 縺06000608 04030406 縺08003 ﾀ05 ｧ 00 C ｧ 0 縺060700308 0 C. 0504 縺06 閨0600 ｧ0504 050600406 0506 縺06 0400605 罍ｧ ﾀ0506009 ｧ 0 000603030 ﾀ ﾀ0503070 ｧ ｧ r 0305 0906005 (a 6 r 6 a) ｧ 04 罍ｧ r =; 6a. ｰ09 0 ｧ06. ｮ0903 縺03 ﾀ0504 04 閨060 ｧ07 050600406 0506 縺03 dt 04 04 縺030403 閨0 ｧ0906 ﾀ03 dr ﾀ05 09 ｧ030 縺0030 ﾀ04 縺06 閨0600060 ｧ060 050600406 0506 縺03 0303 05003 04 縺030403 縺09 ｧ030 00040 0400030 陦ｧ 04030406 縺06 0030503 ﾀ06 060 ｧ07. x. 6ｧ8 ｧ0060800306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 7080006 n (n) + n, (n,) + + n, 0 + n = ( )

3x. ｮ ｧ ﾀ0506003 00 ｧ0060800308 陦ｧ070706 縺06009 ｧ 000400308 06 040060 ｧ08 3 70603 閨003 縺03 閨00306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 7080006 縺0600 0605 ﾀ07 0060503- 閨030 04 罍ｧ 09 ｧ ｧ 0403 ﾀ050307000308. 70600 ｧ0504 06 040060 ｧ07: 70. ) x 3 y 000, 3xy 0 +3y =0; ) x y 00, y =0; 3) x y 00 +xy 0, n(n +)y =0. 7. ) x y 00 +5xy 0 +4y =0; ) x y 00 + xy 0 + y =0. 7. ) xy 00 +y 0 =0x; ) x y 00, 6y =lnx. 73. ) x y 00, xy 0 +y =4x; ) x 3 y 00 +3x y 0 + xy =6lnx. 74. ) x y 00, 4xy 0 +6y = x 5 ; ) x y 00 + xy 0 + y = x. x. ｮ ｧ ﾀ0506003 00 ｧ0060800308 陦ｧ070706 縺06009 ｧ 000400308 06 040060 ｧ08 ﾀ 0403 ﾀ05030700030 ｧ 0903050707 ｧ09 ｧ06005 0 ｧ 70600 ｧ0504 06 040060 ｧ07: 75. dt + y =0,, dy dt dt 76. dt + x, y = et, =3x + y. dy dt, x + y = et. ｰ 09 0 ｧ 06 09 06 75. 7 縺03 陦ｧ070706 縺06009 ｧ 縺0304 04 0406 縺040306 ｧ 06 04006-0 ｧ08 0403 t, ｧ ﾀ090006 060 ｧ 05 縺060806 040060 ｧ08 y ｧ dy dt. 77. 5, dt dy dt +4x, y = e,t, dt +8x, 3y =5e,t. 78. x, 4_x +4x, y =0, y +4_y +4y, 4x =6e t. 70600 ｧ0504 06 040060 ｧ07: 79. _x +3x + y =0, _y, x + y =0; x =;y =04 罍ｧ t =0: 80. _x = y, _y = x +sht.

34 0600.. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 x 3. 6ｧ8 ｧ0060800306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 04 ﾀ0500308 04 縺03 ｧ 0403 閨00308 040503 縺030503 0403 縺07 閨09 (0060503 08 090506 罍ｧ ﾀ05 ｧ09) 8. 7 0805 ｧ 030300606 縺0600060 ｧ06 ( ﾀ03 閨06 縺07 00606 閨0406 04 縺03 ｧ 0403000400306 070600909 ｧ ｧ) 06 040060 ｧ08: ) @ u @x@y =0; ) @ u @y =0; 3) @ u, @u @x@y x @y =0; 4) @ u @x@y =ax y + b. ｰ09 0 ｧ06. 703000307 ｧ0504 @u @y = z. 8. 7 0805 ｧ ﾀ0500306 縺0600060 ｧ06 06 040060 ｧ07 @ z @x @z 0030 06 ﾀ000304 ｧ070: z = y 3, @x = y 04 罍ｧ x =. 83. 7 縺060303 0304 0504 06 040060 ｧ06 @ u, @x 4 @ u u @x@y +3@ @y =0 0909 0030 ｧ 06 ﾀ090308 0703 縺006 ｧ 0 0805 ｧ 060503 030300606 縺0600060 ｧ06. 84. 7 縺060303 0304 0504 06 040060 ｧ06 x @ u @x +xy @ u @x@y + y @ u @y =0 0909 0030 ｧ 06 ﾀ090308 0703 縺006 ｧ 0 0805 ｧ 060503 030300606 縺0600060 ｧ06. = 00403 0 0004-6 5 ﾀ0006 閨06060 ｧ08 陦ｧ070706 縺06009 ｧ 000400308 06 040060 ｧ0708 0 0805 ｧ 03030 ｧ06 縺06-00060 ｧ07, 06 ﾀ00 ｧ 003 0 000400306 06 ﾀ000304 ｧ07, 0503 ｧ ﾀ0500306 縺0600060 ｧ07: 85. @ u @x, 4 @ u @x@y +4@ u @y =0. 86. @ u, @ u @x @y =0; u = sin y, @u = y 04 罍ｧ x =0. @x 87. x @ u @x + y @ u @x@y =0; u =y +, @u = y 04 罍ｧ x =. @x

388. t @ u @t x 3. 03060503 08 090506 罍ｧ ﾀ05 ｧ09 5, x @ u @x =0; u =x, @u @t = x 04 罍ｧ t =. 7 0805 ｧ ﾀ0500306 縺0600060 ｧ07 陦ｧ070706 縺06009 ｧ 000400308 06 040060 ｧ08: 89. @ u @t + @ u @u @x@t + @t =0; u =0, @u =,x, 04 罍ｧ t =0. @t 90. 4a x @ u, @ u @u @x @t +a @x =0; u =0, @u = ax 04 罍ｧ t =0. @t 9. a @ u @x = @ u @t ; u = f (x), @u = F (x) 04 罍ｧ t =0. @t

30600 04 3 6 76 505 76ｧ9 ｭ, ｯ 705 76ｧ9 ｭ 0ｷ3 07 70ｷ36 5056ｧ80ｷ3 7 ｭ 76ｧ9 ｭ 0ｷ3 7 ｯ ｭ06 7 縺6ｧ86ｧ9 x. 6 503 ｧ ﾀ00060 ｧ06 0400030 陦ｧ ﾀ 040300300406 閨0403080030503 ｧ0050605 00. 6 70403080030 ｧ0050605 00030 0305 00604 縺06 縺030400308 070600909 ｧ ｧ F (x; y), ﾀ04 縺03- ﾀ05 00600030 0 0305 0 ｧ 06000606 030300 ﾀ0504 (S) 040003 ﾀ0903 ﾀ05 ｧ xoy, 0 0304 0605 ﾀ07 04 縺06 閨0600 ﾀ030305040605 ﾀ0504060600608 閨0406006 縺00308 ｧ0050605 000400308 ﾀ060003: (S) F (x; y) dy = lim max x i!0 max y k!0 XX i k F (x i ;y k )x i y k ; 05 閨06 x i = x i+, x i, y k = y k+, y k ｧ ﾀ0600 ﾀ04 縺03 ﾀ05 0060 0 0506 0 060 ｧ07 i ｧ k, 閨0007 09030503 縺0308 0503 09 ｧ (x i ; y k ) 04 罍ｧ0 閨000607 05 030300 ﾀ05 ｧ (S). 700030 閨04 S 030300 ﾀ05 ｧ (S) 0304 縺06 閨0600070605 ﾀ07 0703 縺006000308 S = (S) dy:. ｭ ﾀ00 ｧ 030300 ﾀ0504 (S) 0304 縺06 閨0600070605 ﾀ07 006 04060 ﾀ0504 0 ｧ a 6 x 6 b, y (x) 6 6 y 6 y (x), 04 罍ｧ 060 09 07 07 ｧ 00604 縺06 縺030400308 09 罍ｧ040308 y = y (x) ｧ y = = y (x) 0406 縺06 ﾀ0609 0605 ﾀ07 ﾀ 0406 縺05 ｧ09 000406 x = X (x <X<x ) 050300040903 04 03 閨00308 0503 0906, 0503 b y (x) (S) F (x; y) dy = 05 閨06 04 罍ｧ 0403 ｧ ﾀ00060 ｧ ｧ ｧ0050605 00 a y (x) y (x) y (x) F (x; y) dy; F (x; y) dy 040600 ｧ ｧ006 x 040300 05 0605 0403- ﾀ050307000308. 3. ｭ ﾀ00 ｧ 030300 ﾀ0504 (S) 0304 縺06 閨0600070605 ﾀ07 006 04060 ﾀ0504 0 ｧ h 6 y 6 l, x (y) 6 6 x 6 x (y), 04 罍ｧ 060 09 07 07 ｧ 00604 縺06 縺030400308 09 罍ｧ040308 x = x (y) ｧ x = = x (y) 0406 縺06 ﾀ0609 0605 ﾀ07 ﾀ 0503 罍ｧ 03005 000406 y = Y (y <Y <y ) 050300040903 04 03 閨00308 0503 0906, 0503 l x (y) (S) F (x; y) dy = h dy x (y) F (x; y) ;

3x. 6 503 ｧ ﾀ00060 ｧ06 0400030 陦ｧ ﾀ 040300300406 閨0403080030503 ｧ0050605 00 7 05 閨06 04 罍ｧ 0403 ｧ ﾀ00060 ｧ ｧ ｧ0050605 00 x (y) x (y) F (x; y) 040600 ｧ ｧ006 y 040300 05 0605 0403- ﾀ050307000308. 4. ｭ ﾀ00 ｧ 030300 ﾀ0504 (S) 0304 縺06 閨0600060 04 04030007 縺00308 090303 罍陦ｧ0 05 08 006 04060- ﾀ0504 0 ｧ ' 6 ' 6 ', r (') 6 r 6 r ('), 0503 0400030 閨04 05050308 030300 ﾀ05 ｧ ' r (') S = rdrd'= d' rdr: (S) ' r (') 6ｦ0 04 ｧ ﾀ 0504 ﾀ 040300300406 閨04030800308 ｧ0050605 000304 ｧ 0403 ｧ ﾀ00 ｧ0504 040003-0 陦ｧ, 0305 0 ｧ 06000306 00 ｧ0 ｧ070 ｧ: 9. xy =4, y = x, x =4. 93. ) y = x, 4y = x, y =4;)y = x, 4y = x, x =. 94. y =4+x, x +3y =0. 95. ay = x, ax, y = x. 96. y =lnx, x, y = ｧ y =,. 97. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ05 ｧ, 0400030 陦ｧ 09030503 縺0308 0403 07 0605 ﾀ07 ｧ00506-05 00 0 ｧ: ) a x dy; ) a dy a,y ; 3) 0 0 0 a,y 0 x 0ｷ3 0060 ｧ0504 0403 縺07 閨0309 ｧ0050605 罍ｧ 縺0304 0 ｧ07. ｰ09 0 ｧ06. ｪ05030303 04030006 ｧ0504 06 040060 ｧ07 00 ｧ0 ｧ08, 0305 0 ｧ ｧ04 060 ｧ08 030300 ﾀ0504, 00607003 04 縺06 閨060003 ｧ0050605 00 0403 04 罍ｧ 040070504 x, 04 縺06 閨060003 ｧ00506-05 00 0403 dy 04 罍ｧ 040070504 y. 98. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ05 ｧ, 0400030 陦ｧ 09030503 縺0308 0403 07 0605 ﾀ07 ｧ0-,x 0 0 050605 00 0 ｧ: ) 0 x dy; ), dy y,4 a. a,x dy. 0ｷ3 0060 ｧ0504 0403 縺07 閨0309 ｧ0050605 罍ｧ 縺0304 0 ｧ07 ｧ 0403 ｧ ﾀ00 ｧ0504 0400030 陦ｧ. 99. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04, 0305 0 ｧ 06000606 00 ｧ0 ｧ070 ｧ r = = a(, cos ') ｧ r = a ｧ ﾀ040300030706000606 04006 09 縺0605. 300. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04, 0305 0 ｧ 06000606 04 縺0700308 r cos ' = a ｧ 0309 縺0607003 ﾀ050406 r =a. 6 503 ｧ ﾀ00 ｧ0504 0400030 陦ｧ, 0305 0 ｧ 06000306 00 ｧ0 ｧ070 ｧ: 30. xy = a, xy =a, y x =, y =x. ｰ09 0 ｧ06. 6 5 06 30 04030503 閨003 0406 縺060805 ｧ 09 00304030 090303 罍陦ｧ0-05 0 xy = u ｧ y = vx, 0403 ﾀ0006 060503 0400030 閨04 0304 縺06 閨0600070605 ﾀ07 0403 0703 縺0060006

38 0600. 3. 6 704030800306, 05 縺030800306 ｧ 09 罍ｧ040300 ｧ0060800306 ｧ0050605 0003 @x @y jjj du dv, 05 閨06 J @u @u = ｧ0 0304 0605 ﾀ07 07090303 ｧ 0030. 6 5 06 30 @x @y @v @v 0403000307 ｧ0504 y = ux, vy = x 3, 04 06 303 0406 縺060805 ｧ 09 0303030300600030 04030007 縺0030 090303 罍陦ｧ0 05 0 x = r cos 3 ' ｧ y = r sin 3 '. 30. y = ax, y =6ax, ay = x 3, 6ay = x 3. 303. x =3 + y =3 = a =3. 6 503 ｧ ﾀ00 ｧ0504 0400030 陦ｧ, 0305 0 ｧ 06000306 00 ｧ0 ｧ070 ｧ: 304. y = x, y = x +. 305. ax = y, ay ｧ y + x =0. 306. y = sin x, y = cos x ｧ x =0. 307. y = a, ax, y = a + x. 308. r = 4( + cos '), r cos ' =3( ﾀ04 04 0305 04 縺0700308). 309. r = a(, cos '), r = a ｧ ﾀ040300030706000606 04006 09 罍陦ｧ03 ｧ 閨03. 30. xy =, xy =8, y = x, y =8x. 3. 703 ﾀ05 縺03 ｧ0504 030300 ﾀ05 ｧ, 0400030 陦ｧ 09030503 縺0308 0403 07 0605 ﾀ07 ｧ00506-05 00 0 ｧ: ) b a x a dy; ) a 0 dy a,y ay ; 3) 4 0 8,x x dy. 0ｷ3 0060 ｧ0504 0403 縺07 閨0309 ｧ0050605 罍ｧ 縺0304 0 ｧ07 ｧ 0403 ｧ ﾀ00 ｧ0504 0400030 陦ｧ. x. ｨC06005 0 ﾀ ﾀ ｧ 003006005 ｧ006 縺09 ｧ ｧ 0400030 陦ｧ ﾀ 04003006 縺003 ﾀ04 縺06 閨060006000308 0 ﾀ ﾀ0308 (04 罍ｧ 04000305003 ﾀ05 ｧ =) 070303 罍陦ｧ0 0503 0906005 0 ﾀ ﾀ 0400030 陦ｧ S ﾀ 04003006 縺003 ﾀ04 縺06 閨060006000308 0 00608 0 ﾀ ﾀ0308: RR RR xdy ydy x c = ; y c = : () S S 030300600503 ｧ006 縺09 ｧ ｧ 0400030 陦ｧ S: J x = y dy; J y = x dy; J 0 = r dy: () (S) (S) (S)

3x. ｨC06005 0 ﾀ ﾀ ｧ 003006005 ｧ006 縺09 ｧ ｧ 0400030 陦ｧ 9 0504 縺06 閨0600 ｧ0504 0906005 0 ﾀ ﾀ 0400030 陦ｧ, 0305 0 ｧ 06000308 00 ｧ0 ｧ070 ｧ: 3. y =0 ｧ 03 閨00308 0403000604030000308 ﾀ ｧ006 ﾀ03 ｧ 閨03 y = sin x. 33. y = x, x =4, y =0: 34. y = ax ｧ y = x. 35. x + y = a ｧ y =0. 36. 0504 縺06 閨0600 ｧ0504 0906005 0 ﾀ ﾀ 0400030 陦ｧ, 0305 0 ｧ 06000308 ﾀ05 縺03 ｧ- 閨0308 x =3 + y =3 = a =3 ｧ03 ﾀ0406 Ox. ｰ09 0 ｧ06. 706 縺060805 ｧ 09 0303030300600030 04030007 縺0030 090303 罍陦ｧ0 05 0 x = r cos 3 ' ｧ y = r sin 3 ': 37. 0504 縺06 閨0600 ｧ0504 00300600503 ｧ006 縺09 ｧ ｧ J x, J y ｧ J 0 0400030 陦ｧ 04 縺07-00306050300040 ｧ09, 0305 0 ｧ 0600030503 00 ｧ0 ｧ070 ｧ x = 0, x = a, y = 0 ｧ y = b. 38. 0504 縺06 閨0600 ｧ0504 003006005 ｧ006 縺09 ｧ ｧ 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Ox 040003-0 陦ｧ, 0305 0 ｧ 06000308 00 ｧ0 ｧ070 ｧ y = x=, x = a, y = a. 39. 0504 縺06 閨0600 ｧ0504 003006005 ｧ006 縺09 ｧ ｧ 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oy 040003-0 陦ｧ 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(0; a), B(a; 0) ｧ C(a; a). 6 5 08 30{33 0304 縺06 閨0600 ｧ0504 04030007 縺00308 003006005 ｧ006 縺09 ｧ ｧ 0400030 陦ｧ, 0305 0 ｧ 06000308 00 ｧ0 ｧ070 ｧ: 30. x + y = a, x =0, y =0: 3. r = a cos '. 3. 0509 縺0607003 ﾀ050406 r = a. 33. y = ax, x = a. 0504 縺06 閨0600 ｧ0504 0906005 0 ﾀ ﾀ: 34. 7030006 ﾀ0605006005 04 03030003 y = ax, x = a, y = 0 (04 罍ｧ y>0). 35. 7030006050000 ｧ04 ﾀ x a + y =,0305 ﾀ06 0600030503 03 ﾀ0406 Ox. b 36. 0504 縺06 閨0600 ｧ0504 003006005 ｧ006 縺09 ｧ ｧ 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oy 040003-0 陦ｧ, 0305 0 ｧ 06000308 00 ｧ0 ｧ070 ｧ y = a + x, y =x ｧ x =0. a 37. 0504 縺06 閨0600 ｧ0504 003006005 ｧ006 縺09 ｧ ｧ 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Ox 040003-0 陦ｧ 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(; ), B(; ), C(3; 3). 0504 縺06 閨0600 ｧ0504 04030007 縺00308 003006005 ｧ006 縺09 ｧ ｧ 0400030 陦ｧ, 0305 0 ｧ 060-00308 00 ｧ0 ｧ070 ｧ: 38. x a + y b =, x =0, y =0. 39. y =4, x ｧ y =0: 330. r = a(, cos ').

330 0600. 3. 6 704030800306, 05 縺030800306 ｧ 09 罍ｧ040300 ｧ0060800306 ｧ0050605 0003 x 3. 6 503 ｧ ﾀ00060 ｧ06 03030060 ﾀ 040300300406 閨0403080030503 ｧ0050605 00 05030060 050600, 0305 0 ｧ 0600030503 ﾀ0406 縺0806 04030406 縺08003 ﾀ050406 z = F (x; y), ﾀ0 ｧ- 06 040003 ﾀ0903 ﾀ050406 z = 0 ｧ ﾀ 0303090304 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ090308 04030406 縺08003 ﾀ050406, 0403 縺06 0600608 0 040003 ﾀ0903 ﾀ05 ｧ xoy 030300 ﾀ0504 (S), 04060 V = zdy= F (x; y) dy: (S) (S) 6 503 ｧ ﾀ00 ｧ0504 0303006003 050600, 0305 0 ｧ 06000308 04030406 縺08003 ﾀ05070 ｧ: 33. z = x + y, x + y =4, x =0, y =0, z =0. 33. z = x + y + a, y = ax, x = a, z =0, y =0(04 罍ｧ y>0). 333. (x + y) + az = a, x =0, y =0, z =0(04030406 縺08003 ﾀ0504 0403 ﾀ05 縺03 ｧ0504 0403 ﾀ06 060 ｧ070: x = 0, y = 0, z = 0, z = h 6 a; ﾀ0. 06 546). 334. x + y = a, x + z = a ( ﾀ0. 06 55). 335. z = xy, x = a, x =0, y = a, y =0. 336. az = x, y, z =0, x = a. 337. z = xy, x + y = a. 338. x + y + z =3a, x + y = a, z =0. ｰ 09 0 ｧ 06. 6 5 08 338{344 0406 縺060805 ｧ 09 04030007 縺0030 090303 罍陦ｧ0-05 0. 339. z = mx, x + y = a, z =0. 340. az = a, x, y, z =0. 34. x + y + z =4a, x + y = a (04006 09 ｧ00 ｧ0 陦 ). 34. x + y + z = a, x + y ax =0(0400605 罍ｧ 09 ｧ00 ｧ0 陦縺0304). 343. 706 縺04030 04 ｧ0509030 050600 ｧ0903 ｧ y = x tg z 0400605 罍ｧ 09 ｧ00 ｧ0 陦 a x + y = a ｧ 040003 ﾀ0903 ﾀ050406 z =0. 344. z =ax, x + y = ax. 345. z x =, c a, y b, z =0. ｰ 09 0 ｧ 06. 6 5 08 345 ｧ 346 0406 縺060805 ｧ 09 0303030300600030 (050000 ｧ- 0405 ｧ 06 ﾀ09 ｧ0) 04030007 縺0030 090303 罍陦ｧ0 05 0: x = ar cos ', y = br sin '. 346. z = ce (,x =a ),(y =b ) ｧ x a + y b =. 347. x =3 + y =3 + z =3 = a =3 (0403000307 ｧ0504 x = r cos 3 ', y = = r sin 3 ').

3x 4. 700030 陦ｧ 09 罍ｧ040308 04030406 縺08003 ﾀ050608 3 6 503 ｧ ﾀ00 ｧ0504 0303006003 050600, 0305 0 ｧ 06000308 04030406 縺08003 ﾀ05070 ｧ: 348. z = a, x, y = ax ｧ z =0. 349. z = x + y, y = x, y =, z =0. 350. y + z =4ax, y = ax, x =3a (04006 09 ｧ00 ｧ0 陦 ). 35. x a + z b =, x a + y b =. 35. 0703003 ｧ x y + h z = a y 04 罍ｧ 0 6 y 6 h ( ﾀ0. 06 559). 353. x =3 + z =3 = a =3, x =3 + y =3 = a =3. 354. 4z =6, x, y, z =0, x + y =4(04006 09 ｧ00 ｧ0 陦 ). ｰ 09 0 ｧ 06. 6 5 08 354{358 0406 縺060805 ｧ 09 04030007 縺0030 090303 罍陦ｧ0-05 0. 355. z =(x + a), x + y = a. 356. z = 4 x + y, z =0, x + y =, x + y =4. 357. az = x + y, z =0, x + y ax =0. 358. az = a, x, y, z = 0, x + y ax = 0(0400605 罍ｧ 09 ｧ00 ｧ0 陦縺0304). 359. x a + y b + z c =. ｰ09 0 ｧ06. 703000307 ｧ0504 x = ar cos ', y = br sin '. x 4. 700030 陦ｧ 09 罍ｧ040308 04030406 縺08003 ﾀ050608 700030 閨04 ﾀ05 ｧ 04030406 縺08003 ﾀ05 ｧ F (x; y; z) =0, 04 縺03060909 ｧ07 09030503 縺0308 0 040003 ﾀ0903 ﾀ0504 z =00304 縺06 閨0600070605 030300 ﾀ0504 ( z ), 040 (@F=@x) +(@F=@y) +(@F=@z) = dy = j@f=@zj ( z) = ( z) N dy: j@f=@zj 縺0 000305 ｧ 003 04 罍ｧ 04 縺03060905 ｧ 縺0304 0 ｧ ｧ 0 閨0406 陦縺0605 ｧ06 090303 罍陦ｧ0 0500306 040003 ﾀ- 0903 ﾀ05 ｧ 04030006 ｧ0 = ( y ) N dz; j@f=@yj = ( x) N dy dz: j@f=@xj

33 0600. 3. 6 704030800306, 05 縺030800306 ｧ 09 罍ｧ040300 ｧ0060800306 ｧ0050605 0003 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04: 360. 7030406 縺08003 ﾀ05 ｧ 09 ｧ00 ｧ0 陦 z = x,0305 ﾀ06 06000308 040003 ﾀ0903 ﾀ05070 ｧ y = x=, y =x, x =. 36. 7030406 縺08003 ﾀ05 ｧ 0903006 ﾀ z = xy, 0305 ﾀ06 06000308 040003 ﾀ0903 ﾀ05070 ｧ x = a ｧ y = a, 04 罍ｧ x > 0 ｧ y > 0. 36. 7030406 縺08003 ﾀ05 ｧ 0903006 ﾀ y + z = x, ﾀ040300030706000308 0400605 罍ｧ 09 ｧ00 ｧ0 陦 x + y = a. 363. 7030406 縺08003 ﾀ05 ｧ az = xy, ﾀ040300030706000308 0400605 罍ｧ 09 ｧ00 ｧ0 陦 x + y = a. 364. 7030406 縺08003 ﾀ05 ｧ 0903006 ﾀ x + y = z, ﾀ040300030706000308 0400605 罍ｧ 09 ｧ00 ｧ0 陦 z =x. 6 503 ｧ ﾀ00 ｧ0504 0400030 閨04: 365. 7030406 縺08003 ﾀ05 ｧ 09 ｧ00 ｧ0 陦 x + z = a, ﾀ040300030706000308 04006-05 罍ｧ 09 ｧ00 ｧ0 陦 x + y = a. 366. 7030406 縺08003 ﾀ05 ｧ 00 x + y + z = a, ﾀ040300030706000308 04006-05 罍ｧ 09 ｧ00 ｧ0 陦縺0304 x + y ax =0. 367. 7030406 縺08003 ﾀ05 ｧ 04 03030003 ｧ x + y =az, ﾀ040300030706000308 0400605 罍ｧ 09 ｧ00 ｧ0 陦 x + y =3a. 368. ｮ 040300300406 閨0403080030503 ｧ0050605 00 0304 縺06 閨0600 ｧ0504 0400030 閨04 - ﾀ05 ｧ 06000308 04030406 縺08003 ﾀ05 ｧ, 0305 0 ｧ 06000308 006 罍ｧ 陦ｧ 0 0 ｧ 0 ｧ, 050904-0503 縺030 ｧ 04 000006000406. 7 ﾀ ﾀ00305 縺060504 ﾀ0500308 ﾀ0006 08 04 罍ｧ =30, =60. x 5. ｯ 縺030800308 ｧ0050605 00 ｧ 060503 04 罍ｧ000307060 ｧ07 ｯ 縺03080030 ｧ0050605 00030 0305 070600909 ｧ ｧ f(x; y; z), ﾀ04 縺03 ﾀ05 00600030 0 030300 ﾀ0504 (V ),0 0304 0605 ﾀ07 04 縺06 閨0600 ﾀ030305040605 ﾀ0504060600608 05 縺0608006 縺00308 ｧ0050605 0004-00308 ﾀ060003: (V ) F (x; y; z) dy dz = lim max x i!0 max y j!0 max z k!0 XXX i j k F (x i ;y j ;z k )x i y j z k ; 05 閨06 x i = x i+, x i, y j = y j+, y j, z k = z k+, z k ｧ ﾀ0600 ﾀ04 縺03- ﾀ05 0060 0 0506 0 060 ｧ07 j ｧ k, 閨0007 09030503 縺0308 0503 09 ｧ (x i ; y j ; z k ) 04 罍ｧ0 閨0006-07 05 030300 ﾀ05 ｧ (V ). ｭ ﾀ00 ｧ 030300 ﾀ0504 (V ) 0304 縺06 閨0600060 006 04060 ﾀ0504 0 ｧ 0503 a 6 x 6 b; y (x) 6 y 6 y (x); z (x; y) 6 z 6 z (x; y); (V ) F (x; y; z) dy dz = b a y (x) y (x) dy z (x; y) z (x; y) F (x; y; z) dz:

3x 5. ｯ 縺030800308 ｧ0050605 00 ｧ 060503 04 罍ｧ000307060 ｧ07 33 7 罍ｧ F (x; y; z) =04030006 060 03030060 V. 070303 罍陦ｧ0 0503 0906005 0 ﾀ ﾀ 03 閨003- 縺03 閨0030503 050600 03030060030 V 0403 ｧ ﾀ00070605 ﾀ07 0403 0703 縺00600 0: x c = V (V ) xdydz; y c = V (V ) ydydz; z c = V (V ) zdydz: 369. 0504 縺06 閨0600 ｧ0504 03030060 050600, 0305 0 ｧ 0600030503 04030406 縺08003 ﾀ05070 ｧ az = x + y, az = a, x, y. 370. 0504 縺06 閨0600 ｧ0504 03030060 050600, 0305 0 ｧ 0600030503 04030406 縺08003 ﾀ05070 ｧ x + y, z =0, x + y + z = a,0400605 罍ｧ 0903006 ﾀ. 37. 70309 0504, 0503 04030406 縺08003 ﾀ0504 0903006 ﾀ x + y, z =0 閨0600 ｧ05 03030060 00 x + y + z =az 04030500300060 ｧ ｧ 3:. 37. 0504 縺06 閨0600 ｧ0504 0 ﾀ ﾀ06 04 ｧ 0 ｧ 閨03, 0303 0304 000308 040003 ﾀ0903 ﾀ05070 ｧ x + y + z = a, x =0, y =0, z =0, 06 ﾀ00 ｧ 04000305003 ﾀ0504 0409 07 閨0308 0606 0503 0906 040 040400 ｧ09 0506 z 05050308 0503 09 ｧ. 0504 縺06 閨0600 ｧ0504 0906005 0 ﾀ ﾀ 03 閨003 縺03 閨0030503 050600, 0305 0 ｧ 0600030503 0403-0406 縺08003 ﾀ05070 ｧ: 373. x + y + z = a, x =0, y =0, z =0. 374. az = a, x, y, z =0. 0504 縺06 閨0600 ｧ0504 003006005 ｧ006 縺09 ｧ ｧ 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oz 050600, 0305-0 ｧ 0600030503 04030406 縺08003 ﾀ05070 ｧ (04000305003 ﾀ0504 =): 375. x =0, y =0, y = a, z =0, x + z = a. 376. x + y + z = a, x + y = a, z =0. 377. 0504 縺06 閨0600 ｧ0504 03030060 050600, 0305 0 ｧ 0600030503 009006050308 0403-0406 縺08003 ﾀ050406: ) (x + y + z ) = a 3 x; ) (x + y + z ) = az(x + y ). ｰ09 0 ｧ06. 706 縺060805 ｧ 09 ﾀ0706 罍ｧ 06 ﾀ09 ｧ0 090303 罍陦ｧ0 05 0 0403 0703 縺00600 0 x = = r sin cos ', y = r sin sin ', z = r cos '; 050006006005 03030060 dv = = r sin drd'd. 0504 縺06 閨0600 ｧ0504 0303006003 050600, 0305 0 ｧ 06000308 04030406 縺08003 ﾀ05070 ｧ: 378. az = x + y, x + y + z =a. 379. x + y, z =0, z =6, x, y. 380. az = x + y, z = x + y. 38. 0504 縺06 閨0600 ｧ0504 0 ﾀ ﾀ06 050600, 0305 0 ｧ 0600030503 04030406 縺08003 ﾀ05070 ｧ x + y, z = 0 ｧ z = h, 06 ﾀ00 ｧ 04000305003 ﾀ0504 0409 07 閨0308 0503 0906 060503 040 040400 ｧ09 0506 05050308 0503 09 ｧ.

334 0600. 3. 6 704030800306, 05 縺030800306 ｧ 09 罍ｧ040300 ｧ0060800306 ｧ0050605 0003 38. 0504 縺06 閨0600 ｧ0504 0 ﾀ ﾀ06 050600, 0305 0 ｧ 0600030503 04030406 縺08003 ﾀ05070 ｧ x + z =a; x + z = a; y = ax; y =0 (04 罍ｧ y>0), 06 ﾀ00 ｧ 04000305003 ﾀ0504 0409 07 閨0308 060503 0503 0906 040 03 罍陦ｧ0 0506 y 05050308 0503 09 ｧ. 383. 0504 縺06 閨0600 ｧ0504 0906005 0 ﾀ ﾀ 03 閨003 縺03 閨0030503 0403000600 x + y + + z = a, z =0. 384. 0504 縺06 閨0600 ｧ0504 003006005 ｧ006 縺09 ｧ ｧ 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oz 050600, 0305 0 ｧ 0600030503 04030406 縺08003 ﾀ05070 ｧ z =ax, z =0, x + y = ax. 385. 0504 縺06 閨0600 ｧ0504 03030060 050600, 0305 0 ｧ 0600030503 04030406 縺08003 ﾀ050406 (x + y + z ) = axyz (0406 縺060805 ｧ 09 ﾀ0706 罍ｧ 06 ﾀ09 ｧ0 090303 罍陦ｧ0 05 0) ( ﾀ0. 06 377). 386. 0504 縺06 閨0600 ｧ0504 0 ﾀ ﾀ06 ﾀ0706 罍ｧ 06 ﾀ09030503 ﾀ000307 00607 閨06 04030406 縺08003- ﾀ05070 ｧ x + y + z = a ｧ x + y + z = 4a, 06 ﾀ00 ｧ 04000305003 ﾀ0504 0409 07 閨0308 060503 0503 0906 0303 05003 04 縺030403 縺09 ｧ030 00040 ﾀ ﾀ0503070 ｧ06 0305 0503 09 ｧ 閨03 0 00 090303 罍陦ｧ0 05 (0406 縺060805 ｧ 09 ﾀ0706 罍ｧ 06 ﾀ09 ｧ0 090303 罍陦ｧ0 05 0). x 6. 07 罍ｧ040300 ｧ0060800308 ｧ0050605 00. ｱ03 縺00600 06 罍ｧ0. 0504 縺06 閨0600060 ｧ06 09 罍ｧ040300 ｧ006080030503 ｧ0050605 00. 706 ﾀ0504 0 閨060506 AB, ^ ﾀ04 縺07000070600308 09 罍ｧ040308, 0304 縺06 閨0600060 00604 縺06 縺03040 07 070600909 ｧ07 P (x;y;z). 7 030304060 閨060506 0 ﾀ05 ｧ 0503 09 0 ｧ A(x 0 ; y 0 ; z 0 ), M (x ; y ; z ), ::: :::, M n, (x n, ; y n, ; z n, ) ｧ B(x n ; y n ; z n ) ｧ0406 ﾀ0504 x i,x i, =x i. ｯ03-05 lim np x i!0 i= P (x i ;y i ;z i )x i 0 0304 0605 ﾀ07 09 罍ｧ040300 ｧ006080030 ｧ0050605 00030, 04 0705030 0403 閨060506 AB, ^ ｧ 030303 0 0605 ﾀ07 R^AB P (x; y; z). 縺0 000305 ｧ 003 0304 縺06 閨06-00070605 ﾀ07 ｧ0050605 0003 R^AB R R Q(x; y; z) dy, R(x; y; z) dz ｧ P+ Qdy + + Rdz 09 09 ﾀ0600 04 縺06 閨03 閨060 ｧ08 ｧ0050605 000304. 7 090300609, 04 ﾀ05 縺06 0605 ﾀ07 06006 09 罍ｧ040300 ｧ0060800308 ｧ0050605 00 04 ｧ ^ AB P (x; y; z) ds = lim nx s i!0 i= ^ AB ^ AB P (x i ;y i ;z i )s i ; 05 閨06 s i = M i, M i :. 6 503 ｧ ﾀ00060 ｧ06 09 罍ｧ040300 ｧ006080030503 ｧ0050605 00. 706 ﾀ0504 09 罍ｧ- 04 07 AB ^ 0 06 040060 ｧ070 ｧ x = f(t), y = '(t), z = (t), 04 00605 t 04 罍ｧ 0406 縺060060060 ｧ ｧ 0503 09 ｧ M (t) 0403 閨060506 AB ^ 04 03 閨0030 0 04 0400060 ｧ ｧ ｧ 006007-0605 ﾀ07 00300305030003; 050305 ^ AB P (x; y; z) = t B t A P f(t); '(t); (t) f 0 (t) dt;

3x 6. 07 罍ｧ040300 ｧ0060800308 ｧ0050605 00. ｱ03 縺00600 06 罍ｧ0 35 05. 06. 04 ﾀ06 0406 縺06006000306 ｧ 陦ｧ070706 縺06009 ｧ 0003 0403 0 09030 09 罍ｧ040300 ｧ006080030503 ｧ0-050605 00 00607003 0403 ｧ0504 06 縺06 03 閨006 0406 縺06006000606 (t) ｧ 0606 陦ｧ070706 縺06009 ｧ 00 (dt) ｧ 06 040060 ｧ08 09 罍ｧ040308. 3.030608 0 ｧ 06 ﾀ090306 0 060 ｧ06 09 罍ｧ040300 ｧ006080030503 ｧ0050605-00. 0ｷ30050605 00 04 ｧ R^AB P+ Qdy+ Rdz 0304 縺06 閨0600070605 03030506 04 罍ｧ 0406 縺06006-0060 ｧ ｧ 06 陦ｧ0 ｧ0903 0 ﾀ ﾀ03 0403 閨060506 AB 04 04030006, 0303 0304 00030 ﾀ ｧ000308 FfP ; Q; Rg. 4. ｮ0006 08 0403000030503 陦ｧ070706 縺06009 ｧ 00. ｭ ﾀ00 ｧ0400609030503 縺0308 030300 - ﾀ05 ｧ (V ) P+ Qdy + Rdz = du, 0503 R^AB P+ Qdy + Rdz = u B, u A, 05. 06. 04060 003 ﾀ05 ｧ 0 060 ｧ08 070600909 ｧ ｧ u(x; y; z) 040503 09 08 B ｧ A ｧ 006 04 ｧ ﾀ ｧ05 0305 040605 ｧ ｧ0050605 罍ｧ 縺0304 0 ｧ07 AB, 04 0705030503 04 030300 ﾀ05 ｧ (V ). 5. ｱ03 縺00600 06 罍ｧ0 I (C) P+ Qdy = (S) @Q @P @x, dy @y 04 縺060303 060605 09 罍ｧ040300 ｧ0060800308 ｧ0050605 00 0305 P+ Qdy, 04 07050308 (04 縺0305 ｧ04 ﾀ03040308 ﾀ05 縺060009 ｧ) 0403 0090060503006 090300506 縺06 (C), 04 閨04030800308 ｧ0050605 00 0403 030300 ﾀ05 ｧ (S), 0305 0 ｧ 06000308 0505 ｧ0 090300506 縺030. 6. 7 00 03 0 04, 0305 0 ｧ 0600 07 090300506 縺030 (C): S = I (C) xdy, y : 387. 6 7 003 0503 09 ｧ A(; ) ｧ B(; 0). 6 503 ｧ ﾀ00 ｧ0504 (C) (x + y) : ) 0403 04 縺0700308 OA; )0403 閨060506 OA ^ 04 03030003 y = x ; 3) 0403 00030 00308 OBA. 388. 6 7 003 0503 09 ｧ A(4; ) ｧ B(; 0). 6 503 ｧ ﾀ00 ｧ0504 (C) (x + y), xdy : ) 0403 04 縺0700308 ^ OA; ) 0403 00030 00308 OBA. 389. 70600 ｧ0504 06 388 閨0007 ｧ0050605 00 (C) y+ xdy: 703 06006 閨06 ﾀ04 040600 ｧ ｧ0 ｧ0050605 00 006 04 ｧ ﾀ ｧ05 0305 040605 ｧ ｧ0050605 罍ｧ 縺03-04 0 ｧ07?

336 0600. 3. 6 704030800306, 05 縺030800306 ｧ 09 罍ｧ040300 ｧ0060800306 ｧ0050605 0003 390. 6 7 003 0503 09 ｧ A(a; 0;0), B(a; a; 0) ｧ C(a; a; a). 6 503 ｧ- ﾀ00 ｧ0504 ｧ0050605 00 y+ zdy+ xdz 0403 04 縺0700308 OC ｧ 0403 00030 00308 OABC. 39. 7030006 0303 0304 003 ﾀ ｧ000308 FfP ; Qg, 05 閨06 P = x, y, Q = = x. 703 ﾀ05 縺03 ｧ0504 ﾀ ｧ0006 F 0409 07 閨0308 0406 縺00 ｧ006 0904 陦 05 ﾀ03 ﾀ0503 縺030 0 ｧ x = a ｧ y = a ｧ 0403 ｧ ﾀ00 ｧ0504 03030506 04 罍ｧ 0406 縺060060060 ｧ ｧ 06 陦ｧ0 ｧ0903 0 ﾀ ﾀ03 0403 090300506 縺06 0904 陦 05. 39. 7030006 0303 0304 003 ﾀ ｧ000308 FfP ; Qg, 05 閨06 P = x + y, Q = = x. 703 ﾀ05 縺03 ｧ0504 ﾀ ｧ0006 F 040 0006 09 07 閨0308 06050406 縺05 ｧ 0309 縺0607003 ﾀ05 ｧ x = a cos t, y = a sin t ｧ 0403 ｧ ﾀ00 ｧ0504 03030506 04 罍ｧ 0406 縺060060060 ｧ ｧ 06 陦ｧ- 0 ｧ0903 0 ﾀ ﾀ03 0403 0309 縺0607003 ﾀ05 ｧ. 70600 ｧ0504 050506 0706 06 04 罍ｧ 06 ﾀ000304 ｧ ｧ P = x + y, Q = x. 703 06006 閨06 ﾀ04 030305 040 0? 393. 7030006 0303 0304 003 ﾀ ｧ000308 Ffy; ag. 0504 縺06 閨0600 ｧ0504 03030506 04 罍ｧ 0406 縺060060060 ｧ ｧ 0 ﾀ ﾀ03 m 0403 090300506 縺06, 0303 0304 0003006 0403000603 ﾀ070 ｧ 0903-03 罍陦ｧ0 05 ｧ 0406 縺040308 06050406 縺050406 050000 ｧ04 ﾀ x = a cos t, y = b sin t. 394. 7030006 0303 0304 003 ﾀ ｧ000308 Ffx; y; zg. 6 503 ｧ ﾀ00 ｧ0504 03030506 04 罍ｧ 0406 縺060060060 ｧ ｧ 06 陦ｧ0 ｧ0903 0 ﾀ ﾀ03 0403 00030 00308 OABCO, ﾀ0306 陦ｧ0070600608 0503 09 ｧ O(0; 0; 0), A(0; a; 0), B(a; a; 0), C(a; a; a). 395. 7 04 ｧ ﾀ 0504 ｧ 04 縺030406 罍ｧ0504 0703 縺0060006 06 罍ｧ0 閨0007 I (C) (x + y), xdy 0403 090300506 縺06 05 縺0606050300040 ｧ09 ﾀ03 ﾀ0503 縺030 0 ｧ x =0, y =0, x + y = a. 396. 6 503 ｧ ﾀ00 ｧ0504 ｧ0050605 0003: ) ^ 3) AB ^ AB xy + x dy; ) tg y+ x sec ydy ^ AB cos y, x sin ydy; 0403 0006030308 00 ｧ0 ｧ ｧ 0305 0503 09 ｧ A(; =6) 閨03 B(; =4). 397. 7 罍ｧ0060 ｧ04 0703 縺0060006 06 罍ｧ0, 0403 ｧ ﾀ00 ｧ0504 ｧ0050605 00 I (C) y +(x + y) dy 0403 090300506 縺06 4ABC ﾀ 0406 縺00 ｧ0 0 ｧ A(a; 0), B(a; a) ｧ C(0; a).

3x 6. 07 罍ｧ040300 ｧ0060800308 ｧ0050605 00. ｱ03 縺00600 06 罍ｧ0 37 398. 0504 縺06 閨0600 ｧ0504 09 罍ｧ040300 ｧ006080030 ｧ0050605 00030 0400030 閨04 050000 ｧ- 04 ﾀ x = a cos t, y = b sin t. 399. 0504 縺06 閨0600 ｧ0504 09 罍ｧ040300 ｧ006080030 ｧ0050605 00030 0400030 閨04 04060500 ｧ 09 罍ｧ040308 x 3 + x, y =0( ﾀ0. 罍ｧ ﾀ. 48 0 ﾀ. 304). ｰ09 0 ｧ06. 706 縺060805 ｧ 09 04 00605 罍ｧ 06 ﾀ09 ｧ0 06 040060 ｧ070, 0403000307 ｧ04 y = = xt. 400. 0504 縺06 閨0600 ｧ0504 09 罍ｧ040300 ｧ006080030 ｧ0050605 00030 0400030 閨04 04060500 ｧ 閨0609 縺050304 00 ｧ ﾀ05 x 3 + y 3, 3axy =0( ﾀ0. 0609 0 ｧ06 09 06 399 ｧ 罍ｧ ﾀ. 79 0 ﾀ. 334). 40. ｮ09 090308 ﾀ ｧ000308 04 罍ｧ050705 ｧ04 0605 0 ﾀ ﾀ M, 04003006 縺003 ﾀ- 04 縺06 閨06000600 07 0403 0406 縺0800608 040300060309 縺0607003 ﾀ05 ｧ x + y = a, 0 ﾀ ﾀ06 m, ﾀ03 ﾀ 縺06 閨030503 06000606 040 0006 090303 罍陦ｧ0 05? ｰ09 0 ｧ06. 706 ﾀ0504 00 ｧ006080 07 04000305003 ﾀ0504, ds 050006006005 閨00 ｧ003 040300060309 縺0607003 ﾀ05 ｧ, 06050300 陦ｧ06 ﾀ-0406090503 ﾀ 03 ﾀ0406 Ox, X ｧ Y 04 縺03-060909 ｧ ｧ ﾀ ｧ0003 04 罍ｧ050707060 ｧ07. ｯ0305 X = (C) km cos ds r ; Y = 05 閨06 k 05 04 ｧ05 09 ｧ0300 07 0403 ﾀ05030700 07. (C) km sin ds r ; 40. 6 7 003 0503 09 ｧ A(,a; a) ｧ B(a; a). ｮ09 090308 ﾀ ｧ000308 0 ﾀ ﾀ M, 04003006 縺003 ﾀ04 縺06 閨06000600 07 0403 0305 縺06 0906 AB, 04 罍ｧ050705 ｧ04 0605 0 ﾀ ﾀ06 m, ﾀ03 ﾀ 縺06 閨030503 06000606 04 0503 0906 (0; 0). 403. 6 7 003 0503 09 ｧ A(a; 0), B(0; a) ｧ C(,a; 0). ｮ 09 090308 ﾀ ｧ000308 0 ﾀ ﾀ M, 04003006 縺003 ﾀ04 縺06 閨06000600 07 0403 00030 00308 ABC, 04 罍ｧ050705 ｧ- 04 0605 0 ﾀ ﾀ06 m, ﾀ03 ﾀ 縺06 閨030503 06000606 040 0006 090303 罍陦ｧ0 05. 404. 6 7 003 0503 09 ｧ A(0; ), B(; 5) ｧ C(0; 5). 6 503 ｧ ﾀ00 ｧ0504 (C) (x + y), ydy : ) 0403 04 縺0700308 AB; ) 0403 閨060506 AB 04 03030003 y = x +; 3) 0403 00030 00308 ACB. 405. 6 7 003 0503 09 ｧ A(,a; 0) ｧ B(0; a). 6 503 ｧ ﾀ00 ｧ0504 03030506 ﾀ ｧ0003 FfP ; Qg, 05 閨06 P = y ｧ Q = y, x, 04 罍ｧ 0406 縺060060060 ｧ ｧ 06 陦ｧ0 ｧ0903 0 ﾀ ﾀ03: ) 0403 04 縺0700308 AB; ) 0403 00030 00308 AOB; 3)0403 閨060506 AB 04 03030003 y = a, x a. 406. 70309 0504, 0503 I (C) y+(x + y) dy

338 0600. 3. 6 704030800306, 05 縺030800306 ｧ 09 罍ｧ040300 ｧ0060800306 ｧ0050605 0003 0403 00060303006 0090060503006 090300506 縺06 04060 0060006. 7 縺030406 罍ｧ0504, 0403 ｧ ﾀ00 ｧ04 ｧ0050605 00 0403 090300506 縺06 07 ｧ0506 縺03, 0305 0 ｧ 06000308 00 ｧ0 ｧ070 ｧ y = x ｧ y =4. 407. 7 04 ｧ ﾀ 0504 ｧ 04 縺030406 罍ｧ0504 0703 縺0060006 06 罍ｧ0 閨0007 ｧ0050605 00 I (C) y, dy x ; 04 0705030503 0403 090300506 縺06 4ABC ﾀ 0406 縺00 ｧ0 0 ｧ A(; ), B(; ) ｧ C(; ). 408. ｮ 040300300406 09 罍ｧ040300 ｧ006080030503 ｧ0050605 00 0304 縺06 閨0600 ｧ0504 040003-0 閨04 07 ｧ0506 縺03, 0305 0 ｧ 06000308 ﾀ05 縺03 ｧ 閨0308 x = a cos 3 t, y = a sin 3 t. 409. ｮ 040300300406 09 罍ｧ040300 ｧ006080030503 ｧ0050605 00 0304 縺06 閨0600 ｧ0504 040003-0 閨04, 0305 0 ｧ 06000606 09 罍ｧ040308 y + x 4, x =0.( 706 縺060805 ｧ 09 04 006-05 罍ｧ 06 ﾀ09 ｧ0 06 040060 ｧ070, 0403000307 ｧ04 y = xt.) x 7. 7030406 縺08003 ﾀ0500306 ｧ0050605 0003. ｱ03 縺0060003 05 ﾀ05 縺0305 陦ﾀ09030503{06 06 ﾀ ﾀ ｧ ｮ050309 ﾀ. 7030406 縺08003 ﾀ0500306 ｧ0050605 0003. 706 ﾀ0504 F (x; y; z) 00604 縺06- 縺03040 07 070600909 ｧ07 ｧ z = '(x; y) 06 040060 ｧ06 04030406 縺08003 ﾀ05 ｧ S, 04 罍ｧ 060 @'(x; y) @'(x; y) ﾀ06006 ﾀ0504060605 ｧ. @x @y 7030406 縺08003 ﾀ0500308 ｧ0050605 00 0406 縺04030503 05 ｧ04 04 縺06 陦ﾀ05 0400070605 ﾀ03030308 04 縺06 閨0600 ｧ0050605 000400308 ﾀ060003: (S) F (x; y; z) ds = lim n! nx i= F (x i ;y i ;z i )S i ; 05 閨06 S i 0400030 閨04 i-0503 050006006005 04030406 縺08003 ﾀ05 ｧ S, 0503 09 (x i ; y i ; z i ) 04 罍ｧ- 0 閨000607 ｧ05 050503006 05000600600506, 04 罍ｧ 060 0 09 ﾀ ｧ0 000400308 陦ｧ 00605 0500060060050304 03 ｧ060 ｧ07 ﾀ05 縺060 ｧ05 ﾀ07 09 0060006. ｭ ﾀ00 ｧ 04 縺03060909 ｧ07 04030406 縺08003 ﾀ05 ｧ S 0 040003 ﾀ- 0903 ﾀ0504 xoy 03 閨003 0 0, 05. 06. 04 ﾀ0709 07 04 縺070 07, 04 00000600040 07 03 ﾀ ｧ Oz, 0406 縺06- ﾀ0609 0605 04030406 縺08003 ﾀ0504 S 00 ｧ0004 04 03 閨00308 0503 0906, 0503 ﾀ030305040605 ﾀ050406060 ｧ08 04030406 縺08-003 ﾀ0500308 ｧ0050605 00 0406 縺04030503 05 ｧ04 003070605 03030504 0403 ｧ ﾀ00060 0403 0703 縺0060006 (S) F (x; y; z) ds = = () F x; y; '(x; y) s + @'(x; y) @'(x; y) + dy: @x @y ｭ ﾀ00 ｧ P = P (x; y; z), Q = Q(x; y; z), R = R(x; y; z) 00604 縺06- 縺030400306 070600909 ｧ ｧ ｧ S + ﾀ0503 縺030 04030406 縺08003 ﾀ05 ｧ S, 08 090506 罍ｧ 06060 07 0-04 0400060 ｧ060 040060000608 003 縺0 00 ｧ nfcos ; cos ; cos g,0503 ﾀ030305040605 ﾀ050406060 ｧ08

3x 7. 7030406 縺08003 ﾀ0500306 ｧ0050605 0003 39 04030406 縺08003 ﾀ0500308 ｧ0050605 00 040503 縺030503 縺03 0403 07 0605 ﾀ07 ﾀ0006 閨06060 ｧ0 0303 030: Pdydz+ Qdz+ Rdy = (P cos + Q cos + R cos ) ds: (S + ) (S) (S). ｱ03 縺00600 05 ﾀ05 縺0305 陦ﾀ09030503{06 06 ﾀ ﾀ : @P (P cos + Q cos + R cos ) ds = @x + @Q @y + @R @z (V ) dy dz; 05 閨06, ｧ 06050003 040060000608 003 縺0 00 ｧ 009006050308 04030406 縺08003 ﾀ05 ｧ S ｧ 03 ﾀ070 ｧ 090303 罍陦ｧ0 05, V 03030060 050600, 0305 0 ｧ 0600030503 05050308 04030406 縺08003 ﾀ050406. 706-040308 ｧ0050605 00 00307003 04 ｧ ﾀ 0504 04 04 ｧ 閨06 (S z ) P @F @x + Q@F @y + R@F @z dy @F=@z ; 05 閨06 F (x; y; z) = 0 06 040060 ｧ06 04030406 縺08003 ﾀ05 ｧ, S z 04 縺03060909 ｧ07 S 0 040003 ﾀ0903 ﾀ0504 xoy. 3. ｱ03 縺00600 ｮ050309 ﾀ : I (C) = (S) P+ Qdy+ Rdz = @R @Q @P @y, cos + @R @Q @z @z, cos + @P @x @x, cos ds; @y 05 閨06,, 06050003, 0303 0304 000306 03 ﾀ070 ｧ 090303 罍陦ｧ0 05 ﾀ 003 縺0 000406 09 0403-0406 縺08003 ﾀ05 ｧ S, 0 04 040006000308 04 0506 0606 ﾀ0503 縺03006, ﾀ 09030503 縺0308 03030803 090300506 C ﾀ ﾀ0 05 罍ｧ04 0605 ﾀ07 04 縺03 ｧ ﾀ0803 閨070 ｧ0 04 縺0305 ｧ04 ﾀ03040308 ﾀ05 縺060009 ｧ. 40. 6 503 ｧ ﾀ00 ｧ0504 (S) (x cos + y cos + z cos ) ds 0403 0406 縺0800608 04030406 縺08003 ﾀ05 ｧ 040003 ﾀ0903 ﾀ05 ｧ x + y + z = a, ﾀ040300030706000308 04 0406 縺04030 030905 00506. 4. 6 503 ｧ ﾀ00 ｧ0504 (S) x cos (n; i) +y cos (n; j) +z cos (n; k) ds 0403 0406 縺0800608 04030406 縺08003 ﾀ05 ｧ 04 03030003 ｧ x + y +az = a, ﾀ04030003-0706000308 0403 040503 縺030 030905 00506 (05 閨06 x<0, y>0, z>0).

340 0600. 3. 6 704030800306, 05 縺030800306 ｧ 09 罍ｧ040300 ｧ0060800306 ｧ0050605 0003 ｰ 09 0 ｧ 06. 7 罍ｧ0406 閨07 ｧ0050605 00 09 04 ｧ 閨06 (x 3 + y 3 + az dy ) a, 0406- (S z) 縺060805 ｧ 09 04030007 縺0030 090303 罍陦ｧ0 05 0. ｰ050300 ' 0306 閨0605 ｧ 0060070504 ﾀ07 0305 = 閨03. 4. 7 04 ｧ ﾀ 0504 ｧ 04 縺030406 罍ｧ0504 0703 縺0060006 05 ﾀ05 縺0305 陦ﾀ09030503 閨0007 ｧ0-050605 00 (S) x cos (n; i) +y cos (n; j) +z cos (n; k) ds; 04 0705030503 0403 04030406 縺08003 ﾀ05 ｧ 00 x + y + z = a. 43. 7 04 ｧ ﾀ 0504 ｧ 04 縺030406 罍ｧ0504 0703 縺0060006 05 ﾀ05 縺0305 陦ﾀ09030503{06 06 ﾀ ﾀ 閨0007 ｧ0050605 00 (S) x cos (n; i) +y cos (n; j) +z cos (n; k) ds; 04 0705030503 0403 0 縺060700308 04030406 縺08003 ﾀ05 ｧ 050600, 0305 0 ｧ 0600030503 04030406 縺08003- ﾀ05070 ｧ x + y +az = a, x =0, y =0, z =0,0400605 罍ｧ 0406 縺04030503 030905 005. ｰ 09 0 ｧ 06. 6 704030800308 ｧ0050605 00 0403 040003 ﾀ09 ｧ0 05 0070 050600 04060 0, ｧ0303, 0 04 罍ｧ006, 0 040003 ﾀ0903 ﾀ05 ｧ z =0 ｧ cos (n; i) =0 ｧ cos (n; j) =0. 44. 70300 05 07 04 0703 縺0060006 05 ﾀ05 縺0305 陦ﾀ09030503{06 06 ﾀ ﾀ P = x, Q = y, R = z, 04030006 ｧ0504 0703 縺0060006 閨0007 03030060 : V = [x cos + y cos + z cos ] ds: 3 (S) 6 503 ｧ ﾀ00 ｧ0504 040305050308 0703 縺0060006 03030060 050000 ｧ04 ﾀ03 ｧ x a + y b + z c =: 45. 70300 05 07 04 0703 縺0060006 05 ﾀ05 縺0305 陦ﾀ09030503{06 06 ﾀ ﾀ P @u = @x, Q = = @u @y ｧ R = @u (05. 06. 040300 05 07 0406090503 fp ; Q; Rg 040030 grad u), @z 閨0309 0504, 0503 (V ) udydz = (S) du dn ds; 05 閨06 u = @ u @x + @ u @y + @ u 030406 0503 6ｧ8 0400 ﾀ. @z

3x 7. 7030406 縺08003 ﾀ0500306 ｧ0050605 0003 4 46. 7 縺030406 罍ｧ0504 04030006 06000606 04 04 縺06 閨03 閨0600608 06 0703 縺0060006 閨0007 070600909 ｧ ｧ u = x + y + z 0 04030406 縺08003 ﾀ05 ｧ x + y + z = a. 47. 70309 0504 ﾀ 040300300406 0703 縺0060003 ｮ050309 ﾀ, 0503 (C) yz + xz dy + xy dz 0403 00060303006 0090060503006 090300506 縺06 04060 0060006. 7 縺030406 罍ｧ0504 050503 0403- ｧ ﾀ00060 ｧ060 ｧ0050605 00 0403 090300506 縺06 4OAB ﾀ 0406 縺00 ｧ0 0 ｧ O(0; 0; 0), A(; ; 0) ｧ B(; ; ). 48. 7 04 ｧ ﾀ 0504 ｧ 04 縺030406 罍ｧ0504 0703 縺0060006 ｮ050309 ﾀ 閨0007 ｧ0050605 00 I (C) (z, y) +(x, z) dy +(y, x) dz; 04 0705030503 0403 090300506 縺06 4ABC ﾀ 0406 縺00 ｧ0 0 ｧ A(a; 0;0), B(0; a; 0) ｧ C(0; 0; a). ｰ 09 0 ｧ 06. 6 704030800308 ｧ0050605 00 00307003 04 070504 0403 0006030308 04030406 縺08003 ﾀ05 ｧ, 04 縺030803 閨0700608 06 縺06 0406 罍ｧ00605 05 縺0606050300040 ｧ09 ABC, 0 04 罍ｧ006 0403 040003 ﾀ0903 ﾀ05 ｧ x + y + z = a. 49. 7 04 ｧ ﾀ 0504 ｧ 04 縺030406 罍ｧ0504 0703 縺0060006 05 ﾀ05 縺0305 陦ﾀ09030503{06 06 ﾀ ﾀ 閨0007 ｧ0050605 00 (S) x 3 cos (n; i) +y 3 cos (n; j) +z 3 cos (n; k) ds; 04 0705030503 0403 04030406 縺08003 ﾀ05 ｧ 00 x + y + z = a. ｰ 09 0 ｧ 06. ｯ 縺030800308 ｧ0050605 00 04 縺060303 0304 0504 09 ﾀ0706 罍ｧ 06 ﾀ09 ｧ0 090303 - 陦ｧ0 05 0. I 40. 7 04 ｧ ﾀ 0504 ｧ 04 縺030406 罍ｧ0504 0703 縺0060006 ｮ050309 ﾀ 閨0007 ｧ0050605 00 (C) x(z, y) + y(x, z) dy + z(y, x) dz 0403 090300506 縺06 05 縺0606050300040 ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(a; 0; 0), B(0; a; 0) ｧ C(0; 0; a) ( ﾀ0. 0609 0 ｧ06 09 06 48). 4. ｮ 040300300406 0703 縺0060003 05 ﾀ05 縺0305 陦ﾀ09030503{06 06 ﾀ ﾀ 0403 ｧ ﾀ00 ｧ0504 (S) x 3 dy dz + y 3 dz + z 3 dy; 04 07050308 0403 0 縺060700308 04030406 縺08003 ﾀ05 ｧ 04 ｧ 0 ｧ 閨03, 0303 0304 000308 040003 ﾀ- 0903 ﾀ05070 ｧ x + y + z = a, x =0, y =0, z =0.

30600 04 4 7006 76ｧ9 x. ｪ ｧ ﾀ0003040306 縺07 閨03. 707 u + u + u 3 + :::+ u n + ::: 0 0304 0605 ﾀ07 ﾀ0803 閨070 ｧ0 ﾀ07, 06 ﾀ00 ｧ ﾀ0600 S n 060503 n 0406 縺040308 000600304 04 罍ｧ n! ﾀ05 縺060 ｧ05 ﾀ07 09 0903006 003006 04 縺06- 閨060006 S: lim n! S n = S. ｪ ｧ ﾀ0003 S 0 0304 0605 ﾀ07 ﾀ06000308 ﾀ0803 閨070060503 ﾀ07 縺07. 706 ﾀ0803 閨070 ｧ08 ﾀ07 縺07 0 0304 0605 ﾀ07 ﾀ0803 閨070 ｧ0 ﾀ07. 6 70007 ﾀ0803 陦ｧ003 ﾀ05 ｧ 縺07 00603030803 陦ｧ003 (003 006 閨03 ﾀ05 0503 003), 05030303 u n! 0 04 罍ｧ n!.. 0ｷ30050605 000400308 04 罍ｧ 0 09 ﾀ0803 陦ｧ003 ﾀ05 ｧ 縺07 ﾀ 0403000307 ｧ05060004-0030 ｧ 06030304 060 ｧ0 ｧ 00060 0 ｧ: ｭ ﾀ00 ｧ u n = f(n), 05 閨06 f(x) 06030304 060 07 070600909 ｧ07, ｧ A; 0503 縺07 ﾀ0803 陦ｧ05 ﾀ07, f(x) = ; 0503 縺07 ﾀ0803 陦ｧ05 ﾀ07. 3. 7 罍ｧ 0 09 6 7 00 00306 ﾀ0803 陦ｧ003 ﾀ05 ｧ 縺07 ﾀ 0403000307 ｧ050600040030 ｧ 00060 0 ｧ: 06 ﾀ00 ｧ u n+ lim n! u n = r 8 < : < ; 0503 縺07 ﾀ0803 陦ｧ05 ﾀ07, > ; 0503 縺07 ﾀ0803 陦ｧ05 ﾀ07, =; 0503 040304 縺03 ﾀ 03 ﾀ05 0605 ﾀ07 006 縺06000600030. 4. ｮ 040060 ｧ06 縺07 閨0304 ﾀ 0403000307 ｧ050600040030 ｧ 00060 0 ｧ: u + u + u 3 + :::+ u n + ::: ; () v + v + v 3 + :::+ v n + ::: () ) ｭ ﾀ00 ｧ u n 6 v n ｧ 縺07 () ﾀ0803 陦ｧ05 ﾀ07, 0503 ﾀ0803 陦ｧ05 ﾀ07 ｧ 縺07 (). ) ｭ ﾀ00 ｧ u n > v n ｧ 縺07 () ﾀ0803 陦ｧ05 ﾀ07, 0503 ﾀ0803 陦ｧ05 ﾀ07 ｧ 縺07 (). 5. 707 ﾀ 06 縺06 閨06060 ｧ0 ｧ ﾀ07 0 09 0 ｧ u, u + u 3, u 4 + ::: ﾀ0803 陦ｧ05 ﾀ07, 06 ﾀ00 ｧ u >u >u 3 > ::: ｧ lim n! u n =0. 6. 縺03 ﾀ030006050 07 ﾀ0803 陦ｧ003 ﾀ0504. 707 ﾀ0803 陦ｧ05 ﾀ07, 06 ﾀ00 ｧ ﾀ0803 陦ｧ05 ﾀ07 縺07 u + u + u 3 + :::+ u n + ::: (3) ju j + ju j + ju 3 j + :::+ ju n j + ::: (4)

3x. ｪ ｧ ﾀ0003040306 縺07 閨03 43 6 5 0505030 ﾀ0006 06 縺07 (3) 0 0304 0605 ﾀ07 03 ﾀ03000605003 ﾀ0803 閨070 ｧ0 ﾀ07. ｭ ﾀ00 ｧ 0706 縺07 (3) ﾀ0803 陦ｧ05 ﾀ07, 縺07 (4) ﾀ0803 陦ｧ05 ﾀ07, 0503 縺07 (3) 0 0304 0605 ﾀ07 06 ﾀ000304003 (006 03 ﾀ03000605003) ﾀ0803 閨070 ｧ0 ﾀ07. 6 5030403000070605 ﾀ07 00 ｧ 00603030803 陦ｧ00306 06 ﾀ000304 ｧ06 ﾀ0803 陦ｧ003 ﾀ05 ｧ 縺07 : 4. + 3 4 + 5 6 + 7 8 + ::: 43. + 3 + 5 + 7 + ::: 44. 3 + 4 9 + 6 7 + 8 8 + ::: 0ｷ3 ﾀ ﾀ0006 閨0304 0504 0403 ｧ0050605 0004003006 04 罍ｧ 0 0906 ﾀ0803 陦ｧ003 ﾀ0504 縺07 : 45. + 3 + 5 + 7 + ::: 46. + 4 + 7 + 0 + ::: 47. 3 + 3 3 + 3 4 3 + ::: 48. + + + + +3 + ::: 49. + + + + 3 +3 + ::: 430. 3, + 5, + 7, + ::: 43. ln + 3ln 3 + 4ln 4 + ::: 0ｷ3 ﾀ ﾀ0006 閨0304 0504 040304 罍ｧ 0 0906 6 700 00306 ﾀ0803 陦ｧ003 ﾀ0504 縺07 : 43. 3 + 4 9 + 6 7 + 8 8 + ::: 433. +! + 4 3! + 8 4! + ::: 434. + 3 + 3 3 5 + ::: 435. + 3 3 + 3 5 + 33 3 7 + ::: 436. + 3! 4 + 5! 4 6 + 7! 4 6 8 + ::: 5 437. + 3 3 + 9 3 3 3 + 4 3 3 4 + :::

344 0600.4. 707 閨03 ｮ 040060 ｧ060 ﾀ 05 縺0030 ｧ 06 ﾀ09 ｧ0 縺07 閨030 ｧ00 ｧ ﾀ 06030304 0600608 04 縺0305 縺06 ﾀ- ﾀ ｧ0608 ｧ ﾀ ﾀ0006 閨0304 0504 ﾀ0803 陦ｧ003 ﾀ0504 縺07 : 438. + + 3 + 4 + ::: 439. + 5 + 3 5 + 4 5 3 + ::: 440. ln + ln 3 + ln 4 + ln 5 + ::: 44. 03060503 閨030 ﾀ 040060 ｧ07 縺07 閨0304 040309 0504, 0503 縺07 +x + + +x 4 + +x6 + ::: 04 罍ｧ jxj 6 ﾀ0803 陦ｧ05 ﾀ07, 04 罍ｧ jxj > ﾀ0803 陦ｧ05 ﾀ07. ｰ09 0 ｧ06. 6 70007 ﾀ 040060 ｧ07 04 0406 縺04030 ﾀ0006 06 0060 ｧ0504 x, x 4, x 6, ::: 06 陦ｧ0 ｧ09 0 ｧ, 0403 040503 縺030 ﾀ0006 06 030503 縺03 ﾀ ｧ0504 04 0 0060 0506000708 06 陦ｧ0 ｧ0903. 7 0805 ｧ ﾀ060006 縺07 : 44. + 3 + 3 4 + ::: ｰ09 0 ｧ06. 7 000307 ｧ0504 u n 0 050006006005 縺00306 陦縺0303 ｧ. 443. 4 + 4 7 + 7 0 + ::: 0ｷ3 ﾀ ﾀ0006 閨0304 0504 ﾀ0803 陦ｧ003 ﾀ0504 縺07 : 444., + 3, 4 + ::: 445., 3 + 5, 7 + ::: 446. 447. sin, ln 3ln3 +, ::: 4ln4 + sin + sin 3 3 + ::: 448. 70309 0504, 0503 ﾀ0600 S 06 ﾀ000304003 ﾀ0803 閨070060503 ﾀ07 縺07, + +, + :::0600600400 ｧ05 ﾀ07 04 閨040306, 06 ﾀ00 ｧ 0403 ﾀ0006 09 07 閨030503 0403000307 ｧ05060004-3 4 0030503 00060 縺07 0403006 ﾀ05 ｧ0504 閨04 0403 ﾀ0006 閨06060 ｧ08 0305 罍ｧ09 0506000400308, ｧ 06040600 ｧ ｧ05 ﾀ07 04 0403000503, 06 ﾀ00 ｧ 0403 ﾀ0006 09 07 閨0308 閨040608 0403000307 ｧ05060004-00308 000600304 0403006 ﾀ05 ｧ0504 03 陦ｧ0 0305 罍ｧ09 0506000400308.

3x. 7 04003006 縺0 07 ﾀ0803 陦ｧ003 ﾀ0504 070600909 ｧ030 00040030503 縺07 45 0ｷ3 ﾀ ﾀ0006 閨0304 0504 ﾀ0803 陦ｧ003 ﾀ0504 縺07 : 449. + 3 3 + 5 5 + ::: 450. + 0 + 0 + 30 + ::: 45. + 4 + + 4 + 3 +3 4 + ::: 45. + 3 4 + 5 9 + 7 6 + ::: 453. + 4 + 7 + 0 + ::: 454. + 3 + 5 3 + 7 4 + ::: 455. 3 + 4 9 + 6 7 + ::: 456. + 4 3! + 6 5! + ::: 457., 3 + 5, ::: 458., 3 + 3 3, 4 3 + ::: 459., a + 3a 4, 4a 6 + ::: 7 0805 ｧ ﾀ060006 縺07 : 460. 46. 3 + 3 5 + 5 7 + ::: 3 + 3 4 + 3 4 5 + ::: x. 7 04003006 縺0 07 ﾀ0803 陦ｧ003 ﾀ0504 070600909 ｧ030 00040030503 縺07. ｮ030403090604003 ﾀ0504 0 060 ｧ08 x, 04 罍ｧ 09030503 縺0308 070600909 ｧ030 000400308 縺07 u (x)+u (x) +:::+ u n (x) +::: () ﾀ0803 陦ｧ05 ﾀ07, 0 0304 0605 ﾀ07 030300 ﾀ050406 ﾀ0803 陦ｧ003 ﾀ05 ｧ 0505030503 縺07. ｱ0600909 ｧ07 S(x) = lim n! S n(x) 0 0304 0605 ﾀ07 060503 ﾀ06000308, 003 ﾀ0504 R n (x) =S(x), S n (x) 03 ﾀ05 0509030 縺07.

346 0600.4. 707 閨03. 707 () 0 0304 0605 ﾀ07 04003006 縺003 ﾀ0803 閨070 ｧ0 ﾀ07 0 0305 縺06 0906 [a; b], 06 ﾀ00 ｧ 閨0007 04 ﾀ0709030503 ">0 00307003 0 0805 ｧ 05 090308 003006 N, 0503 04 罍ｧ n>n ｧ 000603030 x 0 0305 縺06 0906 [a; b] 0306 閨0605 0403040300006003 006 04060 ﾀ050403 jr n (x)j <". 3. 7 罍ｧ 0 09 04003006 縺00308 ﾀ0803 陦ｧ003 ﾀ05 ｧ. 707 () ﾀ0803 陦ｧ05 ﾀ07 03 ﾀ03000605003 ｧ 04003006 縺003 0 0305 縺06 0906 [a; b], 06 ﾀ00 ｧ ﾀ06006 ﾀ0504060605 ｧ ﾀ0003040308 ﾀ0803 閨070 ｧ08 ﾀ07 縺07 ﾀ 0403000307 ｧ050600040030 ｧ 00060 0 ｧ c + c + c 3 + :::+ c n + ::: 05 090308, 0503 ju n (x)j 6 c n 04 罍ｧ a 6 x 6 b. 46. 0504 縺06 閨0600 ｧ0504 04 罍ｧ jxj < ﾀ060006 ｧ03 ﾀ05 050309 縺07 +x + + x + x 3 + ::: ｧ 040309 0504, 0503 030 ﾀ0803 陦ｧ05 ﾀ07 04003006 縺003 0 0305 縺06 0906 [0; =]. 7 罍ｧ 09 09030 n 03 ﾀ05 050309 jr n (x)j < 0; 00 閨0007 000603030503 x 0 0505030 0305 縺06 0906? 463. 70309 0504, 0503 縺07 x + x(, x) +x(, x) + x(, x) 3 + ::: ﾀ0803 陦ｧ05 ﾀ07 006 04003006 縺003 0 0305 縺06 0906 [0; ] ｧ 04003006 縺003 0 0305 縺06 0906 [=; ]. 7 罍ｧ 09 09030 n 03 ﾀ05 050309 jr n (x)j < 0; 0 閨0007 000603030503 x 0 0305 縺06 0906 [=; ]? 464. 70309 0504, 0503 縺07 x, x + x3, ::: ﾀ0803 陦ｧ05 ﾀ07 04003006 縺003 3 0 0305 縺06 0906 [0; ]. 7 罍ｧ 09 09 ｧ08 n ｧ 000603030 x 0 0505030 0305 縺06 0906 jr n (x)j < < 0;? 465. 70309 0504, 0503 縺07 x 3 + x3 +x 3 + x 3 ( + x 3 ) + ::: ﾀ0803 陦ｧ05 ﾀ07 006 04003006 縺003 04 罍ｧ x>0 ｧ 04003006 縺003 04 罍ｧ x >. 7 罍ｧ 09 09030 n 03 ﾀ05 050309 jr n (x)j < 0; 00 閨0007 000603030503 x >? 466. 70309 0504, 0503 縺07 + +x 3 +3x + 3 +5x + + 3 3 + ::: ﾀ0803 陦ｧ05 ﾀ07 04003006 縺003 04 ｧ00506 縺04 0006 0 6 x<. +7x 7 罍ｧ 09 09030 n ( ｧ 000603030 0060305 罍ｧ09 050600040030 x) 03 ﾀ05 050309 縺07 jr n (x)j < < 0; 0? ｰ 09 0 ｧ 06. ｮ 040 ｧ0504 000308 縺07 ﾀ ｧ ﾀ000304030 ﾀ0803 閨070 ｧ0 ﾀ07 縺07 閨030. 467. 70309 0504, 0503 縺07 x, + x +4 + x, +9 x +::: +6 ﾀ0803 陦ｧ05 ﾀ07 04003006 縺003 0 04 ﾀ0608 ｧ ﾀ0003040308 03 ﾀ ｧ. 7 罍ｧ 09 09030 n ( ｧ 000603030 x) 03 ﾀ05 050309 縺07 jr n (x)j < 0; 000?

3468. 70309 0504, 0503 縺07 x 3. ｮ05060406000306 縺07 閨03 47 x(x +) + (x + )(x +) + (x + )(x +3) + ::: ﾀ0803 陦ｧ05 ﾀ07 04003006 縺003 09 04 ｧ00506 縺04 0006 0 <x<. 7 罍ｧ 09 09030 n ( ｧ x 000603030 x>0) 03 ﾀ05 050309 縺07 jr n (x)j < 0;? 469. 70309 0504, 0503 縺07 +x + +x + 4 +3x + 6 +4x + ::: ﾀ0803 陦ｧ05 ﾀ07 04003006 縺003 04 ｧ00506 縺04 0006 0 6 x<. 7 罍ｧ 09 09030 n 03 ﾀ05 050309 縺07 jr n (x)j < 0; 0? 706 ﾀ0504 0 ﾀ05060406000308 縺07 x 3. ｮ05060406000306 縺07 閨03 a 0 + a x + a x + :::+ a n x n + ::: () ｪ ｧ ﾀ0003 R 0 0304 0605 ﾀ07 陦ｧ06 ﾀ030 ﾀ0803 陦ｧ003 ﾀ05 ｧ 縺07 (), 06 ﾀ00 ｧ 04 罍ｧ jxj <R 縺07 ﾀ0803 陦ｧ05 ﾀ07, 04 罍ｧ jxj >R ﾀ0803 陦ｧ05 ﾀ07. R 00307003 0 0805 ｧ ｧ00 ｧ ｧ ﾀ ﾀ0006- 閨0304 0 ｧ060 03 ﾀ0300060500308 ﾀ0803 陦ｧ003 ﾀ05 ｧ 縺07 () 0403 04 罍ｧ 0 0906 6 7 00 00306, ｧ00 ｧ, 090305 04 ﾀ06 a i 030500 ｧ 003 0305 0060007, 0403 0703 縺0060006 R = lim. 6 5 ﾀ05003 ﾀ05 ｧ, a n+ n! a n 06 ﾀ00 ｧ 05050305 04 縺06 閨0600 04060, 0503 縺07 () 03 ﾀ03000605003 ﾀ0803 陦ｧ05 ﾀ07 0 04 ﾀ0608 03 ﾀ ｧ Ox. ｮ05060406000308 縺07 ﾀ0803 陦ｧ05 ﾀ07 006 050300040903 03 ﾀ03000605003, 003 ｧ 04003006 縺003 0 000603030 0305 縺06 0906 [a; b], 000607 0060 0400605 罍ｧ ｧ00506 縺04 00 ﾀ0803 陦ｧ003 ﾀ05 ｧ (,R; R). 0504 縺06 閨0600 ｧ0504 ｧ00506 縺04 00 ﾀ0803 陦ｧ003 ﾀ05 ｧ 縺07 ｧ ｧ ﾀ ﾀ0006 閨0304 0504 060503 ﾀ0803- 陦ｧ003 ﾀ0504 0 05 0 ｧ09 08 ｧ00506 縺04 00 : 470. + x 3 + x 3 3 + x3 3 3 4 + ::: 47., x 5 + x3 5, x5 3 5 3 4 + ::: 47. + x 3 3 + 4x 5 3 + 8x3 7 3 3 + ::: P x n P 473. n= : n! 474. (,x) n,. n= n P 3 n x n 475. n= (3n,. ) n

348 0600.4. 707 閨03 476. ) P n= xn, n!; ) P n= (x +) 477. (x +)+ + 4 478. x, 3 n!x n (n +) n. (x +)3 (x +)4 3 4 + 4 4 3 + ::: (x, 3) (x, 3)3, +, ::: 3 5 0504 縺06 閨0600 ｧ0504 ｧ00506 縺04 00 ﾀ0803 陦ｧ003 ﾀ05 ｧ 縺07 ｧ 0 0805 ｧ 060503 ﾀ060006: 479. +x +3x +4x 3 + ::: ｰ09 0 ｧ06. 6 70007 0 080307 閨060 ｧ07 ﾀ060003 S 0 0805 ｧ ﾀ0 00 480. x, x3 3 + x5 5, x7 7 + ::: ｰ 09 0 ｧ 06. 7 0805 ｧ ﾀ0 00 ds. x 0 S. 48. +3x +5x +7x 3 + ::: ｰ09 0 ｧ06. 050303 0 ｧ04 ﾀ060006 縺07 06 縺06 S, ﾀ03 ﾀ05 04 ｧ0504 0403 07060 ｧ06 S, Sx 04 04 ｧ 閨06 ﾀ0600 ｧ 縺06060030503 縺07. 48. + m m(m, ) x + x + ｰ09 0 ｧ06. 70309 0504, 0503 S0 09 ｧ 000400306 06 040060 ｧ06. m + S0 x m m(m, )(m, ) x 3 + ::: 3 = S, ｧ 縺0600 ｧ0504 050503 陦ｧ070706 縺060-0504 縺06 閨0600 ｧ0504 ｧ00506 縺04 00 ﾀ0803 陦ｧ003 ﾀ05 ｧ 縺07 ｧ ｧ ﾀ ﾀ0006 閨0304 0504 060503 ﾀ0803- 陦ｧ003 ﾀ0504 0 05 0 ｧ09 08 ｧ00506 縺04 00 : 483. + x + 4x 5 5 9 5 + 3 8x3 5 3 + ::: 484., x x 4 x 6 + 3 3, 3 3 3 3 + ::: 4 4 P 485. n= 487. x, + 0 n x n n : 486. (x, ) + P xn, (,)n, n= n,. (x, )3 3 + :::

3x 4. 707 閨03 ｯ06080003 ｧ 03 090003 縺060 49 x + (x +) (x +)3 488. + + + ::: 4 7 0504 縺06 閨0600 ｧ0504 ｧ00506 縺04 00 ﾀ0803 陦ｧ003 ﾀ05 ｧ 縺07 ｧ 0 0805 ｧ ﾀ060006: 489., 3x +5x 4, 7x 6 + ::: ｰ09 0 ｧ06. 6 70007 0 080307 閨060 ｧ07 ﾀ060003 S 0 0805 ｧ ﾀ0 00 x S. 490. x + x + x3 3 + ::: ｰ 09 0 ｧ 06. 7 0805 ｧ ﾀ0 00 ds. 49., 4x +7x, 0x 3 + ::: ｰ09 0 ｧ06. ｮ03 ﾀ05 04 ｧ0504 0403 07060 ｧ06 S + Sx. 0 x 4. 707 閨03 ｯ06080003 ｧ 03 090003 縺060. ｱ03 縺00600 03 090003 縺060 : f(x) =f(0) + f0 (0)! 05 閨06 R n (x) = xn n! f (n) (x), 0 6 <.. ｱ03 縺00600 ｯ06080003 : f(x) =f(a) + f0 (a)! x + f00 (0) x + :::+ R n (x); ()! (x, a) + f00 (a) (x, a) + :::+ R n (x); ()! (x, a)n 05 閨06 (n) R n (x) = f a + (x, a). n! 3. 707 閨03 03 090003 縺060 ｧ ｯ06080003. ｭ ﾀ00 ｧ 04 0703 縺00600 08 () ｧ () R n (x)! 0 04 罍ｧ n!,0503 ｧ 0505 ｧ08 0703 縺00600 04030006 0605 ﾀ07 0306 ﾀ0903006 00306 縺07 閨03: f(x) =f(0) + f0 (0)! x + f00 (0) x + ::: ; (3)! f(x) =f(a) + f0 (a)! (x, a) + f00 (a) (x, a) + ::: ; (4)! ﾀ0803 閨070 ｧ06 ﾀ07 09 f(x) 04 罍ｧ 050608 0 060 ｧ0708 x, 04 罍ｧ 09030503 縺0308 lim n! R n(x) =0.

350 0600.4. 707 閨03 4. 7 000307060 ｧ06 04 縺07 閨03 050006006005 縺00308 070600909 ｧ08: e x =+ x! + x! + x 3 3! + ::: ; sin x = x, x3 3! + x5 5!, ::: ; cos x =, x! + x4 4!, ::: 9 >= >; ﾀ0803 閨0705 ﾀ07 09 ﾀ030305040605 ﾀ0504060600608 070600909 ｧ ｧ 04 罍ｧ 04 ﾀ0608 0 060 ｧ0708 x; ( + x) m =+ m m(m, ) x + x + ::: 03 ｧ0030 ｧ 000400308 縺07 ; 030 ﾀ0803 陦ｧ05 ﾀ07 09 03 ｧ003006 ( + x) m 04 罍ｧ jxj < ; ln ( + x) =x, x + x3, ::: ﾀ0803 陦ｧ05 ﾀ07 09 ln ( + x) 04 罍ｧ, <x6; 3 arctg x = x, x3 3 + x 5, ::: ﾀ0803 陦ｧ05 ﾀ07 09 arctg x 04 罍ｧ jxj 6. 5 49. 7 000307 ｧ0504 04 縺07 0403 ﾀ050604060070 x 070600909 ｧ ｧ: ) cos (x, ); ) sin x; 3) xe x ; 4) sin mx + ｧ 0 04 ｧ ﾀ 0504 ｧ ｧ ﾀ ﾀ0006 閨0304 0504 0703-3 0060006 03 ﾀ05 0503 0030503 00060. 493. 7 04 ｧ ﾀ 0504 0406 縺040306 05 罍ｧ 00060 000307060 ｧ07 04 縺07 070600909 ｧ ｧ f (x) = ln ( + e kx ). 494. 703 0703 縺0060006 03 090003 縺060 0 04 ｧ ﾀ 0504 000307060 ｧ06 04 縺07 0403 ﾀ0506-04060070 x 03 ｧ0030 陦ｧ05 ﾀ07 04 罍ｧ jxj <a. + x a m ｧ 040309 0504, 0503 04030006 06000308 縺07 ﾀ0803-495. ｮ 040300300406 03 ｧ0030 ｧ 00040030503 縺07 040309 0504, 0503 04 罍ｧ jxj < ( + x) 3 =, 3x +6x, 0x 3 n(n +) + :::= (,x) n, : n= 496. ｮ 040300300406 03 ｧ0030 ｧ 00040030503 縺07 04030006 ｧ0504 000307060 ｧ06 04 縺07 070600909 ｧ ｧ +x =, x + 3! x4, 3 5 3 3! x6 + ::: 04 罍ｧ jxj < : 497. 7 000307 ｧ0504 04 縺07 0403 ﾀ050604060070 x 070600909 ｧ ｧ: ) ln +x, x ; ) ln (, 3x + x ); 3) ln (, x + x ). 498. 0ｷ30050605 罍ｧ 縺0304 0 ｧ060 04030006 0600030503 04 06 496 縺07 0 04 ｧ- ﾀ 0504 縺07 閨0007 ln (x + +x ). X

3x 5. 7 罍ｧ000307060 ｧ07 縺07 閨0304 09 04 罍ｧ0300 ｧ070600030 0403 ｧ ﾀ00060 ｧ070 5 499. 7 000307 ｧ0504 e x=a 04 縺07 0403 ﾀ050604060070 x, a; 0 04 ｧ ﾀ 0504 ｧ ｧ ﾀ- ﾀ0006 閨0304 0504 0703 縺0060006 03 ﾀ05 0503 0030503 00060 縺07. 500. 7 000307 ｧ0504 070600909 ｧ06 f (x) =x 3, 3x 0403 ﾀ050604060070 x,. 50. 7 000307 ｧ0504 x 4 0403 ﾀ050604060070 x +. 50. 7 000307 ｧ0504 04 縺07 0403 ﾀ050604060070 x +070600909 ｧ06 f (x) = x ｧ ｧ ﾀ ﾀ0006 閨0304 0504 ﾀ0803 陦ｧ003 ﾀ0504 縺07 0403 04 罍ｧ 0 0906 6 7 00 00306. 503. 7 000307 ｧ0504 04 縺07 閨03 070600909 ｧ ｧ: ) f (x) = cos x 0403 ﾀ050604060070 x, ;)f(x) = sin 3x 0403 ﾀ050604060070 x + 3. 504. 7 000307 ｧ0504 04 縺07 0403 ﾀ050604060070 x+ 070600909 ｧ06 f (x) = 3 x ｧ ｧ ﾀ ﾀ0006 閨0304 0504 0403 04 罍ｧ 0 0906 6 7 00 00306 ﾀ0803 陦ｧ003 ﾀ0504 04030006 0600030503 縺07. 505. 7 000307 ｧ0504 04 縺07 0403 ﾀ050604060070 x 070600909 ｧ ｧ: ) x ; ) cos mx + ｧ 0 04 ｧ ﾀ 0504 ｧ ｧ ﾀ ﾀ0006 閨0304 0504 0703 縺0060003 03 ﾀ05 0503 00308 4 000600304 000307060 ｧ07. 506. 7 000307 ｧ0504 070600909 ｧ06 f (x) =x 4, 4x 0403 ﾀ050604060070 x +. 507. 7 000307 ｧ0504 04 縺07 0403 ﾀ050604060070 x, 3 070600909 ｧ06 f (x) = cos x ｧ 0 04 ｧ ﾀ 0504 ｧ ｧ ﾀ ﾀ0006 閨0304 0504 0703 縺0060006 03 ﾀ05 0503 0030503 00060 縺07. 508. 7 000307 ｧ0504 04 縺07 0403 ﾀ050604060070 x, 070600909 ｧ06 f (x) = = sin x 3. 509. 7 000307 ｧ0504 04 縺07 0403 ﾀ050604060070 x,4 070600909 ｧ06 f (x) = x ｧ ｧ ﾀ ﾀ0006 閨0304 0504 0403 04 罍ｧ 0 0906 6 7 00 00306 ﾀ0803 陦ｧ003 ﾀ0504 04030006 0600030503 縺07. 50. ｮ 040300300406 03 ｧ0030 ｧ 00040030503 縺07 040309 0504, 0503, x =+ x + 3! x4 + 3 5 3 3! x6 + ::: 04 罍ｧ jxj <. 5. 703 000600030 ｧ0050605 罍ｧ 縺0304 0 ｧ060 縺07, 04030006 0600030503 04 06 50, 0 04 ｧ ﾀ 0504 縺07 閨0007 arcsin x. x 5. 7 罍ｧ000307060 ｧ07 縺07 閨0304 09 04 罍ｧ0300 ｧ070600030 0403 ｧ ﾀ00060 ｧ070 5. 7 04 ｧ ﾀ 0504 03 ｧ0030 ｧ 000400308 縺07 閨0007 +x ｧ 0403 ｧ ﾀ00 ｧ0504 ; 004, 0; 99, 90, 0305 0 ｧ ｧ0400 ｧ ﾀ04 閨0406007 00060 0 ｧ 縺07. 050906-0 ｧ0504 040305 縺0600003 ﾀ0504. 53. 7 04 ｧ ﾀ 0504 03 ｧ0030 ｧ 000400308 縺07 閨0007 3 +x ｧ 0403 ｧ ﾀ00 ｧ0504 3 ; 006, 3 0; 99, 3 30, 0305 0 ｧ ｧ0400 ｧ ﾀ04 閨0406007 00060 0 ｧ 縺07. 050906-0 ｧ0504 040305 縺0600003 ﾀ0504.

35 0600.4. 707 閨03 54. 6 503 ｧ ﾀ00 ｧ0504 sin, 0305 0 ｧ ｧ0400 ｧ ﾀ04 閨0406007 00060 0 ｧ 縺07 閨0007 sin x, ｧ 0309060 ｧ0504 040305 縺0600003 ﾀ0504. ｰ09 0 ｧ06. x =, 04 陦ｧ 0 08 x = =5 = 0; 094. 6 506 縺0800606 05 0 ｧ0906 040305 縺0600003 ﾀ05 ｧ 0304 縺06 閨0600 ｧ0504 ｧ 06 ﾀ000304 ｧ07 x<0; 3. 55. 6 70600060 ｧ060 ｧ ﾀ00 ｧ05060007 陦縺0303 ｧ +x 0 0606 0 0060 05060004 0403- P 0006 ｧ0504 000307060 ｧ06 +x = n= (,)n, x n, ｧ, 04 縺03 ｧ0050605 罍ｧ 縺0304 04 0403 00060003 04030006 06000308 縺07, 0 04 ｧ ﾀ 0504 000307060 ｧ06 04 縺07 arctg x. P 56. 70300 05 07 x = 3 (,) n, x n, 04 000307060 ｧ ｧ arctg x =, n= n, 04030006 ｧ0504 縺07 閨0007 0403 ｧ ﾀ00060 ｧ07 : X (,) n, = 3 (n, )3 n= n, : 57. 6 503 ｧ ﾀ00 ｧ0504, 04 0704 04070504 000600304 縺07 ｧ 56. 58. ｮ 040300300406 04030006 0600030503 04 06 497 縺07 ln +x, x = x + x3 3 + x5 5 + ::: 0403 ｧ ﾀ00 ｧ0504 ln, ln 3, ln 4, ln 6. ｰ09 0 ｧ06. 703000307 ｧ04 +x =, 0 0805 ｧ x ｧ05.., x sin x 59. 0504 縺06 閨0600 ｧ0504 04 04 ｧ 閨06 縺07 閨0304 ｧ0050605 0003 x 50. 0504 縺06 閨0600 ｧ0504 04 04 ｧ 閨06 縺07 070600909 ｧ06 (x) = ｧ e x x x. e,x ｧ 0 0403 ｧ ﾀ00 ｧ0504 (=3), 04 0704 ﾀ050300040903 000600304, ﾀ090300040903 00607003 閨0007 05030503, 05030303 040305 縺0600003 ﾀ0504 030300 0060040006 0,00. 5. 0504 縺06 閨0600 ｧ0504 04 04 ｧ 閨06 縺07 070600909 ｧ06 (x) = 0 ｧ 0403 ｧ ﾀ00 ｧ0504 (=5), 04 0704 ﾀ050300040903 000600304, ﾀ090300040903 00607003 閨0007 05030503, 05030303 040305 縺0600003 ﾀ0504 030300 0060040006 0,0000. 5. 7 0805 ｧ 04 04 ｧ 閨06 縺07 縺0600060 ｧ06 06 040060 ｧ07 y 00 = x y ﾀ 0-00040030 ｧ 06 ﾀ000304 ｧ070 ｧ: 04 罍ｧ x =0, y =, y 0 =. 53. 7 0805 ｧ 0406 縺040306 060503 縺06 00060 縺07, 0304 縺06 閨060007060060503 縺060006-0 ｧ06 06 040060 ｧ07 ( 7 ｧ0909 05 ｧ) y 0 =+x,y ﾀ0 00040030 ｧ 06 ﾀ000304 ｧ070 ｧ: y =04 罍ｧ x =0. x 3 +x

3x 5. 7 罍ｧ000307060 ｧ07 縺07 閨0304 09 04 罍ｧ0300 ｧ070600030 0403 ｧ ﾀ00060 ｧ070 53 54. 7 04 ｧ ﾀ 0504 04 04 ｧ 閨06 縺07 縺0600060 ｧ06 06 040060 ｧ07 706 ﾀ ﾀ060007 xy 00 + + y 0 + xy =0 ﾀ0 00040030 ｧ 06 ﾀ000304 ｧ070 ｧ: y =, y 0 =004 罍ｧ x =0. 55. 6 503 ｧ ﾀ00 ｧ0504 3 ; 005, ; 00, 3 0; 993, 0; 997, 0, 3 70, 5 40, 0305 0 ｧ ｧ0400 ｧ ﾀ04 閨0406007 00060 0 ｧ 03 ｧ0030 ｧ 00040030503 縺07 ( + x) m m(m, )x =+mx + + :::, ｧ 0309060 ｧ0504 040305 縺0600003 ﾀ0504.! 56. 6 503 ｧ ﾀ00 ｧ0504 cos, 0305 0 ｧ ｧ0400 ｧ ﾀ04 閨0406007 00060 0 ｧ - 000307060 ｧ07 04 縺07 cos x. 0509060 ｧ0504 040305 縺0600003 ﾀ0504. 57. 70300 05 0704 000307060 ｧ ｧ 04 縺07 arcsin x ( 5) x = ==, 0403 ｧ ﾀ00 ｧ0504, 0305 0 ｧ ｧ04 07 ﾀ04 05 縺06007 00060 0 ｧ 縺07. ｰ09 0 ｧ06. ｮ0 00 0403 ｧ ﾀ00 ｧ0504 0406 縺040308 ｧ 030503 縺030006000308 000600304, 05060 0403 ｧ0504 閨06 ﾀ0705 ｧ 00308 陦縺03030406 09 07 閨0308 ｧ 0406 縺040308 05 縺0608 000600304 ﾀ 040305 縺0600003 ﾀ050406 006 030300040006 0406 縺04030503 030503 縺03000600030503 00060. 58. 7030004 0607 ﾀ04 050307 閨06 ﾀ0504030 4 = arctg + arctg, 0 04 ｧ ﾀ 0504 3 0403 07060 ｧ06 閨0007 06 縺06 ﾀ060006 閨040608 0306 ﾀ0903006 00308 縺07 閨0304. 59. 70300 05 07 x ==N 04 000307060 ｧ ｧ ln ( + x) 04 縺07, 04030006 ｧ0504 0703 縺0060003: ) ln (N +)=lnn + ) lg (N +)=lgn +0; 4343 N, N + 3N 3, ::: ; N, N + 3N 3, :::. 530. 6ｦ00 07 ln = 0; 693, 0403 ｧ ﾀ00 ｧ0504 ln 5 ｧ ln 0 ｧ 040309 0504, 0503 003 閨060004 M = 0; 4343. ln 0 53. 6 503 ｧ ﾀ00 ｧ0504 lg 0 ｧ lg 0. 53. 0504 縺06 閨0600 ｧ0504 04 04 ｧ 閨06 縺07 閨00 ｧ006 閨0605 ｧ 050000 ｧ04 ﾀ. 0;5 533. 6 503 ｧ ﾀ00 ｧ0504 +x 3, 04 0704 ﾀ050300040903 000600304 縺07, ﾀ090300040903 0 00607003 閨0007 05030503, 05030303 040305 縺0600003 ﾀ0504 030300 0060040006 0,00. 534. 0504 縺06 閨0600 ｧ0504 04 04 ｧ 閨06 縺07 070600909 ｧ06 (x) = 0 0403 ｧ ﾀ00 ｧ0504 ﾀ 0503 003 ﾀ050406 閨03 0,00000. x cos x 4 ｧ

354 0600.4. 707 閨03 535. 7 04 ｧ ﾀ 0504 0406 縺040306 05 罍ｧ 00060 縺07, 0304 縺06 閨060007060 ｧ06 縺0600060 ｧ06 06 040060 ｧ07 y 0 = x + y,06 閨03040006050403 縺070600606 06 ﾀ000304 ｧ06: y = 004 罍ｧ x =0. 536. 7 04 ｧ ﾀ 0504 04 04 ｧ 閨06 縺07 縺0600060 ｧ06 06 040060 ｧ07 y 00 + xy =0 ﾀ 0 00040030 ｧ 06 ﾀ000304 ｧ070 ｧ: 04 罍ｧ x =0, y =, y 0 =0. 537. 7 04 ｧ ﾀ 0504 04 04 ｧ 閨06 縺07 閨0304 06 040060 ｧ07 0406 縺060803 閨00308 09 罍ｧ040308, 04 閨030004 09030503 縺0308 09 罍ｧ04 ｧ 0 k 0 ﾀ05 0605 04 縺030403 縺09 ｧ030 0004003 閨00 ｧ006 閨0605 ｧ s. ｰ09 0 ｧ06. 0ｷ3 06 ﾀ000304 ｧ07 d',05 閨06 C 0403 ﾀ05030700 07, 0 0805 ｧ ' ｧ ds = s C 05060 縺0600 ｧ0504 06 040060 ｧ07 = ds cos ' ｧ dy = ds sin '. x 6. 707 ｯ06080003 閨0007 070600909 ｧ ｧ 閨040608 0406 縺06006000308 ｱ03 縺0060006 ｯ06080003 閨0007 070600909 ｧ ｧ 閨040608 0406 縺06006000308 00307003 0 04 ｧ ﾀ 0504 04 05 縺0608 ﾀ0006 閨06060 ｧ08 04 ｧ 08: F (x + h; y + l) =F (x; y) + +! h @ @x + l @ @y F (x; y) +! F (x; y) =F (a; b)+! (x, a) @ h @ @x + l @ F (x; y) +::: ; @y @x +(y, b) @ F (a; b)+ @y + (x, a) @! @x +(y, b) @ F (a; b) +::: ; @y z = dz! + d z! + :::+ dn z n! x=x0+x y=y 0+y (I) (II) : (III) 538. 7 04 ｧ ﾀ 0504 000307060 ｧ06 070600909 ｧ ｧ F (x +h; y +l) 0403 0703 縺0060006 ｯ06080003 (I), 06 ﾀ00 ｧ F (x; y) =x + xy + y. 539. 7 000307 ｧ0504 070600909 ｧ06 F (x; y) =x 3 +xy 0403 ﾀ050604060070 x, ｧ y, (0703 縺00600 (II)). 540. 7 000307 ｧ0504 070600909 ｧ06 F (x; y) = ln (x, y) 0403 ﾀ050604060070 x ｧ y +, 0 04 ｧ ﾀ 04 0006003-0503 ｧ -0503 0403 縺07 閨090304 ｧ 03 ﾀ05 0503 00308 00060 (0703 縺00600 (II)). 54. 7 000307 ｧ0504 070600909 ｧ06 F (x; y) = sin (mx + ny) 0403 ﾀ050604060070 x ｧ y, 0 04 ｧ ﾀ 04 0006003-0503, -0503 ｧ 3-0503 0403 縺07 閨090304 ｧ 03 ﾀ05 0503 00308 00060 (0703 縺00600 (II) 04 罍ｧ a = b =0). 54. 7 000307 ｧ0504 0403 ﾀ050604060070 x ｧ y 070600909 ｧ06 e,x,y (0703 縺00600 (II) 04 罍ｧ a = b =0). 543. 0504 縺06 閨0600 ｧ0504 04 罍ｧ 0060 ｧ06 z 070600909 ｧ ｧ z = x, xy + y (0703 縺00600 (III)) ｧ 0403 ｧ ﾀ00 ｧ0504 060503 04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 x ｧ 0060070605 ﾀ07 ﾀ 閨03,, y ｧ 0060070605 ﾀ07 ﾀ 3 閨03,8.

3x 7. 707 ｱ06 縺0406. 0ｷ30050605 00 ｱ06 縺0406 55 544. 0504 縺06 閨0600 ｧ0504 04 罍ｧ 0060 ｧ06 z 070600909 ｧ ｧ z = cos (ax, by), 0 04 ｧ ﾀ 04 閨04 00060 0703 縺0060003 (III) ｧ 03 ﾀ05 0503 00308 00060. 545. ｱ0600909 ｧ06 F (x; y) =x y 000307 ｧ0504 0403 ﾀ050604060070 x, ｧ y +(0703 縺00600 (II)). 546. ｱ0600909 ｧ06 F (x; y) = arctg y 000307 ｧ0504 0403 ﾀ050604060070 x, x ｧ y, 0305 0 ｧ ｧ0400 ｧ ﾀ04 00060 0 ｧ -0503 ｧ -0503 0403 縺07 閨090304. 547. 7 000307 ｧ0504 070600909 ｧ06 z = y x 0403 ﾀ050604060070 x, ｧ y,, 0 04 ｧ ﾀ 04 0006003-0503 ｧ -0503 0403 縺07 閨090304, ｧ 0403 ｧ ﾀ00 ｧ0504 ; ;. 548. 0504 縺06 閨0600 ｧ0504 04 罍ｧ 0060 ｧ06 z 閨0007 070600909 ｧ ｧ z = x y, y ｧ 0403 ｧ ﾀ00 ｧ0504 060503 ﾀ 0503 003 ﾀ050406 閨03 0,000 04 罍ｧ 06 ﾀ000304 ｧ ｧ, 0503 x ｧ 006007-0605 ﾀ07 0305 閨03,99, y 0305 5 閨03 5,0. x 7. 707 ｱ06 縺0406. 0ｷ30050605 00 ｱ06 縺0406. 0504 縺06 閨0600060 ｧ06. ｱ0600909 ｧ07 f(x) 0 0304 0605 ﾀ07 06 閨03040006050403 縺070600608 06 ﾀ000304 ｧ070 6 7 ｧ 罍ｧ080006 0 0305 縺06 0906 [a; b], 06 ﾀ00 ｧ 030 0 0505030 0305 縺06 0906: ) ｧ0060605 0903006 00306 ｧ ﾀ0003 縺03040304, 04 罍ｧ 060 04 ﾀ06 030 ｧ 0406 縺04030503 縺03 ; ) ｧ0060605 0903006 00306 ｧ ﾀ0003 0509 ﾀ05 縺0600600304; f(x, 0) + f(x +0) 3) f(x) = 0403 04 ﾀ0608 0503 09 08 (a; b).. ｱ0600909 ｧ07 f(x), 06 閨03040006050403 縺07060 07 06 ﾀ000304 ｧ070 6 7 ｧ 罍ｧ080006 0 0305 縺06 0906 [,l; l],003070605 03030504 0304 縺06 閨0600060 0403 04 ﾀ06080503 09 08 0505030503 0305 縺06 09 縺07 閨030 ｱ06 縺0406: 05 閨06 f(x) = a 0 + X n= ha n cos nx l + b n sin nx l i ; () l a n = l,l f(x) cos nx l ; l b n = l,l f(x) sin nx l : () ｭ ﾀ00 ｧ f(x) =f(,x), 05.06. f(x) 070600909 ｧ07 06050 07, 0503b n =0 ｧ f(x) = a 0 + X n= a n cos nx : (3) l ｭ ﾀ00 ｧ f(x) =,f(,x), 05.06.f(x) 070600909 ｧ07 006 06050 07, 0503a n =0 ｧ f(x) = X n= b n sin nx : (4) l

356 0600.4. 707 閨03 ｭ ﾀ00 ｧ 070600909 ｧ06 f(x), 0304 縺06 閨060006000606 縺07 閨030 () 0 0305 縺06 0906 [,l; l], 04 縺03- 閨030007 ｧ0504 0403 0406 罍ｧ03 陦ｧ 06 ﾀ0903006 0903006 ﾀ 0406 罍ｧ03 閨030 l, 040305 縺06030304 04, 05030303 f(l, 0) + f(l +0) f(l) =, 0503 030 0306 閨0605 0304 縺06 閨0600070504 ﾀ07 縺07 閨030 () ｧ 0 04 ﾀ060 ﾀ0403060 04 縺03 閨030007060 ｧ ｧ. 3. ｭ ﾀ00 ｧ 070600909 ｧ07 f(x) 03 ﾀ03000605003 ｧ0050605 罍ｧ 縺06060 04 04 縺030060706050906 + (,; ) 05. 06. jf(x)j ﾀ0803 陦ｧ05 ﾀ07 ｧ 06 閨03040006050403 縺070605 06 ﾀ000304 ｧ070 6 7 ｧ-, 罍ｧ080006 0 04 ﾀ0709030 0903006 0030 0305 縺06 0906, 0503 030 003070605 03030504 04 縺06 陦ﾀ05 0400060 ｧ0-050605 00030 ｱ06 縺0406: f(x) = + 0 d +, f(t) cos (x, t) dt = = + [a() cos x + b() sin x] d; (5) 05 閨06 + 0 a() = f(t) cos t dt ｧ b() = f(t) sin t dt: (6),, + 7 000307 ｧ0504 04 縺07 閨03 ｱ06 縺0406 ﾀ0006 閨06060 ｧ06 0406 罍ｧ03 陦ｧ 06 ﾀ09 ｧ06 070600909 ｧ ｧ ﾀ 0406 罍ｧ03 閨030 : 549. f (x) =04 罍ｧ 0 <x< ｧ f (,x) =,f (x). ｮ 040300300406 04030006 0600030503 縺07 040309 0504, 0503, 3 + 5, 7 + :::= 4 : 550. f (x) =x 04 罍ｧ 0 6 x 6 ｧ f (,x) =f (x). ｮ 040300300406 04030006 0600030503 縺07 040309 0504, 0503 + 3 + 5 + 7 + :::= : 8 55. f (x) =x 04 罍ｧ, 6 x 6. ｮ 040300300406 04030006 0600030503 縺07 040309 0504, 0503 ), + 3, 4 + :::= ; ) + + 3 + 4 + :::= 6. 55. f (x) = 04 罍ｧ, <x<0;, x 04 罍ｧ 0 6 x 6 :

3x 7. 707 ｱ06 縺0406. 0ｷ30050605 00 ｱ06 縺0406 57 7 000307 ｧ0504 04 縺07 ｱ06 縺0406 0406 罍ｧ03 陦ｧ 06 ﾀ09 ｧ06 070600909 ｧ ｧ ﾀ 0406 罍ｧ03 閨030 l: 553. f (x) =04 罍ｧ 0 <x<l ｧ f (,x) =,f (x). 554. f (x) =, x 04 罍ｧ 0 6 x 6, f (,x) =f (x), l =. 0 04 罍ｧ,l <x6 0; 555. f (x) = x 04 罍ｧ 0 6 x<l: 556. f (x) 04 030300 ﾀ05 ｧ (0; ] 0 05 07 ｧ09030 ( 罍ｧ ﾀ. 35) ｧ 04 縺03- 閨030007060 : ) 0403 0605003006; ) 0403 006 0605003006 0406 罍ｧ03 陦ｧ 06 ﾀ0903006 0903006 ﾀ 0406 罍ｧ03 閨030 l =4. 7 000307 ｧ0504 09 07 閨0606 ｧ 0505 ｧ08 070600909 ｧ08 04 縺07 ｱ06 縺0406. 7 ｧ ﾀ. 35 7 ｧ ﾀ. 36. 557. 7 ﾀ04 縺03 ﾀ05 0060 ｧ06 05060400 04 ﾀ0506 縺07006 閨00 ｧ00308 l 0304 縺06 閨0600070605 ﾀ07 06 040060 ｧ060 @u a @t = @ u @x ; 05 閨06 u(x; t) 050600406 0506, ｧ 06 ﾀ000304 ｧ070 ｧ ) 05 0 ｧ 0030 ｧ: u =004 罍ｧ x =0 ｧ 04 罍ｧ x = l; x 04 罍ｧ x<l=; ) 0 00040030 ｧ: u = 04 罍ｧ t =0. l, x 04 罍ｧ x>l= 0504 縺06 閨0600 ｧ0504 0060503 閨030 ｱ06 縺0406 070600909 ｧ06 u(x; t). 558. 7 縺03 閨03000400306 0903000603 0 ｧ07 ﾀ0506 縺07007 閨00 ｧ00308 l,0609030503 縺030503 03 陦ｧ0 090300609 (04 罍ｧ x =0) 09 縺060400060, 陦縺06050308 (04 罍ｧ x = l) ﾀ04030303 閨060, 0304 縺06- 閨0600070605 ﾀ07 06 040060 ｧ060 @ u a @t = @ u @x ; 05 閨06 u(x; t) 04 縺03 閨03000400306 ﾀ0060060 ｧ06, ｧ 06 ﾀ000304 ｧ070 ｧ @u ) 05 0 ｧ 0030 ｧ: u =004 罍ｧ x =0; =004 罍ｧ x = l; @x @u ) 0 00040030 ｧ: u = f (x), =004 罍ｧ t =0. @t 0504 縺06 閨0600 ｧ0504 0060503 閨030 ｱ06 縺0406 070600909 ｧ06 u(x; t).

358 0600.4. 707 閨03 559. 7030406 縺06 00306 0903000603 0 ｧ07 ﾀ0506 縺07007 閨00 ｧ00306 l ﾀ 09 縺060400060-0030 ｧ 0903009 0 ｧ 0304 縺06 閨0600070605 ﾀ07 06 040060 ｧ060 ｧ06 ﾀ000304 ｧ070 ｧ @ u a @t + @4 u @x 4 =0 ) 05 0 ｧ 0030 ｧ: u =0 ｧ @ u @x =004 罍ｧ x =0 ｧ x = l; ) 0 00040030 ｧ: u = f (x) ｧ @u =004 罍ｧ t =0. @t 0504 縺06 閨0600 ｧ0504 0060503 閨030 ｱ06 縺0406 070600909 ｧ06 u(x; t). 6 5 08 560{56 0 04 ｧ ﾀ 0504 ｧ0050605 00 ｱ06 縺0406 閨0007 070600909 ｧ ｧ: 04 罍ｧ 0 <x<; 560. f (x) = ｧ f (,x) =,f (x). 0 04 罍ｧ x> 56. f (x) =e,x 04 罍ｧ x > 0 ｧ f (,x) =f (x). 56. f (x), 000308 0 0305 縺06 0906 [,; ] 05 07 ｧ09030 0 罍ｧ ﾀ. 36 ｧ 0400308 0060006 04006 0505030503 0305 縺06 09. 7 000307 ｧ0504 04 縺07 閨03 ｱ06 縺0406 070600909 ｧ ｧ: 563. f (x) =, x 04 罍ｧ 0 <x6, f (,x) =f (x); f (x +) =f (x): 564. f (x) = j sin xj; ﾀ 040300300406 04030006 0600030503 縺07 040309 0504, 0503 3 + 3 5 + 5 7 + :::=. x 04 罍ｧ 0 6 x 6 =; 565. f (x) =, x 04 罍ｧ = 6 x 6 566. f (x) =x 04 罍ｧ 0 6 x 6 l, ｧ f (,x) =,f (x). f (,x) =f (x); f (x +l) =f (x): 567. f (x) = 04 罍ｧ, 6 x<0; x 04 罍ｧ 0 <x6 ｧ f (x +)=f (x). 568. f (x) =e x 04 罍ｧ,l <x<l ｧ f (x +l) =f (x).

3x 7. 707 ｱ06 縺0406. 0ｷ30050605 00 ｱ06 縺0406 59 569. 03060503 閨030 ｱ06 縺0406 縺0600 ｧ0504 06 040060 ｧ06 @ u @t = @ u @x 04 罍ｧ 06 ﾀ000304 ｧ0708: @u ) u =004 罍ｧ x =0, =004 罍ｧ x = ; @x ) u = f (x) ｧ @u =004 罍ｧ t =0. @t 570. 7 04 ｧ ﾀ 0504 ｧ0050605 00 ｱ06 縺0406 閨0007 070600909 ｧ ｧ 04 罍ｧ, <x<; f (x) = 0 04 罍ｧ jxj > :

305 ｯ6 5 ｭ ｯ6ｧ9. AB =9, BC =,6, AC =3, 9,6 =3. 3. 5(+ ), 90, 45. 5. 0. 6. 5. 7. (5; 5), (5;,3). 8. B(0; ) ｧ B(0;,4). 9. x = a c, b ; 04 罍ｧ c>jbj 閨0406 0503 09 ｧ, 04 罍ｧ c = jbj 03 閨0, 04 罍ｧ c<jbj 0 ｧ 03 閨00308. 0. M (5; 0).,!,!,!. ｨC06005 (;,), R =5.. 04 x AB =,, 04 繦 AB =,4, j ABj = 5. 3. B(5; 8), j,! ABj = 3. 4. B(4;,3). 5.,4,, 3. 6. 8. 7. (0; ; 9). 8. B(4; 0), B (,8; 0). 9. ｨC06005 (;,), R =5.. X =7, Y =,; 5.. M (; 4). 3. M (3; 6). 4. x = m x + m x. m + m 6. 6 56 ﾀ0 0305 0906005 00 0 ﾀ ﾀ0308 00 05. 7. (; ; 5). 9. OC = 5, OD = 4. 30. (3; 3). 3. 9. 33. 3. 34. (; 3), 06 ﾀ00 ｧ ﾀ ｧ0003 0 04-7 040006003 04 03 閨006 ﾀ0503 縺03006, ｧ (5; 7), 06 ﾀ00 ｧ 04 00306 ﾀ0503 縺03003. 35. (;,). 0 36.. 37. x = x + x + x 3, y = y + y + y 3 37. 38. 3 3 3 7 ; 3. 7 39. C (3; 0), C (,7; 0). 40. M (;,6), N (5; 8), P (,4; ), k = 7=3. 4. x + y, 6x, 8y =0, A ｧ O 000607 05 0 0309 縺0607003 ﾀ05 ｧ. 43. x, y, =0, D ｧ E 000607 05 0 00 ｧ0 ｧ ｧ. 45. x + y =8. 46. y = x. 47. x 5 + y =. 48. y = x 4,x+. 49. y = x. 5. (; 0), (3; 0), (0; 3). 53. y =8(x,). 54. x, y +5 = 0. ｯ03 09 ｧ B ｧ D 000607 05 0 00 ｧ0 ｧ ｧ. 55. x + y =4. 57. y = x 4 +. 58. (x +) +(y +), (x, ) +(y, ) = 4 ｧ00 ｧ xy =; 04 罍ｧ x = =; ; ; 4, y = 4; ; ; =; 04030505 ｧ0 0503 09 0 00307003 0403 ﾀ05 縺03 ｧ0504 09 罍ｧ040606. 59. ) y = x +3; ) y =,x +3. 60. ) y = x 3, 3; )y =,x 3, 3. 6. y =,; 5x. 63. ) k ==3, b =,; )k =,=3, b = 0;3)k =0, b =,3; 4)k =,3=4, b =3. 65. k =, b =, y = x +. 66. ) x 3 + y, =;) x,4=3 + y =. 67. y =0; 4x, 3y =0; y =4; 4x, 3y + = 0. 68. x, y 3 = ｧ00 ｧ, x 4 + y 3 =. 69. 04,!,!,! OxAB = 8,04 Oy AB = 6, j ABj = 0. 70. A ｧ C 0 04 縺0700308, B 080403000609, D 080 ｧ070609 04 縺0700308. 7. 706 04060 ﾀ0504 0304 縺06 閨0600070605: ) 04 ﾀ06 0503 09 ｧ, 000607 0 ｧ06 080403000609 04 縺0700308 y = 3x + (0403-0006040003 ﾀ0903 ﾀ0504); ) 04 ﾀ06 0503 09 ｧ, 000607 0 ｧ06 080 ｧ070609 04 縺0700308 y =3x +; 3) 04 ﾀ06 0503 09 ｧ, 000607 0 ｧ06 080403000609 04 縺0700308 y =4, x ｧ 0 ﾀ 00308 04 縺0700308; 4) 0503 09 ｧ,

3050504060503 6 000607 0 ｧ06 080 ｧ070609 04 縺0700308 y = 4, x. 73. x, y = a. 74. ｪ06 縺06 t ﾀ0609060 090303 罍陦ｧ0 0503 0503 09 ｧ M 0306 閨0605 x = a + mt, y = b + nt. 0ｷ3 ﾀ090006 ｧ04 x, a t, 04030006 ｧ0 06 040060 ｧ06 05 06090503 罍ｧ ｧ: m = y, b n. 75. ) y = x 3, ; ) y =,x 3,. 76. k =, b =5. 77. x +y, 4=0, x,y +4 = 0; y =3, y =0. 78. x 5 y 3 =. 79. x 4 + y 3 = ｧ x, + y =. 80. y = (x+3).,6 8. AB,!,! 3 =4 5,04 Ox AB =4,04 絨y AB =8. 8. ) arctg 4 ;)45 ;3)45 ; 4) 0 ; 5) 90 ; 6) arctg a, b. 86. 5x +y +4 = 0, 5x +y = 5. ab 88. x, 3y + = 0, 5x, y = 4, 3x + y =. 89. 8, 30 0 ｧ 39 30 0. 90. y = 3x ｧ y =, x. 9. x, 5y +6 = 0, 5x + y =,4. 3 9. y =x, 6, y =,x +6. 93. (3;,), (3; 3), (,9=5; 3=5); 45, 7 34 0, 63 6 0. 94. (5=; 5=). 95. AE: x, 5y =,4, AD: x, y =,; 9. 96. A = 8 6 0, B = 6 34 0, C = 35 0. 97. x +y, = 0. 98. tg A = 4=3, tg B = tgc = ; S = 6. 99. (;,), (8=3;,). 00. x + y =,4, x, y =,4, x + y =4. 03. ; 8; 0; ; 4. 05. 3=. 06. k =. 07. 6 70406 04 縺0700306, 04 000006000400306 000308: 4x, 3y 0 = 0. 08. 8x, 5y +6=0, 8x, 5y = 30. 09. x, y =0 ｧ x + y, 4=0. 0. 3x, y = ｧ x +3y = 4.. x + y = ｧ00 ｧ 4x + y, 8 = 0.. 3x +6y =,. 3. x +3y =. 4. 0. 5. 3x, 4y +0= 0; x =. 6. h = 8= 34. 7. 7 縺0700306: x + y = 0 ｧ x, 3y = 0; ﾀ ﾀ0503070 ｧ07: d =, d =0; 4 0. 8. 7 04 縺0700308: x +y =0 ｧ x +y =0. 9. x +3y =0 ｧ3x + y =0. 0. x +y =74.. y =,x= ｧ y =,3x=.. x +y =4. 3. y =0, x +3y =,4; y =,4, x+3y =0; x+y =,; y =,x, tg = 8. 4. 8 6 0, 08 7 0 ; S 4 =b =3. 5. a =5. 6. A =36 5 0, B = 7 5 0. 7. 4( 0+ 5); 0. 8. x,y+6=0, x,4y =4, x,3y+ = 0. 9. y = x+, x,5y =6, y =,x, y = x. 30. 0. 3. ｯ03 09 閨04 ｧ070605 ﾀ07 0403 ﾀ0503 縺030 0 0904 陦 05, 0305 0 ｧ 0600030503 04 縺070030 ｧ x,3y = 5, 3x+y = 5. 33. h = h =6= 5. 34. (3=5; 9=5), (,9=5; 7=5). 35. (4; 5). 36. (0; ), (4; 0), (; 4), (,; 6). 37. y, x =, x +y = 4, x + y = 8. 38. ) B(; ); ) C(,;,5). 39. y =x+6; = 5; DAB 53. 40. x +y +8x,, 6y =0; A ｧ O 0 0309 縺0607003 ﾀ05 ｧ, B 04006 0606. 4. x +y +4x,6y =0. 43. (0; 0), (,; 5; ; 5). 44. (x,) +(y,) = ｧ00 ｧ (x,5) +(y,5) = =5. 45. tg =,; 4, = 37 0. 46. (x +4) +(y +) =5. 47. x + y, 8y = 0. 49. y = 4x=3 ｧ y = 0. 50. y = x(a, x).

36 050504060503 = a y, a 3 5. (x, 3) + y =9. 5. x + 9. 53. x + y = a. 54. x + y = ax. 55. x + y, 6y, 9=0. 56. ) (3;,), R =6; ) (,5=; 7=), R =4;3)(0;,7=), R =7=. 57. x +y +4y =0; (0; 0), (;,), (,;,). 58. x + y + ax + ay =0. 59. y =0, 5x +8y =0. 60. 90. 6. x + y =3. 6. x + y + ax =0. 63. (x, ) + y = = 6. 64. x + y = ax. 65. a = 4, b =, c = 3, e = 3=. 66. ) x 5 + y 9 =;)x 36 + y =. 67. b =; 4; 3; 4; 4; 8; 5; e =0; 96; 7 0; 8; 0; 6; 0; 8; 0. 68. a = 50 0000 090, e ==60. 69. x 6 + y 4 =, 3 e =, r =4, 3, r =4+ 3. 70. x 64 + y 8 =, r =, r =5. 7. 4 3. 7. 0; 4. 73. (=7; 4 3=7). 74. (,5=4; 63=4). 75. x 36 + y 4 x =. 76. 4 + y 3 x =. 78. a + y x = ｧ00 ｧ b b + y a =. 79. x 9 + y x = ｧ00 ｧ 5 5 + y x =. 80. 9 36 + y 9 =; e = 3=, r =3, r =9. 8. (a + b ). 8. (4 =3; =3) ｧ (0;,). 83. (,5; 7). 84. ( 5; ). 85. x +4y =6. 86. x 9 + y 8 =. 87. e = 5=, 53 08 0. 88. r =, r = 9. 89. ) x 6, y 9 = ;)x 0, y 4 =. 90. x, y 4 = ; 3 ｧ 6 3. 9. x 6, y = a. 93. (0; a ); 90. 94. y + = 96. 9 =. 9. x, y = x. 95. b, arctg b a. ab, b>a. 97. ) e =;)e = sec. 98. y 6,3, y<,jxj. b, a 99. x 4, y =. 00. x, y 3 = (04 罍ｧ x> 0). 0. x, y = = a. 0. x a, y x =. 03. b 6, y 9 = ｧ00 ｧ x 9, y 6 =,. 04. (0; 0) ｧ (6; 3). 05. y = 4 3 (x +5). 06. (,9; 6; 3 9=5). 07. ( 6; ). 08. (,4; 3) ｧ (,4=7;,3=7). 09. x 6, y 48 =. 0. x a, y x =(04 罍ｧ x>0).. y =3, 3a x, + y = ; 4.. y =8(x +). 4. ) y =9x; )y =,x. 5. y = a b x. 6.. ; 7. y =, x. 8. (3; 3 ). 9. 40 ﾀ0.. y = x.

3050504060503 63. y =4ax ｧ y =0. 4. y = 8(, x). 5. y = x, x 4, O (; ). 6. ) y =,4x; )y = x. 7. y =,3x. 8. (0; 0), (6; 3). 9. x =0, x + y +=0. 30. y =, 3(x +); 6=3. 3. r =7; 4, d =9; 5. 3. 6 7 ｧ 縺060905 罍ｧ ﾀ x = 3;, e =; 5, r = 0; 5, d = 8;. x 33. 4 + y =. 34. x, y =. 35. ｮ0304 縺070706000308 陦ｧ 00605 y =, x, a = b = 0. 36. ｮ0304 縺070706000308 陦ｧ 00605 4y + x =0; 8. 37. ｰ 040060 ｧ06 陦ｧ 00605 y = b a x, 060503 閨00 ｧ0 (a + b ). 38. y =; 5x. 39. y =. 40. 8x, 9y +5 = 0. 4. y =x +3. 43. ) x 3y =8; ) x y =;3)x y =,. 45. x, y = 5. 46. y = x +6. 47. x + y = a + b. 49. y = y 4. 50. ｰ 040060 ｧ06 003-0 00 ｧ MN: a y 0 x, b x 0 y = c x 0 y 0. 703000307 ｧ0 y =0, 0 08 閨060 03 ﾀ09 ｧ ﾀ ﾀ06 0503 09 ｧ N 0406 縺06 ﾀ06 060 ｧ07 003 縺0 00 ｧ MN ﾀ 03 ﾀ0406 Ox: x = e x 0. ｯ0305 FN = c,e x 0 = er, F N = c+e x 0 = er, 05. 06. 003 縺0 0004 MN 閨0600 ｧ05 FF 04030500300060 ｧ ｧ r : r ｧ 0403050503006 06 ﾀ0504 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ. 5. 703 縺0 0004 09 04 03030006 y =x ｧ0060605 06 040060 ｧ06 y 0 x + y = y 0 ( + x 0 ). 703000307 ｧ04 y =0, 0 08 閨060 x = + x 0, FM = x, = + x 0 = FM,05.06. FMN = FNM. 53. (3; ; ; 4). 54. 6 7 ｧ 00605 縺03 y = x ｧ y =,x=4, 06050300 59 0 0. 55. y = x=4. 56. 4x, y =6. 57. arctg 3 7 3 0. 59. x + y +=0. 60. ) O (; ); ) tg ' = 3 4. 6. 5) X +4Y = 6;6)Y = 4X; 7) X,4Y =4;8)Y = X =. 6. ) X +4Y =6;)X,4Y =6. 63. X, Y =8. 64. ) XY =6;)XY =,6; 3)XY =4;4)XY = =, 6. 68. ｰ 040060 ｧ06 ﾀ05 縺06 ｧ: y = 6(x, x ); y =30 04 罍ｧ x =0; 750. 69. y = b y, x. a 70. x + y +4x =0. 7. ) 45 ;) arctg. 7. y = x tg ', gx v 0 cos '. 73. y =4x +3x (05 ｧ0406 縺030300 ). 75. ) 700-00 ｧ04 ﾀ; ) 05 ｧ0406 縺030300. 76. ) X 5 + Y =, O (3;,); )X, Y =9; 3) Y =X; 4)X =4Y. 77. X +Y =4. ｱ030906 ﾀ03 04 ﾀ05 縺0308 ﾀ ｧ ﾀ0506006 (; ) ｧ (,;,). 78. (x +) + y =4. 79. (x, 3) +(y, 3) = (x, =. 80. x +3y = 0. 8. y ) = 4(x +4). 83. + y 6 =. 84. x + y, ax, by =0. a 5 85.. 86. 05 ﾀ00304 0 ｧ06 AB =a, 0403 ﾀ0305 OD = a, 0400030 閨04 a. 87. 6ｦ0 0 0003 04 罍ｧ0060 0503 0906 O, 閨06000700606 AB 5 5

364 050504060503 04030500300060 ｧ ｧ AO : OB = m, 03 ﾀ04Ox 04 縺0700606 OB; 0406 ﾀ0504 OB = a, 050305 090303 罍陦ｧ0 0503 0503 0609 A ｧ B 0306 閨0605: A(,ma; 0), B(a; 0). ｰ 040060 ｧ06 ｧ ﾀ- 090300308 00 ｧ0 ｧ ｧ: (m,)x +(m,)y =max; 04 罍ｧ m 6= 0309 縺0607003 ﾀ0504: x + y = ma x; 04 罍ｧ m =04 縺070 07: x =0. 88. ｯ03 0906 O 04 罍ｧ0060 0 0003, m, OB 03 ﾀ04Ox. ｰ 040060 ｧ06 ｧ ﾀ090300308 00 ｧ0 ｧ ｧ: (a, b)(x + y )=abx; 04 罍ｧ a 6= b 0309 縺0607003 ﾀ0504: x + y = ab x; 04 罍ｧ a = b 04 縺070 07: x = 0. a, b 89. (k x +y )=a (k +); 050000 ｧ04 ﾀ 04 罍ｧ k 6=,0309 縺0607003 ﾀ0504 x +y = a 04 罍ｧ k =. 90. x +0x + y 5 9 =0. 9. 3a 3. 9. arctg 3 4 36 5 0. 93. (a; a). 94. A( 6; 0), B(;,), C(, ; ); S 4 ABC = = + 3+ 6. 96. ; y = x,. + 3 97.. 98. x, +y = = 9 6. 99. ax, by + jabj a + b =0; d =. 300. 6 503 ｧ05 07 06 04- a + b 0060 ｧ07 0403 00060003, 04030006 ｧ0 4(y, x) =(y + x)(y, x), 0305 ﾀ06 : ) y = x; ) x +y =4; ﾀ0006 閨0304 05060004003, 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 04 030300 000607 05 0 04 縺0700308 y = x ｧ00 ｧ 0 04 縺0700308 x + y =4; 0 08 閨060 x =, x =,6; 閨00 ｧ0 0803 罍閨03 8. 30. 30. 30. x + y (x, ) = a(x + y). 303. + y =(050000 ｧ04 ﾀ ﾀ 4 0906005 縺030 (; 0)). 304. xy =4. 305. y = x, 6x +5. 306. X, Y =4; 8 (x, ; 5) O (;,3). 307., y = (05 ｧ0406 縺030300 ﾀ 0906005 縺030 (; 5; 0)). ; 5 4 308. 706 ﾀ0504 M (x; y) 0503 09 050000 ｧ04 ﾀ. ｯ0305 FM+F M = AF +AF ｧ00 ｧ (x, a) +(y, a) + (x + a) +(y + a) =4a; 3x, xy +3y =8a ; 0403 ﾀ0006 04030403 縺0305 03 ﾀ06080 45 : X +Y =4a. 309. cos ' = +tg ' = = 5, sin ' = 5 ; 003040306 06 040060 ｧ06 X, Y =4. 30. 3x +8xy,, 3y =0; 04030403 縺0305030 03 ﾀ0608 0 06050300 ' = arctg (=) 04 罍ｧ0403 陦ｧ05 ﾀ07 09 04 ｧ 閨06 X, Y =4( ﾀ0. 309). 3. y =x +(e, )x. 33. ) 7 04 縺07-00308 y = x; )0503 09 (0; 0); 3) 00 ｧ0 07 0309 縺0607003 ﾀ0504; 4) 0503 09 (3; 4); 5) 04 04 縺0700308 x = 0, y =,x; 6) 04 04 縺0700308 y = 4; 7) 04 04 縺0700308 y = x ｧ y = x. 34. ) (;,), X = ; ) (; ), 6 + Y 4 X, Y = 9; 3) X +5XY +Y = 8. 35. ) X 4 + Y 4 ) X 4, Y X =. 36. ) 6 8 + Y X =;) 4 = ; 8, Y 4 =. 37. ) Y = = 5X; ) 04 04 縺0700308 x, y =3. 38. ) 3y =x, 7 (x, );

3050504060503 65 ) 0503 09 (;,); 3) 4y =,x, 3. 39. 4X, Y = 8; 0906005 (; 0); tg ' =,=. 30. 5(x, ) +(y, ) =9. 3. 7030406 縺00604 03 ﾀ ｧ 0,45, 04030006 ｧ0 Y = X a + a. ｰ 040060 ｧ06 x + y = a 0304 縺06- 閨0600070605 閨060506 AB 05050308 04 03030003 ( 罍ｧ ﾀ. 87), 0 09030503 縺0308 x 6 a ｧ y 6 a. 3. (x, m) +(y, n), e (x cos + y sin + q) =0; A + C =, e ; =, e. 33. ) 7 04 縺0700308 x y =0;)0503 09 (,; ); 3) 04 04 縺0700308 y = x, x +6y =0. 34. ) X + Y 4 =;)X 0, Y 5 =. 35. ) Y =4 X; ) 04 縺0700306 x + y =. 36. ) y = x, ; ) 3y = x, 5 (x +). 37. ) 7x, xy +7y, 48x, 48y + 44 = 0; ) x +4xy + y +6x +6y, 8 = 0. 38. (x, y), a(x + y) +a =0; Y = a X. 39. x, 4xy, y, 4x +8y, = 0; X, Y =3; 5. 335. ) r = a a sin ;) r =. 336. r = a sin (, ) sin (, '). 337. r = cos ' sin ' =acos '. 338. ) r max =504 罍ｧ ' = 35, 35 ; r min =04 罍ｧ ' =45, 5 ; r =304 罍ｧ ' =0, 90, 80, 70 ;)r max =304 罍ｧ ' =0, 0, 40 ; r min =04 罍ｧ ' =60, 80, 360 ;3)r max =04 罍ｧ ' =90, 0, 330 ; r min =004 罍ｧ ' =30, 50, 70. 339. ) r max = a 04 罍ｧ ' =30, 50, 70 ; r =004 罍ｧ ' =0, 60, 0, 80, 40, 300 ;)r = a 04 罍ｧ ' = 45, 5 ; r =,a 04 罍ｧ ' = 35, 35 ; r = 004 罍ｧ ' = 0, 90, 80, 70 ( ﾀ0. ﾀ. 334, 罍ｧ ﾀ. 83). 340. ) r = a ;)r = a; 3)r = cos ' = cos (', ) ;4)tg ' =;5)r = cos '; 6) r = a cos '. 34. ) x = a; ) x +y =ay; 3)xy = a ;4)x+y =a;5)(x +y,ax) = a (x +y ). 34. ) x 5 + y 9 =;)x 6, y 9 =;3)y =6x. 343. r = a sin ' b. a( sin ') 344. r = OB AB = ｧ00 ｧ 04 閨0609 縺0503040308 090303 罍陦ｧ0 05 08 y = cos ' x(x, a) = a, x. 345. FM = r +a,ra cos ', F M = r +a +ra cos ', FM F M =(r + a ), 4r a cos ' = b 4,0305 ﾀ06 r 4, a r cos ' = = b 4, a 4. 346. r = a( + cos '); (x + y, ax) = a (x + y ). 347. 706 ﾀ0504 C 0906005 0060403 閨04 ｧ070030503 09 縺0605, C 0906005 ﾀ00600600030503 09 縺0605 ｧM ('; r) 050609060 07 0503 09. ｯ 09 09 09 OCC = MC C = ' ｧ CO = C M = a, 0503 OM k CC. ｮ04 縺03060905 ｧ 縺0304 04 00030 00606 COMC 0 CC, 04030006 ｧ0 a cos ' + r + a cos ' = a. 0505 ﾀ06 r = a(, cos '). 348. ) r max =504 罍ｧ ' =0, 80 ; r min =04 罍ｧ ' =90, 70 ;)r max =4

366 050504060503 04 罍ｧ ' =90, 0, 330 ; r min =04 罍ｧ ' =30, 50, 70 ;3)r = a 04 罍ｧ ' = 0, 80 ; r =,a 04 罍ｧ ' = 90, 70 ; r = 004 罍ｧ ' = 45, 35, 5, 35 ab sin (, ) x. 350. r =. 35. ) a sin (', ) +b sin (, ') 4 + y =; ) x 4, y =;3)y = x. 35. r =c cos '; (x + y ) =c (x, y ). 7 罍ｧ ﾀ. 80 040300030706003 c =a. 353. r = b + a cos '. 354. 0ｷ3 4OAM: r = OM = OA cos ', 003 ｧ 4OAB: OA =a sin ', 03050906 r = a sin '. 358. 706 ﾀ0504 0503 09 A 0 03 ﾀ ｧ Ox, 0503 09 B 0 03 ﾀ ｧ Oy ｧ OAB = t. ｯ03-05 x = BM cos t = BC cos t = a cos 3 t, y = AM sin t = AC sin t = = a sin 3 t; ｧ05 09, x = a cos 3 t, y = a sin 3 t,03050906 x =3 + y =3 = a =3. 360. y = x + x. 36. (3y + x ) = 4x (a, y ). 36. 6 5 04030007 縺00308 090303 罍陦ｧ0 05 08 r = OM = AB = BD sin ' = a tg ' sin '; 04 閨0609 縺0503040308 y = x3 a, x ( 罍ｧ ﾀ. 85). 365. 050303 0 ｧ04 06 縺06 t 06050300 0006 OA ﾀ Ox, 0 08- 閨060 x = a ctg t, y = a sin 8a 3 t. 0ｷ3 ﾀ090006 ｧ04 t, 04030006 ｧ0 y = x +4a. 367. x = a(t, sin t), y = a(, cos t). 368. x = a(cos t + t sin t), y = = a(sin t,t cos t). 369. y = x ctg x a y =(R + r) sin t, r sin 37. x = (R, r) cos t + r cos R, r r (R + r)t. 370. x =(R+r) cos t,r cos, r (R + r)t, 05 閨06 t 06050300 04030403 縺0305 00 ｧ0 ｧ ｧ 0906005 縺0304. r t, y = (R, r) sin t, r sin R, r t. 374. X = P X i = 8; Y = P Y i =,; OM = 64+4 = 7. 375. 8+ 3. 379. ) c = a + b ;)a =c, b. 380. c = (a, b). 3 38. m + = n;,! OB =3(m + n),,! BC =3(n, m),,! EO =3(m, n),,! OD = 3(n, m),,! DA =6(m, n).,! 38. AC =(n, m),,,! OM =n + m,,,! ON =3m + n, P,,! MN =m, n. 383. 6 3. 384. X = X + X + X 3 = =, 3, Y = Y i =6, OM = 9+36 = 3 5. 385. ) a =3(c, b); ) c =b, a 3. 386. OM = r =5 ; cos =0; 5, cos =,0; 3, cos = 0; 4. 387. r = 7; cos = =7. 388. 5 ｧ00 ｧ 8. 389. M (3 ;3;,3), r =3( i + j, k). 390. u =i, 6j +3k, u =7. 39.,! OC = i, j + k, OC,! = 6; AB = k, 4j, i, AB = 3. 39. 070300609 B(4;,; 5) ｧ00 ｧ B (4;,;,7), cos ==7; cos =,3=7; cos = 6=7. 393. a = b, 0; 8c. 394. u = 3 5, cos =,. 3 5 395. cos = cos = cos == 3. 396. 45 ｧ00 ｧ 35. 397. D(4; 0; 6). 398. c = b, a. 399. 35. 400. B = C = 45. 40. cos ' = r

3050504060503 67 == 0=0; 36; ' =7 35 0. 40. cos ' == 5=0; 894; ' 6 37 0. 403. 60. 404. arccos 0; 8. 405. 90. 406. 04 b a = 4 =3. 407.. 408. ) + 3;)40. 409. (a + b) = a + b +ab cos ' (050603 縺060 0903 ﾀ ｧ- 006 ﾀ0304); (a + b) +(a, b) =a +b ( ﾀ040308 ﾀ050403 陦ｧ 05030 000608 04 0000060003-05 00 ). 40. 7. 4. R = (a + b + c + d) =0 4+ 5; 3 7. 4. 7 ｧ 3. 43. cos ( [a; m) = (m, n)m (m, n) = 5 ; cos ( da; n) = 7 =, = 7. 44. 5=6.,,! 45. OM =(i + j +k),,,! ON =(i +j + k); cos = 5=6. 46. cos ' = = 7. 47. cos ' = 0; 6 0; ' 34 4 0.,! AB,! CD AB 48. D(,; ; ); ' = 0. 49. 04 a b = =,6. 40. OM = = (n + m) = 7, ON = (3m,,! OM,,! ON + n) = 3; cos ' = OM ON = = 7 9 = 7 9; 08 0; 89, ' =7. 4. 0. 43. 80 6 707, cos = 4 5. 44. a 6. 45. cos ' =,=4. 46. a b 04003: ),6j; ),k; 3) 6i, 4j +6k. 700030 閨04 040 : ) 6; ); 3). 47. 4; 5. 48., h = 4;. 49. ) (k, i); ) a c; 3) a c; 4) 3. 430. 70003-0 閨04 04 000006000305 00, 0403 ﾀ05 縺030600030503 0 陦ｧ 05030 000708 00030503 04 000006-000305 00, 04 閨040306 030300040006 0400030 陦ｧ 00030503 04 000006000305 00. 43. 50. 43. ; 5. 433. 3 7, S 4 =3 7=. 434. S 4 =7 5, BD = =3. 435. ja + bj = ja, bj = 5, S = 6. 437. ; 5. 438. V =5, 000604 07. 439. V =4, H =7 3=3. 44. c =5a + b. 443. =3. 444. V =4, H = 4. 445. c = a+b. 446. V = j(a+b)[(b+c)(a+c)]j =jabcj. 447. (m n) = jm nj cos = sin cos =0; 5 sin. 449. 5. 45. cos = 7, cos = 3 7, cos = 6. 45. x+4y,z =. 453. x+y =. 7 454. x,y +z =. 455. y,3z +7 = 0. 456. 3y +z =0. 457. x+y =0. 458. x a + z c =. 459. x + y + z =4. 460. x 4 + y 3 + z =. 46. cos = 3, cos =, 3, cos = 3 ; =48 0, = 3 49 0, =70 3 0. 463. x,, y, 3z +4 = 0. 464. 3x,4z =0. 465. x +y =4. 466. x + y 4 + z 4 =. 467. ) 45 ;) 78 30 0. 468. x, y, 3z =4. 469. x +3y +4z =3. 470. x+y+z = a. 47. x,y+z =. 47. x,y+z = 5. 473. 3x,y = 0 ｧ x +3y =0. 474. 3. 475. 6. 476.. 477. ) x, y +z =, x,y +z =,; )x+y,z =0 ｧ x+y +z =0. 478. ) x,8y +9z =; ) x, y +z =0 ｧ x, y, z =0. 479. (;,; ). 480. 3x, 4y + z =. 48. y, 5z +0 = 0. 48. ｰ 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ x + y, z =0;06050300

368 050504060503 0606 ﾀ 040003 ﾀ0903 ﾀ050406 z =0: cos ' = 6=3 0; 865, ' =35 5 0. 483. jaj 3. abc 484. y = z. 485.. 486. x +y + z =0 ｧx +y + a b + a c + b c + z+4 = 0. 487. 7x+4y+4=0. 488. ) (5; 4; 0) ｧ (7; 0; );)(0;,4; 0) x, 3 ｧ (; 0; ). 489. x =,z +3, y =,z +5; = y, 5 = z,. 490. x, 4, = y, 3 = z. 49. Pf0; 0; g. 49. ) P = i; )P = i + k; 3) P = j+k. x 493. + = y, = z, 3 3 4,5 ; cos =0; 3, cos =0; 4, cos =,0; 5. 494. x =, z =3. 495. ｪ06 縺06 t ﾀ0609060 090303 罍陦ｧ0 0503 0503 09 ｧ M 0306 閨0605 x = 4+t, y =,3+3t, z =+t; x, 4 = y +3 = z, 3. 496. ) x =,+t, y =,t, z =,+3t; )x =+t, y =,t, z =+t. 497. ) x, a = y, b = z, c, 0503 0 ｧ05 x = a, y = b; )z = c ｧ 0 0 x, a m = y, b n. 498. cos ' =. 499. cos ' =. 50. 7 04 040007060 ｧ08 3 6 0406090503 P = NN = i+3j+5k. ｰ 040060 ｧ07 04 縺0700308: x +4 = y, 3 = z 3 5. 50. 3x +y = 0, z = 4. 503. 0; 3 38. 504. 4 =3. 505. (4; ; 0), (3; 0; ), (0;,6; 8). 506. x =6, 3z, y =,z +4; x, 6,3 = y, 4, = z ; ﾀ0006 閨03: (6; 4; 0), (0; 0; ). 507. 509. Pf; ; g; = = arccos 5. 7 罍ｧ0406 閨060 06 040060 ｧ07 09 09 0030 ｧ 06 ﾀ090308 0703 縺006: ｧ x = y, 4 3 ﾀ 04 06 040060 ｧ07 000308 04 縺0700308 04 04 ｧ 閨06 x, x = y +4 = z. 508. Pf0; ; 0g. 3. 50. y =,3, x, z = 0. 6 x = y +7 = z, 5 = z 0 ; cos ' = 6 0; 95, ' = 7 48 0. 5. 7 04 ｧ- = y = z, =3, 04030006 ｧ0 x + 06 040060 ｧ06 ｧ ﾀ090300308 04 縺0700308: = y, = z +. 53. A(0; ; 0),,,! AMf3;,; 4g, Pf; ; g, d = 7. 54. sin == 6. 55. 6 70007 030306 ｧ08 04 縺0700308 Am + Bn + C =+(,)+(,) 3=0,0030503 09 0406-040308 (,;,; 3) 006 000607 ｧ05 0 040003 ﾀ0903 ﾀ05 ｧ, 0503 09 040503 縺0308 (,;,;,3) 000607 ｧ05 0 040003 ﾀ0903 ﾀ05 ｧ. 56. y + z + = 0 04 ｧ ﾀ 0504 04 04 ｧ 閨06 x, 0 = y, = z 06 040060 ｧ07 04 縺0700308 00307003. 57. x, y + z +5 = 0.

3050504060503 69 58. 8x, 5y + z, = 0. 59. x +y, z =. 50. x 3 = y = z ; 7 33 0. 5. (5; 5;,). 5. (6; 4; 5). 53. (5; 5; 5). 54. (3; 3; 3).,,! AA PP 55. d = jp P j =. 56. x+y,5z =0. 57. x, 3,9 = y, = 7 8. 58. (; ; ); 70. 59. (,; ; ); 30. 530. (6; ; 0). 53. (3;,; ). 53. x,y,z =0. 533. (,; 3; ). 534. x, = y 5,4 = z +. 535. ｯ03 09 ｧ, 0 04 縺0700308: O(0; 0; 0) ｧ A(; ; 0); 0 04 040007060 ｧ06 0406090503 縺03 04 縺0700308:,! OAPP Pf0; 0; g ｧ P f;,; g, d = jp P j = 6. 536. ) C(; 5;,; 5; ), 5 R =; 5 ;)C(0; 0; a), R = a. 537. (x, ) +(y +) +(z, ) =. 538. x + y + z =8x. 539. x + y + z, a(x + y + z) =0. 54. y = =ax,x. 54. x +y =ax, x +z =ax, y +z = a. 544. (; 7; ), R =4. 545. (3Y, ) = (3X,). 546. ) y =0, x = a,az (04-030300 ); ) x =0, y = a,az (04 030300 ); 3) z = h, x+y = a(a, h) 04 縺070 07, 04 00000600040 07 x +y = a ( ﾀ0. 罍ｧ ﾀ. 59 0 ﾀ. 30). ｨC ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 07 04030406 縺08003 ﾀ0504 x +(y, z +) =8, 0703 縺0 05060 ｧ x (y +) + = 4 8 050000 ｧ04 ﾀ. 548. x, y +3z, 7 = 0. 549. x +(y +4) + z = 4. (x, ) (y +4) 550. + =. 553. (x, z) +(y, z) = 4(x, z). 36 8 554. x =4, z y =. 555. x + y a = z c. 556. h x =z[h(y + a), az]. 557. (0; a; 0), 0 04 040007060 07 0309 縺0607003 ﾀ0504 z = a, x +(y, a) = a. 558. 6 506 縺00 ｧ0 (0; 0; 0), 0 04 040007060 07 04 030300 z = h, x = hy. 559. 7 罍ｧ z = 0 x = a; 04 罍ｧ y = h x + y = a ; 04 罍ｧ x = c 04 縺07- a, c 00306 z = y, 05. 06. 04030406 縺08003 ﾀ0504 0303 0304 0 閨04 ｧ07060 ｧ060 04 縺07- h 00308, 04 000006000400308 040003 ﾀ0903 ﾀ05 ｧ yoz ｧ 0406 縺06 ﾀ0609 0600608 0309 縺0607003 ﾀ0504 ABC ( ﾀ0. 罍ｧ ﾀ. 65 0 ﾀ. 3) ｧ 03 ﾀ04 Ox. 560. ) z = x + y ;03) y + z = = x. 56. ) z = e,(x +y ) 4 ;)z = x + y. 56. 9(x + z )=6y. 563. x + z = z(y + a). 564. ) x + z = y ; 03) z = x + y. 565. 7030406 縺00604 03 ﾀ ｧOx ｧ Oy 040309 縺0605 03 ﾀ ｧOz 0 45, 04030006 ｧ0 06 040060 ｧ07 04030406 縺08003 ﾀ05 ｧ ｧ 040003 ﾀ0903 ﾀ05 ｧ 04 04 ｧ 閨06 = X, Y, X = a. 0505 ﾀ06 ﾀ06 060 ｧ06: X = a Y, a + a = 050000 ｧ04 ﾀ ﾀ 0403000603 ﾀ070 ｧ a ｧ a. 566. x + y a + z 45 =. 567. ) 3; 84; 03) c 4. 568. ) x + y, z a c = (03 閨00304030003 ﾀ0500308 05 ｧ0406 縺03030003 ｧ ); 03) x a, y + z c =( 閨040604030003 ﾀ0500308 05 ｧ-

370 050504060503 0406 縺03030003 ｧ ). 570. x 4 + z 6 = 3 + y, x 4, z 6 =3, y ｧ x 4 + z 6 = =, y, x 4, z 6 =+y. 57. x = a c (c, z) cos t +(c + z) cos (t + ), x + y (c, z) sin t +(c + z) sin (t + ),03050906 y = a c = + cos ; 04 罍ｧ =90 x + y a, 3z c =, 04 罍ｧ = 80 x + y, z 4a, z (, cos ) = c a, z c =, 04 罍ｧ = x 0 + y, a c =0(0903006 ﾀ). 57. x + y = az. x 574. x + y =4, x, y = z; x + y =z, x, y =. 575. a + y + z =. a 576. x +y,z =,a ( 閨040604030003 ﾀ0500308 05 ｧ0406 縺03030003 ｧ ). 577. x =, z + y 4a 578. 9x = 3z. 579. 4y = 3z. 580. ) ｮ0706 ﾀ 0906005 縺030 (0; 0; a) ｧ 陦ｧ06 ﾀ030 R = a; ) 04 03030003 ｧ 04 0060 ｧ07 040309 縺0605 Oz; 3) 09 ｧ00 ｧ0 陦; 4) 05 ｧ- 0406 縺030300 ｧ 06 ﾀ09 ｧ08 04 03030003 ｧ ; 5) 0903006 ﾀ; 6) 04 030300 ｧ 06 ﾀ09 ｧ08 09 ｧ00 ｧ0 陦; 7) 0903-006 ﾀ; 8) 04 03030003 ｧ 04 0060 ｧ07; 9) 0903006 ﾀ; 0) 09 ｧ00 ｧ0 陦. 58. x + y =+z, x, y =, z; x + y = 3(z, ), 3(y, x) =z +. 58. x + y = = az. 583. z = a, x + y. 584. y = 3z. 585. 3x +4y = 4, a 3x, 4y =z; z =0, 3x =4y. 586. 6. 587.,38. 588. 7. 589. a. 590.. 59. sin ( + ) sin (, ). 59.,0. 593. 4a. 594.,b. 595.,x. 596.,4a 3. 597. 44. 598. 7. 599. (x, y)(y, z)(x, z). 600.. 60. sin (, ). 60. 0. 603. 6ｧ80607 05 0 04 縺0700308 y = x +. 604. ) x y x y x y = 0; ) x y 3, 5 = 0. 605. 0. 606. amn. 607. a(x, z)(y, z)(y, x). 608. 4 sin sin. 60. ) x =, x =3; ) x =0, x =,. 6. x =5, y =,4. 6. x = 4, y =. 63. x =0, a y =. 64. x = m, y =m, n. 65. 5; 6; 0. 66.,; 0;. 67. 7k, 8k, 3k. 68. 5k,,k,,7k. 69. x = y = z =0. 60. 706 ﾀ0304006 ﾀ050. 6. 7060304 縺06 閨06000600 : x = +5z, y = 5, 7z. 6. 706 ﾀ0304006 ﾀ050. 64. ; 3 3,;,3. 65. ;,;. 66. k, k,,4k. 67. x = y = z =0. 68.,k, 3k, 5k. 69. 7060304 縺06 閨06000600 : y =7, 3x, z =8, 7x. 630. ) +5i; ) a +b ;3)5,i; 4),+i; 5)i; 6)+i. 634. ) cos 3 4 + i sin 3 ; 4 ) sin cos + i sin. 640. ) 3i; )64; 3)4(, i); 4)(3+ )i; 5) 8i. 64. sin 3 = 3 sin cos, cos 3, cos 3 = cos 3, 3 sin cos..

3050504060503 7 64. cos k 3 +i sin k 3 ; k =0; ; :::; 5. 643. ),, i 3 ;),i, i 3 ; 3) i, 3 i ;4)+i,,; 36 + 0; 365i, 0; 365, ; 36i. 644. ) +i ; ) 6 (cos ' + i sin '), ' =45, 65, 85 ;3)( 3+i), (,+i 3). 645. ),, i 3;) i. 646. ) ln + i; ) i ln + 4 ;3)i 4 ; 4) ln x + y +i arctg y x ;5) 3 ln, sin (nx=) sin [(n +)=]x i. 647.. 4 sin (x=) sin (nx=) cos [(n +)=]x 648.. 650. ) 7, 4i ; ) b(3a, b )i. sin (x=) 5 65. ) 4 e i=4 ; ) e i=3 ; 3) e,i=4. 65. ) 5(cos 0 + i sin 0); ) e,i= ;3)e,3i=4. 654. ｯ03 09 ｧ 0400605 罍ｧ 09 縺0605 ﾀ 0906005 縺030 C(z 0 ) ｧ r =5. 655. ) 8i; )5(, i 3); 3),7. 657. ) i ;)cos ' + i sin ', 05 閨06 ' =0, 7, 44, 6, 88. 658. ),,i 3;)i, 3i; 3)3, sin nx, i 5 3i. 659.. 660. ),,, 3; )5,. 66. ) x =3, sin x x = 4, x 3 =,; ) x =, x =,, x 3; 4 = i ; 3) x =,, x ; 3 = =3; 4)x =, x ; 3 = i=. 66. ) =49=4 > 0, u =, v =, z = 3, z ; 3 =,3 i 3 ; ) = 0, z = 4, z = z 3 = =,. 663. ) < 0, ' =60, z = 4 cos 0, z ; 3 = 4 cos (0 0 ). f() f() k k 665.,0 4 4 3 0; 7,0; 3 ; 85 <x<; 86: ; 7 ; 87,3; 0; 36 6 0; 4,0; 0 666. ; 5; 0; 54;,; 66. 667. ) ; 305; )4 ｧ 0; 30; 3),0; 68l; 4)x = =; 494, x =,0; 798 x 0 08 閨06003 0403 0703 縺0060006 x = 4 x +, x 0403 0703 縺0060006 x = x4 +3x,. 668. ),6,, i ; ),; ;. 5 669. ) = 5 > 0, u = 3, v =,, z =, z ; 3 =, 5i 3 ; 4 ) =,4 < 0, ' =45, z = cos 5 =+ 3, z =,, z 3 = =, 3;3)=0, z =,, z ; 3 =; 4) 0403000307 ｧ04 x = z,, 04030006 ｧ0 z 3, 3z +=0; =0; z =,, z = z 3 =; x =,4, x = x 3 =,. 670. ; 76 ｧ,; 5. 67. ) ; 7; )3; 07. 67. ; 67. 675. 0 6 x<. 68. x = 0, x = 4. 683. ) x >,; ),3 6 x 6 3; 3) 0 6 x 6 6 4. 684. ),4 6 x 6 0; ), 6 x 6 3. 685. ) x > 0; )x 6 4.

37 050504060503 686. ) k 6 x 6 (k +); ),4 6 x 6 +4. 687. ) f(0)=, f()=, f(,) = 3, f() = 3, f(a +) = a + a +. 688. ) b + a; ) ah. b + a 689.. b 690. F (4; 3) = 9, F (3; 4) =,5. 69. ) ｪ06050 07; + ab + a ) 006 06050 07; 3) 06050 07; 4) 006 06050 07; 5) 006 06050 07; 6) ｧ 006 06050 07, ｧ 006 f(x )+f(x ) 006 06050 07. 69. > f x + x. 693. log a x. 694. a x. 696. <x6 3. 700. ) jxj 6 ; ), 6 x 6 3; 3), 4 +k 6 x 6 4 +k; 4) jxj >. 70. ) 6x +h ;3)4(, a). 70. jj < 0; 00, 09 09 050300040903 n> 3 3 lg (=") ｧ00 ｧ n> =0; jj <",09 09 050300040903 n>. 703. x =; lg 0; 3 lg 3 ; 5 ; 6 7 ; ; :::! ; jx, j < 0; 0, 09 09 050300040903 n > 50; jx, j <", 9 09 09 050300040903 n>, ". 704. x =4; 3; ; 3; 0; :::! 3+0; x =; ; 9; " ; 99; :::! 3, 0. 705. x = 6; 5; ; 5; 0; :::! 5+0; x = 4; 4; 9; 4; 99; :::! 5, 0; x =,;,; 9;,; 99;,; 999; :::!,+0; x =,3;,; ;,; 0;,; 00; :::!,, 0. 707. = "=. 708. = 0; 0. 7. 7 罍ｧ jxj > 500; 5. 73. 7 罍ｧ jxj > 7; 036. 75. lim n! x 04 0406 縺04030 04 罍ｧ006 縺06 04060, 0403040503 縺030,, 04 06050406 縺05030 0, 04 040705030, 04 0006 ﾀ05030 0, 04 05 縺060504060 006 ﾀ06006 ﾀ0504060605. 76. x 3; ; ; ; 0; :::! +0 3 x, 3; 30; 300; :::! + ; lim x!+0 x ; ; 9; ; 99; :::!, 0 3,3;,30;,300; ::: x,!, ; lim x!,0 3 x, =+; 3 x, =,. 77. x ; 0; ; 0; 0; :::! +0 =x ; 0 ; 00 ; :::! + ; lim x!+0 =x =+; x,;,0; ;,0; 0; :::!,0 =x =; = 0 ; = 00 ; :::! 0 ; lim x!,0 =x =0. 78. ) lim x! x =0;) lim x!+0 x 4) lim x!, 3x = 0; 5) lim x!+0 =+; lim x!0 x =,; 3) lim x!+ 3x = ; lg x =,; 6) lim x!90,0 tg x =,. 74. AB!, CB!, BCD! 0, lim x!90 +0 ACB! 80. tg x = +;

375. x =5; 4; ; 4; 0; 4; 00; :::! 4+0; x =3; 3; 9; 3; 99; 3; 999; :::! 4, 0; 050504060503 73 x =,0; 5;,; 4;,; 49;,; 499; :::!,; 5+0; x =,; 5;,; 6;,; 5;,; 50; :::!,; 5, 0. 79. ｯ0300040903 0406 縺04 07 0403 ﾀ0006 閨0304 05060004003 ﾀ0504 ｧ0060605 04 縺06 閨0600: lim n! x =. 6 503 ﾀ05000400308 04 罍ｧ006 08 lim x 006 ﾀ06006 ﾀ0504060605. 730. ) 0; ); 3); n! 4) 0; 5); 6)0; 7)0 04 罍ｧ a>, = 04 罍ｧ a =, a 04 罍ｧ 0 <a<. 733.. 734. ),0; 6; ). 735. 4. 736.. 737. 3=. 738. =. 739.,=. 740. =3. 74.,= 04 罍ｧ a>0 ｧ 04 罍ｧ a<0. 74. =3. 743. m=3. 744.. 745.,=. 746. ) =3; ),; 5. 747. 0. 748.. 749.,. 750.,3=. 75. =. 75. =6. 753. =4. 754.,. 755.,. j sin xj 756. lim x!+0 sin x, cos x =,=. 757. ; 5. 758. 3. 759.,4. 760.. 76.,=56. 76.,. 763. 4. 764. =3. 765.. 766. =4. 767.. 768. 6. 769. cos x. 770. ) ; ),=. 77. =. 77. =. j sin xj 773. =3. 774. 8. 775. lim =,. 776. 4. 777. m =. 778. 3. x!,0 x 779. =4. 780. ), sin x; ),=. 78.. 78. ; 5. 783. =. 784.. 785. =. 786. =4. 787.,3. 788. =. 789.,. 790.,=4. 79.. 79. 0. 793. =. 794.,=. 795.,. 796. ) =0; ) 3. 797. ) 3=4; ) [0403000307 ｧ0504 04 04 罍ｧ006 縺06 ) x = t, 04 04 罍ｧ006 縺06 ) +x = t 4 ]. 798.,a. 799. ),; ),0;. 800. ) 3; )3=. 80. ) ; ),=. 80. ),; ),0;. 803. ),; 5; ); 5. 804. ), ; ),. 805. ) ; )3. 806. ) 4; ) ; 3) 3. 807.. 809. 7 罍ｧ! 0 ( + ) 3, 3. 80. ) ; 5; )a=b; 3); 5. 8. ｧ 3. 8. ) ; )3; 3). 85. ) 7 罍ｧ x =0;)04 罍ｧx = n, ; 3)04 罍ｧx =. 86. 7 罍ｧ x =0403040300006003 0406 縺040306 05 罍ｧ 06 ﾀ000304 ｧ07 ｧ 006 0403040300006003 06050406 縺050306. ( (, 04 罍ｧ x<,; x, 04 罍ｧ x<,; 87. ) y = ) y = 04 罍ｧ x>,; x + 04 罍ｧ x>,: 7 罍ｧ x =, 070600909 ｧ ｧ ｧ0060605 縺0304 I 縺03 (0403040300006003 050300040903 040503 縺0306 06 ﾀ000304 ｧ06 00604 縺06 縺0304003 ﾀ05 ｧ). 88. 7 罍ｧ x =0006 0403040300006003 050300040903 06050406 縺050306 06 ﾀ000304 ｧ06 ( 罍ｧ ﾀ. 37). 89. 7 縺0304 04 罍ｧ x = 0, lim y =, lim y = 0, x!+0 x!,0 lim y =( 罍ｧ ﾀ. 38). 80. 7 縺030403 04 罍ｧ x =. 8. ) 7 縺0304 I 縺03 x! 04 罍ｧ x =0, 04 罍ｧ 0505030 lim y =0, x!+0 lim y =, x!,0 x!+ lim y =, x!, lim y =

374 050504060503 = ( 罍ｧ ﾀ. 39); ) 縺0304 I 縺03 04 罍ｧ x = a, 04 罍ｧ 0505030 lim x!a,0 y =,, lim y = x!a+0, x! lim x y = 0;3)y = 04 罍ｧ x = 縺0304 I 縺03, 04 罍ｧ 060 04 罍ｧ x> ｧ,x lim y =, x!,0, 04 罍ｧ x< ; lim y = x!+0. 7 ｧ ﾀ. 37 7 ｧ ﾀ. 38 7 ｧ ﾀ. 39 8. ｰ 040060 ｧ06 x, y =00304 縺06 閨0600070605 y 09 09 0306 ﾀ ｧ ﾀ0006000306 00030706 ﾀ050403 03 閨003 0 00308 070600909 ｧ08 x. 0ｷ3 0 ｧ08 閨0406: y = x ｧ y =,x 00604 縺06 縺030400306. 05 ﾀ05 000400306 ( 縺030400306) 0 03 閨0 ｧ08 06 ﾀ0509 08 03 ﾀ ｧ Ox 0304 縺06 閨0600070605 ﾀ07 06 04-0060 ｧ060 y = x, 0 陦縺0605 ｧ08 06 040060 ｧ060 y =,x. ｪ060500606 ﾀ 縺0304 0 ｧ 04 罍ｧ x = ; ; 3; :::00307003 0304 縺06 閨0600 ｧ0504 05 09: (,jxj 04 罍ｧ n, <x<n; y = +jxj 04 罍ｧ n < x < n +; 006 060500606 05 09: (,x 04 罍ｧ n, <x<n; y = +x 04 罍ｧ n < x < n +; 05 閨06 n =0; ; ; 3; ::: 83. 7 縺0304 II 縺03 04 罍ｧ x =,; lim y = +, lim y =,, x!,,0 x!,+0 lim y x! =. 84. 7 罍ｧ x = 0006 0403040300006003 050300040903 06050406 縺050306 06 ﾀ000304 ｧ06

3050504060503 75 00604 縺06 縺0304003 ﾀ05 ｧ; 04 罍ｧ x = 06006 ｧ 05 縺06050406. 85. ｯ03 09 ｧ 縺0304 : ) x = =0;)x =;3)x =0;4)x =0;5)x = ｧ x =0. 86. 706 ﾀ ｧ ﾀ0006000306 00030706 ﾀ050403. 0ｷ3 0 ｧ08: ) 00604 縺06 縺030400306 y = 4, x ｧ y =, 4, x ; ) ｧ ﾀ09030 07 縺03040 07: (, 4, x 04 罍ｧ jxj 6 ; y = + 4, x 04 罍ｧ < jxj 6 : 87. x =0 ｧ y =. 88. ) x =0 ｧ y = x; )x =, ｧ y = x,;3)y =. 89. ) x =0, y =,; )x =0 ｧ y = x, ; 3)x =,n=m ｧ y = a=m. 830. ) x =,= ｧ y =,; ) y = x; 3) y =,x. 83. ) y = x; ) x + y =,a; 3)y = x ; 4)y =,=4. 83. ) y =0;)y = x; 3) x =0 ｧ y = x. 833. 7 03030003: ) y = x 3 =3; )y = x. 834. ) x =0 ｧ y =;)x =0 ｧ y =,x. 835. ) x = =,, y ==;)x = ｧy =, x + ; 3) x =, x =,, y = ( 罍ｧ ﾀ. 40); 4) x =, x =, ｧ y =,x. 836. =e 5. 837. ) e,=3 ;)e 4. 838. ) e ;)e,4. 839. ) e, ;)e,. 840. ) 3; )e 3. 84. = e. 84. ) ; ),; 3)lna. 843. 3 ｧ 4. 844. ) e 6 ; ) e e. 845. ) =e ; ),3. 846. = e. 847. ) =x; ),. 848. ) 3x ;)4x 3 ; 3) x ;4)cos x; 5), x ;6), x x ; 7), x ;8) 3 cos x ;9), 3 x ; 0) ; 4 +x 3 ), (3x +) ; ) x ) x 3, x. 85. ) + ;), x, 7 ｧ ﾀ. 40. 849. ) (x, +x ) ;) b a. 850. ) (x, ) ; r a 30 +x +3. 85. ), x x 4 ;),x. x 4 853. ), ; ) x 3 3. 854. ) x 3, x 3. 855. ), x x x ;) 4 x ), tg x(sin x, x) x. 857. ) x( cos x,x sin x); ) sin x + cos x ; ) x 3 3, x 4 ; ) x 3 3x 3, x 4. 856. ) sin x x ;. 858. ), x sin x x 3 + x (x +). 859. ) (, 4x) ; ) 4x, sin x 4x x cos x.

376 050504060503 860. ), sin x ;) 866. ) x, x 6 ; ) x x( x +). 86. ) gt; ) a t sin. 86. ; 3 3 x 3 +. x 0; 4. 863. 8; 5. 864.,90. 865. ),6bx(a, bx ) ;) x, 3 x. 867. ) cos x ; ), ctg x. x(sin x + x) cos x, x sin x 868. ) x( sin x + x cos x); ). cos 869. ) x ; x ) ds dt = + t. 870. ) (x +) 4x ;). x 4 (x 87. ), +) x 3 + x 3 ; x ), + sin x ( + sin x). 87.,=3. 873.,,,=9,,=5. 874. ) 6 cos 6x; cos x, sin x ; ), sin x 3. ) b sin(a, bx). 875. ) 876. ),0(, 5x) 3 ; ) 3),tg4x cos 4x. 878. 3 4+3x. 877. ) 0x (, x ) ;), x ; 6, x sin x x, sin x. 879. 4 sin 3 x cos x. 880. ) sin x; ), sin x; 3) tgxsec 3 x. 88. sin x sin x,. 88. 3tg 4 x. 4, sin x 883. 4 cos x. 884. 4 ( + cos x) 3 x. 885. (, sin x + + sin x); 0 09 08+09 04 罍ｧ cos x >0, 0 09 08,09 04 罍ｧ cos x <0, 04 罍ｧ cos x =0 y 0 006 ﾀ06006 ﾀ0504060605 ( lim y 0 =, x!=4,0 lim y 0 =, ). 886. x!=4+0 0 sin 4x ( + cos 4x) 6. 887., ctg (x=3). sin (x=3) 888. sin x( + sec x). 889. x, x,. 890., x x x,. 89., sin t dr a sin '. 89. ) =, ;) dr a d' cos ' d' = sin ' ' + cos ('. + =4) 893. f 0 894. 898., = 3. 895. ab a + b, f0 () = 0, f 0 3 4 cos x 4x + sin 4x. 896. ab =, a + b. x(, 3x ). 897., sin 4x., x sin 6x 3( + cos 6x). 899. ) sec6 x;)3x sin x 3. 900. 4 cos x (, sin x) 3. 90. ds dt = sin (t=4) t=,. sin (t=) 90. dr d' = (3x +) cos '. 903., x 3 4x. + r 904.,. 905. k = tg = 4. 906. y = 8, 4x, x, 4y =. 6

3050504060503 77 907. y = x + 3. 908. y = 0, y = (3x, ). 909. y =,x +. 90. y =, x. 9. 45 ｧ 35. 9. arctg 4 3. 93. ) 5,,, 5; ) 3, 3 3 3, 3,. 95. y = x, 3x +4. 7 00605 b 0 0803 陦ｧ05 ﾀ07 ｧ 06 ﾀ000304 ｧ07 y 0 =x + b =4+b =, c ｧ 06 ﾀ000304 ｧ07, 0503 (; ) 0503 09 09 - ﾀ 0 ｧ07. 96. y =,4x +8, y =, 5 x, ; ' = arctg 4 8 6. 97. y =4x, y =,4x +6. 98. x 4y =8. 99. y = (3x +8) ｧ y =0. 90. 4= 7. 9. 40 54 0 ｧ00 ｧ 39 6 0. 9. (,;,4). 93. (=; 7=4). 94. ; ; ;. 95. 00 ｧ 7 7 0. 96. y, 0 =,, y0 + =. 97. y0, =, =; y 0, + ==. 98. y = x ｧ y =,x. 99. y = x ; 09 30 0. 930. x =0. 93. x =. 93. x =0. 933. x =. 934. y, = x,. 935. x =,. 4 936. y = 4x; 8. 937. ) ln x+;), ln x 0; 4343 (x +) ;3). 938. ) ; x x x 3 (x +) ) x(x +). 939. ), tg x ;)ctg x cos x. 940. x + x. 94. 4a x a 4, x. 4 94. x(, x ). 943. cos x. 944., 4x. 945. a + x. 946. + x. 947. ), ctg x sin x ; ). 948. y = x,. 949. 07 ﾀ 0605 ﾀ07 04 x, ax5 0503 0906 ( e;=). 950. ) x +3 x ln 3; )(x + x ln ) x ;3)x( + x)e x. 95. ) a sin x cos x ln a; ),xe,x ;3)x(, x)e,x. 95. e x= + e,x=. 953. +. 954.. x e x e x (, e x ). 955. a ex=a cos x a, sin x a 956. ),e,x sin x; ), x (x, ). 957. +x 959., ln a. 960. 6 35 0. 96. ) x x (ln x +);)x sin x x +. 958. a(eax, e,ax ).. cos x ln x + sin x x 963., tg x sin x. 964., x, x. 965., x +x. 966. cos x. + sin x 967. x(, x ). 968. ctg x. 969. ctg x, sin x. 970. tg x + cos x. x 97.,. 97.,x ax + x a e,x=a. 973. (ex=a,e,x=a 4 ). 974., (e x, e,x ). e x 975.. e 4x + 976. e 4x +., ln x 977. x=x. x 978. 6. 979. y = =, x. 980. r, x +x. 98. x +x. 98., x, 4x. 983. a jaj a, x.

378 050504060503 a 984. a + x. 985.. 986., x, x +x. 987. ), x ; 3e 3x )., e 6x 988., x. 4 989. x x,. 990. arctg x a. 99. x, x. 99. x 6x,. 993. ) x jxj, x ; ) x + x. 4 994. ex, e x. 995. arccos x. 996. 4e x, e 8x. 997. r 4 t,. 998. r x, 4. 999. (, 4)=4. 000. ) sh x; ) th x; 3) ch x +. 00. ; 5. 00. ) th x; ),4= sh x. 003. ) cth x;)= sh x. 004. ) = ch x; )4sh4x., x 005. x +; 75y =; 85a. 006. y =3; 76x +3; 89. 008. ) x x, ; 4x, ) tg 3 x. 009.. 00. = et (e t, ) x dt e t +. 0. x x, 4x. 0. ds dt =tg5 t. 03. a. 04. ) x + a x(x, a ;)cos (ln x). 05. ) 5. 07., 3a. 0. ) cos x; ) tgx sec x; 3) ( + x) 3=. 0. ),4 sin x; ), 4 x ; 3),(x 5 cos x + 3 sin x). 03. ), x ; ) e,t (3,t); 3) a(3x, a ) (x + a ). 3 04.,, (, t). 3= 05. ) n e,x=a ; a (,) n, (n, )! (,) n, 3 5 ::: (n, 3) ) x n ; 3) n. 06. ) n!; x n, ) sin x + n ; 3) n, cos x + n. 08. ),e x sin x; ) xa x (x ln a + 6x ln a + 6); 3) sin x + 4x cos x, x sin x. 09. ) e,x (sin x + cos x); )=x; 3)x sin x, 3 cos x. 030. f 000 (x) = = x +3a e x=a, f (n) a 3 (x) = x + na a n e x=a, f (n) (0) = n. 03., m, an, m(m, ), m(m, )(m, ), :::, m(m, ) ::: (m, n + ). 035. ) e,x (x, ); ) ctg x sin x ;3) x (4, x ). 3= 036. ) ax (ln a) n ; ) (,) n n n! ( + x) ; 3) n+,n, cos x + n. 037. 3 6,, 6, 7 3 36. 038. ) e x (x 3 +9x +8x +6);) 6a a 3 cos x a, 6ax sin x a, x cos x ; a 3),xf IV (a,x). 04. 703 0703 縺0060006 6ｧ80608030 ｧ09 f (n) (x) =x e,x=a, a n + + n xe,x=a, n, n(n, ) + e,x=a, n,. 0505 ﾀ06 f a a (n) (0) =

3050504060503 79 n(n, ) = a n, (,) n, n(n, ) = a n, (,) n. 04. f 0 (x)=,xe,x =,xf(x). 6 7 000606 0403 0703 縺0060006 6ｧ80608030 ｧ09 f (n) (x)=,xf(x) (n,) ｧ05.. 044. ), x y ; r ) y ;3) b x x + y x, y y. a 045. ), ;) y x +y x, y. 046. ), 3 x ;) e,x + y e y + x. 047., ex sin y + e,y sin x e x cos y + e,y cos x. 048. +. 049. =3. 050. ),a y y ; 3 (y, a) + n)y ) ;3)m(m. (x, b) n x 05., b. 05. y =3, x ｧ y = x,. a 053. (40=9; 40=9) ｧ (40; 40). 054. ) xx 0 a + yy 0 b = ; ) yy 0 = = (x + x 0 ). 055. x + y = a. 056. arctg 3. 057. ), b x a y ; ) x, ay. 058. ),a ax, y y ;), R 3 (y, ) ;3), ( + y ) ;4), 6a 3 y 5 (x +y). 3 059. y =, x, 3 ｧ y = x +. 060. x +y =4. 06., e. 06. e(e, ). 063.. 064. ) dy = nx n, ; )dy =3(x, ). 065. ) dy = x +x ;)ds = gt dt. 066. ) dr = 4 sin 'd';) = =, dt t 3. 067.) sin tdt;)sin udu. 068.), a3 x (a + x ) +)d ;)( ; 3), sin ' d'; 4), x. 070. ) 0; 04; )0; 05. 07. ) dv = x, =3x =0; 75; dv 3bds =0; x 3 006 ｧ00 ｧ 0; 6%;)d =. 07. ) 6 0; 8f 6 5x < 0; 005; ) 陦ｧ06 ﾀ 00607003 ｧ 006 罍ｧ0504 ﾀ 040305 縺0600003 ﾀ050406 006 0303000606 x 3 %. 073. ) S = R, S ds = R dr; )V = 4 3 R3, V dv =4R (, x) dr. 074. ) ;)b x 3 sin (a, b') d'; 3), tdt., t du 075. ), tg x;) u 4u, ;3),e,t dt. 076. ) x ;)tg d; 3) b( + e,bt ) dt. 077. ) y = 3x x +3xx +x 3 =,0; 376, dy =3x =,0; 4; )dl =, 4 4; 46 ﾀ0; 3) jj 6 x 0; 6 0; 006. 4 x. 078. ) 4y = x 3 ;)y = x 3, x 079. ) a + y b =;)x=3 + + y =3 = a =3. 080. ) x, y 3at =;)y =. 08. x = +x +t, 3

380 050504060503 y = 3at (4, )a. 083. y = x+. 084. x+y = a. 085. ), +t3 a sin 3 t ; ) t + 4t ;3), 3 4a. sin 4 (t=) 086. ) y =,x, x; )(y +) 3 = x 3. 087. x + y = a ) 3t, 4t 3 ;3) t = a g, x = a g 095. v = dt ; dv dt =av, 03050906 w = dv d x dt = 3 +. 088. y = x, a 3 gt. 090. x = at, 4et ;. 089. ), 4 sin 3 t ; dt = a, gt; d x =,g; 06 縺06 dt (0403 ﾀ00 07 0503 09 ). 09. dt = t, 4t +3; t =, t =3. = w; 0406 縺06000307 ｧ0 0403 00060003. dt = a. =,g. 6 5 0 ｧ0403 ﾀ000608 0503 0906 dt a h(r, h) = a r. 099. dt dv 096. v dt =a dt = gt 097. x =0+0t, ; dt =0; t = 0 g = k(a, x). 00. =0, gt; ; 04 ﾀ. 098. dh dt = d(! ) =!d!, d(! ) d! d! dt =! =! d' d' dt d' =!" =". 0. 0703 縺0 ｧ 070600909 ｧ ｧ ; 3.! 0703 縺06004 04 縺03 ｧ 0403 閨00308 f 0 (x) =x, 4 04060 ; < < 3. 0. 706 04 罍ｧ- 0060 ｧ0, ｧ0303 04 罍ｧ x =000605 04 縺03 ｧ 0403 閨00308. 03. 7030503006, 0503 0503 09 x =0 0605000304 07 ( 閨0406 09 ﾀ 0506000400306). 04. 7 0900030 0803 罍閨03 (AB): k = 9, 3+ =; f 0 (x) =x =, x =;040503 0906 x =09 ﾀ 050600040 07 04 00000600040 0803 罍閨06. 05. f(b) = b, f(a) = a, f 0 (c) = c; 0403 陦ﾀ05 04 ｧ0 050503 04 0703 縺0060006 6ｧ8-05 007 b, a =(b, a) c; 03050906 c = b + a. 06. c = 9 4. 08. 7 閨060506 06 ﾀ0504 0605000304 07 0503 09 04 罍ｧ x =,0409030503 縺0308 070600909 ｧ07 006 ｧ0060605 04 縺03- ｧ 0403 閨00308. 09. ｱ0600909 ｧ07 00604 縺06 縺03040 ｧ ｧ0060605 04 縺03 ｧ 0403 閨00606 0400605 罍ｧ 0305- 縺06 09 [0; ], 003 縺03040 0 060503 04 04030 090300906. 0. 706 ﾀ0504 s = f(t) 06 040060 ｧ06 閨04 ｧ07060 ｧ07, t ｧ t 0 000400308 ｧ 0903006 00308 00300600503 閨04 ｧ- 07060 ｧ07. 703 050603 縺06006 6ｧ8 05 007 00607 閨06 t ｧ t 0 08 閨0605 ﾀ07 t 3, 04 罍ｧ 09030503 縺030 f(t ), f(t ) = f 0 (t 3 ),05.06.40 = f 0 (t 3 )= ds t, t dt 04 003006005 t 3.. 0 (x) = f 0 (x) 0 b f(b). ｯ 09 09 09 (b) =(a) =0 ｧ 04 ｧ00506 縺04 0006 a f(a) (a; b) ｧ0060605 ﾀ07 04 縺03 ｧ 0403 閨0 07 0 (x), 0403 050603 縺06006 703000007 00607 閨06 a ｧ b 0 08 閨0605 ﾀ07

3x = c, 04 罍ｧ 09030503 縺030 0 (c) = 0,05.06. 050504060503 8 f 0 (c) 0 b f(b) = 0,03050906 f(b), a f(a), f(a) =(b, a)f 0 (c). ｱ0600909 ｧ07 (x) 06 ﾀ0504 06 閨04030600 07 0400030 閨04 4AMB, 05 閨06 M 000603 07 0503 09 0 閨060506 AB.. b3, a 3 b, a = 3c c,03050906 c = = (a + ab + b ). 3. ｰ050003040308 0903050707 ｧ09 ｧ06005 09 ﾀ 0506000400308 dy 3(a + b) = f0 (t) ' 0 (t), 040503 0906 t = c k = f0 (c) ' 0 (c). ｰ050003040308 0903050707 ｧ09 ｧ06005 ﾀ06090600608 k = y, y = x, x f(b), f(a) = ; 0403 050603 縺06006 070300 ｧ 00607 閨06 a ｧ b 0 08 閨0605 ﾀ07 t = c, 04 罍ｧ 09030503 縺030 '(b), '(a) k = k, 05.06.09 ﾀ 050600040 07 04 00000600040 0803 罍閨06. 7 罍ｧ 0505030 05 09 09 09 ' 0 (t) 6= 0, 0503 '(a) <'(c) <'(b) ( ｧ00 ｧ 0 030303 縺0305), ｧ 0503 09 09 ﾀ 0 ｧ07 0 0803 陦ｧ05 ﾀ07 0400605 罍ｧ r r r a + ab + b 4 閨0605 ｧ. 7. c =. 8. ) 3, ; ), 4 ;3) s ln. 9. ) 5 4 ;) 3 ; 4. 0. ｱ0600909 ｧ07 y = jx, j 006 ｧ0060605 4 04 縺03 ｧ 0403 閨00308 04 罍ｧ x =.. 6 50503 0906 x =,=.. 3. 3. =. 4. na. n, 5.. 6. a =b. 7. =. 8. =6. 9. 3. 30. ) ; ) 0. 3. 0. 3. 0. 33. 3. 34.. 35. 0. 36. 0. 37.. 38.. 39. e 3. 40. -0503 0403 縺07 閨09. 44. a, b. 45. =3. 46. =8. 47. ln a b. 48. = 3. 49.. 50.. 5.,=3. 5.,. 53. =e. 54. =6. 55. e 3. 60. 7 罍ｧ x =, y min =. 6. 7 罍ｧ x =, y min =,6=3; 04 罍ｧ x = y max = +6=3; 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ Ox: x =0, x ; 3 = 3 3; 4. 6. 7 罍ｧ x =, y max = 5=3, 04 罍ｧ x =3 y min =,9; 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ Ox: x =0, x ; 3 ; 5 3; 3. 63. 7 罍ｧ x = y max =5, 04 罍ｧ x =0 y min =, 04 罍ｧ y =0 x ; 9. 64. 7 罍ｧ x =0 y =0 0406 縺0605 ｧ03; 04 罍ｧ x =3 y min = =,7=4. 65. 7 罍ｧ x =, y max =,, 04 罍ｧ x = y min =; ﾀ ｧ00405030503 x =0 ｧ y = x=. 66. 7 罍ｧ x =0 y min =, (0503 09 0403 04 05 ) 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ0406 Ox: x =. 67. 7 罍ｧ x =0 y max =, 04 罍ｧ x! y! 0, 05. 06. y =0 ﾀ ｧ004050305. 07 罍ｧ04 07 ﾀ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 03 ﾀ ｧ Oy (0403 06006?). 68. 7 罍ｧ x = y max =,4, 04 罍ｧ x =5 y min =4; ﾀ ｧ00405030503 x =3 ｧ y = x, 3. 69. 7 罍ｧ x =0 y min =0, 04 罍ｧ x ==3 y max =4=7. 70. 7 罍ｧ x =4 y max =, 04 罍ｧ y =0 x =3 ｧ00 ｧ x =5, 04 罍ｧ

38 050504060503 y =,3 x =,4 ｧ00 ｧ x =. 7. 7 罍ｧ x =0 y max =; ﾀ ｧ004050305 y =0. ｮ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 Oy. 7. 7 罍ｧ x = y max = + 3 ;, 04 罍ｧ x = 5 y min 0; 4. 73. 7 罍ｧ x = 3 y max = 4 3, 3 ; 45, 04 罍ｧ x =, 3 y min = 3, 4 3,; 45. 罍ﾀ ｧ00405030503 x =. 74. 7 罍ｧ x = y max =, 04 罍ｧ x! 0 y!,; 04 罍ｧ x! y! 0. 罍ﾀ ｧ00405030503 x =0 ｧ y =0. ｯ03 09 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ0406 Ox: +lnx =0, ln x =,, x = e, 0; 4. 75. 7 罍ｧ x = y min =, 4,0; 8, 04 罍ｧ x =, y max 0; 8. 罍ﾀ ｧ00405030503 y = x. 76. ) 7 罍ｧ x = y max ==e. 罍ﾀ ｧ004050305 y =0. ) 7 罍ｧ x ==e y min =,=e; lim y = x!+0 =0 09030090604 07 0503 09 ; 04 罍ｧ x = y =0. 77. ) 7 罍ｧ x =0 y min =0 r 4n + (0605000304 07 0503 09 ), 04 罍ｧ x = y max =;)04 罍ｧx =0 y min =0 (0605000304 07 0503 09 ). 78. y min == 04 罍ｧ x = =3; 3=4; 5=4; :::; y max = 04 罍ｧ x =0;=; ; 3=; ::: 79. 050300 ﾀ0504 ﾀ0403000307060 ｧ07 09 罍ｧ040308 x 6 ; y max = 04 罍ｧ x = ; y =004 罍ｧ x =0 ｧ x =. 80. 7 罍ｧ x = y max = ; 030300 ﾀ0504 ﾀ0403000307060 ｧ07 09 罍ｧ040308 x>0. 8. 罍ﾀ ｧ00405030503 x = ｧ x = 4( 縺030403) y min =,=9 04 罍ｧ x =,, y max =, 04 罍ｧ x =. 8. 7 罍ｧ x = y min =; 5. 07 罍ｧ04 07 ﾀ ｧ004050305 ｧ 06 ﾀ09 ｧ 04 罍ｧ0300 ｧ07 0605 ﾀ07 09 04 03030006 y = x = ｧ0903 ﾀ ｧOy. 83. 7 罍ｧ x =0 ｧ x = y min = 3 4 ; 6, 04 罍ｧ x = y max =(04 0503 09 08 0 ｧ0 ｧ0060 0503 09 ｧ 0403 04 05 ). 84. 7 罍ｧ x =0 y0406 縺0605 =0, 04 罍ｧ x = y max =0;, 04 罍ｧ x =3 y min =,5; 4. 85. 7 罍ｧ x =, y max =0, 04 罍ｧ x =,; y min,;, 04 罍ｧ x =0 y0406 縺0605 =0. 86. 7 罍ｧ x = y max =, 04 罍ｧ y =0 x =; ﾀ ｧ00405030503 03 ﾀ ｧ 090303 罍陦ｧ0 05. 87. 7 罍ｧ x =,3 y max =,4; 5, 04 罍ｧ x =0 y0406 縺0605 =0, 04 罍ｧ x = 3 y min = +4; 5; ﾀ ｧ00405030503 y = x, x = 3. 88. 7 罍ｧ x = 4 +k y max =, 04 罍ｧ x = +k 縺030403. 89. 7 罍ｧ x = 4 +k y max = 4 +k, ln. 90. ) 7 罍ｧ x = y min = ln, 4 ;)04 罍ｧx = =, y max =, 04 罍ｧ x =0 y min =0(0605000304 07 0503 09 ﾀ 0 0900030 0 ｧ k = ). 9. 7 罍ｧ x =0 y min =0; 04 罍ｧ x = y max =4=e =; ﾀ ｧ004050305 y =0. 9. 7 罍ｧ x =, 0503 09 0403 04 05 y min =, 04 罍ｧ x =0 y max =3, 04 罍ｧ y =0 x 4. 93. 7 罍ｧ x = y max =4; 04 罍ｧ y =0 x =0, x =4.

3050504060503 83 94. 7 罍ｧ x =, y min =,4; 04 罍ｧ y =0 x =, x =,3. 95. 7 罍ｧ x =0 y min =0, 04 罍ｧ x =, y max =4=3; 04 罍ｧ y =0 x =0, x =,3. 96. 7 罍ｧ x =, y min =,4, 04 罍ｧ x =,3 y max =0. 97. 7 罍ｧ x =0 y max = 0, 04 罍ｧ x = y =, 04 罍ｧ x =4 y min =8; ﾀ ｧ00405030503 x = ｧ y = x + ( 罍ｧ ﾀ. 4). 98. 7 罍ｧ x =,3 7 ｧ ﾀ. 4 7 ｧ ﾀ. 4 y min =,6; 75, 04 罍ｧ x =0 y0406 縺0605 =0; 04 罍ｧ y =0 x =0, x =,4 ( 罍ｧ ﾀ. 4). 99. 7 罍ｧ x = y min =,4, 04 罍ｧ x =0 y max =0; 04 罍ｧ y =0 x =0, x ; 3 = 8 ; 8. 00. 7 罍ｧ x =00503 09 0403 04 05 y max =0, 04 罍ｧ 0. 7 罍ｧ x = y min = =,; 04 罍ｧ y =0 x =0, x =7=8 ( 罍ｧ ﾀ. 43). x =, y max =, 04 罍ｧ x = y min =0, 04 罍ｧ x =0 y =. 罍ﾀ ｧ004050305 y =. 0. 7 罍ｧ x =, y min =,= e,0; 6, 04 罍ｧ x = y max 0; 6; 03 ﾀ04Ox ﾀ ｧ004050305. 03. 7 罍ｧ x = y min = (, ln ) 0; 6; 03 ﾀ04Oy ﾀ ｧ004050305 04 罍ｧ x = y =; 04 罍ｧ x = e 7; 4 y 3; 4. 04. 7 罍ｧ x = 0 0503 09 0403 04 05 7 ｧ ﾀ. 43 y max =0, 04 罍ｧ x = y min =,3 3 4,4; 8, 04 罍ｧ x =5 y =0. 06 07 ｧ09 0403 閨0303060 05 07 ｧ0906 0 罍ｧ ﾀ. 43. 05. 7 罍ｧ x =+ 6 y max 0; 34, 04 罍ｧ x =, 6 y min,0; 34, 04 罍ｧ x = y = = ; 57.

384 050504060503 06. 7 罍ｧ x = 4 y min = + ; 57, 04 罍ｧ x = 3 4 y max = +3; 7; ﾀ ｧ00405030503 x = 0 ｧ x =. 07. 7 罍ｧ x =, y max =, + 3 4 ; 85, 04 罍ｧ x = y min ; 8, 04 罍ｧ x = 0 y = =. 罍ﾀ ｧ004050305 y = x. 08. 7 罍ｧ x = 0503 09 0403 04 05 y min =, 04 罍ｧ x = 0y =, 04 罍ｧ x = y =. 09. 7 罍ｧ x = =6 ｧ x = 5=6 y max = ; 5, 04 罍ｧ x = = y min =. 0. 7 罍ｧ x =0 y min =0, 04 罍ｧ x = y0406 縺0605 =.. 7 罍ｧ x = e y max ==e 0; 4, 04 罍ｧ y =0 x =. 罍ﾀ ｧ00405030503 x =0 ｧ y =0.. 7 罍ｧ x =,3 y min =6, 04 罍ｧ x =, y = ( 縺0304), 04 罍ｧ x =, y max =. ｯ03 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ: x =0, y =; 5; y =0, x = 3 ; 7. 罍ﾀ ｧ00405030503 x =, ｧ y =, x. 3. 7 罍ｧ x = y min =, 04 罍ｧ x =, y max =,, 04 罍ｧ x =0 縺0304. 罍ﾀ ｧ00405030503 y = x ｧ x =0. 4. ) 7 罍ｧ x =0 y = a. ｯ03 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ0406 Ox: x = + k. 709 ﾀ05 縺060060: 04 罍ｧ x = 3 +k 0 ｧ0 ｧ0060, 04 罍ｧ 4 x = 7 +k 0 09 ﾀ ｧ0060. 07 罍ｧ04 07 05 07 ｧ09 050608 060 ｧ08 0903000603-4 0 ｧ08; 030 0404 ｧ ﾀ 0 04 09 罍ｧ040306 y = ae,x,0 09030503 縺0308 ｧ 0 0803 閨0705 ﾀ07 0503 09 ｧ 0509 ﾀ05 縺060060. 703 ﾀ05 縺03060 ｧ06 00607003 0 0504 ﾀ09 罍ｧ040308 y = ae,x. 05 ﾀ04 Ox ﾀ ｧ004050305. ) 7 罍ｧ x =, y max =, 04 罍ｧ x =0 0503 09 0406 縺0605 ｧ03, 04 罍ｧ x = y min =,; 04 罍ｧ y =0x =0, x ; 3 ; 3. 5. 7 罍ｧ x = y min =3, 04 罍ｧ x = y = ( 縺0304), 04 罍ｧ x =4 y0406 縺0605 =0, 04 罍ｧ x =0 y 3; 6. 6. 7 罍ｧ x =, y min =0, 04 罍ｧ x =,4 y max =0; 8, 04 罍ｧ x = y max ; 8; 03 ﾀ04Ox ﾀ ｧ004050305. 7. 7 罍ｧ x = y max =; 04 罍ｧ y =0 x = = 0; 7. 罍ﾀ ｧ00405030503 03 ﾀ ｧOx ｧ Oy. 8. 7 罍ｧ x =0 y max =, 04 罍ｧ x = y min =0; 04 罍ｧ y =0 x =. 9. 7 罍ｧ x =, y min ==3, 04 罍ｧ x = y max =3, 04 罍ｧ x =0 y =; ﾀ ｧ004050305 y =. 0. 7 罍ｧ x =, y max =; 04 罍ｧ y =0 x =0, x =,4; 030300 ﾀ0504 ﾀ0403000307060 ｧ07 09 罍ｧ040308 x 6 0.. ) 7 罍ｧ x =, y = ( 縺0304), 04 罍ｧ x =,3 y0406 縺0605 =0, 04 罍ｧ x =0 y min 7=4; ﾀ ｧ00405030503 x =, ｧ y = x +5;)y min =004 罍ｧ x =n, y max = 04 罍ｧ x =(n +). 6 50503 09 08 0 ｧ0 ｧ0060 y 0 006 ﾀ06006 ﾀ0504060605 (06050003040306 0503 09 ｧ).. 30 060 0. 3. 5 ｧ 5. 4. ah=4. 5. a=6. 6. 4 04 0 0. 7. 0 ﾀ0. 8. 60 8. 9. +4 ; 5. 30. cos = (03 閨0 0903 04 罍ｧ 06 ﾀ0003m 04 ｧ ｧ, 0503 a m 6,05 閨06 a 04 縺03060909 ｧ07 AB 0 0 04 0400060 ｧ06 07060006 00308 AB

3050504060503 85 閨03 縺0305 ｧ). 3. 6 580 0305 0303000606 ﾀ ｧ00040030503 ｧ ﾀ0503 0 ｧ09 ﾀ040605. 3. ｪ06 縺06 a ﾀ0304 0 ｧ006004000606 ﾀ ﾀ0503070 ｧ06 0306 閨0605 04003 a= 090. 33. x = D=, v y = D 3=. 34. 6 5 3 ; 7. 35. l 5; 6 0; 0304 縺06 閨0600070605 ﾀ07 09 09 0 09 ﾀ ｧ0060 070600909 ｧ ｧ l = ; 4 sin + ; 6 cos. 36. v max = 8 閨0 3 04 罍ｧ 9 0403 ﾀ030506 x = 閨0. 37. S max = R 04 罍ｧ 0403 ﾀ030506 x = R. 38. (; ). 39. ab. 40. 7 罍ｧ x =0. 4. 4 ﾀ0 ｧ 3 ; 7 ﾀ0. 4. x =; 5. 43. ｮ06 060 ｧ06 0904 陦 05 ﾀ03 ﾀ0503 縺0300308 D=. 44. 7 罍ｧ = =3 陦ｧ 00304 94 P. 45. F = cos + sin ; tg = =0; 5, 4. 46. ) y = x, y 00 => 0, 09 罍ｧ04 07 04 ﾀ06 閨06 040304060900 08040 ｧ 09; ) y = x 3, y 00 = 6x, 09 罍ｧ04 07 040304060900 08040 ｧ 09 04 罍ｧ x > 0 ｧ 08040406 縺0809 04 罍ｧ x < 0, x = 0 0503 09 0406 縺0605 ｧ03 ; 3) y = e x, y 00 = e x > 0, 09 罍ｧ04 07 04 ﾀ06 閨06 0403-04060900 08040 ｧ 09, (0; ) 0503 09 0406 縺06 ﾀ06 060 ｧ07 ﾀ Oy; 4) y = lnx (x > 0), y 00 =, < 0, 09 罍ｧ04 07 04 ﾀ06 閨06 040304060900 08040406 縺0809, x (; 0) 0503 09 0406 縺06- ﾀ06 060 ｧ07 ﾀ Ox; 5) (0; 0) 0503 09 0406 縺0605 ｧ03. 47. ｯ03 09 ｧ 0406 縺0605 ｧ03 09 罍ｧ- 040308: ) (;,8=3); )(= ; e,= );3)( 3; 3=) ｧ (0; 0); 4)04 罍ｧ x =, ln,0; 35. 5. 050300 ﾀ0504 ﾀ0403000307060 ｧ07 x>,. ｯ03 09 ｧ 0406- 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ (,; 0) ｧ (0; ln ). y 04 ﾀ06 閨06 04 ﾀ05 0605, 09 罍ｧ04 07 0403-04060900 08040406 縺0809. 罍ﾀ ｧ004050305 x =,. 53. y > 0, y = 0 ﾀ ｧ004050305. 54. ) ｮ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 Ox. 050300 ﾀ0504 ﾀ0403000307060 ｧ07 x > 0. 6 506 縺0800707 0406050404 040304060900 08040 ｧ 09, 0 ｧ0700707 08040406 縺0809. 050306 04060504 ｧ 09 ﾀ 0605 ﾀ07 Ox 04 0503 0906 (0; 0). 07 罍ｧ04 07 0 0304 0605 ﾀ07 0804030006090603 ｧ 06 ﾀ090308 04 030300030809 (04006- ﾀ0506 ﾀ 03 ﾀ0406 Oy 0303 060605 0306090406 07); ) 05 09 07 0706, 09 09 04 縺06 閨03 閨060 07 09 罍ｧ04 07, 003 ﾀ 閨04 ｧ00605 0400060403 0 3 06 陦ｧ0 ｧ0903. 55. ) 7 罍ｧ x =0 y max =,, ﾀ ｧ004-05030503 x =,, x = ｧ y = 0(05 罍ｧ 04060504 ｧ); ) 04 罍ｧ x = y max =, 04 罍ｧ x =, y min =,, 0406 縺06 ﾀ0609 0605 ﾀ07 ﾀ Ox 04 罍ｧ x = 3, 0406 縺0605 ｧ03 04 罍ｧ x =, ﾀ ｧ00405030503 03 ﾀ ｧOx ｧ Oy. 56. ) 050300 ﾀ0504 ﾀ040300030706-0 ｧ07 x>0, 04 罍ｧ y =0 x =, ﾀ ｧ00405030503 03 ﾀ ｧOx ｧ Oy, 04 罍ｧ x = e y max =;)04 罍ｧx = y max =, 04 罍ｧ x = y0406 縺0605 ==e =3, 03 ﾀ04 Ox ﾀ ｧ004050305, 04 罍ｧ x = 0y = 0. 57. ) 7 罍ｧ x = 0 y min =, ﾀ ｧ00405030503 x =, ｧ x, y =0; ) ﾀ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 Oy, 04 罍ｧ y =0 x = = 0; 7, 04 罍ｧ x = y min =,, ﾀ ｧ004050305 03 ﾀ04 Oy. 58. ) 050300 ﾀ0504 ﾀ0403000307060 ｧ07 x> 0, 04 罍ｧ x = y min =, 0403-04060900 08040 ｧ 09; ﾀ ｧ004050305 03 ﾀ04Oy; ) Oy 03 ﾀ04 ﾀ ｧ000605 罍ｧ ｧ, 04 罍ｧ x =0

386 050504060503 y min = a, 04 ﾀ06 閨06 040304060900 08040 ｧ 09; 09 罍ｧ04 07 0 0304 0605 ﾀ07 09060400308 00 ｧ0 ｧ0608. 59. ) 7 罍ｧ x = 0 y max = 0, 04 罍ｧ x = 3 4 ; 6 y min ;, 04 罍ｧ x =, 3,; 3 y0406 縺0605,0; 8, ﾀ ｧ00405030503 x = ｧ y = x; ) 04 罍ｧ x =, y min =,3, 04 罍ｧ y = 0x =, 3 0; 5,0; 6, ﾀ ｧ00405030503 03 ﾀ ｧ Ox ｧ Oy. 60. ) ｮ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 Ox ｧ Oy, 030300 ﾀ0504 ﾀ0403000307060 ｧ07 jxj <, 04 罍ｧ x = y 05 =, 04 罍ｧ y = 0 x = 0 ｧ00 ｧ x = ;)0 04060504 ｧ y = x + y min = 304 罍ｧ x =,0406050404 x y = x, x 0406 縺06 ﾀ0609 0605 Ox 04 罍ｧ x = 3 4 ; 6, 03030604060504 ｧ ｧ0060605 ﾀ ｧ004-05030503 y = x ｧ x = 0. 6. 7 罍ｧ x =, y min =, 3 6,; 5, 04 罍ｧ x = y max ; 5 (030306 0503 09 ｧ 0403 04 05 ), 03 ﾀ04 Ox ﾀ ｧ004050305, 8x ｧ0303 y =! 0, 090305 x!. (x +) 4=3 +(x, 4) =3 +(x, ) 4=3 6. ｮ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 Ox, 030300 ﾀ0504 ﾀ0403000307060 ｧ07 x > 0, ﾀ ｧ004-050305 03 ﾀ04Ox ( lim x! y =0), 04 罍ｧ x =0509 ﾀ05 縺060060 y 05 = =e 0; 3. 64. ) x3 3 + x +lnjxj + C; ) x 5,, x + C. 65. ) x3 x + C; ) x +lnjxj, x + C. 66. ) x 3 x + 3 x + C; ) x, 3 4, 4 4 x + C. 67. ) x x, 3x +6 x, ln jxj + C; ) 3 3 4 (x, 4) 3 x + C. 68. ) e x + ax +C;) x ln a, +C. 69. ),ctg x,tg x+c;), ctg x, x sin,x + C. 70. ) sin x cos x = x + cos x =tgx, ctg x+ sin x cos x +C; )3tgx + ctg x + C. 7. ) x, sin x + C; ) x + sin x + C. 7. ) arctg x,3 arcsin x+c;) x3 3,x+arctg x+c. 73. ) x4,, x, lnjxj+c; )3 3 x + x +C. 74. ) (x +) x +C; )4lnjxj, 8 x,, x + C. 75. ) ln jxj, x, x + C; )x + cos x + C. 76. ) ex + + tgx + C; ) ax ln a, + C. 77. cos x, ctg x + C. 78. tg x, 4x4, x + C. 79. 3 sin 3x + C. 80., cos x + C. 8., 3 e,3x + C. 8. 5 tg 5x + C. 83. (ex=, e,x= )+C. 84. 6 (4x, )3= + C.

3050504060503 87 (3, x)5 85., + C. 86., 0 8 (5, 6x)4=3 + C. 87., 3, x + C. 88. b cos (a, bx)+c. 89. ln (x, 5x +7)+C. 90. ln (x +)+C. 9.,0; lnj, 0xj+C. 9., 6 ln j, 3ex j+c. 93. ln j sin xj + C. 94., ln j cos xj+c. 95. ln j sin xj+c. 96., ln j + 3 cos xj + C. 3 97. ln j+sinxj + C. 98. ln j +lnxj + C. 99. sin3 x + C. 3 300., cos4 x + C. 30., 4 3 sin 3 x + C. 30. cos x + C. 303., cos x sin x +C. 304. +C. 305.,e cos x +C. 306. sin x 3 ex3 +C. 307., e,x + C. 308. e x + C. 309. (x +) 3 + C. 3 30. 4 3 (x3, 8) 4 + C. 3. 0; 5 3 ( + x 3 ) + C. 3.,, x + + C. 33., + cos x + C. 34. ( + ln x) 3 + C. 35. 3 6 ( + + 4 sin x) 3= + C. 36., 40 (,6x5 ) 4=3 +C. 37. x+ (ex,e,x )+C. 38. sin4 x 4 + C. 39.,, 4x + C. 30., sin (a, bx) +C. b 3. 4 ( + 3x)4=3 + C. 3., 7 (, x3 ) 7=6 + C. 33. +x + C. sin x, 34. + C. 35. lnjsin xj,ctg x + C. 36. e sin x + C. cos x 37., 3 ln j, x3 j+c. 38. b(a, bx) +C. 330. ) 0; ln x, 5 x +5 + + C; ) 3 arctg x 3 + C. 33. ) arcsin x + C; )ln (x + x +5)+C. 33. ) ln jx + x, 4j+C; ) arctg x +C. 333. ) arcsin x +C; 3 3 5 ab ln bx, a ) x3 arctg 6 + C. 334. ) x arcsin + C; ) 3 bx + a + C. 335. ) x arcsin + C; ) 3 4 ln (x4 + x 8, ) + C. 336. ) ; 5 ln (x +4),arctg x +C;) 3 ln jx,4j,ln x, x + +C. 337.) x + + + ln(x + x +)+C; ),, x + arcsin x + C. 338. x, arctg x + + C. 339. x3 3 +3x + 3 3 ln x, 3 x + 3 + C. 340. arctg (x +)+C.

388 050504060503 34. arctg x, 3 + C. 34. ln (x ++ x +x +3) + C. 343. arcsin x + + C. 344. arcsin x, x +3 + C. 345. arctg + 3 3 + C. 346. arcsin 4x, 3 +C. 347. ln j3x, + 9x, 6x +3j + 5 3 + C. 348. 3 arctg +ln 3 x, 3! x + C. 349. arcsin + 3 x + +ln(x + +x ) + C. 350. ln(x +5), 5 arctg x + C. 5 35. x + ln x, x + + C. 35. x 3 3, x + arctg x + C. 353. arcsin (e x )+C. 354. arctg (x )+C. 355. 0; arctg x + + C. 5 356. arctg x, +C. 357. arcsin x + +C. 358. 3 ln (x + x +),, x + arctg + C. 359. 3 3 ln (x ++ 4x +4x, 3) + C. 360. x ln jxj,x + C. 36. x ln jx, j, x + x +lnjx, j + C. 36. ex x, + C. 363. x + arctg x, x + C. 364. x sin x + +x cos x, sin x + C. 365. ex (sin x, cos x) +C. 367. x (ln jxj,, ) ln jxj + + + C. 368.,x ctg x +lnjsin xj + C. 369., + C. x 370. +xarcsin x +4, x + C. 37. x arcsin x +, x + C. 37.,e,x (x 3 +3x +6x +6)+C. 373. x ln (x +), x + arctg x + + C. 374. x (cos ln x + sin ln x) +C. 375. 5 x 3 ln jxj, 3 + C. 376.,e,x= (x +4x +8)+C. 377. x arctg x, ln ( + x )+C. x sin x + ctg x + C. 378. x tg x+lnj cos xj+c. 379. 0; 5e x (sin x+cos x)+c. 380. 4 +x,,, x arcsin x + C. 38., 38. x x, arctg x,, + C. 384. 3x + 4 sin x + sin x + C. sin 4x + cos x, +C. 386. 3x sin x sin 4x + + 8 8 4 3 +C. 387. x 8, 385. 3x sin 4x, 3 +C. 388. 3x 8 sin 4x sin 8x, + 8 04 +C. 389. x sin 4x, 6 64 + sin 3 x + 48

3+ C. 390., cos x + 3 cos3 x, cos5 x 5 050504060503 89 + C. 39. sin3 x 3, sin5 x 5 + C. 39. 4 sin4 x, 6 sin6 x + C. 393. sin x, sin 3 x + 3 5 sin5 x, 7 sin7 x + C. 394. 7x + 4 sin x + 3 sin x, 8 sin3 x + C. 395.,, sin x + C. 3 sin x 396. cos x + cos x + C. 397. ln j tg xj + C. 398. ) tg ln x + x + C; )lntg + h 4 + C. 399. lntg x x +lntg + i + C. 4 400. sin x, cos x = sin x, sin (=, x) = sin (x, =4) = x, 8 + C. 40. tg x = ctg lntg +lnj cos xj+c. 40.,,, ln j sin xj+c. 403., 8 (cos 4x+ cos x)+c. 404. sin (m + n)x + m + n sin (m, n)x + + C 04 罍ｧ m 6= n ｧ x m, n + sin mx + C 04 罍ｧ m = n. 4m 405. ) 4 sin x, sin (m, n)x 6 sin 8x + C; ) sin (m + n)x, + C m, n m + n 04 罍ｧ m 6= n ｧ x, sin mx + C 04 罍ｧ m = n. 406., cos 6x, 4m, 8 sin 4x + C. 407. ) 5 sin 5 6 x, cos x x + 5 sin3 x + 5 sin x + C. 4 6 6 cos x 408. ) sin x + tg ln x sin x + C; ) cos x + tg x ln + 4 409. x + 3 sin x + 9 8 sin 4x + C. 40. 3 8 x, 4 sin x + sin 4x + C. 3 x sin 4x 4.,, sin3 x + C. 4. sin x, sin3 x + sin5 x + C. 6 64 48 3 5 43. cos5 x, cos3 x + C. 44. 7x, 4 cos x, 3 sin x + 8 cos3 x + C. 5 3 3 45. ln j tg xj,x + C. 46. 8 ( sin x, sin 4x) +C. 47. cos x + + cos x +tgx + C. 48., 4 cos x + 4 x3 + x + C. 49. ) 6 3 + x + +4x+8 lnjx, j+c;) x3 3,a x+a 3 arctg x +C; 3)x3 a 3 + a3 3 ln jx3, a 3 j + C(x, ) + C. 40. ln. 4. ln (x, )3 x, 3 x + + C. 4. ln Cx3 (x, ). x + 43. x + 4x +ln(x, )8 jxj + C. 44. x +ln x, x x + C. + C.

390 050504060503 45. a ln x, a x + x, a ax + C. 46. ln Cx(x, ) + x,. 47. ln x, x +, x + + C. 48. 5 ln (x +x, 0), arctg x + + C. 3 49. ln(x, 0; x +0; 7), 5 arctg 0x, + C. 4 430. ln jx +j x, x x +5 +4+C. 43. 3ln + arctg x, + jxj (x +) + C. 43. ln 4 x, x +4 + 4 3 arctg x, x +C. + 433. ln, 3 jx +j, x + + arctg x + C. 434. ) b 3 arctg x b + bx x + b + C; x(5b +3x ) ) 8b 4 (x + b ) + 3 b arctg x x +9 + C. 435. ), b 8(x +x +5), 6 arctg x + + C; ) (x, 3)(3x, 8x + 3) 8 (x, 6x + 0) + 3 arctg (x, 3) + C. x + 436. ln + x, jx +j x + + C. 437. x, 4(x +) + 8 arctg x + C. 438. a ln x 44. x + a +C. 439. 0 3 ln x, 3 x + 3, + C. 443. 4 a, b ln x + b x + a +C. 440. ln, x 5 arctg x + C. 44. x + ln x, x + 4+x, x x(4 + x ) = 4 ln jxj 4+x 445. ln C(x, ) x +3. 446. ln,, ). 448. lnc(x, x + x + arctg (x, ) + C. 450. 45. 3 jx +j ln x + + 3 arctg x + C. +C. + C(x, )3 + C. 444. ln. x, C(x, ) 3, ). (x +) 447. 3lnC(x, (x, ) x + x,. 449. ln jxj x, x + + a ln x + a (x, ) 45. ln 4 x +x +4, x + arctg + C. 4 3 3 453., x + x +x + + arctg (x +) + C. 454. 455. x +3, x 3 x (x +3) =, 3x, 3 3 arctg x + C. 3 jxj + a arctg x a + C. 5 ln x x +5 + C.

3456. 457. 3 (x + )(x, ) = 4 ln x, x + x +, (x, ) x +, (x, ) (x + )(x, ) = 050504060503 9 6 ln x,, arctg x + C. x +, arctg x + C. 3 x + 458. 3 x (3x +) + + C. 459. ( x +, 3) + C. 5 x 3 x 460. 6 3, + 6 x, ln ( + 6 x) + C. 46. 5 (3x, ax, a ) a, x + C. " 3 46. 3 (x 4 +), 3 x 4 ++ln( 3 x 4 ++) 4 # + C. (x, 4) x + 463. + C. 464. arcsin + C (, 04 罍ｧ x > 0 ｧ 3 x r Cx + 04 罍ｧ x< 0). 465. ln x ++ x +x. + 466., a, x + a x C(x h +) + C. 467. ln + x +x. + 468. x a, x + a arcsin x i +C. a x 469. 4 4+x + C. 470. arcsin x, x 4 (, x ) 4, x + C. 47. x 3 3a (a + x ) 3 +C. 47. 4, (x, ) 縺0600 060 0403 陦ﾀ05 00304-090308 x, = sin t, 473. 4, 4 sin t cos tdt= arcsin x, x, x, arcsin x + C., (x, ) 3+x, x + C. 474. (x +5) x +x +, 3; 5ln(x ++ x +x +)+C. 475., 3, x, x, arcsin x + + C. 477. x, a ax, x + a arcsin x, a + C. a 478. 4 3 ln +x 3, 4 +x 3 + + 3 arctg 4 +x 3 +C. 479., 3 (, x 3 ) 4x +C.

39 050504060503 480. m +,+ + = + 3 n 040070605 ﾀ07 09060003006 ｧ ﾀ0006; 0403000307 ｧ04 x, + = = t, 04030006 ｧ0 x, x,3 (x, +) 3= =, t, dt t =, +x x +x + C. 48. m + = 3+ 040070605 ﾀ07 09060003006 ｧ ﾀ0006; 0403000307 ｧ04 a, bx = t, n 04030006 ｧ0 t, a a, bx (x, ) x, dt b t = b + C. 48. + C. a, bx 3 (3x +)=3 483. +(3x +) =3 +lnj(3x +) =3, j +C. 484. x, x + + ln ( x +)+C. 485.,0; 3(x 3 +3a) (a, x) + C. r 486. x, x, + arctg + C. 487. 3(x +) 3! (x 3 +) x + + + + C. 5 4 3 488. ln ( + +x )+ + +x + C. 489. x + 3 (4, x ) 3 + C; 040505030 04 罍ｧ006 縺06 04030503 閨003 ﾀ0 00 03 ﾀ04030303- 陦ｧ0504 ﾀ07 r0305 ｧ 罍 09 ｧ030 0004003 ﾀ05 ｧ 04 0 0060 05060006. x + 490. x +C (, 04 罍ｧ x>0 ｧ + 04 罍ｧ x<,). 49. arccos r x, + + C. 49. arcsin x, x x 4, x +C. 493. arcsin, x, x +C. 494. +x 4x + x, lnjx ++ 4x + x j + C. 495., x +6 5+4x, x + 7 arcsin x, + C. 3 +x 496., x + +x ln + + C. jxj +x 497., + C. 498. 703000307 ｧ04, x 3 = t, 0 08 閨060 x x x 3, x = dt 3 3 t, =, x3 3 ln,, x 3 + + C. 499. 703000307 ｧ04 x ==t, 0 08 閨060 dt, =, dt 3, t, t 4, (t +) = arccos x + x + C.

3050504060503 93 500. ln (ex +), arctg (e x )+C. 50. 3 tg3 x, tg x + x + C. 50. ex, ex + 4 ln (e x +)+C. 503. lntg x + C. 504. arctg tg x 506., ctg3 x 3 + C. 505. 5 ln tg(x=)+ tg (x=),, ctg x + C. 507. arctg tg x + C. + C. 508. e x +lnje x, j + C. 509. tg4 x, tg x, ln j cos xj + C. 50. e x + 4 + ln e x, e x + + C. 5. tg (x=) arctg + C. 5. tg3 x + 3 + tgx + C. 53. arctg ( tg x) +C. 54. tg 4 ln x + x 8 tg + + C. 55. tg ln x, x 4 ctg + C. 56. lnjex, j,x + C. 57. sh 6x (tg x +lnjtg xj)+c. 58. ), x +C; )x sh 4x +chx+ + 8 + C. 59. ) sh x + sh3 x + C. 50. ln j ch xj + C. 5.,, ch x 3 sh x + C. x sh x 5., + + sh x + C. 53. ｧ 54. ｮ0. 06 366. 4 3 x 55. 4 4+x + C. 56., x ch 3x + C. 57., ch3x + 5 x, 5 9 3 sh 4x + C. 58., x 3 8 + C. 59. sh5 x + C. 530. x, cth x + C. 5 53. ch x, +C (0403 ｧ0050605 00030 06000307 ｧ0504 ﾀ0 00 ｧ ﾀ00 ｧ05060004 ｧ 0 0060 05060004 0 ch x, ). 53. sh x, ch x +C. 533. 3 ln jx + x, 3j + + x x, 3+C. 534. ln jx x + x +3j, +3 +C. 535. x ++ x + ln x +, +x (arctg x) x + C. 536. + C. 537. a ln x + a x,, x ax + C. 538. tg, + C. 539. arcsin x + C (0403000307 ｧ0504 4 b x = sin t). 540. ab arctg a tg x + C. x + x sin x + cos x 54. 4 543. r, x +x = + C. 54. ln C(e x +), x, e,x., x, x = arcsin x +, x + C.

394 050504060503 + C. 545. x tg x +lnjcos xj, x + C. tg 546. ln x + + cos x + C. 547., cos x arctg + C. 548. 3x =3, x =6 + b b +4 ln (x =6 b, 3ax +)+C. 549. + C (0403000307 ｧ0504 ax + b = t). 6a(ax + b) 3 544., ctg3 x 3 550., x + arctg x + C. 55., tg x + ( 閨0600 ｧ0504 ｧ ﾀ00 ｧ05060004 ｧ 0 0060 05060004 0 cos x ｧ 0403000307 ｧ0504 tg x = t). 55. b a + b ln x + C. 553. 3b(n, )(a, bx 3 ) n, + C 04 罍ｧ n 6= ｧ, 3b ln ja, bx3 j + C 04 罍ｧ n =. 554. 6 503 閨0600 ｧ04 0403 0903 縺0060 04030000308 0904 陦 05, 0403000307 ｧ0504 x + + = sin t ( ｧ00 ｧ 0706 縺0600 ｧ0504 0060503 閨030 0060304 縺06 閨060006000308 0903050707 ｧ09 ｧ- 060050304); x +, x, x + arcsin x + + C. 555., x + ( x +) + C. 556. ln x +x, arctg x +C. 557. x arctg ex, x+ 4 ln (4 + ex )+C. 558. ln C x + + x +. 559. x + ctg x, ctg3 x + C. 3 4, x 560.,, arcsin x x + C. 56. ) + C = 3, ctg x 3 ln sin (x + =6) sin (x, =6) = + C; ) 0303 陦ｧ0504 ﾀ07 0305 ｧ 罍 09 ｧ030 0004003 ﾀ05 ｧ 04 0 0060 05060006; 3 ln 3 + ctg x 3+tgx 3 ln + C. 56. ) 05 ﾀ0403-3, tg x (x + a) 3=, x 3= + C; 3a ) x x ++ln(x + x +) + x + C. 563. + ln C(x, ). 564., x 3 3= x + + C 565. 3 arctg x 3, +C (0403000307 ｧ0504 x 3, =t ). x x + x + x + 0403000307 ｧ0504 x =. t 566. 0; 5[x +lnjsin x + cos xj]+c. 567. x arcsin x +, x + C. 568. tg x + C ｧ00 ｧ cos x + C. cos x, sin x 569. sin 4 =, ctg xd(ctg x) + d(ctg x) = ctg x, x, ctg3 x + C. 570., ctg x ln j cos xj,x+c. 57. e,x + 3 ln e x, e x + +C. 57. 4 tg4 x + C (0403000307 ｧ0504 tg x = t). 573. ln jxj, x + ln jx +j + C. x

3574., sin x = 050504060503 95 cos x + sin x = + sin x + C (+ 04 罍ｧ cos x>0 ｧ, 04 罍ｧ cos x<0). 575. 576. d(x ) (x + )(x, ) = 6 arctg ( tgx) +C. x +, (x, ) (x + )(x, ) d(x )= = 6 ln jx, j x + + C. 577.,e,x ( x +)+C. 578. x arctg x,, ln j+xj + C. 579. tg x + C (0403000307 ｧ0504 tg x = t). 580. ln jxj,, x + x ln (x +) + C. 58. ln a arctg (ax )+C. 58. ( x + (x +7) j x +, j + cos x)+c. 583. x ++ ln + C (0403-3 x ++ 000307 ｧ0504 x + = t ). 584. x, x,, x arcsin x + C. 585. x 0403000307 ｧ0504 x =. 586., 3x +3x + t 3(x +) 3 + C (0403000307 ｧ0504 x + = = t). 587. ax + x, a ln jx + a + ax + x j + C ( ﾀ. 53, 04. 4 ). 588. ln (x, ) jx + xj + C. 589., + cos x + sin x sin x + C. 590. 6 ln C(x +x +) + x, x + 8 arctg x ( 0 0060 05060004 00 05 0605 ﾀ07 0, x 000307 ｧ050600 ｧ 05 09: x 4 +4=x 4 +4x +4, 4x =(x +), 4x ｧ 05..). 59. s 5 = 0; 646, S 5 = 0; 746, x =0; 693. 593. 0. 594. 8. 595. 4 3. 596. 6. 597. a. 598. 3(e, ). 599. ln ( + ). 600.. 60. 703000307 ｧ04 x = t ｧ ｧ 0060 ｧ04 ﾀ030305040605 ﾀ0504060003 04 縺06 閨060003, 04030006- ｧ0 3 tdt t, =, t + ln (t, ) 3, 3 = ( + ln ). 60.. 603.,, ln. 604. 3 3, e a(, ). 605. ln. 606. (0403000307 ｧ0504 x = e + 4 = a sin t). 607. a +ln(+ ). 608.. 609. ln,. 60.. 3 6 3, 6.. 6. ln 3. 63. ) ;) 3 4 ;3) 3 5 4 6.

396 050504060503 64., a3 6. 65. 3,. 66.. 67.. 68. ln; 5, 6 3. 69. arctg e, 4 0; 433. 60. 7 6. 6., 4. 6.,. 63., ln. 64. ) ;) 3 4 ;3) 3 5 4 6. 65. 3 3. 66. ab. 67. 3 04 縺03 ｧ 0406 閨060 ｧ07 03 ﾀ00304 0 ｧ07 ( h) 0 0403 ﾀ030506 h. 68. 3 3. 69. 8ln. 630.. 63. 6. 63. 9;. 633. 5; 6. 3 634. 8 5. 635. 8 5. 636. 3 6. 637. a ( ﾀ0. 罍ｧ ﾀ. 56 0 ﾀ. 309). (4, )a 638. 0; 8 ( ﾀ0. 罍ｧ ﾀ. 53 0 ﾀ. 307). 639. ; 0403000307 ｧ0504 x =asin t ( 罍ｧ ﾀ. 84, ﾀ. 335). 640. a sh = a (e, e, ) ; 35a. 64. 3a. 64. 3a 8. 643. a. 644. 3a. 645. r max = 404 罍ｧ ' = 90 + + 360 n, 05. 06. 04 罍ｧ ' =45 + 80 n =45, 5 ; r min = 04 罍ｧ ' = =, 90 + 360 n,05.06. 04 罍ｧ' =,45 + 80 n = 35, 35. ｮ00607-00306 0509 ﾀ05 縺060 000400306 陦ｧ06 ﾀ-0406090503 縺03 04 罍ｧ 45 ｧ 35. 0ｷ3 ﾀ09030 07 0400030 閨04 040 3=4 =4 (3 + sin ') d' = 9 649. r = a(sin ' + cos ') =a cos 8. 3 646. 4 ', 4 a a. 647.. 648. 4. ; r max = a 04 罍ｧ ', 4 = = 0, ' = 4 ; r min = 004 罍ｧ ', 4 =, ' =, 4 ｧ 3. 700030 閨04 4 3=4 S = (a ) cos ', d' = a.0505040605 04030006 0605 ﾀ07 04 縺03006, 06 ﾀ00 ｧ 4,=4 0406 縺060805 ｧ 09 閨0609 縺050304030 090303 罍陦ｧ0 05 0: x + y = a(x + y) 0309 縺0607003 ﾀ0504. 7a 650. 4. 65. (0 +7 3)a. 65. 3a. 653. 36. 654.. 64 655. 3 3. 656. 4 3 660. 7; 5, 6ln6. 66. 4 ( ﾀ0. 罍ｧ ﾀ. 5 0 ﾀ. 307). 657. 3. 658.. 659. 6 3. 0,,x x + = 8 ( ﾀ0. 罍ｧ ﾀ. 49 0 ﾀ. 307). 5 66. r max =4, 090305 ' = 80 +360 n, ' =90 +80 n =90 ｧ00 ｧ 70 ; r min =, 090305 ' =0 + 360 n, ' = 80 n =0 ｧ00 ｧ 80. 700030 閨04

3 = 050504060503 97 S = (3 + cos ') d' = 9 8. 663. 3 a a. 664.. 665. 4 4. 0 a 666. 4 (e, e, ) = a sh. 667. 4ab arctg b a. 668. 8 a. 669. h 8a b. 670.. 67.. 67. 58; 5. 673. a b. 3 sh 674. a 3 +. 675. 5 5. 676. 7 6 a3. 677. 3. 678. 5 7.! 3 679. 4 5 3 +. 680. a3 6. 68. 6. 68. 64 3 ( +). 683.. 4 4 3a 3 684. 3 a b. 685. 05. 686. 9;. 687. 8a 3 3. 688. V = = 8 3. 689. 5 a 3. 690. 7. 69. 670. 693. 6a. 694. 7 7. 695. 8a. 696. ｯ03 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ 04 罍ｧ t = 0 ｧ t = 4 8, 4 8 s = t 4 +t 3 dt = 3 3. 697. 6 + ln ( + 3). 698. a sh 0 ; 35a. 699. s = = t + ln t, t + =5 3=4 ;6 ;5 03 ﾀ070 ｧ 04 罍ｧ x =0 ｧ x = 3 ; s = =3 0 cos x = =3 0 +x ; 040300 05 060 +x = t ; s = x 3=5 5=4 t dt t, = =; 35 + ln ; 043. 700. ｯ03 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ cos x cos x = =3 0 d(sin x), sin x =ln(+ 3) ; 3. 70. ) 4 3;)0; 5ln(ch) ; 009. 70. ) 8a; )a +4 + + a ln ( + +4 ). 3a 8 703.. 705.. 706. ln 3. 707. ln3,. 3 708. +ln(+ ) ; 9. 709. 4 3. 7. 4 3. 7. a (sh +). 73. + 4. 74. +ln ( + ). 75. 64 3 3 3 a. 76. 3. 77. 4 34 7, 6 ab. 78.. 79. 9 3. 70. ; 4a. 7. 9; 6. 7. ; 44 0 6 7; 0 0 ｧ0700606 0403000304 ｧ006 ﾀ ｧ00 0400060 ｧ07 ; 08 0 6 7. 73. ah 6. 74. 3 R3. 75. ; 4 0 6 7.

398 050504060503 76. J x = ab3 3, J y = a3 b 3. 77. J x = ab3, J y = a3 b. 78. 6; 4. 79. M x = M y = a3 6 ; x c = y c = a a 3. 730. M y x = y=0; ab, a a M y = xy = 4 ba ; S = 0 = 0, y c = ) ; 5 0 3 R 4 6 707. 733. ar 0 b y= ab 3 ; x c = 3 4 a, y c =0; 3b. 73. x c = (y=)y 0; 5a = 4 3 a 4 9 a. 73. ) ; 04 6 707; R+h R mgr x = mgrh R + h. 734. 000R H 6 0 6 707. 735. 406 707. 736. 0; 4 6 707. 737. t = = 00 ﾀ. 738. t = R 0; 6r H g H+h h 0 H 0 S 0; 6s gx = x x,05 閨06 h 0403 ﾀ0305 閨03040300-0 ｧ050600040030503 0903006 ﾀ. 6 503 ｧ ﾀ00 ｧ04, 04030006 ｧ0 t 4 ﾀ. 739. ah 3. 740. 7 5. 74. = = 0 h. 74. ; 4 0 4 70 09 07 閨0606 ﾀ050600906. 743. I x = a 4 sin t cos tdt= a4 6. 744. x c = 0, y c = H R 0 R 0 " V0 a 0 y y y xdy = = 8 5. 745. R 000 (H, H x) x 300 6 707. 746., 0V 0,, V 0 r 5 980 6 707. 747. t = 4R R 5 s 0; 8 g = 400 49 ﾀ. 748. ) ; 3 ｧ0050605 0003 ) ｧ 3) ﾀ0803 閨0705 ﾀ07; 4) 04 罍ｧ n > ; ﾀ0803- x n = n, 陦ｧ05 ﾀ07 04 罍ｧ n 6. 749. ) ; ) =; 3) =4; 4) ; 5) ln ; 6) 6. 750. ) 6 ;) 4 + ln ;3),. 75. ) 6 3 ;) ﾀ0803 陦ｧ05 ﾀ07; 3) 6. 8 #

3050504060503 99 75. ) ﾀ0803 陦ｧ05 ﾀ07, ｧ0303 < +x 3 +x 3 x, 3= 0 ( ﾀ0. 06 748); ) ｧ0303 x ﾀ0803 陦ｧ05 ﾀ07; 3) 0 x 3= ﾀ0803 陦ｧ05 ﾀ07 3 x3, ﾀ0803 陦ｧ05 ﾀ07, ｧ0303 3 x3, > x, e,x x e,x ﾀ0803 陦ｧ05 ﾀ07 ( ﾀ0. 06 749); 4) j sin xj x 6 x, ﾀ0803 陦ｧ05 ﾀ07, ｧ0303 6) 0 x >, e,x = 04 罍ｧ n > ; ) x x 4 + > 0 e,x + ﾀ0803 陦ｧ05 ﾀ07, ｧ0303 e,x x 6 e,x 04 罍ｧ x >, sin x x ﾀ0803 陦ｧ05 ﾀ07 ( ﾀ0. 06 748); 5) x x x 4 + x 04 罍ｧ x>, 4 e,x ﾀ0803 陦ｧ05 ﾀ07. 753.) b a 03 ﾀ03000605003 ﾀ0803 陦ｧ05 ﾀ07, x x 4 + x ﾀ0803 陦ｧ05 ﾀ07; e,x ﾀ0803 陦ｧ05 ﾀ07, ｧ0303 e,x 6 e,x 04 罍ｧ 0 x n = 04 罍ｧ n<, ﾀ0803 陦ｧ05 ﾀ07, n (b, a),n = 04 罍ｧ n<, ﾀ0803 陦ｧ05 ﾀ07 04 罍ｧ n >. (b, x) n, n 754.. 755.. 756. 3a. 757. a 3. 758. + ln ( + ). 759. 4 3. 76. ) ;) 3 ;3); 4) ﾀ0803 陦ｧ05 ﾀ07. 76. ) ln ( + ); ) ; 3), 4. 763.. 764. 6. 765.. 766. ) 3ln ;) ; + ab + b 3) e, ;4)a ;5) 4. 768. ) "(h) =0;)j"(h)j 6 < 0; 3. 3 4 5 770. 55 6 8; 8 閨03. 77. ln = 0; 693; j"(h)j 6 0,4 < 0; 00. 5 773. 8; 6. 777. 7 罍ｧ0300 ｧ07060003 ;. 778. R =. 779. R =. 780. 6 5 0406 縺00 ｧ006 (; 0) R = ; 04 0406 縺00 ｧ006 (0; ) R =4. 78. R = =4a. 78. y max = 04 罍ｧ x =; R = e. 783. (4; 4). 784. (3;,). e

3300 050504060503 785. (0; ). 786. 7X +8Y 3 = 0. 787. (X) =3 + Y =3 = 3 =3. 788. X =3,Y =3 =(a) =3. 789. X = a cos t, Y = a sin t ｧ00 ｧ X +Y = = a. 790. k = e x ( + e x ),3= ; k max = ln 040503 0906 x =, 3 3, 0; 347. 79. ) R = a r3 ar; ) ;3) 3 3r a. 793.. 794.. 795.. 796.. 797. (,; 3). 798. 0;, 4. 799., 3 ; 6. 3 800. X = 4, 3,0; 7; Y =,,; 4. 80. 8X 3, 7Y =0. 80. X =,t + t, Y = 4t + t ; 閨0007 0403 ﾀ05 縺03060 ｧ07 09 罍ｧ040308 ｧ 3 05040300060503 ﾀ03 ﾀ05 04 ｧ0504 05 0300 ｧ0906 0 060 ｧ08 x, y, X, Y 閨0007 t =0;; 3=. 803. (X +Y ) =3,(X,Y ) =3 =4. 804. (X +Y ) =3 +(X,Y ) =3 =a =3 ; 04 罍ｧ 04030403 縺030506 03 ﾀ0608 0 45 050503 06 040060 ｧ06 04 罍ｧ00605 04 ｧ x =3 +y =3 =(a) =3, 05. 06. 050403000605 ﾀ05 縺03 ｧ 閨03 06 ﾀ0504 05030706 ﾀ05 縺03 ｧ ﾀ 06040600 ｧ 0600030 ｧ 04 閨040306-006 0 ｧ ｧ 04030406 縺00605 07 0 45. 806.. 807. 5t. 808. 7; 5. 809.. 80. sh ; 35. 8. 3 + ln 83. y = 4 3 x, x 9 ; dr dt w = 4jt, j 4t, 6t +5, w n = w n = ;. 85. 86. x, t 88. x, x a = y, t t = y, 3. 8. 3x +4y =0; dr dt = 3i + (, t)j. 84. w = d r + y b =4i, 3j. =,j, dt 6 4t, 6t ; 04 罍ｧ t +5 = 0 w = ; 6; = ; v =,a sin ti + b cos tj, w =,r. = z, t3 3t. 87. X, x = Y, x x =, x =( x).,4 = z, 4. 89. _r =,i + k, B = i + k, N =,j; 3 =,i + k ; = i + k, =,j. 80. B = _r r = 6i, 6j +k, N = (_r r) _r =,i, 6j +8k, 06 040060 ｧ07 0500 0400308 003 縺0 00 ｧ: x, = y, = z, x, ; 03 ｧ003 縺0 00 ｧ: = y, 8,9 3,3 = z, ｧ ﾀ0304 罍ｧ09 ﾀ - 0600608 ﾀ07 040003 ﾀ0903 ﾀ05 ｧ: 3x, 3y + z =. 8. N =3(i + j), B =,i + j +k. ｰ 040060 ｧ07 0500 0400308 003 縺0 00 ｧ: x = y, z =0; 03 ｧ003 縺0 00 ｧ: x,, = y, = = z. 8. 0ｷ3 ﾀ090006 ｧ04 t, 04030006 ｧ0 x + y = z 06 040060 ｧ06 09030 ｧ 06- ﾀ090308 04030406 縺08003 ﾀ05 ｧ. _r = (cos t, t sin t)i + (sin t + t cos t)j + k = i + k; r = (, sin t, t cos t)i + ( cos t, t sin t)j = j; B = _r r = i +k, N = 4j. 07 ﾀ 050600040 07: x = z ｧ y = 0;0500 040 07 003 縺0 0004: 03 ﾀ04 Oy; 03 ｧ-

3050504060503 30 003 縺0 0004: x + z = 0 ｧ y = 0. 83. 7 罍ｧ t = x,a = z, b= ; b a b 4 4ab y = a. 84. cos =, cos =, cos = ; a + b a + b a + b 04030303 0 09 04 ｧ ﾀ ｧ05 0305 04030303 0 04 0400060 ｧ07 0 09 07 閨0308 04060504 ｧ 09 罍ｧ040308. 85. ｰ 040060 ｧ07 04 ｧ00503040308 00 ｧ0 ｧ ｧ: x = sin t, y =, cos t, z = t, 05 閨06 t 06050300 04030403 縺0305 ( 罍ｧ ﾀ. 44). ｭ 陦ｧ0 ｧ 00308 03 ｧ003 縺0 000400308 0406090503 040503 0906 C (04 罍ｧ t = =): = i + j + k +. 86. 7 罍ｧ t = v = 7 ｧ ﾀ. 44 = a(i + j), w = ai. 87. x, = y, = z, 8 8. 88. x, = = y, ｧ z = 3. 89. x,, 83. N =,6i, 3j +k, B =6i, j +k; x, = y, 6 3 x, = y, 8,6 = z,. 83. N =,4j, 4k, B =j, k. ｰ 040060 ｧ07 0500 0400308 003 縺0 00 ｧ: x =, z = y +; 03 ｧ003 縺0 00 ｧ: x =, y + z = 6. 834. v = _r = i +(, t)j, w = r =,j, R = jv wj v 3 = v ; 3 v =, 4t +4t ; w =_v = 835. v = _r =,4 sin ti + 3 cos tj = R = v, v 3 = 6 sin t + 9 cos t, _v = = y = z,. 830. 0, 60, 45. = z,,, 4t, =,, w n = v, 4t +4t R = v =.,4i +3j 4i +3j, w = r =, ; 7 sin t ; 04 罍ｧ t = v 4 v = 5,

330 050504060503 w = _v = 7, w n = v 5 R = v = 5 =; 4. 836. v = = _r = i +tj +t k, w = j +4tk; v = t +, jv wj R = v 3 = = (t +) = 9, w =_v =4t =4, w n = v R = (t +) =(04 0006030308 (t +) 0503 0906). 837. ｮ0 00 ﾀ03 ﾀ05 04 ｧ0 0 05 罍ｧ0906 090303 罍陦ｧ0 05 0406090503 縺0304 r _r r... r _r r t t t 3 t 3t 0 6t 0 0 6 6t,6t 6ｦ0 05060 0 08 閨060: ) j_rj = +4t +9t 4 ; ) j_r rj = 9t 4 +9t +; 3) _rr... r =;4) R = 9t 4 +9t + ( + 4t +9t 4 ) =;5) 3 = 4(9t 4 +9t +) =3. 838. R = (x + y) = 4, =, 4. 839. R = 3, = 3. 840. 7 0804 04030809 04 ｧ00503040308 00 ｧ0 ｧ ｧ: = b a + b, 0 08000604030809: =, b a + b. 84. R = t (t +) = 9, =, t (t +) =, y. 84. r = 9 i + yj + + y4 4 k; R = 9y4 +4y 6 + (y ++y 6 ) 3 = 4 7, =, 3 7. 843. R = 3, =, 3. 844. 3) 6 5 ﾀ07 040003 ﾀ0903 ﾀ0504, 09 縺03006 0503 09 ｧ (0; 0); 4)x + y 6 a ;5)xy > 0 (0406 縺040308 ｧ 05 縺0605 ｧ08 0904 陦 00503); 6) x + y < ; 7) 04 ﾀ07 040003 ﾀ0903 ﾀ0504, 09 縺03006 04 縺0700308 y = x. ｰ 040060 ｧ07 ) ｧ ) 0304 縺06 閨0600070605 04 03030003 ｧ 閨03 04 0060 ｧ07; 3) 04030406 縺08003 ﾀ0504 04 0060 ｧ07 040309 縺0605 03 ﾀ ｧOz 09 罍ｧ040308 z = 4 x ｧ y = 0 ( 罍ｧ ﾀ. 45); 4) 04030006 ﾀ0706 縺06; 5) 0903006 ﾀ, 閨0007 ｧ 0303 07060 ｧ07 09030503 縺030503 0403 040060 ﾀ06 060 ｧ07: x = a, z = ay ｧ y = b, z = bx 04 03030003 ( 罍ｧ ﾀ. 46); 6) 0403-0406 縺08003 ﾀ0504 04 0060 ｧ07 09 罍ｧ040308 z =, y = 0040309 縺0605 Oz; 7) 0903006 ﾀ, x ﾀ 0303 06060 ｧ0 ｧ y = kx, z = kx ｧ 0 04 040007060 ｧ0 ｧ 04003 ﾀ0503 縺030k, 0 ｧ0 ｧ 05 ｧ0406 縺030300 0 ｧ y = h, (x, h)(z + h) =,h, ｧ006060 ｧ0 ｧ 0406 縺00 ｧ003 0 03 ﾀ ｧ Oy ｧ 03 閨006 ｧ ﾀ ｧ004050305 0 040003 ﾀ0903 ﾀ05 ｧ y = x (x = h, y = h); 05-09 ｧ06 0706 05 ｧ0406 縺03030003 04030006 0605 ﾀ07 04 ﾀ06 060 ｧ0708 x = h ｧ00 ｧ z = h ( 罍ｧ ﾀ. 47). :

3050504060503 303 845. s = (, x)(, y)(x + y, ). 050300 ﾀ0504 ﾀ06006 ﾀ05040304 0 ｧ07 0706009-09 ｧ ｧ: 0 <x<, 0 <y< ｧ x + y>, 05. 06. 00030706 ﾀ050403 0503 0609 0400605 罍ｧ 05 縺0606050300040 ｧ09, 0305 0 ｧ 0600030503 00 ｧ0 ｧ070 ｧ x =, y = ｧ x + y =. 7 ｧ ﾀ. 45 7 ｧ ﾀ. 46 848. x z =(x, y +x)x =0;, y z =(y, x +y)y =,0; 9, z = x z + y z, xy =0; 03. 849. 70604 縺06 縺030400306 ｧ 03 閨003 0 00306 7 ｧ ﾀ. 47 04 030300 ﾀ05 ｧ jyj 6 jxj 070600909 ｧ ｧ z = + x, y ｧ z =, x, y ｧ 03-03 07 0605 ﾀ07 0406 縺0800608 ｧ 0 ｧ0700608 04030406 縺08003 ﾀ05070 ｧ 09 縺06050304030503 0903006 ﾀ ( ﾀ 03 ﾀ0406 Ox). 7 罍ｧ006 縺030 縺030400308 070600909 ｧ ｧ, 0304 縺06 閨0600070600308 06 040060 ｧ060 z =

3304 050504060503 = x, y,003070605 ﾀ000607 ｧ0504 070600909 ｧ07 z = 8 >< >: + x, y 04 罍ｧ 0 6 x<;, x, y 04 罍ｧ 6 x<; + x, y 04 罍ｧ 6 x<3 ｧ05.. 7 縺0700306 x =, x = ｧ 05.. 00 ｧ0 ｧ ｧ 縺0304. 0ｷ3 0303 0706-0 ｧ060 0306 閨0605 06 縺06 閨06060 ｧ06 ﾀ07 04030003 ﾀ03 0406 縺0800608 ｧ 0 ｧ0700608 04030406 縺08003 ﾀ050608 0903-006 ﾀ. 050300 ﾀ0504 0304 縺06 閨0600060 ｧ07 05050308 070600909 ｧ ｧ jyj 6 jxj, 05. 06. 00030706 ﾀ050403 0503 0609 0400605 罍ｧ 03 ﾀ05 縺030503 060500 00607 閨06 04 縺070030 ｧ y = x ｧ0 0505 ｧ08 04 縺0700308. 854. ) 6 5 ﾀ07 040003 ﾀ0903 ﾀ0504, 09 縺03006 04 縺0700308 y =,x; 3)0503 09 ｧ 0400605 罍ｧ 050000 ｧ04- ﾀ x a + y = ｧ0 050000 ｧ04 ﾀ06; 4) 04 ﾀ07 040003 ﾀ0903 ﾀ0504; 5) 0503 09 ｧ 0400605 罍ｧ 060500 b jyj 6 jxj ｧ 0 060503 ﾀ0503 縺030 08; 6) 0904 陦 005 040003 ﾀ0903 ﾀ05 ｧ x > 0 ｧ y > 0. 7030406 縺08003 ﾀ0504 ) 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 07 ﾀ 0303 06060 ｧ0 ｧ z = h, x + y =4=h ｧ 0 04 0400070600608 z =4=x, y =0( 罍ｧ ﾀ. 48). 7030406 縺08003 ﾀ05 ｧ 5) { 6) 09030 ｧ 06 ﾀ09 ｧ06; 7 ｧ ﾀ. 48 04030406 縺08003 ﾀ0504 4) 04 03030003 ｧ. 858. 3x(x+y), 3(x,y ). 860., y x, x.,y 86. x + y, x x + y. 86., y (x, y), x 3 t (x, y). 863. 3x( 3 x, 3, t) 3 x @c 3t( 3 t, 3. 864. x) @a = a, b cos, @c c @b = b, a cos, @c ab sin =. c @ c 866. @u @x = e,xy (, xy), @u @y =,x e,xy. 867. @u @x = 5t (x +t), @u @t =

3050504060503 305 r 5x @ =,. 868. (x +t) @x = t x, x t, @ x @t = @z. 874., xt @x = @z =,a sin (ax, by), = b sin (ax, by). @y 875. @z @x =, yjxj x x, y, @z @y = jxj x x, y. 876. @z @x = 3y (3y, x), @z @y =, 3x (3y, x). @u 877. @x = ctg (x, t), @u =, ctg (x, t). 878. @u = sin y @t @x cos (x + y), @u @y = sin x cos (x +y). 885. ) 0; 075; ),0; e, 0; 739. 887.,0;. 888. ; 閨0 3. 889. 0; 3 ﾀ0. 890. ) dz = =, y x + y + x + x y dy; )ds =lnt+ xdt. 89. z = t =0; 043, dz =0; 04. 89. 0; 5. 893.,30 ﾀ0 3. 895. dz dt =,(et + + e,t )=,cht. 897. dz = ey + xe y dy. @z 899. @u = x, x, y y @z @v =,x 4+ x. 900. ) @z y y @x = @z @u @u @x + @z @v @v @x = m @z @u + @z @v, @z @y = n @z @u + q @z @z ;) @v @x = y @z @u, y @z x @v, @z @y = x @z @u + @z @u. 90. x @v @r = = @u @u @u cos ' + sin ', @x @y @' =, @u @u sin ' + @x @y cos ' r. 903. ) @z @t = = (Ax+By) cos t,(bx+cy) sin t =(A,C) sin t+b cos t; ) dz dt = = et e 4t +. = @z y @u @z 906. ) @x = @z @u + @z @v, @z @y = @z x +@z @v, @z @y = @z @u x y +@z @v @u, @z @z ; ) @v @x r = @y. 907. @x =, x y y +3. 908. ), 3 x ; ) yex, e y xe y, e. x 90. 3. 9.,. 9. ) (,; 3) ｧ (,;,); 4 @z ) (; ) ｧ (,3; ). 93. @x = 3, x @z, z @y =,y z. @z 94. @x = y z, @z @y = x z. @z 95. @x = a c, @z @y = b c. 98. dy = x 4y. 99., y x. 90. x + xy + y. 9. xy. 9. 4 5, @z. 93. 5 @x =, @z @y = y x, z. 96. 6; ; 0; 6. 99., 6y x 4 ; x 3 ; 0; 0. 93. xy (x + y ), y, x (x + y ),,xy (x + y ). 938. ) x 4 (3y, 4xy dy +x dy );), y(, xdy) xy.

3306 050504060503 94. @ z @x = 3 @ @u + @ @v @ z @x @y = 3 @ @u + z @ @v @ @u + @ z =3 @ z @v =9 @ z @u +6 @ z @u @v + @ z @v ; @u +4 @ z @u @v + @ z @v ;,4 @ z @y = @ @u + @ @v z = @ z @u + @ z, @ z @u @v @v ; 3 @ z, @x 4 @ z z @x @y +3@ @y =,4 @ z @u @v : 943. 6ｦ0 04 ｧ ﾀ0304 07 05 09 0706, 09 09 ｧ 04 06 94, 04030006 ｧ0 4 @ z @v. 945. @ z @x = @ z, y @u y @ z x @u @v + y @ z x 4 @v + y @z x 3 @v ; x @ z @y = @ z x @u + @ z @u @v + @ z x @v ;,y 946. x @ z, @x y @ z @y = @ z,4y @u @v + y x @ z @x + @ z @y. 8x (, y). 3 948. 0; 0; 4 @z @v : 947. @ z @x =, y, @ z @x@y = 4x (, y), @ z @y = = 9t 3 t ;, 8x 7t 3 3 t. 953. d u =, y x + + x dy, d3 u = y x 3 3, 3 x dy. 954. 4a @ z. 955. @ z,v @u@v @u@v + + v @z x. 959. u = u @v + x ln y, cos y + C. 96. u = x z + x +lny,, arctg z + C. 963. u = xy, x + 3y + C. 964. u = x sin y + + y ln cos x + y + C. 965. u = xy + sin y + y + C. 966. u = x x ( + t +)+C. 967. u = x ln y, x cos z + yz + C. 968. u = x, 3y + C. 969. y = x +x; 030300 ﾀ0504 ﾀ0403000307060 ｧ07: +x > 0; z x >,. ｯ03 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ Ox: y =0, x =0 ｧ00 ｧ x =,. 05 ﾀ0303 07 0503 09 O(0; 0) 06 0600. 709 ﾀ05 縺060060 y 04 罍ｧ x =, 3, y 05 = 3 3 ( 罍ｧ ﾀ. 49). 5 970. y = (x +) x +; x >, 030300 ﾀ0504 ﾀ0403000307060 ｧ07. 05 ﾀ0303 07 0503 09 : (,; 0) 0503 09 0403 04 05. ｯ03 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ: 04 罍ｧ x =0 y = ; 04 罍ｧ y =0 x =, ( 罍ｧ ﾀ. 50). 97. y = x x,. 050300 ﾀ0504

3050504060503 307 ﾀ0403000307060 ｧ07 x >, x =0, y =0 03 ﾀ0303 07 ｧ 0300 ｧ 縺0304 00 07 0503 09. 7 罍ｧ x = y =0, 04 罍ｧ x = y =. ｯ03 09 0406 縺0605 ｧ03 : x = 4 3, y = 4 3 3 ( 罍ｧ ﾀ. 5). 97. y = x, x ; 030300 ﾀ0504 ﾀ0403000307060 ｧ07 jxj 6, ｧ00 ｧ 7 ｧ ﾀ. 49 7 ｧ ﾀ. 50 7 ｧ ﾀ. 5, 6 x 6. ｯ03 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ: 04 罍ｧ y =0 x =0, x =, x 3 = =,. 05 ﾀ0303 07 0503 09 O(0; 0) 06 0600. 709 ﾀ05 縺06006003 04 罍ｧ x = 0; 7 y 05 = ( 罍ｧ ﾀ. 5). 973. y = x x x. 050300 ﾀ0504 ﾀ0403000307060 ｧ07 x > > 0; 0503 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ: 04 罍ｧ y = 0 x = 0 ｧ00 ｧ x = ; 03 ﾀ0303 07 0503 09 O(0; 0) 0503 09 0403 04 05 0406-04030503 縺03 ﾀ 09 ﾀ 0506000400308 y = x. 709 ﾀ- 7 ｧ ﾀ. 5 7 ｧ ﾀ. 53 05 縺060060 ｧ0060605 070600909 ｧ07 y = x, x x; 04 罍ｧ x = 4 9 y max = 4 ( 罍ｧ ﾀ. 53). 7 974. y = (x, ) x; 030300 ﾀ0504 ﾀ0403000307060 ｧ07 x > 0; 04 罍ｧ y =0 x =0 ｧ00 ｧ

3308 050504060503 x =; 03 ﾀ0303 07 0503 09 (; 0) 06 0600. 07 罍ｧ04 07 ｧ0060605 05 090308 0706 04 ｧ, 09 09 ｧ 0 罍ｧ ﾀ. 5, 003 ﾀ 閨04 ｧ00605 0404 0403. 975. y = (x + a), x +a ;09 罍ｧ04 07 x ﾀ0403000307060 04 050308 030300 ﾀ05 ｧ, 05 閨06 x ｧ x +a ｧ0060605 00306 0 09 ｧ, 05. 06. 04 罍ｧ,a 6 x<0. 05 ﾀ0303 07 0503 09 (,a; 0) 0503 09 0403 04 05 ; x =0 ﾀ ｧ004-050305. 07 罍ｧ04 07 09 ｧ ﾀ ﾀ03 ｧ r, 05 09 07 0706, 09 09 0 罍ｧ ﾀ. 85, 003 ﾀ00600600 07 0 a x 3, y 3 0400060403. 976. y = ; 030300 ﾀ0504 ﾀ0403000307060 ｧ07 y 6 x. ｯ03 09 ｧ 0406 縺06-3 ﾀ06 060 ｧ07 ﾀ 03 ﾀ070 ｧ: 04 罍ｧ x =0 y =0 ｧ00 ｧ y =,3. 05 ﾀ0303 07 0503 09 (0; 0) 0503 09 0403 04 05. 7 08 閨060 ﾀ ｧ00405030506 04 ｧ y = kx + b. 7 閨0600 ｧ0 0006003 y 3 y 06 040060 ｧ07 0 x 3 : =, 3 =0. 0505 ﾀ06 k = x x x x! lim y x =,,3y b = lim (y, x) = lim x! x! x =,. 0ｷ305 09, ﾀ ｧ004050305 y = x,. + xy + y 709 ﾀ05 縺060060 070600909 ｧ ｧ x = '(y) = 3 y 3 +3y : 04 罍ｧ y =, x 05 = 3 4 ; 6; 04 罍ｧ x =0 y =,3 0406 縺0605 ｧ03 ( 罍ｧ ﾀ. 54). 977. x 3 + y 3, 3axy =0 閨0609 縺050304 00 ｧ ﾀ05 ( ﾀ0. 06 366). 05 ﾀ0303 07 0503 09 O(0; 0) 06 0600 ﾀ09 ﾀ - 050600040030 ｧ y = 0 ｧx = 0. 7 08 閨060 ﾀ ｧ00405030506 y = kx + b. 7 罍ｧ0406 閨060 y 3 y y 06 040060 ｧ06 09 04 ｧ 閨06 +, 3a =0; 03050906 k = lim =,, x x x x! x r b = lim (y + x) = lim x! x! 7 ｧ ﾀ. 54 7 ｧ ﾀ. 55 3axy x =,a. 0ｷ305 09, y =,x, a ﾀ ｧ004-, xy + y x 050305 ( 罍ｧ ﾀ. 79). 978. y =. ｮ ｧ000605 罍ｧ 0 0305003 ﾀ ｧ05060004003 Ox x, a

3050504060503 309 ｧ Oy. 050300 ﾀ0504 ﾀ0403000307060 ｧ07 jxj > a ｧ jyj > jxj. O(0; 0) 03 ﾀ0303 07 ｧ 0300 ｧ 縺0304 00 07 0503 09. 7 罍ｧ x = a 0509 ﾀ05 縺060060 y = a. 罍ﾀ ｧ00405030503 x = a ｧ y = x ( 罍ｧ ﾀ. 55). 979. y = x, x; 030300 ﾀ0504 ﾀ040300030706-0 ｧ07 x 6. ｯ03 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ 03 ﾀ0406 Ox: 04 罍ｧ y =0 x =0, x =. 05 ﾀ0303 07 0503 09 (0; 0) 06 0600. 709 ﾀ05 縺06006003 y: 04 罍ｧ x = 4 3 y 05 = 4 3 3 = = ; 08. (07 罍ｧ04 07 ｧ0060605 05 090606 0706 0703 縺006, 09 09 0 罍ｧ ﾀ. 49.) 980. y = = x a a, (x, a) ; 030300 ﾀ0504 ﾀ0403000307060 ｧ07 jx, aj 6 a, ｧ00 ｧ,a 6 6 x, a 6 a, ｧ00 ｧ 0 6 x 6 a. 7 罍ｧ y =0 x =0, x =a. ｯ03 09 (0; 0) 03 ﾀ0303 07 (0503 09 0403 04 05 ). 7 罍ｧ y 0 x(a, =0,05.06. ax, x x) + = ax, x =0, x = 3a, y 05 = 3 3 4 a 5 4 a ( 罍ｧ ﾀ. 56). 98. y = (x +) x. 050300 ﾀ0504 ﾀ0403000307060 ｧ07 x > 0 ｧ 06006 ｧ 0300 ｧ 縺0304 00 07 0503 09 (,; 0). ｯ03 09 0406 縺0605 ｧ03 04 罍ｧ x ==3. 07 罍ｧ04 07 05 09 07 0706, 09 09 0 罍ｧ ﾀ. 5, 003 ﾀ0060060 0400060403. 7 ｧ ﾀ. 56 7 ｧ ﾀ. 57 98. 6 70406 030300 ﾀ05 ｧ ﾀ0403000307060 ｧ07: ) x > 0; ) x<,a. 罍ﾀ ｧ00405030503: y = x + 3a, y =,x, 3a ｧ x =0. ｯ03 09 0403 04 05 (,a; 0). 709 ﾀ05 縺06006003 y 04 罍ｧ x = a y 05 = 3 3a ; 6a. 983. y = x x +5; 030300 ﾀ0504 ﾀ0403000307060 ｧ07 x >,5. 05 ﾀ0303 07 0503 09 (0; 0) 0503 09 ﾀ 00304 罍ｧ0903 ﾀ00304060 ｧ07. 709 ﾀ05 縺06006003 y: 04 罍ｧ x =,4 jyj max =8, 04 罍ｧ x =0 jyj min =0( 罍ｧ ﾀ. 57).

330 050504060503 984. y = x x,. 050300 ﾀ05 ｧ ﾀ0403000307060 ｧ07 jxj > ﾀ ｧ 0300 ｧ 縺0304 000308 0503 090308 O(0; 0). 06 07 ｧ09 05 090308 0706, 09 09 ｧ 0 罍ｧ ﾀ. 5, ﾀ 閨0303 0400060 ｧ060 06006 閨03 ﾀ ｧ000605 罍ｧ ｧ 09 罍ｧ040308 ﾀ000604. 985. 7 罍ｧ y =0 x =0 ｧ x =,4; 04 罍ｧ x =0 y =0, y =,. 05 ﾀ0303 07 0503 09 (0; 0) 06 0600 ﾀ 0 0900030030 09 ﾀ 0506000400308 k =. 7 罍ｧ x =,8=3 y max =; 8 ｧ 04 罍ｧ x =0 y min =,. 罍ﾀ ｧ004050305 y = x+. 07 罍ｧ04 07 0406 縺06 ﾀ0609 0605 ﾀ ｧ00405030506 04 罍ｧ x =,0; 4 ｧ r05060 0304 ｧ ﾀ0304 0605 x 0406050006, 04 縺0308 閨07 06 縺06 (0; 0) ｧ (0;,). 986. ) y = (x,a) a, x ;09 罍ｧ- 04 07 ﾀ0403000307060 05 0, 05 閨06 x ｧ a, x ｧ0060605 03 陦ｧ0 0903040306 0 09 ｧ, 05. 06. 04 罍ｧ 0 6 x 6 a. ｯ03 09 (a; 0)03 ﾀ0303 07 06 0600 ﾀ 0 0900030030 09 ﾀ 0506000400308 k =. ax 罍ﾀ ｧ004050305 x =a ( 罍ｧ ﾀ. 84). ) y = ; 030300 ﾀ0504 ﾀ0403000307060 ｧ07 x, a jxj >a ｧ jyj >a ﾀ ｧ 0300 ｧ 縺0304 000308 0503 090308 (0; 0). 罍ﾀ ｧ00405030503 x = a ｧ y = a. 030607 閨06 09 07 閨0308 04 縺0308 0505 ｧ08 ﾀ ｧ004050305 0503 0609 09 罍ｧ040308, 09 縺03006 03 ﾀ03030308, 00605, ｧ0303 jxj >a ｧjyj >a. 07 罍ｧ04 07 ﾀ03 ﾀ0503 ｧ05 ｧ 060503 縺0608 ﾀ ｧ000605 罍ｧ 00308 040605040608, 04 罍ｧ0300 ｧ07 060 ｧ08 ﾀ07 09 ﾀ ｧ004050305 0 x = a ｧ y = a. 987. ) y = r a, x = x ; 030300 ﾀ0504 ﾀ0403000307060 ｧ07,a <x6a. ｯ03 09 ｧ 0406 縺06 ﾀ06 060 ｧ07 ﾀ x + a 03 ﾀ0406 Ox: y =0, x =0, x = a. 05 ﾀ0303 07 0503 09 (0; 0) 06 0600. 罍ﾀ ｧ004050305 x =,a. 07 罍ｧ04 07 ﾀ05 縺030703 ｧ ｧ 04030006 0605 ﾀ07 0406 縺0605 ｧ03 0 ｧ060 罍ｧ ﾀ. 84 0403 03 ﾀ ｧ Oy ｧ ﾀ0060060 ｧ060 05060 03 ﾀ ｧ Oy 0400060403 0 a. ) 050300 ﾀ05 ｧ ﾀ0403000307060 ｧ07: x > a; x<,a ｧ x =0. ｯ03 09 (0; 0) ｧ 0300 ｧ 縺0304 00 07. 罍ﾀ ｧ00405030503 x =,a, y = a, x ｧ y = x, a. 7 罍ｧ x =, a( 5+),; 6a y 05 3;3a. 988. ) y =,x =4; )y = x. 989. ) y = R; )y =0 ｧ y =,x. 990. ) y =;)y = 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 0403 04 05, 003 006 0305 ｧ03 060 07; 3) y = ｧ 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 0403 04 05 ｧ 0305 ｧ03 06-0 07; 4) y = x, 4=3 0305 ｧ03 060 07, y = x 05060300605 罍ｧ 06 ﾀ090306 006 ﾀ0503 0503 0609 0403 04 05. 99. x =3 +y =3 = a =3. 99. y =, x3 x +. 993. (x +y ) = = 4a xy. 994. ｮ0600608 ﾀ050403 05 06090503 罍ｧ08 y = x tg, gx b cos. 0ｷ308 0305 ｧ03 060 07 (04 030300 080306 0304 ﾀ003 ﾀ05 ｧ09) y = b g, gx b. 995. ) x + y = = ;)y =4x; 3)y =. 996. y =4(x +). 997. x =3 + y =3 = = l =3. 998. y =,4x =3. 999. x +4y, z =3. 000. xy 0 + yx 0 = =zz 0. 00. xy 0 z 0 + yx 0 z 0 + zx 0 y 0 =3a 3. 00. xx 0 a + yy 0 b, zz 0 c =. 003. x + y, z = 9. 004. x, 3 = y, 4 = z, 5 ;040503 0906 (0; 0; 0). 3 4,5

3050504060503 3 005. cos =, cos = cos = 3. 006. y = 0, x + z + = 0; 04030406 縺08003 ﾀ0504 ｧ 0303 07060 0 罍ｧ ﾀ. 45, ﾀ. 303. 009. 07 ﾀ 050600040 07 040003 ﾀ- 0903 ﾀ0504 x, y +z = a. ｭ06 ﾀ ﾀ0503070 ｧ06 0305 0 00 04003 a. 060600 ｧ- 6 0903 ｧ 04030406 縺08003 ﾀ0504 0800 ｧ00608 05 0709. 7 縺0700306 00 ｧ0 ｧ ｧ 04030006 0605 ﾀ07 04 ﾀ06 06-0 ｧ0708 z = h. 7 罍ｧ z = 0 y = 0, 04 罍ｧ z = a x =0, 04 罍ｧ z = 3a 4 = 0, x + y, z = a. x, 0. 4 4 y = x, 04 罍ｧ z = a y =,x, 04 罍ｧ z = a y =0( 罍ｧ ﾀ. 58). 00. z = 4 = y, 3 3 = z 5. 03. cos = 3 ; cos =, 3 ; cos =, 3. 04. 70003 ﾀ0903 ﾀ0504 z + y, x = a, = a 3. 06. ) z =4;)x +y +z =6. 07. grad z =,xi,yj =,(i +j). 08. ) grad z =,i + j ; ) grad z = x = i + j x. 09. grad h = s,x i, j. x + y 00. tg ' = jgrad zj = = 4xy 0 du = 0; 79. 0. = 4 dl. 0. du dl =+ ; grad u =i +j + +k, jgrad uj = 3. 03. grad u = 6 = 4i. 04. a + b + c. 05. grad z =0; 3i,0; 64j, jgrad zj = =0; 3 5. 06. du yz + xz + xy = = dl 3 7 ｧ ﾀ. 58 = 5 3. 07. grad u =(xi + yj, zk), jgrad uj =z. 08. grad u = xi + yj + zk 3 =, jgrad uj = 04 0006030308 0503 0906. 09., u a + b + c. 030. z min =, 04 罍ｧ x =,4, y =. 03. z max = 04 罍ｧ x = = y = 4. 03. z min = 0 04 罍ｧ x =, y =,. 033. 70605 0509 ﾀ05 縺06-0060. 034. z min =, e 04 罍ｧ x =,, y =0. 035. z max = 3 3 04 罍ｧ x = y = 3. 036. z min = 04 罍ｧ x = y =. 037. z max =,4 04 罍ｧ x = y =, ｧ z min =404 罍ｧ x = y =. 038. x = y = 3 V, z =0; 5 3 V. 039. (8=5; 3=5), (,8=5;,3=5). 040. 70607003 0 0805 ｧ 0 ｧ0 ｧ0060 070600909 ｧ ｧ

33 050504060503 z = d = x +(y, ) 04 罍ｧ 06 ﾀ000304 ｧ ｧ x, y, 4=0. 0ｷ3 ﾀ09030 07 0503 09 ( 5; ). 04. R =, H =. 04. ) 6 506 縺00 ｧ003 (3;,) ｧ (0; ); ) 0006 閨030007060 04 縺030805 ｧ 05 09, 05030303 sin : sin = v : v,09 09 050503 ｧ 04 縺03 ｧ ﾀ- 0803 陦ｧ05 04 04 罍ｧ 縺03 閨06. 043. z min =904 罍ｧ x =0 ｧ y =3. 044. z min =004 罍ｧ x = y =. 045. z min =004 罍ｧ x =0 ｧ y =0. 046. z min =004 罍ｧ x =, y =4. 047. z max =04 罍ｧ x = y =, z min =, 04 罍ｧ x =,y =. 048. V =8. 049. ) 70607003 0 0805 ｧ 0 ｧ0 ｧ0060 d = x, y +4 ｧ00 ｧ 0 ｧ0 ｧ- 0060 z = x r, y +404 罍ｧ 06 ﾀ000304 ｧ ｧ 4x, y =0; ｧ ﾀ09030 07 0503 09 (; ); )ab. S 050. R = x3. 05. ｰ 040060 ｧ07 ｧ0050605 000400308 09 罍ｧ040308: ) y = 3 3 ; ) y = x 3 ;3)y =, x3 3. 053. xy0 = y. 054. ) y, x = xyy 0 ; ) x + y = xy 0. 057. y = Cx, y =,x. 058. xy = C, xy =,8. 059. x + y = C, x + y =0. 060. y = Ce x, y =4e x+. 06. y = = Ce =x. 06. x+y =lnc(x+)(y+). 063. r = Ce =' +a. 064. s = = t, +Ct. 065. y = Ce x, y = e x,. 066. y = C sin x, ; t y = sin x,. 067. x + = C; y =,x. 068. 05030 ｧ06 ｧ0050605 0003: y ) y = C(x, 4); )y = C cos x. 6 5 ﾀ06 ｧ0050605 000400306 09 罍ｧ040306 0406 縺04030503 06 04-0060 ｧ07 0406 縺06 ﾀ0609 0605 03 ﾀ04Ox 04 罍ｧ x =, 040503 縺030503 04 罍ｧ x =(n, ) x x (03 ﾀ03030306 0503 09 ｧ). 069. y = x3 3. 070. y= a +y 0, 03050906 r y y = a +y 0, y 0 =, ; 0403000307 ｧ0 y = a ch u, 050305 a sh u a u 0 = sh u. 0505 ﾀ06 : ) sh u =0, ch u =, y = a; )adu =, au = (x + C), y = a ch u = a ch x + C ; 04 罍ｧ x =0 y = a ｧ C =0.0ｷ305 09, a ｧ00 ｧ y = a ch x a 0906040 07 00 ｧ0 ｧ07, ｧ00 ｧ y = a 04 縺070 07. 07. y = ax. 07. y =4(x +). 073. 6ｦ0 40 0 ｧ0. 7 06 00 06 0 ｧ 06. 706 ﾀ0504 06 縺06 t ﾀ0609060 050600406 0506 050600 0306 閨0605 T ; dt dt =,k(t, 0 C), 05 閨06 k 040309 006 ｧ 0406 ﾀ05-00308 0903050707 ｧ09 ｧ06005 04 縺030403 縺09 ｧ030 0004003 ﾀ05 ｧ; ln (T, 0 C) =,kt + C; 04 罍ｧ t =0 T = 00 C, 0403050503006 C =ln80 80 C C, kt =ln T, 0 ; 0403 陦ﾀ05 04 ｧ04 C ﾀ06 T =5 C ｧ T =60 C ｧ 閨0600 ｧ04 0403 00060003, ｧ ﾀ090006 ｧ0 006 ｧ 0406 ﾀ05kt 00306 k: k 0 0 = ln 6 ln, t = 400 ｧ0. 074. P X i =,H + T cos = 0, 0

3050504060503 33 P Yi =,x + T sin = 0,03050906 tg = dy = x H, y = H x + C (04 030300 ). 075. ｰ 040060 ｧ06 09 ﾀ 0506000400308 Y, y = y 0 (X, x). 7030003-07 ｧ04 Y =0, 0 08 閨060 03 ﾀ09 ｧ ﾀ ﾀ06 0503 09 ｧ A 0406 縺06 ﾀ06 060 ｧ07 09 ﾀ 0506000400308 ﾀ 03 ﾀ0406 Ox: X A = x, y y 0. 70306 ﾀ000304 ｧ06 X A =x, x =, y y0 ; 縺0600 ｧ04 050503 陦ｧ070706- 縺06009 ｧ 000400306 06 040060 ｧ06, 0 08 閨060 ｧ ﾀ090300606 09 罍ｧ040606 xy =,a (05 ｧ0406 縺030300 ). 076. x +y = C. 077. y, x = C. 078. x +3y = 3a. 079. y = Cx 4. 080. y = Ce,=x. 08. y = Cx,. 08. y = ( + x) = C(x + x + a ). 083. y = C, x. 084. r = C cos ', r =, cos '. +Cx 085. y = x ln x, x + C, y = x ln x, x +. 086. y = C +x x + +x, +x y = x + +x. 087. xy =,. 088. y = aex=a. 089. y = x, x. 090. x y = C. 09. 7 陦ｧ06 ﾀ-0406090503 OM = x + y,0305 縺06 0309 003-0 00 ｧ MN = y cos = y +tg = y +y 0. 0ｷ3 ﾀ09030 07 09 罍ｧ04 07 ｧ00 ｧ x +y = C (0309 縺0607003 ﾀ0504), ｧ00 ｧ x,y = C (05 ｧ0406 縺030300 ). 09. y = Cx. 093. y, x = Ce x=(y,x). 094. x, y = Cx. 095. s = t ln C t. 096. y = Cx 3, x. 097. y = C, e,x. x 098. y = 099. y = x ln Cx. 00. y = = C, cos x cos x. ex x + C. 0. sin y +lnx = C. 0. y = x x C, ln x. 03. y = lnx + C x. 04. y3 = 3 x + C x 3. 05. y = = x,. 06. s = Ct + t, s =t + t. 07. y = xecx, y = xe,x=. 08. (x,y) = Cy. 09. x +y =Cy. 0. i = kt R + kl R (e,rt=l,).. 703000307 ｧ04 X = 0 0406 040060 ｧ ｧ 09 ﾀ 0506000400308 Y, y = y 0 (X, x), 0 08 閨060 Y 0 =,ON = y, xy 0, ON = xy 0, y = OM = x + y. 0505 ﾀ06 y = x, C. 6ｦ006 縺09 0003 閨030007003 03030504 04 03030003 ｧ 閨030 04 0060 ｧ07. C. y = Cxe,y=x. 3. y = ln C(x + a + x ). 4. 7 罍ｧ x>0 r a + x y r x = ln C y x, 04 罍ｧ x< 0 x, C = lncx. 5. y = +. x 3 x + ln C tg (x=) 6. y =+. 7. s = t 3 (ln t, ) + Ct. 8. y = cos x

334 050504060503 = +Ce x. 9. y = (sin x, ) + Ce, sin x. 0. y = = x, 3x.. y3 = x+ce,x, y 3 = x,e,x.. y = 3. (x, a) + y = a. 4. y = 6. xy = y4 4 + C. x 7. y + y 9. s = x, Cx, y = 3, x,. ln Cx x. 5. y = x(cy, ). = C. 8. y = cos x + C sin x. t C + t, t ln t. 30. x y +lnx = C. 3. s = Ct, t. 3. y = x + Cx. 33. sin y = x + C x. 34. y = x C +e,x=. 35. 4x + y = Cx. 36. x 3 e y, y = C. 37. y + xe,y = C. 38. x cos y + y = C. 39. = x ; x + y = C. 40. ln = ln cos y; x x sin y +0; 5 cos y = C. 4. = e,x ; y =(C, x)e x. 4. = = sin y ; x sin y +x3 = C. 43. x 3 +xy,3y = C. 44. x 3 y,x y +3y 4 = = C. 45. x cos y + x = C. 46. = y ; xy, ln y =0. 47. = x ; 4 y = Cx 3 +x. 48. = e,y ; e,y cos x = C +x. 49. ln =, ln x, = = x ; x sin y +y ln x = C. 50. y =(C x). ｪ06 縺06 0503 0906 M (; 4) 04 縺030803- M ;! 閨0705 09 罍ｧ040306 y =(+x) ｧ y =(3,x). 5. y = sin (C x). ｪ06 縺06 0503 0906 04 縺030803 閨0705 09 罍ｧ040306 y = sin x, ｧ y = sin 4 3 4, x. 5. y = Cx + ; 03 ﾀ03030306 ｧ0050605 0003 y = x. 53. ) y = x + C C ｧ x + y = C ;)x r + y x = C ｧ00 ｧ (y, C) =4Cx. 05 ﾀ03-030306 ｧ0050605 0003 x =0 ｧ y =,x. 050300 ﾀ0504 ﾀ0403000307060 ｧ07 04 030300: 04 罍ｧ x > 0 y >,x, 04 罍ｧ x < 0 y <,x. 7 03030003 09 ﾀ 0605 ﾀ07 03 ﾀ ｧ Oy ｧ (x, C) 04 縺0700308 y =,x. 54. ) y = + ; 03 ﾀ03030308 ｧ0050605 00 y =; 4 ) x =,, y =, + C. 55. ) y = (C + x +) ;03 ﾀ03-030308 ｧ0050605 00 y = 0; ) x = Ct, t 3, y = Ct, 3t, 05 閨06 t = ; 3) Cy =(x, C) ; 03 ﾀ03030306 ｧ0050605 0003 y =0 ｧ y =,4x. 56. ) y = = Cx, C ; 03 ﾀ03030308 ｧ0050605 00 y = x 4 ;)y = Cx, a +C ; 03 ﾀ03030308 ｧ0050605 00 x + y = a ;3)y = Cx + C ; 03 ﾀ03030308 ｧ0050605 00 y =; 5x=3.

3(x + C) 57. y =, ; 06 縺06 M 4 050504060503 35 ; 3 4 04 縺0308 閨0605 閨040609 罍ｧ040306: y =, x ｧ y = x, x 4. 58. ) x =+ 3 +C, y = + 3 ;)x +(y+c) = a. 59. y =, x 4 + Cx+ C ; y =, x. 60. ) y = Cx+ ; 03 ﾀ03030308 ｧ00506- C (x 05 00 y =4x; )y = C(x +)+C +) ; y =,. 6. 0505 縺06 09 ｧ 09 ﾀ - 4 0506000400308 Y,y = y 0 (X,x) 0 03 ﾀ0708 090303 罍陦ｧ0 05: X A = x, y y 0, Y B = y,xy 0. 703 06 ﾀ000304 ｧ06 X A Y B =a ; (y, xy 0 ) =,4a y 0, y = xy 0,4a y 0 06 040060 ｧ06 070006 縺03. 6ｧ80603 07 04 縺070 07 ﾀ0600608 ﾀ0504 y =,Cx a C, 05 090706 09 罍ｧ04 07, 0304 縺06 閨060007060 07 03 ﾀ0303030 ｧ0050605 00030 xy = a, 0605 縺0600060 ｧ06 ｧ. 6. 7 030300 (y, x, a) =4ax. 63. ) y =3lnx +x, 6x +6; ) y =, cos x; 3)y = C x + x arctg x, ln +x + C. 64. y = = x + C ln x + C. 65. y =,C x + C. 66. y = C sin x, x,, sin x + C. 67. y 3 + C y + C =3x. 68. y = C x(ln x, ) + C. 69. ctg y = C, C x. 70. ) y = e x (x, ) + C x + C ;)y = = x arctg + C (04 罍ｧ C > 0), ln C C,C x,,c 4 x +,C + C (04 罍ｧ C < 0), C, x (04 罍ｧ C =0). 7. y 00 = P (l, x). 7 罍ｧ x =0 y =0 ｧ EI y 0 =0, y = P lx, x3 06 040060 ｧ06 09 罍ｧ040308 ｧ 05 ｧ03. 7. C y = EI 3 = (C x + C ) (x, b) +. 73. y = a ch = a, e (x,b)=a + e,(x,b)=a. 4 a 74. y = x3 6. 75. y = C x + C, ln cos x; ﾀ0500308 ｧ0050605 00 y = =, ln cos x. 76. y = x3, x 4 + C arctg x + C. 77. C y = =+(C x + C ). 78. y =(C x + C ). 79. s =, t 4 + C ln t + + C. 80. 4(C y, ) = (C x + C ). 8. y = C, C cos x,, x. 8. ｮ0. 77. 83. y =, ln cos x. 84. y = C e x + C e 3x. 85. y =(C + C x)e x. 86. y = e x (A cos 3x + B sin 3x). 87. y = = C e x + C e,x = A chx + B sh x. 88. y = A cos x + B sin x = = a sin (x + '). 89. y = C + C e,4x. 90. x = C e t + C e,4t. 9. = A cos ' + B sin '. 9. s = e,t (A cos t + B sin t); s =

336 050504060503 = e,t (cos t+ sin t). 93. y = C e x +(C +C 3 x)e x. 94. y = C chx + + C sh x + C 3 cos x + C 4 sin x. 95. y = C e x + e,x (C cos x 3+ + C 3 sin x 3). 96. y =(C +C x+c 3 x )e,ax. 97. y = A sin x sh x + + B sin x ch x + C cos x sh x + D cos x ch x. 98. y = A ch x + B sh x + r + C cos x + D sin x g. 99. 05050900030060 ｧ06 x = a sin l (t, t 0), 0406 罍ｧ03 r r l g r a T = g. 00. x = a cos a t, 0406 罍ｧ03 T = g r. 0. x = g = ae,kt sin (!t + '), 05 閨06! = l, k 4. 0. y = C e,x + C e,x. 03. y =(C x + C )e ax. 04. y = e,x (C cos x + C sin x). 05. x = = C e 3t + C e,t. 06. x = C cos!t + C sin!t. 07. s = C + + C e,at. 08. x = e,t (A cos t +B sin t ). 09. y = C e,x + +(C x+c 3 )e x. 0. y = C e x +C e,x +C 3 cos x+c 4 sin x.. y = = (C + C x) cos x +(C 3 + C 4 x) sin x.. y = ex, e,x = shx. 4. y = C e x + C e,x, x 3, 3x. 5. y = C e,x + C e,x + +0; 5 cos 4, x. 6. y = C cos x + C sin x + x + e x. 7. y = = C + C e,3x + 3 x, x. 8. y = e,x (C cos x + C sin x) +x,, 8x+7. 9. y = C e x +(C,x)e x. 0. x = A sin k(t, t 0 ),t cos kt.. y = C e x + C e,x, (x, )e,x.. y = C + C e x, x3 6. 3. y = e,x + xe,x + C e,x + C e,3x. 4. x = e,kt (C cos kt + + C sin kt)+sin kt, cos kt. 5. y = C +C x+(c 3 +x)e,x +x 3,3x. C, x e,3x + C 3 cos 3x + C 4 sin 3x. 7. x = C + 4 6. y = C e 3x + + C cos t + C 3 sin t + t 3, 6t. 8. y = C + x + C 3 sin x 3)e x. 9. ) x = C + C t + t e,x +(C cos x 3+ e,t ;)x = A cos t a + + B sin t a + a. 30. 6ｦ0 閨06 ﾀ04 y = cos x, y = sin x, w =; A =, x +C ; B = 4 ln sin x + C ｧ y = C, x cos x + C + 4 ln sin x sin x. 3. y = (C + ln cos x) cos x +(C +x) sin x e x. 3. y =(C, ln x + x + 4 + C x)e x. 33. y = C cos x+c sin x,cos x ln tg. 34. ) y = = C + C e,x, ( + e,x )ln(+e x )+x; )y = e,x C + C x +. x 35. x = a(e,t + t, ). 36. y = C e x + C e,x, 3(x + x +; 5).

3050504060503 37 37. y = C e x +C e 3x + 6 (5 cos 3x,sin 3x). 38. y =(C x + C )e,x + + 4 ex. 39. y = e,x= C cos 3x + C sin 3x, 6 cos x + 8 sin x. C, x e,x. 4. s = e,t (C cos t + C sin t)+(t, ) 3. 43. ) y = e mx (C + C x)+ 40. y = C e x= + C e,x=, x 3. 4. y = C e x + cos mx + m ;)y = C e x=n + C e,x=n, n + C cos x + D sin x, x cos x. 45. y =. 44. y = A cos x + B sin x + C + C x + C 3 x + x3 e x. 6 x ln x 46. y =, 3x 4 + C + C x e,x. 47. ) y = C sin x + + C cos x + cos x ;)y =(C, ln j sin xj) cos x + C, x, ctg x sin x. 48. y = C + 4, x + x arcsin x + C x e x. 49. y = C, (x +)e,x x + =. 50. y =+Ccos x. 5. y = x( + C, x ), x 00 ｧ0060800306. 5. y = C +. 53. s = et + C. +x t 54. y = = Cx,. 55. Cy = x(c x,). 56. y = x ln x,x+c ln x+c. 57. y(c, C x)=. 58. y = C e mx + C, x m = lnx + C ln x. 60. y = xec=x,. 6. y = e,mx. 59. y = x + Ce x. 6. y = =(C + C x)e x + C 3 + x3 3 +x +6x. 63. C y =+C e Cx. 64. s = = C e t +e,t (C +C 3 t), sin t. 65. ) s =(t +C)tg t ;)y = Cx,. sin x + C cos x 66. ) y = ;)y = e,x C + x + C e x= cos x 3 + x 3 + C 3 e x= sin x 3. 67. ) y =(C, ln +e x )e x +(C + arctg e x )e x ; ) y = C e cx + C e, cx ｧ y = C x + C. 68. a g x = A cos 0 0g t + B sin 0 0g d x + 000x =0, dt t, 0406 罍ｧ03 T = a a a 5 dt. 69. 0g dr = =, k 4r ; T = k 8r + C; k ｧ C 0 0803 陦ｧ0 ｧ 06 ﾀ000304 ｧ08: 0 k C= 8 a + C ｧ 00 C= k 60 C a + C; T =, 60 C=40 C. 70. ) y = 8 a r = C x + C x, + C 3 x 3 ;)y = C x + C x ;3)y = C x n + C x,(n+).

338 050504060503 7. ) y = x, (C +C ln x); )y = C cos ln x+c sin ln x. 7. ) y = = 5x + C x, + C ;) y = C x 3 + C 3 x, lnx +. 73. ) y = 3 = C x + C x, 4x ln x; ) y = C + C ln x +ln 3 x. 74. ) y = x x 3 = 6 + C x + C x ;)y = x +C cos ln x+c sin ln x. 75.x = C e t + + C e,3t, y =, dt = C e t,3c e,3t. 76. x = e t +C +C e,t, y = e t + + C,C e,t. 77. x =e,t +C e t +C e,t, y =3e,t +3C e t +C e,t. 78. x = e t + C e 3t + C e,3t + C 3 cos (t + '). 79. x = e,t (, t). 80. x = C e t + C e,t + t ch t. 8. ) u = '(x)+ (y); )u = y'(x) + + (x); 3)u = x'(y)+ (x); 4)u = ax ln y+bxy+'(x)+ (y). 8. z = = y (x+y,). 83. ｪ05030303 06 040060 ｧ06 A @ u @x +B @ u @x@y +C @ u @y = F 04 罍ｧ- 0406 ﾀ05 ｧ 0909 0030 ｧ 06 ﾀ0903006 04 ｧ 閨06, 00607003 縺0600 ｧ0504 08 090506 罍ｧ ﾀ05 ｧ 06 ﾀ090306 06 04006-0 ｧ06 Ady, Bdy+ C =0;04 閨040608 060503 ｧ0050605 00 08 '(x; y) = ｧ (x; y) = 04 縺03 ｧ 0403000400306 0403 ﾀ050307000306 ｧ 04 罍ｧ0070504 003040306 0406 縺06-006000306 ｧ 04 縺060303 0304 0504 09 0505 ｧ0 00304030 0406 縺0600600030 000306 06 040060 ｧ06 ( ﾀ0. ｧ 94 ｧ 94). 6 5 0 00060 04 罍ｧ006 縺06 00607003 縺0600 ｧ0504 06 040060 ｧ06 +4 dy +3dy =0,03050906 dy + =0, dy +3 =0, y + x =, y +3x =. 6 5 003040308 0406 縺06006000308 06 040060 ｧ06 04 罍ｧ00605 04 ｧ @ u @ @ = 0. 0505 ﾀ06 u = '() + () ='(y + x) + (y +3x). 84. 6ｦ 090506 罍ｧ ﾀ05 ｧ 06- ﾀ090306 06 040060 ｧ06 x dy, xy dy + y =0, ｧ00 ｧ (xdy, y) =0, y y ｧ00 ｧ d = 0; =. 70600060 ｧ07 0400306; 04 罍ｧ0 ｧ0 060 y. 0ｷ305 09, x x y 08 090506 罍ｧ ﾀ05 ｧ09 ｧ: = ｧ y =. ｰ 040060 ｧ06 04 罍ｧ00605 04 ｧ ( ﾀ0. ｧ x 944 ｧ 945) @ u =0; u @ = '() + (), ｧ00 ｧ u = y'(y=x) + (y=x). 85. u = y'(y +x)+ (y +x). 86. u = xy + sin y cos x. 87. ( ｮ0. 06 944.) u = y ln x +y +. 88. u = xt' x t + (xt); ﾀ05-00306 縺0600060 ｧ06 u = x ( + t 3 ). 89. u = e,x '(x, t) + (x); ﾀ05t 00306 縺0600060 ｧ06 u = (x, t)e,t, x. 90. ｪ ﾀ0500306 縺0600060 ｧ06 u = xat + + f(x, 3 a3 t 3 at) +f(x + at). 9. u = + a, 4ln 3; 8. 93. ) 3 3 x+at x,at F (z) dz. 9. 6, 5 9a ;)4. 94.. 95. 6. 96., e.

3 a x a a 97. ) dy = dy ) a 0 dy 0 0 a,y a,y = 0 a 0 y 050504060503 39 = a ; a,x a,x dy = a, 4 ; 3) a 4.,x y,y 98. ) dy = dy + dy = 6 ; ) 0, dy 0 0 y,4 x = 0,4 0 閨04 00600400060503 ﾀ0605006005 0 0, 4+x 0 dy = 6 4 3, 3 0 3. 99. 4 + a. a ; 457a. 30. 300. 70003-3a ln. 30. 868 5 a. 303. 3 8 a. 304. 4; 5. 305. a 6. 306.,. 307. 9 a. 308. 8 +9 3. 309., b x a. 30. 7ln. 3. ) dy = 4 a a b b (b, a a,y a x =a a) = dy = ; ) dy = dy+ ay a a y a,x 0 + dy = a (3, ) ; 3) dy = dy + a 0 0 x 0 0 8 8,y + dy = 40 3. 3. ; a. 33. (3; 4; 8). 34. 8 5 ; a. 4 0 35. 0; 4a. 36. 0; 56a. 38. 7a4 a4 a4. 39.. 30. 3 35 96 4 6. 3. a4 a4 88a4 3a. 3.. 33. 8 05. 34. 5 ; 3a. 35. 0; 4b. 8 3 36. a 4 30. 37. 3. 38. ab(a + b ) 4 0 8,x 0 4. 39. 47; 5. 330. y =4 35a 4 6. 33. 4 3. 33. 79 60 a3. 333. ｮ06 060 ｧ06 040003 ﾀ0903 ﾀ050406 z = h, x + y = = a(a, h) 04 000006000400306 04 縺0700306, 05. 06. 04030406 縺08003 ﾀ0504 09 ｧ00 ｧ0 陦罍ｧ-

330 050504060503 06 ﾀ09 07 ( 罍ｧ ﾀ. 59). 0ｷ3 ﾀ090300308 03030060 V = ( 罍ｧ ﾀ. 60). 335. ( ｮ0. 罍ｧ ﾀ. 46, ﾀ. 303.) a 0 a,x 0 zdy= a3. 334. 6 3 a3 8 9 a3. 336. a3 a3. 337. 3. 7 ｧ ﾀ. 59 7 ｧ ﾀ. 60 338. 3a 3. 339. V =4 = 340. a3. 34. 4 3a 3. 34. 4a3 0 m cos 'd' 344. 6 5 a3. 345. abc. 346. abc, e a 0 r dr = 4m3 3 ( 罍ｧ ﾀ. 6). 9 (3, 4) ( 罍ｧ ﾀ. 6). 343. a 3 ( 罍ｧ ﾀ. 58).. 347. 4a3 35. 348. 8 5 a3. 7 ｧ ﾀ. 6 7 ｧ ﾀ. 6

3349. V = 350. V =4 0 3a x ax 050504060503 3 88 zdy= ( 罍ｧ ﾀ. 63). 05 4ax, y dy =3a 3 (4, 3 3) ( 罍ｧ ﾀ. 64). 0 ax a 35. V =8 35. V =4 7 ｧ ﾀ. 63 7 ｧ ﾀ. 64 0 a 0 b (b=a) a,x a, x a h 0 0 dy = 6ab. 3 ya h, x dy = a h 04060 0400030 陦ｧ 03 ﾀ00304 0 ｧ07 0903-7 ｧ ﾀ. 65 003 ｧ, 0600030706000308 0 0403000304 ｧ006 0403 ﾀ030503 ( 罍ｧ ﾀ. 65). 353. 8 05 a3. 354. 8. 355. a 3. 356. 8 ln ( ﾀ0. 罍ｧ ﾀ. 45, ﾀ. 303). 357. 3 6 a3. 358. 5a3 6.

33 050504060503 359. 4abc. 360. 3. 36. 8 3 3 a. 36. a. 363. a 3 (, ). 364.. 365. 8a. 366. 4a 4 (, ). 367. 3 a. x 368. + y + z = dy = z 80 R sin a; 04 罍ｧ =60 ｧ =30 (S) = R a3. 369. 陦ｧ06 ﾀ ﾀ06 060 ｧ07 r = 3 a. 370. a3 6 3 (, ). 37. a a,x dy a,x,y 0 0 0 a 5 375. 4. 376. a 5. 379. zdz= a4 4. 373. a 4 ; a 4 ; a 4 377. ) a3 3. 374. 0; 0; a 3 a3 a3 ; ). 378. 60 6 (8, 7). 3 3. 380. a 3 6. 38. h 4 4. 38. a 4 0;. 383. 0; 3a. 8 a 3 3 a 5 384.. 385. 35 360. 386. 6ka,05 閨06 k 0903050707 ｧ09 ｧ06005 04 縺030403-09 ｧ030 0004003 ﾀ05 ｧ. 8 4 0403 04 縺0700308 OA; >< 0 387. (x + y) = 0403 閨060506 OA; 3 >: 0403 00030 00308 OBA: 388. ) 8; ) 4. 389. (xdy+ y)=804 030303 ｧ08 ﾀ0006 0708. 70503 04030503006, 0503 閨06 ﾀ04 @Q @x = @P @y. 390. ) ; 5a ; ) a. 39. 8a. 39. a. 393. mab. 394. 0. 396. ) 5 4 6 ; ), ; 3),. 397. a3 3 3. 398. ab. 399. = kmm kmm a. 403. Y = 8 5. 400. 3 a. 40. X =0, Y = kmm. a 40. Y =. a 404. ),6; ), 5 3. ;3),. 405. ) 3a ; ) a a ;3) 6. 408. 3 8 a. 409. 4 a3 a4. 40.. 4.. 4. 07 07-3 48 07 ｧ ﾀ050608 0703 縺0060003 040 4a 3. 43. 07 07 07 ｧ ﾀ050608 0703 縺0060003 040 a4 4 3 5 +. 49. 07 07 07 ｧ ﾀ050608 0703 縺0060003 040 6 5 a5. 4. 0; 5a 5. 4. 70605. 43. 6 7. 44. 6 7. 45. 7 ﾀ0803 陦ｧ05 ﾀ07. x 46. 7 ﾀ0803 陦ｧ05 ﾀ07. 47. ｮ0803 陦ｧ05 ﾀ07, ｧ0303 (x +) 3 = 3. 48. ｮ0803 陦ｧ05 ﾀ07, 8

3 ｧ0303 陦ｧ05 ﾀ07, ｧ0303 +x =. 49. 7 ﾀ0803 陦ｧ05 ﾀ07, ｧ0303 4 (x +), = 4 ln x x + 050504060503 33 x =. 430. ｮ0803- +x = ln. 43. ｮ0803 陦ｧ05 ﾀ07. 4 u n+ 43. ｮ0803 陦ｧ05 ﾀ07. 433. ｮ0803 陦ｧ05 ﾀ07. 434. ｮ0803 陦ｧ05 ﾀ07, ｧ0303 lim = n! u n < <. 435. 7 ﾀ0803 陦ｧ05 ﾀ07. 436. 7 ﾀ0803 陦ｧ05 ﾀ07. 437. ｮ0803 陦ｧ05 ﾀ07. 438. 7 ﾀ0803- 陦ｧ05 ﾀ07. 439. ｮ0803 陦ｧ05 ﾀ07. 440. 7 ﾀ0803 陦ｧ05 ﾀ07. 44.. 443.. 444. ｮ0803-3 陦ｧ05 ﾀ07 006 03 ﾀ03000605003. 445. ｮ0803 陦ｧ05 ﾀ07 03 ﾀ03000605003. 446. ｮ0803 陦ｧ05 ﾀ07 006 03 ﾀ03000605003. 447. ｮ0803 陦ｧ05 ﾀ07 03 ﾀ03000605003. 448. 703 ﾀ0006 0406 縺040308 0406 縺06 ﾀ05-0030409 ｧ 000600304 0 04 ｧ00060 縺07 04 04 ｧ 閨06,, 4 + 3,, 6 8 + + 5,, +:::6 5030403000 ｧ04 閨0608 ﾀ0504 ｧ07 04 ﾀ09030309 08, 04030006 ｧ0 縺07, 0006003 0 09030503 縺030503 04 閨040306 0060040006 000600304 00030503 縺07. 703 ﾀ0006 040503 縺0308 0406 縺06 ﾀ05-0030409 ｧ 000600304 04 縺060303 06060 n-06 05 縺03080906 000600304: 4n, 3 + 4n,, n = = 4n, 3, 4n, + 4n,, 4n + 4n,, ; 04 罍ｧ n =; ; 3; ::: 4n 0406 縺040306 060503 縺06 00060 0303 060605 000308 縺07 ﾀ ﾀ06000308 S, 0403 ﾀ0006 閨0 ｧ06 閨04 縺07 ﾀ ﾀ06000308 S. 449. ｮ0803 陦ｧ05 ﾀ07. 450. 7 ﾀ0803 陦ｧ05 ﾀ07, ｧ0303 00x, 99 =. 陦ｧ05 ﾀ07, ｧ0303 45. ｮ0803 陦ｧ05 ﾀ07, ｧ0303 x +x 4 =. 45. 7 ﾀ0803-8 x, x =. 453. ｮ0803 陦ｧ05 ﾀ07. 454. ｮ0803 陦ｧ05 ﾀ07, ｧ0303 u n+ lim = <. 455. ｮ0803 陦ｧ05 ﾀ07, ｧ0303 lim n! u n n! u n+ u n 0n + = lim n! 3(0n +) = = <. 456. ｮ0803 陦ｧ05 ﾀ07. 457. ｮ0803 陦ｧ05 ﾀ07 006 03 ﾀ03000605003. 458. ｮ0803 陦ｧ05 ﾀ07 3 03 ﾀ03000605003. 459. 7 罍ｧ a> ﾀ0803 陦ｧ05 ﾀ07 03 ﾀ03000605003, 04 罍ｧ a = ﾀ0803 陦ｧ05 ﾀ07 006 03 ﾀ03000605003, 04 罍ｧ a< ﾀ0803 陦ｧ05 ﾀ07. 460.. 46.. 46. ｮ0600 4 縺07 S(x) =, x 04 罍ｧ x<, 03 ﾀ05 050309 R n = S, S n = xn, x. 7 0305- 縺06 0906 [0; =] jr n j < lg 000 < 0; 00, 09 09 050300040903 n, > n, lg ; n >.

334 050504060503 463. 707 ｧ0060605 ( x ﾀ060006 S =, (, x) = 04 罍ｧ 0 <x6 ; 0 04 罍ｧ x =0 ( (, x) n 04 罍ｧ 0 <x6 ; ｧ03 ﾀ05 050309 R n = 0 04 罍ｧ x =0: 7 罍ｧ 000603030 n 03 ﾀ05 050309 R n 0306 閨0605 030300040006, 0 04 罍ｧ006, 0;9,09 09 050300040903 x<,, n 0;9, 05.06. 0 0305 縺06 0906 [0; ] 縺07 ﾀ0803 陦ｧ05 ﾀ07 006 04003006 縺003. 7030 0305 縺06 0906 [=; ] 030 ﾀ0803 陦ｧ05 ﾀ07 04003006 縺003, ｧ0303 050305 04 罍ｧ 000603030 xjr n j < n <", 09 09 050300040903 n>, lg " lg ; 04 ﾀ05003 ﾀ05 ｧ, jr nj < 0; 0 04 罍ｧ n > 7. 464. 05 ﾀ05-050309 0 0903 06 縺06 閨06060060503 ﾀ07 縺07 0060040006 0403 003 閨060006 0406 縺04030503 030503 縺03000600030503 00060. 703050503006 0 0305 縺06 0906 [0; ] jr n (x)j < xn+ n + < 6 0;, 09 09 n + 050300040903 n +> 0 ｧ00 ｧ n > 9. 465. 707 ｧ0060605 ﾀ060006 S = ｧ03 ﾀ05 050309 R n = ( +x 3 04 罍ｧ x>0; 8 >< >: 0 04 罍ｧ x =0 ( + x 3 ) n, 04 罍ｧ x>0; 0 04 罍ｧ x =0: 7 罍ｧ 000603030 n 03 ﾀ05 050309 R n 0306 閨0605 030300040006, 0 04 罍ｧ006, 0;, 09 09 050300040903 x 3 < < n, 0,, 05.06. 04 罍ｧ x > 0 縺07 ﾀ0803 陦ｧ05 ﾀ07 006 04003006 縺003. 703 04 罍ｧ x > 030 ﾀ0803 陦ｧ05 ﾀ07 060706 04003006 縺003, ｧ0303 050305 04 罍ｧ 000603030 x > jr n j 6 6, lg " <",09 09 050300040903 n, > n, lg ; 04 ﾀ05003 ﾀ05 ｧ, jr nj < 0;00 04 罍ｧ n > >. 466. 7 罍ｧ 000603030 0060305 罍ｧ09 050600040030 x 0006003 00030503 縺07 0060040006 ( ｧ00 ｧ 04003) 000600304 ｧ ﾀ000304030503 ﾀ0803 閨070060503 ﾀ07 縺07 + 3 + 3 + 3 3 + ::: ｮ0006 閨0304 05060004003, 縺07 ﾀ0803 陦ｧ05 ﾀ07 04003006 縺003 閨0007 04 ﾀ0608 x > 0, R n (x) 0060040006 03 ﾀ05 0509 ｧ ﾀ000304030503 縺07, 05. 06. R n (x) < (=3)n, =3 = < 0;0, 09 09 3n, 050300040903 3 n, > 50, ｧ00 ｧ n > 5, 04 罍ｧ 000603030 x > 0. 467. jr n (x)j < n 6 6 0;000, 09 09 050300040903 n > 00, 04 罍ｧ 000603030 x. 468. u n =, x + n. 703050503006 S n = x, x + n ; S = n! lim S n = x x + n,, 04 罍ｧ 000603030 x 6= 0.

3050504060503 35 6 5 ﾀ05003 ﾀ05 ｧ, 04 罍ｧ x>0 R n (x) = x + n < 6 0;, 09 09 050300040903 n > 0. n 469. 7 罍ｧ 000603030 0060305 罍ｧ09 050600040030 x 0006003 00030503 縺07 0060040006 ( ｧ00 ｧ 04003) 000600304 ｧ ﾀ000304030503 ﾀ0803 閨070060503 ﾀ07 縺07 + + 4 + +::: 703050503006 縺07 8 ﾀ0803 陦ｧ05 ﾀ07 04003006 縺003 閨0007 04 ﾀ0608 x > 0, R n (x) < (=)n, = = < 0; 0, n, 09 09 050300040903 n, > 00, ｧ00 ｧ n > 8. 470.,3 6 x<3. 47., 5 6 3 3 6 x 6 5. 47., 6 x 6. 473. 縺03 ﾀ03000605003 ﾀ0803 陦ｧ05 ﾀ07 0 04 ﾀ0608 ｧ ﾀ0003040308 03 ﾀ ｧ. 474., <x6. 475., 3 <x<. 476. ) R =0; 3 ) R = e. 477.,5 6 x<3. 478. <x6. 479. 04 罍ｧ jxj <. (, x) +x 480. arctg x 04 罍ｧ jxj 6. 48. (, x) 04 罍ｧ jxj <. 48. ( + x)m. 5 5 483., <x6. 484., 3 6 x 6 3. 485.,0; 6 x<0;., x 486., 6 x 6. 487., 6 x<3. 488., 6 x<0. 489. ( + x ), x 04 罍ｧ jxj <. 490., ln (, x) 04 罍ｧ, 6 x<. 49. 04 罍ｧ jxj <. ( + x) 49. ) cos (x, ) = sin jr n (x)j = + 5 x 6 6! 4) sin xn n! cos x, + n x!, x3 3! + x5 5!, ::: +, :::; 3) xe x = x + x! + x3! + x4 mx + 3 =, m x + m4 x 4 3! 4! + cos ; ) sin x = x + 493. ln ( + e kx )=ln+ kx + k x!, k4 x 4 497. ) ln +x, x = x + x3 3 + x5 5 + ::: ; 3! + :::;, x! + x4 4! + ::: ;!, 3 x 4 + 4!, ::: + mx, m3 x 3 + m5 x 5, ::: 3! 5! 4! 3 + ::: :

336 050504060503 ) ln (, 3x + x )=ln(, x)(, x) =ln, X n= ( +,n ) xn n ; 3) ln (, x + x )=ln +x3 +x = =, x, x, x3 3, x4 4 + x5 5 + x6 6 + ::: =, 498. ln (x + +x )=x + 499. e x=a = e + x, a!a X n= X n= x n+ n 3 ::: (n, ) (,) n n! n +. (x, a) (x, a)3 +!a + 3!a 3 + ::: (x, a)n R n (x) = n!a n e +(x=a,). 500. x 3, 3x =,+3(x, ) +(x, ) 3. 50. x 4 =, 4(x +)+6(x +), 4(x +) 3 +(x +) 4. 50. x =, (, (x +)=) = =, + x + 503. ) cos x = (x +) (x +)3 + + 4 8 + ::: (x, =) (x, =),,!! + ::: = X n=, cos n x n 3 n : 04 罍ｧ, 4 <x<0: = (x, =) n, (n, ) cos ; (n, )!n, 4 04 罍ｧ0 ｧ0 07 0! 06 ﾀ000304003 040030 ( ﾀ0. ﾀ003 ﾀ0906 0 ﾀ. 74 09 06 760); ) sin 3x = 504. X n= n (3x + )n, (,). (n, )! 3 x =, 3, (x +)=, + x + (x +) + 3! 3 +! 5(x +)3 + 3 3 + :::=, + x + 3! 3 + + X n= 505. ) x =+ x ln! 5 8 ::: (3n, ) (x 3 n+ +) n+ 04 罍ｧ, <x<0: (n + )! + x ln! + :::, jr n j = xn ln n x ; n!

3 ) cos mx + =, mx 4! X = 050504060503 37 n=, m x! + ::: = (mx) n, (n, )! cos (n, ) 4 506. x 4, 4x =(x +) 4, 8(x +) 3 + 0(x +), 6(x +). (040300 05 07 0! = ): 507. cos x = = 4, 3 3 x, =3, (x, =3) 3! 3! (x, =3) + X!, 3 (x, =3) 4 4! 3 + n + 4 (x, =3) 5 5!, ::: + + 5 (x, =3) 6 6! 508. sin x 3 = n (x, ) n 3 n sin (040300 05 07 0! = ). n! n=0 509. x = + x, 4 (x, 4) 3(x, 4)3, 3! 6 +,! 9 3! 5. arcsin x = x + x3 3 + 3! x5 5 + 3 5 3 3! x7, ::: 3 5(x, 4)4, + ::: 4! r : : 7 +::: 5. 0; 99 = =, 0; 008, 0; 004 = 0; 996; 90 = 8 + 9 = 9 9 + = 9; 5. 53. 8 r = 3 5 + 5 = 5 3 + 5 5 + 75 3 0; 99 = 3, 0; 009 0; 997; + 9 3 30 = = 76 5. 55. arctg x = x,, x3 3 + x5 5, ::: 57. = 3, 3 3 + 5 3, 7 3 3 + sin x = ; 84 3 3; 4. 59. ) x 9 3 4 = x3 = C + x, 3!3 + x5 5!5, :::; e x ) x = C +lnx + x! + x! + x3 + ::: 50. (x) = 3!3 = x, x3!3 + x5!5, x7 3!7 + :::; 3 3, 3 4 x 0 e,x = 0; 49 ﾀ 040305 縺0600003 ﾀ050406

338 050504060503 < x. 5. (x) = 430 + 5 3 3 3! x7 7, :::; 5 < 0 3 +x = x + 3 x3 3, 3! x5 5 + 5 + 0; 008 ﾀ 040305 縺0600003 ﾀ050406 3 3 53 < 0; 000. 3 5. 7 縺03 陦ｧ070706 縺06009 ｧ 縺0304 04 06 040060 ｧ06 n ｧ 0403-56 ﾀ05 04 ｧ04 x =0, 04030006 ｧ0 y (n+) 0 = n(n, )y (n,) 0. 0505 ﾀ06 y 00 0 = y000 0 =0, y IV 0 =, y V 0 =3, yvi 0 =0 ｧ05.. 703 陦ﾀ05 04 ｧ04 0505 ｧ 0 060 ｧ07 04 0703 縺0060006 03 090003 縺060 y = y 0 + y0 0! x + y00 0! x + :::, 0 0803 陦ｧ0 y = + x + x4 3 4 + + x5 4 5 + x 8 + ::: 53. y =+x 3 4 7 8, x3 3 + x4, ::: 54. 7060006-6 0 ｧ060 070400070605 ﾀ07 08070600909 ｧ07 706 ﾀ ﾀ060007 006000604030503 0403 縺07 閨09 09: I 0 (x) =, x + + x4 4, x 6 4 6 + ::: 55. 3 ;005 ;005; ;00 ;0004; 0;993 0;9965; 3 0;997 0;999; 0 = 00+0 0 =0;5; 57. =6 58. = 53. s =4 3 70 4 + =4; 5; 5 40 + =;. 3 0 + 3 3 + 3! 5 5 + :::, = 0 + 4 3 + 0 3( + 0; 047 + 0; 0047) 3; 4: 3 + 5, 4 7 6 + ::: +, 3 3 + 5 3, 4 7 3 6 + ::: = X = 0 3 + a sin t + b cos tdt=4a =a ", e 3, 4 = 0 e4 n= (,) n n, 4 n + 9 n 3, e cos tdt= 3, 3 5 4 6 = : e6 5, ::: # ; 05 閨06 e 0509 ﾀ0906005 罍ｧ ﾀ ｧ050605 050000 ｧ04 ﾀ, a 060503 0303000400 07 0403000603 ﾀ04 ( ﾀ0. 00 64

3 ｧ 060503 0305040605). 533. 0;5 0 +x 3 = 050504060503 39 x + x4 4, x7!7 + ::: 0;5 0 = + + 4 65, ::: 0; 508 ﾀ 040305 縺0600003 ﾀ050406 <. 534. (x) = 4 8 7 0 = x, x 5! 4 5 + x 9 4! 4 4 9, :::; =, + ::: 0; 499805 5 0 x3 ﾀ 040305 縺0600003 ﾀ050406 < 7 0. 535. y = 3 + x7 3 7 + x 3 7 + ::: 536. 7 縺03 陦ｧ070706 縺06009 ｧ 縺0304 04 06 040060 ｧ06 n ｧ 0403 陦ﾀ05 04 ｧ04 x =0, 04030006 ｧ0 y (n+) 0 =,ny (n,) 0,0305 ﾀ06 y 0 =, y 0 0 = 0, y00 0 = 0, y000 0 =,, y0 IV = = y0 V = 0, yvi 0 = 4, :::, y =, x3 s 537. x = cos s C ds = s s 4, s y = 0 0 sin s s3 ds = C C 3! + 4 x6 6!!(C) 5 + :::,, 4 7 x9 9! + ::: 3, s 4 3!(C) 7 + :::,05 閨06 0403 ﾀ05030700 07 C = RL, R 陦ｧ06 ﾀ 09 縺060503040308 09 罍ｧ040308 ｧ L 閨00 ｧ0 0406 縺060803 閨00308 09 罍ｧ040308. 07 罍ｧ04 07 0 0304 0605 ﾀ07 0900030503 ｧ 閨0308 ( 罍ｧ ﾀ. 88, ﾀ. 336). 538. F (x + h; y + l) =x + + xy + y + h(x + y) +l(y + x) +h + hl + l. 539. x 3 +xy = = 9+(x,)+8(y,)+3(x,) +8(x,)(y,)+(y,) +(x,) 3 + +(x,)(y,). 540. ln (x, y) =x,(y+), x (y +) +x(y+), + (x, y, ) 3 + R 3,05 閨06 R 3 =. 54. sin (mx + ny) =mx + ny, 3[x +, (y + )] 3 (mx + ny)3 (mx + ny)4, + sin (mx + ny). 543. =0;, dy =,0;, 3! 4! z =(x, y) +(y, x) dy +, dy + dy =,0; 63. 544. z = =, (a, bdy) sin (ax, by),! (a, bdy) cos (ax, by) +R 3,05 閨06 R 3 = 3! (a, bdy)3 sin a(x + ), b(y + dy). 545. x y =,,, (x,)+(y+),(x,) +(x,)(y+). 546. arctg y x = y,(x,)y+::: 547. y x = + (y, )+(x, )(y, ) + + 0; +0; 0; + 0; (y, ) + :::; ; ; =; 5. 548. =,0; 0, dy =0; 0; z =yx +(x, y) dy + y + xdy, dy + 3 dy,0; 407.

3330 050504060503 549. X 4 X n= sin (n, )x. 550. n,, 4 X n= cos (n, )x. (n, ) 55. 3 + n cos nx 3 sin x +4 (,). n 55. 4, sin x sin 3x, +, ::: + 3 n= cos x cos 3x cos 5x 4 + 3 + 5 + :::. 553. sin x + 3x sin + l 3 l + 5 sin 5x l + :::. 554. + 4 cos x cos 3x + 3 + :::. 555. l 4, l cos x + 3x cos + ::: + l l 3 l 556. ) 3 4 + 4 cos x, x cos + 3x cos 33 + ) sin x + x sin + 3x sin 3 + ::: + + 4 557. u = 4l 558. u = 559. u = X n=0 X n= 560. f(x) = X n= 56. f(x) = a n cos n sin n b n sin nx l 0 n + at sin l, cos cos x + d. 0 sin x, x sin + ::: l l + 5x cos 5, 6x cos 6 + ::: ; sin x, 3x sin 33 + 5x sin 5, ::: nx sin e,(n a t)=l. l n + x; l 05 閨06 a n = l l 0 f() sin cos a n t l,05 閨06 b l n = f() sin n d. l l 0 sin x d.. : n + d: l

3 56. f(x) = 4 (, cos ) sin sin x d. 0 563. + 4 cos 3x cos 5x cos x + 3 + 5 + :::. 564. j sin xj =, 4 cos x cos 4x + 3 3 5 565. 4 cos x, sin 3x sin 5x 3 +, ::: 5 566. l, 4l cos (x=l) cos (3x=l) + 567. 3 4, cos x cos 3x + 3 + ::: cos (x=l) 568. sh l l, l cos (x=l), + l X 050504060503 33. + cos 6x 5 7 + :::. 3 + :::., sin x cos x + + l + ::: + sin (x=l) +, + l + ::: : sin (x=l) + l + ::: : n + n + 569. u = a n cos t sin x, n=0 05 閨06 a n = n + f() sin d. 0 570. f(x) = sin cos x d. 0

305066ｧ8 縺6 56ｧ8 ｭ 70ｷ3 ｭ 0ｷ36ｦ0 7 7 ｭ6 70ｷ3 ｮ6ｧ8056 50ｷ300 縺6 5 ｯ05 7 07 ｯ 7 ｭ ｯ04 ｭ03 ｰ 0ｷ36ｦ06 7 繖 70ｷ306... 8 0505 縺06 0909 ｧ ｧ... 8 0600 04. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 0 040003 ﾀ0903 ﾀ05 ｧ... 9 x. 070303 罍陦ｧ0 05030503 09 ｧ 0 04 縺0700308 ｧ 0 040003 ﾀ0903 ﾀ05 ｧ. 7 ﾀ ﾀ0503070 ｧ06 00607 閨06 閨0406007 0503 09 0 ｧ... 9 x. 6 70600060 ｧ06 0305 縺06 09 04 00030 030500300060 ｧ ｧ. 700030 閨04 05 縺060605030004-0 ｧ09 ｧ 0003050306050300040 ｧ09... x 3. ｰ 040060 ｧ06 00 ｧ0 ｧ ｧ 09 09 05060300605 罍ｧ 06 ﾀ09030503 006 ﾀ05 0503 0609... x 4. ｰ 040060 ｧ06 04 縺0700308: ) ﾀ 0605000304030 0903050707 ｧ09 ｧ06005030, ) 0303-00606, 3) 04 0305 縺06 09 080 03 ﾀ0708... 4 x 5. ｰ050300 00607 閨06 04 縺070030 ｧ. ｰ 040060 ｧ06 0406 09 04 縺0700308, 04 縺030803- 閨070 ｧ08 06 縺06 000606 0503 0906. ｰ 040060 ｧ06 04 縺0700308, 04 縺030803- 閨0700608 06 縺06 閨0406 000306 0503 09 ｧ. ｯ03 09 0406 縺06 ﾀ06 060 ｧ07 閨040608 04 縺0700308... 6 x 6. 703 縺0 000400306 06 040060 ｧ06 04 縺0700308. 7 ﾀ ﾀ0503070 ｧ06 0305 0503 09 ｧ 閨03 04 縺0700308. ｰ 040060 ｧ07 03 ｧ ﾀ ﾀ060905 罍ｧ ﾀ. ｰ 040060 ｧ06 0406 09 04 縺07-00308, 04 縺030803 閨070 ｧ08 06 縺06 0503 0906 0406 縺06 ﾀ06 060 ｧ07 閨040608 000308 04 縺0700308... 9 x 7. ｮ00600 000306 ｧ 0 04 縺0700606... x 8. 0509 縺0607003 ﾀ0504... x 9. 70000 ｧ04 ﾀ... 4 x 0. 06 ｧ0406 縺030300... 6 x. 7 030300... 9 x. 6 7 ｧ 縺060905 罍ｧ ﾀ03, 陦ｧ 00605 縺03 ｧ 09 ﾀ 0506000400306 09 09 罍ｧ04030 040503 縺030503 0403 縺07 閨09... 3 x 3. 7 縺060303 0304 0 ｧ06 閨0609 縺0503040308 090303 罍陦ｧ0 05. 7 03030003 06 = = 08 + b08 + c ｧ x = ay + by + c. 06 ｧ0406 縺030300 xy = k... 35 x 4. ｮ00600 000306 ｧ 0 09 罍ｧ040306 040503 縺030503 0403 縺07 閨09... 38 x 5. 050300606 06 040060 ｧ06 00 ｧ0 ｧ ｧ 040503 縺030503 0403 縺07 閨09... 40 x 6. 7030007 縺00306 090303 罍陦ｧ0 0503... 44 x 7. 縺00050603 ｧ 06 ﾀ09 ｧ06 09 罍ｧ040306 05 縺060504060503 ｧ 0403 ﾀ00 ｧ08 0403 縺07 閨090304.. 48 x 8. ｯ 0 ﾀ09060 閨0600500306 09 罍ｧ040306... 49 0600 04. 6 506090503 縺0 07 00050603... 5 x. ｮ000307060 ｧ06 0406090503 縺0304. ｰ000307060 ｧ06 0406090503 0 ﾀ09 0007... 5 x. 7 縺07003060503000400306 090303 罍陦ｧ0 0503 0503 09 ｧ ｧ 0406090503 04 04 縺03 ﾀ05 0 ﾀ050406... 53 x 3. ｮ09 0007 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 閨040608 0406090503 縺0304... 55 x 4. 6 506090503 縺00306 04 縺03 ｧ 0406 閨060 ｧ06 閨040608 0406090503 縺0304... 58 x 5. ｮ00600 000306 04 縺03 ｧ 0406 閨060 ｧ06 05 縺0608 0406090503 縺0304... 60

34 050500 0400060 ｧ06 0600 04 3. 縺0 00 ｧ05 ｧ 06 ﾀ09 07 05060300605 罍ｧ07 04 04 縺03 ﾀ05 0 ﾀ050406... 6 x. ｰ 040060 ｧ06 040003 ﾀ0903 ﾀ05 ｧ... 6 x. 05 ﾀ0030400306 ｧ 0 040003 ﾀ0903 ﾀ0504... 63 x 3. ｰ 040060 ｧ07 04 縺0700308... 65 x 4. 7 縺070 07 ｧ 040003 ﾀ0903 ﾀ0504... 68 x 5. ｮ0706 罍ｧ 06 ﾀ09 ｧ06 ｧ 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ06 04030406 縺08003 ﾀ05 ｧ... 70 x 6. 07030 ｧ 06 ﾀ09 ｧ06 04030406 縺08003 ﾀ05 ｧ ｧ 04030406 縺08003 ﾀ05 ｧ 04 0060 ｧ07... 7 x 7. 70000 ｧ04 ﾀ03 ｧ, 05 ｧ0406 縺03030003 ｧ 閨03 ｧ 04 03030003 ｧ 閨03... 74 0600 04 4. 6 503 ﾀ00 07 00050603... 78 x. 0504 縺06 閨0600 ｧ050600 ｧ... 78 x. ｮ ｧ ﾀ0506003 00 ｧ0060800308 06 040060 ｧ08.... 80 x 3. 0703004000609 ﾀ00306 ｧ ﾀ00... 83 x 4. ｰ 040060 ｧ07 0403 ﾀ00 ｧ08 ﾀ0506040600608 ｧ 04 罍ｧ0300 ｧ0706000306 縺0600060 ｧ06 06 040060 ｧ08... 86 0600 04 5. 6 50406 閨060 ｧ06 04 0 00 ｧ... 90 x. 706 縺06006000306 040600 ｧ ｧ003 ｧ 070600909 ｧ ｧ... 90 x. 7 縺06 閨060003 0403 ﾀ0006 閨0304 05060004003 ﾀ05 ｧ ｧ070600909 ｧ ｧ. 706 ﾀ0903006 003 0-000306 ｧ 0306 ﾀ0903006 003 0303000400 ｧ06... 93 x 3. ｮ040308 ﾀ0504 04 縺06 閨06000304. 7 ﾀ09 縺0305 ｧ06 0060304 縺06 閨0600060003 ﾀ050608 04 ｧ 0 ｧ... 0 97 x 4. 7 縺06 閨0600 030500300060 ｧ07 sin 04 罍ｧ! 0... 98 x 5. 7060304 縺06 閨0600060003 ﾀ05 ｧ 04 ｧ, ｧ 0... 99 x 6. ｮ00600 000306 04 罍ｧ006 縺03 0 0403 ｧ ﾀ00060 ｧ06 04 縺06 閨06000304... 00 x 7. ｮ 040060 ｧ06 0306 ﾀ0903006 003 0 000308..... 0 x 8. 70604 縺06 縺0304003 ﾀ0504 070600909 ｧ ｧ... 0 x 9. 罍ﾀ ｧ00405030503... 05 x 0. ｪ ｧ ﾀ0003 e... 06 0600 04 6. 7 縺03 ｧ 0403 閨0 07 ｧ 陦ｧ070706 縺06009 ｧ 00... 08 x. 7 縺03 ｧ 0403 閨00306 00050603 ｧ 06 ﾀ09 ｧ08 ｧ 05 罍ｧ050300300605 罍ｧ 06 ﾀ09 ｧ08 070600909 ｧ08... 08 x. 7 縺03 ｧ 0403 閨0 07 ﾀ00030700308 070600909 ｧ ｧ.... 0 x 3. 07 ﾀ 050600040 07 ｧ 003 縺0 0004 09 040003 ﾀ090308 09 罍ｧ040308... x 4. ｮ0006 ｧ 006 陦ｧ070706 縺06009 ｧ 縺0606003 ﾀ05 ｧ 00604 縺06 縺030400308 070600909 ｧ ｧ.. 3 x 5. 7 縺03 ｧ 0403 閨00306 000305 罍ｧ070 ｧ 06 ﾀ09 ｧ08 ｧ 040309 0506000400308 070600909 ｧ08 4 x 6. 7 縺03 ｧ 0403 閨00306 0303 0500308 05 罍ｧ050300300605 罍ｧ 06 ﾀ09 ｧ08 070600909 ｧ08.. 6 x 7. 7 縺03 ｧ 0403 閨00306 05 ｧ0406 縺030300 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08... 7 x 8. ｮ00600 000306 04 罍ｧ006 縺03 ｧ ｧ 0 陦ｧ070706 縺06009 ｧ 縺0304 0 ｧ06. 8 x 9. 7 縺03 ｧ 0403 閨00306 0403 ﾀ00 ｧ08 0403 縺07 閨090304.... 9 x 0. 7 縺03 ｧ 0403 閨0 07 006070400308 070600909 ｧ ｧ....

3050500 0400060 ｧ06 5 x. 6 7 ｧ070706 縺06009 ｧ 00 070600909 ｧ ｧ... 3 x. 7 00605 罍ｧ 06 ﾀ09 ｧ06 06 040060 ｧ07 09 罍ｧ040308... 4 0600 04 7. 7 罍ｧ000307060 ｧ07 04 縺03 ｧ 0403 閨00308... 7 x. ｮ0903 縺03 ﾀ0504 ｧ06 ﾀ0903 縺060 ｧ06... 7 x. ｯ0603 縺06003 03 ﾀ 縺06 閨0060... 8 x 3. 7 ﾀ09 縺0305 ｧ06 0060304 縺06 閨0600060003 ﾀ050608. 7 04 ｧ0003 6ｧ80304 ｧ05 0007... 3 x 4. 6 503 ﾀ05 0 ｧ06 ｧ 06030304 0 ｧ06 070600909 ｧ ｧ. 03 09 ﾀ ｧ0060 ｧ 0 ｧ0 ｧ0060 33 x 5. 6ｦ0 ｧ 03 0 ｧ0303000400 ｧ08 ｧ 0 ｧ00600400 ｧ08 0 060 ｧ0708 040600 ｧ ｧ0 36 x 6. 7 04 0400060 ｧ06 04030406090003 ﾀ05 ｧ ｧ0503 09 ｧ 0406 縺0605 ｧ03 09 罍ｧ040308. 703 ﾀ05 縺03060 ｧ06 09 罍ｧ040308... 38 0600 04 8. 7060304 縺06 閨060006000308 ｧ0050605 00... 40 x. 7060304 縺06 閨060006000308 ｧ0050605 00. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 000307060 ｧ060 40 x. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 0403 陦ﾀ05 00304090308 ｧ 0060403 ﾀ 縺06 陦ﾀ050406000306... 4 x 3. 0ｷ30050605 0003 04 ｧ x, a a,, x x + k ｧ 09 0 ｧ0 04 罍ｧ0403 閨070 ｧ06 ﾀ07... 45 x 4. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 0403 ﾀ05070... 47 x 5. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 05 罍ｧ050300300605 罍ｧ 06 ﾀ09 ｧ08 070600909 ｧ08... 48 x 6. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 09 ｧ030 000400308 00050603 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08 50 x 7. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 00609030503 縺0308 ｧ 罍 09 ｧ030 000400308 00050603 ｧ 06- ﾀ09 ｧ08 070600909 ｧ08... 5 x 8. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 00609030503 縺0308 05 0 ﾀ09060 閨0600500308 070600909 ｧ08 55 x 9. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 05 ｧ0406 縺030300 ｧ 06 ﾀ09 ｧ08 070600909 ｧ08. 06 ｧ0406 縺030300 ｧ- 06 ﾀ09 ｧ06 0403 陦ﾀ05 0030409 ｧ... 56 x 0. ｮ00600 000306 04 罍ｧ006 縺03 0 ｧ0050605 罍ｧ 縺0304 0 ｧ06... 57 0600 04 9. 0504 縺06 閨060006000308 ｧ0050605 00... 60 x. 6 503 ｧ ﾀ00060 ｧ06 0304 縺06 閨06000600030503 ｧ0050605 00... 60 x. 6 503 ｧ ﾀ00060 ｧ06 0400030 閨0608... 63 x 3. 05030060 050600 04 0060 ｧ07... 65 x 4. 6 700 ｧ0 閨0605 ｧ 040003 ﾀ090308 09 罍ｧ040308... 67 x 5. 700030 閨04 04030406 縺08003 ﾀ05 ｧ 04 0060 ｧ07... 69 x 6. 6ｦ0 ｧ ｧ 07 ｧ ｧ09 ｧ... 70 x 7. 706 ﾀ0303 ﾀ050406000306 ｧ0050605 0003... 7 x 8. ｮ 縺06 閨00606 0 060 ｧ06 070600909 ｧ ｧ... 75 x 9. ｱ03 縺00600 05 040609 ｧ08 ｧ 0703 縺00600 ｮ ｧ004 ﾀ030... 76 0600 04 0. 07 罍ｧ04 ｧ 0 040003 ﾀ090308 ｧ 04 縺03 ﾀ05 0 ﾀ050406000308 09 罍ｧ040308... 78 x. 07 罍ｧ04 ｧ 0 040003 ﾀ090308 09 罍ｧ040308. ｨC06005 ｧ 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003. 70403000605... 78 x. 6 700 ｧ0 閨0605 ｧ 09 罍ｧ040308 04 04 縺03 ﾀ05 0 ﾀ050406... 80

36 050500 0400060 ｧ06 x 3. 7 縺03 ｧ 0403 閨0 07 0406090503-070600909 ｧ ｧ 0403 ﾀ09 0007 縺06 ｧ060600608 0 ｧ 06- ﾀ090306 ｧ 05060300605 罍ｧ 06 ﾀ090306 0 060 ｧ06. ｭ ﾀ0506 ﾀ050406000308 05 縺0608-05 00 ｧ09 09 罍ｧ040308... 80 x 4. 07 罍ｧ04 ｧ 0 ｧ 09 縺06 060 ｧ06 04 縺03 ﾀ05 0 ﾀ050406000308 09 罍ｧ040308... 83 0600 04. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306, 04030000306 陦ｧ070706 縺06009 ｧ 0003 ｧ ｧ08 04 罍ｧ000307060 ｧ07... 85 x. ｱ0600909 ｧ ｧ 閨040608 0406 縺06006000308 ｧ ｧ08 05060300605 罍ｧ 06 ﾀ090306 ｧ 0303-07060 ｧ06... 85 x. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306 0406 縺04030503 0403 縺07 閨09... 87 x 3. 7030000308 陦ｧ070706 縺06009 ｧ 00 0406 縺04030503 0403 縺07 閨09... 89 x 4. 7 縺03 ｧ 0403 閨00306 ﾀ00030700308 070600909 ｧ08... 9 x 5. 7 縺03 ｧ 0403 閨00306 006070400308 070600909 ｧ08.... 9 x 6. ｪ ﾀ0500306 04 縺03 ｧ 0403 閨00306 ｧ 04030000306 陦ｧ070706 縺06009 ｧ 0003 0403 ﾀ00 ｧ08 0403 縺07 閨090304... 94 x 7. 0ｷ30050605 罍ｧ 縺0304 0 ｧ06 04030000308 陦ｧ070706 縺06009 ｧ 000304... 98 x 8. 05 ﾀ03030306 0503 09 ｧ 040003 ﾀ090308 09 罍ｧ040308... 99 x 9. 0505 ｧ03 060 07 ﾀ0600608 ﾀ0504 040003 ﾀ09 ｧ08 09 罍ｧ040308... 00 x 0. 07 ﾀ 050600040 07 040003 ﾀ0903 ﾀ0504 ｧ 003 縺0 0004 09 04030406 縺08003 ﾀ05 ｧ... 0 x. ｮ09 0007 縺00306 04030006. 6ｧ8 ｧ0 ｧ ｧ ｧ 04030406 縺08003 ﾀ05 ｧ 06 縺030400608. 7 縺03 ｧ 0403 閨0 07 04 00030 0 04 0400060 ｧ ｧ. 06 陦ｧ06005... 03 x. 709 ﾀ05 縺060060 070600909 ｧ ｧ 閨040608 0406 縺06006000308... 05 0600 04. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07... 07 x. 70300705 ｧ06 03 陦ｧ070706 縺06009 ｧ 00040030 06 040060 ｧ ｧ... 07 x. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 0406 縺04030503 0403 縺07 閨09 ﾀ 閨060007-060 ｧ0 ｧ ﾀ07 0406 縺0600600030 ｧ. 05 縺050305030 000400306 05 06090503 罍ｧ ｧ... 08 x 3. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0406 縺04030503 0403 縺07 閨09 : ) 03 閨003 縺03 閨00306, ) 00 ｧ0060800306, 3) 706 縺0060000 ｧ... x 4. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07, ﾀ03 閨06 縺07 0 ｧ06 陦ｧ070706 縺060-09 ｧ 0003 04 縺03 ｧ 0406 閨060 ｧ07 ｧ ﾀ050030503.... 3 x 5. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0406 縺04030503 0403 縺07 閨09 04 04030000308 陦ｧ070706 縺06009 ｧ 00 08. 0ｷ30050605 罍ｧ 縺06060 ｧ08 000307 ｧ05060004... 3 x 6. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0406 縺04030503 0403 縺07 閨09, 006 - 縺060006000306 0305003 ﾀ ｧ05060004003 04 縺03 ｧ 0403 閨00308. ｰ 040060 ｧ07 6ｧ8 05 007 ｧ 070006 縺03... 5 x 7. 6 7 ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 0403 ﾀ00 ｧ08 0403 縺07 閨090304, 閨030406 ﾀ- 09 060 ｧ06 04030 ｧ07060 ｧ06 0403 縺07 閨09... 7 x 8. 6ｧ8 ｧ0060800306 03 閨003 縺03 閨00306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 ﾀ 0403 ﾀ05030700030 ｧ 0903050707 ｧ09 ｧ06005 0 ｧ... 8 x 9. 6ｧ8 ｧ0060800306 00603 閨003 縺03 閨00306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 ﾀ 0403 ﾀ05030700030 ｧ 0903050707 ｧ09 ｧ06005 0 ｧ... 9 x 0. 7 罍ｧ006 縺03 陦ｧ070706 縺06009 ｧ 000400308 06 040060 ｧ08 00308 05 ｧ040304 x. 6ｧ8 ｧ0060800306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ06 7080006 x n y (n) + + ax n, y (n,) + :::+ an,xy 0 + any = f(x)...

3050500 0400060 ｧ06 7 x. ｮ ｧ ﾀ0506003 00 ｧ0060800308 陦ｧ070706 縺06009 ｧ 000400308 06 040060 ｧ08 ﾀ 0403- ﾀ05030700030 ｧ 0903050707 ｧ09 ｧ06005 0 ｧ... 3 x 3. 6ｧ8 ｧ0060800306 陦ｧ070706 縺06009 ｧ 000400306 06 040060 ｧ07 04 ﾀ0500308 04 縺03- ｧ 0403 閨00308 040503 縺030503 0403 縺07 閨09 (0060503 08 090506 罍ｧ ﾀ05 ｧ09)... 4 0600 04 3. 6 704030800306, 05 縺030800306 ｧ 09 罍ｧ040300 ｧ0060800306 ｧ0050605 0003.. 6 x. 6 503 ｧ ﾀ00060 ｧ06 0400030 陦ｧ ﾀ 040300300406 閨0403080030503 ｧ0050605 00.. 6 x. ｨC06005 0 ﾀ ﾀ ｧ 003006005 ｧ006 縺09 ｧ ｧ 0400030 陦ｧ ﾀ 04003006 縺003 ﾀ- 04 縺06 閨060006000308 0 ﾀ ﾀ0308 (04 罍ｧ 04000305003 ﾀ05 ｧ =)... 8 x 3. 6 503 ｧ ﾀ00060 ｧ06 03030060 ﾀ 040300300406 閨0403080030503 ｧ0050605 00... 30 x 4. 700030 陦ｧ 09 罍ｧ040308 04030406 縺08003 ﾀ050608.... 3 x 5. ｯ 縺030800308 ｧ0050605 00 ｧ 060503 04 罍ｧ000307060 ｧ07... 3 x 6. 07 罍ｧ040300 ｧ0060800308 ｧ0050605 00. ｱ03 縺00600 06 罍ｧ0... 34 x 7. 7030406 縺08003 ﾀ0500306 ｧ0050605 0003. ｱ03 縺0060003 05 ﾀ05 縺0305 陦ﾀ09030503{06 06 ﾀ ﾀ ｧ ｮ050309 ﾀ... 38 0600 04 4. 707 閨03... 4 x. ｪ ｧ ﾀ0003040306 縺07 閨03... 4 x. 7 04003006 縺0 07 ﾀ0803 陦ｧ003 ﾀ0504 070600909 ｧ030 00040030503 縺07... 45 x 3. ｮ05060406000306 縺07 閨03... 47 x 4. 707 閨03 ｯ06080003 ｧ 03 090003 縺060...... 49 x 5. 7 罍ｧ000307060 ｧ07 縺07 閨0304 09 04 罍ｧ0300 ｧ070600030 0403 ｧ ﾀ00060 ｧ070... 5 x 6. 707 ｯ06080003 閨0007 070600909 ｧ ｧ 閨040608 0406 縺06006000308... 54 x 7. 707 ｱ06 縺0406. 0ｷ30050605 00 ｱ06 縺0406... 55 050504060503... 60 7 罍ｧ000307060 ｧ06. 70609030503 縺0306 09 罍ｧ040306 ( 閨0007 ﾀ04 040309)... 33

3 70609030503 縺0306 09 罍ｧ040306 ( 閨0007 ﾀ04 040309) 7 罍ｧ000307060 ｧ06

3 7 罍ｧ000307060 ｧ06 333

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