繖 7 縺6ｧ80ｷ3 ｯ0ｷ3 ｪｭｮ07 縺00 06 ｭ0503 ｭｯ 70ｷｧ805 ｮ0705 ｮｯ0ｷ3 x 罍陦ｧｧ 0 04 縺ｧﾀ0903 ﾀ05 ｧ. 7

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1 30ｷ36ｦ0 7 7 ｭ6 70ｷ3 ｮ6ｧｷ300 縺6 5 ｯｯ 7 ｭｯ04 ｭ03 ｰ 0ｷ36ｦ06 7 繖 70ｷﾀｮ0303 縺0 ｧ閨ｧ陦ｧ 06 ﾀ09 ｧﾀ04 縺06 閨ｧｧ 04 罍ｧ006 縺ｧ05 ｧ 06 ﾀ罍ｧｧｧｧ- 06 ﾀｧ閨罍ｧ0406 閨縺 , 0304 縺06 閨ｧ07 ｧ陦縺0605 ｧｧﾀ0060 ｧ罍ｧｧ, 陦ｧ00306 閨0007 縺ｧﾀ0006 閨06060 ｧ閨ｮ0303 縺0 ｧ罍ｧ0406 閨06003 (0403 ﾀ ) ｧ閨縺060 ｧ07, ﾀ03 ﾀｧ閨縺0605 ｧ 04 ﾀ罍ｧｮ0303 縺0 ｧﾀﾀ縺閨0303 縺06 閨ﾀﾀ06 ｧ閨0007 閨ｧ08 0 ｧ08 ｧ00 ｧ閨縺060 ｧ縺縺ｧｧ. 07 縺 , 04 罍ｧﾀ04 縺06 閨ｧｧ縺06 閨0600 ｧｧ0 ｧ0060, 陦ｧ閨ﾀｧ罍ﾀ, 縺縺閨ｧｧ00 ｧ閨縺0 ｧｮ0303 縺0 ｧｧﾀ閨縺陦ﾀ縺 , ｧ閨0007 ﾀ 003 ﾀｧｧ罍ﾀ 0403 ﾀｧ09 ｧ , ｧ 04 ﾀ06 ｧｧ , 縺0306 ｧ縺ｧ07 ｧ, 09 縺 , ｧﾀ0506 ｧ00 ｧｧ07 09 ｧ08 縺ｧﾀ0403 ﾀ0303 ﾀｧﾀ0060 ｧ罍ｧｧ. 05 ｯ 7 ｭ6 7 縺07ｨC0ｷ30ｷ3 0ｷ3 0 ｧ06 0 ﾀｮ0303 縺0 ｧﾀ06006 ﾀﾀ0407 ｧﾀｧﾀｧｧ, 0403 ﾀｧ0400 ｧ0 ｧ陦縺06 ﾀｧ- 06 ﾀ09 ｧ陦, 03 ｧ0300 ｧ , ﾀ0506 閨ｧ 04 縺ｧﾀﾀｧｧ. 6 5 ﾀ0407 ｧﾀ 05060, , 6 5 ﾀｧ00 ｧｧ 03 ｧ003 罍ﾀ , , ﾀ 0 0 ｧ, 縺ｧ07 04 縺06 閨縺ﾀ00 ﾀﾀ0506, 03 ﾀ06006 ﾀ0504 ｧ04 00 ｧ , 0406 罍ﾀｧ縺ｧ06, ﾀ07 04 罍ｧｧ09 09 ｧ08 ﾀ06006 ﾀｧｧﾀ05, 09 縺03006 ｧﾀｧﾀｧﾀ0403 ｧ0 04 罍ｧ閨罍ｧｧｧ08 ﾀｧ縺ｧ 0406 ﾀｧ 06 ﾀｧｧ 08 ｮ0303 縺0 ｧ0909, , 縺06 閨03 閨ｧ 0 ｧ

2 繖 7 縺6ｧ80ｷ3 ｯ0ｷ3 ｪｭｮ07 縺00 06 ｭ0503 ｭｯ 70ｷｧ805 ｮ0705 ｮｯ0ｷ3 x 罍陦ｧｧ 0 04 縺ｧﾀ0903 ﾀ05 ｧ. 7 ﾀﾀｧ閨06 閨ｧ. 7 ﾀﾀｧ06 d 閨ｧ A(x ) ｧ B(x ) 0 03 ﾀｧ: d = jx, x j = (x, x ) : () : ｧｧ0 AB ( ｧ 06 ﾀ09 07) 縺ﾀｧ: AB = x, x : () 3. 7 ﾀﾀｧ06 d 閨ｧ A(x ; y ) ｧ B(x ; y ) ﾀﾀ05 ｧ: d = (x, x ) +(y, y ) : (3) ｧｧ 0 03 ﾀｧ罍陦ｧ縺06 09, ｧ00 ｧ ,! AB ﾀ0903 ﾀ05 ｧﾀ A(x ; y ) ｧ B(x ; y ): 04 x,! AB = X = x, x ; 04 y,! AB = Y = y, y : (4). 703 ﾀ05 縺03 ｧｧﾀﾀｧｧ A(,5), B(+4) ｧ C(,) ｧｧｧｧ003 AB, BC ｧ AC 0305 縺ﾀｧ. 7 縺罍ｧ0504, 0503 AB + BC = AC ｧ縺06 閨03 閨ｧ06 閨 A(+), B(,4) ｧ C(+5) ﾀ05 縺03 ｧ縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(,4; ), B(0;,) ｧ C(3; 3) ｧ 0304 縺06 閨0600 ｧ罍ｧ00605 ｧ , 縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(,3;,), B(0;,) ｧ C(,; 5) 04 縺ﾀ05 縺03 ｧｧ A(,4; 0), B(,; 4) ｧｧ A, B, ﾀｧ罍ｧﾀｧﾀｧ Oy ｧﾀ00 ｧ罍ｧｧｧ ABB A. 6. ｯ03 09 B ﾀｧ罍ｧ 0 A(4;,) ﾀｧｧﾀﾀ罍ｧﾀ縺罍陦ｧｧ閨00 ｧ006 AB.

3 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧｧ , 陦ｧ0 ｧｧ A(; ), ｧﾀｧOy ﾀｧ 03 罍陦ｧｧ , ｧ A(4;,) 陦ｧ0 ｧﾀ0 ｧﾀ05 縺03060 ｧ060, ﾀ07 閨04 縺ｧﾀｧ 03 ﾀ09 ｧﾀﾀｧ , ｧ A(a; b) 0 c 06 陦ｧ0 ｧ09. 0ｷ3 ﾀﾀ0006 閨縺ｧ06 04 罍ｧ c>jbj, c = jbj ｧ c<jbj ﾀｧ Ox ｧ , 03 陦ｧ罍陦ｧ0 05 ｧｧ A(8; 4) ｧｧ陦ｧ06 ﾀ 09 縺0605, 0304 ｧﾀ縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(4; 3), B(,3; ) ｧ C(;,6) ｧ A(; 6) ｧ B(0; ); 0403 ﾀ05 縺03 ｧ ,! AB, ,!,! ﾀ0708 ｧ 0403 ｧﾀ00 ｧ x AB, 04 繦 AB ｧ閨00 ｧ006 AB A(; 5) 04 罍ｧﾀｧ00, 04 縺ｧｧ縺ﾀｧ罍陦ｧ : X =3 ｧ Y = 縺06 閨0600 ｧ ,! AB, ｧﾀｧ0006, ｧｧｧ006 ﾀｧ A(,3;,) 04 罍ｧﾀｧ00, 04 縺ｧ縺0308 Y = =,, 04 縺ｧ07 X ｧ縺06 閨0600 ｧ ,! AB, ｧﾀｧ0006, 06 ﾀ00 ｧｧｧ ). 7 ｧﾀﾀｧ 0403 ﾀ05 縺03 ｧｧ A(), B(,3) ｧ C(,) ｧｧｧｧ003 AB, BC ｧ CA 0305 縺ﾀｧ. 7 縺罍ｧ0504, 0503 AB + BC + CA = ﾀ0903 ﾀ05 ｧ 0403 ﾀ05 縺03 ｧｧ A(,7; 0) ｧ B(0; ) ｧｧ A ｧ B, ﾀｧ罍ｧ A ｧ B ﾀｧｧﾀﾀ罍ｧﾀ縺ｧ 05 縺罍陦ｧｧﾀ00 ｧ罍ｧｧｧ ABB A ﾀｧ 03 罍陦ｧｧ , 03 陦ｧ罍陦ｧ0 05 ｧｧ A(,; 5) ﾀｧ 03 ﾀ09 ｧﾀﾀｧ , ｧ A(,; 3) 陦ｧ0 ｧ縺06 閨0600 ｧｧ陦ｧ06 ﾀ 09 縺0605, 0304 ｧﾀ縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(,3;,), B(5; 3) ｧ C(6;,4) ｧ A(x ; y ) ｧ B(x ; y ) 罍陦ｧ罍ｧﾀｧ0003, ｧｧ,! OA ｧ,! OB. 703 ﾀ05 縺03 ｧ0504 ｧ閨0608 ﾀ ,! OC ｧ閨 , 縺ｧ閨0608 ﾀ罍陦ｧﾀﾀ縺ｧ08 ﾀ03 ﾀｧﾀｧ A(; ), B(3; 5), C(5; ) ｧ D(;,) A 04 罍ｧﾀｧ0003,! AB,,! AC ｧ,! AD ｧ 04 縺ｧｧ 0 03 ﾀｧ陦ｧ閨0608 ﾀﾀｧ0003 ｧｧｧ006. ) 閨ﾀ縺罍ｧ0406 閨06003 ｧ, 縺0306 縺閨ﾀ07 閨ｧ07 0 閨030 ｧ00 ｧ閨縺060 ｧ08.

4 3x ｧ縺ｧｧ x ｧ縺ｧｧ閨04 05 縺ｧ09 ｧｧｧ縺ｧｧｧ A(x ; y ) ｧ B(x ; y ) 罍陦ｧｧ M (x; y), 閨縺 AB ｧｧ AM : MB =, 0304 縺06 閨ﾀ縺 : x = x + x + ; y = y + y + : () 6 5 ﾀ05003 ﾀ05 ｧ, 04 罍ｧ閨ｧｧ , ｧｧ =:=, x = x + x ; y = y + y : () 閨ｧ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) ｧ06 04 ｧ x y x y x y, x y ｧﾀ縺06 閨0600 ｧ縺縺07 閨09 ) ﾀ05 縺03 ｧｧ A(,; ) ｧ B(3; 6) ｧｧ M (x; y), 閨 AB ｧｧ AM : MB =3: ｧ A(,; ) ｧ B(3; 6). 7 閨0600 ｧ縺 AB ｧｧ AM : MB =,3 : A(x ) ｧ B(x ) 03 ﾀｧ Ox ﾀﾀ03 m ｧ m ｧﾀﾀﾀｧﾀ A(x ), B(x ) ｧ C(x 3 ) 03 ﾀｧ Ox ﾀﾀﾀﾀ03 m, m ｧ m , ﾀﾀﾀｧﾀ閨 x = m x + m x + m 3 x 3 m + m + m 閨003 縺03 閨ﾀ0506 縺07007 閨00 ｧﾀ0 ｧ 0 ﾀﾀﾀ縺03 0 ﾀﾀｧ縺06 閨0600 ｧﾀﾀﾀｧﾀ A(,; 4), B(3;,) ｧ C(; 3) ﾀﾀﾀﾀ , ｧ縺06 閨0600 ｧﾀﾀﾀｧ- ﾀ縺06 閨0600 ｧ0504 ﾀ06 縺06 陦ｧ003 ﾀ0503 縺縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(;,), B(4; 3) ｧ C(,; ) 縺ｧ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 ｧ陦ｧ 003 OC ｧ 03 ｧﾀﾀ罍ｧﾀ03 OD. ) 縺06 閨0600 ｧ陦縺ｧ , x.

5 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧｧﾀﾀ 05 縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(;,), B(6; 4) ｧ C(; 6). ｰ09 0 ｧ06. ｨC ﾀﾀ 05 縺ｧ陦ｧ05 ﾀ縺06 ﾀｧ陦ｧｧﾀ00 ｧ閨04 05 縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(; 0), B(5; 3) ｧ C(; 6) , ｧ A(; ), B(,; 7) ｧ C(0; 4) 閨縺ｧﾀ00 ｧ閨縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(3; ), B(4; 6), C(6; 3) ｧ D(5;,) A(,3;,) ｧ B(4; 6) 04 罍ｧﾀｧ0003, ﾀﾀｧ縺 AB ｧ罍ｧｧ閨0608 ﾀ O(0; 0), A(;,5) ｧ B(4; ) ﾀﾀﾀﾀ , ｧ縺06 閨0600 ｧﾀﾀﾀｧ- ﾀ縺ｧ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 ｧｧﾀﾀ罍ｧﾀ03 AE ｧﾀﾀ 05 縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(x ; y ), B(x ; y ) ｧ C(x 3 ; y 3 ) ｧﾀﾀ縺閨003 縺03 閨00308 閨03 ﾀ09 ｧﾀｧ0 0 ｧ A(,; ), B(3; 6), C(5; ) ｧ D(0;,6). ｰ09 0 ｧ縺 , , ｧ縺03 0 ﾀﾀ 05 縺ｧ ABC ｧ ADC ｧ閨0600 ｧ0504 ﾀﾀｧ閨06 0 ｧ0 ｧｧｧ, ｧ閨縺ｧｧ A(; ) ｧ B(4; 4) ﾀｧOx 0304 縺06 閨0600 ｧ C 05 09, 閨04 4ABC , ｧ 0403 ﾀ05 縺03 ｧ0504 4ABC 縺ｧ0906 ﾀ 0406 縺00 ｧ0 0 ｧ A(,; ), B(;,4) ｧ C(4; 5) ﾀ0503 縺縺03 閨ｧｧ罍ｧ縺03-05 ｧ04 ﾀﾀ05 縺ｧ 0 03 閨縺ﾀ閨00 ｧ縺06 閨0600 ｧ M, N ｧ P 04 縺03 閨ｧ08 ﾀ0503 縺030 ｧｧｧ06 k 陦ｧ 4MNP 陦ｧ 4ABC. x 3. ｰｧ06 00 ｧ0 ｧｧ罍ｧ 06 ﾀﾀｰｧｧ0 ｧｧﾀｧ06 ﾀ 0406 縺ｧ x ｧ y; 縺閨縺罍陦ｧｧｧ0 ｧｧｧｧ閨070 ｧｧ06 00 ｧ0 ｧｧ 0406 縺 x ｧ y ﾀｧ0 ｧ罍陦ｧｧ, ﾀｧ罍ｧ006, ｧｧ 0309 縺ﾀ05 ｧ ( 4) x + y = R 0406 縺 x ｧ y ｧ罍陦ｧ0 0503, 0403 ﾀ R ｪﾀ03 ﾀ05 04 ｧｧ06 00 ｧ0 ｧｧ罍ｧ 06 ﾀﾀ , ｧ08 03 陦ｧﾀﾀ , :

6 3x 3. ｰｧ06 00 ｧ0 ｧｧ罍ｧ 06 ﾀﾀ ) 縺03 ｧ ( ) M (x; y) 00 ｧ0 ｧｧ; ) 04 ｧﾀﾀﾀﾀﾀ M 00 ｧ0 ｧｧ; 3) 閨070 ｧﾀ縺06 09 ｧ ( ｧ ) 0403 ｧ縺ｧ罍陦ｧｧ M (x; y) ｧ 06 縺 , ｧ縺ﾀ05 ｧﾀ陦ｧ06 ﾀ030 R ｧﾀ縺罍陦ｧ閨0605 x + y = R ｧﾀｧ縺ﾀ05 ｧﾀ縺030 C(3; 4) ｧ陦ｧ06- ﾀ030 R =5.6ｧｧ縺ﾀ05 ｧｧ A(,; ), B(; 3), O(0; 0) ｧ D(4; )? ｧﾀｧ06 00 ｧ0 ｧｧ, 縺0308 閨04 ｧﾀ M (x; y), A(0; ) ｧ B(4;,). 6ｧｧｧ0 ｧｧｧ C(,; ), D(;,), E(0;,) ｧ F (; )? ｧﾀｧ罍ｧｧｧ M (x; y), 罍ｧﾀ閨04 ｧ07060 ｧｧ 03 ﾀﾀ縺ｧ A(0; 9), ｧ B(0; ) ｧﾀｧ罍ｧｧｧ M (x; y), 罍ｧﾀ閨04 ｧ07060 ｧｧ 03 ﾀﾀ07 04 閨ｧ A(,; ), B(,4; 4) ｧﾀｧ07 03 ｧﾀﾀ罍ｧﾀ罍陦ｧｧﾀｧ罍ｧ 06 ﾀﾀ , ﾀ0600 ﾀﾀｧ閨0308 ｧ縺0308 閨 F (; 0) ｧ F (,; 0) ﾀ05 縺03 ｧｧ0 ｧｧｧﾀｧ罍ｧ 06 ﾀﾀ , ｧ F (; ) ｧﾀｧOx. 703 ﾀ05 縺03 ｧｧ0 ｧｧｧﾀｧ06 00 ｧ0 ｧｧ, 縺0308 閨04 ｧﾀ M (x; y), 03 ﾀﾀ04 04 閨ﾀｧOx, ﾀｧOy ﾀ05 縺03 ｧｧ0 ｧｧ: ) y =x +5;)y =7, x; 3)y =x; 4) y =4;5)y =4, x 縺06 閨0600 ｧｧ 0406 縺06 ﾀｧ07 00 ｧ0 ｧｧ y = x, 4x +3 ﾀ 03 ﾀ070 ｧ罍陦ｧ0 05 ｧ 0403 ﾀ05 縺03 ｧ縺06 閨0600 ｧｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ罍陦ｧｧ0 ｧ08: ) 3x, y =;)y = x +4x; 3)y =x ﾀ05 縺03 ｧｧ 00 ｧ0 ｧｧｧﾀｧ罍ｧ 06 ﾀﾀ , ﾀｧOy ｧｧ F (4; 0), ｧ 0403 ﾀ05 縺03 ｧｧ0 ｧｧｧﾀｧ06 00 ｧ0 ｧｧ, 縺0308 閨04 ｧﾀ M (x; y), 罍陦ｧ0 05 ｧｧ A(,4; ). 6ｧｧｧ0 ｧｧｧ B(,; ), C(; 3), D(; 7)?

7 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧｧﾀｧ罍ｧｧｧ M (x; y), 罍ｧﾀ閨04 ｧ07060 ｧｧ 03 ﾀﾀ07 04 閨ｧ A(0;,), B(0; 4). 703 ﾀ05 縺03 ｧ罍ｧ06 閨04 ｧ07060 ｧ縺06 閨0600 ｧｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ罍陦ｧｧ0 ｧ08: ) x +5y +0=0;)y =3, x, x ;3)y =4, x. 703 ﾀ05 縺03 ｧｧ0 ｧｧｧﾀｧ罍ｧ 06 ﾀﾀ , ﾀｧOx ｧｧ F (0; ), ｧ 0403 ﾀ05 縺03 ｧｧ0 ｧｧｧﾀｧ罍ｧ 06 ﾀﾀ , 003 ﾀ0504 ﾀﾀｧ閨0308 ｧ縺0308 閨 F (,;,) ｧ F (; ) ﾀ05 縺03 ｧｧ0 ｧｧ06. x 4. ｰｧ06 04 縺 : ) ﾀｧ09 ｧ , ) , 3) 縺ﾀ0708. ｰｧ06 04 縺ﾀｧ09 ｧ y = kx + b: () k ﾀ縺ﾀｧ Ox (k = = tg ) ｧﾀｧ09 ｧ , ｧ00 ｧｧ縺 b ｧｧ縺ﾀｧOy, ｧ00 ｧ罍陦ｧｧ06 04 縺 Ax + By + C =0: () 05 ﾀﾀ0006 ｧ: ) 04 罍ｧ C =0 y =, A x 04 縺縺陦ｧ05 06 縺罍陦ｧ0 05; B 03) 04 罍ｧ B =0 x =, C A = a 04 縺ﾀｧ Oy; 04) 04 罍ｧ A =0 y =, C = b 04 縺ﾀｧ 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. ｰｧ06 04 縺縺ﾀ0708 x a + y =; (3) b 05 閨06 a ｧ b ｧｧ縺 , 0305 ﾀ縺ﾀ罍陦ｧﾀ05 縺03 ｧ縺 , 0305 ﾀﾀｧOy 0305 縺 b = =3 ｧﾀ03 ﾀﾀ03 ﾀ0406 Ox : ) 45 ;) ｧﾀｧｧ08 04 縺ﾀ05 縺03 ｧ縺 , 0305 ﾀﾀｧOy 0305 縺 b = =, 3 ｧﾀ03 ﾀﾀ03 ﾀ0406 Ox : ) 60 ;) ｧﾀｧｧ08 04 縺

8 3x 4. ｰｧ06 04 縺ｧﾀｧ06 04 縺 , 04 縺閨縺罍陦ｧ0 05 ｧﾀ03 ﾀﾀ 03 ﾀ0406 Ox : ) 45 ;)60 ;3)90 ; 4) 0 ;5) ﾀ05 縺03 ｧ縺 , 04 縺閨縺罍陦ｧ0 05 ｧ 06 縺 (,; 3), ｧ 0 04 ｧﾀｧ縺06 閨0600 ｧ縺03 k ｧ b 閨閨0308 ｧ 04 縺 : ) x, 3y =6; ) x +3y =0; 3) y =,3; 4) x 4 + y 3 = ﾀ05 縺03 ｧ縺 : ) 3x +4y =; ) 3x, 4y =0; 3) x, 5=0; 4) y +5= 縺06 閨0600 ｧ縺03 k ｧ b 04 縺 , 04 縺閨縺 A(; 3) ｧﾀ03 ﾀﾀ Ox ｧﾀｧ縺ｰｧ07 04 縺 : ) x, 3y = 6;) 3x, y +4 = 0 04 罍ｧ0406 ﾀ05 ｧｧ閨縺ﾀｧ O(0; 0) ｧ A(,3; 0) 縺 OA 0403 ﾀ05 縺 , 陦ｧｧ縺縺06 ﾀﾀ B(0; ) ｧﾀｧ07 ﾀ0503 縺030 ｧ陦ｧｧﾀｧ06 04 縺 , 04 縺閨縺 A(4; 3) ｧ 0305 ﾀ罍陦ｧ縺ｧ閨0406, 縺 y =, ｧ y =40406 縺06 ﾀ縺 x, 4y, 5=0 ﾀﾀ A ｧ B. 703 ﾀ05 縺03 ｧ ,! AB, 0304 縺06 閨0600 ｧ閨00 ｧ006 ｧ縺ｧｧ 0 03 ﾀｧ罍陦ｧｧｧｧ A(3; 5), B(; 7), C(,;,3) ｧ D(,;,6) 0 04 縺 y =x, ｧ00 ｧｧｧ00 ｧ 080 ｧ縺 ? 罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀﾀ0504: ) y>3x +;)y<3x +;3)x + y, 4 > 0; 4)x + y, 4 < 0? ﾀ05 縺03 ｧﾀ05 ｧ ), 罍陦ｧ縺閨縺ﾀ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 ｧﾀ , ﾀ陦ﾀ- ﾀｧ閨 A(,a; a) ｧ B(a;,a) 03 ﾀﾀ a ｧﾀｧ罍ｧｧ. ) ｮﾀ閨06 ﾀﾀﾀ0903 ﾀ05 ｧ xoy, 罍陦ｧ閨ｧ縺閨縺縺ﾀｧ070 (0 04 罍ｧ006, ﾀ0504 0) ﾀﾀ , 06 ﾀ00 ｧｧ, ｧｧﾀ05 ｧ縺0305 ｧﾀﾀﾀ縺

9 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧｧﾀｧ罍ｧｧｧ M (x; y), 04 縺ｧ縺ﾀ04 Ox 閨04 ｧﾀ07 ﾀ03 ﾀ0903 縺03 ﾀ m 06 / ﾀ, 0 03 ﾀ04 Oy ﾀ03 ﾀ0903 縺03 ﾀ n 06 / ﾀｧｧ M 0 (a; b) ﾀ05 縺03 ｧ縺 , ｧ: ) b =,, ' = =60 ｧ)b =,, ' = 0, ｧ 0 04 ｧﾀ 0504 ｧｧ縺06 閨0600 ｧ縺03 k ｧ b 04 縺 , 04 縺閨縺 (,; 3) ｧﾀ03 ﾀﾀ Ox ﾀ05 縺03 ｧ縺ｧ 0 04 ｧﾀｧ陦縺ｧ07 ﾀ 03 ﾀｧ070 ｧ 8 ﾀ0 ｧﾀ0 ｧﾀ05 縺ｧﾀｧ07 ﾀ0503 縺ｧｧ, 04 罍ｧﾀ04 Ox ﾀｧ06 ｧ 03 ﾀ04 Oy 03 ﾀ04 ﾀｧ罍ｧｧｧｧｧﾀｧ07 ﾀ0503 縺030 縺03003 ﾀ陦ｧｧ 0 ﾀ0 ｧ 6 ﾀ0, 04 罍ｧ陦ｧﾀ04 Ox ｧﾀ04 Oy ｧﾀｧ06 04 縺 , 04 縺閨縺 (,4; 6) ｧ0305 ﾀﾀ罍陦ｧ縺ｧ閨ｧﾀｧ06 00 ｧ0 ｧｧ, 縺0308 閨04 ｧﾀ M (x; y),03 ﾀﾀ04 04 閨ﾀｧOx, 縺 x =, 縺 x =, ｧ x =30406 縺06 ﾀ縺 y =x A ｧ B 縺06 閨0600 ｧ0504 閨00 ｧ ,! AB ｧ縺ｧｧ 0 03 ﾀｧ罍陦ｧ0 05. x 5. ｰ閨06 04 縺ｧ. ｰｧ縺 , 04 縺閨070 ｧ08 06 縺ｰｧ06 04 縺 , 04 縺閨縺06 閨ｧ. ｯ縺06 ﾀｧ07 閨縺ｰ ', 0305 ﾀｧ縺0305 ｧ04 ﾀﾀ05 縺ｧ縺 y = = k x + b 閨03 04 縺 y = k x + b, 0304 縺06 閨ﾀ縺縺 , ｧ070 ｧ 0703 縺00600 () 04 罍ｧｧ 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 : ｰﾀｧﾀ05 ｧ: k = k ｧ00 ｧ A A = B B. ｰﾀｧ縺04060 陦ｧ縺003 ﾀ05 ｧ: k =, k ｧ00 ｧ A A + B B =0.

10 3x 5. ｰ閨06 04 縺ｧ. ｰｧ縺ｰｧ縺 ,04 縺閨070 ｧ08 06 縺 A(x ; y ): y, y = k(x, x ): () 3. ｰｧ06 04 縺 , 04 縺閨縺06 閨ｧ A(x ; y ) ｧ B(x ; y ): y, y y, y = x, x x, x : (3) 4. ｪｧ縺06 ﾀｧ縺 A x + B y + C =0 ｧ A x + B y + C =0, 縺0600 ｧ0504 ﾀﾀ05003 ｧｧｧ0:,C B,C B A,C A x = A,C ; y = B A B A : B A B 縺06 閨0600 ｧ閨06 04 縺ｧ: ) 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 x,y+7 = 0, 6x,4y,9 = 0, 6x+4y,5 = 0, x +3y, 6= ｧ 0406 縺04060 陦ｧ縺ｧﾀｧ縺 , 04 縺閨070 ｧ08 06 縺 A(; 3) ｧ縺 , ﾀ03 ﾀｧ06 ﾀ 03 ﾀ0406 Ox : ) 45 ;)60 ;3)35 ;4)0, ｧ 0403 ﾀ05 縺03 ｧ0504 ｧﾀ05 縺03 ｧ A(,; 5) ｧ 04 縺 x, y = ｧ- ﾀｧ縺 , 04 縺閨070 ｧ08 06 縺06 A, ｧｧ : ) 04 縺 , ; ) 04 縺 , 0406 縺04060 陦ｧ縺縺06 ﾀｧ07 04 縺 x, 5y, 0 = 0 ﾀ 03 ﾀ070 ｧ罍陦ｧﾀﾀ縺04060 陦ｧ縺縺ｧﾀ 0504 ｧｧｧﾀｧ06 04 縺 , 04 縺閨縺ｧ A(,; 3) ｧ B(4;,).

11 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧ縺ｧ0906 ﾀ 0406 縺00 ｧ0 0 ｧ A(,; 0), B(; 6) ｧ C(4; ) 04 縺閨ﾀ0305 BD ｧ 006 陦ｧ 0 BE ｧﾀｧ07 ﾀ0503- 縺03003 AC, 006 陦ｧ 003 BE ｧ 0403 ﾀ BD ｧ縺0600 ｧ縺ｧ09, ﾀ0503 縺縺ｧ070 ｧ x +y =0, x +4y, 6=0, x, 4y, 6=0. ｰ09 0 ｧ06. ｪｧ縺0600 ｧ縺ｧ09, ｧ09 ｧﾀ0503 縺ｧﾀ縺07 閨ｧ07: k >k > k, k 3 +k k 3, >k 3, ｧﾀﾀ縺 k, k, +k k k 3, k. ｰ0306 陦ｧ0504 ﾀｧ 06 縺050607, ﾀ05 ｧ04 03 閨006 ｧ 0406 縺00 ｧ0 04 +k k 罍陦ｧｧﾀｧ07 04 縺 , 04 縺閨070 ｧ08 06 縺罍陦ｧ縺 y =4, x ｧﾀｧ07 04 縺 , 04 縺閨070 ｧ08 06 縺 A(,; ) 縺 x +3y = ｷｧ A(5; 4) 陦ｧﾀ ' = arctg 0903 ﾀｧOx ｧﾀｧﾀｧｧ縺06 閨0600 ｧ縺00 ｧ003 ｧ縺ｧ09, ﾀ0503 縺縺ｧ070 ｧ x +3y =0, x =3, x, y +3= 縺縺 x +y = 6, 0305 ﾀﾀ070 ｧ陦ｧ0 05, ﾀｧ05 05 ｧ陦縺縺縺ｧｧ 0406 縺00 ｧ縺 , 06 ﾀ00 ｧｧ 0406 ﾀ05003, ｧ縺縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(,; 0), B(; 4) ｧ C(4; 0) ｧﾀｧ07 ﾀ0503 縺縺ｧ09, 006 陦ｧ 003 AE, 0403 ﾀ AD ｧｧ閨00 ｧ陦ｧ 003 AE ｧﾀｧ07 ﾀ0503 縺030 ｧｧ縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(0; 7), B(6;,) ｧ C(; ) 縺 x, y +8 = 縺06 ﾀﾀｧOx ｧ Oy A ｧ B. ｯ03 09 M 閨0600 ｧ05 AB ｧｧ AM : MB =3: ｧﾀｧ縺04060 陦ｧ , 0403 ﾀﾀ M 縺 AB ﾀ05 縺03 ｧ縺ｧ09, ﾀ0503 縺縺ｧ070 ｧ x+y =4, 3x,y =0, x,3y,8 =0; ｧｧ閨04 05 縺ｧｧ縺06 ﾀｧ陦ｧ 0 ｧ縺06 ﾀｧﾀ縺ｧ09, 0406 縺00 ｧ縺 A(,4; ), B(;,5) ｧ C(5; 0) ｷｧ A(,5; 6) 陦ｧﾀ ' = = arctg (,) 0903 ﾀｧOx ｧﾀﾀｧOx, ﾀｧ Oy ｧﾀｧ07 04 ﾀ縺

12 3x 縺ｧ06 04 縺 x 縺ｧ06 04 縺ﾀﾀｧｧ閨03 04 縺ｰｧ07 03 ｧﾀﾀ罍ｧﾀ. ｰｧ縺 , 04 縺閨070 ｧ08 06 縺縺06 ﾀｧ07 閨縺縺ｧ06 04 縺 x cos + y sin, =0; () 05 閨06 閨00 ｧ縺04060 陦ｧ (003 縺0 00 ｧ), ｧ陦ｧ縺 , 縺04060 陦ｧﾀｧ Ox. ｪ罍ｧ0406 ﾀ05 ｧｧ06 04 縺 Ax + By + C = 縺ｧ閨06, ﾀｧ縺0 ｧ縺06060 ｧｧ M =, ﾀ , 04 縺0305 ｧﾀ A + B C.. 7 ﾀﾀｧ06 d ｧ (x 0 ; y 0 ) 閨03 04 縺閨060; 06 ﾀ00 ｧﾀ縺ｧ07 04 縺ﾀｧ罍陦ｧ陦ﾀ05 04 ｧ罍陦ｧ (x 0 ; y 0 ) ｧｧﾀﾀｧｧ006: d = jx 0 cos + y 0 sin, j; () ｧ00 ｧ d = jax 0 + By 0 + Cj A + B : ( 0 ) 3. ｰｧ07 03 ｧﾀﾀ罍ｧﾀ閨06 04 縺ｧ 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. ｰｧ縺 ,04 縺閨070 ｧ08 06 縺縺06- ﾀｧ07 閨縺 : (Ax + By + C) +(A x + B y + C )=0: (4) ｧ0504 =, ｧﾀｧｧ0 ｧ (4) 縺0606 ｧ縺罍ｧ0406 ﾀ05 ｧ縺ｧ閨ｧ07 04 縺 : ) 3x, 4y, 0 = 0; ) x + y +3=0; 3) y = kx + b ﾀ05 縺03 ｧ縺 , 06 ﾀ00 ｧ閨00 ｧ0 003 縺0 00 ｧ =, ﾀｧ Ox 04060: ) 45 ;) 35 ;3) 5 ;4) ｧﾀｧｧ08 04 縺ｧﾀﾀｧ A(4; 3), B(; ) ｧ C(; 0) 閨03 04 縺 x +4y, 0= ﾀ05 縺03 ｧｧｧ 04 縺

13 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧｧﾀﾀｧ罍陦ｧ0 05 閨03 04 縺 x,, 5y +39= , 縺 x, 3y =6 ｧ 4x, 6y = , ｧｧﾀﾀｧ閨06 0 ｧ0 ｧ. ｰ 09 0 ｧ閨00308 ｧ 04 縺縺03 ｧｧｧﾀﾀｧ閨03 陦縺縺ｧ k ｧ 06 ﾀｧ07, 縺 y = kx 罍陦ｧﾀﾀｧ06 d = ｧﾀｧ罍ｧ 06 ﾀﾀ , 縺 x, 3y =00 ﾀﾀｧ06 d = ｮ03 ﾀ05 04 ｧｧ06 04 縺 , ｧ A(4;,) 0 ﾀﾀｧ06 d =4 ｧ縺 x, 5y = ｧﾀｧ07 03 ｧﾀﾀ罍ｧﾀ閨06 04 縺ｧ x +3y =0 ｧ 3x +y = ｧﾀｧ07 03 ｧﾀﾀ罍ｧﾀ閨06 04 縺ｧ 3x +4y = ｧ y = ｧﾀｧ罍ｧｧｧ M (x; y), 罍ｧﾀ閨04 ｧ07060 ｧｧ 03 ﾀﾀ縺縺 y =x, 4, 縺 y =4, x ｧﾀｧ06 04 縺 , 04 縺閨縺 M 0406 縺06 ﾀｧ07 04 縺 x + y +6= 0 ｧ 3x +5y, 5 = 0 ｧ 06 縺 N (;,) ( 閨ｧ M) ｧﾀｧ06 04 縺 , 04 縺閨縺 M 0406 縺06 ﾀｧ07 04 縺 x, y +0=0 ｧ 8x +4y +9=0 ｧ縺 x +3y =0( 閨ｧ M) ｧ閨00 ｧﾀ BD 0405 縺ｧ0906 ﾀ 0406 縺00 ｧ0 0 ｧ A(,3; 0), B(; 5) ｧ C(3; ) ｧﾀｧ06 04 縺 , 04 縺閨縺 A(; 4) ｧ罍陦ｧﾀﾀｧ06 d = 縺罍ｧ0504, ｧ A(,4;,3), B(,5; 0), C(5; 6) ｧ D(; 0) ﾀ縺00 ｧ0 0 ｧｧｧ, ｧｧﾀｪ06 縺罍陦ｧ縺閨縺陦ｧﾀﾀｧｧ A(; ) ｧ B(4; 0) ｧﾀﾀｧｧﾀｧ罍ｧ 06 ﾀﾀ , 縺 x +y, 5=00 ﾀﾀｧ06, ｧﾀｧ罍ｧｧｧ M (x; y), 罍ｧﾀ閨04 ｧ07060 ｧｧ 03 ﾀﾀ07 04 閨縺 y = x, 縺 y =,x ｧﾀｧ06 04 縺 , 04 縺閨縺 M 0406 縺06 ﾀｧ07 04 縺 x, 3y +5=0 ｧ 3x + y, 7=0 ｧ 0406 縺04060 陦ｧ縺縺 y =x ( 閨ｧ M).

14 3x 7. ｮｧ 0 04 縺 x 7. ｮｧ 0 04 縺ｪ06 縺罍陦ｧ縺ﾀ05 ｧ 04 縺 , ﾀ 04 縺ｧ x + y = a ｧ x =005 縺ｧ閨0406 a ｧ A(,4; 0) ｧ B(0; 6). ｪ06 縺06 ﾀ06 縺06 陦ｧ縺06 09 AB 04 縺ﾀ05 ｧ 04 縺 , 0305 ﾀﾀｧOx 0305 縺 , 04 閨ｧ08, ﾀｧ Oy ｧ A(,; 0) ｧ B(;,) 縺 OA 0403 ﾀ05 縺 OACD, 陦ｧｧ縺縺06 ﾀﾀ B ｧﾀｧ07 ﾀ0503 縺030, 陦ｧｧｧ CAD ｧｧ閨04 05 縺ｧ09, 縺ｧ y =x, y =,x ｧ y = x + b. 5. 0ｷ罍陦ｧ縺閨06003 閨ｧ縺04060 陦ｧ縺縺 , ｧ06 ﾀ 04 縺 x +y = a 陦縺縺ｧｧ閨縺ｧｧ縺0600 ｧ縺ｧ09, 06 ﾀ00 ｧｧﾀ0503 縺030: (AB) x, 3y +3 = 0 ｧ (AC) x +3y +3 = 0 ｧ 03 ﾀｧ06 D(,; 3) 0403 ﾀ AD ｧﾀ0503 縺陦縺縺ｧ09 3x + y =0 ｧ x, 3y =0 ｧ (5; 0) ﾀｧｧｧ 0406 罍ｧ00605 ｧ閨04 05 縺ｧ縺ｧ0906 ABC 003: ) ｧ06 ﾀ0503 縺03003 (AB) 3x +y =;) ｧﾀ (BM) x +y =4;3) ｧﾀ (AM ) 4x + y = 6, 05 閨06 M 縺06 ﾀｧﾀｧﾀｧ07 ﾀ0503 縺030 AC, BC ｧ 0403 ﾀ CM ﾀ0503 縺ｧ070 ｧ y = = x, ｧ 5y = x ｧｧ縺06 ﾀﾀ陦ｧｧﾀｧ07 閨陦縺0605 ｧ08 ﾀ0503 縺ｧ陦ｧ縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(0;,4), B(3; 0) ｧ C(0; 6) ｧﾀﾀｧ縺00 ｧ003 C ｧﾀﾀ罍ｧﾀ A ｧﾀｧ罍ｧｧｧ M (x; y), 閨04 ｧﾀ , 0503 ﾀ0600 ﾀﾀｧ閨03 04 縺 y = x ｧ y =,x= 03 ﾀﾀﾀｧﾀ05 縺03 ｧﾀ05 ｧ, 罍陦ｧ縺閨縺ﾀ0504 0: ) x, <y<0 ｧ x>0; ), 6 y 6 x 6 ; 3) < x + y<8, x>0 ｧ y> ｮ0503 縺03003 AB ｧ BC ｧ070 ｧ x,y +5=0 ｧ x,y +4 = 0, 陦ｧｧ縺06 ﾀﾀ M (; 4) ｧ閨00 ｧﾀｧ 0406 縺00 ｧ縺陦縺縺ｧ09, 06 ﾀ00 ｧ縺00 ｧ0 04 縺 C(3;,) ｧｧ06 05 ｧ x, y +=0.

15 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧ閨縺00 ｧ縺ｧ09 A(,4; 3) ｧ B(4;,) ｧ縺06 ﾀｧﾀ0305 M (3; 3) ｧ 05 縺縺00 ｧ006 C ｧﾀ00 ｧ罍陦ｧ縺00 ｧ003 縺03003, 06 ﾀ00 ｧｧ 0406 ﾀｧ07 閨ﾀ0503 縺030: x +y =4 ｧ x +y =0, ｧｧ06 03 閨00308 ｧ陦ｧ : y = x ｮ03 ﾀ05 04 ｧｧ07 ﾀ0503 縺縺ｧ09, 閨縺00 ｧ006 A(0; ) ｧｧﾀ0305: (BM) x + y = 4 ｧ (CM) y =x, 05 閨06 M 縺06 ﾀｧﾀ縺 x +y, 4 = 0 ｧ A(5; 7) ｧ: ) 04 縺ｧ06 B ｧ A 縺 ; ) ｧ06 C ｧ A 縺ｰ 09 0 ｧｧﾀｧ縺04060 陦ｧ AB ｧ縺0600 ｧﾀﾀ05003 ﾀｧ縺 , 0 08 閨 B, ﾀ0504 ﾀ06 縺06 陦ｧ0 AC 縺 x + y, 6=0 ｧ閨ｧ A ｧ B ﾀ 03 罍陦ｧｧ y A =6 ｧ y B =, ｧﾀｧﾀ AD 05 縺ｧ09 AOB, ｧ 0606 閨00 ｧ006 ｧ DAB. x 縺ﾀ0504 ｰｧ縺ﾀ05 ｧﾀ縺 C(a; b) ｧ陦ｧ06 ﾀ030, R: (x, a) +(y, b) = R : () ｭﾀ00 ｧｧｧ () ﾀ09 縺ﾀｧ, 罍ｧｧ x + y + mx + ny + =0: () ｪｧ07 () 縺ｧｧ06 04 ｧ (), ﾀ05 ｧｧ07 () 0403 閨0600 ｧ陦 0503: x + m + y + n = m 4 + n, : (3) ｧﾀｧ縺ﾀ05 ｧﾀ縺030 C(,4; 3), - 陦ｧ06 ﾀ030 R =5 ｧ 0403 ﾀ05 縺03 ｧｧｧ縺ﾀ05 ｧｧ A(,;,), B(3; ), O(0; 0)? (,4; 6) ｧﾀｧ縺ﾀ05 ｧ, 陦ｧ縺縺0308 ﾀｧ縺 OA ﾀ05 縺03 ｧ縺ﾀ05 ｧ: ) x + y, 4x +6y, 3 = 0; ) x + y, 8x =0; 3) x + y +4y = ﾀ05 縺03 ｧ縺ﾀ0504 x + y +5x =0, 04 縺 x + y =0 ｧｧｧｧ縺06 ﾀｧｧﾀｧ縺ﾀ05 ｧ, 09 ﾀﾀ07 03 ﾀ陦ｧ0 05 ｧ 04 縺閨縺 A(; ).

16 3x 縺ﾀｧ閨06 陦ｧ06 ﾀ 0 ｧ 0309 縺ﾀ05 ｧ x + y +4x,, 6y =0, 04 縺閨ｧｧ 0406 縺06 ﾀｧﾀ 03 ﾀ0406 Oy ｧﾀｧ縺ﾀ05 ｧ, 04 縺閨縺ｧ A(,; 3), B(0; ) ｧ C(;,). ｰ 09 0 ｧｧﾀｧ06 ｧﾀ縺ﾀ05 ｧｧ閨06 x + + y + mx + ny + =0, 0403 陦ﾀ05 04 ｧ罍陦ｧ閨ｧｧｧ m, n ｧｧﾀｧ縺ﾀ05 ｧ, 04 縺閨縺ｧ 0406 縺06 ﾀｧ縺ﾀ05 ｧ x + y +4x, 4y =0 ﾀ 04 縺 y =,x ｧ 06 縺 A(4; 4) 縺06 閨0600 ｧﾀ0504 ﾀｧ07 09 罍ｧ y =,,x, 4x. 703 ﾀ05 縺03 ｧ罍ｧｧﾀｧ06 09 ﾀ縺ﾀ05 ｧ x + y,, 8x, 4y +6=0, 04 縺閨ｧ罍陦ｧ A(a; 0). ｯ03 09 M 閨04 ｧﾀ , OMA OMA 03 ﾀﾀ07 04 縺縺06 閨0600 ｧ罍ｧ06 閨04 ｧ07060 ｧｧ M ｧ A(,6; 0) ｧ B(; 0) ｧ罍ｧ 06 ﾀﾀ , ｧ縺縺06 09 ｧ OA ｧ OB 04 ｧ閨ｧｧ縺06 閨0600 ｧ罍ｧｧ M (x; y), 閨04 ｧﾀ , 0503 ﾀ陦ﾀﾀｧ閨 A(,a; 0), B(0; a) ｧ C(a; 0)03 ﾀﾀ a 縺06 閨0600 ｧ罍ｧｧ M (x; y), 閨04 ｧﾀ , 0503 ﾀ陦ﾀﾀｧ閨03 03 ｧﾀﾀ罍ｧﾀ罍陦ｧﾀﾀ a 縺ﾀ0504 x + y = a. 0ｷｧ A(a; 0)04 縺閨ﾀ罍閨縺06 閨0600 ｧ罍ｧ 06 ﾀﾀ0503 ﾀ06 縺06 陦ｧｧ罍ｧ A(,3; 0) ｧ B(3; 6) ｧﾀｧ縺ﾀ05 ｧ, 陦ｧ縺縺0308 ﾀｧ縺 AB ｧ縺03 ｧ陦ｧ06 ﾀ縺ﾀ050608: ) x + y, 6x + +4y, 3 = 0; )x + y +5x, 7y +; 5=0;3)x + y +7y = ﾀ05 縺03 ｧ縺ﾀ05 ｧ縺ﾀﾀ 0605 ﾀ07 03 ﾀｧ Ox 罍陦ｧ0 05 ｧ 04 縺陦ｧ05 06 縺 A(0;,4) ｧﾀｧ縺ﾀ05 ｧｧｧｧ 0406 縺06 ﾀｧﾀ 03 ｧﾀﾀ罍ｧﾀ 0 ｧ罍陦ｧｧﾀｧ縺ﾀ05 ｧ, 04 縺閨縺罍陦ｧ0 05 ｧ 06 縺ｧ 0406 縺06 ﾀｧ07 04 縺 x + y + a =0 ﾀ 0309 縺ﾀ x + y = a.

17 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧｧﾀｧ07 09 ﾀ , 04 縺閨ｧ罍陦ｧ縺ﾀ05 ｧ, 04 縺閨縺ｧ A(;,), B(0;,) ｧ C(,3; 0) ｧ閨06 陦ｧ06 ﾀ 0 ｧ 0309 縺ﾀ05 ｧ x + y, 4x + +6y, 5=0, 04 縺閨ｧｧ 0406 縺06 ﾀｧﾀ 03 ﾀ0406 Ox , A(3; 0) ｧ罍ｧ 0309 縺ﾀ05 ｧ x + + y, 4x +y +=0, ｧ 0 04 ｧﾀｧ罍閨03, 閨ﾀ A ｰ09 0 ｧ06. 0ｷ3 ﾀ罍 0406 縺04060 陦ｧ縺0 09 CA,05 閨06 C 縺ﾀ05 ｧ. 6. ｯ03 09 M (x; y) 閨04 ｧﾀ , 0503 ﾀ陦ﾀﾀｧ閨罍陦ｧ0 05 ｧ閨ｧ A(,a; 0) 03 ﾀﾀ a 縺06 閨0600 ｧ罍ｧ06 閨04 ｧ07060 ｧｧ M 縺ﾀ0504 x + y =4.0ｷｧ 0606 A(,; 0) 04 縺閨罍 AB ｧ 04 縺03 閨ﾀﾀｧ06 BM = AB 縺06- 閨0600 ｧ罍ｧ 06 ﾀﾀ M 縺 AM = a 0406 縺ﾀﾀ0903 ﾀ05 ｧ xoy, 03 ﾀﾀ Ox, 05 09, A ﾀｧ縺ﾀ05 ｧ x + y = a 縺06 閨0600 ｧ罍ｧ06 閨04 ｧ07060 ｧｧ M. x ｧ04 ﾀｧ04 ﾀﾀ罍ｧ 06 ﾀﾀ ; ﾀ0600 ﾀ- ﾀｧ閨0308 ｧ縺0308 閨03 閨 F ｧ F ( ﾀ0304) 06 ﾀﾀｧｧ0 a; F F ｧ 06 ﾀ (04 縺03 ﾀ ) ｧｧ04 ﾀ x a + y =: () b ｧ04 ﾀ, ｧ060 (), ﾀｧ罍ｧﾀｧﾀ罍陦ｧ0 05 ( 罍ｧﾀ. ) 縺03 a ｧ b ﾀﾀ070 ｧｧ04 ﾀ. 7 ｧﾀ. 706 ﾀ0504 a>b, ﾀ03 F ｧ F 閨0705 ﾀﾀｧ Ox 0 ﾀﾀｧｧ c = a, b ｧ06 c = e< ﾀﾀ罍ｧ-

18 3x ｧ04 ﾀ 5 ﾀｧｧ04 ﾀ. 7 ﾀﾀｧｧ M (x; y) ｧ04 ﾀ閨ﾀ0304 ( 陦ｧ06 ﾀ縺03) 0304 縺06 閨ﾀ縺ｧ r = a, ex; r = a + ex: () ｭﾀ00 ｧ 0706 a<b, ﾀ閨0705 ﾀﾀｧ Oy, c = b, a, e = c b, r = b ey ﾀ05 縺03 ｧｧ04 ﾀ x +4y =6, ｧﾀ03 ｧ 0509 ﾀ罍ｧﾀｧｧﾀｧ 06 ﾀｧｧ04 ﾀ, 0 07, 0503: ) ﾀﾀｧ閨ﾀ 0 ｧ , ﾀ04 b = 3; ) ﾀ04 a =6, 0509 ﾀ罍ｧﾀｧ e =0; ｧﾀ04 b ｧ 0509 ﾀ罍ｧﾀｧ e ｧ04 ﾀ, ｧﾀ04 a =5 ｧ c, : ) 4,8; ) 4; 3) 3; 4),4; 5) ﾀ05 縺03 ｧ閨0308 ｧｧ04 ﾀｦ閨04 ｧﾀｧ04 ﾀ06, 閨0030 ｧﾀ縺陦ｧ05 ﾀ07 ｮｧﾀﾀｧｦｧ閨03 ｮ罍ｧ0300 ｧｧ , , 0 ｧ , ｧﾀ04 ｧ 0509 ﾀ罍ｧﾀｧ縺03 ｧ0503 6ｦｧｧ04 ﾀ, ﾀｧ罍ｧﾀｧﾀ罍陦ｧ0 05, 04 縺陦ｧ05 06 縺ｧ M (; 3) ｧ B(0; ) ｧﾀｧ06 ｧｧﾀﾀｧｧ M 閨ﾀｧ04 ﾀ, ﾀｧ罍ｧﾀｧﾀ罍陦ｧ0 05, ﾀ縺閨0705 ﾀﾀｧ Ox, 04 縺陦ｧ05 06 縺 M (,4; ) ｧｧﾀ罍ｧﾀｧ e = ｧﾀｧｧ04 ﾀｧｧ陦ｧ06 ﾀ縺ｧ M ｧ閨00 ｧ罍閨ｧ04 ﾀ x +y =8, 閨閨06 03 ﾀ070 ｧｧ 0509 ﾀ罍ｧﾀｧｧ04 ﾀ, 06 ﾀ00 ｧﾀﾀｧ閨ﾀ 0 ｧﾀﾀｧ閨ｧｧﾀｧ04 ﾀ x +4y =40404 ｧﾀｧ縺ｧ09, 03 閨0 ｧ 0406 縺00 ｧ縺ﾀﾀﾀｧ縺06 閨0600 ｧ罍陦ｧ閨陦縺0605 ｧ縺00 ｧ0 05 縺ｧ09. ｰ 09 0 ｧｧﾀｧ06 03 閨00308 ｧﾀ0503 縺030, ｧ k = tg 30, ｧｧｧ縺06 ﾀｧ07 ﾀｧ04 ﾀｧ04 ﾀ06 9x +5y = ｧ , ﾀﾀｧ縺0308 閨ﾀ縺ﾀﾀｧ閨ﾀ罍陦ｧﾀ縺ﾀ05 ｧ x + y =36 ﾀ縺ｧﾀｧ罍ｧ縺06 閨0600 ｧ罍ｧｧ M, 罍ｧﾀ閨04 ｧｧｧ 03 ﾀﾀ07 04 閨ｧ F (,; 0), 縺 x =,4.

19 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧ縺 AB 0403 ﾀ閨00 ｧ003 a + b 閨04 ｧﾀ , A ﾀｧﾀｧ Ox, B ﾀｧOy 縺06- 閨0600 ｧ罍ｧ06 閨04 ｧ07060 ｧｧ M 0305 縺06 09, 閨ﾀ05 ｧ BM = a ｧ MA = b ( ｧ0405 ｧ 06 ﾀ09 ｧ08 09 ｧ縺ｧ罍閨ｧ0 ｧ) 縺ﾀ05 ｧ x + y = b ｧ x + y = a (b<a). 7 縺03 ｧ OBA 0406 縺06 ﾀｧ08 ﾀﾀ B ｧ A, ｧ縺縺閨縺 , ﾀ罍陦ｧ- 0 05, 閨縺06 ﾀｧ07 ｧ M 縺06 閨0600 ｧ罍ｧ 06 ﾀﾀ M ｧﾀ縺03 ﾀｧｧ04 ﾀ, 縺ﾀ- ﾀｧ閨ｧﾀ0304 閨ﾀｧｧｧ04 ﾀ, ﾀｧ罍ｧﾀｧﾀ罍陦ｧ0 05, 04 縺陦ｧ05 06 縺ｧ M ( 3; 6) ｧ A(6; 0) ｧﾀｧ06, ｧ 0509 ﾀ罍ｧﾀｧｧﾀﾀｧｧ M 閨ﾀｧ閨00 ｧ罍閨ｧ04 ﾀ x a + y =, b 0403 陦ｧｧ 04 縺ｧ09, 0403 ﾀ05 縺ﾀｧ04 ﾀｧｧｧｧ04 ﾀ x +4y =4 ｧ0309 縺ﾀ05 ｧ, 04 縺閨縺ﾀｧ04 ﾀｧｧ縺縺00 ｧ縺 x =, ｧ , 03 陦ｧﾀｧ縺縺00 ｧｧ04 ﾀ x +5y = ｧ04 ﾀ06 x +5y = ｧ , 陦ｧ06 ﾀ縺縺縺04060 陦ｧ縺003. ｰ09 0 ｧ06. 0ｷ3 ﾀｧﾀｧ 0406 縺06 ﾀｧ07 ﾀｧ04 ﾀ縺ﾀ05 ｧ, 04 縺閨縺ﾀｧ04 ﾀｧｧ罍陦ｧ縺03 ﾀ09 ｧﾀﾀ縺ﾀ05 ｧ x + y = ｧ閨縺06 閨0600 ｧ罍ｧ縺06 閨0600 ｧ罍ｧｧ M, 罍ｧﾀ閨04 ｧｧｧ 03 ﾀﾀ縺ｧ A(; 0), 縺 x =9. x ｧ0406 縺ｧ0406 縺ﾀ罍ｧ 06 ﾀﾀ ; 003 ﾀ0504 ﾀﾀｧ閨0308 ｧ縺0308 閨03 閨 F ｧ F ( ﾀ0304) 06 ﾀﾀｧｧ0 a (0 < a <F F ) ｧ 06 ﾀ (04 縺03 ﾀ ) ｧ06 05 ｧ0406 縺 x a, y =: () b

20 3x ｧ0406 縺ｧ0406 縺030300, ｧ060 (), ﾀｧ罍ｧﾀｧﾀ罍陦ｧ0 05 ( 罍ｧﾀ. ) 縺06 ﾀﾀ04Ox A(a; 0) ｧ A (,a; 0) 0406 縺00 ｧｧ0406 縺ｧ縺06 ﾀﾀ04Oy a ﾀﾀﾀ0406, b 00 ｧﾀ c = a + b 7 ｧﾀ. 06 ﾀ0504 ﾀﾀｧﾀ閨ｧ06 c a = e> ﾀﾀ罍ｧﾀｧｧ0406 縺縺 y = b x ﾀ07 ﾀｧ004a ｧ 05 ｧ0406 縺ﾀﾀｧｧ M (x; y) 05 ｧ0406 縺閨ﾀ0304 ( 陦ｧ06 ﾀ縺03) 0304 縺06 閨ﾀ縺ｧ r = jex, aj; r = jex + aj: () 06 ｧ0406 縺030300, 縺0308 a = b, ﾀﾀ0503 縺 , ｧ06 x,y = a, ｧ07 ﾀｧ y = x. 06 ｧ0406 縺 x a, y b = ｧ y b, x = ﾀ07 ﾀ0304 縺ｧ. a ﾀ05 縺03 ｧｧ0406 縺 x,4y =6 ｧ 0606 ﾀｧｧﾀ03, 0509 ﾀ罍ｧﾀｧｧ閨06 ﾀｧｧｧ0406 縺 x, 4y = M ﾀ 03 罍陦ｧ , ｧﾀﾀｧ閨ﾀｧﾀｧ 06 ﾀｧ06 05 ｧ0406 縺 , 0 07, 0503: ) ﾀﾀｧ閨ﾀ 0 ｧ c =0, 閨縺00 ｧ0 0 ｧ a = = 8; ) ﾀﾀ04 a = 5, 0509 ﾀ罍ｧﾀｧ e = = ; ｧ0406 縺ﾀｧ罍ｧﾀｧﾀ罍陦ｧ0 05, 04 縺陦ｧ05 06 縺 M (6;, ) ｧｧｧﾀ04 b = ｧﾀｧ06 ｧｧﾀﾀｧｧ M 閨ﾀ0304.

21 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧｧﾀｧ06 05 ｧ0406 縺 , ｧ縺00 ｧﾀ 08, ﾀ縺00 ｧｧ04 ﾀ x 5 + y = ｧﾀｧ06 05 ｧ0406 縺 , ｧﾀ罍ｧﾀｧ e =,04 縺閨縺 (a; a 3) ｧﾀｧ罍ｧﾀｧﾀ罍陦ｧﾀ05 縺03 ｧｧ0406 縺 y = a + x, ｧ罍陦ｧﾀ0304 ｧ閨06 ﾀｧｧｧﾀｧ07 09 ﾀｧ0406 縺 x, 4y = =6, 04 縺閨ｧｧ A(0;,) ｧﾀﾀｧﾀ 05 ｧ0406 縺 x a, y = 閨 b ﾀｧｧ閨06 ﾀｧｧ., y ｧﾀ0503 縺陦 05, 0404 ｧﾀｧ0406 縺 x a, =, ｧｧﾀﾀ0006 閨 , ｧ06 05 ｧ0406 縺ｧﾀ陦 05. b ｧ 0509 ﾀ罍ｧﾀｧｧ0406 縺 , ﾀｧ縺0308 ﾀ03- ﾀﾀﾀﾀ : ) 60 ;) 縺06 閨0600 ｧﾀ0504 ﾀｧ07 09 罍ｧ y =, 9+x. 703 ﾀ05 縺03 ｧ罍ｧ縺06 閨0600 ｧ罍ｧｧ M (x; y), 罍ｧﾀ閨04 ｧ07060 ｧｧ 03 ﾀﾀ07 04 閨ｧ縺 x =, F (4; 0) ｧ A(,; 0) ｧ B(; 0). ｯ03 09 M 閨04 ｧﾀ , AMB B 03 ﾀﾀ07 04 閨 A 縺06 閨0600 ｧ罍ｧ06 閨04 ｧ07060 ｧ A(a; 0) ﾀｧOy 閨04 ｧﾀ B 縺 BE, Ox, 閨ﾀ縺06 09 ｧ 7 ｧﾀ. 3 BM ｧ BM, AB 縺06 閨0600 ｧ罍ｧ 06 ﾀﾀ M ｧ M 縺 x = b ｧ x = a (b <a). 7 縺03 ｧ OA ( 罍ｧﾀ. 3) 0406 縺06 ﾀ縺 x = b ( ｧ00 ｧ x =,b) B ｧ 04 縺 x = a ( ｧ00 ｧ x =,a) A. 7 陦ｧ06 ﾀ030 OA 0304 ｧﾀ 0 閨0605, 0406 縺06 ﾀ Ox C. 0ｷ B ｧ C 04 縺閨縺 , ﾀﾀ Ox ｧ Oy, 閨縺06 ﾀｧ M 縺06 閨0600 ｧ罍ｧ 06 ﾀﾀ M ｧﾀｧ 06 ﾀｧ06 05 ｧ0406 縺 , 0 07, 0503 ﾀﾀｧ閨00308 ｧ縺00 ｧ0 閨ﾀｧ.

22 3x ｧｧ 0406 縺06 ﾀｧ07 ﾀｧｧ0406 縺 x, 3y = = ﾀ 0309 縺ﾀ050406, ｧﾀ06 05 ｧ0406 縺ｧ04 縺閨縺罍陦ｧｧ0406 縺縺陦ｧ05 06 縺 M (6; 3 5=), ﾀｧ罍ｧﾀｧﾀ罍陦ｧ0 05 ｧｧﾀﾀ04 a = ｧﾀｧ縺04060 陦ｧ縺0304, ｧﾀ 05 ｧ0406 縺ﾀｧｧ0406 縺 x, 6y = ｧ , ﾀﾀｧ縺0308 閨ﾀ 04 閨 , 060 閨ｧ0406 縺 x, y = ｧ , 陦ｧ06 ﾀ縺縺縺04060 陦ｧ縺003 ( ﾀｧ ). 08. ｯ03 09 M 閨0600 ｧ05 ﾀﾀｧ閨ﾀ 0 ｧ 05 ｧ0406 縺 x, 6y = ｧｧ F M : MF =:3,05 閨06 F ﾀ05 ｧ0406 縺ｪ06 縺 M 04 縺閨縺ﾀｧOx ｧｧ 0406 縺06 ﾀｧ縺ﾀﾀｧｧ 05 ｧ0406 縺縺06 閨0600 ｧ罍ｧｧ M, 閨04 ｧﾀ , ﾀﾀ07 04 閨ｧ F (,8; 0), 縺 x =, ｧ A(,a; 0) ｧ B(a; 0). ｯ03 09 M 閨04 ｧﾀ , MAB 03 ﾀﾀ縺 AMC 05 縺ｧ09 AM B 縺06 閨0600 ｧ罍ｧ06 閨04 ｧ07060 ｧｧ M. x ﾀ罍ｧ 06 ﾀﾀ ; 03 陦ｧｧ ( ﾀ ) ｧ縺 ( 陦ｧ縺罍ｧﾀ03) ｧ 06 ﾀｧｧ閨04 04 ｧ : ) y =x ﾀｧ罍ｧﾀｧﾀｧ Ox ( 罍ｧﾀ. 4); ) x =y ﾀｧ罍ｧﾀｧﾀｧ Oy ( 罍ｧﾀ. 5). 7 ｧﾀ. 4 7 ｧﾀｧ08 ﾀ縺00 ｧ , , ﾀｧﾀｧ罍ｧｧ, 陦ｧ05 ﾀ罍陦ｧ0 05.

23 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧｧﾀ F ; ｧ M (x; y) r = x +. y =x ｧ陦ｧ縺罍ｧﾀ06 x =, ; 陦ｧ06 ﾀ x =y ｧﾀ F 0; ｧ陦ｧ縺罍ｧﾀ06 y =, ; 陦ｧ06 ﾀｧ M (x; y) r = y +.. ｮ03 ﾀ05 04 ｧｧ罍ｧ 06 ﾀﾀ , 03 陦ｧｧ F (0; ) ｧ縺 y = ｧｧ 0406 縺06 ﾀｧ罍ｧﾀ 03 ﾀ070 ｧ罍陦ｧ0 05 ｧ 0403 ﾀ05 縺03 ｧｮ03 ﾀ05 04 ｧｧ罍ｧ 06 ﾀﾀ , 03 陦ｧ罍陦ｧ0 05 ｧ縺 x =, ｧｧ 0406 縺06 ﾀｧ罍ｧﾀ 03 ﾀ070 ｧ罍陦ｧ0 05 ｧ 0403 ﾀ05 縺03 ｧﾀ05 縺03 ｧ , ｧ070 ｧ: ) y =4x; ) y =,4x; 3) x = 4y; 4) x =,4y, ｧﾀ03 ｧ陦ｧ縺罍ｧﾀ03 ｧ 0 04 ｧﾀｧ07 陦ｧ縺罍ｧﾀｧﾀｧ : ) 04 縺閨縺ｧ (0; 0) ｧ (;,3) ｧﾀｧ罍ｧﾀｧﾀｧ Ox; )04 縺閨縺ｧ (0; 0) ｧ (;,4) ｧﾀｧ罍ｧﾀｧﾀｧ Oy 閨0406 ﾀﾀ05 ｧ縺 ( 罍ｧﾀ. 6) ｧﾀｧﾀｧ縺ﾀ0608, 06 ﾀ00 ｧ 04 縺0305 ｧ OA = a, 閨00 ｧ0 04 縺 BC = b ｧﾀｧ縺ﾀ05 ｧ, ｧﾀ y = x ｧ09 ﾀﾀ陦ｧ縺罍ｧﾀｧｧ 0406 縺06 ﾀｧｧ 0309 縺ﾀ05 ｧｧﾀｧｧ 0606 陦ｧ縺罍ｧﾀ03, 06 ﾀ00 ｧ縺陦ｧ05 06 縺ｧ 0406 縺06 ﾀｧ07 04 縺 x + y = 0 ｧ 7 ｧﾀ縺ﾀ05 ｧ x + y +4y =0 ｧﾀｧ罍ｧﾀｧﾀｧ Oy. 703 ﾀ05 縺03 ｧ縺ﾀ0504, 04 縺ｧ

24 3x y = 6x ｧ , 陦ｧ06 ﾀ縺 , ｦ006 縺縺08003 ﾀ縺ｧ縺ﾀｧﾀｧ罍ｧｧ. 6 7 ｧ縺ﾀ0, ｧﾀﾀﾀｧｧ縺00 ｧﾀ05 ｧ0504 ｧﾀｧ09 ﾀ040605, 06 ﾀ00 ｧ閨ｧ閨ﾀ ? 縺06 閨0600 ｧﾀ0504 ﾀｧ07 09 罍ｧ y =,,x. 703 ﾀ05 縺03- ｧ罍ｧｷ縺00 ｧ y = =x 04 縺閨ﾀ閨ｧﾀｧ罍ｧ 06- ﾀﾀ05 ﾀ06 縺06 陦ｧｧ罍縺06 閨0600 ｧ罍ｧ 06 ﾀﾀ縺縺ﾀ050608, 09 ﾀ 06-0 ｧ08 ﾀ縺ﾀ05 ｧ x + y =ax ｧ 03 ﾀｧ Oy ｧ A(0; a) ｧ B(a; a) 縺06 09 ｧ OA ｧ AB 閨 n 7 ｧﾀﾀｧ A, A, A 3, ::: ｧ B, B, B 3, ::: ( 罍ｧﾀ. 7). 706 ﾀ0504 M k 縺06 ﾀｧ OB k ﾀ 04 縺 A k M k kox , ｧｧ M k y = ax. 703 ﾀ05 縺03 ｧｧ0 04 罍ｧ y =4x, y =5x, y =3x. 4. ｮ03 ﾀ05 04 ｧｧ罍ｧ 06 ﾀﾀ , 03 陦ｧ罍陦ｧ0 05 ｧ縺 x = ｧｧ 0406 縺06 ﾀｧ罍ｧﾀ 03 ﾀ070 ｧ罍陦ｧ0 05 ｧ 0403 ﾀ05 縺03 ｧｮ03 ﾀ05 04 ｧｧ罍ｧ 06 ﾀﾀ , 03 陦ｧｧ F (; 0) ｧ縺 y = ｧ 0406 縺00 ｧ , ｧ 0406 縺06 ﾀｧﾀ Ox ｧ 0403 ﾀ05 縺03 ｧｧﾀｧ : ) 04 縺閨縺ｧ (0; 0) ｧ (,; ) ｧﾀｧ罍ｧﾀｧﾀｧ Ox; )04 縺閨縺ｧ (0; 0) ｧ (; 4) ｧﾀｧ罍ｧﾀｧﾀｧ Oy ｧﾀｧｧ 0606 陦ｧ縺罍ｧﾀ03, 06 ﾀ00 ｧ縺陦ｧ05 06 縺ｧ 0406 縺06 ﾀｧ07 04 縺 y = x ｧ 0309 縺ﾀ05 ｧ x + y +6x =0 ｧﾀｧ罍ｧﾀｧﾀｧ Ox ﾀ05 縺03 ｧ縺 , 0309 縺ﾀ0504 ｧ y = x 0404 ｧﾀｧ縺ｧ縺06 閨0600 ｧ縺00 ｧ003 ( ﾀｧ ) ｧﾀｧ07 09 ﾀ y = 8x, 04 縺閨ｧｧ A(0;,).

25 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧ 30. ｪ06 縺ﾀ y =,4x 04 縺閨縺ﾀｧOx ｧﾀｧ06 04 縺ｧｧ閨00 ｧﾀ罍閨03. x. 6 7 ｧ縺罍ｧﾀ03, 陦ｧ縺03 ｧ 09 ﾀ罍ｧ縺縺07 閨ｧ縺罍ｧﾀ 0 ｧｧ04 ﾀ x a + y =(04 罍ｧ a>b) ｧ05 ｧ0406 縺 b x a, y = ﾀ07 04 縺 , ﾀｧ Oy ｧ0305 ﾀｧ b ﾀﾀｧ06 a,05 閨06 e 0509 ﾀ罍ｧﾀｧ罍ｧ e ｰｧ07 陦ｧ縺罍ｧﾀ: x = a e : () ｮﾀ陦ｧ縺罍ｧﾀ: ｧ06 ﾀﾀｧｧ 09 罍ｧ閨ﾀｧ閨03 ﾀﾀ陦ｧ縺罍ｧﾀﾀ罍ｧ- ﾀｧ罍ｧ r = e: () d. 6 7 ｧ縺罍ｧ縺縺07 閨ﾀ罍ｧ 06 ﾀﾀ0503 ﾀ06 縺06 陦ｧ罍. 6 7 ｧｧｧ04 ﾀｧ 05 ｧ0406 縺ﾀ縺06 09 ｧｧ 0006 ｧ 04 縺 , 04 縺閨070 ｧ08 06 縺 , 陦ｧｧｧ, ﾀｧ. ｰｧ06 ｧ , 閨罍閨03 ﾀ tg = k, 0306 閨0605 閨罍ｧ x a y b =: y = b a x; (3) k 閨 y =x: y = k : (4) 陦ｧｧ04 ﾀｧ 05 ｧ0406 縺 , ｧ縺閨0308 閨0600 ｧ罍閨03, 陦縺 , ﾀ07 04 ｧ0003 ﾀ0304 縺ｧ. 0ｷｧ09 ｧ k ｧ k ﾀｧﾀｧ003 ﾀ kk = =, b a ( ｧ04 ﾀ ) ｧ kk = b (06 05 ｧ0406 縺 ). a 3. ｰｧ07 09 ﾀ : x ｧ04 ﾀ06 a + y xx0 yy b = a + 0 b =; x 0905 ｧ0406 縺 a, y xx0 b = a, yy 0 b =; (y =x) yy 0 = (x +x 0 ),05 閨06 (x 0 ; y 0 ) ﾀ 0 ｧ07.

26 3x. 6 7 ｧ縺罍ｧﾀ03, 陦ｧ縺03 ｧ 09 ﾀ罍ｧ縺07 閨ﾀ05 縺03 ｧｧ04 ﾀ x 5 + y =, 陦ｧ縺罍ｧﾀ03 ｧｧ 9 ﾀﾀｧｧｧ04 ﾀﾀ 03 ﾀ09 ｧﾀﾀ0308 x =,3 閨ﾀｧ陦ｧ縺罍ｧﾀﾀ05 縺03 ｧｧ0406 縺 x 6, y =, 0606 陦ｧ縺罍ｧﾀ03 ｧｧ 9 ﾀﾀｧｧ 05 ｧ0406 縺ﾀ 03 ﾀ09 ｧﾀﾀ0308 x =5 閨ﾀｧ陦ｧ縺罍ｧﾀｧﾀｧ 06 ﾀｧｧ04 ﾀ, 陦ｧ縺罍ｧﾀ 0 ｧ縺ﾀ縺 x = 4 3 ｧﾀ縺ｧﾀｧ06 05 ｧ0406 縺 , ﾀｧ縺0308 y = = x, 陦ｧ縺罍ｧﾀ03 x = ﾀ05 縺03 ｧｧ04 ﾀ x +4y =6, 陦ｧ y = x ｧﾀ0304 縺陦ｧｧｧ閨00 ｧ003 a ｧ b 0403 ﾀ05 縺陦ｧ縺ﾀ05 縺03 ｧｧ0406 縺 x, 4y = 4, 陦ｧ y =,x ｧﾀ0304 縺陦ｧｧｧ閨06 陦ｧｧｧ閨00 ｧ陦ｧｧ04 ﾀ x a + y b =, 縺ﾀﾀ0304 縺陦ｧ縺罍ﾀｧｧ0406 縺 x a, y = ﾀ03 ﾀﾀ03 ﾀ0406 Ox b ｧﾀｧ06 陦ｧ 00605, ﾀ0304 縺ﾀ陦ｧ縺030 y = x 縺03 ｧ縺 a, 0403 ﾀ05 縺03 ｧ罍ｧ040606, 陦ｧ縺03 ｧ 0803 罍閨03, 陦ｧ縺縺06 閨0600 ｧ罍ｧ 06 ﾀﾀ0503 ﾀ06 縺06 陦ｧ罍 y =4x, ﾀ03 ﾀｧ08 ﾀ Ox ｧ04 ﾀ x 9 + y =. ｪ06 縺 (,; ) 04 縺ﾀ05 ｧ罍閨06, 閨ﾀ y =,4x. ｪ06 縺 (,;,) 04 縺ﾀ05 ｧ 0803 罍閨06, 閨ﾀ罍ｧ006 縺06 ｧ縺罍ｧ縺06006 縺ｧ07: a + b = a + b ｧ a b sin ' = ab, 05 閨06 a ｧ b 閨00 ｧ003 ﾀ03-04 縺陦ｧ縺0304, a ｧ b ﾀｧｧ04 ﾀ, ' 閨06 ﾀ0304 縺ｧ陦ｧｧｧﾀｧ07 09 ﾀ罍ｧ04030: ) x +4y =6;)3x,y =3;3)y =x ﾀ 03 ﾀ09 ｧﾀﾀ0308 x 0 =.

27 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧ , ﾀ00 ｧ 04 縺 Ax + By + C =006 ﾀﾀｧ04 ﾀ06 x a + y b =,0503A a + B b = C. ｰ09 0 ｧ06. 0ｷ3 04 縺縺09 ｧﾀ05 ｧｧ09 ｧｧ08 xx 0 a + yy 0 b = ｧ Ax + By + C = 縺06 閨0600 ｧ0504 x 0 ｧ y 0 ｧ 0403 陦ﾀ05 04 ｧ0504 ｧｧ06 x a + y b = ｧﾀｧ07 09 ﾀｧ04 ﾀ06 x +4y =0, ｧﾀﾀ罍ｧﾀ縺罍陦ｧｧﾀｧ07 09 ﾀｧ04 ﾀ06 x +y =8, 04 縺閨ｧｧ (0; 6) ｧﾀｧ06 09 ﾀｧ04 ﾀ06 x a + y b =, 0305 ﾀﾀ罍陦ｧｧ縺06 09 ｧ , ﾀ00 ｧ 04 縺 Ax+By+C =006 ﾀﾀｧ0406 縺 x a, y b =,0503A a, B b = C ( ﾀｧ ) ｧﾀｧ07 09 ﾀｧ0406 縺 x, 9y = =36, 0406 縺04060 陦ｧ縺縺 x +y = , 縺ｧ04 ﾀ06 06 ﾀｧﾀﾀ罍ｧﾀ閨06 陦ｧ06 ﾀｧﾀﾀｧｧ04 ﾀ , ﾀｧ0406 縺ﾀｧﾀﾀ罍ｧﾀ閨06 陦ｧ06 ﾀｧｧ 09 ﾀ 0 ｧ , ｧ, 閨070 ｧ06 ｧﾀ , ﾀ縺070030, ﾀｧ. ｰ09 0 ｧｧﾀｧ縺0 00 ｧ MN, ｧ N 0406 縺06 ﾀｧﾀ 03 ﾀｧ閨 , 0503 FM = FN,05 閨06 F ﾀｧｧ 0406 縺06 ﾀｧ07 ﾀｧｧ0406 縺 x 6, y 9 = ﾀ 0606 陦ｧ縺罍ｧﾀ 0 ｧﾀ05 縺03 ｧｧ04 ﾀ x +4y = 6, 陦ｧ y = x ｧﾀ0304 縺陦ｧｧｧ閨ｧ0 ｧ陦ｧｧ縺06 閨0600 ｧ罍ｧ 06 ﾀﾀ0503 ﾀ06 縺06 陦ｧ罍 05 ｧ0406 縺 x, 4y =6, ﾀ03 ﾀｧﾀ03 ﾀ0406 Ox ｧ0406 縺 x, y =4. ｪ06 縺 (; ) 04 縺ﾀ05 ｧ 0803 罍閨06, 閨ﾀｧ04 ﾀ06 x +y = M ﾀ 03 罍陦ｧｧ0305 罍ｧﾀ09 ｧﾀﾀｧﾀｧ04 ﾀ M ﾀ 04 縺 OM.

28 3x 3. 7 縺ｧ06 閨0609 縺罍陦ｧ , ﾀ00 ｧ 04 縺 Ax + By + C =006 ﾀﾀ y = x, 0503B = AC ( ﾀｧ ) ｧﾀｧ06 09 ﾀ y =8x, 縺 x + y =0. x 3. 7 縺ｧ06 閨0609 縺罍陦ｧ = + + ｧ = ｧ0406 縺 = 罍陦ｧ (x; y) ﾀｧﾀ縺ﾀ罍陦ｧ (X; Y ) ﾀｧﾀ縺 : ) 04 罍ｧﾀ閨04 ｧﾀ0608 ｧ 0406 縺06006 ﾀ060 ｧｧ罍陦ｧ O (; ) x = X + ; y = Y + ; () ) 04 罍ｧ縺ﾀ ' x = X cos ', Y sin '; y = X sin ' + Y cos ': (). ｰｧ06 y = a(x, ) 縺06003 ﾀ罍陦ｧ O (; ) 04 罍ｧ0403 陦ｧ05 ﾀｧ閨06 Y = ax ｧ, ﾀ0006 閨 , 0304 縺06 閨ﾀ 0406 縺00 ｧ00308 O (; ) ｧ03 ﾀ0406 ﾀｧ罍ｧｧ, Oy ( 罍ｧﾀ. 8). ｰｧ06 y = ax + bx + c 0403 閨ｧﾀ05 ｧ陦罍ｧ0403 陦ｧ05 ﾀ縺06 閨03 閨ｧ縺06 閨罍ｧ a > 縺00 ｧ縺0809, 04 罍ｧ a< ｧｰｧ06 xy = k 04 罍ｧ縺ﾀ罍陦ｧ ' =45 04 罍ｧ0403 陦ｧ05 ﾀｧ閨06 X, Y =k ｧ, ﾀ0006 閨 , 0304 縺06 閨ｧﾀ. 8 7 ｧﾀﾀ0503 縺ｧ0406 縺 , ﾀｧｧ縺0308 ﾀﾀｧ罍陦ｧ ( 罍ｧﾀ. 9). ｰｧ06 (x, )(y, ) =k 0406 縺06003 ﾀ罍陦ｧ O (; ) 04 罍ｧ0403 陦ｧ05 ﾀｧ閨06 XY = k ｧ縺06 閨ﾀ0503 縺ｧ0406 縺

29 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧ 60. ) ｯ03 09 A(3; ) 04 罍ｧﾀ閨04 ｧﾀ罍陦ｧｧ罍陦ｧ (;,). 703 ﾀ05 縺03 ｧｧﾀﾀｧ罍陦ｧ0 05 ｧ A. ) ｧ 03 ﾀ05 縺縺ﾀ罍陦ｧ0 05, 04 罍ｧ縺 A(; 4) ｧﾀ09 ｧﾀﾀﾀ05 縺03 ｧﾀｧﾀ罍陦ｧ0 05 ｧ A 縺06006 ﾀ060 ｧ罍陦ｧ縺03 ﾀ05 ｧｧ07: (x, ) (x ) +(y +) +3) (y, ) =; ), =; ) (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 ｧﾀｧ罍陦ｧ0 05 ｧ 09 罍ｧ縺ﾀ罍陦ｧ縺03 ﾀ05 ｧｧ07: ) 5x, 6xy +5y =3; ) 3x, 0xy +3y +3= ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧﾀｧ罍陦ｧ0 05 ｧ 09 罍ｧﾀ05 縺03 ｧ罍ｧ xy =,4 ｧ縺ﾀ ' =,45 04 縺ｧ縺06003 ﾀ罍陦ｧ罍ｧ0406 ﾀ05 ｧｧ閨06 xy = k 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. ｰｧ06 xy + Ax + By + C = ｧﾀｧ閨06 (x + B)(y + A) =AB, C ﾀ05 縺03 ｧ : ) y =(x, ) ; ) y =(x, ) +3; 3) y =(x +) ; 4) y =(x +), 3: ﾀ05 縺03 ｧ : ) y = x, 4x +5; ) y = x +x +3; 3) y =,x +x,, 0403 閨0600 ｧﾀｧ陦ﾀ05 縺03 ｧ : ) y =4x, x ｧ)y =3+x, x, 0 08 閨07 ｧｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ0406 Ox. 68. ｮ05 縺閨閨03 ﾀ05 ｧｧﾀﾀﾀｧｧ 0, 縺05 ｧ09 00 ｧ, 04 縺閨縺 O ﾀ05 縺06 ｧｧ 0403 ﾀﾀ05 縺06 ｧ罍ｧ Ox 0 ﾀﾀｧｧ 0, ｧ O.

30 3x 3. 7 縺ｧ06 閨0609 縺罍陦ｧｮ03 ﾀ05 04 ｧｧ , ﾀｧ罍ｧﾀｧﾀｧ Oy ｧ 0305 ﾀ縺 b, 0 03 ﾀｧ Ox 0305 縺06 09 ｧ a ｧ,a. ｰ09 0 ｧｧｧｧ y = Ax + Bx + C ﾀ05 04 ｧ罍陦ｧ (,a; 0), (a; 0) ｧ (0; b) ｧｧ A, B ｧ C y = ax + bx + c 04 縺陦ｧ05 06 縺ｧ O(0; 0), A(,;,3) ｧ B(,;,4) ｧﾀｧ縺ﾀ05 ｧ, 陦ｧ縺縺0308 ﾀｧ縺ﾀｧ Ox, 0305 ﾀ縺ﾀｧ罍陦ｧ0 05, ｧﾀ , ﾀ03 閨06 縺07 0 ｧ08 xy, ｧ0708: ) x, xy + y, 3=0; ) 5x, 4xy +y, 4 = 0? 703 ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧﾀｧ罍陦ｧ0 05 ｧ 09 罍ｧ縺06 閨0600 ｧ罍ｧ06 閨04 ｧ07060 ｧｧ, 03 縺 ' 罍ｧﾀﾀ0903 縺03 ﾀ v 縺06 閨0600 ｧﾀｧｧ 0 ｧ0403 ﾀ罍ｧｧ ( ﾀ0304 縺0305 ｧｧ閨縺縺06 04) ｧﾀｧ罍ｧ 06 ﾀﾀ M (x; y), ｧ06 ﾀﾀｧ縺0308 閨ｧ F (4; 0) 09 ﾀﾀｧ070 閨03 04 縺 x =, , 縺06003 ﾀ罍陦ｧｧｧ04 ﾀ x a + y b = ｧ00 ｧ縺00 ｧｧ0406 縺 x a, y b = ｧ07 04 罍ｧ0403 閨0705 ﾀ陦ｧｧ閨06: y = =x + qx,05 閨06 = b a, q = e, 縺ｧ縺06 閨0600 ｧﾀ罍ｧﾀｧｧ 05 ｧ04 09 罍ｧ040308: ) y = x, 4 x ;)y = x+ 4 x ;3)y = x. 703 ﾀ05 縺03 ｧ罍ｧ040306, 0 08 閨07 閨縺閨ｧ 0406 縺06 ﾀｧ07 ｧ08 ﾀ 03 ﾀ0406 Ox ｧ縺03 a ｧ b 閨ｧ陦ｧ 0406 縺06003 ﾀ陦ｧ縺03 ﾀ05 ｧｧ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, ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧﾀｧｧ 09 罍ｧ

31 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧ縺ﾀ罍陦ｧ縺03 ﾀ05 ｧｧ06 3x, xy +3y, 8= 縺06 閨0600 ｧ罍陦ｧﾀﾀ05 縺0308 ﾀｧﾀ罍陦ｧｧﾀｧ縺ﾀ05 ｧ, 陦ｧ縺縺0308 ﾀｧ縺 , 0305 ﾀﾀｧ Ox y =3, x, x. 703 ﾀ05 縺03 ｧ罍ｧｧﾀｧ縺ﾀ05 ｧ, 陦ｧ縺縺0308 ﾀｧ縺縺 x + y = 6,0305 ﾀｧ0406 縺 xy = ﾀ05 縺03 ｧﾀ06 05 罍ｧ 00 ｧ0 ｧｧ. 80. ｯ03 09 A 0406 縺00 ｧ y = x +6x +5, B 縺06 ﾀｧﾀ 03 ﾀ0406 Oy ｧﾀｧ縺04060 陦ｧ , 0403 ﾀﾀｧﾀ06 縺06 陦ｧ縺06 09 AB. 8. ｮ03 ﾀ05 04 ｧｧ , ﾀｧ罍ｧﾀｧﾀｧ Ox ｧ 0305 ﾀ縺 ,4, 0 03 ﾀｧ Oy 0305 縺06 09 ｧ 4 ｧ,4. ｰ09 0 ｧ06. ｰｧ閨ｧｧ x = ay + c ( ?) ﾀ05 縺03 ｧ縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ罍陦ｧ : ) 3y =9, x ; ) y =9, 3x; 3) y =4+x; 4) x =4+y: ｧﾀｧ罍ｧ 06 ﾀﾀ M (x; y), ｧ06 ﾀﾀｧ縺0308 閨ｧ F (4; 0) 09 ﾀﾀｧ070 閨03 04 縺 x = =. x 4. ｮｧ 0 09 罍ｧ縺縺07 閨ｧﾀｧ縺ﾀ05 ｧ, 陦ｧ縺縺0308 ﾀｧ縺縺 x a + y =,0305 ﾀﾀ070 ｧ罍陦ｧ0 05. b ｧﾀﾀｧ縺ﾀ05 ｧ x + y + ay =0 閨03 04 縺 y =(a, x). 86. ｪ06 縺縺ﾀ05 ｧ x + y =ax 04 縺閨縺070 07, 縺 x +y = 0 ｧ 0406 縺06 ﾀ縺ﾀ A ｧ B ｧ閨04 4AOB , 罍ｧ 06 ﾀﾀ M, 縺 m ｧ A, 陦縺ｧ B, 06 ﾀ縺罍ｧ m = ｧ0309 縺ﾀ罍ｧ m 6= 縺 AB 閨ﾀ05 ｧ AO = a ｧ OB = b , 罍ｧ 06 ﾀﾀ , ｧ縺縺06 09 ｧ AO ｧ OB 04 ｧ閨ｧｧ, 06 ﾀ縺罍ｧ a = b ｧ0309 縺ﾀ罍ｧ a 6= b ( ｧ縺ﾀ0504).

32 3x 4. ｮｧ 0 09 罍ｧ縺縺07 閨縺06 閨0600 ｧ罍ｧｧ M (x; y), 閨04 ｧﾀ , 0503 ﾀ陦ﾀﾀｧ閨03 04 縺 y = kx ｧ y = =, kx 03 ﾀﾀﾀｧ a ｧ04 ﾀ, ﾀｧ罍ｧﾀｧﾀｧ Ox ｧ 04 縺 x =,5, 04 縺陦ｧ05 06 縺ｧ (,; ; 8) ｧ (,5; 3) ｧﾀｧｧ04 ﾀｧ 0403 ﾀ05 縺03 ｧｧ閨ﾀ0503 縺縺ｧ09, 0404 ｧﾀｧ0406 縺 x, y = a ｧ閨06 陦ｧｧ 04 縺ｧ09, 0406 縺00 ｧ縺閨0705 ﾀ縺06 ﾀｧｧ04 ﾀ x +3y =l ｧ05 ｧ0406 縺 x, 3y =6l 縺ﾀ0504 ﾀ縺罍陦ｧ縺陦ｧ縺ﾀ03 05 ｧ0406 縺 x, y = a ｧｧ 0406 縺06 ﾀｧ縺ﾀ05 ｧﾀﾀｧｧ 05 ｧ0406 縺ﾀ05 縺03 ｧｧ0406 縺 xy =,4 ｧ x, y = 6 ｧｧ閨04 4ABC, 05 閨06 A ｧ B 0406 縺00 ｧ003 閨縺06 ﾀｧ08 ﾀｧ0406 縺030300, C 縺06 ﾀｧ07 閨陦縺0605 ｧｧ0406 縺 , 縺03 ｧ 0406 閨060 ｧ06 ﾀﾀｧｧ 05 ｧ縺ﾀｧﾀｧｧﾀ , a b c ｧ閨00 ｧ006 ｧｧ縺04060 陦ｧ , ｧﾀ y =, x 0 04 縺 , 0305 ﾀﾀ罍陦ｧ縺06 09 ｧ a = b = ﾀ05 縺03 ｧｧ04 ﾀ x +4y = 4 ｧ x = 6y ｧｧ閨ｧｧ, 03 ﾀｧ070 ｧ縺0308 ﾀﾀｧ04 ﾀｧ罍ｧ04 ﾀｧｷﾀ y = x, ｧ , 0304 ｧﾀ縺ﾀ , 罍 09 罍ｧ陦ｧ縺00 ｧ003 ｧﾀｧﾀｧ縺ﾀ05 ｧｧ閨00 ｧ006 ｧｧ縺04060 陦ｧ , ｧ 0406 縺00 ｧ by = x +ax+a +b 0 04 縺 , 0305 ﾀﾀ罍陦ｧ縺06 09 ｧ a ｧ b ﾀ05 縺03 ｧ縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ罍陦ｧ y =, x ｧ 4x =, y ｧｧ閨00 ｧ006 ｧ罍閨ｧ閨縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ縺06 ﾀｧ y =4, x ﾀ03 ﾀ0406 Ox ｧﾀ 04 縺 y =3x ｧﾀｧ縺ﾀ05 ｧ, 04 縺閨縺罍陦ｧ0 05 ｧ 06 縺ｧ 0406 縺06 ﾀｧ y = x a,, x + a ﾀ 03 ﾀ070 ｧ罍陦ｧ0 05.

33 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧｧ04 ﾀ x +4y =6. 0ｷ縺00 ｧ003 A(4; 0) 04 縺閨ﾀ罍閨縺06 閨0600 ｧ罍ｧ 06 ﾀﾀ0503 ﾀ06 縺06 陦ｧｧ罍ｧ 0403 ﾀ05 縺03 ｧ罍ｧ縺06 閨0600 ｧ罍ｧｧ M (x; y), 閨04 ｧﾀ , ﾀ陦ﾀﾀｧ閨03 03 ｧﾀﾀ罍ｧﾀ罍陦ｧﾀﾀｮ03 ﾀ05 04 ｧｧ罍ｧ 06 ﾀﾀ縺縺ﾀ050608, 04 縺閨070 ｧ08 06 縺 A(3; 4) ｧ09 ﾀ 060 ｧ08 ﾀ07 03 ﾀｧ Ox 閨ｧ陦ｧ 0406 縺06003 ﾀ縺03- ﾀ05 ｧｧ06 00 ｧ0 ｧｧ x,y,4x,6y,9 = ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧﾀｧ罍陦ｧ0 05 ｧ 09 罍ｧｧ罍ｧ 06 ﾀﾀ0503 ﾀ06 縺06 陦ｧ陦ｧ06 ﾀ縺0304, 04 縺閨ｧﾀﾀｧ x 9, y 6 = ｧﾀｧｧ04 ﾀ, 04 縺閨縺 A(a;,a), 06 ﾀ00 ｧﾀ閨0705 ﾀ F (a; a) ｧ F (,a;,a). ｰ04 縺03 ﾀ05 ｧｧ縺ﾀ罍陦ｧ縺ﾀ罍陦ｧ ' = arctg 0604 縺03 ﾀ05 ｧｧ06 00 ｧ0 ｧｧ 3x +8xy, 3y = ﾀ05 縺03 ｧ0504 ﾀ05 縺0306 ｧﾀｧ罍陦ｧ0 05 ｧ 09 罍ｧｧﾀｧ罍ｧ 06 ﾀﾀ , ﾀ陦ﾀﾀｧ縺0308 閨03 04 縺 x +4y = 0 ｧ閨0303 ﾀｧOx 03 ﾀﾀﾀｧ ; ｧﾀｧ罍ｧ 06 ﾀﾀ M (x; y), ｧ06 ﾀﾀｧ縺0308 閨ｧ F e + ;0 09 ﾀ- ﾀｧ070 閨03 04 縺 x =, e. e(e +) ﾀ05 縺03 ｧﾀ05 ｧ, 罍陦ｧ縺閨縺ﾀ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 ｧ06 00 ｧ0 ｧｧ縺縺07 閨09.6ｧ8 ｧ0 ｧ縺縺07 閨ﾀ07 00 ｧ0 ｧ07, 0304 縺06 閨ｧﾀｧ, 縺ｧ閨ｧﾀ : Ax +Bxy + Cy +Dx +Ey + F =0: ()

34 3x ｧ06 00 ｧ0 ｧｧ縺縺07 閨09 4 ｮ03 ﾀ05 04 ｧ0 ｧｧ09 ｧｧ07 () 閨縺06 閨0600 ｧ : = B A B A B D C ; = B C E D E F : 0504 縺06 閨0600 ｧﾀ07 陦ｧﾀ09 罍ｧ0 ｧｧ07 (), 陦ｧﾀ09 罍ｧ0 ｧﾀ05 縺00 ｧｧﾀｧ003 ﾀ05 ｧｧ08 ｧｧ06 () 0304 縺06 閨ﾀ0006 閨06060 ｧ罍ｧ 06 ﾀ09 ｧ : 6= 0 =0 > ｧ04 ﾀ ( 閨0608 ﾀ0504 ｧｧ00 ｧ 00 ｧ00308) ｯ03 09 <0 06 ｧ0406 縺縺06 ﾀｧ08 ﾀ07 04 縺 = 縺 ( 閨0608 ﾀ0504 ｧｧ00 ｧ 00 ｧ ). 7 縺ｧｧ07 () 縺06. ｭﾀ00 ｧ = = B A B C 6= 0, ｧ0 ｧ07 ｧ , 罍陦ｧ縺閨0705 ﾀ07 ｧｧ08 0 (x; y) =0; (x; y) =0; () x 0 y 05 閨06 (x; y) ﾀｧ07 (). 706 縺06006 ﾀ O (x 0 ; y 0 ) ( 罍ｧﾀ. 0), 04 罍ｧ0406 閨ｧ06 () ｧ閨06 05 閨06 Ax +Bx y + Cy + F =0; (3) F = Dx 0 + Ey 0 + F = : (4) 3. 7 縺ｧｧ07 (3) ﾀ070 ﾀｧ罍ｧｧ縺ﾀ0608 O x ｧ O y 縺 ' ( 罍ｧﾀ. 0) ｧ06 (3) 04 罍ｧ0403 陦ｧ05 ﾀｧ 06 ﾀｧ閨06: A X + C Y + F =0: (5) ｧ09 ｧ A ｧ C ﾀ縺0070 ｧｧ07 ｰ縺0305 ' 陦ｧ05 ﾀ縺 , (A + C) + =0: (6) tg ' = B A, C : (7)

35 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧ 4. 7 縺ｧｧ07 00 ｧ0 ｧｧ縺縺07-09, 006 ｧｭﾀ00 ｧ =0, ｧ0 ｧｧｧ00 ｧ 006 ｧ縺06 閨ｭｧｧﾀｧ閨06 (x + y) +Dx +Ey + F =0: (8) ｮ D ｧ E 04 縺縺09 ｧｧ : D = m, E = m. ｰｧ06 () 04 罍ｧｧ (x + y) +m(x + y) +F =0, x + y =,m m, F 縺 : ｮ D ｧ E 縺縺09 ｧｧ. ｰｧ06 (8) 縺0604 ｧﾀｧ閨06 (x + y + n) +m(x, y + q) =0: (9) 縺03 m, n ｧ q 0 08 閨0605 ﾀ07 ﾀｧｧ09 ｧｧ0708 (8) ｧ (9) , 04 罍ｧﾀ04 O X 04 縺 x + y + n =0, x + y + n 03 ﾀ04 O Y 04 縺 x, y + q =0( 罍ｧﾀ. ), 0 08 閨060: Y = +, x, y + q X = ﾀｧ06 (9) 04 罍ｧｧ Y =X, 05 閨06 = 7 ｧﾀ. 0 7 ｧﾀ. jmj ﾀ04 O X ﾀﾀ0903 ﾀ0504, 縺0308 x, y + q ｧ , 04 縺0305 ｧ m, ﾀ0006 閨ｧｧ07 (9) ﾀ0 ｧ罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀｧ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.

36 3x ｧ06 00 ｧ0 ｧｧ縺縺07 閨ｧ縺03 ｧ 04 縺縺ｧ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= 縺ﾀ罍陦ｧ縺ｧｧ 06 ﾀｧ閨06 ｧ 0403 ﾀ05 縺03 ｧ罍ｧ040306: ) 5x, 4xy +y = 4; ) x +4xy, y = 縺ｧ 06 ﾀｧ閨ｧ07 ｧ 0403 ﾀ05 縺03- ｧ罍ｧ040306: ) 3x, xy +3y, 4x, 4y, =0; ) x, 6xy + y, 4x, 4y += 縺ｧ 06 ﾀｧ閨ｧ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 ｧ陦ｧﾀ09 罍ｧ0 ｧｧ 0304 縺06 閨0600 ｧ罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀｧ08: ) x, 4xy +3y, 8x +4y +5=0; ) x +xy +4y, x +4y +4=0; 3) x +4xy +4y +3x +6y += ｧ縺ｧ 05 縺ｧﾀｧ y, 0403 ﾀ05 縺03 ｧｧ- 0 ｧｧ, 0304 縺06 閨ｧ0 ｧｧ070 ｧ罍ｧ0406 ﾀ05 ｧｧ 06 ﾀｧ閨ｧ06 09 罍ｧ y = = 3x, x +4 ｧ 0403 ﾀ05 縺03 ｧ x, ｧﾀｧ06 09 罍ｧ縺縺07 閨09, ｧ縺 O (; ) ｧ04 縺閨縺罍陦ｧ0 05 ｧ 06 縺ｧ (0; 4) ｧ (;,) , ｧ06 x + y = a 0304 縺06 閨閨 , 0403 ﾀ05 縺03 ｧｧｧ縺00 ｧ006. ｰ09 0 ｧ縺ﾀｧ罍陦ｧ ' =, ｧﾀｧ罍ｧ 06 ﾀﾀ M (x; y), ｧ06 ﾀﾀｧ閨0308 ｧ縺0308 閨ｧ F (m; n) 09 ﾀﾀｧ閨03 04 縺 x cos + y sin, q = e ｧｧ09 ｧｧ07 06 縺06 A, B, C, :::, 0304 縺06 閨0600 ｧ0504 ｧ004 罍ｧ A + C ｧ A B B C.

37 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧﾀ0 ｧ罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀｧ08: ) x, 4y =0; ) x +y +4x, 8y +=0; 3) x +5xy, 6y = 縺ｧ 06 ﾀｧ閨ｧ07 ｧ 0403 ﾀ05 縺03- ｧ罍ｧ040306: ) x, xy + y, x, y, =0; ) 3x +0xy +3y, x, y +4= 縺ｧ 06 ﾀｧ閨ｧ07: ) x, xy + y, 0x, 6y +5=0; ) x +xy + y, 4x, 4y +3=0 ｧ 0403 ﾀ05 縺03 ｧｧ0 ｧｧ, ｧｧ0 ｧ陦ｧﾀ09 罍ｧ0 ｧｧ 0304 縺06 閨0600 ｧ罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀｧ08: ) x, xy + y, 4x +4y +3=0; ) x, xy, 3y +6x +0y, 7= ｧ閨ｧﾀｧ y, 0403 ﾀ05 縺03 ｧｧ0 ｧ06, 0304 縺06- 閨ｧｧﾀｧ罍ｧ 06 ﾀﾀ M (x; y), ｧ06 ﾀﾀｧ縺0308 閨ｧ F (3; 3) 09 ﾀﾀｧ070 閨03 04 縺 x + y = : ) e = ;)e = ｧﾀｧ罍ｧ 06 ﾀﾀ M (x; y), 03 陦ｧｧ F (a=; a=) ｧ縺 x + y =0, ｧ 04 罍ｧ0406 ﾀ05 ｧｧ 06 ﾀｧ閨ｧﾀｧ罍ｧ 06 ﾀﾀ , ﾀ陦ﾀﾀｧ縺0308 閨03 04 縺 x, y = ｧ閨0303 ﾀｧOx 03 ﾀﾀﾀｧ ,. 7 縺ｧ 06 ﾀｧ閨06 ｧ 0403 ﾀ05 縺03 ｧ罍ｧ x 縺罍陦ｧﾀﾀ0903 ﾀ05 ｧ O ﾀｧ 0006 OP 縺ﾀ04 ( 罍ｧﾀ. ). ｯｧｧ M ﾀ0903 ﾀ05 ｧ 0304 縺06 閨0600 ｧ05 ﾀ07: ) 縺 ' = MOP; ) 閨00 ｧ00308 r 陦ｧ06 ﾀ ,,! OM: r = j,,! OMj. 7 罍ｧｧｧｧｧ08, ﾀｧ08 r ｧ ', ﾀ- ﾀ0 05 罍ｧ縺罍陦ｧ ' ｧ r 04 罍ｧ0 ｧ0 060 ｧ0 ｧｧ閨ｧｧ 0305 罍ｧｧ07. 7 罍ｧ罍ｧ ' 0305 ﾀｧﾀﾀﾀ05 縺 , 0305 罍ｧ r 閨ﾀ , 縺03 閨ｧﾀ.

38 3x 縺罍陦ｧｭﾀ00 ｧ 04 罍ｧﾀ閨0609 縺縺罍陦ｧ- 0 05, 縺ﾀ04OP 03 ﾀ04Ox, 0503 閨0609 縺罍陦ｧ (x; y) ｧ M ｧ縺罍陦ｧ ('; r) 閨0605 ﾀｧﾀｧ003 ﾀ x = r cos '; y = r sin '; () r = x + y ; tg ' = y x : () ｭﾀ00 ｧ 04 罍ｧﾀｧ04 ﾀ, 05 ｧ0406 縺ｧﾀ, ﾀ04 ﾀｧ罍ｧｧ縺ﾀ04, ﾀｧﾀ. 縺03006, 04 縺0305 ｧｧ縺00 ｧ006, ｧ06 04 ﾀ縺罍ｧ縺罍陦ｧ閨陦ｧ : r =, e cos ' ; (3) 05 閨06 e 0509 ﾀ罍ｧﾀｧ050605, ｧ04 ﾀｧ 05 ｧ0406 縺 = b a 縺00308 ﾀｧﾀ罍陦ｧ0 05 ('; r) 0403 ﾀ05 縺03 ｧｧ A(0; 3), B(=4; ), C(=; 3), D(; ), E(3=; 3) ﾀ05 縺03 ｧｧ A(=;,), B(,=; 3), C(,=4;,4), D(=3;,3) ﾀ05 縺03 ｧｧ0 ｧ06 r =+cos'. ｰ09 0 ｧ06. ｮ03 ﾀ05 04 ｧｧｧ08 r 閨0007 ' =0; =3; =; =3; ﾀ05 縺03 ｧｧ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 ' ( ｧﾀ09 05 ); 4) r = a=' (05 ｧ0406 縺ｧ 06 ﾀ09 07 ﾀ04 ｧ 0004); 5) r = a( + cos ') (0600 ｧﾀ ): ﾀ05 縺03 ｧｧ0 ｧｧ: ) r = a; )' = 4 ;3)r = b sin ' ｧﾀ縺罍陦ｧｧ07: ) 04 縺 , 0305 ﾀ縺ﾀｧ 0305 縺 a ｧ 0406 縺04060 陦ｧ縺 ; ) 04 縺 , 04 縺閨縺 A(; a) ｧ縺ﾀｧｧﾀ縺罍陦ｧｧ06 04 縺 , 04 縺閨縺 A(; a) ｧﾀ03 ﾀﾀ縺ﾀｧﾀ縺罍陦ｧｧ縺ﾀ05 ｧﾀ縺 C(0; a) ｧ陦ｧ06 ﾀ030, a.

39 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧﾀ05 縺03 ｧ罍ｧ040306: ) r =3, sin '; ) r = + cos 3'; 3) r =, sin 3'. ｰ 09 0 ｧ縺06 閨0600 ｧ , 04 罍ｧ縺0308 ｧ r max ｧ r min ﾀ05 縺03 ｧｧ0 ｧｧ ( ﾀ0. ﾀ. 334, 罍ｧﾀ. 8 ｧ 83): ) r = a sin 3' (05 縺ﾀ縺03 ); ) r = a sin ' ( 縺ﾀ縺03 ) 縺縺罍陦ｧｧ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 ) 縺閨0609 縺罍陦ｧｧ07 00 ｧ- 0 ｧ08 ｧ 0403 ﾀ05 縺03 ｧｧ0 ｧｧ: ) r cos ' = a; ) r =a sin '; 3) r sin ' =a ; 4) r sin ' + = a ; 4 5) r = a( + cos ') ｧﾀｧ 06 ﾀ09 ｧｧ07 09 罍ｧ縺縺07 閨09 : ) r = 3) r = 9 5, 4 cos ' ; ) r = 9 4, 5 cos ' ; 3, cos ' ｧ. ｪ06 縺 A(=; a) 04 縺閨縺070 07, 縺ﾀｧ. 7 縺03 ｧ OB 0406 縺06 ﾀ縺 B ｧ B ﾀ0503 縺縺06 09 ｧ BM = BM = b 縺06 閨0600 ｧ罍ｧ 06 ﾀﾀ M ｧ M 縺罍陦ｧｧ 0403 ﾀ05 縺03 ｧ罍ｧｮｧ. 7 縺 x = a 0406 縺06 ﾀﾀ04Ox A ｧ04 縺03 ｧ OB B ｧ B ﾀ0503 縺縺06 09 ｧ BM ｧ BM, AB ｧﾀｧ罍ｧ 06 ﾀﾀ M ｧ M 縺00308 ｧ閨0609 縺罍陦ｧ ( 罍ｧﾀ. 84, ﾀ. 335).

40 3x 縺罍陦ｧﾀﾀｧ0 ｧ. ｯ03 09 M ('; r) 閨04 ｧﾀ , 縺03- ｧ 0406 閨060 ｧ06 ﾀﾀｧ閨 F (0; a) ｧ F (; a) 03 ﾀﾀ b ｧﾀｧ罍ｧｧ閨04 ｧ07060 ｧｧ M 縺罍陦ｧ罍陦ｧ03 ｧ縺03 ｧ OA ｧ A 縺06 ﾀｧﾀ 0309 縺ﾀ r = a cos ' 閨ﾀﾀ0503 縺縺 AM = AM = a. ｮ03 ﾀ05 04 ｧｧ罍ｧ- 06 ﾀﾀ M ｧ M 縺00308 ｧ閨0609 縺罍陦ｧ罍陦ｧ03 ｧ (0504 ｧ09 ｧｧ ). 07 縺0605 陦ｧ a ｧ05 ﾀﾀｧ縺陦ｧﾀ0 縺0607 ｧｧﾀｧ06 09 罍ｧ040308, 0304 ｧﾀ M ﾀ縺ﾀ05 ｧ, 06 ﾀ00 ｧﾀｧｧｧ M 04 罍ｧﾀ 0 ｧ07 09 縺 , 縺ﾀ04 04 縺ﾀ05 ｧ 06 縺縺03 09 縺 ( ｧｧ) ﾀ05 縺03 ｧ罍ｧ040306: ) r = 3 + cos '; )r =3, sin 3'; 3) r = a cos ' ( ﾀｧ ) ﾀ05 縺03 ｧ0504: ) r = 4( + cos '); )r =, sin ' ｧﾀ縺罍陦ｧｧ06 04 縺 , 04 縺閨縺ｧ A(; a) ｧ B(; b). ｰ09 0 ｧ06. 7 ﾀﾀ00305 縺ｧﾀｧ003 ﾀ閨閨070 ｧ 05 縺ｧ AOM, BOM ｧ AOB, 05 閨06 M ('; r) 04 縺03 ｧ縺ｧﾀｧ 06 ﾀ09 ｧｧ07 09 罍ｧ縺縺07 閨09 : ) r =, 3 cos ' ; ) r = 3) r =, 5 cos ' ;, cos ' ｧ80600 ｧﾀ縺ｧ. ｯ03 09 M ('; r) 閨04 ｧﾀ , 縺03 ｧ 0406 閨060 ｧﾀﾀｧ F (0; c) ｧ F (; c) 03 ﾀﾀ c ｧﾀｧ罍ｧｧ閨04 ｧ07060 ｧ縺00308 ｧ閨0609 縺罍陦ｧｰ09 0 ｧ縺ﾀｧ006 ﾀ0304 FM = r + c, rc cos ' ｧ F M = r + c +rc cos ', 04 罍ｧﾀｧ06 FM F M = c 4.

41 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧ 353. ｰ00 ｧﾀ縺03 ｧ OA ｧ A 0406 縺06 ﾀｧﾀ 0309 縺ﾀ r = a cos ' ﾀ0503 縺縺06 09 ｧ AM = AM = b. ｮ03 ﾀ05 04 ｧｧ罍ｧ 06- ﾀﾀ M 縺罍陦ｧｪ縺ﾀ縺06 09 AB = = a ﾀﾀ070 閨0609 縺罍陦ｧｷ陦ｧ AB 0406 縺04060 陦ｧ OM ｧﾀｧ罍ｧ 06 ﾀﾀ M (x; y) 04 罍ｧ 04 ﾀｧ縺06 09 AB. x 7. 縺ｧ 06 ﾀ09 ｧ06 09 罍ｧ縺ｧ 0403 ﾀ00 ｧ縺07 閨ﾀ05 縺03 ｧ罍ｧ ( ﾀ0. ﾀ. 33, 罍ｧﾀ. 66{69): ) y = x 3 =3 ( ｧ 06 ﾀ ); ) y = x 3 ( ｧ 06 ﾀ ); 3) y 3 = x 4) y = x(x, 4) ( ): ﾀ05 縺03 ｧ罍ｧ040306: ) x =3 + y =3 = a =3 ( ﾀ05 縺03 ｧﾀ0503 縺 ); x =3 y =3 ) + =, b 6= a ( ﾀ05 縺03 ｧﾀ0503 縺 ). a b ｰ 09 0 ｧｧｧ 0406 縺06 ﾀｧ07 09 罍ｧﾀ 03 ﾀ070 ｧ Ox ｧ Oy ｧ 0406 縺罍ｧﾀ 04 縺ｧ y = x, 縺0308 ﾀ 04 縺ｧ y = b a x ( 罍ｧﾀﾀ. 334) ﾀ05 縺03 ｧ縺 [,; ] 09 罍ｧ040306: ) y = x n+ ;)y = = x n ; 3) x n + y n = 04 罍ｧ n = ; ; ｧ罍ｧ0300 ｧﾀｧ 09 罍ｧ040306, n!? ｰ 09 0 ｧｧｧ 0406 縺06 ﾀｧ縺罍ｧﾀ 04 縺 y = = x, 縺罍ｧﾀ 04 縺 y = ｧ05 縺罍ｧﾀ 04 縺 y = x. n n 6ｦ0 06 陦ｧ0 ｧﾀ罍ｧｧ罍ﾀ05 縺03 ｧ縺06 09 AB = a ﾀﾀ070 閨0609 縺罍陦ｧ縺 AC ｧ BC, ﾀ罍陦ｧ0 05, 0406 縺06 ﾀﾀ C. 0ｷ3 C AB 0406 縺陦ｧ CM ｧﾀｧ罍ｧ 06 ﾀﾀ M (x; y) 04 罍ｧ 04 ﾀｧ縺06 09 AB.

42 ﾀ05 縺03 ｧ罍ｧ040306: x 8. ｯ 0 ﾀ09060 閨罍ｧ ) y = x3 a, x (09 ｧﾀﾀ03 ｧ, 罍ｧﾀ. 85, ﾀ. 335); ) y = 8a3 x +4a ( , 罍ｧﾀ. 76, ﾀ. 333): P (x 0 ; y 0 ) y =x ﾀﾀｧ Ox 0 ﾀﾀｧ06 PM = OP ｧ罍ｧ- 06 ﾀﾀ M. 36. ｮ0506 縺 OA = a ﾀ縺罍陦ｧ0 05 O A 罍ｧ09 縺縺0 ｧ縺030 ﾀ0506 縺 AB =a, 縺ﾀｧ Ox ｧﾀｧ06 00 ｧ0 ｧｧ, 縺閨ｧﾀ罍ｧﾀ06 縺06 陦ｧ0 M 0305 縺06 09 AB. 36. ｨC ｧﾀﾀ03 ｧ. 7 縺03 ｧ OA ( 罍ｧﾀ. 85, ﾀ. 335) 縺06 ﾀ縺ﾀ0504 x + y = ax A ｧ 04 縺 x = a B 閨ﾀ縺 OM = AB. ｮ03 ﾀ05 04 ｧｧ罍ｧ 06 ﾀﾀ M 縺03 ｧ OB ( 罍ｧﾀ. 85) 0406 縺06 ﾀ縺 x = a B. C 04 縺ｧｧ B 0 03 ﾀ04 Oy ｧ M 04 縺ｧｧ C 0 04 縺 OB , 罍ｧ 06 ﾀﾀ M 06 ﾀｧﾀﾀ03 ｧｭﾀ00 ｧｧ 0406 縺00 ｧ y =,4ax ﾀ縺04060 陦ｧ縺ﾀ罍ｧ040308, 罍ｧ 06 ﾀ09 ｧ0 006 ﾀﾀｧ縺04060 陦ｧ縺閨ｧﾀﾀ03 ｧｧ縺03 ｧ OA 0406 縺06 ﾀ縺ﾀ0504 x + y = ay ｧ 04 縺 y = a A ｧ B, ｧ縺縺閨縺 , ﾀﾀﾀｧ Ox ｧ 03 ﾀｧ Oy 閨縺06 ﾀｧ M 縺06 閨0600 ｧ罍ｧ 06 ﾀﾀ M 縺ｧﾀ05 x 3 + y 3, 3axy = , ｧ縺ﾀ罍陦ｧ罍ｧ0403 陦ｧ05 ﾀｧ閨06 Y X 3 (3b, X) = 3(b + X),05 閨06 b a =. 703 ﾀ05 縺03 ｧ罍ｧ040606, 0304 縺06 閨0600 ｧﾀｧﾀ罍陦ｧﾀ0504 ﾀｧ07 09 罍ｧｧ 0606 ﾀｧ罍ｧ06, ｧ 0406 縺06 ﾀｧ07 ﾀ 04 縺 y = x ( ﾀﾀ0406 OX) ｧﾀｧ , ｧ06 ﾀｧﾀｧﾀ罍陦ｧ閨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 閨罍ｧｨC ｧｧ. 07 縺0605 陦ｧ06 ﾀ a ｧ05 ﾀ縺 OX 0306 ﾀｧ07. ｮ03 ﾀ05 04 ｧ罍ｧ 06 ﾀ09 ｧｧ07 09 罍ｧ040308, 0304 ｧ- ﾀ M 0309 縺ﾀ05 ｧ, 04 罍ｧ t 縺0305

43 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧﾀ07 09 縺0605 ｧｧ04, 罍ｧ t = M 陦ｧ05 ﾀ罍陦ｧ縺ｧ0504, 縺ﾀ0504 x + y = a, ﾀ07, 03 ﾀﾀｮ03 ﾀ05 04 ｧ罍ｧ 06 ﾀ09 ｧｧ07 09 罍ｧ040308, 0304 ｧﾀｧ05 ｧ, 06 ﾀ00 ｧｧ05 ｧ陦ｧ05 ﾀ (a; 0). 6ｦ t 04 罍ｧ閨00 ｧ006 ﾀ閨0605 ｧ (04 陦ｧ06 ﾀ 08) 陦 05 罍ｧﾀ. 7 縺03 ｧ OM, ﾀ03 ﾀｧ08 ﾀ03 ﾀ0406 Oy t (04 陦ｧ 0 08), 0406 縺06 ﾀ縺 x = at M ｧﾀｧ罍ｧ 06 ﾀﾀ M ｧ09 ｧｧ. 07 縺0605 陦ｧ06 ﾀ r ｧ05 ﾀﾀｧ縺陦ｧ06 ﾀ R ﾀ0 縺0607 ｧｮ03 ﾀ05 04 ｧ罍ｧ 06 ﾀ09 ｧｧ07 09 罍ｧ040308, 0304 ｧﾀ M ﾀ縺ﾀ05 ｧ. ( 7 罍ｧ r = R 0504 ｧ09 ｧｧﾀｧ 03 ｧ 06. ｮ ) ｧｧｧ. 07 縺0605 陦ｧ06 ﾀ r ｧ05 ﾀﾀｧ縺陦ｧ06 ﾀ R>r 罍ｧｮ03 ﾀ05 04 ｧ罍ｧ 06- ﾀ09 ｧｧ07 09 罍ｧ040308, 0304 ｧﾀ M ﾀ縺ﾀ05 ｧ. ( 7 罍ｧ r = R=4 05 ｧｧｧﾀ0704 ﾀ05 縺03 ｧ閨06 x =3 + + y =3 = a =3.)

44 ｭ07 ｯ縺00 縺6ｧ806 ｭ 7 7 x. ｮｧ縺0304. ｰｧﾀ縺06 閨ｧ縺ﾀ縺 ,! AB ( 罍ｧﾀ. 3), 縺 A ﾀﾀ0 05 罍ｧﾀ , B ﾀ07 ｧ00 ｧｧｧ ,! AB ﾀ03 ﾀ05 縺縺0806, ｧ00 ｧ 03 閨ｧ0306 閨 , 0403 閨ｧ縺罍ｧ , 0 04 罍ｧ006 a 閨 ( 閨00 ｧ0 ) ﾀ07 j,! ABj, ｧ00 ｧ jaj, ｧ00 ｧ AB, ｧ00 ｧ a 縺03, 閨縺 , ﾀｧ006 縺0030 ｧ縺03, 閨ﾀ0903 ﾀ05 ｧ, ﾀ縺0030 ｧ a ｧ b ( 罍ｧﾀ. 3) ﾀｧ, 06 ﾀ00 ｧ 030 ｧ: ) ｧ閨0600 ｧ; ) ｧ006 縺003; 3) 閨006 ﾀ0503 縺ｰｧﾀ縺03 ｧ 0406 閨060 ｧ a 0 ｧﾀ0003 ( ﾀ ) m ﾀ , ｧｧ08 閨00 ｧ006 ajmj ｧ陦ｧﾀ a (04 罍ｧ m>0) ｧ00 ｧ 04 縺0305 ｧ a (04 罍ｧ m<0). 3. ｮｧ縺0304. ｮ縺0304 a+b+c ﾀ R =,! OC ( 罍ｧﾀ. 4), ｧ OABC, 0403 ﾀ05 縺ｧﾀ. 3 7 ｧﾀ. 4 ｧ縺ﾀ05003 ﾀ05 ｧ, , 0403 ﾀ05 縺 ,! OA = a ｧ,! OB = b, 03 閨陦ｧ ,! OC 06 ﾀ0504 ﾀ0600 a + b, 陦縺 ,! BA 06 ﾀﾀ0504 a, b 縺縺ｧﾀﾀ a ﾀ03 ﾀ ' ﾀ 03 ﾀ0406 Ox. ｯ縺ｧﾀ縺06 閨ﾀ縺 x a = jaj cos ' = a cos ( a;ox):

45 縺縺ｧ07 ﾀ縺ﾀﾀ縺ｧ08 ﾀ03 ﾀｧ縺ﾀ04: 04 x (a + b) =04 x a + 04 x b: ﾀ0503 縺030 0 OA ｧ OB 04 縺ｧ09 OACB 陦ｧ0 ｧ縺03 i ｧ j ( 罍ｧﾀ. 5) ｧ縺06 i ｧ j 縺03,! OA,,! AC,,! CB,,! BO,,! OC ｧ,! BA, 06 ﾀ00 ｧ OA =3 ｧ OB = ﾀ罍ｧﾀ. 5 M ﾀ06 縺06 陦ｧ0 BC ｧ N ﾀ06 縺06 陦ｧ0 AC 縺06 閨0600 ｧ縺03,,! OM, ON,,! ｧ MN,,! 04 罍ｧ OA =3 ｧ OB = ﾀ0903 ﾀ05 ｧｧ A(0;,), B(4; ) ｧ C(4;,) 罍陦ｧ罍ｧﾀｧ0003,! OA,,! OB ｧ,! OC. 703 ﾀ05 縺03 ｧ0504 ｧ閨0608 ﾀ ,,! OM, ｧ縺ｧｧ 0 03 ﾀｧ罍陦ｧ0 05 ｧｧｧｧ0504 ﾀｧ- 7 ｧﾀ ,! OA,,! OB,,! OC ｧ,,! OM 06 縺06 06 陦ｧ0 ｧ縺03 i ｧ j 罍陦ｧﾀ罍ｧ縺陦ｧ0 ｧ m, n ｧ, 04 罍ｧ 060 ( [m; n) =30 ｧ ( dn; ) = ﾀ05 縺03 ｧ u = m + +n, 3 ｧ 0403 ｧﾀ00 ｧ閨ｰ09 0 ｧ , 0403 ﾀ05 縺ｧ縺0304 m, n ｧ,3, 04 縺03 閨ｧ縺閨縺06 ﾀｧ07 ﾀ 05 縺ｧ縺罍ｧｧ05 ｧ 06 ﾀ09 ｧｧ罍ｧ 06 ﾀ09 ｧ縺閨06 ﾀ0504 : ) a b, a a + = + b ; ) a, a + b a, b = 縺縺 ,! OA = a,,! OB = b ｧ,! OC = c 0403 ﾀ05 縺ｧ0406. ｰ陦ｧｧ, 縺0306 ﾀﾀ a + b, c, a, b + c, a, b, c ｧ b, a, c ｮ縺ｧ縺罍ｧ縺06006 ﾀ05 ｧﾀﾀ縺00308 ﾀ a + b, c = a, c + b = b + a, c = b, c + a: 縺03,! OA = a ｧ,! OB = b ,! OC = c 006 陦ｧ- 0 4OAB ｧｧ05 ｧ 06 ﾀ09 ｧｧ罍ｧ 06 ﾀ09 ｧ: ) c a ｧ b; ) a b ｧ c 縺ｧ0906 OACB ( 罍ｧﾀ. 5) M ｧ N ﾀ06 縺06 陦ｧ003 ﾀ0503 縺030 BC =3 ｧ AC = ｧ罍ｧ 06 ﾀ09 ｧｧ 0 00 ｧ05 ｧ- 06 ﾀ09 ｧ ,! OC = c ,,! OM = a ｧ ON,,! = b. ｰ09 0 ｧﾀｧ06 c = ma + nb 0403 陦ﾀ05 04 ｧｧ07 a, b ｧ c 06 縺06 i ｧ j ｧﾀ 040 ｧｧ09 ｧﾀｧﾀ罍ｧ i ｧ j.

46 3x. 7 縺罍陦ｧｧｧ縺03 ﾀ05 0 ﾀｧﾀ05 ｧｧ09 OABCDE ﾀ03 ﾀ0503 縺 OA = ｧ04 06 陦ｧ0 ｧ縺ｧ08,! OA,,! AB,,! BC 06 縺06 m, n ｧ, 06 ﾀｧｧﾀｧ003 ﾀ閨06 0 ｧ0 ｧ (0 04 罍ｧ- 006, ﾀﾀ00305 縺060 ｧｧｧ OABC) ｧ縺06 m ｧ n 縺03,! OB,,! BC,,! EO,,! OD ｧ,! DA 陦縺ｧｧ OACB ( 罍ｧﾀ. 6) BOA = = 60, OB = BC = CA =, M ｧ N ﾀ06 縺06 陦ｧ003 ﾀ0503 縺030 BC ｧ AC ｧ縺03,! AC,,,! OM,,,! ON ｧ,,! MN 06 縺06 m ｧ n 06 陦ｧ0 ｧ縺ｧ08,! OA ｧ,! OB 縺03 a ｧ b, 閨縺030 ｧﾀ05 縺03 ｧ c = a, ; 5b ｧ 0304 縺06 閨0600 ｧ閨060004, 06 ﾀ00 ｧ a =3 ｧ b =4. 7 ｧﾀﾀ0903 ﾀ05 ｧｧ A(3; 3), B(,3; 3) ｧ C(,3; 0) 罍陦ｧ罍ｧﾀｧ0003,! OA,,! OB ｧ,! OC. 703 ﾀ05 縺03 ｧ閨0608 ﾀ ,,! OM, ｧ縺ｧｧ 0 03 ﾀｧ罍陦ｧ0 05 ｧｧｧｧ0504 ﾀｧ0003,! OA,,! OB,,! OC ｧ,,! OM 06 縺06 06 陦ｧ0 ｧ縺03 i ｧ j 罍陦ｧﾀ ) ｧｧ OACB ｧ BC = OA=3 ｧ BCkOA ｧ罍ｧ 06 ﾀ09 ｧｧ 0 00 ｧ05 ｧ 06 ﾀ09 ｧ ,! OA = a ,! OC = c ｧ,! OB = b. ｰ09 0 ｧ06. 0ｷ3 4OBC c 0403 ｧ縺06 b ｧ a ｧ縺0600 ｧｧﾀｧ a. ) ｯ03 09 B 閨0600 ｧ05 閨縺ﾀ05 ｧ ^ AC= ｧｧ :. O 縺ﾀ05 ｧｧ ,! OC = c ,! OA = a ｧ,! OB = b. x. 7 縺罍陦ｧｧｧ縺03 ﾀ05 0 ﾀ縺06 閨ｧﾀ罍ｧ 04 ｧ縺04060 陦ｧ縺罍陦ｧﾀｧﾀｧ0,,! O ｧ M ( 罍ｧﾀ. 7). 7 縺ｧｧ 0606 陦ｧ06 ﾀ OM = r 0 03 ﾀｧ罍陦ｧ0 05 OM = x, OM = y ｧ OM 3 = z ﾀ07 04 縺ｧ罍陦ｧｧｧ M ｧ00 ｧ r,,! = OM.. 7 陦ｧ06 ﾀｧ縺03 ﾀ05 0 ﾀ閨ｧ00 ｧ閨00 ｧ0,,! 陦ｧ06 ﾀ OM = r: r = x + y + z : ()

47 縺ｭ陦ｧ0 ｧ縺罍陦ｧﾀ0608 i, j ｧ k ﾀ07 03 縺05 0 ｧ. 7 陦ｧ06 ﾀﾀ07 06 縺06 03 縺0503: 7 ｧﾀ. 7 r = xi + yj + zk: () , 陦ｧｧ 0 00 ｧﾀｧ A(x ; y ; z ) ｧ B(x ; y ; z ). 7 縺ｧｧ u,! = AB 0 03 ﾀｧ陦ｧ閨0605:,! 04 x AB = X = x, x ;,! 04 y AB = Y = y, y ; (3),! 04 z AB = = z, z : ｧﾀ縺 , ｧ縺 (), (): u = X + Y + = (x, x ) +(y, y ) +(z, z ) ; (4) u =,! AB = Xi + Y j + k: (5) ｭﾀ00 ｧ, ｧ u =,! AB ﾀ 03 ﾀ070 ｧ罍陦ｧ0 05, 0503 cos = X u ; cos = Y u ; cos = u ; (6) 04 罍ｧ 060 cos + cos + cos =; (7) ﾀ陦ｧﾀｧ006 ﾀｷ縺00600 (4){(6) ﾀ0006 閨060605, u 縺06 閨ﾀ07 05 縺06007 ｧﾀ00 0 ｧ: X, Y ｧ縺ｧ070 ｧｧ00 ｧ罍陦ｧｧｧｧｧ00 ｧ縺0705: ufx; Y ; g ﾀ05 縺03 ｧ M (5;,3; 4) ｧ 0304 縺06 閨0600 ｧ0504 閨00 ｧ006 ｧｧ陦ｧ06 ﾀﾀ05 縺03 ｧ r,,! = OM = i +3j +6k ｧ 0304 縺06 閨0600 ｧ閨00 ｧ006 ｧｧ06 (04 縺罍ｧ縺 cos + cos + + cos =) ﾀ03 ﾀﾀ 03 ﾀ070 ｧ Ox ｧ Oz ｧｧﾀ 03 ﾀ0406 Oy 陦ｧ06 ﾀｧ M ﾀ03 ﾀﾀ03 ﾀ0406 Ox ｧﾀ03 ﾀ0406 Oy ｧ r = 縺06 閨0600 ｧ罍陦ｧｧ M, 06 ﾀ00 ｧ罍陦ｧ0 05 z 0305 罍ｧ , ｧ 0403 ｧ ,,! OM = r 06 縺06 03 縺0503 i, j, k ｧ A(; ; 3) ｧ B(3;,4; 6). 703 ﾀ05 縺03 ｧ ,! AB = u, 縺ｧｧ 0 03 ﾀｧ罍陦ｧ0 05 ｧ 0304 縺06 閨0600 ｧ0504 閨00 ｧ006 ｧｧﾀ05 縺03 ｧ u ﾀ 03 ﾀ070 ｧ罍陦ｧ0 05.

48 3x 3. ｮ縺縺03 ｧ 0406 閨060 ｧ06 閨縺ﾀ05 縺03 ｧ ,! OA = i+j ｧ,! OB = = k, 3j ｧ 0304 縺06 閨0600 ｧ陦ｧｧ A(; ;,) 04 罍ｧﾀｧ00 R = 7. 6ｦ00 07 閨罍陦ｧﾀｧ0003 X = ｧ Y =,3, 0304 縺06 閨0600 ｧｧ06 ｧ , ｧﾀｧﾀ0903 ﾀ05 ｧ xoy ｧ A(4; ), B(; 3) ｧ C(0; 5) ｧ 0403 ﾀ05 縺縺03,! OA = a,,! OB = b ｧ,! OC = c ｧ罍ｧ 06 ﾀ09 ｧｧ 0 00 ｧ05 ｧ 06 ﾀ09 ｧ a b ｧ c ｧ A(; ; 0) ｧ B(0;,; 5). 703 ﾀ05 縺03 ｧ ,! AB = u ｧ 0304 縺06 閨0600 ｧ閨00 ｧ006 ｧｧ ,,! OM = r ﾀ03 ﾀﾀ 03 ﾀ070 ｧ罍陦ｧﾀ05 縺縺06 閨0600 ｧｧｧ 0403 ﾀ05 縺03 ｧ r, 06 ﾀ00 ｧ閨00 ｧﾀ03 ﾀﾀ 03 ﾀ070 ｧ Oy ｧ Oz ｧﾀ03 ﾀﾀ03 ﾀ0406 Ox? 罍ｧ 0403 ﾀ0006 閨縺00 ｧ A(;,; 3), B(3; ; ) ｧ C(6; 4; 4) ｧ縺ｧ006 D. ｰ09 0 ｧ06. 0ｷﾀ0504,! AD,! = BC ﾀ0006 閨060605, ｧｧ陦ｧ0 0503: x, =6, 3 ｧﾀ0903 ﾀ05 ｧ xoy 0403 ﾀ05 縺03 ｧ縺03,! OA = a =i,,! OB = = b = 3i +3j ｧ,! OC = c = i +6j ｧ罍ｧ 06 ﾀ09 ｧｧ 0 00 ｧ05 ｧ 06 ﾀ09 ｧ c a ｧ b. x 3. ｮ縺縺03 ｧ 0406 閨060 ｧ06 閨縺縺06 閨ｧ06. ｮ縺縺03 ｧ 0406 閨060 ｧ060 閨縺ﾀ07 04 縺03 ｧ 0406 閨060 ｧ06 ｧ閨 ; ﾀｧ006 ﾀ閨06 0 ｧ0 ｧ. ｮ縺縺03 ｧ 0406 閨060 ｧ a b ﾀ07 a b. 0ｷ305 09, a b = ab cos ': () 0ｷ3 罍ｧﾀｧ閨003, 0503 b cos ' = 04 a b a b = ab cos ' = a 04 a b = b 04 b a: (). ｮﾀ0504 ﾀ縺縺03 ｧ閨060 ｧ07: 7 ｧﾀ. 8 I. a b = b a 0406 縺06006 ﾀ05 ｧ II. a (b + c) =ab + a c ﾀ04 縺06 閨0600 ｧ 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)

49 縺 IV. ｭﾀ00 ｧ a? b, 0503a b = ab cos 90 =0. V. ｮ縺縺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 ｧ縺03 a ｧ b 罍陦ｧｧ afa x ;a y ;a z g ｧ bfb x ;b y ;b z g, ｰ閨ｧ: 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 ｰﾀｧﾀ05 ｧ: b = ma ｧ00 ｧ bx b b y z = = = m. a x a y a z ｰﾀｧ縺04060 陦ｧ縺003 ﾀ05 ｧ: a b =0 ｧ00 ｧ a x b x + a y b y + a z b z = 縺06 閨0600 ｧ閨ｧ a =,i + j ｧ b = i,, j +k 縺06 閨0600 ｧ ABC ﾀ 0406 縺00 ｧ0 0 ｧ A(;,; 3), B(; ; ) ｧ C(0; 0; 5) ｧ A(a; 0;0), B(0; 0; a) ｧ C(a; 0;a). 703 ﾀ05 縺03 ｧ縺03,! OC ｧ,! AB ｧｧ閨06 0 ｧ0 ｧﾀ0903 ﾀ05 ｧ 0 05 縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ O(0; 0), A(a; 0) ｧ B(a;,a) ｧ , ﾀ0503 縺 OB ｧ 006 陦ｧ OM 縺ｧｧ閨06 03 ｧﾀﾀ罍ｧﾀ 0 ｧ xoy ｧ yoz ｷ縺00 ｧ陦縺閨縺 , 閨ｧ06 04 縺0305 ｧﾀ0503 縺ｧ閨ｧ0 ｧ 04 縺ｧｧ閨06 陦ｧｧ , 0403 ﾀ05 縺 a =i + j ｧ b =,j + k 縺03 a = i + j +k ｧ b = i, j +4k 縺06 閨0600 ｧ b a ｧ04 a b ﾀ09 縺ﾀｧｧｧ (i, j) j +(j, k) k +(i, k) : ｧﾀ00 ｧ0504: ) (m+n), 06 ﾀ00 ｧ m ｧ n 06 陦ｧ0 ｧ縺03 ﾀ閨06 0 ｧ0 ｧ 30 ; ) (a, b), 06 ﾀ00 ｧ a =, b = 4 ｧ (d a; b) = ﾀ09 縺ﾀｧｧ0708: ) (a + b) ; ) (a + b) +(a, b) ｧﾀ0 ｧ罍ｧ 06 ﾀ09 ｧ08 ﾀ003 ﾀ縺縺縺03 a, b ｧ c, 04 罍ｧ 060 a =3, b =, c =5, (d a; b) =60 ｧ (d b; c) = ﾀ05 縺03 ｧ u = a + b, c

50 3x 3. ｮ縺縺03 ｧ 0406 閨060 ｧ06 閨縺ｧ 0403 ｧﾀ00 ｧ閨縺 u = (a + b, c) : ｧｧｧ閨0608 ﾀ縺ﾀｧ00, 04 罍ｧ O, 06 ﾀ00 ｧｧｧ閨0308 ﾀｧ , 閨06 閨ﾀ0006 閨ｧﾀｧ00 0 ｧ縺06 閨0600 ｧ0504 閨00 ｧ003 陦ｧ , 0403 ﾀ05 縺 a = m + n ｧ b = m, n, 05 閨06 m ｧ n 06 陦ｧ0 ｧ縺03, 閨縺030 ｧ a =m, n, 05 閨06 m ｧ n 06 陦ｧ0 ｧ縺03 ﾀ閨06 0 ｧ0 ｧｧ cos ( [a; m) ｧ cos (da; n) 縺06 閨0600 ｧ閨06 03 ｧﾀﾀ罍ｧﾀ 0 ｧ閨ﾀ09 ｧｧ陦, 04 縺閨ｧｧ 03 閨縺00 ｧ003. ｰ09 0 ｧ06. ｭﾀ00 ｧ m, n ｧ 06 陦ｧ0 ｧ縺03 縺060306, 0503 m + n ｧ m 縺03, ｧﾀﾀ罍ｧﾀﾀ0708 Ox, Oy ｧ Oz ｧ縺06 09 ｧ a =4 ｧ 0 0 ｧﾀ05 縺03 ｧﾀ0504 M 縺ｧ, N ｧ縺06 閨0600 ｧ縺03,,! OM ｧ,,! ON ｧ閨06 0 ｧ0 ｧ縺03,! OA = a ｧ,! OB = b, 04 罍ｧ 060 a =, b =4, (d a; b) = 縺06 閨0600 ｧ閨陦ｧ 00308,,! OM 05 縺ｧ09 AOB ｧﾀ0503 縺 ,! OA ｷ縺00 ｧ縺ｧ09 ﾀ03 ﾀ0503 縺030 0 ｧ 6 ﾀ0 ｧ 4 ﾀ0 04 縺閨縺 , 閨ｧ06 04 縺0305 ｧﾀ0503 縺ｧ ' 閨06 0 ｧ0 ｧ罍ｧ 0403 ﾀ0006 閨縺00 ｧ : A(,3;,; 0), B(3;,3; ) ｧ C(5; 0; ) ｧ縺ｧ006 D ｧ閨ｧ,! AC ｧ,! BD ｧ A(3; 3;,), B(0;,3; 4), C(0;,3; 0) ｧ D(0; ;,4). 703 ﾀ05 縺03 ｧ縺03,! AB = a ｧ,! CD = b ｧｧ 04 a b 陦縺ｧｧ 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 縺ｧｧ縺06 閨0600 ｧ閨ｧ,,! OM ｧ ON.,,! ｧ閨ｧ a =m +4n ｧ b = m, n, 05 閨06 m ｧ n 06 陦ｧ0 ｧ縺03, ｧ , 閨06 陦ｧｧ 04 縺ｧ09, 0403 ﾀ05 縺 a ｧ b (a? b), 0304 縺06 閨ﾀ縺 cos ' = a, b a + b.

51 縺縺ｧｧ 0406 縺ｧ07 s 閨04 ｧﾀｧ 0 03 ﾀｧ陦ｧ0 05 s x = 0, s y = 0, s z =, 0. 7 縺ｧｧ閨0608 ﾀﾀｧ0003 F 0 03 ﾀｧ罍陦ｧ F x =5 7, F y =4 7 ｧF z = ｧﾀ00 ｧ A ﾀｧ0003 F (A = F s) ｧ閨06 ﾀｧ F ｧ 0406 縺ｧ060 s 縺00 ｧｧ陦ﾀ縺0603 縺030 a 04 罍ｧ罍ｧﾀｧ0003, ｧ縺ｧ縺06 閨0600 ｧｧ- ｧ閨0608 ﾀｰ09 0 ｧ06. 0ｷ3 ﾀｧｧ0 040 a (m + n + ),05 閨06 m, n ｧ 06 陦ｧ0 ｧ縺ﾀｧ陦 05 閨罍ｧﾀ03 03 陦ｧｧ罍ｧ003 ｧﾀ0406 縺ｧ縺罍ｧｧ閨06 閨ﾀｧｧ , 罍ｧ陦ｧ陦 05. x 縺縺03 ｧ 0406 閨060 ｧ06 閨縺縺06 閨ｧ縺縺03 ｧ 0406 閨060 ｧ a b ﾀ縺0605 ｧ c ( 罍ｧﾀ. 9), 縺0308: ) ｧ閨060004, ｧﾀ陦ｧ , ﾀ05 縺 a ｧ b; ) 0406 縺04060 陦ｧ縺ﾀ0903 ﾀ05 ｧ ; 3) ﾀ0503 縺03006, ﾀ縺ｧ a 09 b ﾀﾀ0 05 罍ｧﾀ07 ﾀ縺ｧ0 ﾀ07 04 縺0305 ｧ04 ﾀﾀ05 縺ｧ. ｯﾀｧ縺0304 a, b ｧ c ﾀﾀ縺縺03 ｧ 0406 閨060 ｧﾀ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 ﾀﾀｮﾀ縺縺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 III. ｭﾀ00 ｧ akb, 0503a b =0; 04 ﾀ05003 ﾀ05 ｧ, a a = 縺縺03 ｧ 0406 閨060 ｧ07 03 縺050304: i j = k; j k = i; k i = j: () 縺03 ｧ 0406 閨060 ｧ閨ﾀ縺ﾀ0006 閨ﾀ05 ｧ 0605 ﾀ0006 閨06060 ｧﾀ , ﾀ0006 閨ﾀ05 ｧﾀ ,.

52 3x 縺縺03 ｧ 0406 閨060 ｧ06 閨縺ｧ縺縺03 ｧ 0406 閨060 ｧ07 06 縺陦ｧﾀｧ 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 : () 閨 , 0403 ﾀ05 縺 a ｧ b: S = ja bj; (3) 閨04 05 縺ｧ09, 0403 ﾀ05 縺 a ｧ b: S4 = ja bj: (4) 縺06 閨0600 ｧ0504 ｧ 0403 ﾀ05 縺03 ｧ c = a b, 06 ﾀ00 ｧ: ) a =3i, b = k; )a = i + j, b = i, j; 3) a = i +3j, b = 3j +k ｧ閨030 ﾀ閨 , 0403 ﾀ05 縺 a ｧ b ｧﾀ00 ｧ閨04 05 縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(7; 3; 4), B(; 0; 6) ｧ C(4; 5;,) ﾀ05 縺03 ｧ a =j + k ｧ b = = i +k ｧ 0403 ｧﾀ00 ｧ閨04 ｧ 0403 ﾀﾀ09 縺ﾀｧｧ 0604 縺03 ﾀ05 ｧｧ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) , 0503 (a, b) (a + b) = a b, ｧﾀ0 ｧ罍ｧ 06 ﾀｧ閨06 ﾀ縺03 a ｧ b ﾀ03 ﾀｧ閨04 05 縺ｧ09, 0403 ﾀ05 縺 a, b ｧ 3a +b, 06 ﾀ00 ｧ jaj = jbj = ｧ閨 , 陦ｧｧ縺ﾀ縺03 m, n ｧ 4m, 5n, 05 閨06 m ｧ n 06 陦ｧ0 ｧ縺03, ｧｰ09 0 ｧ06. 0ｷ a + b =m, n ｧ a, b =4m, 5n, 05 閨06 a ｧ b 縺03- ﾀ0503 縺縺ｧ04, 0 08 閨 b a, 003 閨縺ｧ閨ｧﾀ陦ｧﾀ05 縺03 ｧ縺03 a =3k,j, b =3i,j ｧ c = ab ｧﾀ00 ｧ閨 c ｧ閨04 05 縺ｧ09, 0403 ﾀ05 縺 a ｧ b ﾀ05 縺03 ｧ縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(;,; 8), B(0; 0; 4) ｧ C(6; ; 0) ｧﾀ00 ｧ閨04 ｧ 0403 ﾀ BD.

53 縺ｧﾀ00 ｧ0504 陦ｧｧｧ閨 , ﾀ05 縺 a = k, j ｧ b = i + j + k , 0503 (a + b) (a +b) =3a b ｧ閨 , 0403 ﾀ05 縺 a = m +n ｧ b =m + n, 05 閨06 m ｧ n 06 陦ｧ0 ｧ縺03, ｧ x 5. ｮ縺03 ｧ 0406 閨060 ｧ06 05 縺縺縺06 閨ｧ06. ｮ縺03 ｧ 0406 閨060 ｧ縺0304 a, b ｧ c ﾀｧ06 04 ｧ (a b) c. ｭﾀ00 ｧ縺03 a, b ｧ c 003 ﾀ0403 ｧ0 ｧ罍陦ｧｧ, 0503 (a b) c = a x a y a z b x b y b z c x c y c z : (). ｮﾀ0504 ﾀ縺03 ｧ 0406 閨060 ｧ07. I 縺06 ﾀｧ閨ﾀｧﾀ縺03 ｧ閨060 ｧ : (a b) c =,(a c) b =,(c b) a: () II. ｭﾀ00 ｧ閨04 ｧ 05 縺縺ｧ00 ｧ , 0503 ｧ08 ﾀ縺03 ｧ 0406 閨060 ｧ III. 6ｦ00 09 ｧｧｧ0809 縺06 ﾀﾀ05 0 ｧ, (a b) c = a (b c); ﾀ縺03 ｧ 0406 閨060 ｧ06 04 罍ｧｧﾀｧ閨06 abc, 閨0608 ﾀ0504 ｧ08 ｧ 0306 ﾀｧ0406,0403 ﾀ05 縺 a, b ｧ c. V = abc (+ 04 罍ｧﾀ ,, 04 罍ｧﾀ ) ｧ 0 ｧ閨03, 0403 ﾀ05 縺 a, b, c: V04 ｧ = 6 abc: 4. ｰﾀｧ縺003 ﾀ05 ｧ. ｭﾀ00 ｧ a, b ｧ c 縺003, 0503 abc =0, ｧ罍ｧ閨06 a, b ｧ c ﾀ06006 ﾀｧｧﾀｧ003 ﾀｧ c = ma + nb ﾀ05 縺03 ｧｧ a = 3i +4j, b = =, 3j + k, c =j +5k ｧ 0403 ｧﾀ00 ｧｧ00 ｧ閨0605 ﾀ縺0304 (a; b; c)? ﾀ05 縺03 ｧｧ 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 , 閨ｧ ABC ｧ 0403 ﾀｧ 0 ｧ閨03, , ｧ A(;,;,), B(; ; ), C(; 3; 0) ｧ D(5; 0;,6) 閨ﾀ0903 ﾀ05 ｧ , 縺03 a =,i +3j +k, b =i, 3j, 4k, c =,3i+j+6k 縺003, ｧｧ c a ｧ b.

54 3x 5. ｮ縺03 ｧ 0406 閨060 ｧ06 05 縺縺 , 0503: ) (a + b) (a + c) b =,abc; ) (a +b, c) (a, b) (a, b, c) =3abc ｧ陦, 0403 ﾀ05 縺 ,! OA,,! OB ｧ,! OC, 06 ﾀ00 ｧ 0505 ｧ縺ｧﾀﾀ罍ｧﾀ罍陦ｧｧ閨00 ｧ閨ﾀ05 縺03 ｧｧ 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 ｧｧ 0403 ﾀ030506, ABC ﾀ05 縺03 ｧ縺03 a = i+j+4k, b = i,j ｧ c =3i,3j+4k, , ｧ縺003, ｧｧ 00 ｧｧﾀｧ003 ﾀ閨06 0 ｧ0 ｧ , ｧ0406, 0403 ﾀ05 縺陦ｧｧ0406, 閨ｧ陦ｧ0 ｧ縺03 m, n ｧ. ｰ ( [m; n) = X X ; (m n) sin. = = , (m n) = 罍ｧ a, b ｧ c 縺03 a, b, b, c ｧ c, a 縺ｧ05 ｧ 06 ﾀ09 ｧｧ罍ｧ 06 ﾀ09 ｧ ( ﾀ- ﾀ00305 縺060 ｧｧ0406, 0403 ﾀ05 縺 a, b ｧ c) ｧﾀ00 ｧｧ0406 OABCO A B C, 縺罍ｧ 0406 縺00 ｧ003 0 ｧﾀｧ07 O(0; 0; 0), A(;,3; 0) ｧ C(3; ; 0) ｧ 0406 縺00 ｧ縺ﾀｧ07 B (3; 0; 4), 縺0603 縺06 BB,04 縺0305 ｧ縺0603 縺06 OO.

55 繖 7 縺6ｧ80ｷ3 ｯ0ｷ3 ｪｭｮ07 縺00 06 ｭ0503 ｭｯ 70ｷｮｯ 7 繖 7 ｮｯ6 5 ｭ x. ｰｧﾀ0903 ﾀ05 ｧ. ｰｧﾀ0903 ﾀ05 ｧ,04 縺閨縺 M (x ; y ; z ) ｧ 0406 縺04060 陦ｧ縺縺06 NfA; B; Cg. 706 ﾀ0504 M (x; y; z) 04 縺03 ｧﾀ0903 ﾀ05 ｧ ( 罍ｧﾀ. 0). ｯ0305,,,! M M?? N ｧﾀｧ縺04060 陦ｧ縺003 ﾀ05 ｧ縺0304 A(x, x )+B(y, y )+C(z, z )=0: () ｧﾀﾀ05 ｧ: Ax + By + Cz + D =0: () NfA; B; Cg ﾀ縺ﾀ0903 ﾀ05 ｧ () 7 ｧﾀ. 0 ｧ00 ｧ () ﾀﾀ0006 ｧｧ07 Ax + By + Cz + D =0: I. D = 0, Ax + By + Cz = ﾀ0903 ﾀ縺陦ｧ05 06 縺罍陦ｧ0 05. II. C =0, Ax + By + D = ﾀ0903 ﾀﾀｧ Oz. III. C = D =0, Ax + By = ﾀ0903 ﾀ縺陦ｧ05 06 縺06 03 ﾀ04Oz. IV. B = C = 0, Ax + D = ﾀ0903 ﾀﾀ0903 ﾀ05 ｧ yoz. V. ｰｧ罍陦ｧﾀ0903 ﾀ050608: x =0, y =0, z =0. 4. ｰｧﾀ0903 ﾀ05 ｧ縺ﾀ0708: x a + y b + z c =: (3) ﾀ05 縺03 ｧﾀ0903 ﾀ05 ｧ: ) 5x, y +3z, 0 = 0; )3x + +y, z =0;3)3x +z =6;4)z, 7= ﾀ05 縺03 ｧﾀ0903 ﾀ0504 x +3y +6z, = 0 ｧｧ縺0 00 ｧﾀ0903 ﾀ05 ｧﾀ 03 ﾀ070 ｧ罍陦ｧｧ M (0;,; 3) ｧ M (; 3; 5) ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺 M ｧ 0406 縺04060 陦ｧ縺縺06 N,,,,! = M M.

56 3x. 05 ﾀｧﾀ0903 ﾀｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺 M (a; a; 0) ｧ 0406 縺04060 陦ｧ縺縺06,,! OM. 703 ﾀ05 縺03 ｧﾀﾀｧﾀｧ罍ｧ 06 ﾀﾀ , A(a;,a=; a) ｧ B(0; a=; 0) ｧﾀｧﾀ0903 ﾀ05 ｧ, ﾀｧ Ox ｧ 04 縺閨縺ｧ M (0; ; 3) ｧ M (; 4; 5) ｧ 0403 ﾀ05 縺03 ｧｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺06 03 ﾀ04Ox ｧ M (0;,; 3). 703 ﾀ05 縺03 ｧﾀ0903 ﾀｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺06 03 ﾀ04Oz ｧ M (;,4; 3). 703 ﾀ05 縺03 ｧﾀ0903 ﾀｧﾀｧﾀ0903 ﾀ05 ｧ, ﾀｧ Oy ｧ 0305 ﾀﾀ0708 Ox ｧ Oz 0305 縺06 09 ｧ a ｧ c. 703 ﾀ05 縺03 ｧｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺 M (;,; 3) ｧ0305 ﾀﾀ罍陦ｧ縺06 09 ｧｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺 M (,4; 0; 4) ｧ0305 ﾀﾀ0708 Ox ｧ Oy 0305 縺06 09 ｧ a = 4 ｧ b = ﾀ05 縺03 ｧﾀ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 = ﾀ05 縺03 ｧﾀ0903 ﾀ0504 x, y + z, 6=0 ｧｧ縺0 00 ｧﾀ 03 ﾀ070 ｧ罍陦ｧｪ06 縺 M (,; ; 3) 04 縺閨ﾀ0903 ﾀ0504, 0406 縺04060 陦ｧ縺 OM ｧﾀｧｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺06 03 ﾀ04Oy ｧ 06 縺 (4; 0; 3). 703 ﾀ05 縺03 ｧﾀ0903 ﾀｧﾀｧﾀ0903 ﾀ05 ｧ, ﾀｧ Oz ｧ 04 縺閨縺ｧ M (; ; 0) ｧ M (4; 0; 0). 703 ﾀ05 縺03 ｧﾀﾀｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺 M (;,3; 5) ｧ0305 ﾀﾀ0708 Oy ｧ Oz 04 閨ｧ縺06 09 ｧ, ﾀｧ Ox. x. 05 ﾀｧﾀ0903 ﾀ0504. ｰ050300, 閨ﾀ0903 ﾀ05070 ｧ: cos ' = N N NN = AA + BB + CC NN ; () 05 閨06 N ｧ N 003 縺縺ﾀ0903 ﾀ05070 Ax + By + Cz + D =0 ｧ A x + B y + C z + D =0. ｰﾀｧﾀ05 ｧ: A A = B B = C C : ()

57 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧ縺03 ﾀ05 0 ﾀｰﾀｧ縺04060 陦ｧ縺003 ﾀ05 ｧ: AA + BB + CC =0: (3). 7 ﾀﾀｧｧ M 0 (x 0 ; y 0 ; z 0 ) 閨ﾀ0903 ﾀ05 ｧ Ax + + By + Cz + D =0: d jax 0 + By 0 + Cz 0 + Dj = : (4) N 3. ｰｧﾀﾀ0903 ﾀ050608, 04 縺閨070 ｧ08 06 縺06 00 ｧ0 ｧ縺06 ﾀｧ07 閨ﾀ0903 ﾀ050608: (Ax + By + Cz + D) +(A x + B y + C z + D )=0: (5) ｧ0504 =, ｧﾀｧｧ0 ｧ (5) 縺0606 ｧﾀ0903 ﾀｧ閨ﾀ0903 ﾀ05070 ｧ: ) x, y +z, 8=0 ｧ x + z, 6=0; ) x +z, 6=0 ｧ x +y, 4=0: ｧﾀ0903 ﾀ0504, 04 縺閨縺 (; ;,) ｧﾀ0903 ﾀ05 ｧ x, y, 3z = ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺 (,;,; ) ｧ 0406 縺04060 陦ｧ縺ﾀ0903 ﾀ05070 x, y + z, 4=0 ｧ x +y, z +4= ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺 (0; 0; a) ｧ 0406 縺04060 陦ｧ縺ﾀ0903 ﾀ05070 x, y, z =0 ｧ y = x ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺ｧ M (,;,; 0) ｧ M (; ; ) ｧ 0406 縺04060 陦ｧ縺ﾀ0903 ﾀ05 ｧ x + +y +z, 4= ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺ｧ M (;,; ), M (; ; ) ｧ M 3 (; ; 4) ｪ06 縺06 03 ﾀ04 Oz 04 縺ﾀ05 ｧﾀ0903 ﾀ0504, ﾀ03 ﾀﾀﾀﾀ x + y, 5z = ｧﾀﾀｧｧ (5; ;,) 閨ﾀ0903 ﾀ05 ｧ x,, y, z +4= ｧﾀﾀｧｧ (4; 3; 0) 閨ﾀ0903 ﾀ05 ｧ, 04 縺閨縺ｧ M (; 3; 0), M (4;,; ) ｧ M 3 (3; 0; ) ｧﾀﾀｧ閨ｧﾀ0903 ﾀ05070 ｧ 4x +3y, 5z, 8=0 ｧ 4x +3y, 5z +=0: ｰ09 0 ｧ縺ﾀ0903 ﾀ05 ｧ , 0 04 罍ｧ006 (; 0; 0), ｧｧ 0606 ﾀﾀｧ陦縺ﾀ0903 ﾀ05 ｧ ) 7 04 ｧﾀｧﾀ0903 ﾀ050608, ﾀﾀ05 ｧ x, y +z, 5 = 0 ｧﾀﾀｧ06,

58 3x 3. ｰｧ07 04 縺 ) 7 04 ｧﾀｧﾀ0903 ﾀ050608, 閨ｧ閨 , ﾀ0903 ﾀ05070 ｧ x +y = z ｧ z = 0, ｧ 0403 ﾀ05 縺03 ｧｧｧﾀﾀ0903 ﾀ05 ｧ ) 7 04 ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺06 00 ｧ- 0 ｧ縺06 ﾀｧﾀ0903 ﾀ x, y +3z, 6=0, x +y, z +3=0 ｧ 06 縺 (; ; 4). ) ｧ閨ｧ縺04060 陦ｧ縺ﾀ0903 ﾀ05 ｧ, 04 縺閨07-0 ｧ06 06 縺06 04 縺縺06 ﾀｧﾀ0903 ﾀ x = y ｧ z = 0, 06 ﾀ00 ｧ 03 閨0 ｧｧﾀﾀ0903 ﾀ縺陦ｧ05 06 縺 (0; 4; ) ﾀ05 縺03 ｧ縺ｧｧﾀﾀ0903 ﾀ05 ｧｧ縺06 ﾀｧﾀ0903 ﾀ x, y +3z, 9=0, x +y +z, 3=0 ｧ 3x + y, 4z +6= ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺 (;,; ) ｧ 0406 縺04060 陦ｧ縺ﾀ0903 ﾀ x +y, z +4=0 ｧ x + y + z, 3= ﾀ05 縺03 ｧｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺ｧ (0;,5; 0) ｧ (0; 0; ) ｧ 0406 縺04060 陦ｧ縺ﾀ0903 ﾀ05 ｧ x +5y + +z, 0 = ﾀ05 縺03 ｧｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺ｧ O(0; 0; 0), M (a;,a; 0) ｧ M (a; a; a), ﾀﾀ0903 ﾀ xoy ｧﾀﾀｧ罍陦ｧ0 05 閨ﾀ0903 ﾀ05 ｧ, 04 縺閨縺ｧ M (a; 0;0), M (0; a; 0) ｧ M 3 (a; a; a) ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺06 03 ﾀ04Ox ｧﾀ03 ﾀﾀﾀ0903 ﾀ y = x ｧﾀﾀｧｧ (a; b; c) 閨ﾀ0903 ﾀ05 ｧ, 0305 ﾀﾀ罍陦ｧ縺06 09 ｧ a, b ｧ c ｧﾀｧﾀ0903 ﾀ050608, ﾀ0903 ﾀ05 ｧ x +y + z, 8=0 ｧﾀﾀｧ06 d = ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺06 00 ｧ0 ｧ縺06 ﾀｧﾀ0903 ﾀ x, y +3z, 6=0 ｧ x +5y, z +0=0 ｧ 0406 縺04060 陦ｧ縺ﾀ0903 ﾀ05 ｧ x, y +5z, 5=0. x 3. ｰｧ07 04 縺ｰｧ07 04 縺 ,04 縺閨縺 A(a; b; c) ｧ縺06 Pfm; n; g. 706 ﾀ0504 M (x; y; z) 04 縺03 ｧ縺 ( 罍ｧﾀ. ), ,,! AM k P ｧﾀｧﾀ05 ｧ縺0304 x, a m = y, b n = z, c : () ｰｧ07 () ﾀｧ 06 ﾀ09 ｧ0 ｧｧ070 ｧ 04 縺 Pfm; n; g ﾀｧ縺縺

59 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧ縺03 ﾀ05 0 ﾀ罍ｧ 06 ﾀ09 ｧｧ07 04 縺ｧ0, 04 罍ｧ閨0306 ｧｧ08 () 縺06 t: 7 ｧﾀ. x = mt + a; y = nt + b; z = t + c: () 3. ｰｧ07 04 縺 , 04 縺閨縺06 閨ｧ: x, x x, x = y, y y, y = z, z z, z : (3) ｧｧ07 04 縺 : Ax + By + Cz + D =0; A x + B y + C z + D =0: 5. ｰｧ07 04 縺縺ｧｧ0, ｧﾀｧ04 ｧｧｧ08 (4) 03 陦ｧ0 y, 陦縺 x: (4) x = mz + a; y = nz + b: (5) ｰｧ07 (5) ｧﾀｧ 06 ﾀ縺006: x, a m ｧﾀ0006 閨03 04 縺 : = y, b n = z, 0 : ) x = z +5, y =4, z ｧ ) x, 3 = y, = z, ﾀ0903 ﾀ xoy ｧ xoz ｧ 0403 ﾀ05 縺03 ｧ縺ｰ09 0 ｧｧｧ縺 : ) z =0;) y = ｰｧ07 04 縺 x +y +3z, 3 = 0, 3x + y +4z, 4 = =00 04 ｧﾀ 0504: ) 縺ｧ0708; ) ｧ 06 ﾀ縺ｧﾀ0006 閨03 04 縺罍陦ｧﾀ0903 ﾀ050708, 0403 ﾀ05 縺03 ｧ縺ｧ縺ｧｧｧﾀｧ07 04 縺 , 04 縺閨縺 A(4; 3; 0) ｧ縺06 Pf,; ; g ｧﾀ縺ﾀ0903 ﾀ05 ｧ yoz ｧ 0403 ﾀ05 縺03 ｧ縺ﾀ05 縺03 ｧ縺 x =4, y =3 ｧｧｧﾀ05 縺03 ｧ縺 : ) y =3, z =; ) y =, z = x +; 3) x =4, z = y ｧ 0304 縺06 閨0600 ｧ0504 ｧｧ縺ｧﾀｧ07 04 縺 , 04 縺閨縺ｧ A(,; ; 3) ｧ B(; 6;,), ｧｧｧﾀｧ006 ﾀﾀ05 縺03 ｧ縺 , 04 縺閨縺ｧ A(;,; 3) ｧ B(; 3; 3), ｧ 0 04 ｧﾀｧ07.

60 3x 3. ｰｧ07 04 縺ｧﾀｧ罍ｧｧｧ M (x; y; z), , 閨07 ｧｧ A(4;,3; ), 閨04 ｧﾀ07 ﾀ03 ﾀ0903 縺03 ﾀ vf; 3; g ｧﾀ罍ｧ 06 ﾀ09 ｧｧ07 04 縺 : ) 04 縺閨縺 (,; ;,) ｧ縺06 Pf;,; 3g; ) 04 縺閨縺ｧ A(3;,; 4) ｧ B(; ; ) ｧﾀｧ07 04 縺 , 04 縺閨縺 (a; b; c): ) ﾀｧ Oz; ) 0406 縺04060 陦ｧ縺ﾀｧ Oz ｧ縺 x = z,, y =,z + ﾀ 04 縺 , 04 縺閨縺罍陦ｧ0 05 ｧ 06 縺 (;,;,) ｧ閨06 04 縺ｧ: x, y + z, 4=0, x + y,, z +5=0 ｧ x + y + z, 4=0, x +3y, z, 6=0. ｰ 09 0 ｧｧ閨0308 ｧ 04 縺縺06- 閨0600 ｧ縺縺03 ｧ 0406 閨060 ｧ縺縺ﾀ0903 ﾀ (P = N N ) , 縺 x = y 3 = z 0406 縺04060 陦ｧ縺縺 x = z +, y =, z ｧﾀｧ07 04 縺 , 04 縺閨縺 (,4; 3; 0) ｧ縺 x, y + z =4, x + y, z = ｧﾀｧ縺04060 陦ｧ , ｧｧ (;,3; 4) 0 03 ﾀ04 Oz. ｰ09 0 ｧ06. 0ｷ3 ﾀ縺縺陦ｧ縺 (0; 0; 4) ｧﾀﾀｧｧ M (;,; 3) 閨03 04 縺 x + = y 3 = + z, =. 4 5 ｰ09 0 ｧ06. ｯ03 09 A(,;,; ) ｧ縺 ; Pf3; 4; 5g ｧ縺ｯ0305,,!,,! AMjP AMj jp AMj d = AM sin = = : P AM P ｧﾀﾀｧ閨ｧ 04 縺ｧ x, = y + = z +3 ｧ x, = y, = z + : ｧﾀ0006 閨03 04 縺 x, 4 = y, = z, 罍陦ｧﾀ0903 ﾀｧ 0403 ﾀ05 縺03 ｧ縺ｰｧ07 04 縺 x + y +8z, 6 = 0, x, y, z += =00 04 ｧﾀ 0504: ) 縺ｧ0708; ) ｧ 06 ﾀ縺ｧﾀ0006 閨03 04 縺罍陦ｧﾀ0903 ﾀ050708, 0403 ﾀ05 縺03 ｧ縺ｧ縺ｧｧ.

61 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧ縺03 ﾀ05 0 ﾀｧﾀｧ07 04 縺 , 04 縺閨縺 A(0;,4; 0) ｧ縺06 Pf; ; 3g, ｧﾀ縺ﾀ0903 ﾀ05 ｧ xoz ｧ 0403 ﾀ05 縺03 ｧ縺ﾀ05 縺03 ｧ縺 x =3, z =5 ｧｧｧｧｧ縺 x + y, z =0, y = x ｧ縺ﾀ 03 ﾀ070 ｧ罍陦ｧ0 05 ( ﾀｧ ) ｧﾀｧ縺04060 陦ｧ , ｧｧ (;,3; 4) 0 03 ﾀ04 Oy ｧ閨06 04 縺ｧ x, y, 7=0, x, z +5 = 0 ｧ 3x, y +8=0, z =3x ｧﾀｧ07 04 縺 , 04 縺閨縺 (,; ;,) ｧ縺 x, y =, y =z ｧﾀﾀｧｧ M (3; 0; 4) 閨03 04 縺 y =x + +, z =x ( ﾀ ). x 4. 7 縺ｧﾀ0903 ﾀ0504. ｰ閨06 04 縺 x, a m 0903 ﾀ Ax + By + Cz + D =0: = y, b n = z, c ｧﾀ- sin = ｰﾀｧ06 ｧﾀ05 ｧ (N k P): jn Pj jam + Bn + Cj = : () NP NP ｰﾀｧ06 ｧ縺04060 陦ｧ縺003 ﾀ05 ｧ (N? P): Am + Bn + C =0: () A m = B n = C : (3). ｯ縺06 ﾀｧ07 04 縺ｧﾀ0903 ﾀ05 ｧｧﾀ罍ｧ 06 ﾀ09 ｧｧ07 04 縺 x = mt + a, y = nt + b, z = t + c, 0403 陦ﾀ05 04 ｧｧﾀ0903 ﾀ05 ｧ Ax + By + Cz + D = ﾀ0503 x, y, z ｧｧ07 06 縺06 t 閨060 t 0, x 0, y 0, z 罍陦ｧｧ 0406 縺06 ﾀｧｰﾀｧ06 ﾀｧ07 閨縺閨ﾀﾀ05 ｧ: a, a b, b c, c m n =0: (4) m n ｧ縺 y =3x,, z =,3x + ﾀﾀ0903 ﾀ x + y + z, 4=0.

62 3x 4. 7 縺ｧﾀ0903 ﾀ , 縺 x + y = + z,, = ﾀ0903 ﾀ05 ｧ x + y, z =0, 04 縺 x + y = + z, = ｧﾀ0903 ﾀ05 ｧｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺 (,; ;,3) ｧ 0406 縺04060 陦ｧ縺縺 x =, y, z = ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺06 04 縺 x, y, = 3 z = + 3 ｧ (3; 4; 0) ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺06 04 縺 x, y = + ｧ 0406 縺04060 陦ｧ縺ﾀ0903 ﾀ05 ｧ x +3y, = z +, z = ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺縺 x, 3 y z, = = ｧ x + y, = z = ｧﾀｧ07 04 縺 , 04 縺閨縺罍陦ｧ0 05 ｧﾀ03 ﾀﾀﾀ0903 ﾀ05070 ｧ 4y =3x, y =0 ｧ z = ｧ 0505 ｧｧ縺06 ﾀｧ07 04 縺 x = t,, y = t +, z =, t ﾀﾀ0903 ﾀ x, y + z = ｧ縺06 ﾀｧ07 04 縺 x y, = = z + ﾀﾀ0903 ﾀ x +y +3z, 9 = ｧ 04 縺ｧｧ (3; ;,) ﾀ0903 ﾀ0504 x +y + +3z, 30 = ｧ 04 縺ｧｧ (; 3; 4) 0 04 縺 x = y = z ｧﾀﾀｧ閨ｧ 04 縺ｧ: ) 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 閨縺ﾀﾀ09 縺060 ｧｧ- 0 ｧﾀ07, 0 罍ｧﾀﾀ0903 ﾀ05 ｧ, 縺ｧﾀｷ A(a; b; c) ｧ A (a ; b ; c ) 04 縺閨縺03,! AB,,,! = A B = = Pfm; n; g ｧ,! AC,,,! = A C = P fm ; n ; g ﾀ罍ｧ 003 ABCA B C ｧ 040 ｧﾀﾀﾀｧ , 縺 x = z,, y =z + ｧ x, y, = 4 = y, b n = z, c 3 = z, 0406 縺06 ﾀﾀ07, ｧ 0 04 ｧﾀｧﾀ0903 ﾀ05 ｧ, 縺ｧﾀ

63 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧ縺03 ﾀ05 0 ﾀｧﾀｧ縺04060 陦ｧ , ｧｧ (; ; 0) 0 04 縺 x =3z,, y =z ﾀ05 縺03 ｧﾀ0903 ﾀ0504 x + y, z = 0 ｧ 04 縺 , 04 縺閨縺ｧ A(0; 0; 4) ｧ B(; ; 0) ｧ縺06 ﾀｧ07 04 縺ﾀﾀ0903 ﾀｧ閨06 0 ｧ0 ｧﾀ05 縺03 ｧﾀ0903 ﾀ0504 y = z, 04 縺 x =,z +, y = ｧｧ: ) ｧ縺06 ﾀｧ07; ) 閨06 0 ｧ0 ｧｧ 04 縺ｧｧ (3; ;,) ﾀ0903 ﾀ0504 3x + y + + z, 0= ｧ 04 縺ｧｧ (; ; 8) 0 04 縺 x, y = =z., ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺縺 x, y = + z,, = ｧ x y 3 = + z,, = , 縺 x +3 y = + z = + ｧ x =3z,4, y = z 縺06 ﾀﾀ07, ｧｧ縺06 ﾀｧｧﾀｧ縺04060 陦ｧ , ｧｧ (; 0;,) 0 04 縺 x + y, = z =., ｧﾀﾀｧ閨06 04 縺ｧ 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 ｧ縺08003 ﾀ05 ｧ. ｰｧ06 ﾀ0706 罍ｧ 06 ﾀ縺08003 ﾀ05 ｧﾀ縺030 C(a; b; c) ｧ陦ｧ06 ﾀ030 R: (x, a) +(y, b) +(z, c) = R : (). ｰｧ06 F (x; y) =0, 006 ﾀ03 閨06 縺 z, 0304 縺06 閨ｧ00 ｧ0- 陦罍ｧ 06 ﾀ縺08003 ﾀ0504 ﾀ ; ﾀｧ Oz. 縺ｧ閨0306 ｧｧ08: ) F (y; z) = 0 ｧ)F (x; z) = 縺06 閨ｧ00 ｧ0 陦罍ｧ 06 ﾀ縺08003 ﾀ0504 ﾀ , : ) Ox; ) Oy. 3. ｰｧ06 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ縺08003 ﾀ05 ｧﾀ F (x; y) = 0, z = 0 ｧﾀ , 縺06 Pfm; n; g. ｰｧ06 04 縺03 ｧ閨0605 x, x 0 = y, y m 0 z = = n,05 閨06 (x 0; y 0 ;0) 縺06 閨0600 ｧﾀ06 x 0 ｧ y 0 ｧ 0403 陦ﾀ05 04 ｧ04 ｧｧ , ｧｧ06 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ縺08003 ﾀ05 ｧ: x, m z; y, n z F =0: ()

64 3x 5. ｮ0706 罍ｧ 06 ﾀ09 ｧ06 ｧ 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ縺08003 ﾀ05 ｧｧｧ陦ｧ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 ｧｧ縺0308 ﾀ0706 縺ｧﾀｧ06 ﾀ0706 罍ｧ 06 ﾀ縺08003 ﾀ05 ｧ, 0404 ｧﾀ陦, ﾀ0903 ﾀ05070 ｧ 3x, y +6z, 8 = 0; x =0; y =0; z =0: ｧﾀｧ罍ｧ 06 ﾀﾀ , ﾀ閨ｧ A(; 0; 0), B(,4; 0; 0) ｧﾀｧ06 ﾀ0706 罍ｧ 06 ﾀ縺08003 ﾀ05 ｧ, 04 縺閨縺縺ﾀ0504 x + y + z = a, x + y + z = a ｧ 06 縺 (a; a; a). ｰ09 0 ｧ06. 0ｷ3 ﾀｧ06 閨ｧｧ x + y + z, a + (x + y + z, a) =0: ﾀ05 縺03 ｧﾀｧﾀ罍陦ｧ縺08003 ﾀ05 ｧ: ) y + z =4; ) y = ax; 3) xz =4; 4) x + y = ax ｧﾀｧ罍ｧ 06 ﾀﾀ , 03 陦ｧ縺 x = a, y =0 ｧﾀ0903 ﾀ05 ｧ yoz. 703 ﾀ05 縺03- ｧ縺08003 ﾀｧﾀｧ07 05 縺ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ縺08003 ﾀ050608, 0304 ｧﾀﾀ0706 縺03 x + y + z, ax = 0 ﾀｧ0 ｧ, ｧﾀﾀ : ) 03 ﾀｧ Ox; )03 ﾀｧOy; 3)03 ﾀｧOz 罍ｧﾀ縺ﾀｧﾀ罍陦ｧ罍ｧｧ04 ｧ 0 ｧ: x + y + z =6; x + y =4x; 0403 ﾀ05 縺03 ｧｧ 04 罍ｧ x =0; ｧ , 縺ｧ07 09 罍ｧﾀ0903 ﾀ0504 xoz 06 ﾀｧｧ陦ｧ06 ﾀ 0309 縺ﾀ05 ｧ x + y + z =0y; x +y +z, 9=0: ｰ09 0 ｧ06. ｨC 縺ﾀ05 ｧ 06 ﾀ縺ｧﾀﾀ0504 ( ﾀ ) ｧﾀｧ06 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ縺08003 ﾀ05 ｧﾀ y =4x, z =0 ｧﾀ , 縺06 Pf; ; 3g ﾀ05 縺03 ｧ縺縺08003 ﾀ0504 (x+y) +az = a 0403 ﾀｧﾀ0903 ﾀ05070 ｧ x = 0, y = 0, z = 0, z = h 6 a ｧ , 縺08003 ﾀｧ00 ｧ0 陦罍ｧ 06 ﾀ09 07 ﾀｧ0 ｧ, ｧ 04 縺 x + y = a, z =0.

65 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧ縺03 ﾀ05 0 ﾀｹ0 x + y + z =4z 03 ﾀｧ, ｧ 04 縺 x =0, y = z ｧ 0703 縺ｧﾀ0903 ﾀ05 ｧ xoy. ｰ09 0 ｧｧﾀｧ06 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ縺ﾀ05 ｧ, ｧ, 09 ﾀｧ縺06. 6ｦ罍ｧｧ0 ｧ06 ﾀｧﾀ0903 ﾀ050406, 04 縺閨縺ｧ 0406 縺04060 陦ｧ縺ｧﾀｧﾀ0903 ﾀ05 ｧ, 04 縺閨縺 C 縺08003 ﾀ05 ｧ x + y + z, x + y, 3z =0 ｧ 0406 縺04060 陦ｧ縺縺 OC ｧﾀｧ罍ｧ 06 ﾀﾀ , 閨罍陦ｧ0 05, ｧ (0;,3; 0) ｧ 04 縺ｧﾀ0903 ﾀ0504 z = 0 ﾀｧ07 00 縺縺08003 ﾀ05 ｧ x + y + z =4(x, y, z) ﾀ0903 ﾀ050406, 04 縺閨縺ｧ 0406 縺04060 陦ｧ縺縺 x =0, y + z = ﾀｧﾀ罍陦ｧﾀ05 縺03 ｧ縺08003 ﾀ05 ｧ: ) z =4, x ;)y + z =4z; 3)y = x ﾀ05 縺03 ｧ縺ﾀｧﾀ罍陦ｧ罍ｧ縺06 ﾀｧ07 09 ｧ00 ｧ0 陦縺0304 x + z = a ｧ x + y = a. ｰ09 0 ｧﾀ05 縺03 ｧﾀ0903 ﾀ xoz ｧ xoy 縺05 ｧｧ縺ﾀ050608, 閨0600 ｧ0504 ｧ08 04 罍ｧ0300 ｧﾀ05 ｧ (0 04 罍ｧ- 006, 0 4) ｧ 06 縺ｧ閨ｧ07 04 縺ﾀ05 ｧｧ06 09 ｧ00 ｧ0 陦縺0304 閨03 ｧ縺06 ﾀｧ07 ( ﾀ0. 罍ｧﾀ. 60, ﾀ. 30) ｧﾀｧ06 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ縺08003 ﾀ05 ｧﾀ , 縺06 Pf; ; g, ｧﾀ x + + y =4x, z = ﾀ05 縺03 ｧ , ｧ縺08003 ﾀ05070 ｧ y = x, z = 0, z = 4, x = 4, ｧ 0 04 ｧﾀｧ07 陦ｧｧ, ﾀ0903 ﾀ05 ｧ x =4. x ｧ 06 ﾀ09 ｧ縺08003 ﾀ05 ｧｧ縺08003 ﾀ05 ｧｧｧ 06 ﾀ09 ｧ縺08003 ﾀ05 ｧ. 706 ﾀｧ 06 ﾀ縺08003 ﾀ0504 ｧ縺00 ｧ罍陦ｧ0 05, F (x; y) = 00 x ﾀ0903 ﾀ05 ｧ z = h. ｰｧ閨0605: y z = = x 0 y 0 h,05 閨06 (x 0 ; y 0 ; h) 縺06 閨0600 ｧﾀ06 x 0 ｧ y 0 ｧ 0403 陦ﾀ05-04 ｧ04 ｧｧ06 F (x; y) =0, ｧｧｧ 06 ﾀ縺ﾀ05 ｧﾀ 0406 縺00 ｧ罍陦ｧ0 05: xh F z ; yh =0: () z

66 3x ｧ 06 ﾀ09 ｧ縺08003 ﾀ05 ｧｧ縺08003 ﾀ05 ｧｧ07 73 ｭﾀ00 ｧ 0406 縺00 ｧﾀ 0306 閨 (a; b; c), ｧ06 04 罍ｧｧ (x, a)(h, c) (y, b)(h, c) F + a; + b =0: () z, c z, c ｰｧ06 () 03 閨003 縺03 閨ﾀｧ x, y, z, ｧ06 () 縺03 閨ﾀｧ x, a, y, b ｧ z, c 閨003 縺03 閨003 ﾀ05 ｧｧｧｧ 06 ﾀ縺08003 ﾀ05 ｧ縺08003 ﾀ05 ｧｧ07: ｰｧ07 09 罍ｧﾀｧ07 ｰｧ縺08003 ﾀ05 ｧｧ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)= ｧﾀｧｧ 06 ﾀ縺08003 ﾀ05 ｧﾀ 0406 縺00 ｧ罍陦ｧ0 05 ｧ x + y = a, z = c. 703 ﾀ05 縺03- ｧ0504 ｧｧ縺08003 ﾀ05 ｧｧﾀｧｧ 06 ﾀ縺08003 ﾀ05 ｧﾀ 0406 縺00 ｧ A(0;,a; 0) ｧ x =y, z = h. 703 ﾀ05 縺03 ｧ0504 ｧｧ縺08003 ﾀ05 ｧ縺06 閨0600 ｧ縺00 ｧﾀ x +(y, a), z = 0, ﾀ0903 ﾀ05 ｧ z = a ｧ 0403 ﾀ05 縺03 ｧﾀ縺06 閨0600 ｧ縺00 ｧﾀ x =yz, ﾀ0903 ﾀ05 ｧ z = h ｧ 0403 ﾀ05 縺03 ｧﾀｷ3 ﾀﾀ0006 閨縺08003 ﾀｧ ) ｧ00 ｧ 0900 ｧ0 (a, x ) y = h z 0403 ﾀｧﾀ0903 ﾀ05070 ｧ z =0, y = h, x = c (c 6 a) ｧ 0403 ﾀ05 縺03 ｧｧﾀ05 ｧ z > ｧﾀｧ縺08003 ﾀ05 ｧ, ｧ罍ｧ z = x, y = 0: ) 縺ﾀｧOz; ) 縺ﾀｧ Ox. 703 ﾀ05 縺03 ｧ縺08003 ﾀ05 ｧ. ) ｧ閨ﾀ縺08003 ﾀ0504, 閨04 ｧ07060 ｧ縺 , ﾀ0903 ﾀ05 ｧｧ 0406 縺06 ﾀ罍ｧｧ縺

67 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧ縺03 ﾀ05 0 ﾀｧﾀｧ縺08003 ﾀ05 ｧ, ｧ縺ﾀｧOz: )09 罍ｧ z = e,x, y =0;)09 罍ｧ z = 4 x, y = ﾀ05 縺03 ｧ縺08003 ﾀ05 ｧ ( ﾀｧﾀ罍陦ｧ0 05) ｧﾀｧｧ 06 ﾀ縺08003 ﾀ05 ｧﾀ 0406 縺00 ｧ00308 O(0; 0; 0), x +(y,6) +z =5, y =3 ｧ 0 罍ｧﾀ縺08003 ﾀｧﾀｧｧ 06 ﾀ縺08003 ﾀ05 ｧﾀ 0406 縺00 ｧ00308 C(0;,a; 0), x + y + z =5, y =3 ｧ 0 罍ｧﾀ縺08003 ﾀｧﾀｧ縺08003 ﾀ05 ｧ, ｧ縺 z = y, x = 0: ) 縺ﾀｧOy; ) 縺0605 Oz, ｧ 0 罍ｧﾀ縺08003 ﾀ05 ｧ , 0503 ﾀｧﾀ z = xy ﾀ0903 ﾀ x + y = =a 06 ﾀｧ04 ﾀ, ｧｧﾀｧ. x ｧ04 ﾀ03 ｧ, 05 ｧ0406 縺ｧ閨03 ｧｧ閨ｧ 06 ﾀ09 ｧｧ縺ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ08, ﾀﾀﾀﾀｧ閨縺08003 ﾀ縺縺07 閨09, 0304 縺06- 閨ﾀ0006 閨06060 ｧ0 ｧｧ 06 ﾀ09 ｧ0 ｧ (04 縺03 ﾀｧ0 ｧ) ｧ070 ｧ: I ｧ04 ﾀ03 ｧ : x II. 06 ｧ0406 縺ｧ閨03: a + y b + z c =. 8 >< >: x a + y x a + y III ﾀ縺縺07 閨09 : b, z = 03 閨ﾀ ; c b, z =, 閨ﾀ c x 8 >< IV ｧ閨03 (04 罍ｧ q > 0): >: a + y b, z c =0. x + y q x =z ｧ0405 ｧ 06 ﾀ09 ｧ08; =z 05 ｧ0406 縺ｧ 06 ﾀ09 ｧ08., y q. 7 縺ｧｧ06. ｪ06 縺閨ﾀｧ0406 縺ｧ 04 縺閨0705 閨縺ｧｧ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 :

68 3x ｧ04 ﾀ03 ｧ, 05 ｧ0406 縺ｧ閨03 ｧｧ閨03 75 ｪ06 縺閨ｧ0406 縺ｧ 06 ﾀｧ縺閨0705 閨縺ｧｧ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 縺ﾀｧﾀ縺08003 ﾀ050708, ｧｧｧ04-05 ｧ 06 ﾀ09 ｧ06 ﾀｧ07, ｧﾀｧ 09 縺ﾀｧ07. 7 ｧｧ06 09 縺ﾀｧｧ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 縺ﾀｧｧ0405 ｧ 06 ﾀｧ x + y q ﾀ0706 縺06 x + y + z =z (04 罍ｧ >q). = z, 04 縺閨070 ｧ06 06 縺縺00 ｧ006, 閨0705 ﾀｧﾀｧ縺08003 ﾀ05 ｧ, ｧｧ04 ﾀ x a + z =, y = 縺ﾀｧOz. c ﾀ05 縺03 ｧ縺08003 ﾀ0504 x 9 + y 4 + z = ｧｧ陦ｧﾀｧﾀ0903 ﾀ05070 ｧ: ) z =3;)y = ｧﾀｧ縺08003 ﾀ05 ｧ, ｧ罍ｧ x a, z =, y =0:) 縺0605 c 03 ﾀｧ Oz; ) 縺ﾀｧOx. 703 ﾀ05 縺03 ｧ縺08003 ﾀ05 ｧ ( ﾀｧﾀ罍陦ｧ0 05) ﾀ05 縺03 ｧ縺08003 ﾀ05 ｧ: ) x + y, z =4; ) x, y + z +4=0., z ﾀ05 縺03 ｧｧ0406 縺ｧ x 6 + y 4, = ｧｧｧ06, 04 縺閨070 ｧ06 06 縺 (4; ;,3) ｧ閨ｧ00 ｧ0 陦縺縺縺縺 ( 罍ｧﾀ. ) 縺06 閨0600 ｧｧｧ縺08003 ﾀ05 ｧ, 06 ﾀ00 ｧｧﾀ.

69 縺0 00 ｧ05 ｧ 06 ﾀ罍ｧ縺03 ﾀ05 0 ﾀ縺ﾀ05 ｧﾀｧﾀ0903 ﾀ z = c, ｧ縺ﾀｧ Oz, ｧ08 陦ｧ06 ﾀ a. 7 ﾀﾀ00305 縺ﾀﾀ0006 ｧ 04 罍ｧ =90, 0, 80. ｰ09 0 ｧ06. ｯ03 09 M (x; y; z) 閨0600 ｧ05 ﾀﾀｧ閨ｧ A(a cos t; a sin t;,c); B(a cos (t + ); a sin (t + ); c) ｧｧ AM : MB =(c + z) :(c, z) ｧﾀｧ縺08003 ﾀ05 ｧ, ｧ az = x, y = 縺ﾀｧOz. 703 ﾀ05 縺03 ｧ縺08003 ﾀﾀｧﾀ0903 ﾀ05070 ｧ: z = a, x =0, y = ﾀ05 縺03 ｧ縺08003 ﾀ05 ｧ: ) z = x + y ; ) z = c, x a, y. b ﾀ05 縺03 ｧ0504 ( ﾀｧﾀ罍陦ｧ0 05) 縺08003 ﾀ0504 x,, y =4z ｧｧｧ06, 04 縺閨070 ｧ06 06 縺 (3; ; ) ｧﾀｧ罍ｧ 06 ﾀﾀ , ｧ06 ﾀﾀｧ閨0308 ｧ縺0308 閨ﾀ0903 ﾀ05 ｧ x = a 09 ﾀﾀｧ070 閨ｧ F (a; 0;0) ﾀ05 縺03 ｧ縺ﾀｧﾀｧ罍ｧ 06 ﾀﾀ , ｧ06 ﾀﾀｧ閨0308 ｧ縺0308 閨ｧ F (0; 0; a) ｧ閨ﾀ0903 ﾀ05 ｧ z = a ﾀ05 縺03 ｧ縺08003 ﾀｧﾀｧ罍ｧ 06 ﾀﾀ , 03 陦ｧｧ F (,a; 0;0) ｧﾀ0903 ﾀ05 ｧ x = a ﾀ05 縺03 ｧ縺08003 ﾀｧ 0 ｧｧ06 09 縺ﾀｧｧ04 ﾀ03 ｧ x y 5 + z 9 =. x 縺06 閨0600 ｧ縺ﾀｧｧ0405 ｧ 06 ﾀｧ 5 + y 9 = z, 04 縺閨070 ｧ06 06 縺罍陦ｧｧ 0403 ﾀ05 縺03 ｧ閨0606 ｧ縺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 :

70 3x ｧ04 ﾀ03 ｧ, 05 ｧ0406 縺ｧ閨03 ｧｧ閨ｧﾀｧ07 04 縺ｧｧ08 05 ｧｧ x, y + z =4,04 縺閨070 ｧ08 06 縺 (; 4; 4) ｧﾀｧ罍ｧ 06 ﾀﾀ , 03 陦ｧｧ F (0; 0; a=) ｧﾀ0903 ﾀ05 ｧ z =,a=. 703 ﾀ05 縺03 ｧ縺08003 ﾀｧﾀｧ罍ｧ 06 ﾀﾀ , 03 陦ｧｧ F (0; 0; a=) ｧﾀ0903 ﾀ05 ｧ z = 3a=. 703 ﾀ05 縺03 ｧ縺08003 ﾀｧ 0 ｧｧ06 09 縺ﾀｧ07 05 ｧ0406 縺ｧ x 5 + y 9, 3z 5 =: ｧﾀｧ07 04 縺ｧｧ08 05 ｧｧ 06 ﾀｧ x (4; 3; 0)., y 6 9 =z, 04 縺閨070 ｧ08 06 縺

71 ｧ9 ｮ0ｹ0 縺00 縺6ｧ806 ｭ 7 7 x 縺06 閨0600 ｧｧ縺06 閨0600 ｧｧ縺06 閨0600 ｧ縺縺07 閨ﾀ07 ｧﾀ0003, ﾀｧ a b a b ｧ 0304 縺06 閨ﾀ a b = a b a b, a b : () 0504 縺06 閨0600 ｧ縺縺07 閨ﾀ07 ｧﾀ0003, a b c ﾀｧ a b c ｧ 0304 縺06 閨ﾀ 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 () 縺06 閨0600 ｧｧ縺縺07 閨09, 閨070 ｧﾀﾀ0504 (), ﾀ07 ｧ縺06 閨0600 ｧ縺縺07 閨縺09 ｧ04-0 ｧ閨00308 ﾀ05 縺0309 ｧｧ 03 閨ﾀｧﾀｧ003 0 ｧ. ｱ () ﾀ縺ｧ縺06 閨0600 ｧ縺縺07 閨縺ﾀ05 縺0309 ｧ.. ｮﾀ縺06 閨0600 ｧ : I ｧｧ縺06 閨0600 ｧｧ 0060 ｧ05 ﾀﾀ05 縺0309 ﾀｧ. II ｧｧ縺06 閨0600 ｧ縺06 ﾀｧ閨縺07 閨ｷ3 ﾀﾀ0504 I ｧ II ﾀ0006 閨060605, 縺06 閨0600 ｧｧ縺07, 縺ﾀ閨縺ﾀ05 縺 III 縺06 閨0600 ｧﾀ閨陦ｧｧｧ縺07 0 ｧ IV ｧｧ閨縺ﾀ05 ｧ縺06 閨0600 ｧ V ｧｧ縺06 閨0600 ｧｧ 0060 ｧ05 ﾀ07, 06 ﾀ00 ｧ閨縺07 04 罍ｧ03 04 ｧ縺07, 縺03 ｧ陦ｧｧﾀ罍ｧ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 :

72 3x 縺06 閨0600 ｧｧ 79 ｮﾀﾀ縺07 閨縺06 閨0600 ｧ縺縺07 閨09 ﾀ閨閨 , 縺03 ﾀ05 ｧ05 ﾀｧ縺06 閨0600 ｧ縺閨04 05 縺ｧ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) ｧﾀ00 ｧ縺06 閨0600 ｧｧ: , : , : ,0, a, sin cos : 590. a a, cos sin. 59. sin sin cos cos ｧﾀ00 ｧ縺06 閨0600 ｧｧ, ｧ04 ｧ縺ﾀ : 3 4 a a 59. 5, 3 : 593., a a, a ｧﾀ00 ｧ縺06 閨0600 ｧｧ, ｧ04 ｧ縺07, 縺0308 ﾀ03 閨06 縺07 ｧ05 0 ｧｧﾀ : b,x x b 0 b 0,b : ,x, x,x. ｰ04 縺03 ﾀ05 ｧ0504 ｧ 0403 ｧﾀ00 ｧ縺06 閨0600 ｧｧ: a,a a a a a,a,a : ,4 7,3, ,4 x x : 599. y y z z + cos + sin, sin + cos. cos sin cos sin. 0.

73 ﾀｧ閨04 05 縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(; 3); B(4;,) ｧ C(6; 5): ｧｧ 0 03 閨縺ｧ A(; 3); B(; 4) ｧ C(3; 5)? ｧﾀ 0504 ﾀ縺06 閨0600 ｧ縺縺07 閨ｧ06 04 縺 , 04 縺閨縺ｧ: ) (x ; y ) ｧ (x ; y ); ) (; 3) ｧ (,; 5). ｰ04 縺03 ﾀ05 ｧ0504 ｧ 0403 ｧﾀ00 ｧ縺06 閨0600 ｧｧ: ,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 ｧﾀ05 ｧ a 縺06 閨0600 ｧ , ｧ 0406 縺ｧ縺0308 ﾀ05 縺ﾀ縺ｧﾀ05 ｧ (x, z) ｧ (y, z) 縺06 閨0600 ｧ , 0503 x + x y + y x, x y, y x y ｧ x ｧｧ08: ) x 4 9 x 3 =0; ) x 3 x, 0 4 =0 x y : = x y ｧ 04 縺罍ｧ陦ﾀ縺縺06 閨0600 ｧ x. ｮｧﾀｧｧ08. ｮｧﾀ05060 閨ｧｧ08 ﾀ閨ｧﾀｧ a x + b y = c ; a x + b y = c ().

74 3x. ｮｧﾀｧｧ08 8 ｧ縺ｧ06 c b c b a c a x = a c ; y = b a b a () b a b 04 罍ｧ 06 ﾀｧｧ, 縺06 閨0600 ｧﾀｧﾀ = a b a b 6= 0.. ｮｧﾀ05060 閨閨003 縺03 閨ｧｧ08 ﾀ 05 縺ｧ 0406 ﾀｧ a x + b y + c z =0; a x + b y + c z =0 ｧ縺ｧ07, 0304 縺06 閨縺ｧ 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 ｧｧﾀｮｧﾀ縺閨003 縺03 閨ｧｧ08 ﾀ 05 縺ｧ 0406 ﾀｧ a x + b y + c z =0; a x + b y + c z =0; a 3 x + b 3 y + c 3 z =0 ｧｧ縺ｧ07, 06 ﾀ00 ｧ 0304 縺06 閨0600 ｧﾀｧﾀ a b c = a b c a 3 b 3 c 3 =0; ｧｮｧﾀ縺ｧｧ08 ﾀ閨ｧﾀｧ a x + b y = c ; a x + b y = c ; (6) a 3 x + b 3 y = c 3 a b c ﾀﾀ050, = a b c =0 ｧﾀｧﾀﾀ03 閨06 縺07 ｧ縺003 a 3 b 3 c 3 04 縺0305 ｧ0403 縺06 ｧｧｮｧﾀ縺ｧｧ08 ﾀ 05 縺ｧﾀｧ (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)

75 ﾀ罍ｧ 06 ﾀｧｧ, 縺06 閨0600 ｧﾀｧﾀ a b c = a b c a 3 b 3 c 3 6=0; ｧﾀ0006 閨陦ｧ0 ﾀ縺ｧ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 : ﾀﾀｧ縺06 閨ﾀｧﾀｧﾀ05 ｧｧ08 (7) 06 縺06 X, X ｧ X ﾀ縺06 閨0600 ｧﾀｧﾀ (7) =0. 7 罍ｧ閨04 04 縺06 閨ｧ07. I 閨ﾀ05 縺縺06 閨0600 ｧ縺縺09 ｧ , 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 ﾀｧﾀﾀﾀ050 (0406 縺閨ｧ07 04 縺0305 ｧ0403 縺06 ｧ0403); ) 06 ﾀ00 ｧ d = md,0503 ﾀｧﾀ縺06 閨 (06 ﾀ00 ｧ 0406 縺ｧ 05 縺ｧ縺0305 ｧ0403 縺06 ｧ0403). II 縺06 閨0600 ｧﾀ05 縺0309 ﾀ 04 縺縺09 ｧｧｧ. ｯ0305 ﾀ06006 ﾀｧｧﾀ00 m ｧ n ｧ06, 04 罍ｧ縺0308 mx + + nx = X 3, ｧ: ) 06 ﾀ00 ｧ md + nd 6= d 3,0503 ﾀｧﾀﾀﾀ050 ; ) 06 ﾀ00 ｧ md + nd = d 3,0503 ﾀｧﾀ縺06 閨ｪｧﾀ00 m ｧ n 閨ｧ00 ｧｧｧ08 ｧｧ08 a m + + a n = a 3, b m + b n = b 3, c m + c n = c ｧ0504 ﾀ縺06 閨0600 ｧﾀｧﾀｧ08: n n 3x +y =7; ax, 3y =; x, 5y =40: ax, y =: n 5x +y =4; mx, ny =(m, n) ; x +4y =8: x, y = n (04 罍ｧ m 6= n): ｧ0504 ﾀｧﾀｧ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: ( x, 4y +3z =; x, y +4z =3; 3x, y +5z =: ( 3x +y, z =0; x, y +3z =0; x +3y, 4z =0;

76 ( 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 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: 縺06 ﾀﾀ07 00 ｧ閨縺 : ) x, 3y =6, 3x + y =9, x +4y =3; ) x, 3y =6, x +y =4, x, 5y =5? ｧｧ08 ﾀﾀ05 縺03060 ｧｧ0504 ﾀｧﾀｧｧ08: ( x, y + z = ; 64. 3x +y +z =,; 65. x, y + z = : n 3x +y +z =0; 5x +y +3z =0: ( x, y +3z =0; x +y, 5z =0; 3x + y, z =0: x 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 =: 縺06 閨ｧﾀ0030 ｧﾀﾀｧ06 04 ｧ x + yi, 縺030 x ｧ y ﾀｧﾀ00, i 縺0308 ﾀｧ , 06 ﾀ00 ｧ 04 罍ｧ罍ｧﾀｧ07: ) x +0i = x, 0+yi = yi ｧ i = i, (,)i =,i; ) x + yi = x + y i ｧ , 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 ﾀｧ08 ) ｧ 4) ﾀ07 ﾀｧｧﾀ00 i: i =,; i 3 =,i; i 4 =; i 5 = i ｧ05.. () ﾀ00306 ｧﾀ0003 x + yi, 縺030 y 6= 0, ﾀ07 00 ｧ0030 ｧﾀｪｧﾀ0003 i ﾀ07 00 ｧ陦ｧ0 ｧﾀ0504 ｧﾀ0030 ｧｧﾀ00 0 ｧ. ｮｧ06, 0403 ｧ05 0 ｧ06, ｧ06 ｧ閨060 ｧ06 04 ﾀﾀ00308 ｧﾀｧｧ08 閨0608 ﾀ0504 ｧｧﾀﾀｧﾀ00 i 縺 () ｧﾀ00308 ｧﾀ0600 ｧｧｧ縺007 ｧﾀｧ- ﾀ縺06 閨ﾀ閨0608 ﾀ0504 ｧ

77 ﾀｯ罍ｧ罍ｧ 06 ﾀ縺ﾀｧﾀﾀ00306 ｧﾀ0003 x+yi 0304 縺06 閨ﾀ07 04 縺ﾀｧﾀ0600 (x; y) ｧｧﾀ M (x; y) ﾀ0903 ﾀ05 ｧｧ00 ｧ 0606 陦ｧ06 ﾀ縺030,,! r = OM ( ﾀ0. 罍ｧﾀ. ) ｧ r = x + y ﾀ閨ﾀｧﾀ00, ' ﾀ03 ﾀ0406 Ox ﾀ07 縺ﾀｧﾀ00. ｯ x = r cos ', y = r sin ', 0503 x + yi = r(cos ' + i sin '): () ﾀ0504 ｧﾀ0030 ｧｧﾀ00 0 ｧ罍ｧ罍ｧ 06 ﾀ縺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 (5) ｧ (6) ﾀ縺ｧｱ03 縺 : 6. 6ｧ80305 罍ｧﾀｧﾀ00 : e i' = cos ' + i sin ': (7) ln z =lnr + i' 0 +ki; (8) 05 閨06 ' ｧ06 縺 ', 06 閨縺ﾀ0504 0, < <' ｧ06 ln r + i' ﾀｧ罍ｧｧ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 ｧｧ07: ) x +5= 0; ) x, x +5= 0; 3) x +4x +3= 0 ｧ 04 縺罍ｧ陦ﾀ縺ｧ06. ｮ0006 閨06060 ｧﾀ00306 ｧﾀ00 ｧ 0303 ｧｧ, 0304 縺06 閨06-00 ｧ0504 ｧ閨0600 ｧｧ縺ｧ 04 ｧﾀ罍ｧ罍ｧ 06 ﾀ縺006: 63. ) z =3; ) z =,; 3) z =3i; 4) z =,i ) z =, i; ) z =+i 3; 3) z =, 3, i.

78 3x ﾀ00306 ｧﾀ ), +i ; ) sin + i(, cos ) ｪｧﾀ00, {634, 04 ｧﾀ縺006 re 'i (04 罍ｧ, <'6 ) ﾀ05 縺03 ｧﾀ05 ｧ z ﾀｧ070: ) jzj < 3; ) jzj < ｧ = <'<; 3) < jzj < 4 ｧ, <'<,= , 0503 jz, z j 06 ﾀ0504 ﾀﾀｧ閨ｧ z ｧ z z 0 =, +3i. 703 ﾀ05 縺03 ｧﾀ z, 閨縺0308 jz, z 0 j < ｪｧﾀ0003, ﾀ0304 縺ﾀ z, ﾀ07 06 縺06 z , 0503 z z = jzj ｧﾀ00 ｧ縺 : ) ( + i) 0 ; ) (, i 3) 6 ; 3) (, +i) 5 ; 4) + cos 4 + i sin 4 4 ; 5), 3+i ｧ0504 sin 3 ｧ cos 3 06 縺ｧｧ , ｧﾀ閨06 ﾀ (cos + i sin ) 3 = cos 3 + i sin ｧ 04 ﾀｧ07 z = 6 ｧｧ 0303 ｧ0504 ｧ08 陦ｧ06 ﾀｧ, 0403 ﾀ05 縺03 ｧ04 09 縺0605 陦ｧ06 ﾀ, ｧ: ) 3 ; ) 3 i; 3) 6,; 4) 3,+i ｧ: ) i; ) 3,+i; 3) 4,8+8i ｧ0504 閨ｧ07: ) x 3 +8=0;)x 4 +4 = ｧｧ罍ｧ070 : ) ln (,); )ln ( + i); 3) ln i; 4) ln (x + yi); 5) ln (, i) ｧﾀ sin x + sin x + sin 3x + :::+ sin nx. ｰ09 0 ｧ縺ｧ0504 sin x = exi, e,xi ｧﾀ cos x + cos x + cos 3x + :::+ cos nx 閨06 ﾀ x 5, =(x, )(x, x cos 7 + )(x, x cos 44 +): i ｧｧﾀ00 ｧ0504: ) 4, 3i 4+3i ; ) (a + bi)3, (a, bi) 3. ｮ0006 閨06060 ｧﾀ00306 ｧﾀ00 ｧ 0303 ｧｧ, 0304 縺06 閨06-00 ｧ0504 ｧ閨0600 ｧｧ縺ｧ 04 ｧﾀ罍ｧ罍ｧ 06 ﾀ縺006 ｧ縺006 re 'i (04 罍ｧ, <'6 ):

79 ﾀ ) z =4+4i; ) z =, +i 3; 3) z =, i. 65. ) z =5; ) z =,i; 3) z =,,, ﾀ05 縺03 ｧﾀ z ﾀｧ070 < jzj < 3 ｧ =4 <'<3=4: z 0 = 3, 4i. 703 ﾀ05 縺03 ｧﾀ z, 閨縺0308 jz, z 0 j < ｧﾀ00 ｧ縺 : ) (, i) 6 ; ) ( + i ) 5 ; 3) + cos 3 + i sin ｧ0504 sin 4 ｧ cos 4 06 縺ｧｧ , ｧﾀ閨06 ﾀ (cos + i sin ) 4 = cos 4 + i sin ｧ 04 ﾀｧ縺00608: ) 4,; ) 5 ｧｧ 0303 ｧ0504 ｧ08 陦ｧ06 ﾀｧｧｧ07: ) x 3, 8=0; ) x 6 +64=0; 3) x 4, 8 = ｧﾀ ( ﾀ ) cos x + cos 3x + cos 5x + :::+ cos (n, )x: x 4. ｰｧﾀ00 ｧ08 ﾀｧ 04 罍ｧ0300 ｧ縺ｧｧｧ 06 ﾀｧ06: x 3 + ax + bx + c =0: () ｭﾀ00 ｧ x, x, x 縺0 ｧｧ07 (), ｧｧﾀｧ閨06 (x, x )(x, x )(x, x 3 ) = ﾀ06 a =,(x + x + x 3 ), b = x x + x x 3 + x x 3, c =,x x x 3. ｰｧ06 x 3 + ax + bx + c =004 罍ｧ0403 陦ｧ05 ﾀｧ閨06 z 3 + z + q = 陦ﾀ x = z, a 3. ｰｧ06 z3 + z + q =0 縺ﾀ縺罍 003: s r z = 3, q + q s, q, r q = u + v: I. ｭﾀ00 ｧ = q > 0, 0503 z = u + v, z ; 3 =, u + v i 3,05 閨06 u ｧ v ﾀｧ縺00608 u ｧ v. u, v II. ｭﾀ00 ｧ = q =0,0503z = 3q, z = z 3 =, 3t =, z.

80 3x 4. ｰｧﾀ00 ｧ08 ﾀ r III. ｭﾀ00 ｧ = q < 0, 0503z =, r 3 cos ', 3 z ; 3 =, 3 ' cos,05 閨 cos ' =, q r, 閨ｧﾀ縺ｧ07 f(x) = = 閨06 a ｧ b ﾀ03 閨06 縺07 ｧ05 ﾀ07 06 陦ｧ0 ﾀ縺ｧ07 f(x) =0, 06 ﾀ00 ｧ f(a) ｧ f(b) ｧｧ, f(x) 縺06 縺縺 [a; b] ｧ罍ｧｧ縺03 ｧ 0403 閨00606 f 0 (x) 6= 閨060 ﾀｧ , 縺ｧ f 00 (x) 6= ｮ0403 ﾀ罍 04 罍ｧ0300 ｧ縺ｧｧ07 f(x) = ﾀｧ縺06 09 [a; b], 0305 閨縺06004, 7 ｧﾀ縺030 f( 0 )f 00 ( 0 ) < 0. ｯ罍ｧ0300 ｧ07060 ｧ縺006 x 0306 閨縺06 ﾀｧ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 ﾀﾀ ( ﾀ0403 ﾀ ). 706 ﾀｧ縺06 09 [a; b], 縺030 f( 0 )f 00 ( 0 ) > 0. ｯ罍ｧ0300 ｧ07060 ｧ縺006 x 0306 閨縺06 ﾀｧ07 ﾀ Ox 09 ﾀ罍ｧ y = f(x) [ 0 ;f( 0 )] ( 罍ｧﾀ. 3): = 0, f( 0) k ; 05 閨06 k = f 0 ( 0 ). 7 罍ｧ縺003 ﾀ0403 ﾀ罍陦ｧ09 ﾀ , ﾀ03 ﾀ05 04 ｧｧ0906 j j f() j f() j k j k j j j :: j :: j :: j :: j :: j :: j :: j :: j ; ()

81 ﾀ閨06 k ｧ k 罍ｧ 09 ﾀ , =, f() ｧ =, f() : k k 703 ﾀ0006 閨ﾀ05 ｧｧ0906 () ｧ08 ｧﾀ0803- 閨0705 ﾀ07 09 ｧﾀ縺ｮ0403 ﾀ0303 ｧｧ08. ｭﾀ00 ｧｧ06 f(x) = 罍ｧ0406 ﾀ05 ｧｧ閨06 x = '(x), 04 罍ｧ縺縺06 ﾀ05003 ﾀ05 ｧ 0903 縺007 j' 0 (x)j < 0 6 ｧ x ｧﾀ縺06 ﾀ05003 ﾀ05 ｧ, 0503 ﾀ0803 閨ﾀﾀ0006 閨ﾀ罍ｧ0300 ｧ縺ｧ閨0605 x = '(x 0 ); x = '(x ); x 3 = '(x );::: ｧ , 66 ﾀ縺06 陦ｧｧﾀ閨陦ｧ縺06004, 閨0600 ｧﾀ x,x ｧｧ 03 ﾀ縺0 ｧ: 660. ) x 3, 4x + x +6= 0; ) x 3, 4x, 4x, 5= ｧ06 04 縺罍ｧ0504 ﾀ03 ﾀｧｧ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, = ｧ縺罍 003 ﾀ0006 閨06060 ｧｧ07: 66. ) z 3, 6z, 9=0; ) z 3, z, 6 = ) z 3, z, 8=0; ) z 3 +6z, 7= x 3 +9x +8x +9= ｧ06 f (x) =x 4, x, 0 = 0. ｮ03 ﾀ05 04 ｧｧ f (x) 04 罍ｧ x =0; ; ; :::, 0304 縺06 閨0600 ｧｧｧ縺007 ｧ 0403 ｧﾀ00 ｧﾀﾀ閨03 0, ﾀ0403 ﾀ罍ｧ 09 ﾀﾀ05 縺03 ｧｧｧ08 y = x3, 0304 縺06 閨0600 ｧｧ 06-3 ﾀ09 ｧ 05 0 ｧ縺ｧ07 x 3, 6x +3 = 0 ｧ 0403 ｧﾀ00 ｧ縺0 ｧﾀﾀ閨03 06 陦ｧ0 ｧ縺ﾀ0403 ﾀｧｧ08 (0403 ﾀ0006 閨罍ｧ0300 ｧ07060 ｧ08) ｧﾀ縺0 ｧｧ08: ) x 3 +60x, 80 = 0; ) x =4x; 3) x 3 + l x + l 3 =0; 4) x 4, x, = 閨0303 縺閨縺007 ﾀ縺06 陦ｧｧﾀ縺0600 ｧｧ07: ) x 3 +8x +5x +8=0; ) x 3, 3x +4= 縺縺09 ｧﾀ03 ﾀ05 04 ｧｧ07 P x i, P x i x j ｧ x x x 3.

82 3x 4. ｰｧﾀ00 ｧ08 ﾀ縺罍 003 縺0600 ｧｧ07: ) z 3 +8z, 9=0; ) z 3, 6z, 4=0; 3) z 3, 3z +=0; 4) x 3 +6x +9x +4=0: ﾀ05 縺03 ｧｧｧｧ y = x4, 0304 縺06 閨0600 ｧｧ縺ｧ07 x 4 +3x,5 = 0 ｧ 0403 ｧﾀ00 ｧ縺0 ｧﾀﾀ閨03 0, ｧﾀﾀ閨03 0, ｧ縺0 ｧｧ08: ) x 3 +50x, 60 = 0; ) x 3 + x, 3 = ﾀ0403 ﾀｧｧｧﾀ縺ｧ07 x 3 +x, 8=0, 0403 ｧﾀﾀ0006 閨罍ｧ0300 ｧ07060 ｧ縺 x = 3 8, x.

83 ｭ6 7 ｭ 70ｷ3 ｭ 6 5 繖 7 縺6ｧ80ｷ36ｦ0 x. 706 縺ｧｧ003 ｧｧｧ縺06 09 ｧｧｧ00506 縺ﾀｧﾀ0600 x, 06 閨縺ｧﾀ a<x<b, ﾀ07 ｧ00506 縺ｧﾀ07 (a; b) ﾀｧﾀ0600 x, 06 閨縺07060 ｧﾀ a 6 x 6 b, ﾀ縺ｧﾀ07 [a; b] ｧﾀ0504 (04 罍ｧ a>0) x <a ; ｧ00 ｧ jxj <a; ｧ00 ｧ, a<x<a 0304 縺06 閨ｧ00506 縺04 00, ﾀｧ罍ｧﾀｧ縺ｧｧ003 ｧｧｧ. ｭﾀ00 ｧ閨ｧ縺 x 0403 ﾀﾀﾀ0504 ｧ06 03 閨003 ｧﾀ0003, 縺 y, 0304 縺06 閨ﾀﾀｧ08 ｧﾀ0600, ﾀ07 03 閨ｧ0608 x. 706 縺 x ﾀ07 04 罍ｧ縺 , ﾀﾀｧ08 縺ﾀ縺06 閨ｧｧｧ. ｯ03, 0503 y 06 ﾀｧ07 x, ﾀｧｧ 06 ﾀ09 ｧ 04 ｧﾀｧ閨06 y = = f(x), ｧ00 ｧ y = F (x), ｧ00 ｧ y = '(x) ｧｮｧ f(x) ｧ00 ｧ F (x) ｧﾀﾀ0504 ｧ縺 x ｧ y, 04 ﾀ05003 ﾀ05 ｧ, ﾀﾀ0504 閨0608 ﾀ0504 ｧ08 ｧ00 ｧｧ08, 縺ｧ x, ｧ0504 ﾀﾀｧ06 y ﾀ05 縺03 ｧﾀ05 ｧｧｧ縺 x, 06 閨縺ﾀ0504 0: ) jxj < 4; ) x 6 9; 3) jx, 4j < ; 4), <x, 3 6 ; 5) x > 9; 6) (x, ) ｦ0 04 ｧﾀﾀｧｧ 0403 ﾀ05 縺03 ｧ0504 ｧ00506 縺ｧｧ縺 : [,; 3]; (0; 4); [,; ] 縺06 閨0600 ｧﾀ0504 ｧｧ縺 x =, t,05 閨06 t 04 罍ｧ0 ｧｧ { ﾀ05 縺03 ｧ縺 jxj ｧ09 ｧｧ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, :

84 3x. 706 縺ｧｧ003 ｧｧｧﾀ05 縺03 ｧｧ09 ｧｧ08: ) y = 6 x ;)y =x ;3)y = = log x ﾀﾀﾀｧｧ 0505 ｧ08 09 罍ｧﾀｧﾀ罍陦ｧｧ0504? ﾀ05 縺03 ｧ閨縺ｧ09 ｧｧ08: ) y = = sin x; )y = cos x , 縺0308 y ｧｧ , 0 ｧｧｧ07. ｮｧ罍陦ｧｧ08 09 罍ｧﾀ05 縺03 ｧ縺ｧｧｧ y = sin x + cos x ｧ 0903 縺0 ｧ x ｧ x ｧｧ y =4x, x ｧ 0403 ﾀ05 縺03 ｧｧ縺 [x, ; x +] ﾀ05 縺03 ｧｧ09 ｧｧ08: ) y = jxj; ) y =,jx, j; 3) y = jxj,x { ｧﾀ05 ｧ 0304 縺06 閨ｧﾀｧｧ08 ｧ 0403 ﾀ05 縺03 ｧ0504 ｧｧ09 ｧ ) y = x +; ) y = 9, x ; 3) y = 4x, x ) y =,x + 4+x; ) y = arcsin x, : 685. ) y = x( x) ; ) y = x 4, x: ) 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. ｱｧ07 f (x) ﾀ , 06 ﾀ00 ｧ f (,x) =f (x); , 06 ﾀ00 ｧ f (,x) =,f (x). ｰ , ｧ06 ｧﾀ0006 閨06060 ｧｧｧ09 09 ｧ : ) 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..

85 閨060 ｧｧ 69. ｮ06 縺06 陦ｧ罍閨ｧ縺ｧｧ f (x) ｧｧｧｧ. 6ｦ0 04 ｧﾀﾀﾀｧｧﾀ縺罍ｧ0504, ｧ0 ﾀﾀｧ07 f (x) =x ｧ縺ｧﾀﾀｧ f () = 0, f (a) =, f (xy) =f (x)+f (y)? ｧ縺ｧﾀﾀｧ f (0) =, f () = a, f (x + y) =f (x)f (y)? ﾀ05 縺03 ｧﾀ05 ｧｧｧ縺 x, 06 閨縺ﾀ0504 0: ) jxj < 3; ) x 6 4; 3) jx, j < ; 4) (x, ) 縺06 閨0600 ｧﾀ0504 ｧｧ縺 x =+ t,05 閨06 t 04 罍ｧ0 ｧｧﾀ05 縺03 ｧｧ09 ｧｧ08: ) y =4, x 縺 jxj 6 ; ) y =3; 5+3x, x 閨ｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀﾀ09 ｧﾀﾀﾀ05 縺03 ｧｧ09 ｧｧ08: ) y = x, 4+jx, j 縺 [,; 5]; ) y =, cos x 縺 jxj ﾀ05 縺03 ｧｧ09 ｧｧ08: ) y =, 4 x ; ) y =,x ｧﾀ05 ｧ 0304 縺06 閨ｧﾀｧｧ08: ) y = 4, x ; ) y = x +, 3, x; 3) y =, cos x; 4) y = + ｧ 0403 ﾀ05 縺03 ｧ0504 ｧｧ09 ｧ. 70. ) ｧｧ f (x) = 4 x, 4 x + x ｧﾀ00 ｧ0504 f (0), f (,), f (,=), f (x, ), f (=); '(x ) 閨ｧｧ '(x) =x 3 + h), '(x, h) 0403 ｧﾀ00 ｧ0504 ; h 3) 閨ｧｧ f (x) =4x, x 0403 ｧﾀ00 ｧ0504 f (a +), f (a, ).

86 3x. 7 縺06 閨ﾀ0006 閨ﾀ05 ｧｧｧｧ 93 x. 7 縺06 閨ﾀ0006 閨ﾀ05 ｧｧｧｧ. 706 ﾀｧ 0306 ﾀｧ06. ｪｧﾀﾀ0006 閨ﾀﾀ閨ｧﾀ0006 n =; ; 3;::: 縺ﾀﾀﾀ0504 ｧ06 ｧﾀ0003 x n. ｯ縺0705, ｧ縺06 閨ﾀ0006 閨ﾀ0504 ｧﾀ0600 x, x, x 3, ::: ｧ00 ｧ, 0903 縺03 06, 0403 ﾀ0006 閨ﾀ0504 fx n g = = fx ;x ;x 3 ; :::g 閨ｧﾀﾀ0006 閨ﾀ05 ｧ fx n g ﾀｧ縺 , 縺 x n 04 縺ｧﾀ0006 閨ﾀ05 ｧ fx n g.. 7 縺06 閨ﾀ0006 閨ﾀ05 ｧ (04 縺06 閨縺 ). ｪｧﾀ0003 a ﾀ07 04 縺06 閨ﾀ0006 閨ﾀ05 ｧ fx n g, ｧ00 ｧ 04 縺06 閨縺 x n ( ﾀ07 x n! a), 06 ﾀ00 ｧ閨ﾀ "> 閨0605 ﾀ07 04 ｧﾀ " ｧﾀ0003 n , 0503 jx n, aj <" 閨ﾀ n>n 0. 0ｷ縺04 00 (a, "; a + ") ﾀ07 "-0309 縺06 ﾀ05003 ﾀｧﾀ00 a ( ｧ00 ｧｧ a). ｯ 09 ｧ , x n! a , 0503 閨ﾀ "> 閨0605 ﾀｧﾀ0003 n 0, 0503 閨ﾀ0608 n>n 0 ｧﾀ00 x n 0306 閨陦ｧ0504 ﾀ07 04 "-0309 縺06 ﾀ05003 ﾀ05 ｧｧﾀ00 a 縺06 閨ｧｧ. 706 ﾀｧ07 f(x) 0304 縺06 閨縺0308 "-0309 縺06 ﾀ05003 ﾀ05 ｧｧ a, ｧﾀｧﾀｧ a 縺0705, 0503 ｧﾀ0003 b ﾀ07 04 縺06 閨ｧｧ f(x) 04 罍ｧ x! a (04 ｧ f(x)! b 04 罍ｧ x! a ｧ00 ｧ lim f(x) =b), 06 ﾀ00 ｧ閨 ">0 ﾀ06006 ﾀ x!a 04 ｧﾀ " ｧﾀ0003 > , 0503 jf(x), bj <"04 罍ｧ 0 < jx, aj <. 縺ｧ 003, lim f(x) = b, 06 ﾀ00 ｧ閨ﾀ ">0 ﾀ06006 ﾀｧﾀ07- x!a " ｧﾀ0003 N , 0503 jf(x), bj <"04 罍ｧ jxj >N. ｰ縺ﾀｧﾀ04 lim x!a f(x) =, , 0503 閨ﾀｧﾀ00 A> 0 ﾀ06006 ﾀｧﾀ A ｧﾀ , 0503 jf(x)j >A04 罍ｧ 0 < jx, aj <. ｭﾀ00 ｧ x! a ｧ 04 罍ｧ x<a, ｧ x! a, 0; ｧ 003, 06 ﾀ00 ｧ x! a ｧ 04 罍ｧ x>a, ｧ 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) ﾀ07 ﾀﾀ縺06- 閨ﾀｧｧ f(x) a ｧ 04 縺06 閨ﾀｧｧ f(x) a ﾀ06006 ﾀｧ07 04 縺06 閨ｧｧ f(x) 04 罍ｧ x! a 陦ｧ003 ｧ閨03 ﾀ , f(a, 0) = f(a +0) ﾀ0503 x! 0, 0 ｧ x! ｧ x!,0 ｧ x! +0 ﾀﾀﾀｭﾀ00 ｧ lim (x) =0, ﾀ00 ｧ j(x)j <" x!a 04 罍ｧ 0 < jx, aj <("), ｧ07 (x) ﾀﾀ罍ｧ x! a. 縺ｧ縺06 閨ﾀﾀ (x) 04 罍ｧ x! ﾀｧ06. ｭﾀ00 ｧ閨ﾀ閨ｧﾀ00 N ﾀ06006 ﾀ (N ), 罍ｧ 0 < jx, aj <(N ) ﾀ jf(x)j >N, ｧ07 f(x) ﾀﾀ罍ｧ x! a. 縺ｧ縺06 閨ﾀﾀ f(x) 04 罍ｧ x!.

87 閨060 ｧｧ n =0; ; ; 3; :::, 0 04 ｧﾀﾀ0006 閨ﾀ05 ｧｧ縺 : = ; =, n ; =, n : n 7 ｧ0 07 ﾀ n 003 閨閨0308 ｧ 0406 縺ﾀ閨ﾀ07 ｧ閨ﾀﾀ ,00, ｧ "? ｧﾀﾀ0006 閨ﾀｧ縺 x = =+ (,)n. 7 ｧ0 07 ﾀ n 003 閨ﾀ05 ｧ x, ﾀ閨ﾀ07 ｧ n 閨ﾀﾀ ,0, ｧ "? 罍ｧ ( ｧ00 ｧ 0403 ｧ05 07 ｧ 3) ﾀ0 00, ,, ,0 ｧ 05.., 04 ｧﾀ閨06 ﾀ0705 ｧ 0030 ｧﾀ0006 閨ﾀ05070 ｧ 04 罍ｧ0300 ｧ07060 ｧ縺縺06 閨060006: x n! 3+0, x n! 3, ｦ0 04 ｧﾀ閨06 ﾀ0705 ｧ 0030 ｧﾀ0006 閨ﾀ05070 ｧ 04 罍ｧ0300 ｧｧ縺縺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! ;, , 0503 lim x = ﾀ0 ｧｧ09 0 ｧｧ08 x! x ｧ x , 0503 lim (x, )= ｧﾀ0006 ">0 x! ｧ 0 ｧｧﾀ0003 > , 罍ｧ x ｧ縺06 ﾀ05003 ﾀ05 ｧｧﾀｧｧｧ y =x, ﾀ04 04 "-0309 縺06 ﾀ05003 ﾀ05 ｧｧﾀﾀ0 ｧｧ 06 ﾀ09 ｧ , 0503 lim x!, 縺06 ﾀ05003 ﾀ05 ｧｧﾀ00, ｧ06 x, ｧｧｧ y =3, x, x ｧ 0003 ﾀ縺06 閨 " =0; 000? , 0503 sin 06 ﾀﾀ罍ｧ! 0. ｰ 09 0 ｧ 06. ｮ閨縺ｧ , 0503 j sin j < jj , 0503 lim sin x = sin a. x!a (3, x, x )=4. 0ｷｧｰ09 0 ｧｧ04 x = a +, ﾀ03 ﾀ05 04 ｧﾀ0504 sin x, sin a ｧｧ0504! , 0503 lim 060 ｧ08 x ｧ 3x +4 x 3x +4 = ﾀ0 ｧｧ09 0 ｧ 0 - x! x 04 罍ｧ x =; 0; 00; 000; ::: 4x, , 0503 lim x! x + =. 7 罍ｧｧ08 x ｧｧｧ 0306 閨ｧ 0504 ﾀﾀ縺06 閨 ,00?

88 3x. 7 縺06 閨ﾀ0006 閨ﾀ05 ｧｧｧｧ 95, x , 0503 lim =,0;5. 7 罍ｧｧ08 x ｧ07 x! +4x ｧｧ 0306 閨ｧ 0504 ﾀﾀ縺06 閨 ,0? , 0503 lim 0; 333:::3 n! {z } =, ﾀ03 ﾀ05 04 ｧﾀ05 ｧ 3 3, n , 0;3; 3, 0;33; 3, 0;333; :::;, 0; 333:::3 3 {z }. n ｧﾀﾀ0006 閨ﾀ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 ﾀｧ lim x n 閨罍ｧ006 縺06 ｧ ? n! ｧ lim x!+0 3 x, ｧ lim x!,0 3 x, ｧﾀ0 ｧｧ09 0 ｧｧ lim =x ｧ lim =x ｧﾀ0 ｧｧ09 0 ｧ. x!0+0 x!0, ﾀ0 ｧﾀ003 ﾀﾀｧﾀ0608: ) =0; ) 0 = ; 3) 3 = ; 4) 3, =0; 5) lg 0 =,; 6) tg90 = , 0503 lim x! ﾀ05 ｧｧ08 sin x: sin x 006 ﾀ06006 ﾀ , ﾀ03 ﾀ05 04 ｧﾀ0006 閨03- ) 04 罍ｧ x = n; ) 04 罍ｧ x = +n; 3) 04 罍ｧ x =, +n (n =0; ; ; 3; 4; :::) , 0503 lim sin x!0 x 006 ﾀ06006 ﾀ , 0503 lim x sin =004 罍ｧﾀ0403 ﾀ罍ｧ0300 ｧx!0 x ｧ07 x 縺0605 陦ｧ06 ﾀ R 0404 ｧﾀｧｧ09 ﾀｧﾀ00030 ﾀ0503 縺030 n ｧﾀ0503 縺 a n ｧﾀ縺陦 05, , 0503 a n <", n> 8R=", a n! 0, n! ﾀ0504 r n ｧ , 0404 ｧﾀ縺0605 n ｧ , 0503 lim r n = R, 05 閨06 R 陦ｧ06 ﾀ 09 縺0605. n!

89 閨060 ｧｧ縺00 ｧ0 B 05 縺ｧ09 ABC 0406 縺ﾀ縺 BE k AC, ｧﾀ閨罍ｧｧﾀ07 ﾀ0503 縺縺ｧ09, 閨04, 縺0600 ｧｧｧ BCD? ｧﾀ閨06 ﾀ0705 ｧﾀ0006 閨ﾀ05 ｧ 04 罍ｧ0300 ｧ07060 ｧ縺縺06 閨0600 0: x n! 4+0; x n! 4, 0; x n!,;5 +0; x n!,;5, , 0503: ) lim x 3 =7; x! , 0503 lim ) lim (x +x) =3. x! 5x + = ;5, , ﾀ0504 x! x 5x +, ;5 06 ﾀﾀ罍ｧ x 0306 ﾀ x ﾀ0 ｧｧ090608, x =; 0; 00; 000; ::: , 0503 lim cos x = cos a ( ﾀ ). x!a ｧﾀﾀ0006 閨ﾀ05 ｧｧ縺 : ) x n =+, n ; ) x n =(,) n + n ; 3) x n =(,) n (n + ); 4) x n = n sin (n=) : n ｧ 0403 ﾀ0006 閨ﾀｧ縺06 閨罍ｧ n! +? 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! , 0503 lim n!, 0; 6; 3, 0; 66; :::;, 0; 666:::6 3 {z }. n a +a x. + tg x ; 0; 666:::6 {z } =, ﾀ03 ﾀ05 04 ｧﾀ05 ｧ, 3 3 n ﾀ0504 n 縺0600 ｧｧ n ｧ , 0503 lim n =. n! 縺03 閨ｧｧ 0305 縺06 09 AB = a ﾀ M 0 ﾀﾀｧｧ BM = x ｧ lim x! AM.

90 3x 3. ｮﾀ縺06 閨 x 3. ｮﾀ縺06 閨ﾀ09 縺0305 ｧ縺06 閨ﾀｧ 0 0 ｧ. 7 縺06 閨ﾀﾀﾀ : lim(u + v) = limu + limv; 3 06 ﾀ00 ｧ lim u ｧ limv ﾀ06006 ﾀ : : lim(uv) = limu limv; 4. lim u v = limu, 06 ﾀ00 ｧ limu ｧ limv ﾀ06006 ﾀｧlimv 6= 0. limv 5. ｭﾀ00 ｧ閨ﾀｧ08 x 縺縺06 ﾀ05003 ﾀ05 ｧｧ a, 09 縺03006, , x = a, ｧｧ f(x) ｧ '(x) ｧ 03 閨0 ｧ 0 ｧ08 ｧ縺06 閨罍ｧ x! a, 0503 ｧｧ縺06 閨ﾀﾀ罍ｧﾀ07 04 罍ｧﾀ09 縺0305 ｧｧ縺06 閨ﾀｧ 0 0 ｧ罍ｧ006, x, a = x + a 04 罍ｧ x, 09 縺03006 x = a. 703 ﾀ040308x, a ﾀ lim x!a x, a x, a = lim (x + a) =a. x!a ｧ 04 縺06 閨060003: x, 4x + + sin x 734. ) lim ; ) lim x! x + x!=4, cos 4x. x, lim x! x, ( ﾀ0 ｧｧ090608). x, 736. lim x! x, 3x. x, lim + x!3 x, x,. 3 ｰ09 0 ｧｧ閨ﾀ0403 ﾀｧ: ) x = =+; ) ｧｧ. 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 ｧｧ0504 x = t 6, mx = t 3. +x,, x 744. lim. x!0 x, tg x, +tg x lim x!

91 閨060 ｧｧ x, 746. ) lim x! 3x, 4x ; 5x 3, 7x ) lim x!, x. 3 ｰ09 0 ｧ縺0600 ｧ0504 閨ﾀ0403 ﾀｧ: ) 閨0600 ｧ04 ｧﾀ00 ｧｧ x ﾀﾀｧ; ) ｧ04 x == 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 ｧ 04 縺06 閨060003: 3x 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 lim : 758. lim x! x +4x + n! 3n : + 5x 759. lim x!, x +=x lim n! +3+5+:::+(n, ). ++3+:::+ n, x, lim x!7 x, 49 : 76. lim x!=4 sin x, cos x,. cos x, sin x x 4. 7 縺06 閨ｧ07 sin 04 罍ｧ! 0 ｭﾀ00 ｧ陦ｧ 0 08, 0503 sin lim =; lim!0!0 sin =:

92 3x 縺06 閨ﾀ05 ｧ 04 ｧ, ｧｧ 04 縺06 閨060003: sin 4x sin (x=3) 763. lim : 764. lim. x!0 x x!0 x ｰ 09 0 ｧｧ0504 ｧﾀ00 ｧｧ ( ｧ00 ｧｧ0504 4x = ). tg x 765. lim x!0 x : 768. lim x! ) 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 ｧｧ罍ｧ006 縺06 ) arctg x =, 罍ｧ006 縺06 ) arcsin (, x) =., cos x tg x, sin x 77. lim : 77. lim. x!0 x x!0 x 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,) ( ｧ0504 x =+). x! x, ) 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 縺06 閨ﾀ05 ｧ 04 ｧ, ｧｧ 04 縺06 閨060003: 78. lim x!+ ( x +3x, x) lim x! x,,. x,

93 閨060 ｧｧ 784. lim x!+ ( x + x +, x, x) lim x! x,,. x 3, lim x!0 sin x,. 4 sin (x=) +3+:::+(n, ) 787. lim, n. n! n lim (, x)tg x ( ｧ0504 x =, ). x! 789. lim ( x +, x, 4x). x!, 790. lim x!, x x, 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 lim, tg x ( ｧ0504 x = x!= + ). x 6. ｮ罍ｧ006 縺ｧﾀ00060 ｧ06 04 縺06 閨ｧ 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! lim x!, ( x + ax, x, ax). +xsin x, cos x : sin x x 4 +x, :

94 3x 7. ｮｧﾀ , 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 ) 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. ｮｧﾀ縺06 閨ｧﾀ罍ｧ x! a ｧｧ (x) ｧ (x) ﾀﾀｧ. ｯ0305 : I. ｭﾀ00 ｧ lim = 0, ﾀﾀﾀ x!a 0403 縺07 閨ﾀｧ II. ｭﾀ00 ｧ lim = A ( ｧｧ ), ﾀ07 x!a n 0306 ﾀ n 縺07 閨ﾀｧ III. ｭﾀ00 ｧ lim =,0503 ｧﾀｧｧ 0306 ﾀ0903x!a ｧｧﾀｧﾀﾀ :.. ｮﾀｧﾀ : ) ﾀ0504 閨ｧﾀﾀﾀﾀ縺07 閨ﾀｧ閨0308 ｧ 0 ｧ08. ) ｭﾀ00 ｧｧﾀﾀｧﾀ縺07 閨縺03 ﾀｧﾀﾀ00 ｧ縺07 閨090304, ﾀﾀ07 ﾀ0504, , ｧﾀ0608 ﾀｷ縺ﾀﾀ0504 ﾀ0006 閨060605, ｧﾀﾀ閨ﾀ07 04 罍ｧ0300 ｧｧﾀ03 ﾀ閨ﾀｧ縺ﾀ罍ｧ閨ｧｧﾀ05 ｧ 0306 ﾀ , ｧ閨ｧﾀｧ 04 罍ｧ0300 ｧﾀ0504 ｧ08 閨03 ﾀｧ縺06 閨0600 ｧ縺07 閨09 ｧ 0306 ﾀ : ), cos x; ) tg x, sin x ﾀｧﾀ x , 罍ｧｧｧ x 04 閨ｧｧ0, cos x ﾀ07 04 罍ｧ0300 ｧｧ縺06, ｧｧ0 tg x, sin x 04 罍ｧ0300 ｧｧﾀ06004.

95 閨060 ｧｧ縺06 閨0600 ｧ縺07 閨09 ｧ 0306 ﾀ : ) sin 4 x, x 5 ; ) sin x + x 4 ; 3) +x 3, ﾀｧﾀ x 縺06 閨0600 ｧ縺07 閨ﾀ05 ｧ 08 ﾀ05 縺縺ﾀﾀｧﾀ閨0605 ｧﾀ , 罍ｧ x! 0: ) sin mx mx; ) tgmx mx; 3) 3 +x, 3 x ｧ09 ｧﾀ00 ｧ縺060 ｧ罍ｧ0 ｧﾀ07 04 罍ｧ0300 ｧ縺ｧ09 ｧｧﾀ- 00 ｧ縺060 ｧｧﾀ05 ｧｧﾀﾀ ? 縺06006 lim = lim, 06 ﾀ00 ｧ, ｧ 03 陦ｧ0 ｧ 04 縺06 閨ﾀ06006 ﾀ , 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 ; 閨03 ｧﾀ04 縺ﾀ , 陦ｧ06 ﾀﾀ05 縺060 ｧ05 ﾀ縺06 閨0600 ｧ縺07 閨09 ｧ 0306 ﾀ縺08003 ﾀ05 ｧｧｧﾀｧ陦ｧ06 ﾀ縺06 閨0600 ｧ縺07 閨09 ｧ 0306 ﾀ : ) +x, ; ) sin x, sin x; 3), cos ﾀｧﾀ x , 罍ｧ x! 0: ) arctg mx mx; ) +x, x; 3), cos3 x ; 5 sin x. x + 3 x 縺06 縺ﾀｧｧ縺06 閨ｧ06. ｱｧ07 f(x) ﾀ縺06 縺罍ｧ x = a, 06 ﾀ00 ｧ縺06 閨縺縺06 ﾀ05003 ﾀ05 ｧ a ｧ lim f(x) =f(a): x!a 縺06 閨ｧ06 ﾀ03 閨06 縺07 ｧｧ縺06 06 ﾀｧ縺06 縺ﾀ05 ｧ: ) f(x) 閨縺06 閨縺縺06 ﾀ05003 ﾀ05 ｧ a; ) 閨ﾀ06006 ﾀ縺06 閨 lim f(x) ｧ lim x!a,0 3) 0505 ｧ 04 縺06 閨 ( ﾀｧﾀ04 04 ) 閨陦ｧｧ; 4) 0505 ｧ 04 縺06 閨閨 f(a). x!a+0 f(x);

96 3x 縺06 縺ﾀｧｧ 03 ｱｧﾀ縺06 縺縺 [x ;x ], 06 ﾀ00 ｧ縺06 縺閨罍ｧ 0305 縺06 09, ｧ09 08 lim f(x) = x!x+0 = f(x ) ｧ lim f(x) =f(x ). x!x, 縺ｧｧ: ﾀ x n, a x, 罍ｧ07-0 ｧ 06 ﾀ09 07, 05 罍ｧ罍ｧ 06 ﾀ09 ｧ06 ｧｧ , ｧ08 ﾀ0600, 04 縺03- ｧ 0406 閨060 ｧ06, ﾀ縺06 縺罍ｧ 04 ﾀ x, 04 罍ｧ縺ｧｧ縺06 閨ｧ縺ｧｧ. ｱｧ07 ｧ縺罍ｧ x = a, 06 ﾀ00 ｧ縺06 閨ﾀｧﾀ a, a 006 ﾀ閨閨003 ｧ 06 ﾀｧ縺06 縺ﾀ05 ｧｧ 0605 閨04 03 ﾀｧ縺0304. ) 7 縺0304 I 縺ﾀ06006 ﾀ縺06 閨 lim f(x) x!a,0 ｧ lim x!a+0 f(x), 縺ﾀｧ縺06 縺ﾀ05 ｧｧﾀ ( ｧ00 ｧ閨003 ｧ 0 ｧ08) 罍ｧ006, ｧ07 y = x, a, , 04 罍ｧ x<a ｧ + 04 罍ｧ jx, aj x>a, ｧ罍ｧ x = a 縺0304 I 縺03 ( 罍ｧﾀ. 4), ﾀ06006 ﾀ縺06 閨 y =, ｧ y =+, ｧ 04 縺06 閨 lim x!a,0 lim x!a+0 ) 7 縺0304 II 縺 lim f(x) ﾀｧ00 ｧﾀ x!a 7 04 罍ｧ006, ｧ07 y = f(x) = a ( 罍ｧﾀ. 5) ｧ罍ｧ x = a x, a 縺0304 II 縺ﾀ06 陦縺ｧｧ, 縺罍ｧ x = a 7 ｧﾀ. 4 7 ｧﾀ , ｧﾀ00 ｧ , ｧ罍ｧ x = a 縺0304 II 縺03. ｱｧ07 f(x) = =x ( 89, 罍ｧﾀﾀ. 74) ｧ罍ｧ x =0 縺0304 II 縺03, lim f(x) =0,003 lim f(x) =. x!,0 x! ｰ縺ｧｧ y = 4 x,, ｧ y ｧ 0403 ﾀ05 縺03 ｧ罍ｧ lim y, lim x!+0 x! x =,; 0; ; 3; 4 ｧ 6: lim y, x!,0

97 閨060 ｧｧｧｧ縺0304 ｧ 0403 ﾀ05 縺03 ｧｧ09 ｧｧ08: ) y =, 6 4 ; ) y = tg x; 3) y = x 4, x ﾀ05 縺03 ｧｧｧｧ ( x= 04 罍ｧ x 6= ; y = 0 04 罍ｧ x = ｧ縺ｧ06 ｧ縺ﾀｧ縺06 縺ﾀ05 ｧｧｧ ? ﾀ05 縺03 ｧｧ09 ｧｧ08: ) y = x + jx +j ｧ)y = = x x ｧ06 ｧ 06 ﾀｧ縺06 縺ﾀ05 ｧ縺0304 jx +j 0505 ｧｧｧｧ ? ﾀ05 縺03 ｧｧｧｧ 8 < sin x 04 罍ｧ x 6= 0; y = f (x) = x : 04 罍ｧ x =0 ｧ縺ｧ06 ｧ 06 ﾀｧ縺06 縺ﾀ05 ｧｧｧ ? 89. ｰ縺ｧｧ y = =x, ｧ lim y, x!,0 lim y, lim y ｧ 0403 ﾀ05 縺03 ｧｧｧｧｧ06 06 ﾀｧ07 006x!+0 x! 04 縺06 縺ﾀ05 ｧ縺 ? ﾀ05 縺03 ｧｧｧｧ 8 >< 0; 5x 04 罍ｧ jxj < ; y = f (x) = ; 5 04 罍ｧ jxj =; >: 3 04 罍ｧ jxj > ｧｧ 0606 縺ｧｧ縺0304 ｧ 0403 ﾀ05 縺03 ｧｧ09 ｧｧ08 ) y = ; ) y + = arctg a =x x, a ; 3) y = x3, x jx, j. 8. ｮ閨ｧｧ060 x,, y =0? 0504 縺06 閨0600 ｧ0504 ｧ 0 ｧ08: ) ｧ06; ) ｧ , ｧｧ00600 ｧ縺 (I 縺03 ) 04 罍ｧ x = ; ; 3; :::, ｧ 0403 ﾀ05 縺03 ｧ0504 ｧｧ09 ｧ.

98 3x 9. 罍ﾀｧｰ縺ｧｧ y = x x, ｧ + lim y, x!,,0 y ｧ 0403 ﾀ05 縺03 ｧｧ x =,6;,4;,3, lim y, lim x!,+0 x!, ; 0; ﾀ05 縺03 ｧｧｧｧ < 04 罍ｧ x =0 ｧ x = ; y = f (x) = 4, x 04 罍ｧ 0 < jxj < ; : 4 04 罍ｧ jxj > ｧｧ縺ｧ06 06 ﾀｧ縺06 縺ﾀ05 ｧ縺0304 ｧｧ ? ｧｧ縺0304 ｧ 0403 ﾀ05 縺03 ｧｧ09 ｧｧ08: ) y =, jxj x ; ) y ==(x,) ; 3) y =, =x ; 4) y = x3 + x ; 5) y = 4, x jxj j4x, x 3 j : 86. ｮ閨ｧｧ060 x + + y = 4? 0504 縺06 閨0600 ｧ0504 ｧ 0 ｧ08: ) 閨縺06 縺縺 jxj 6 ; ) 0506 ｧ 0 ｧ08, 罍ｧ縺 jxj 6 6 ｧｧ閨ﾀﾀ閨ﾀ05 ｧｧ08 x. 703 ﾀ05 縺03 ｧｧ09 ｧ縺ﾀ0006 閨ｧｧ. x 9. 罍ﾀｧ罍ﾀｧ罍ｧﾀ07 04 縺070 07, 縺ｧ罍ｧ0300 ｧﾀ罍ｧ罍ｧｧｧ罍ｧﾀﾀ0504. I. ｭﾀ00 ｧ lim f(x) =, 縺 x = a 06 ﾀ0504 ﾀｧ罍ｧ x!a y = f(x) 罍ｧ006, 09 罍ｧ04 07 y = a ｧﾀｧ x = a ( 罍ｧﾀ. 5). x, a II. ｭﾀ00 ｧﾀ05 ｧｧ07 09 罍ｧ y = f(x) 閨0600 ｧｧﾀ0504 y = f(x) =kx + b + (x) 05 09, ﾀﾀ07 ﾀ0504 (x)! 0, x!, 縺 y = kx + b 06 ﾀ0504 ﾀｧ罍ｧ罍ｧ006 縺03: ) 09 罍ｧ04 07 y = x3 + x + = x ++ ｧﾀｧ x x y = x +( ｧﾀｧ x =0); ) 09 罍ｧ04 07 y = a x, a =0+ a x, a ｧﾀｧ y =0( 罍ｧﾀ. 5). f(x) III. ｭﾀ00 ｧﾀ06006 ﾀ縺06 閨 lim = k ｧ lim x!+ ｧ00 ｧ, x!+ ｧ00 ｧ, x [f(x), kx] =k, 縺 y = kx + b 06 ﾀ0504 ﾀｧ縺06 閨0600 ｧ0504 ﾀｧ罍ｧ y =, 4 x ｧ 0403 ﾀ05 縺03 ｧ罍ｧ x = ; ; 4.

99 閨060 ｧｧ { ｧﾀｧ罍ｧ040308, 0403 閨0600 ｧ04 ｧ陦縺0303 ｧ 00 ｧﾀ0504; 0403 ﾀ05 縺03 ｧ0504 ﾀｧｧ 09 罍ｧ040306: 88. ) y = x + ; ) y = x x ; 3) y = x x + x : ) y = jxj, ; ) y = x, x, ; 3) y = ax + b x mx + n : 830. ) y =, 4x x3 ; ) y = +x x + ; ｧﾀｧ罍ｧｧ 0403 ﾀ05 縺03 ｧ罍ｧ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 : ﾀ05 縺03 ｧ罍ｧ040306: ) y = x4 + 3x ; ) y = x3 + x, ｧ x , 縺ｧ 09 罍ｧﾀｧｧ 06 ﾀ09 ｧ 04 罍ｧ0300 ｧﾀｧﾀｧ罍ｧ040308: ) y =, x ;)y =,x + + x ｧ 0403 ﾀ05 縺03 ｧ罍ｧ x = ; ; ｧﾀｧ罍ｧｧ 0403 ﾀ05 縺03 ｧ罍ｧ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 ﾀ07 04 縺06 閨0600 lim n! + n n = lim n!, + n n = lim ( + ) = = e:! ｧﾀ0003 ｧ罍 09 ｧｧ 04 罍ｧ0300 ｧ e = ;788::: 6ｧ罍ｧ07003 ﾀ 03 ﾀｧ060 e ﾀｧｧﾀ07 log e x =lnx ﾀ0705 ｧ罍ｧ070: lg x = M ln x, 05 閨06 M =0;4349:::

100 3x 0. ｪｧﾀ0003 e ｧ 04 縺06 閨060003: 836. lim, 5 n ( ｧ0504, 5 n! n n = ) ) lim, n ; ) lim + 4 n+3 : n! 3n n!, n 838. ) lim ( + x) =x ; ) lim x!0 x! ) lim n! n n + n ; ) lim x! (, 4x) (,x)=x : x, x : x ) lim n[ln (n +3), ln n]; ) lim ( + 3 tg x) ctg x. n! x!0 84. lim (cos x) ctg x ( ｧ0504 sin x = ). x!0 ln ( + ) e,x, 84. ) lim ; ) lim!0 x!0 x ｰ 09 0 ｧ罍ｧ006 縺06 ) ｧ0504 e,x, =. ; 3) lim x!0 a x,. x ｧ 0403 ﾀ0006 閨ｧﾀ00, 閨縺030 ｧﾀ03 閨06 縺07 ｧ05 ﾀｧ06 6(, ; 0,00 ) 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 ( ｧ0504 cos x = ). x!=4 t 847. ) lim t!0 ln ( + xt) ; ) lim n! n ln n, ln (n +).

101 ｷ36ｦ縺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 閨ｧ 06 ﾀ09 ｧ08 ｧ 05 罍ｧ罍ｧ 06 ﾀ09 ｧｧ縺06 閨ｧ07. 7 縺03 ｧ 0403 閨ｧｧ y = f(x) x ﾀ07 04 縺06 閨0600 f(x +x), f(x) y lim = lim x!0 x x!0 x : () ｭﾀ00 ｧ縺06 閨 , ｧ07 f(x) ﾀ07 陦ｧ縺ｧ縺 x; 04 罍ｧﾀｧ縺06 縺ｭﾀ00 ｧ縺06 閨0600 () ( ｧ00 ｧ,), 閨罍ｧ0504, ｧ07 f(x) ｧ x 0306 ﾀ縺03 ｧ 0403 閨00606, 03 閨罍ｧ閨ｧﾀｧｧ, ｧ縺06 縺縺03 ｧ 0403 閨ﾀ07 y 0 ｧ00 ｧ f 0 (x), ｧ00 ｧ dy, ｧ00 ｧ df(x) 閨060 ｧ06 04 縺03 ｧ 0403 閨ﾀ07 陦ｧ縺06009 ｧ縺ｧｧｧ.. 05 ﾀ縺陦ｧ縺06009 ｧ縺ｧ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 ｧﾀ00060 ｧ060 lim x!0 x ｧ 04 縺03 ｧ 0403 閨ｧ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.

102 3x. 7 縺03 ｧ 0403 閨ｧ 06 ﾀ09 ｧ08 ｧ 05 罍ｧ罍ｧ 06 ﾀ09 ｧｧｧ縺縺03 ｧ 0403 閨ｧ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 (,) f (x) =x, x ; 0403 ｧﾀ00 ｧ0504 f 0 (), f 0 (,) f (x) = ( x, ) ; 0403 ｧﾀ00 ｧ0504 0; 0 f 0 (0; 0). x ｧ 04 縺03 ｧ 0403 閨ｧ08: 865. ) y =(a, bx ) 3 ; ) y =(+ 3 x) : 866. ) y = 0x 5, 4x 4 ; ) y = 3 3 x, x :

103 縺03 ｧ 0403 閨0 07 ｧ陦ｧ縺06009 ｧ ) 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 ; ｧ f 0 (,8) f (x) = x x, ; ｧ f 0 (0), f 0 () ｧ f 0 (,). x. 7 縺03 ｧ 0403 閨0 07 ﾀｧｧｭﾀ00 ｧ y = f(u), u = '(x), 0503y ﾀｧｧｧｧ00 ｧﾀｧ x. ｯ0305 dy = dy du du ｧ00 ｧ y0 = f 0 (u) u 0 : () ｱ03 縺縺06 閨03 閨罍ｧ縺ｧ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 : ｧ 04 縺03 ｧ 0403 閨ｧ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 y = x, sin x: 879. y = sin 4 x = (sin x) ) 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:

104 3x ﾀｧ 003 縺ﾀ罍ｧ y = + sin x,, sin x y = ( + cos 4x) : 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 ' f (t) = a + b, ab cos t; 0403 ｧﾀ00 ｧ0504 f 0 (=); f 0 (); f f (x) = x + x; ｧ f 0 (). : ｧ 04 縺03 ｧ 0403 閨ｧ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 ; ｧ f 0. x ﾀｧ 003 縺ﾀ罍ｧｰｧ09 ｧﾀ罍ｧ y = f(x) 罍ｧ (x 0 ; y 0 ) ｧ06 04 縺03 ｧ 0403 閨ｧｧ f(x) x 0 : k = tg ' = f 0 (x 0 )=y 0 j x=x0 : () ｪｧﾀ0003 k ｧ罍ｧ (x 0 ; y 0 ). ｰｧ06 09 ﾀ M (x 0 ; y 0 ) 0 09 罍ｧ ( 罍ｧﾀ. 6): y, y 0 = k(x, x 0 ): ()

105 縺03 ｧ 0403 閨0 07 ｧ陦ｧ縺06009 ｧ 00 ｰｧ縺0 00 ｧ: y, y 0 =, k (x, x 0); (3) 05 閨06 k 0304 縺06 閨ﾀ縺 () 縺06 09 ｧ TA = y 0 ctg ', AN = y 0 tg ' ( 罍ｧﾀ. 6) ﾀ07 ﾀﾀ閨09 ﾀｧ 0403 閨003 縺 , 閨00 ｧ縺 MT ｧ MN 閨00 ｧ0 0 ｧ 09 ﾀｧｧ. 7 ｧﾀｧ y = x x = ｧﾀｧ06 09 ﾀｧ 003 縺0 00 ｧ y = =4, x 縺06 ﾀｧﾀ 03 ﾀ0406 Ox (04 罍ｧ x>0) ｧ 0403 ﾀ05 縺03 ｧ , 09 ﾀｧ 003 縺 { ｧﾀｧ07 09 ﾀ罍ｧ04030 ｧ 0403 ﾀ05 縺03 ｧ罍ｧｧ 09 ﾀ : 罍ｧ y = x x =, 罍ｧ y = x x =0 ｧ x = y = x =. 4+x ﾀｧ006 ﾀ03 ｧ閨06 y = sin x x = ｧ罍ｧ04 07 y = sin x 0406 縺06 ﾀﾀ04Ox? ｧ縺06 ﾀﾀ07 09 罍ｧ y = x ｧ y =8, x? ｧ閨00 ｧ閨09 ﾀ , 0403 閨003 縺0 00 ｧ, 09 ﾀｧ 003 縺0 00 ｧ 09 罍ｧ040308: ) y = x ; ) y = x x = , 閨09 ﾀ y = x 閨ﾀ09 ｧﾀﾀｧ 09 ﾀ 0 ｧ07, 0403 閨003 縺ｧｧ y = x + bx + c 0304 縺06 閨0600 ｧ0504 b ｧ c, 06 ﾀ00 ｧﾀ 0605 ﾀ07 04 縺 y = x x = ｧﾀｧ07 09 ﾀｧ0406 縺 xy = x = ｧ x =,4 ｧｧ閨06 09 ﾀｧ. 703 ﾀ05 縺03 ｧ罍ｧｧ09 ﾀ

106 3x 4. ｮ0006 ｧ 006 陦ｧ縺06009 ｧ縺ﾀ05 ｧ縺06 縺ｧｧ { ｧﾀｧ07 09 ﾀ罍ｧ04030 ｧ 0403 ﾀ05 縺03 ｧ罍ｧｧ 09 ﾀｧ0: 97. y =4x, x 縺06 ﾀｧ07 ﾀ 03 ﾀ0406 Ox. 98. y =4, x 縺06 ﾀｧ07 ﾀ 03 ﾀ0406 Oy. 99. y =(4+x) 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ Ox ｧ Oy ｧﾀﾀｧ縺00 ｧ y = x, 4x ﾀ縺06 ﾀｧﾀ 03 ﾀ0406 Oy ｧ縺 y = 0; 縺06 ﾀ罍ｧ y = cos x? ﾀ y = x +4x ﾀｧ Ox? y = x, x 縺ﾀ05 ｧ 09 ﾀ , 縺04060 陦ｧ縺ｧﾀﾀ罍ｧﾀ縺罍陦ｧ ? ｧ閨00 ｧ閨09 ﾀ , 0403 閨003 縺0 00 ｧ, 09 ﾀｧ 003 縺0 00 ｧ 09 罍ｧ y = x =. +x ｧ y = x ﾀ09 ｧﾀﾀ縺ｧ 4? x 4. ｮ0006 ｧ 006 陦ｧ縺06009 ｧ縺ﾀ05 ｧ縺06 縺ｧｧ 4 ﾀ罍閨0308, 03-. ｰｯ03 09 A(x ; y ) 09 罍ｧ y = f(x) ( 罍ｧﾀ. 7) ﾀ , 06 ﾀ00 ｧ縺03 ｧ 0403 閨0 07 y ﾀ06006 ﾀ , 003 ﾀ06006 ﾀｧｧ縺03 ｧ 0403 閨00306: lim x!,0 y x = k y lim x!+0 x = k. 0ｷｧ閨0705 閨04 09 ﾀﾀｧ k ｧ k.. ｯﾀ 0406 縺05 ｧﾀｯ03 09 B(x ; y ) ( 罍ｧﾀ. 7) ﾀﾀ 0406 縺05 ｧﾀ , 06 ﾀ00 ｧ縺03 ｧ 0403 閨0 07 y ﾀﾀ , 003 ﾀ06006 ﾀｧﾀ縺03 ｧ 0403 閨ｧ 7 ｧﾀ (+ ｧ,). ｯﾀ07 ﾀﾀｷ陦ｧ05 03 陦ｧ縺05 ｧﾀｧ00 ｧ, ﾀｧ , 0503 ｧ閨0705 閨04 ﾀ00 ｧ0400 ｧ08 ﾀ07 09 ﾀｯ縺0605 ｧ03 ﾀ 0406 縺05 ｧﾀｯ03-09 C(x 3 ; y 3 ) ( 罍ｧﾀ. 7) ﾀ縺0605 ｧ03 ﾀ 0406 縺05 ｧﾀ , 06 ﾀ00 ｧﾀ06006 ﾀﾀ縺03 ｧ 0403 閨0 07

107 縺03 ｧ 0403 閨0 07 ｧ陦ｧ縺06009 ｧ 00 y lim x!,0 lim y x!+0 x y 0 = x = =+ ( ｧ00 ｧ,) ﾀ06006 ﾀ縺05 ｧﾀ A ｧ B ｧ07 y = f(x) 006 ｧ縺03 ｧ 0403 閨00308; C 030 ｧﾀ縺03 ｧ 0403 閨ﾀ縺ｧ縺06 縺03040, 陦ｧ縺06009 ｧ縺ﾀ05 縺03 ｧｧｧｧ y = x ( ｧ00 ｧ y = jxj) ｧｧ y, 0 ｧ y 縺03 ｧ 0403 閨ｧ縺 [0; 4] 0403 ﾀ05 縺03 ｧｧｧｧ y = 0; 5 (x, ) ｧｧ y, 0 ｧ y 縺03 ｧ 0403 閨ｧｧｧ. sin x 縺 [,; ] 0403 ﾀ05 縺03 ｧｧｧｧ y = ｧ 0 04 ｧﾀｧ07 09 ﾀ罍ｧ縺 [0; ]0403 ﾀ05 縺03 ｧｧｧｧ y = +cos x, 0 04 ｧﾀｧ07 09 ﾀ罍ｧｧｧ閨06 0 ｧ0 ｧ縺 [,; ] 0403 ﾀ05 縺03 ｧｧｧｧ y = 3 x ｧ 0 04 ｧﾀｧ06 09 ﾀ x = 縺 [0; 4] 0403 ﾀ05 縺03 ｧｧｧｧ y =,, 3 (x, ) ｧ 0 04 ｧﾀｧ06 09 ﾀ x = 縺 [,; ] 0403 ﾀ05 縺03 ｧ罍ｧ y 3 =4x ｧ 0 04 ｧﾀｧ06 09 ﾀ x = 縺 [0; 4] 0403 ﾀ05 縺03 ｧ罍ｧ y 3 = 4(, x) ｧ 0 04 ｧ- ﾀｧ06 09 ﾀ x = 縺 [0; ] 0403 ﾀ05 縺03 ｧｧｧｧ y =,, cos x ｧ 0 04 ｧﾀｧ07 09 ﾀ罍ｧ縺 [,; 0] 0403 ﾀ05 縺03 ｧｧｧｧ y = = 3 (x +), ｧ 0 04 ｧﾀｧ06 09 ﾀ罍ｧ x =, 縺 [,; 5] 0403 ﾀ05 縺03 ｧｧｧｧ y = j4x,x j ｧ 0 04 ｧﾀｧ07 09 ﾀ x =0 ｧｧ閨06 0 ｧ0 ｧ. x 5. 7 縺03 ｧ 0403 閨罍ｧ070 ｧ 06 ﾀ09 ｧ08 ｧｧ08 05 ﾀ縺 : (ln u) 0 = u0 u ; (eu ) 0 = e u u 0 ; (a u ) 0 = a u ln a u 0 :

108 3x 5. 7 縺03 ｧ 0403 閨罍ｧ070 ｧ 06 ﾀ09 ｧ08 ｧｧｧ 04 縺03 ｧ 0403 閨ｧ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 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 ) y = x, 4ln(+ x) ) y = cos x sin x +lntg x ; ) y =ln x,. ax ｧﾀｧ06 09 ﾀ罍ｧ y =lnx 縺06 ﾀｧﾀ 03 ﾀ0406 Ox. 703 ﾀ05 縺03 ｧ罍ｧｧ09 ﾀ , y = x 09 ﾀ 0605 ﾀ07 09 罍ｧ y =lnx, e ｧｧﾀ 0 ｧﾀ05 縺03 ｧ罍ｧｧ 04 縺03 ｧ 0403 閨ｧ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 ) f (t) =ln(+a,t ); ｧ f 0 (0) ｧ罍ｧ04 07 y = e x 0406 縺06 ﾀﾀ04Oy? , 0503 閨00 ｧ閨09 ﾀ罍ｧ y = e x=a 040 a.

109 縺03 ｧ 0403 閨0 07 ｧ陦ｧ縺06009 ｧ縺06 閨04 罍ｧ罍ｧ070 ｧ縺ｧｧ 04 縺03 ｧｧ08: ) y = x x ;)y = x sin x ｧ 04 縺03 ｧ 0403 閨ｧ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) 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 ; ｧ f 0 (=3) ｧﾀｧ06 09 ﾀ罍ｧ y =, e x= 縺06 ﾀｧﾀ 03 ﾀ0406 Oy. 703 ﾀ05 縺03 ｧ罍ｧ040606, 09 ﾀｧﾀｧ罍ｧ x 6. 7 縺03 ｧ 0403 閨罍ｧ罍ｧ 06 ﾀ09 ｧｧ08 (arcsin u) 0 = (arctg u) 0 = ｧ 04 縺03 ｧ 0403 閨ｧ08: 980. y =, x + arcsin x. u 0, u ; (arccos u)0 =, u0, u ; u0 +u ; (arcctg u)0 =, u0 +u :

110 3x 7. 7 縺03 ｧ 0403 閨ｧ0406 縺ｧ 06 ﾀ09 ｧｧ 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 ) ｧ 04 縺03 ｧ 0403 閨ｧ08: 99. y = arcsin x: 99. y = arctg 6x, ) y = arccos (, x ); ) y = arcctg x, x y = e x, e x + arcsin e x y = x arccos x,, x. r 996. y = arctg e x e x + +ln e x,. t 997. s = 4t, t + 4 arcsin y = arccos, x + x, 4x f (z) =(z +)arctg e,z ; ｧ f 0 (0). x 7. 7 縺03 ｧ 0403 閨ｧ0406 縺ｧ 06 ﾀ09 ｧｧ08, 縺06 閨ｧｧ07 ex e,x, ex + e,x ｧｧｧﾀ07 ﾀﾀｧ0406 縺ｧ 06 ﾀ09 ｧ0 ｧﾀｧ006 ﾀ030; 0903 ﾀｧ006 ﾀ030; ﾀ030 ｧﾀ030 ｧﾀ07 sh x = ex, e,x ; ch x = ex + e,x ; th x = sh x ch x ; ch x cth x = sh x :. ｮﾀｧ0406 縺ｧ 06 ﾀ09 ｧｧ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 :

111 縺03 ｧ 0403 閨0 07 ｧ陦ｧ縺06009 ｧｧ 04 縺03 ｧ 0403 閨ｧ08: 000. ) y =sh x; ) y = x, th x; 3) y = ch x,. 00. f (x) =sh x +ch x ; 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: ｧ8 ｧ0 ｧ07 y = a (ex=a + e,x=a )=ach x ﾀ a 7 04 ｧﾀｧ縺0 00 ｧｧ0 ｧｧ x = a. 703 ﾀ05 縺03- ｧ罍ｧｧ 003 縺ｧﾀｧ06 09 ﾀ罍ｧ y = shx x =,. 703 ﾀ05 縺03 ｧ罍ｧｧ09 ﾀ , 縺ｧ07 03 罍陦ｧｧｧ0 ｧｧ y = a ch x 縺ﾀｧｧﾀ , 04- a 0 07 a. x 8. ｮ罍ｧ006 縺03 ｧｧ 0 陦ｧ縺06009 ｧ縺ｧｧ 04 縺03 ｧ 0403 閨ｧ08: x, 008. ) y = + arcsin x x ; ) y = tg x + ln cos x 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; ｧ f 0 (a). 04. ) y =ln x, a 05. f (x) = arcsin x, x ; ｧ f 0 (5). x ; ) y = x(cos ln x + sin ln x). 06. '(u) =e,u=a cos u a ; , 0503 '(0) + a'0 (0) = 0.

112 3x 9. 7 縺03 ｧ 0403 閨ﾀ00 ｧ縺07 閨 f (y) =arctg y a, ln 4 y 4, a 4 ; ｧ f 0 (a). 08. F (z) = cos z + sin z ; , 0503 F 4, 3F 0 4 = , ｧ07 s = 06 閨縺陦ｧ070706t ln ct 縺06009 ｧｧ06 t ds dt + s =,ts , ｧ07 x = t, e,t 06 閨縺陦ｧ07- t 0706 縺06009 ｧｧ06 t dt +x = e,t. x 9. 7 縺03 ｧ 0403 閨ﾀ00 ｧ縺07 閨ﾀｧ閨ｧｧ y = f(x) 縺03 ｧ 0403 閨00606 y 0 = f 0 (x). 7 縺03 ｧ 0403 閨縺03 ｧ 0403 閨ﾀ07 04 縺03 ｧ 0403 閨縺縺07 閨ｧｧ f(x) ｧﾀ07 y 00 ｧ00 ｧ f 00 (x) ｧ00 ｧ d y. 縺ｧ縺06 閨ﾀ07 ｧﾀ07 04 縺03 ｧ 0403 閨縺縺07 閨09 y 000 = f 000 (x) = d3 y 3, 04 縺03 ｧ 0403 閨縺縺07 閨09 y IV = f IV (x) = d4 y 4, ｧ縺03 ｧ 0403 閨0 07 n 縺07 閨09 y (n) = f (n) (x) = dn y n ｧ 04 縺03 ｧ 0403 閨縺縺07 閨ｧｧ: ) y = sin x; ) y = tg x; 3) y = +x ｧ 04 縺03 ｧ 0403 閨縺縺07 閨ｧｧ: ) y = cos x; ) y = ; 3) y = x sin x. x ｧ 04 縺03 ｧ 0403 閨縺縺07 閨ｧｧ: ) y = x ln x; ) s = te,t ; 3) y = arctg x a. 04. s = t, t + arcsin t ; ｧ d3 s dt ｧ 04 縺03 ｧ 0403 閨00606 n 縺07 閨ｧｧ: 05. ) e,x=a ; ) ln x; 3) x: 06. ) x n ; ) sin x; 3) cos x:

113 縺03 ｧ 0403 閨0 07 ｧ陦ｧ縺06009 ｧﾀ0006 閨陦ｧ縺06009 ｧ縺ｧﾀ05 ｧｧｧ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 uv IV ｧ縺ｧｧｧ 04 縺03 ｧ 0403 閨縺縺07 閨ｧｧ: ) y = e x cos x; ) y = a x x 3 ; 3) y = x sin x 縺ｧｧｧ 04 縺03 ｧ 0403 閨縺縺07 閨ｧｧ: ) y = e,x sin x; ) y = x ln x; 3) y = x cos x f (x) =xe x=a ; ｧ f 000 (x), f (n) (x), f (n) (0). 03. f (x) = ( + x) m ; ｧ f (0), f 0 (0), f 00 (0), f 000 (0), ::: :::;f (n) (0). x 03. f (x) = ; , 罍ｧ n > +x f (n) n, 3 5 ::: (n, 3) (0) = (,) n, n: 033. f (x) = ; , 0503, x n! 04 罍ｧ n =m; f (n) (0) = 0 04 罍ｧ n =m, : ｰ 09 0 ｧ罍ｧ0060 ｧ閨06 ﾀ050403, x = +x + :, x 縺03 陦ｧ縺06009 ｧ縺閨06 ﾀ (x, )(x + x 3 + ::: :::+ x n )=x n+, x 05 罍ｧ 0403 x ｧｧ x =, ｧ P ﾀ n (n +)n(n, ) k(k, ) = ｧﾀ陦 k= 3 ｧﾀ縺07 nx k= k = + + :::+ n n(n + )(n +) = : ｧ 04 縺03 ｧ 0403 閨縺縺07 閨ｧｧ: ) y = e,x ; ) y = ctg x; 3) y = arcsin x.

114 3x 0. 7 縺03 ｧ 0403 閨ｧｧｧ 04 縺03 ｧ 0403 閨00606 n 縺07 閨ｧｧ: ) y = a x ; ) y = +x ; 3) y = sin x f (x) = arcsin x ; ｧ f (), f 0 () ｧ f 00 () 縺ｧｧｧ 04 縺03 ｧ 0403 閨縺縺07 閨ｧｧ: ) y = x 3 e x ; ) y = x sin x a ; 3) y = xf 0 (a, x)+3f (a, x) , ｧ07 y = e x cos x 06 閨縺陦ｧ縺06009 ｧｧ06 y IV +4y = , ｧ07 y = xe,=x 06 閨縺ｧ06 x 3 y 00, xy 0 + y = f (x) =x e,x=a ; , 0503 f (n) n(n, )(,)n (0) =. a n, 04. f (x) =e,x ; , 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 ; , 0503 f () + f 0 () + f 00 ()! + :::+ f (n) () n! = n : x 0. 7 縺03 ｧ 0403 閨ｧｧｭﾀ00 ｧｧ06 F (x; y) =0, 006 縺ﾀｧ y, 0304 縺06 閨 y 閨ｧ06 x, 0503y ﾀｧ0608 x. ｪｧ 04 縺03 ｧ 0403 閨00606 y ｧｧ, ﾀ05 ｧｧ07 F (x; y) =004 縺03 陦ｧ縺06009 ｧ縺 x, ﾀﾀ0 05 罍ｧ04 07 y ｧ x. 0ｷｧ閨060 ｧﾀ縺03 ｧ 0403 閨00606 y 0. ｪｧ y 00, ｧ06 F (x; y) =0 閨04 07 閨03 04 縺03 陦ｧ縺06009 ｧ縺 x ｧｧ y 0 ｧｧ08: 044. ) x + y = a ; ) y =x; 3) x a, y b = ) 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.

115 縺03 ｧ 0403 閨0 07 ｧ陦ｧ縺06009 ｧ x = y + arcctg y e xy, x + y 3 =0; ｧ dy 04 罍ｧ x = ｧ y 00 ｧｧ08: ) x + y = a ; ) ax + by, xy = c; 3) x m y n =. 05. x a + y b =; ｧ y (0; b) ｧﾀｧ07 09 ﾀ罍ｧ x +y +4x,, y, 3= 縺06 ﾀｧﾀ 03 ﾀ0406 Oy ｧｧ 0406 縺06 ﾀｧ縺0 00 ｧ 05 ｧ0406 縺 x,y =9, 04 縺閨ｧｧ (5; 4), ﾀﾀｧｧｧﾀｧ06 09 ﾀ (x 0 ; y 0 ) 0909 罍ｧ : ) x a + y b =; ) y =x ｧﾀｧ07 09 ﾀﾀ05 縺03 ｧ閨06 x =3 +y =3 = = a = 縺06 ﾀｧﾀ 04 縺 y = x ｧ縺06 ﾀﾀ07 09 罍ｧ x + y =5 ｧ y =4x? ｧ y 0 ｧｧ08: ) x a + y b =; ) x3 + y 3, 3axy = ｧ y 00 ｧｧ08: ) x, y = a ; ) (x, ) +(y, ) = R ; 3) arctg y = x + y; 4) x + xy + y = a : ｧﾀｧ07 09 ﾀ縺ﾀ05 ｧ x +y + +4x, 4y +3= 縺06 ﾀｧﾀ 03 ﾀ0406 Ox. 703 ﾀ05 縺03 ｧ縺ﾀ0504 ｧ09 ﾀｧﾀｧ06 09 ﾀｧ04 ﾀ06 x +4y = , 縺0308 閨0600 ｧ05 ﾀ縺ﾀ , 0305 ﾀﾀ070 ｧ罍陦ｧ0 05, ｧｧ縺縺05 ｧ. 06. te,s= + se,t= =; ｧ ds 04 罍ｧ t =0. dt 06. t ln x, x ln t =; ｧ 04 罍ｧ t =. dt 063. x sin y, cos y + cos y =0; ｧ y 0 04 罍ｧ y = =.

116 3x. 6 7 ｧ縺06009 ｧｧｧ 3 x. 6 7 ｧ縺06009 ｧｧｧｭﾀ00 ｧｧ07 y = f(x) 陦ｧ縺06009 ｧ縺 x, ｧ縺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: () ﾀ0504 y 0 x 04 罍ｧ 0060 ｧ07 y ｧｧ, 00 ｧﾀｧ x, ﾀ07 陦ｧ縺06009 ｧｧｧｧﾀ07 dy: dy = y 0 x: () ｧ縺 () y = x, ｧ0 = x 0 x = x =x, ｧ dy = y 0 : (3) ｱ03 縺00600 (3) 0406 縺0 ｧﾀ , 06 ﾀ00 ｧ x 06 ﾀｧ縺 t. 0ｷ3 () ﾀ0006 閨060605, 0503 y dy, 罍ｧ閨03 ﾀ =x 04 罍ｧ 0060 ｧｧｧ 04 罍ｧ0300 ｧ陦ｧ縺06009 ｧﾀ05003 ﾀ05 ｧ, 閨ｧｧｧ y = ax + b ｧ006060: y = dy ｧ陦ｧ縺06009 ｧｧ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 閨060 ｧ060 陦ｧ縺06009 ｧ閨ｧｧ dy ｧｧ08: ) x + y = a ; ) xy = a ; 3) x, xy, y = ) y = x ; ｧ 04 罍ｧ0300 ｧｧｧ06 y (y dy), x ｧﾀ閨03,0; ) y = x; ｧ 04 罍ｧ0300 ｧｧｧ06 y, x ｧﾀ閨 ) ｮ0503 縺 x = 5 0 0; 縺06 閨0600 ｧﾀｧﾀｧ縺ﾀ罍ｧ 0403 ｧﾀ00060 ｧｧ

117 縺03 ｧ 0403 閨0 07 ｧ陦ｧ縺06009 ｧ 00 ) ｧ縺 s = b + f 3b,05 閨06 b ﾀﾀｧ閨ｧ 0403 閨0406 ﾀ, f 0 ｧｧ08 04 縺0305 ｧ03. 7 ﾀｧｧ05 ﾀ07 04 縺0305 ｧ03 f, 縺縺ｧ07 06 閨00 ｧ- 0 ｧ05 ﾀ07 0 ds? 07. ) ｮﾀｧ 006 罍ｧﾀ09 ｧﾀﾀ06 09 罍ｧ y = x x 04 罍ｧ x 6, 罍ｧ 0403 ｧﾀ00060 ｧｧ罍陦ｧ閨ﾀ05 ｧ縺ﾀ ,? ) ｮﾀｧﾀｧ 006 罍ｧ0504 陦ｧ06 ﾀ 00, 罍ｧ 0403 ｧﾀ00060 ｧｧ閨ﾀ05 ｧ縺ﾀ %? 縺06 閨0600 ｧ罍ｧ0300 ｧ : ) 閨04 09 縺 ; ) ﾀ0706 罍ｧ 06 ﾀｧ. ｮ 040 ｧ0504 ﾀｧｧｧ- 070 ｧｧ陦ｧ縺06009 ｧｧ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 罍ｧｧｧｧ x 0305 閨03,98. ) 706 罍ｧｧｧ09 T = l=980 ﾀ, 05 閨06 l 閨00 ｧｧ09 04 ﾀ 005 ｧｧ 0060 ｧ0504 閨00 ｧｧ09 l =0 ﾀ0, 罍ｧｧｧ00 ﾀ07 0 0, ﾀ? 3) ｮﾀｧ 006 罍ｧﾀ09 ｧﾀﾀ06 09 罍ｧ xy = =404 罍ｧ x > 0; 5, 罍ｧ 0403 ｧﾀ00060 ｧｧ罍陦ｧ閨ﾀ05 ｧ縺ﾀ ,? x 罍ｧ 06 ﾀ09 ｧｧ07 09 罍ｧﾀ罍ｧ罍ｧ 06 ﾀ09 ｧ0 ｧｧ070 ｧ x = f(t) ｧ y = '(t) ｧ 04 縺03 ｧ 0403 閨縺06, 0 08 閨060: dy = _y _x ; d y d(_y= _x) = = y _x, x _y _x 3 : ﾀ05 縺03 ｧ罍ｧ罍ｧ 06 ﾀ09 ｧｧ070: ) x = t ; y = t3 ; ) x = t ; y = t3 3, t. 0ｷ3 ﾀｧ04 ｧｧ08 t, 0 04 ｧﾀｧ閨罍ｧｧ閨06: F (x; y) =0.

118 3x 罍ｧ 06 ﾀ09 ｧｧ07 09 罍ｧ罍ｧ0406 ﾀ05 ｧｧ閨06 F (x; y) =0( ｧ00 ｧ y = f (x)) ｧ07 09 罍ｧ , 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 ) x = et + e,t ; y = et, e,t ; ) x = tg t; y = cos t ﾀ05 縺03 ｧ縺 , ｧ00 ｧ , 09 縺0605 ( ﾀ ) x = a(cos t + t sin t); y = a(sin t, t cos t); t ｧ07: 0, =,, 3=, ｧ04 y = xt, ｧ罍ｧ 06 ﾀ09 ｧｧ07 08 閨0609 縺ｧﾀ05 09 x 3 +y 3,3axy =0( ﾀ ) ｧｧﾀﾀ0006 閨閨04 ｧ07060 ｧｧ罍ｧ罍ｧｧｧｧ t: ) 閨03 +; ) 閨03,; 3)0305, 閨03, ｧﾀｧ06 09 ﾀｧｧ閨06 ( ﾀ ) x = a(t,sin t), y = a(,cos t) , 05 閨06 t = =. 703 ﾀ05 縺03 ｧ罍ｧｧ09 ﾀｧﾀｧ06 09 ﾀｧｧｧ閨06 ( ﾀ05 縺03- ｧ閨06) x = a cos 3 t, y = a sin 3 t t = = ﾀ05 縺03 ｧ罍ｧｧ 09 ﾀｰ09 0 ｧﾀ05 縺03060 ｧ07 09 罍ｧﾀ03 ﾀ05 04 ｧｧｧ08 x ｧ y 04 罍ｧ t =0;=4; =; 3=4 ｧｧ d y ｧｧ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) ﾀ05 縺03 ｧ罍ｧ040306, 罍ｧ 06 ﾀ09 ｧ0 ｧｧ070 ｧ: ) x =t, ; y =, 4t ; ) x = t 3 ; y = t,, 0 08 閨ｧ 0406 縺06 ﾀｧ07 ｧ08 ﾀ 03 ﾀ070 ｧ罍陦ｧ0 05 ｧｧ04, 0503 閨縺罍ｧ dy = 04 罍ｧ t = ｧﾀｧ07 09 罍ｧｧ閨06 F (x; y) =0.

119 縺03 ｧ 0403 閨0 07 ｧ陦ｧ縺06009 ｧｧﾀｧ06 09 ﾀｧｧ閨06 x = a(t, sin t); y = a(, cos t) t =3=. 703 ﾀ05 縺03 ｧ罍ｧｧ09 ﾀｧﾀｧ06 09 ﾀ縺縺 t = =4. x = a(cos t + t sin t); y = a(sin t, t cos t) ｧ d y ｧｧ08: ) x = cos t; y = sin t; ) x = t ; y = t + t 3 ; 3) x = e t ; y = e 3t.

120 ｷ36ｧｭ 70ｷｷ36ｦ x. ｮ0903 縺03 ﾀ0504 ｧ06 ﾀ0903 縺060 ｧﾀ閨04 ｧﾀﾀｧ Ox ｧ t ｧ罍陦ｧ x = f(t). ｯ 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 ｦ0060 ｧﾀ0 縺縺縺05 ｧ縺08 ﾀﾀ0903 縺03 ﾀ a 0/ ﾀﾀ x 閨縺06 t ﾀ ? 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 ﾀﾀﾀ0 縺07 閨03 ﾀ05 ｧｧ0403 ﾀｧｧﾀ- ﾀｧｧｧ? 09. ｯ閨04 ｧﾀ縺 Ox x = t3 3, t +3t 縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀ0504 ｧ06 ﾀ0903 縺060 ｧ06 閨04 ｧ07060 ｧｧｧ06 閨04 ｧ07060 ｧ07? 閨04 ｧ07060 ｧ罍ｧｧﾀ縺ﾀ x = a cos!t 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 x = a ｧ x = , ﾀ0903 縺060 ｧ06 d x ｧｧ06 x ﾀ陦ｧ縺06009 ｧｧ060 dt d x dt =,! x ﾀﾀ03, 閨06 縺07 ｧ縺 , t ﾀｧﾀ ' = a + bt, ct,05 閨06 a, b ｧ c ｧﾀ縺06 閨0600 ｧﾀ0903 縺03 ﾀ0504 ｧ 06 ﾀ0903 縺060 ｧｧﾀ03 03 ﾀｧ05 ﾀ07? ﾀ03 陦ｧ06 ﾀ a ｧ05 ﾀ縺ｰ ' 縺ﾀ t ﾀ縺06 閨ﾀｧ060 ' = t+ t 縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀ0504 ｧ 06 ﾀ0903 縺060 ｧ06 閨04 ｧ07060 ｧﾀﾀ0504 v ﾀ0903 縺03 ﾀ0504 ｧw 06 ﾀ0903 縺060 ｧｧ, 閨04 ｧﾀﾀｧ Ox , 0503 w= vdv.

121 罍ｧｧ07 04 縺03 ｧ 0403 閨ｯ03 09 閨04 ｧﾀ07 04 縺ｧ , 0503 v =ax,05 閨06 v ﾀ0903 縺03 ﾀ0504, x 04 縺0308 閨ｧ a 0403 ﾀ縺06 閨0600 ｧﾀ0903 縺060 ｧ06 閨04 ｧ07060 ｧｯﾀ 0403 ﾀ縺縺05 ｧ縺08 ﾀﾀ0903 縺03 ﾀ / ﾀﾀ x 閨縺06 t ﾀ ? 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 ﾀﾀ閨03 ﾀ05 ｧｧ0403 ﾀｧｧﾀ030506? 098. ｮ03 ﾀ縺陦ｧ06 ﾀ R ﾀﾀ閨0308 ﾀ 0403 ﾀﾀ0903 縺03 ﾀ a 00/ ﾀ縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀｧ07 06 縺ﾀ縺 h ﾀ0 ｧ , 縺ｧ陦ｧﾀ閨縺08003 ﾀ05 ｧ 07 ｧ閨0903 ﾀ05 ｧ. ｰ09 0 ｧ縺ﾀ V = h R, h 3 ﾀ05 ｧﾀ縺03 陦ｧ縺06009 ｧ縺 t, 04 罍ｧ 060 dv = dt a ( ﾀｧ06) ｦ0 04 ｧﾀｧ003 ﾀ閨ｧ 06 ﾀ x ﾀ0504, 縺ｧ0 ｧ 06 ﾀ縺ｧｧ, ｧ 04 縺 t ﾀｧ060 x = A(, e,kt ) 縺06 閨0600 ｧ0504 ﾀ0903 縺03 ﾀ0504 縺ｧｧﾀﾀ0903 縺03 ﾀ0504 d' d! =!, ﾀ0903 縺060 ｧ06 dt dt = = " , 0503 d(! ) d' =". x. ｯ0603 縺ﾀ縺06 閨0060. ｯ0603 縺ｭﾀ00 ｧ f(x): ) 縺06 縺縺 [a; b], ) ｧ縺03 ｧ 0403 閨罍ｧ , 3) f(a) =f(b), 閨06 a ｧ b 閨0605 ﾀ x = c, 04 罍ｧ縺030 f 0 (c) =0: (). ｯ0603 縺060 6ｧｭﾀ00 ｧ f(x): ) 縺06 縺縺 [a; b], ) ｧ縺03 ｧ 0403 閨罍ｧ , 閨06 a ｧ b 0 08 閨0605 ﾀ x = c, 04 罍ｧ縺030 f(b), f(a) =(b, a)f 0 (c): () 3. ｯ0603 縺ｧ. ｭﾀ00 ｧ f(x) ｧ '(x): ) 縺06 縺縺 [a; b], ) ｧ縺03 ｧ 0403 閨罍ｧ , 04 罍ｧ 060 ' 0 (x) 6= 0, 閨06 a ｧ b 0 08 閨0605 ﾀ x = c, 04 罍ｧ縺030 f(b), f(a) '(b), '(a) = f 0 (c) ' 0 (c) : (3)

122 3x. ｯ0603 縺ﾀ縺06 閨ｧ縺ﾀｧ縺ﾀ縺06 閨 , ｧ罍ｧ05 ﾀ縺ｧｧ x = c, ﾀ縺06 閨閨06 a ｧ b 罍ｧ 06 ﾀ09 ｧ縺ｧ 6ｧ縺 , 閨 AB 縺06 縺罍ｧ y = f(x), ｧ閨縺06 閨ﾀｧ 006 ｧ , 0 08 閨0605 ﾀ縺 , 09 ﾀ縺罍閨06 AB. 7 閨 , ﾀ03 閨06 縺07 0 ｧｧｧ00 ｧｧ , 06 ﾀｧ縺ﾀ縺06 閨0060, ｧ閨003, ｯ0603 縺ﾀﾀ罍ｧ f(b) =f(a) =00703 縺00600 ｧ縺 : 閨06 閨縺0070 ｧ a ｧ b ｧｧ f(x) 0 08 閨0605 ﾀ縺06 03 陦ｧ縺縺03 ｧ 0403 閨00308 f 0 (x), 06 ﾀ00 ｧ f(x) 縺06 縺縺 [a; b] ｧｧ縺03 ｧ 0403 閨罍ｧ縺罍ｧ0504, 閨縺0070 ｧｧｧ f (x) =x,4x 陦ｧ05 ﾀ縺縺03 ｧ 0403 閨ﾀ0 ｧｧ 06 ﾀ09 ｧ罍ｧ0060 ｧ0 00 ｧ縺ｧｧ f (x) =, 3 x 縺 [,; ]? ﾀ0 ｧｧ 06 ﾀ09 ｧﾀ05 縺03 ｧ0504 閨 AB ^ 09 罍ｧ y = j sin xj 縺 [,=; =] 閨ﾀ , 罍閨06 AB? ｧ 06 ﾀｧ縺閨06 ﾀ ? ﾀ y = x 罍閨06, ﾀｧｧ A(,; ) ｧ B(3; 9)? ﾀ0 ｧｧ 06 ﾀ09 ｧｧﾀ縺ｧ閨ｧｧ f (x) =x 縺 [a; b] ｧｧ c ﾀ0 ｧｧ 06 ﾀ09 ｧｧﾀ縺ｧ閨ｧｧ f (x) = x 縺 [; 4] ｧｧ c , 縺 [,; ] 縺060 6ｧ罍ｧ0060 ｧｧ070 4 x ｧ, 3 x ﾀ0 ｧｧ 06 ﾀ09 ｧﾀ05 縺03 ｧ0504 AB ^ 09 罍ｧ y = j cos xj 縺 [0; =3] 閨ﾀ , 罍閨06 AB? ｧ 06 ﾀｧ縺ｧ閨06 ﾀ ? ﾀ0504 f (x) = x 04 罍ｧ jxj < ; 703 ﾀ05 縺03 ｧｧ罍ｧ jxj > : ｧｧｧ, ｧ O(0; 0) ｧ B(; ), , 閨06 O ｧ B ｧｧ, 09 ﾀ縺 OB ｧ06 06 ﾀｧ縺ｧ閨ｧｧ縺 [0; ] ｧｧ ? 縺ﾀﾀｧ閨06 ﾀｧ070 ｧﾀ03 ﾀ縺06 閨00608 ﾀ0903 縺03 ﾀ v 0 =40090/. ｯ0603 縺060 6ｧ縺 , 閨04 ｧ07060 ｧ07, 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 / dt

123 罍ｧｧ07 04 縺03 ｧ 0403 閨 , 0503 f (x) 縺06 縺縺 [a; b] ｧｧ縺03- ｧ 0403 閨閨罍ｧ罍ｧ0060 ｧ縺ｧｧ x f (x) (x) = b f (b) a f (a) ; ｧ縺ｧﾀ0 ｧ罍ｧ 06 ﾀｧｧｧ (x). f (b), f (a) ｧﾀ縺ｧ '(b), '(a) = f 0 (c) ' 0 閨 (c) 09 ｧ08 f (x) =x 3 ｧ '(x) =x ｧｧ c 罍ｧ 06 ﾀ09 ｧ縺ｧ縺 , 閨罍ｧ x = '(t), y = f (t) 閨ｧ08 t 縺 a 6 t 6 b 0 08 閨0605 ﾀ縺 , 縺ﾀ罍閨06, 06 ﾀ00 ｧｧｧ '(t) ｧ f (t) 縺 [a; b] 06 閨縺ﾀｧ縺ｧｧﾀ縺ｧｧ閨06 f (x +x), f (x) = = xf 0 (x + x), 05 閨06 0 <<, 閨ｧ08: ) f (x) =x ; ) f (x) =x 3, ｧ , 0503 閨縺ｧｧｧﾀｧ x, 閨縺ｧﾀｧ x ｧ x , = 0 + 0; ｮ縺ｧ閨 , ﾀ00 ｧ閨06 0 <<: f (0) = f 0 (0) = f 00 (0) = :::= f (n,) (0) = 0; f (x) x n = f (n) (x) ; n! ｧﾀ縺ｧ f (b), f (a) =(b, a)f 0 (c) 閨ｧｧ f (x) =x 3 ｧｧ c ｧﾀ縺ｧｧｧ c 閨ｧ08: ) f (x) =arctg x 縺 [0; ]; ) f (x) = arcsin x 縺 [0; ]; 3) f (x) =lnx 縺 [; ]:

124 3x 3. 7 ﾀ09 縺0305 ｧ縺06 閨ﾀｧ0003 6ｧ80304 ｧｧﾀ縺ｧｧｧ c 閨ｧ08: ) sin x ｧ cos x 縺 [0; =]; ) x ｧ x 縺 [; 4]: ﾀ05 縺03 ｧｧｧｧ y = jx, j 縺 [0; 3] 閨06 ﾀ縺ﾀ05 ｧ 09 ﾀ , 罍閨06? ｧ 06 ﾀｧ縺ｧ閨06 ﾀ ? ﾀ罍ｧ y =4, x 罍閨06, ﾀｧｧ A(,; 0) ｧ B(; 3)? ﾀ0 ｧｧ 06 ﾀ09 ｧ. x 3. 7 ﾀ09 縺0305 ｧ縺06 閨ﾀｧ0003 6ｧ80304 ｧ縺06 閨ﾀ0504 ｭﾀ00 ｧ lim x!a 0 ｧ08 ﾀ06006 ﾀ f(x) = lim '(x) =0,0503 lim x!a x!a 縺ｧ0003 6ｧ80304 ｧ f(x) '(x) = lim f 0 (x) x!a ' 0, ﾀ (x) 縺06 閨ﾀ縺ｧ0003 6ｧ80304 ｧ 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, ﾀ (x) 0 ｧ08 ﾀ06006 ﾀ縺06 閨ﾀ05 ｧ 0,,, ｧ 0 0 ﾀ0403 閨0705 ﾀ縺06 閨ﾀｧｧ 06 ﾀ09 ｧ08 04 縺ｧｧ 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 :

125 罍ｧｧ07 04 縺03 ｧ 0403 閨 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 : 縺06 閨0600 ｧ縺07 閨ﾀ xe x, sin x ﾀｧ x! , 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 : , 0503 (04 罍ｧ x! 0) x,sin x x3 ｧ0305 ﾀ06 sin x 6 x ﾀ縺ﾀ050406, 04 罍ｧ0300 ｧ x 3 = ｧﾀ00 ｧ0504 sin ｧ sin 6 ｧｧ縺ﾀ , 0503 (04 罍ｧ! 0) 3 +,, 3, 9 ｧ0305 ﾀﾀ縺ﾀｧﾀ00 ｧ ; 006, 3 0; 99, 3 65, 3 0 ｧｧ縺ﾀｧ 04 縺06 閨060003: e ax, e bx x, arctg x 44. lim : 45. lim : x!0 sin x x!0 x 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! , 罍ｧ x! 0 arcsin x, x x3 6.

126 3x ﾀ05 0 ｧ06 ｧｧｧｧ , 0503 (04 罍ｧ! 0) +,,, 8 ｧ 0305 ﾀﾀ縺ﾀ050406, 04 罍ｧ0300 ｧｧﾀ00 ｧ0504 ; 006, ; 004, 0; 998, 0; 994, 65, 85 ｧｧ縺ﾀ0504. x ﾀ05 0 ｧ06 ｧｧｧｧﾀｧ0060 ｧ 0 ｧ0 ｧ縺06 閨ｧ07: I. ｱｧ07 f(x) ﾀﾀ x 0, 06 ﾀ00 ｧ縺0308 "-0309 縺06 ﾀ05003 ﾀ05 ｧｧ f(x 0, h) <f(x 0 ) <f(x 0 + h) 04 罍ｧｧ h<". II. ｱｧ07 f(x) ﾀﾀ縺 [a; b], 06 ﾀ00 ｧ閨 x ｧ x 縺 f(x ) <f(x ), x <x. 縺ｧ縺06 閨ﾀｧｧｧｧ縺 III. ｱｧ07 f(x) ﾀ07 ｧﾀ05 縺 (0 09 ﾀｧ0060 ｧ00 ｧ 0 ｧ0 ｧ0060) x 0, 06 ﾀ00 ｧ f(x 0 ) ﾀ07 0 ｧｧ0 ｧ00 ｧ 0 ｧｧｧｧｧ縺0308 閨0406 ﾀ0503 縺縺06 ﾀ05003 ﾀ05 ｧｧﾀ罍ｧ 0 09 ｧ 0403 ﾀ05 0 ｧ07 ｧｧｧｧ y = f(x) ( ｧ縺 ): 06 ﾀ00 ｧ y 0 > 0, ｧﾀ ; 06 ﾀ00 ｧ y 0 < 0, ｧ陦ｧﾀｧﾀ05 縺ｱｧ07 y = f(x) ｧﾀ05 縺 , 05 閨06 y 0 =0 ｧ00 ｧ 006 ﾀ06006 ﾀｯ 09 ｧｧﾀ07 09 罍ｧ05 ｧ 06 ﾀ09 ｧ0 ｧｧ08 09 ﾀｧ00 ｧ 0503 罍ｧ (y 0 =0), ｧ00 ｧ 0406 縺05 ｧ ( ), ｧ00 ｧ縺06- 閨ﾀ (0 04 罍ｧ006, ). 6 5 閨ﾀ0006 閨0 ｧ08 ﾀ y ﾀ06006 ﾀﾀﾀｧﾀ05 縺ｭﾀ00 ｧｧ07 f(x) 縺06 縺 x 0 ｧｧ縺縺06 ﾀ05003 ﾀ05 ｧ x 0,09 縺03006, , ｧ x 0, 縺03 ｧ 0403 閨00606 ｧ 06 ﾀ00 ｧ 04 罍ｧ 0406 縺閨06 x 06 縺06 x 0 : y ﾀ + 0,, 0503f(x 0 )=y max, y ﾀ, 0 +, 0503f(x 0 )=y min, y , ﾀ05 縺ｯ縺0605 ｧ08 ﾀｧﾀ (04 罍ｧ y 0 > 0 ｧ00 ｧ y 0 < 0), 縺0605 ｧ03 ｧｷ305 09, ｧ 0509 ﾀ05 縺ｧｧ, : ) ｧ y 0 ｧ09 罍ｧ05 ｧ 06 ﾀ09 ｧｧ, 縺0308 y 0 =0 ｧ00 ｧ 006 ﾀﾀ ) 0504 縺06 閨0600 ｧ y 0 ﾀｧﾀ閨罍ｧ05 ｧ 06 ﾀｧ, ﾀ03 ﾀ05 04 ｧｧ0906, 0 04 罍ｧ006, 04 ｧ

127 罍ｧｧ07 04 縺03 ｧ 0403 閨ｧ y max ｧ y min ｧ 0403 ﾀ05 縺03 ｧ罍ｧ罍ｧﾀﾀ05 縺罍ｧ04 07, ﾀﾀ罍ｧ0406 閨ｧﾀﾀｧﾀ05 縺 ( 縺0308 ﾀ0403 ﾀ0303 ｧﾀﾀ0006- 閨ｧ07). ｭﾀ00 ｧ縺 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, 縺03 ﾀ 03 ﾀﾀ縺ｧｧﾀｧ0504 ﾀ縺ﾀ0403 ﾀｧﾀﾀ0006 閨ｧ07. 0ｷ3 ﾀﾀ0006 閨ﾀ05 0 ｧ06 ｧｧｧ08: 58. ) y = x ; ) y = x 3 ; 3) y = ; 4) y =lnx: x 59. ) y = tg x; ) y = e x ; 3) y =4x, x : ｧ 0509 ﾀ05 縺ｧｧｧ 0403 ﾀ05 縺03 ｧｧ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, y =, 3 (x, 4) : 7. y = e,x : ) , 68, 73 ｧ縺0308 陦縺0605 ｧ08 閨ﾀ05 縺03060 ｧ07 09 罍ｧｧ 0606 ﾀｧ ( ﾀ , x 9).

128 3x ﾀ05 0 ｧ06 ｧｧｧｧ y = x + cos x 04 ｧ00506 縺 (0; ). 73. x =4x, tg x 04 ｧ00506 縺 , ;. 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, : 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 ｧ 0509 ﾀ05 縺ｧｧｧ 0403 ﾀ05 縺03 ｧｧ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):

129 罍ｧｧ07 04 縺03 ｧ 0403 閨 y = sin x, x 04 ｧ00506 縺 (,=; =). 06. y =x + ctg x 04 ｧ00506 縺 (0; ). 07. y = x + arcctg x. 08. y =+ 3 (x, ). 09. y = sin x + cos x 04 ｧ00506 縺 (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 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 ｧｧ08 ｧ 0 ｧｧｧｧｧ閨00 ｧ縺03 陦ｧ罍ｧ閨縺閨ｧ陦ｧ縺06 閨0600 ｧ縺03 04 縺閨09 ｧｧ0504 ｧﾀ閨04 ﾀ , 縺03 ｧ閨060 ｧ06 ｧｧｧ縺ｧ09 ﾀ 03 ﾀｧ060 a ｧ 0403 ﾀ h 0404 ｧﾀ 0 04 縺ｧ09 0 ｧ陦ｧ縺06 閨0600 ｧ閨04 04 縺ｧｷ陦ｧﾀ05 09 縺05030 ﾀ03 ﾀ0503 縺 a 0403 縺ﾀ陦ｧ陦 0503 ｧｧ 03 ﾀﾀ07 ﾀ05 ｧﾀｧﾀ07 04 縺縺閨ﾀ0503 縺縺陦 05, 縺ｧｧｧ0? 縺06 閨0600 ｧ縺縺ﾀﾀ06080 ﾀ 0904 陦閨 , ｧﾀ05060 ｧ閨ｧｧ 06 ﾀ罍ｧﾀ0503 縺03003 ｧﾀｧｧｧﾀ縺06 閨0600 ｧﾀｧ , 閨ｧｧｧ

130 3x 5. 6ｦ0 ｧ 03 0 ｧｧ08 ｧ 0 ｧｧｧｧｧ縺ｧﾀｧ07, 03 ﾀｧ縺ﾀ0504 陦ｧ縺縺06 閨0600 ｧｧｧ 04 罍ｧ 03 ﾀｧｧ 05 09, 閨ｧｧｧｮｧｧ縺縺ｧ09, 縺罍ｧ00605 ﾀｧ罍ｧ陦ｧ06 ﾀ縺閨04 ﾀｧ閨ｧ ? ｧｧ 0403 A 04 縺陦ｧ05 ﾀ縺縺03 閨06 B 閨03 縺ｧ縺ｧ縺閨03 縺縺ﾀ05 ｧ 0003 ﾀﾀ06 ﾀ 0403 A, 閨03 ﾀ縺ｧ A 04 B ｧ閨 , 06 ﾀ00 ｧﾀ0503 ｧ003 ﾀｧ罍ｧ 0406 縺ﾀﾀ06 04 m 閨03 縺030706, 閨03 縺030506? ｧﾀｧ09 ﾀﾀ陦縺陦縺縺 , ﾀ0306 陦ｧｧ08, ｧ 0 ｧﾀ , 06 ﾀ00 ｧﾀｧ0003 ﾀｧﾀｧﾀ0705 ﾀ07, : ﾀ閨ﾀ0903 ﾀ05 ｧｧ 04 縺ｧﾀ 03 陦ｧﾀ0903 縺03 ﾀ v 090/ 縺陦ｧ0 ﾀ罍ｧ縺06 ﾀｧ07 00 ｧ0 ｧ08 閨04 ｧ07060 ｧ07, 縺閨閨 a 090. ｪ06 縺縺06007 ﾀﾀｧ閨06 ﾀｧ 0306 閨ｧｧ0 ｧﾀﾀｧ06? 縺ﾀｧ07 ﾀ03 ﾀ閨縺05030 ｧｧ縺縺ﾀ0608 閨00 ｧ006. ｮ05 縺縺0305 ｧ縺縺09 ｧｧ006 縺09 ｧｧﾀｧｧ I = = xy3,05 閨06 x ｧ y 006 縺ｧ縺06 閨0600 ｧ縺ｧ 04 罍ｧ 0 ｧﾀ05 縺縺0305 ｧ03, 06 ﾀ00 ｧ縺06 0 ｧ 09 縺縺06040 ﾀ陦ｧ縺030 D ﾀｧｧ00 ｧ0 陦, 0404 ｧﾀ ? 罍ｧ閨03 00 ｧ罍ｧ00308,4 0 ｧ, 縺06 ﾀﾀ縺縺06 閨0600 ｧｧ閨00 ｧﾀ050 ｧ0903, 縺縺06006 ﾀ05 ｧ (0503 罍ｧ ) ｧ 03 閨罍ｧ閨03 04 陦縺ﾀﾀ陦ｧ06 ﾀ030 4 閨0 ｧ 0403 ﾀ閨ｧﾀ 0 09 ｧ00 ｧ0 陦 0 ｧｧ縺0605 陦ｧ06 ﾀ R 0404 ｧﾀ 0 04 縺ｧ09 0 ｧ陦ｧ縺06 閨0600 ｧ縺 y = x ｧ , 0 ｧ縺 y =x, 縺05 ｧﾀｧ070 ｧ b ﾀ0, 0406 縺080 ｧ08 0 a ﾀﾀﾀｧｧ 0305 ﾀ閨ﾀ , ﾀﾀ00305 縺縺05 ｧｧｧ ?

131 罍ｧｧ07 04 縺03 ｧ 0403 閨閨00 ｧ0 ﾀ05060 ｧ閨030 ( 罍ｧﾀ. 9) 閨罍ｧｧ罍ｧ006 x 0903 罍ｧ閨閨04 05 縺ﾀ閨ｧ ? 縺縺ｧ09 ﾀ 05 ｧﾀ0 ｧｧﾀ 0 04 縺ｧ09, 03 ﾀｧ縺ﾀｧ閨縺03 04 縺ｧ09, 閨ｧ ? ｧ A(0; 3) ｧ B(4; 5) ﾀｧ Ox ｧ M 05 09, ﾀﾀｧ06 S = AM + MB ｧｧｮ0304 縺0305 ｧｧｧ 04 縺03 閨ﾀ07 05 ｧ06 04 縺縺09 ｧ陦ｧ縺ﾀｧ縺06 閨0600 ｧ縺ｧ, 0403 縺ｧ 09 縺縺06040 陦ｧ縺030 D, 05 09, ﾀ0304 縺0305 ｧｧﾀ07 05 ｧｧｧｷ3 09 縺縺ﾀ07 ﾀ , ﾀ03 閨06 縺07 0 ｧ , ﾀﾀ0406 縺ﾀﾀ. 7 罍ｧｧｧﾀ 0306 閨ｧｧ0? 縺ﾀ030 P, ｧ罍ｧﾀ0903 ﾀ05 ｧ, ﾀ閨04 ｧ罍ｧﾀｧ F ( 罍ｧﾀ. 30) ｧ0 7 ｧﾀ. 9 7 ｧﾀ罍ｧｧ0504 ﾀｧ0006 F, ｧｧ09 ｧ縺060 ｧ07 =0; 5. x ｧﾀ05 ｧｧｧ 0406 縺0605 ｧ03 09 罍ｧﾀ05 縺03060 ｧ06 09 罍ｧﾀ罍ｧﾀ縺0809 (08040 ｧ 09) x = x 0, 06 ﾀ00 ｧ縺縺06 ﾀ05003 ﾀ05 ｧｧ ( ﾀｧﾀ04 04 ) 09 罍ｧ04 07 ﾀｧ ( ) 09 ﾀｭﾀ00 ｧ x = x 0 : ) y 00 > 0, 罍ｧｧ 09; ) y 00 < 0, 罍ｧ縺ｯ縺0605 ｧﾀ , 縺罍ｧ縺陦ｧ05 ﾀ 03 閨00308 ﾀ0503 縺ﾀ陦縺 ( ｧ, ﾀ0006 閨 , ｧﾀ05 ｧ) 陦ｧﾀｧｧ 0406 縺0605 ｧﾀ , y 00 =0 ｧ00 ｧ 006 ﾀ06006 ﾀ , 閨03 ﾀ , 0503 y 罍ｧ

132 3x ｧﾀ05 ｧｧｧ 0406 縺0605 ｧ03 09 罍ｧﾀ05 縺03060 ｧ07 09 罍ｧ縺閨ﾀ縺06 閨0600 ｧ0504: ) ﾀｧ罍ｧ06; ) ﾀ0504 ﾀｧ07; 3) ｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ Ox ｧ Oy; 4) ｧ縺ｧｧ y = '(x) ｧ00 ｧ x = f(y) ｧﾀｧ ; 5) 0403 ﾀ05 0 ｧ06 ｧ00 ｧｧ06 y ｧ00 ｧ x ｧ 0509 ﾀ05 縺ｧ; 6) ｧﾀ05 ｧｧｧ 0406 縺0605 ｧｷ3 ﾀﾀ0006 閨ｧﾀ05 ｧｧ 0403 ﾀ05 縺03 ｧ罍ｧ : ) y = x ; ) y = x 3 ; 3) y = e x ; 4) y =lnx; 5) y = x 5=3 : 縺06 閨0600 ｧﾀ05 縺ｧｧｧ 0406 縺0605 ｧ03 09 罍ｧｧ 0403 ﾀ05 縺03 ｧ罍ｧ040306: ) y = x3 6, x ; ) y = e,x ; 3) y = x +x ; 4) y ==x. 7 罍ｧ縺0306 ｧｧ , 0403 ﾀ05 縺03 ｧ罍ｧ040306, { ｧ070 ｧ: 48. y =x +9: 49. y =,x, 4x. ｰ 09 0 ｧ縺06 閨0600 ｧ0504 ﾀｧ罍ｧ06, ﾀ0504 ﾀｧ07 ｧｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ, ﾀ05 縺ｧｧ 0406 縺06 ﾀｧ07 ﾀ Ox. 50. y = sin x; y = cos x: 5. y =shx; y =chx. ｰ 09 0 ｧ , 縺06 閨0600 ｧｧ 0509 ﾀ05 縺ｧ 0406 縺0605 ｧ y =ln(x +): 53. y = e,x. ｰ 09 0 ｧ , 縺06 閨0600 ｧﾀ0504 ﾀｧｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ, ﾀｧｧｧﾀ05 ｧ. 54. ) y = x 3 ; ) y =(x +3) 3 : 55. ) y =+ x, 4 ; ) y = 3 x, x : ) 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 : ) y =x, x 4 ; ) x(y, x) =4: 6. y =(x +) =3, (x, ) =3. 6. y = xe,x.

133 ｭｭ6 7 ｭ6ｧ8 ｭ 7 76ｧ9 0ｷ3 7 ｯｭ06 7 縺6ｧ8 x 縺06 閨ｧｷ罍ｧ縺ｧｧ縺06 閨ｧ R f (x) ﾀｧ07 F (x) +C, ﾀ03 閨06 縺縺03 ｧﾀ C, 陦ｧ縺06009 ｧ縺閨ｧ06 f(x), f(x) = F (x) +C; 06 ﾀ00 ｧ d[f (x) +C] =f (x) :. ｯ 0300 ｧ09 03 ﾀｧ : : 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. ｮﾀ縺06 閨ｧ : I: d u= u : II: du = u + C: III: Au = A u : IV: (u + v) = u+ v :

134 3x 縺06 閨ｧｷ罍ｧ縺ｧｧｷ罍ｧ縺ｧｧﾀ罍ｧ0406 閨060 ｧｧ (0403 ﾀﾀ IV)09 ﾀ縺03 ﾀｧﾀ0006 閨06060 ｧﾀｧ縺ﾀ05 : ) d( ) = x; ) d( ) = x 3 ; 3) d( ) = cos x; 4) d( ) = x ; 5) d( ) = ; 6) d( ) = cos x +x : ｧｧ x, x 3 ｧｧｧ : 64. ) x +x + ; ) x 0x 8 +3 : x ) 66. ) x, ; ) x 3 ( x + 3 x) ; ) (x +) : x 3 x, : 4 x ) 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 :

135 縺06 閨ｧｧｧ : (x, ) 73. ) x 3 ; ), 3 x x : x 74. ) x, x ; ) ) x + x + ; ) x ) 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ｷ罍ｧ縺ｧ陦ﾀｧﾀ縺06 陦ﾀｧ04 x = '(u), = ' 0 (u) du, ｧ0 f(x) = f '(u) ' 0 (u) du: () ｯ縺ｧ06 ｧﾀ07 ｧ罍ｧ縺ｧﾀ縺03 ﾀﾀ閨060 ｧ縺 u 縺閨ﾀ , 04 罍ｧﾀ0006 閨06060 ｧ06 04 縺ｧ07 陦ｧ縺06009 ｧ 00 : = a d(ax + b); x= d(x ); cos x= d(sin x); x = d(ln x) ｧ ; ｧﾀｧ06 04 ﾀ縺06 u. ｯ罍ｧ060 ｧ罍ｧ縺ｧﾀ縺06 陦ﾀｧｧ : 79. cos 3x : 80. sin x. ｰ 09 0 ｧ 06. 6ｦ縺0600 ｧ0504 閨ﾀ0403 ﾀｧ: ) ｧ04 3x = u, x = u=3, = du=3; ) 04 罍ｧ0406 閨07 ｧｧ閨06 R 3 cos 3xd(3x).

136 3x. 0ｷ罍ｧ縺ｧ陦ﾀｧﾀ縺06 陦ﾀ e,3x : 8. cos 5x : 83. e x= + e,x= : 84. 4x, : (3, x) 4 : 86. 3, x : 88. x, 5 : 90. x, 5x , 6x : sin (a, bx) : x x + : ｰ 09 0 ｧ 06. 6ｦ0 ｧ 89{98 縺ﾀ縺 u 0 u = du =lnjuj + C; u ﾀ00 ｧｧﾀ00 ｧ閨陦縺0303 ｧ 06 ﾀ縺03 ｧ 0403 閨 ; 0503 ｧ罍ｧ , 0x : 9. ctg x : 94. cos x : 96. sin x cos x cos x : 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ｦ縺0600 ｧ陦ﾀ sin x = u ｧ00 ｧﾀ縺06 陦ﾀ , 0060 ｧ04 cos x 06 縺06 d(sin x) cos x sin 4 x : 30., cos x : 304. sin x sin x cos 3 x : sin x cos x :

137 縺06 閨ｧ e cos x sin x : 306. e x3 x : ｰ 09 0 ｧ 06. 6ｦ縺0600 ｧ陦ﾀ x 3 = u ｧ00 ｧﾀ縺06 陦ﾀ , 0060 ｧ04 x 06 縺06 3 d(x3 ) e,x x : 308. x +x : 30. e x x : 3 x 3, 8x : ｰ 09 0 ｧ 06. 6ｦ縺0600 ｧ陦ﾀ x + =u ｧ00 ｧﾀ縺06 陦ﾀ , 04 ｧﾀ 04 ｧｧ閨06 R (x +) = d(x +). x x 3. : 3. : +x 3, x sin x + cos x : 34. +lnx : x sin x cos x : 36., 6x 5 x 4 : ｧｧ : (e x + e,x ) : 38., 4x : x: 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 : (a, : bx) 3

138 3x 3. 0ｷﾀｧｧ 45 x 3. 0ｷｧ,,, + ｧ 09 0 ｧ0 04 罍ｧ0403 閨070 ｧ06 ﾀ , 0503: ) a + x = a arctg x + C, ｧ04 x = a tg t; a ) x = arcsin + C, ｧ04 x = a sin t; a, x a 3) x, a = a ln x, a x + a + C, 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, 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 :

139 縺06 閨ｧ x 338. x : x4 x, : 3 ｰ09 0 ｧ , ｧ 0403 閨ｧ陦縺0303 ｧｧﾀｧｧ x +4x +5 : 34. x +6x +3 : ｰ 09 0 ｧ { ｧ 0904 陦縺閨0600 ｧ陦 x +x +3: x, x : x, x : 347., x, x : x +3x +3 : : 3x, x, ｧｧ : x x, , x +x 4x, 5 x 350. : 35. x +5 x, : x4 35. x : e x :, e x x x 4 +0; 5 : 355. x, x +5 : 357. x x + x + : 359. x +4x +9 : 5, 4x, x : : 4x +4x +3

140 3x 4. 0ｷ罍ｧ縺ｧﾀ x 4. 0ｷ罍ｧ縺ｧﾀｷ縺陦ｧ縺06009 ｧ縺03 ｧ 0406 閨060 ｧ07 d(uv) = udv+ vdu ﾀ縺00600 ｧ罍ｧ縺ｧﾀ05070 udv = uv, v du: 縺ﾀ罍ｧﾀ , ｧｧﾀ07 04 縺03 ｧ 0406 閨060 ｧｧ 06 ﾀｧ 05 0 ﾀ09060 閨ｧ08, 0-04 罍ｧ006 R x e x ｧ00 ｧ R x ln x. 7 罍ｧ u 04 罍ｧ0 ｧﾀｧ07, 陦ｧ縺06009 ｧ縺ｧ縺ﾀ07, dv 05 ﾀ閨ｧ07, ﾀ03 閨06 縺 , ｧ縺0308 ｧ 0406 ﾀ05060 ｧ00 ｧ閨060. 0ｷﾀ09060 閨ｧ08 u 罍ｧ0 ｧﾀ07 ln x, arctg x ｧ arcsin x 罍ｧ006, 04 ｧ R x ln x u 罍ｧ ln x ( 006 x ), 04 ｧ R x e x u 罍ｧ x ( 006 e x ) ｧｧ : ln x : 36. xe x : 363. x cos x : 365. x ln (x, ) : x arctg x : e x sin x : , 0503 x + k = x x + k + k ln (x + (x + k)) + C: ｧｧ : 367. (ln x) : 368. ln x 369. : 370. x 37. arcsin x : ln (x +): 374. x sin x : arcsin x +x : x 3 e,x : cos (ln x) :

141 縺06 閨ｧｧｧ : 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ｷ罍ｧ縺ｧ06 05 罍ｧ罍ｧ 06 ﾀ09 ｧｧ08. 0ｷ陦ｧ陦縺0605 ｧﾀﾀｧ- 006 ﾀｧ 0903 ﾀｧ006 ﾀ閨0705, 04 罍ｧﾀ0006 閨06060 ｧ縺ｧｧ07 ﾀｧ: sin, cos x x = ; cos + cos x sin x x = ; sin x cos x = :.0ｷｧ陦縺0605 ｧﾀﾀｧ 006 ﾀｧ 0903 ﾀｧ006 ﾀ閨0705, 0305 閨ﾀｧ 03 陦ｧｧｧｧ縺 u. 0ｷ R cos m x sin n x 陦ｧ05 ﾀｧ0006, 06 ﾀ00 ｧ m ｧ n , ｧｧ0006, 06 ﾀ00 ｧ m ｧ00 ｧ n ｧｧ : 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 :

142 3x 5. 0ｷ罍ｧ縺ｧ06 05 罍ｧ罍ｧ 06 ﾀ09 ｧｧ 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 ｧｧ0504 tgx = t, x = arctg t sin 3x cos x : 404. cos mx cos nx. ｰ 09 0 ｧ { 罍ｧ0060 ｧ縺 sin cos = sin ( + ) + sin (, ) ; cos cos = cos ( + ) + cos (, ) ; sin sin = cos (, ), cos ( + ) : 405. ) 406. sin 3x sin 5x ; ) sin 5x, cos x sin mx sin nx ｷ罍ｧ縺ﾀ05070, ﾀ05 ｧ 0703 縺ｧ07060 ｧ07 ﾀｧ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 ｧｧ縺ｧ: ) sin 6 x; ) cos 6 x ｧｧ : ) sin 3 x ; ) cos 3 x. ｰ 09 0 ｧ罍ｧ0060 ｧ縺ｧｧｧ cos x. sin x

143 縺06 閨ｧｧｧ : 409. ( + 3 cos x) : sin 4 x cos x : sin 3 x cos x : (sin x, cos x) : sin x sin3 x cos x : 48. sin 4 x : cos 5 x : ( + sin x) 3 : sin 3x sin x : sin x + cos x : 6 x 6. 0ｷ罍ｧ縺ｧ06 09 ｧｧ 06 ﾀ09 ｧｧ08. ｭﾀ00 ｧ 0403 閨陦縺ｧ , ｧﾀｧ0504 ｧｧ06.. 6ｦｧ陦縺0303 ｧﾀｧｧ 04 ｧ (x, a) ｧ (x + x + q), ｧ陦縺ﾀ07 0 ﾀ縺00308 陦縺ﾀ0006 閨06060 ｧ : 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) ｧ0030 ﾀｧ 0 ｧ0706 ﾀｧｧｧ : 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 x +x, 3 : 43. x 3, x x + : 45. x 3, x (x +) 3 x, x : 3x, a x 4, ax 3 :

144 3x 6. 0ｷ罍ｧ縺ｧ06 09 ｧｧ 06 ﾀ09 ｧｧ08 5 x, 5x + 5x, 46. : 47. x 3, x + x x 3, 3x, : 5x + 4x, ; : 49. x +x +0 x, 0; x +0; : 7 ｰ 09 0 ｧ閨0600 ｧ陦 05 ｧｧ0504 x +=t ) x + x +4 : 43. x 3 + x +4x +4 x : (x + b ) ; ) 7x, 5 x 3, x +5x : 3x +x + (x +) (x +) : (x + b ) 3 : ｰ09 0 ｧｧ0504 x = b tg t ｧ ( 縺罍ｧ006 縺06) ｧﾀ縺 ) ｧ ) 436. (x +) (x +x +5) ; ) 4x ( + x)( + x ) : 437. (x, 6x : + 0) 3 x + x 4 +4x : ｧｧ , 罍ｧ縺06 閨ｧ09 ｧ : 438. x(x : a) (x + a)(x + b). ｰ09 0 ｧ { ｧﾀ00 ｧ閨陦縺0303 ｧ 0 04 ｧﾀﾀｧ , 閨0600 ｧ04 ｧﾀﾀｧﾀ x, x : 44. x 4, x : 443. (x, 3)(x +) : x 3 +4x :

145 縺06 閨ｧｧｧ : x, 444. : 445. (x, )(x, ) 5x, : 447. x 3, x, 4x +4 5x, : 449. x 3, 4x +4x x, a 450. : 45. x 3 + a x 45. x 3, : x + x + x, 3 : x +6 (x, )(x +) : x + x 3, x +x : x 3 + x +x + : x (x +x +) : { ｧ0504 ｧ罍ｧ縺ｧ06, 罍ｧ閨縺06 閨ｧ09 ｧ : 454. x +5x : 455. x 4 +3x : 456. x 4, : 457. x 4, x, : x 7. 0ｷ罍ｧ縺ｧ縺0308 ｧ罍 09 ｧｧ 06 ﾀ09 ｧｧ08 n. 0ｷ R x; ax + b, 05 閨06 R(x; y) 09 ｧｧ07, 陦ｧ05 ﾀ陦ﾀ ax + b = t n, ｧｧ R x m n ; ax m + b x m, 0403 陦ﾀ ax m + b = t n. Mx + N.0ｷ陦ｧ05 ﾀ陦ﾀ (x, ) ax + bx + c x, = t. 3. ｯ罍ｧ罍ｧ 06 ﾀ09 ｧ陦ﾀｧｧ罍ｧ罍ｧ 06 ﾀｧ閨06 04 罍ｧ0403 閨0705 ﾀ07 ｧ R x; a, x 0403 陦ﾀ x = a sin t; R x; a + x 0403 陦ﾀ x = a tg t:

146 3x 7. 0ｷ罍ｧ縺ｧ06 ｧ罍 09 ｧｧ 06 ﾀ09 ｧｧ a0 x m + a x m, + :::+ a m. 0ｷ3 ｧ閨06ax + bx + c 00 ｧｧ 06 ﾀﾀ縺 a0 x m + :::+ a m =(A 0 x m, + :::+ A m, )W + A m W W ; 05 閨06 W = ax + bx + c ｧ09 ｧ A 閨0705 ﾀﾀ0006 陦ｧ縺ｧ縺ｧﾀ0504 ｧ 03 ﾀ閨060 ｧﾀｧｧ09 ｧﾀｧﾀ罍ｧ 03 陦ｧﾀ x. 5. 0ｷ陦ｧ縺06009 ｧｧ0030 x m (a + bx n ) 0306 縺0605 ﾀｧ閨縺0608 ﾀ : ) ｧﾀ0003, ｧ060; ) m ｧﾀ0003, 0403 陦ﾀ05 - n a + bx n = t s ;3) m ｧﾀ0003, 0403 陦ﾀ n ax,n + b = t s,05 閨06 s 陦縺0303 ｧ. 0ｷ3 ﾀ陦ﾀｧ 04., ｧｧ : x + : x + : 46. x + x 3 x : 463. x 4 + x x ++ : x a, x : x 3 x +: 0ｷ3 ﾀ陦ﾀ , ｧｧ : 464. x : 465. x, x : x +x x : 467. : ax, x (x +) x +x ｧｧ , ｧﾀ陦ﾀｧ : a, x : 469. x 4, x : x, x : 473. (4 + x ) 3 : x (a + x ) 5 : x (, x ) 3 :

147 縺06 閨ｧｧｧ , 04 罍ｧｧ : 474. x +4x x +x +: 475. x 3, x, x : 476. x + k : 477. ax, x : ｧｧ陦ｧ縺06009 ｧｧ : 478. x 4 : x 3 x 3 : 3, x 3 x : 48. x ( + x ) 3 (a, bx : ) 3= ｧｧ : x, 48. : 483. x, x 484. : 485. x + : 3x +, 3 x : a, x x x x 3 : 487. : 3 x, x +, x x : 489. : x ++ +x + 4, x 490. x x +x : 49. (x, ) x, x : r x x 49. : , x, x : ｰ 09 0 ｧｧ0504 x = sin t. x x + x : 495. : 5 +4x, x x 3 +x : 497. x, x 3 : x +x : x : 3x, x,

148 3x 8. 0ｷ罍ｧ縺ｧ縺ﾀ09060 閨ｧ08 55 x 8. 0ｷ罍ｧ縺ｧ縺ﾀ09060 閨ｧｧｧ 06 ﾀｧ閨06 04 罍ｧ0403 閨0705 ﾀ07 ｧ : R(e x ) 0403 陦ﾀ e x = t, x =lnt, dt = t ; R(tg x) 0403 陦ﾀ tg x = t, x = arctg t, dt = +t ; R(sin x; cos x) 0403 陦ﾀ tg x = t, sin x t = =, t dt, = +t +t. +t, cos x = ｧｧ : e x, e x 500. e x + : 50. e 3x 50. e : x cos x : sin 4 x : 507. tg 4 x : sin x : 3 sin x + 4 cos x : + 3 cos x : ｰ 09 0 ｧ , 507, 5, 53, 05 閨ｧ sin x ｧ cos x ﾀ03 閨06 縺07 05 ﾀﾀｧ, 罍ｧﾀｧ tg x = t; sin x = t +t ; cos x = dt ; = +t +t : ｧｧ : e x 508. e x, : e 3x e x, : 5. cos 4 x : 53. tg 5 x : 3 + cos x : + 3 sin x :

149 縺06 閨ｧ sin x + sin x : + cos x 55. sin 3 x : e x + +tg x 56. : 57. sin x : e x, x 9. 0ｷ罍ｧ縺ｧ06 05 ｧ0406 縺ｧ 06 ﾀ09 ｧｧｧ0406 縺ｧ 06 ﾀ09 ｧ陦ﾀｧ : ch x =shx + C: : sh x=chx + C: 3: =thx + C: 4: =, cth x + C: ch x sh x 0ｷ陦ｧ陦縺0605 ｧﾀ ch x ｧ sh x 閨0705 ﾀ07 04 罍ｧｧ縺00600: ch x = chx + ; sh x chx =, ; sh x ch x = sh x : 0ｷﾀ sh x ｧ ch x 閨0705 ﾀﾀ0403- ﾀ , 0503 ｧｧﾀ sin x ｧ cos x. 06 ｧ0406 縺ｧ 06 ﾀ09 ｧ陦ﾀｧｧ罍ｧﾀ07 04 罍ｧ閨060 ｧｧｧｧ R x; x, a 0403 陦ﾀ x = a ch t; R x; x + a 0403 陦ﾀ x = a sh t: x + x 7 罍ｧ : 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 ｧｧ : 58. ) sh 3x; ) ch 3 x : 50. ch x + : 5. (+shx) : th x : th x, :

150 3x 0. ｮ罍ｧ006 縺03 0 ｧ罍ｧ縺ｧ x + a : 54. (x +4) 3 : 56. x, a : (x, 5) 3 : ｧｧ : 57. sh 3 3x : sh 4 x ch x : 530. sh x ch x : cth x : ch x +: 53. x x, 3: shx ch x : x +3 : x x 0. ｮ罍ｧ006 縺03 0 ｧ罍ｧ縺ｧｧｧ : +x arctg x 535. : 536. : x +x 537. x 3 + ax : 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. : x sin 4 x : 545. x tg cos x x : 546. sin x : sin x b + cos x : 548. ax, b : 550. (ax + b) 4 x + x : 3 x 4 + x :

151 縺06 閨ｧ (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. : ) 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 +) 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 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 : x

152 3x 0. ｮ罍ｧ006 縺03 0 ｧ罍ｧ縺ｧ 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 :

153 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 ｧﾀ00060 ｧ縺06 閨ｧﾀ縺 [a; b] 0304 縺06 閨ｧ07 f(x) 縺 [a; b] 0 n ﾀｧ a = x 0 <x <x <:::<x n = b. 0ｷ閨ｧ00506 縺04 00 (x i, ;x i ) 縺03 ｧ i ｧﾀ03 ﾀ05 04 ｧ0 ﾀ np P f( i )x i,05 閨06 x i = x i, x i,. ｮｧ n f( i )x i ﾀ07 i= ｧﾀ , 縺06 閨罍ｧ max x i! 0, 06 ﾀ00 ｧ 030 ﾀ06006 ﾀｧ , ﾀ縺06 閨ｧｧｧ f(x) 縺06 閨 a 閨03 b ｧﾀ07 b a f(x) = lim nx max x i!0 i= i= f( i )x i : () ｱｧ07 f(x) ﾀﾀ07 ｧ罍ｧ縺縺 [a; b] ｧ罍ｧ縺ﾀ05 ｧ閨03 ﾀ , 縺 [a; b] ｧ縺06 縺03040 ｧ00 ｧ 0706 ｧｧﾀ縺ﾀ0504 f(x) 縺06 縺 [a; b]. ｯ縺ﾀ06006 ﾀ縺06 閨ｧ f(x) = F (x)+c () ｧｧﾀ縺00600 b a f(x) = F (b), F (a) = f(x) b a ; (3) 縺06 閨ｧ縺06 縺ｧｧﾀ05 ｧｧ縺ｧｧ ( ｧ00 ｧ縺06 閨ｧ ) 04 罍ｧ 0406 縺ｧ 0 ｧ縺06 閨ｱ03 縺00600 (3) ﾀ縺 {6ｧｧ09.

154 3x ｧﾀ00060 ｧ縺06 閨ｧｮ03 ﾀｧ060 ｧﾀ0600 ｧ 0406 縺閨縺06 閨ｧｧ : a a a ) 0 x; ) 0 x ; 3) 0 e x ; 4) 0 sin x. ｰ 09 0 ｧ罍ｧ縺ｧｧ縺ｧ縺罍ｧ006 縺ﾀﾀ07 縺ｧ 034 ｧｧﾀ00 ｧｧｧ縺ｧﾀ s 5 ｧ S 5 閨0007 ｧ , 03 ｧ縺 [; ] x ﾀｮ 040 ｧ0504 ﾀｧ060 ｧｰ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 , M i 0 ｧｧ閨ｧｧ 04 i-0 ﾀ05 ｧ縺ｧﾀ00 ｧ0504: x 3 : 594. x + x4 : x : 596. a 3 a a + x : , x : e x=3 : =4 : x sin 4x : : 60. x, 4 0 =3 =4 +tg x ( + tg x) : ｰ 09 0 ｧ罍ｧ0060 ｧ陦ﾀ x = t ; 04 罍ｧ縺06 閨ｧｧﾀ07, ｧﾀﾀｧ閨ｧ0903 x j 4 j 9. 縺ｧ罍ｧｧ罍ｧ縺ｧｧ 0403 陦ﾀ t j j tg x = t ﾀﾀｧ 0060 ｧ縺06 閨

155 縺06 閨ｧ = + x + : 604. e x + : 606. sin x cos x : a= 0 a x 4, x : r x a, x : x a, x : ln (x +): x : ( + x : 6. ) 3 x + x : 63. 0ｷ縺ｧｧ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 ｧﾀ00 ｧ0504: a (x, ax) : 65. x 4, x : 67. ( + x) : =6 =8 0 x : cos x : e x +e x :

156 3x ｧﾀ00060 ｧ閨 x 4x +5 : 6., x : 6. = x cos x : 63. =4 tg 3 x : ｷ縺ｧｧ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 x ｧﾀ00060 ｧ閨閨04 09 罍ｧｧｧｧ A ABB, 04 罍ｧﾀｧ Ox ( 罍ｧﾀ. 3): S = x X lim yx = x!0 y : () x 6 7 ｧ縺06009 ｧ縺陦ｧ A AMM ds = y. 7 ｧﾀ. 3 7 ｧﾀ. 3 ｭﾀ00 ｧ 09 罍ｧｧ070 ｧ x = f(t) ｧ y = '(t), 0503 ds = = '(t) f 0 (t) dt.

157 縺06 閨ｧ閨04 09 罍ｧｧｧｧ, 04 罍ｧﾀｧ Oy: S = X y lim xy = y!0 xdy: () y 6 7 ｧ縺06009 ｧ縺陦ｧ ds = xdy 閨04 ﾀ OAB ( 罍ｧﾀ. 3) 09 ｧ , 縺罍陦ｧ : S = X lim '!0 r ' = ' 6 7 ｧ縺06009 ｧ縺陦ｧ ds = r d'. ' r d': (3) ｧﾀ00 ｧ閨04, ｧｧ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: 罍ｧ (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 =: ﾀ05 縺ｧ閨03 y (a, x) =x(x, a) ｨC ｧ0 ｧ0608 y = a (ex=a + e,x=a ), x = a ｧ y = 閨00308 縺ｧｧ閨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 ｧ80600 ｧﾀ r = a cos ' 罍陦ｧ03 ｧ閨0308 r = a(, cos ') r = 3 + sin ' 閨06 ﾀｧ 0 ｧｧ0 ｧ 0 ｧｧ0 陦ｧ06 ﾀｧ r =, cos 3' 閨06 ﾀｧ 0 ｧｧ0 ｧ 0 ｧｧ0 陦ｧ06 ﾀｧ.

158 3x ｧ r = a cos ': 648. r = a sin 3': 649. r = a(sin ' + cos '): 650. r = a ' ; 4 6 ' 6 : 65. r = a sin 3 ', ｧ縺ﾀｧ閨0609 縺ｧﾀ05 x 3 + y 3, 3axy =0( ﾀ0. 罍ｧﾀﾀ. 334) (0406 縺ｧ縺罍陦ｧ0 05 0). sin ' cos 'd' ｰ 09 0 ｧｧｧ0504 tg' (sin 3 = u, ' + cos 3 ') 閨0600 ｧ04 ﾀ0 00 ｧﾀ00 ｧｧ cos 6 ' ｧﾀ00 ｧ閨04, ｧｧ0 ｧ070 ｧ: 653. y =6x, x, y = y = x 3, y =8, x = y =, x ｧ x =, y + x 4 = x y = x +4x +5, x =0, y =0 ｧ 0 ｧ0 ｧ罍陦ｧ閨ﾀｧ006 ﾀ03 ｧ閨03 y = sin x ｧ y = y = x ｧ y =4x xy =6 ｧ x + y, 7= 罍ｧ x 3 + x, y = r =3, cos ' 閨06 ﾀｧ 0 ｧｧ0 ｧ 0 ｧｧ0 陦ｧ06 ﾀｧ r = + sin 3' 閨06 ﾀｧ 0 ｧｧ0 ｧ 0 ｧｧ0 陦ｧ06 ﾀｧ r = a sin ': 665. r = a cos 3' r = ae ' 0305 ' =, 閨03 ' = ﾀ05 ｧｧ04 ﾀ0304 x a + y x = ｧ b b + y a =(0406 縺ｧ縺罍陦ｧ0 05 0) r = a( + sin ') ｧ r = a. x ｧ , ｧ縺ﾀｧOx 09 罍ｧｧｧｧ A ABB ( 罍ｧﾀ. 33), 05 閨06 AB ^ 閨罍ｧ y = f(x), 0304 縺06 閨ﾀ縺 x X V = lim y x = y : () x!0 x 6 7 ｧ縺06009 ｧ縺 dv = y.

159 縺06 閨ｧ , ｧ縺ﾀｧOy 09 罍ｧｧｧｧ, 04 罍ｧﾀｧ Oy, 0304 縺06 閨ﾀ縺 V = X y lim x y = y!0 x dy: () y 7 ｧﾀｧ縺06009 ｧ縺 dv = = x dy 縺06 閨0600 ｧ , ｧｧ0506 縺03, ｧｧ0 ｧ070 ｧ: 669. y =x ｧ x = h 縺ﾀｧOx x a, y = ｧ y = b 縺ﾀｧOy. b 67. xy =4, x =, x =4, y = 縺ﾀｧOx. 67. y =(x +4) 3 ｧ x = 縺ﾀｧOy x + y = a 縺縺 x = b>a. ｰ09 0 ｧ06. dv = (b + x) dy, (b, x) dy =4bx dy y = a ch x, x = a, y = 縺ﾀｧOx. a 675. y =4, x, x = 縺ﾀｧOy (y, a) = ax, x =0, y =a 縺ﾀｧOx y = cos x ｧ y =, 縺縺 y =, 04 罍ｧ, 6 x y = x,x, x =,4 ｧ y = 縺ﾀｧOy y = cos x,, x =0, y =0(04 罍ｧ x>0) 縺ﾀｧOx y = a, x a ｧ x + y = a 縺ﾀｧOy 縺06 閨0600 ｧ , ｧｧ0506 縺03, ｧｧ0 ｧ070 ｧ: 68. y = sin x (03 閨 ), y = 縺ﾀｧOx. 68. x, y =4, y = 縺ﾀｧOy y =, x =, y = 縺ﾀｧOx. +x

160 3x ｧ0 閨0605 ｧﾀ罍ｧ x a + y = 縺ﾀｧOy. b 685. x =3 + y =3 = a = 縺ﾀｧOx y = x 3, x =0, y = 縺ﾀｧOy x, y = a, x = a 縺ﾀｧOx y = x, y = 縺縺 x =. ｰ09 0 ｧ06. dv = ( + x) dy, (, x) dy 閨00308 縺09 ｧ 09 ｧｧ閨03 x = a(t, sin t), y = a(, cos t) 縺ﾀｧOx (y, 3) +3x =0, x =, 縺ﾀｧOx. x ｧ0 閨0605 ｧﾀ罍ｧｧ0 閨0605 ｧ AB ^ 09 罍ｧ y = f(x): x B s = +y 0 : () x A +y 0 = + dy ｧ0 閨0605 ｧ AB ^ 09 罍ｧ x = f(t), y = '(t): t B 6 7 ｧ縺06009 ｧ 00 閨0605 ｧ ds = s = ｧ0 閨0605 ｧ AB ^ 09 罍ｧ 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 ﾀ縺 x =4= ﾀ罍ｧ x + y = a ﾀ罍ｧ x =3 + y =3 = a = y =(x +) 3,0305 ﾀ縺 x = 閨00308 縺09 ｧ 09 ｧｧ閨03 x = a(t, sin t); y = a(, cos t): 696. x = t6 6, y =, t 罍陦ｧ閨ｧ 0406 縺06 ﾀｧ07 03 ﾀ070 ｧ

161 縺06 閨ｧ y = x,, 0305 ﾀﾀ0406 Ox. ｰ09 0 ｧ06. ﾀ縺ｧ x ｧ00 ｧｧ 0403 ﾀ05070, ｧ00 ｧ 0 04 ｧ y = a (ex=a + e,x=a )=a ch x a 閨06 04 縺ｧ x = a y =lnx 0305 x =3=4 閨03 x ==5. +x ｰ09 0 ｧ06. 0ｷ陦ｧ05 ﾀ陦ﾀ x +x = t y = ln ( cos x) 閨06 ﾀｧｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ罍陦ｧ0 05 Oy ｧ Ox. 70. ) 9y = x(x,3) 閨ｧ 0406 縺06 ﾀｧ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 縺ｧ0509 ﾀ04 ｧ 00 ｧ r = a' ﾀ罍ｧ r = a sin 3 ' ｧｧ閨 A ｧ B, 閨070 ｧ08 ﾀ閨ﾀﾀﾀｧｧ AB =b, ｧｧﾀ05 縺縺0305 ｧ03 f. ｮｧ縺006 0 ｧ05 ｧ , , 0503 閨00 ｧ0 0 ｧ05 ｧ s b + f 3 b 04 罍ｧ閨03 ﾀ f b. ｰ 09 0 ｧ罍ｧ0060 ｧ罍ｧ0300 ｧ縺ｧ縺06 閨0600 ｧ0504 閨00 ｧ006 閨0605 ｧ 09 罍ｧ040308: 705. y = 4 9 (, x)3,0305 ﾀ縺 x =, y = ln (sin x) 0305 x = =3 閨03 x == y =ln(, x ) 0305 x =,= 閨03 x == y =x, 0305 ﾀ縺 x = = x = t, y = t 3 (t, 3) 閨ｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ0406 Ox.

162 3x 閨縺08003 ﾀ05 ｧｧ07 69 x 閨縺08003 ﾀ05 ｧｧ閨縺08003 ﾀ05 ｧ, ｧ縺ﾀｧ Ox 閨0605 ｧ ^ AB 09 罍ｧ y = f(x): P x = ^ AB y ds; 05 閨06 ds = + dy : 閨縺08003 ﾀ05 ｧ, ｧ縺ﾀｧ Oy 閨0605 ｧ ^ AB 09 罍ｧ x = '(y): P y = ^ AB x ds; 05 閨06 ds = + dy : 0504 縺06 閨0600 ｧ閨縺08003 ﾀ05 ｧ, ｧ罍ｧ : 70. x + y = R 縺ﾀｧOx. 7. y = x =, 0305 ﾀ縺 y =; 5, 縺ﾀｧOy. 7. y = a ch x a 閨06 x = a 縺ﾀｧOx x + y = 縺ﾀｧOy. ｰ 09 0 ｧ罍ｧ00704 y ｧﾀｧ縺 , ｧ0, 0503 ｧﾀ閨04 P = 6, 3y dy 罍ｧ陦ﾀ y = 4 3 sin t 閨罍ｧ y = sin x 縺ﾀｧOx 閨00308 縺09 ｧ 09 ｧｧ閨03 x = a(t, sin t), y = a(, cos t) 縺ﾀｧOx ｧ 09 罍ｧ x = t, y = t 3 (t, 3) 縺ﾀｧOx. 77. x + y = a 縺縺 x = b>a. ｰ09 0 ｧ06. dp =(b + x) ds +(b, x) ds 縺06 閨0600 ｧ閨縺08003 ﾀ05 ｧ, ｧ縺0605 Ox: ｧ 09 罍ｧ y = x x =, 閨03 x =. 3

163 縺06 閨ｧｧ 09 罍ｧ y =4+x, 0305 ﾀ縺 x = ﾀ罍ｧ x = a cos 3 t, y = a sin 3 t ｧ 09 罍ｧ x = t3 3, y =4, t 060 ｧ07 ﾀ 03 ﾀ070 ｧ罍陦ｧ閨ｧ 0406 縺06 ﾀ06- x 6. 6ｦ0 ｧｧ 07 ｧｧ09 ｧ縺06 閨0600 ｧ0504 ﾀｧｧ閨縺05 ｧ縺ﾀ 03 ﾀｧｧ 0403 ﾀ縺06 閨0600 ｧﾀｧｧｧｧ縺06 閨0600 ｧ0504 ﾀｧｧ閨縺05 ｧ縺閨0906, 03 ﾀｧ縺0308 a ﾀ縺ﾀ05 ｧ 0403 閨03, 0403 ﾀ h 縺06 閨0600 ｧ0504 ﾀｧｧ閨縺05 ｧ縺0605, 陦ｧ縺 R ﾀ縺08003 ﾀ05 ｧ 0403 閨ｧ0 ｧ縺ｧｧﾀ 0406 縺080 ｧ0 03 ﾀｧ , 0 ｧ070 ｧ0 0 0 ｧ 0403 ﾀ縺06 閨0600 ｧ0504 ﾀｧｧ閨ｧｧｧ006 縺09 ｧｧﾀｧﾀ0608 Ox ｧ Oy 陦ｧ 04 縺ｧ09, ｧｧ0 ｧ070 ｧ x = 0, x = a, y =0 ｧ y = b. ｰ09 0 ｧｧ04 04 縺ｧ罍ｧ閨09 ｧ, ｧ閨閨陦ﾀﾀｧﾀｧOx, y. ｮ0600 ｧ縺0607 ｧ 0406 縺0608 閨縺06 閨060006, ｧ0 X b J x = lim ayy = ay dy: y!0 0 a 縺ｧ 003 J y = bx ｧｧ006 縺09 ｧｧﾀｧﾀ0608 Ox ｧ Oy 陦ｧ 05 縺ｧ09, ｧｧ0 ｧ070 ｧ x = 0, y = 0 ｧ x a + y b = ｧｧ006 縺09 ｧｧﾀｧﾀｧ Oy 陦ｧ, ｧｧ0 ｧ070 ｧ x =, y = x ｧ y = ｧﾀ05 05 ｧ 06 ﾀ09 ｧﾀｧ Ox ｧ Oy ｧ罍陦ｧﾀﾀ 05 縺ｧ09, 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 : M x = xy dy, M y = xy 罍陦ｧﾀﾀ: x c = M y S, y c = M x,05 閨06 S 閨04 07 ｧ0506 縺03. S 0 0

164 3x 6. 6ｦ0 ｧｧ 07 ｧｧ09 ｧｧﾀﾀ陦ｧ, ｧｧ0 ｧ070 ｧ a y = = bx, x = a ｧ y = ｧﾀﾀ縺0605 x + y = a, 0305 ﾀﾀ0406 Ox. 73. ) ｧﾀ00 ｧ , 縺ｧｧ04 0 ｧ閨03 ｧ 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀﾀﾀ06080 ﾀ陦ｧ06 ﾀﾀｧ07 0,5 0, 06 ﾀ00 ｧ縺閨ﾀﾀ ,8 0 ｧ 0 0, 0 0 ｧﾀ閨罍ﾀ05 ｧｧ00 ｧ0 陦縺06. ) ｧﾀ00 ｧ , 縺ｧｧ04-0 ｧ閨03 ｧ陦ｧ06 ﾀ030 R 縺06 閨0600 ｧ , 縺ｧ0504, ﾀﾀ06 m ﾀ縺08003 ﾀ05 ｧｧﾀ h. ｰ 09 0 ｧ 06. ｮｧ00 F 罍ｧｧ07 0 ﾀﾀｧｧ x ｧ 0304 縺06 閨ﾀ07 ｧ 04 縺縺09 ｧｧ F : mg = R : x,05 閨06 R 陦ｧ06 ﾀｧ縺ｧｧｧ00308 H = = 0; 5 0 ｧ陦ｧ06 ﾀﾀｧ07 R = 0; 縺06 閨0600 ｧ , 縺ｧｧ04 0 ｧ閨03 ｧｧ00 ｧ0 陦縺縺陦ｧ05 ﾀ閨 V 0 = =0; 0 3 ﾀｧ060 0 =; 縺06 閨0600 ｧｧ縺0 ｧ 06 ﾀﾀ07 05 ｧ閨0608 閨 V =0; ( {03 罍ｧ V = 0 V 0.) ｧﾀ00 ｧﾀｧ07 0 0, 閨縺ｧ閨00 ｧﾀ陦ｧ06 ﾀ030 ﾀｧｰ 09 0 ｧ 06. ｮｧ00 F ｧ07 04 縺ｧ閨00 ｧ00308 l 0 ｧ閨0406 ﾀｧ07 s 罍ｧ 06 閨00 ｧ0060 ｧｧ x 縺06 閨ﾀ縺 F = = E sx,05 閨06 E 003 閨縺ﾀ05 ｧ陦ｧ罍ｧ E l ; 0 5 7/ ｦ縺 , ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ08 ﾀ03- ﾀ06 ﾀ閨ﾀｧ07 S = 40 ﾀ0 ｧ 0403 ﾀ H = 40 ﾀ0, 縺罍ﾀ05 ｧ06 0 閨閨0406 s = ﾀ0? ｰ09 0 ｧ06. ｮ0903 縺03 ﾀ0504 ｧﾀｧ07 07 ｧ閨0903 ﾀ05 ｧ 04 罍ｧ 06 縺ﾀ x ﾀ縺06 閨ﾀ縺 v = gx, 05 閨ｧ09 ｧ06005, 04 ｧ- ﾀ070 ｧﾀ05 ｧ 07 ｧ閨0903 ﾀ05 ｧ, 0703 縺003 ﾀ03 ﾀ06 ｧ罍ﾀ05 ｧ罍ｧ0060 閨06 ﾀ04, ｧ , =0; ｦ縺ｧｧ 06 ﾀ縺03009 ｧﾀ H =40 ﾀ0, 陦ｧ06 ﾀ030 0 ｧﾀｧ07 r =0; 3 ﾀ0 ｧ 0406 縺 R =6 ﾀ0 ( ﾀｧ )? 縺06 閨0600 ｧ0504 ﾀｧｧ閨縺05 ｧ縺閨ﾀ h, 03 ﾀｧ縺0308 a

165 縺06 閨ｧ縺08003 ﾀ05 ｧ 0403 閨03, 04 縺0305 ｧ縺00 ｧ陦ｧ05 ﾀ縺08003 ﾀ05 ｧ 0403 閨縺06 閨0600 ｧ0504 ﾀｧｧ閨縺05 ｧｧ 06 ﾀ09 ｧ08 ﾀ , 03 ﾀｧ縺ｧﾀ縺08003 ﾀ05 ｧ 0403 閨03, 0406 縺00 ｧ陦ｧ05 ﾀｧｧｧ006 x, 縺縺ﾀ h 閨0600 ｧ05 ﾀ罍ｧｧ06 閨0406 ﾀ05 ｧ, ｧｧ0 ﾀｧｧ縺陦ｧｨC ｧ00 ｧ0 陦罍ｧ 06 ﾀｧﾀ0506 縺0 ﾀ 0503 罍ｧﾀｧﾀ00030 ( ﾀ0504 0,9) 縺06 閨0600 ｧ0504 ﾀｧｧ07 0 ﾀ閨0606 ｧﾀ09 ｧ08 ﾀｧ00 ｧ0 陦, 06 ﾀ00 ｧ陦ｧ06 ﾀ縺06 閨0600 ｧｧ006 縺09 ｧｧﾀｧ Ox 陦ｧ縺05 ｧ 09 縺0605 x = a cos t, y = a sin t ｧ罍陦ｧﾀﾀ陦ｧ, ｧｧ0 ｧ070 ｧ y =4, x ｧ y = ｧﾀ00 ｧ , 陦ｧ00606 閨ｧ04 0 ｧ閨03 ｧ 07003, ｧ縺ﾀ ( ﾀ 0406 縺00 ｧ閨006), 0403 ﾀ縺 H =0, 陦ｧ06 ﾀ 03 ﾀｧ07 R =0; 縺06 閨0600 ｧ陦ｧｧ 06 ﾀﾀ07 05 ｧ閨 V 0 = 0; 0 3 ｧﾀｧ060 0 = ; 閨 V =0; ( 罍陦ｧｧ 06 ﾀﾀ07 05 ｧ06 04 縺03 ｧﾀ0803 陦ｧﾀ- ﾀ030 : V = 0 V 0,05 閨06 ; 4.) ｦ縺 , 縺陦ｧ06 ﾀﾀ0, ｧ罍ﾀ05 ｧ07 0 閨閨0406 ﾀ0? ( ｮｧ ; ｧｧ09 ｧﾀ05 ｧ =0; 8.) x ﾀ0303 ﾀｧ縺06 閨ｧ07: + b I. 0ｷ f(x) ﾀ07 lim f(x), 06 ﾀ00 ｧ b!+ a a 04 縺06 閨0600 ﾀ06006 ﾀｧ縺ｧ縺06 閨ﾀ07 ｧ b + f(x) ｧ f(x).,, II. ｭﾀ00 ｧ f(x) 縺06 縺03040 閨ﾀｧ08 x 0305 縺06 09 [a; b], 09 縺ｧ c, 縺0308 f(x) ｧ縺0304 II 縺03, 0503 ｧ f(x) 縺06 閨 a 閨03 b 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 閨ﾀ06006 ﾀｧ

166 3x ﾀ0303 ﾀｧｷﾀ 0306 ﾀｧ 04 縺06 閨ｧｧｧ縺 ( ｧ ) ｧﾀﾀ0303 ﾀｧ. ｭﾀ00 ｧ 04 罍ｧ0406 閨縺06 閨 , 縺0705, ﾀ0303 ﾀｧﾀ0803 閨0705 ﾀ07, 06 ﾀ00 ｧ 00605, 0503 ﾀ0803 閨0705 ﾀ07.. ｮ0803 陦ｧ003 ﾀﾀ0303 ﾀｧﾀﾀｧﾀ閨030 ﾀｧ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). 縺ｧ罍ｧ 0 09 ﾀ0803 陦ｧ003 ﾀ05 ｧｧ閨0007 ｧ縺ｧｧｧﾀ00 ｧ0504 ｧ : 748. ) 6 x ; ) x ; 3) x ; 4) x n ) e,x ; ) xe,x ; 3) +x ; 0 0 4) x x, ; 5) x + x ; 6) x e,x= : ) x x, ; ) arctg x x ; 3) (x +) : 75. ) (4, x) ; ) 8 0 (x, ) ; 3) 75. 0ｷ3 ﾀﾀ0006 閨ﾀ0803 陦ｧ003 ﾀ0504 ｧ : 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

167 縺06 閨ｧ ) 0 b x n ; ) a (04 罍ｧ b>a). (b, x) n ｰ09 0 ｧ06. 7 ﾀﾀ00305 縺罍ｧﾀ : n =, <, n = ｧ n =+> ｧﾀ00 ｧ閨04, 閨 y = = ｧﾀｧ罍ｧ x ｧﾀ00 ｧ閨04, 閨06 09 罍ｧ y = = xe,x = ｧ 0606 ﾀｧ (04 罍ｧ x>0) ｧﾀ00 ｧ閨04, 閨06 09 ｧﾀﾀ03 ｧ閨0308 y = = x3 ｧ 0606 ﾀｧ a, x ｰ09 0 ｧｧ04 x = a sin t, 0406 縺ｧ罍ｧ 06 ﾀ09 ｧｧｧ , ｧｧﾀﾀ03 ｧ閨03 x3 y = 縺ﾀｧ ( ﾀ ). a, x 縺06 閨0600 ｧ閨縺08003 ﾀ05 ｧ, ｧ縺ﾀｧOx 0306 ﾀ閨0605 ｧ 09 罍ｧ y = e,x 04 罍ｧｧ x ｧ , ｧ縺ﾀｧ Ox 0306 ﾀｧ 09 罍ｧ y = x, x 04 罍ｧ x > , 罍ｧ m ｧｧ ): ) e,x x m = m!; ) e,x x m+ = m! ｧﾀ00 ｧ0504 ｧ : ) x ; ) 0 ) ｱｧ07 x e,x3 ; 3) e ln x x ; 4) x ln x. e,x x t, =,(t) ﾀｧ t. 7 罍ｧ t>, ﾀ0006 閨ｧｧ 760, ),,(t) =(t, )! 閨06 ﾀ04 t =, ｧ0 06 ﾀ ! =,() = 0 e,x x 0 = 罍ｧﾀｧ ! =.

168 3x 8. ｮ縺06 閨ｧｧｧ 75 ｰ09 0 ｧ罍ｧ006 縺06 3) 04 罍ｧ閨060 ｧｧ lim ｧ0003 6ｧ80304 ｧ x! ln x x 04 罍ｧ0060 ｧ ) x +x ; ) ( + x) ; 3) 3 x + x ｧﾀ00 ｧ閨04, 閨06 09 罍ｧ y = = e,x ｧ 03 ﾀ070 ｧ罍陦ｧ0 05 (04 罍ｧ x>0) ｧ , ｧ縺ﾀｧ Oy 陦ｧ 0306 ﾀ閨00 ｧ003, 閨06 00 ｧ0 ｧ070 ｧ: xy =4; y =; x =0: 縺06 閨0600 ｧ , ｧ罍ｧ y = xe,x= (04 罍ｧ x>0) 縺ﾀｧ x 8. ｮ縺06 閨ｧｧｧｯ0603 縺ﾀ縺06 閨0060. ｭﾀ00 ｧ縺 [a; b] ｧ07 f(x) 縺06- b 縺03040, 閨06 04 縺06 閨ｧｧ f(x) 0 08 閨0605 ﾀ x = c, a 04 罍ｧ縺030 b f(x) =(b, a)f(c): () a 6ｦｧｧｧ R b f(x) a y m = f(c) = () b, a ﾀ07 ﾀ縺06 閨0 ｧｧｧｧ f(x) 縺 [a; b] 縺06 閨0600 ｧ0504 ﾀ縺06 閨ｧｧｧ: ) y = sin x 縺 [0; ]; ) y = tg x 縺 [0; =3]; 3) y =lnx 縺 [; e]; 4) y = x 縺 [a; b]; 5) y = 縺 [,; ]. +x ｰ縺ﾀ縺06 閨ｧｧｧ閨罍ｧ006 縺06.

169 縺06 閨ｧ x 9. ｱ03 縺ｧ08 ｧ 0703 縺00600 ｮｧ004 ﾀ030. ｱ03 縺ｧ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 ﾀｧ06 03 罍陦ｧ罍ｧ y = f(x) 縺 [a; b] 縺ﾀ縺 (I): (b, a)h "(h) 6 jy 00 j max : () ｧ 06 ﾀ縺00600 ｮｧ004 ﾀ030 閨0007 閨ﾀ: 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 ﾀ: b a f(x) h 3 " y 0 + y n +4 nx i= 05 閨06 h =(b, a)=n 縺ﾀ縺00600 (II) ｧ (III): (II) # n, X y i, + y i ; (III) i= (b, a)h4 "(h) 6 jy IV j max ; () 縺00600 (II) ﾀ閨縺0308 ｧ 05 縺ﾀ : y = a + bx + cx ｧﾀ00 ｧ縺ｧ08 ln = 縺ﾀ縺 (). ｧ縺ｮｧ004 ﾀ030 (III) 0403 ｧﾀ00 ｧ0504 ｧｧ0504 ﾀｧｧ070 ｧｧ x ｧｧ x 3 x 4, ｧ縺ﾀ縺 () ｧ縺ﾀ 04-

170 3x 9. ｱ03 縺ｧ08 ｧ 0703 縺00600 ｮｧ004 ﾀ縺ｮｧ004 ﾀ030 (III) 0403 ｧﾀ00 ｧ0504 ｧ : ) 0 4 +x 3 (n = 4); ) = 0 3, cos x (n =6); 3) +x 4 (n =4) ｧｧ縺ﾀ0504, () 04 罍ｧ0300 ｧ h 4 jy IV j max j 4 yj max ｧ縺ｮｧ004 ﾀ030 (II) ｧ 0403 ﾀﾀ0 ﾀ陦ｧ縺閨閨0 0 ﾀ0 ｧﾀ陦ｧ縺030 ﾀ縺06 閨ﾀｧ07 30 ﾀﾀ05 ｧ 0703 縺ｧ 0 ｧ閨03 ｧ 00 ｧ 0703 縺ｮｧ004 ﾀ030 (II) ｧﾀ00 ｧ0504 ln = 縺ｮｧ004 ﾀ030 (III) x (04 罍ｧ n =0) ｧｧ縺ﾀ縺 () ｧ閨00 ｧ006 閨0605 ｧｧ04 ﾀ x = 5 cos t, y = 3 sin t, 04 罍ｧｧ04 09 ｧ , 0304 縺06 閨縺縺ﾀ0608 閨0605 ｧ, 0703 縺ｮｧ004 ﾀ030 (II) ｧﾀ00 ｧ罍ｧ0300 ｧ =6 4, x, 04 罍ｧ0060 ｧ04 09 ｧ縺ｮｧ004 ﾀ030 (II) ｧﾀ00 ｧ = 縺ｮｧ004 ﾀ030 +x 0 (III) (04 罍ｧ n =0) ｧｧ縺ﾀ0504, 縺 () 04 罍ｧ0300 ｧ h 4 jy IV j max j 4 yj max ﾀﾀ0 05 罍ｧ閨04 ﾀ05 ｧ 09 縺0605, ｧ罍ｧ x + y =3, , , x =4 +8; ｧ, ｧﾀｧ縺ｮｧ004 ﾀ030 (04 罍ｧ n =4) ｧﾀ00 ｧ縺ｮｧ004 ﾀ030 (III) 閨00 ｧ006 閨0605 ｧﾀｧ006 ﾀ03 ｧ閨03 y = sin x, 03 ｧ縺 [0; ] ﾀﾀ

171 3. 07 罍ｧ04 ｧ 0 : ｷ36 50ｷ36ｦ0 7 76ｧ805 ｮ0705 0ｷｮｯ 7 繖 7 ｮｯ6 5 ｭｷ x. 07 罍ｧ04 ｧﾀ罍ｧｨC06005 ｧ陦ｧ06 ﾀ 09 罍ｧ04 ｧ陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003: k = d' ds = y 00 : () ( + y 0 ) 3= R = ( + y0 ) 3= jy 00 j 罍陦ｧ罍ｧ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) 罍ｧ 06 ﾀﾀ縺罍ｧ04 ｧ 003 C(X; Y ) ﾀｰｧ07 (3) ｧ 0306 閨罍ｧ 06 ﾀ09 ｧ0 ｧｧ070 ｧ陦ｧ06 ﾀ 09 罍ｧ04 ｧ罍ｧ r = f('), 05 閨06 r ｧ ' 縺罍陦ｧ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 ｧ罍ｧｧ 09 縺罍ｧ- 04 ｧ罍ｧ縺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 ｧ罍陦ｧ罍ｧ04 ｧ 003 ｧ 0403 ﾀ05 縺03 ｧ罍ｧｧ09 縺罍ｧ04 ｧ罍ｧ040308: 783. xy = x =.

172 3803. xy =4: 804. x = a cos 3 t; y = a sin 3 t: x. 07 罍ｧ04 ｧﾀ罍ｧ y =lnx 縺06 ﾀｧ07 ﾀ Ox y =, x 縺06 ﾀｧ07 ﾀ Ox ｧﾀｧ罍ｧｧ 0403 ﾀ05 縺03 ｧ罍ｧｧ : 786. y =, x : 787. x = cos t, y = sin t x, y = a ( ｧ00 ｧ x = a ch t ｧ y = a sh t) x = a(cos t + t sin t), y = a(sin t, t cos t) ｧ 0 09 ﾀｧ罍ｧ04 ｧ罍ｧ y = e x , 0503 陦ｧ06 ﾀ 09 罍ｧ04 ｧｧ0 ｧｧ y = a ch x a y 03 ﾀ0406 Ox. a ｧ縺縺0 00 ｧ閨06 09 罍ｧｧ縺06 閨0600 ｧ0504 陦ｧ06 ﾀ 09 罍ｧ04 ｧ縺03 ｧ罍ｧ : ) r = a(, cos '); ) r = a cos '; 3) r = a cos ' 縺06 閨0600 ｧ0504 陦ｧ06 ﾀ 09 罍ｧ04 ｧ 003 ｧ 0403 ﾀ05 縺03 ｧ罍ｧｧ 09 縺罍ｧ- 04 ｧ罍ｧ縺00 ｧ006: 793. y = +x : 794. x, y =4: 795. y = sin x: 796. y = x +4x: 0504 縺06 閨0600 ｧ罍陦ｧ罍ｧ04 ｧ 003 ｧ 0403 ﾀ05 縺03 ｧ罍ｧｧ09 縺罍ｧ04 ｧ罍ｧ040308: 797. y = e x 縺06 ﾀｧﾀ Oy y = x (,;,=3) y = x (; ) y = cos x x = ｧﾀｧ罍ｧｧ 0403 ﾀ05 縺03 ｧ罍ｧｧ : 80. y =(x +): 80. x = t ; y = t3 3 :

173 罍ｧ04 ｧﾀｧ 04 縺03 ﾀ05 0 ﾀ罍ｧ , ﾀ05 縺03 ｧ閨03 x =3 + y =3 = a =3 陦ｧ06 ﾀ 09 罍ｧ04 ｧ ajxyj. x ｧ0 閨0605 ｧ 09 罍ｧ縺03 ﾀ05 0 ﾀｧ縺06009 ｧ 00 閨0605 ｧ: ds = + dy + dz, ｧ00 ｧ ds = _x +_y +_z dt. t ｧ0 閨0605 ｧ: s = _x +_y +_z dt. t ｧ閨00 ｧ006 閨0605 ｧ 09 罍ｧ040308: 806. x = t, y = t, z = t t =0 閨03 t = x = 3 cos t, y = 3 sin t, z =4t 0305 t =0 閨03 04 縺03 ｧ t y = x, z = x x =0 閨03 x = 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 閨ｧｧ 0403 ﾀ縺06 ｧｧ 06 ﾀｧ罍ｧ 06 ﾀｧ06. ｭﾀ0506 ﾀ縺ｧ09 09 罍ｧ陦ｧ06 ﾀ r = xi + yj + zk ｧ 09 罍ｧ x = x(t), y = y(t), z = = z(t) 06 ﾀｧ07 ﾀ t. 7 縺03 ｧ 0403 閨0 07 _r = _xi+ _yj+ _zk 06 ﾀｧｧｧ閨 j_rj = _x +_y +_z = _s = ds dt , 06 ﾀ00 ｧ t 04 縺06007, 09 罍ｧ罍ｧ07 閨04 ｧ07060 ｧ07, 0503 _r = v 06 ﾀﾀ0903 縺03 ﾀ05 ｧ, r = w ﾀ0903 縺060 ｧ07. ｪ06 縺 M (x; y; z) 09 罍ｧ ( 罍ｧﾀ. 34) 04 縺閨罍ｧﾀ0903 ﾀ05 ｧ: ) 0406 縺04060 陦ｧ縺 _r; ﾀ縺 ; ) ﾀ03 閨06 縺 _r ｧ r; ﾀ07 ﾀ0304 罍ｧ09 ﾀﾀ07; 3) 0406 縺04060 陦ｧ縺縺04030 閨ｧﾀ0506 ﾀ縺ｧ09 (05 罍ｧ05 陦) 09 罍ｧ

174 3x 3. 7 縺03 ｧ 0403 閨ｧｧ 0403 ﾀ縺縺06 ﾀｧｧﾀ0903 ﾀｧ罍ｧ 04 縺 : 09 ﾀ ; 03 ｧｧ縺0 0004, 0304 縺06 閨ｧ: ) _r ｧ , ) B = _r r 03 ｧ003 縺 , 3) N = B _r 縺ｭ陦ｧ0 ｧ縺ｧｧｧ0,, ; 030 ｧﾀｧﾀｧ003 ﾀ d ds = d ds ｧ =. 706 ﾀ0504 M (X; Y ; ) ﾀ ( 罍ｧﾀ. 34). ｯ0305,,,! MM k_r ｧｧ 06 ﾀｧﾀ05 ｧ縺ｧｧ07 09 ﾀ X, x _x = Y, y _y =, z : (I) _z 706 ﾀ0504 M (X; Y ; ) 縺ﾀ0903 ﾀ05 ｧ. ｯ0305,,,! MM? _r ｧｧ 06 ﾀｧ縺04060 陦ｧ縺003 ﾀ05 ｧ縺ｧｧ縺ﾀ0903 ﾀ05 ｧ: _x(x, x) + _y(y, y) + +_z(, z) =0: (II) ｰｧ07 03 ｧ003 縺0 00 ｧｧ縺0 00 ｧｧ0, 0060 ｧｧﾀｧ0708 (I) _x, _y, _z ﾀﾀ B x, B y, B z ｧ00 ｧ 0 N x, N y, N z. ｰｧ06 ﾀ0304 罍ｧ09 ﾀﾀﾀ0903 ﾀ05 ｧｧ0, 0060 ｧｧｧ (II) _x, _y, _z 0 B x, B y, B z 陦ｧ06 ﾀ閨04 ｧﾀｧ t ｧ060 r =4ti, 3tj 縺06 閨0600 ｧ罍ｧ06, ﾀ0903 縺03 ﾀ0504 ｧ 06 ﾀ0903 縺06-0 ｧ06 閨04 ｧ07060 ｧｰｧ06 閨04 ｧ07060 ｧ07 r =3ti +(4t, t )j 縺06 閨0600 ｧ罍ｧ06 ｧﾀ0903 縺03 ﾀ0504 閨04 ｧ07060 ｧﾀ05 縺03 ｧ罍ｧ06 ｧ縺03 ﾀ0903 縺03 ﾀ05 ｧ t =0; ; ｧ3 ﾀ縺06 閨0600 ｧﾀ0903 縺060 ｧ06 w 閨04 ｧ07060 ｧ07 ｧｧ w = dv ｧ 003 縺 dt w n = w, w ﾀ03 ﾀｧ t ｧ 04 罍ｧ t = ｰｧ06 閨04 ｧ07060 ｧ07 r = a cos t i + b sin t j 縺06 閨0600 ｧ罍ｧ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 ｧ縺03 ﾀ0903 縺03 ﾀ05 ｧｧ06 ﾀ0903 縺060 ｧ t =0;=4; = { ｧﾀｧ07 09 ﾀ縺ｧ 003 縺ﾀ0903 ﾀ05 ｧ 09 罍ｧ040308: 86. x = t, y = t, z = t ｧ 04 罍ｧ t =.

175 罍ｧ04 ｧﾀｧ 04 縺03 ﾀ05 0 ﾀ罍ｧ y = x, z = x (x > 0) ｧ 04 罍ｧ x = x + y =0, y + z = (; 3; 4). ｰ09 0 ｧ陦ｧ縺06009 ｧｧﾀ閨ｧ07, ｧｧ07 : dy : dz ｧｧ _r, 03 ｧ003 縺 B ｧ縺 N 縺03 09 罍ｧ x =, sin t, y = cos t, z = t t = ｧ , ｧｧﾀｧ縺0 00 ｧ, 03 ｧ003 縺0 00 ｧｧﾀ03-04 罍ｧ09 ﾀﾀﾀ0903 ﾀ05 ｧ 0909 罍ｧ x = t, y = t, z = t t = ｧﾀｧ縺0 00 ｧｧ 03 ｧ003 縺0 00 ｧ 09 罍ｧ x = e t, y = e,t, z = t t = , ｧ07 x = t cos t, y = t sin t, z = t 0304 縺06 閨ｧ 06 ﾀｧｧ0 ｧ06, ｧ 0 04 ｧﾀｧ縺0 00 ｧ, 03 ｧ003 縺0 00 ｧｧ 09 ﾀ陦ｧｧﾀｧ07 09 ﾀｧｧ0 ｧｧ x = = a cos t, y = a sin t, z = bt ｧ 04 罍ｧ t = = , ｧｧ0 ｧ縺06 ﾀｧ06 09 ｧ00 ｧ0 陦 x +y = a 陦ｧ = arccos b a + b ｧﾀ 03 ﾀ070 ｧ罍陦ｧｧ罍ｧ x =az ｧ y =bz z = ab ﾀ0903 ﾀ0504 y =0, 縺罍ｧ04 07 z = x, y =0, 0 09 縺06 ｧﾀｧ00 ｧ0 陦 x + y =y ｧﾀ罍ｧ 06- ﾀ09 ｧｧ罍ｧｧ005 ｧ 0304 縺06 閨0600 ｧｧ罍ｧｧ t = =, 05 閨06 t 縺ﾀ0903 ﾀ05 ｧ陦ｧ06 ﾀ閨04 ｧﾀｧ t ｧ060 r = a(t, sin t)i + a(, cos t)j 縺06 閨0600 ｧ0504 ｧ 0403 ﾀ05 縺03 ｧ縺03 ﾀ0903 縺03 ﾀ05 ｧｧ06 ﾀ0903 縺060 ｧ07 04 罍ｧ t = = ｧ t = { ｧﾀｧ07 09 ﾀ罍ｧ : 87. y = x, z =x x =. 88. x + y + z =4, x +y, z = (; ; 3) ( ﾀ ). 89. x =t, y =lnt, z = t t =.

176 3x 罍ｧ04 ｧ 0 ｧ 09 縺ｧ06 04 縺03 ﾀ05 0 ﾀ罍ｧ r = e t i + e,t j + t k ｧﾀ 03 ﾀ070 ｧ罍陦ｧｧ003 縺 b t = ｧﾀｧ縺0 00 ｧｧ 03 ｧ003 縺0 00 ｧ 09 罍ｧ y = x, z = y x = ｧﾀｧ縺0 00 ｧｧ 03 ｧ003 縺0 00 ｧ 09 罍ｧ x = t, sin t, y =, cos t, z = 4 sin t t =. x 罍ｧ04 ｧ 0 ｧ 09 縺ｧ06 04 縺03 ﾀ05 0 ﾀ罍ｧ罍ｧ04 ｧ 0 =R 06 ﾀ縺06 閨ｧ ' 縺ﾀ閨00 ｧ006 閨0605 ｧ s, s! 縺ｧ06 = 06 ﾀ縺06 閨ｧ縺ｧ003 縺0 00 ｧ 09 s, s! 0. ｯ ' jj ｧ jj, 0503=R ｧ = ｧﾀﾀ閨ｧ縺0304: d ds = R ; d ds =, : () ｭﾀ00 ｧ 09 罍ｧｧ060 r = r(t), 0503 j_r rj = R j_rj ; 3 = _r r... r j_r rj : () 縺03 陦ｧ縺06009 ｧ縺ﾀ v = v 0403 t, ﾀ縺縺 () ｧｧ06 06 ﾀ0903 縺060 ｧ07 w ｧｧ 003 縺 : w = v + v R : 834. ｯ03 09 閨04 ｧﾀ x = t, y = t, t,05 閨06 t 04 縺06007 閨04 ｧ07060 ｧ縺06 閨0600 ｧ罍ｧ04 ｧ 006 =R 罍ｧｧｧｧｧ 003 縺ﾀ0903 縺060 ｧ t ｧ 04 罍ｧ t = ｯ03 09 閨04 ｧﾀｧ04 ﾀ06 x = 4 cos t, y = 3 sin t, 05 閨06 t 04 縺06007 閨04 ｧ07060 ｧ縺06 閨0600 ｧ罍ｧ04 ｧ 006 =R 罍ｧｧｧｧｧ 003 縺ﾀ0903 縺060 ｧ07 04 罍ｧ t = 閨04 ｧ07060 ｧ07 ﾀｧ060 r = ti + t j + 3 t3 k 0304 縺06- 閨0600 ｧ罍ｧ04 ｧ 006 =R 罍ｧｧｧｧｧ 003 縺ﾀ0903 縺060 ｧ t ｧ 04 罍ｧ t = 縺06 閨0600 ｧ罍ｧ04 ｧ 006 =R ｧ 09 縺ｧ06 = 09 罍ｧ040308: 837. x = t, y = t, z = t ｧ 04 罍ｧ t = x = e t, y = e,t, z = t ｧ 04 罍ｧ t = y = x, z = x ｧ 04 罍ｧ x =.

177 罍ｧ04 ｧﾀｧ 04 縺03 ﾀ05 0 ﾀ罍ｧ , ｧ00506 (x = a cos t, y = a sin t, z = bt) 09 縺ｧｧ , (x = a cos t, y = =,a sin t, z = bt) 0305 罍ｧ縺06 閨0600 ｧ罍ｧ04 ｧ 006 =R ｧ 09 縺ｧ06 = 09 罍ｧ040308: 84. x =t, y =lnt, z = t ｧ 04 罍ｧ t =. 84. x = y, z = x ｧ 04 罍ｧ y = x = e t sin t, y = e t cos t, z = e t t =0.

178 ｪ罍ｮｯ 76ｧ9 ｭｷ36ｦｧ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ｧｭ 70ｷ300 x. ｱｧｧ閨縺ｧｧ罍ｧ 06 ﾀｧｧ縺06 閨ｧ縺 z ﾀ07 03 閨ｧ縺 x ｧ y, 06 ﾀ00 ｧ閨縺ｧ08 x ｧ y 縺ﾀ05 ｧｧ08 ｧｧﾀﾀﾀ0504 ｧ06 03 閨ｧ06 z. ｱｧｧﾀｧ003 ﾀ0504 z 0305 x ｧ y 04 ｧﾀｧ閨06 z = F (x; y): () 罍ｧ 06 ﾀｧｧ06. ｰｧ06 () 罍ｧ- 06 ﾀ09 ｧ 0304 縺06 閨縺縺08003 ﾀｧ08 x ｧ y 0304 縺06 閨ﾀ0903 ﾀ05 ｧ xoy P (x; y), z = F (x; y) ｧﾀﾀｧ M (x; y; z) 縺08003 ﾀ05 ｧ縺0705, 0503 z 06 ﾀｧｧ P (x; y), ｧ 04 ｧ z = F (P ) 縺06 閨ｧｧ lim F (P )=A, 06 ﾀ00 ｧ 003 ﾀ0504 F (P ), A P!P0 06 ﾀﾀ , = P 0 P! 0 04 罍ｧﾀ0403 ﾀ罍ｧ0300 ｧｧ07 P 09 P 0 (0 04 罍ｧ006, ｧ0 ｧｧ) 縺06 縺ﾀｧｧ. ｱｧ07 F (x; y) ﾀ縺06 縺 P 0, 06 ﾀ00 ｧ lim P!P0 F (P )=F (P 0 ). 0ｷ縺07, ｧ07 F (x; y) 縺06 縺縺 (x; y), 06 ﾀ00 ｧ lim x!0 y!0 F (x +x; y +y) =F (x; y): 844. ｰﾀ05 ｧｧｧ08 x ｧ y, 閨縺0308 ﾀ0006 閨ｧｧｧｧﾀｧ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 ｧ罍ｧ 06 ﾀ09 ｧ06 ｧｧｧﾀｧ縺08003 ﾀﾀ0903 ﾀ05070 ｧ x =0, y =0, z =0 ｧ z = h.

179 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧ罍ｧ縺ｧ縺06 閨0600 ｧ閨04 S 05 縺ｧｧ06 閨ﾀ0503 縺030 x ｧ y 縺06 閨0600 ｧ0504 ｧ 0403 ﾀ05 縺03 ｧﾀｧ08 x ｧ y 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) = , ﾀ00 ｧ F (x; y)= = t F (x; y) z = x, xy = y 0304 縺06 閨0600 ｧ0504 x z, y z ｧ z ｧﾀ00 ｧ0504 x z, y z, z, 06 ﾀ00 ｧ x ｧﾀ閨03,, y ｧﾀ閨03, , ｧ06 x, y, z = 縺06 閨 z ﾀｧﾀﾀ閨ｧ08 x ｧ y, ｧ縺0308 閨縺06 縺ｰﾀ縺06 閨ｧ07 04 ﾀｧｧ08 ｧ 0403 ﾀ05 縺03 ｧ罍ｧ 06 ﾀｧｧｧ縺06 縺ｧｧ. 7 罍ｧ0406 ﾀ05 ｧ 04 罍ｧ閨 , 縺ｧｧ z = F (x; y), 0304 縺06 閨ｧ060 x, y = z ﾀ05 縺03 ｧｧ0 ｧｧ 06 縺 (04 罍ｧ z =0; ; ｧ 05..) ｧ08: ) z = r, x 4, y ; ) z = x, y; 3) z = x, y ; 4) z = xy: , 罍ｧ x! 0 ｧ y! ｧ06 u = y x, y ﾀ05 縺060 ｧ0504 ﾀ縺06 閨罍ｧ0406 ﾀ05 ｧ 04 罍ｧ006 縺罍ｧ0300 ｧ07060 ｧｧ (x; y) (0; 0), 04 罍ｧ縺030 lim u =3, lim u =, lim u =, lim u =0, lim u =,. ｰ09 0 ｧ06. 7 ﾀﾀ00305 縺ｧｧ06 x ｧ y 04 閨縺 y = kx , 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 罍ｧﾀ0403 ﾀ罍ｧ0300 ｧ07060 ｧｧ (x; y) (0; 0). ｰ09 0 ｧｧ0504 xy =.

180 3x. ｪﾀ縺03 ｧ 0403 閨縺縺07 閨ｷｧ罍ｧ 06 ﾀ09 ｧｧ06 z = F (x; y) = ｧｧ0 ｧｧ 0606 縺 < : 04 罍ｧ xy > 0; 0 04 罍ｧ xy =0;, 04 罍ｧ xy < ｰﾀ05 ｧ 0304 縺06 閨ｧｧ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 ｧ罍ｧ 06 ﾀ09 ｧ06 ｧｧｧｧ , ﾀ00 ｧ F (x; y)= x,0503f(a; b)+f (b; a)=. x, y , ｧ06 z 4 = 4, x, y 0304 縺06 閨 z ﾀｧﾀﾀ閨ｧ08 x ｧ y, ｧ縺0308 閨縺06 縺ｰﾀ縺06 閨ｧ07 04 ﾀｧｧ08 ｧ 0403 ﾀ05 縺03 ｧ罍ｧ 06 ﾀｧｧｧｧ, ｧﾀ05 ｧ x + y 6 ｧ0305 罍ｧﾀ05 縺03 ｧ罍ｧ 06 ﾀｧｧ06 03 閨ｧｧ z = F (x; y), 0304 縺06 閨ｧ060 x + y + z = a, ｧﾀ05 ｧ x + y 6 a ｰｧ0 ｧ縺ｧ0305 罍ｧ x. ｪﾀ縺03 ｧ 0403 閨縺縺07 閨09 7 縺03 ｧ 0403 閨ｧｧ z = F (x; y) 0403 x, 0 08 閨縺06 閨ｧｧ, 0503 y 03 ﾀﾀﾀ , ﾀ07 ﾀ縺03 ｧ 0403 閨00308 z 0403 x ｧｧ00 ｧ F 0 (x; y). 縺ｧ縺06 閨ﾀ07 ｧﾀ07 ﾀ縺03 ｧ 0403 閨0 07 z 0403 = F 0 (x; y). y ｧﾀ縺03 ｧ 0403 閨ｧ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

181 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧ c = a + b, ab cos u = y x + z y, x z : 866. u = xe,yx : 867. u = x, t x +t : 868. = arcsin (t x): , ﾀ00 ｧ, z =ln x + = : , ﾀ00 ｧ z = x sin y = z : , ﾀ00 ｧ u = e =0: , ﾀ00 ｧ u = x y,0503 ln =u: ｧ0706, , 0306 閨0605 閨ﾀ0006 閨縺 : ｭﾀ00 ｧ z = F (x; y) 03 閨003 縺03 閨ｧ07 ﾀｧ n, = nz. 7 縺罍ｧ縺閨ｧ08: ) z = x 3 + xy, y 3 ; ) z = x + xy + y ; 3) z = x 3, y 3 ; 4) z = ex=y : ｧﾀ縺03 ｧ 0403 閨ｧ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.

182 3x 陦ｧ縺06009 ｧ縺縺07 閨 , ﾀ00 ｧ u = x + y @u + @z , ﾀ00 ｧ z = e x=y ln y, = z ln y : r , ﾀ00 ｧ T =,0503 l g , ﾀ00 ｧ z = e x= sin ex sin y =0., 縺罍ｧ縺閨003 縺03 閨ｧ0708 ( ﾀ ) 閨ｧ08: ) z = x3 x, y ; ) z = x + y ; 3) z = arctg y x. x 陦ｧ縺06009 ｧ縺縺07 閨09 ｭﾀ00 ｧｧ07 z = F (x; y) ｧ (x; y) 縺06 縺ﾀ縺03 ｧ 0403 閨00306, 罍ｧ 0060 ｧ縺06 陦ﾀｧ閨06 x + y 閨06 "! 0 04 罍ｧ = jxj + jyj! 0. ｯ @y y 06 ﾀﾀ罍ｧ 0060 ｧ07 z; ﾀ陦ｧ縺06009 ｧｧｧｧﾀ07 dz: x 縺 () z : ) x; )y, 0 08 閨060: =x, dy =y

183 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧｷ3 () ﾀ0006 閨060605, 0503 z dz; (4) 罍ｧ閨03 ﾀ x ｧ y 罍ｧ 0060 ｧｧｧ 04 罍ｧｧ陦ｧ縺06009 ｧ 0006 ( , x 7). ｱｧ07 F (x; y) ﾀ07 陦ｧ縺06009 ｧ縺 (x; y), 06 ﾀ00 ｧ 030 ｧ陦ｧ縺06009 ｧｧ陦ｧ縺06009 ｧｧ08: ) z = x y; ) z = xy x, y ; 3) u = es=t ; 4) z = x + y ｧｧ陦ｧ縺06009 ｧｧｧ: ) z = y x 04 罍ｧ x =, y =, =0;, dy =0; ; ) u = e xy 04 罍ｧ x =, y =, =,0;, dy =0; ｧﾀ00 ｧ0504 dz ｧ z 閨ｧｧ z = xy 04 罍ｧ x = 5, y =4, x =0;, y =,0; 陦ﾀｧ罍ｧ0300 ｧｧｧｧｧ ' = arctg y x, x ｧﾀ閨03,, y 閨03, 罍ｧ閨縺0 09 ｧｧ 09 ｧ00 ｧ0 陦陦ｧ06 ﾀ R ｧｧ00 ﾀ07 ﾀ 0 ﾀ0 閨03 0,5 ﾀ0, 0403 ﾀ0305 H ｧ00 ﾀ04 ﾀ03 00 ﾀ0 閨03 98 ﾀｧ 04 罍ｧ0300 ｧｧｧ V 縺 V dv 縺縺ｧ09, ｧ 006 縺ﾀﾀ閨030; ﾀ0, ｧﾀｧ 7;5 ﾀ0 ｧ 8 ﾀ縺06 閨0600 ｧﾀ縺ﾀ罍ｧ 0403 ｧﾀ00060 ｧｧ 05 ｧｧ陦ｧ縺06009 ｧｧ08: ) z = y x, x y ; ) s = x ln t; 3) u = x + y + z ｧｧ06 dz ｧ z 閨ｧｧ z =ln(x + y ), x ｧﾀ閨03,, y 0305 閨03 0, 陦ﾀｧ罍ｧ0300 ｧｧｧｧｧ z = arcsin y x, x ｧﾀ閨03 4,5, y 閨03 3, 罍ｧ閨縺0 09 ｧｧﾀ陦ｧ06 ﾀ R ｧｧ00 ﾀ07 ﾀ 30 ﾀ0 閨03 30, ﾀ0, 0403 ﾀ0305 H ｧ00 ﾀ04 ﾀ 60 ﾀ0 閨03 59,5 ﾀｧ 04 罍ｧ0300 ｧｧｧ縺 V dv.

184 3x 4. 7 縺03 ｧ 0403 閨00306 ﾀｧ08 9 x 4. 7 縺03 ｧ 0403 閨00306 ﾀｧ08. ｭﾀ00 ｧ z = F (x; y), x = f(t), y = '(t), 0503z ﾀ07 ﾀｧ t. 7 罍ｧ dz dt dt ; () 06 ﾀ00 ｧｧｧ F, f ｧ ' 陦ｧ縺06009 ｧ縺ｭﾀ00 ｧ z = F (x; y), 05 閨06 x = f(u; v), y = '(u; v), ｧ 06 ﾀ00 ｧｧｧ F, f ｧ ' 陦ｧ縺06009 ｧ @x @x @v : () ｧ縺 () dz ｧｧ08: dt ) z = x + xy + y, x = t, y = t; ) z = x + y, x = sin t, y = cos t. 7 縺罍ｧ縺06 閨04 罍ｧ陦ﾀｧ08 x ｧ y ｧ06 閨ｧｧ z ｧ dz dt, 06 ﾀ00 ｧ z = y x, x =, et y =, e t ｧ dz, 06 ﾀ00 ｧ z =,05 閨06 uv u ｧ v ｧｧ 0305 x ｧ dz, 06 ﾀ00 ｧ z =,05 閨06 xey y ｧ x ｱｧ07 z = F (x; y) ﾀ07 03 閨003 縺03 閨00308, 06 ﾀ00 ｧ F (xt; yt) =t n F (x; y). 6 7 ｧ縺06009 ｧ縺ﾀ05 ｧﾀ t ｧ縺 t =, 閨縺閨003 縺03 閨 = nz x, 06 ﾀ00 ｧ z =,05 閨06 x = u, v, y = v y ﾀ0504 z = F (x; y). 6 @z @v, 06 ﾀ00 ｧ: ) u = mx + ny, v = x + qy; ) u = xy, v = y=x ﾀ0504 u = F (x; y), 05 閨06 x = r cos ', y = r sin '. 6 @u ｧ @u + = +

185 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧﾀ0504 z = y + F (u), 05 閨06 u = x, y , + x@z = x 閨陦ｧ縺06009 ｧ縺ｧｧ F ｧ dz ｧｧ08: dt ) z = Ax +Bxy + Cy, x = sin t, y = cos t; ) z = arctg y x, x = et +, y = e t, , ﾀ00 ｧ z = xy + xf (u), 05 閨06 u = y=x, = z + xy: , ﾀ00 ｧ z = y'(u), 05 閨06 u = x, = z y : ﾀ0504 z = F (x; y). 6 @z @v, 06 ﾀ00 ｧ: ) u = x +y, v = x, y; ) u = xy, v = x + y. x 5. 7 縺03 ｧ 0403 閨ｧ08. ｰｧ06 F (x; y) =0, ｧ縺ｧ06 (x 0 ; y 0 ), 0304 縺06 閨縺06 ﾀ05003 ﾀ05 ｧ x 縺 y 縺06 縺ｧ06 x 04 罍ｧ 06 ﾀｧｧ, 縺03 ｧ 0403 閨0 6=0 ｧ縺06 縺縺縺06 ﾀ05 ｧｧ (x 0 ; y 0 ). ｭﾀ00 ｧﾀ0406 縺 , 縺06 ﾀ05003 ﾀ05 ｧｧ (x 0 ; y 0 ) ﾀ06006 ﾀｧ縺06- 縺縺03 ｧ 0403 閨0 dy, ｧ07 ｧ縺03 ｧ縺06 閨縺 dy : (). ｰｧ06 F (x; y; z) =004 罍ｧｧﾀｧ縺06- 閨 z ｧ06 x ｧ y, ｧﾀ縺03 ｧ ; ｧ dy ｧｧ08: 907. x + y, 4x : ()

186 3x 5. 7 縺03 ｧ 0403 閨ｧ ) x =3 + y =3 = a =3 ; ) xe y, ye x = Ax +Bxy + Cy +Dx +Ey + F = ｧｧ09 ｧﾀ罍ｧ040308: 90. x + y =0y 縺06 ﾀｧﾀ 04 縺 x =3. 9. x 3 + y 3, axy = x = y = a ｧｧ, 縺ﾀ罍ｧ x + y + +x, y = : ) Ox; ) Oy. 7 @y ｧｧ08: 93. x + y + z, 6x =0: 94. z = xy. 95. cos (ax + by, cz) =k(ax + by, cz) , ﾀ00 ｧ xyz = a =,z: , 0503 陦ｧ縺06009 ｧ = z 06 閨縺ｧ07 z, 0304 縺06 閨ｧ060 ( ｧ 06 ﾀ09 ｧ縺08003 ﾀ050608) z=x = '(y=x) ｧ dy ｧｧ08: 98. x, 4y =4: 99. xy +lny +lnx =0: 90. x + y = e y=x : 9. cos (x, y) =y, x: ｧｧ09 ｧﾀ罍ｧ y,, xy = 縺06 ﾀｧﾀ 04 縺 x = ﾀ0504 x + y + z, zx = a. 7 @y. 94. sin (x +y, 3z) =x +y, 3z @y =: , 0503 陦ｧ縺06009 ｧ = 06 閨縺ｧ07 z, 0304 縺06 閨ｧ060 (09 ｧ- 00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ縺08003 ﾀ050608): x, mz = '(y, nz).

187 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧ 0003 x 6. ｪﾀ縺03 ｧ 0403 閨00306 ｧ陦ｧ縺06009 ｧﾀ00 ｧ縺07 閨ﾀｧ07 z = F (x; y), ｧﾀ縺03 ｧｪﾀ縺03 ｧ 0403 閨ｧ08 04 縺03 ｧ 0403 閨ﾀｧ 04 縺03 ｧ 0403 閨0030 ｧ縺縺07 閨ｧ ; F @y ; : 縺ｧ縺06 閨ﾀ07 ｧﾀ07 ﾀ縺03 ｧ 0403 閨縺縺07 閨09 ｧ陦縺0605 ｧﾀ00 ｧ縺07 閨ｮ縺03 ｧ 0403 閨00306, ｧ 060 ｧ06 ﾀ縺07 閨09030 陦ｧ縺06009 ｧ縺ｧ07, 04003, 06 ﾀ00 ｧ 030 ｧ縺06 F F @x ｧｧ0 ﾀ0006 閨ｧ縺03 ｧ 0403 閨ﾀ00 ｧ縺07 閨090304: 縺縺07 F F ; 05 縺縺07 F 3 @ 3 ｧ陦ｧ縺06009 ｧﾀ00 ｧ縺07 閨縺06 閨ﾀ : d z dy dy : ｮｧｧ 06 ﾀ09 ｧﾀｧﾀ d z + dy z: 縺ｧ 003 ｧ05.. d 3 dy z ｧﾀ縺03 ｧ 0403 閨縺縺07 閨ｧｧ z = x 3 + x y + y z 縺 z 閨0007 ) z = sin (ax, by); ) z = x =y ; 3) z =ln(x, y).

188 3x 6. ｪﾀ縺03 ｧ 0403 閨ﾀ00 ｧ縺07 閨ｧﾀ縺03 ｧ 0403 閨縺縺07 閨ｧｧ u = x 4 +3x y, y ｧﾀ縺03 ｧ 0403 閨縺縺07 閨ｧｧ u = y=x ﾀ0504 s =ln x, ; 04 縺罍ｧ0504, = x ｧﾀ縺03 ｧ 0403 閨縺縺07 閨ｧｧ z = arctg y x. x , ﾀ00 ｧ z = sin a, y,0503 z =, + a, b , ﾀ00 ｧ u = arctg (x, t), u =0: , ﾀ00 ｧ s = 3 ax + bt, + s =, 9 : , ｧ07 u = xe,y=x 06 閨縺陦ｧ縺06009 ｧｧ06 u @y = : , ﾀ00 ｧ z = F (x; y) 03 閨003 縺03 閨ｧ07 n-0503 ｧ 006 縺060 ｧ07, 0503 ｧ00 ｧﾀｧｧ 06 ﾀ09 ｧ z + z = n(n, x + z = n(n, ｰ09 0 ｧ = nz ( ﾀ ) 04 縺縺06009 ｧ縺 : ) 0403 x; ) 0403 y, ｧ縺 , ﾀﾀ x ｧ0 y, ﾀｧ

189 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧ縺罍ｧ z = n(n, )z 03 閨003 縺03 閨ｧ08: ) z = x + xy + y ; ) z = y x ; 3) z = = y x, y ; 4) z =ln x, ｧ d u, 06 ﾀ00 ｧ: ) u = y x ;) u = x ln y x , ﾀ00 ｧ z = cos (mx + ny), 0503 d z = =,z(m+ ndy) , ﾀ00 ｧ z =ln(ax + by), 0503: ) d 3 z =dz 3 ; ) d n z =(,) n, (n, )! dz n , ﾀ00 ｧ z = F (u; v), 05 閨06 u = mx + ny ｧ v = = @ @y 縺縺 u =3x + y ｧ v = x + y ( ﾀ ) 縺縺 u =x + y ｧ v = y ( ﾀ ) , ﾀ00 ｧ z = F (u; v), 05 閨06 u ｧ v ｧｧ 0305 x ｧ y, z = v0 z + u v00 : 0504 縺06 閨0600 ｧｧ縺ｧ06 z z y u = xy ｧ v = y=x ( ﾀ ) 縺 z 縺 x = r cos ' ｧ y = r sin ' ( ﾀ ).

190 3z = x 6. ｪﾀ縺03 ｧ 0403 閨ﾀ00 ｧ縺07 閨ｧﾀ縺03 ｧ 0403 閨縺縺07 閨ｧｧ x, y ｧﾀ縺03 ｧ 0403 閨縺縺07 閨ｧｧ u = x 3 t , ﾀ00 ｧ z = xy x, z = x, y : , ﾀ00 ｧ s =ln(ax, bt), + 3 s , ﾀ00 ｧ z = cos x, z =0:, , ﾀ00 ｧ z = e z @x ﾀ0504 u =lnx ｧ d u ｧ d 3 u 縺 z z a u = ax + y ｧ v = ax, y ( ﾀ ) 縺ｧ06 + z u = y ｧ v = y ( ﾀ ). x xf (x) y , ｧ07 u = + ' 04 罍ｧ y x 閨04 07 閨03 陦ｧ縺06009 ｧ縺ｧ0708 f ｧ ' 06 閨縺陦ｧ縺06009 ｧｧ06 u + +y =0:

191 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧ 0003 x 7. 0ｷ罍ｧ縺ｧ陦ｧ縺06009 ｧｪｧ06 P+ Qdy,05 閨06 P ｧ Q 陦ｧ縺06009 ｧ縺ｧｧ x ｧ y, 陦ｧ縺06009 ｧ du, 陦ｧ003 ｧ閨03 ﾀｧ @x 閨060 ｧ07 u ｧ 06 = P = Q ｧ0 u = = P+ ' (y), u R = Qdy+ ' (x) ｧﾀ 04 ｧ 0406 縺ｧ07 04 ﾀ06 ｧ 0406 ﾀ , ｧ縺ﾀ y, 006 閨03 ﾀｧ縺04030, ｧｧ06 u.. ｪｧ06 P+ Qdy + Rdz,05 閨06 P, Q ｧ R 陦ｧ縺06009 ｧ縺ｧｧ 0305 x, y ｧ z, 陦ｧ縺06009 ｧ du, 陦ｧ003 ｧ閨03 ﾀｧ @y : 閨060 ｧ07 u ｧ006060: u = P+' (y; z); u = Qdy+' (x; z); u = Rdz+' 3 (x; y): ｧﾀ 04 ｧ 0406 縺ｧ07 04 ﾀ06 ｧ 0406 ﾀ , ｧ縺ｧ 05 縺閨03 ﾀｧﾀ y ｧ z, ｧｧ06 u 閨060 ｧｧｧ陦ｧ縺06009 ｧﾀ07 ｧ罍ｧ縺ｧ陦ｧ縺06009 ｧ縺罍ｧ0504, 0503 ﾀ0006 閨ｧﾀ陦ｧ縺06009 ｧ du, ｧｧ u: 957. (x + y) +(x, y, 3) dy x sin y+ x cos ydy (x +lny) 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 縺罍ｧ0504, 0503 ﾀ0006 閨ｧﾀ陦ｧ縺06009 ｧ du, ｧｧ u: 963. (y, ) +(xy +3y) dy (sin y, y tg x) +(x cos y + ln cos x +y) dy.

192 3x ﾀｧﾀ罍ｧ sin y x + + dy. x 965. y, sin y x + r x t t dt (ln y, cos z) , 3 dy z + 3y, z z t +. x x y + z dy +(y +x sin z) dz. dz. x ﾀｧﾀ罍ｧｯ罍ｧ F (x; y) = ﾀ07 03 ﾀ , 06 ﾀ00 ｧ =0. ｰｧ09 ｧ06005 k = y 0 09 ﾀ陦ｧ05 ﾀ07 ｧｧ07 A +Bk + Ck =0,05 閨06 A, B ｧ C ｧ07 04 縺03 ｧ 0403 F F ﾀ罍ｧ罍ｧﾀ0006 ) B, AC > 0 閨ﾀ ; ﾀ ; ) B, AC < ﾀ ; ｧ 0300 ｧ縺 ; 3) B, AC =0 ｧ00 ｧｧ 0300 ｧ縺 , ｧ00 ｧ , ｧ00 ｧﾀ 003 ﾀ0304 罍ｧ0903 ﾀｧ07; ｧﾀ 003 ﾀ0304 罍ｧ0903 ﾀｧ07 ﾀ06006 ﾀ閨ﾀ閨罍ｧｪ縺 , ﾀ0300 ｧ , ﾀ縺0600 ｧ縺03 ﾀ , , ｧﾀ07 00 ｧｧ 09 罍ｧﾀ閨縺06 ﾀﾀ05 ｧｧﾀﾀ0006 閨ｧ縺06 閨0600 ｧﾀ05 ｧﾀｧ07, ｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ罍陦ｧ0 05, 03 ﾀｧ 09 罍ｧｧ 0403 ﾀ05 縺03 ｧ罍ｧ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 ｧﾀ05 ｧﾀｧ07, 03 ﾀｧｧﾀｧ罍ｧｧ 0403 ﾀ05 縺03 ｧ罍ｧ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 :

193 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧ縺06 閨0600 ｧﾀ05 ｧﾀｧ07, ｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ罍陦ｧ0 05, 03 ﾀｧ 09 罍ｧｧ 0403 ﾀ05 縺03 ｧ罍ｧ040306: 979. y + x 3, x =0: 980. a y = x (ax, x ): 98. y = x(x +) : 98. xy =(x + a) 3 : y = x 5 +5x 4 : 984. y, x 4 + x =0: ｧｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ罍陦ｧ0 05, y max,03 ﾀｧﾀｧ罍ｧ x, y + x 3, y 3 =0 ｧ 0403 ﾀ05 縺03 ｧ罍ｧ縺06 閨0600 ｧﾀ05 ｧﾀｧ07, 03 ﾀｧｧﾀｧ罍ｧ040308: 986. ) y (a, x) =x(x, a) ( ﾀ05 縺ｧ ); ) a (x + y )=x y ) x(x + y )=a(x, y ); ) a(x + y )=x(x, y ). x ｧﾀﾀﾀ09 ｧ08 09 罍ｧ罍ｧﾀｧﾀﾀ罍ｧ F (x; y; ) = 0, 06 ﾀ00 ｧ: ) ﾀ 0605 ﾀ閨罍ｧﾀﾀ0504 ; ) ﾀﾀ 0 ｧ07 ﾀ 09 罍ｧﾀﾀ0504, ｧﾀｧﾀﾀ罍ｧ F (x; y; ) =0, 06 ﾀ00 ｧ 030 ﾀ06006 ﾀ , 陦ｧ05 ﾀ07 ｧﾀｧｧｧ08 F (x; y; ) =0 ｧ F 0 (x; y; ) =0: , 03 閨0 0903, ﾀ0006 ｧ0504 ﾀ07, ｧ0 ﾀ0403 ﾀ罍ｧ閨ｧ , 罍ｧ 06 ﾀ09 ｧ0 006 ﾀﾀ罍ｧﾀﾀ0504 ( ﾀ , )) ｧ 0305 ｧﾀﾀ罍ｧｧ 0403 ﾀ05 縺03 ｧｧｧ 09 罍ｧﾀﾀ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 : 縺ﾀ閨00 ｧ003 a ﾀｧ05 ﾀ0403 ｧ0 ｧｧ罍陦ｧﾀｧ 0305 ｧﾀﾀｧ縺ｧ 0305 ｧﾀﾀ縺ﾀ050608, 04 縺閨070 ｧ08 06 縺罍陦ｧ0 05 ｧｧｧ y =4x ｧ0305 ｧﾀﾀ縺ﾀ050608, ｧｧ08 陦ｧｧ陦ｧ06 ﾀ縺ｧ0406 縺 xy = a.

194 3x ﾀﾀ0903 ﾀ0504 ｧ 003 縺縺08003 ﾀ05 ｧｷ罍陦ｧﾀﾀ07 ﾀ0 縺07 ﾀﾀ0903 縺03 ﾀ b ﾀｧOx ｧ 0305 ｧﾀﾀ罍ｧ08 04 罍ｧ 00 ｧｧ0305 ｧ : ) ﾀﾀ縺 x cos +y sin,, = 004 罍ｧ 0403 ﾀ ; ) ﾀﾀ縺 y = ax + a ; 3) ﾀﾀｧ 06 ﾀ09 ｧ y, =(x, a) ｧ 0305 ｧﾀﾀ縺ﾀﾀｧ 0 03 ﾀｧ Ox, 陦ｧ06 ﾀ 0 ｧ縺0308 ﾀﾀﾀｧ06 03 罍陦ｧ y =4x ｧ 0305 ｧﾀﾀｧ04 ﾀ0304 x a + y b =04 罍ｧ 06 ﾀｧｧ, 0503 ﾀﾀ0608 ｧﾀ閨00 ｧ006 l ｧ 0305 ｧﾀﾀ , ｧｧ08 03 ﾀ04 ﾀｧ罍ｧｧ, ﾀｧOy, ｧ04 縺閨070 ｧ08 06 縺ｧ (,a; 0); (3a; 0) ｧ (0; 3a ) 04 罍ｧ 00 ｧ a. x ﾀﾀ0903 ﾀ0504 ｧ 003 縺縺08003 ﾀ05 ｧ 706 ﾀ縺08003 ﾀｧ060 F (x; y; z) =0; M (x; y; z). ｰｧ縺0 00 ｧ縺08003 ﾀ05 ｧ閨0605 ｰｧ06 09 ﾀﾀ0903 ﾀ05 ｧ: X, = Y, =, : (X, x)+@f (Y, y) +@F (, z) @z ｧ0708 () ｧ () X, Y, ｧ罍陦ｧ縺0 00 ｧｧ00 ｧ 09 ﾀﾀ0903 ﾀ ; 縺縺ﾀ05 ｧ. ｭﾀ00 ｧ縺08003 ﾀ05 ｧ 06 ﾀ , 04 = =0, ﾀ07 03 ﾀｧ09 ﾀﾀ0903 ﾀ05 ｧ, 0 ｧ 003 縺0 00 ｧ縺08003 ﾀ05 ｧｧﾀｧ07 09 ﾀﾀ0903 ﾀ05 ｧ縺08003 ﾀ05 ｧ: 999. z = x +y (; ; 3).

195 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧ xy = z (x 0 ; y 0 ; z 0 ). 00. xyz = a (x 0 ; y 0 ; z 0 ). 00. x a + y b, z c = (x 0; y 0 ; z 0 ) ｧ (a; b; c) 縺06 閨0600 ｧﾀ0903 ﾀ0504, 09 ﾀ縺08003 ﾀ05 ｧ x + +4y + z =36 ｧﾀ0903 ﾀ05 ｧ x + y, z = ｧﾀｧ縺0 00 ｧ (3; 4; 5) 縺ﾀ05 ｧﾀ x + y = z ﾀ 003 縺縺06- 閨 ? ｧﾀ 03 ﾀ070 ｧ罍陦ｧ縺0 00 ｧ縺08003 ﾀ05 ｧ x + y, xz, yz = (0; ; ) ｧﾀｧ縺0 00 ｧ縺08003 ﾀ05 ｧ x z+y z = (,; 0; ). 703 ﾀ05 縺03 ｧ縺ｧ縺08003 ﾀ , ﾀﾀ0903 ﾀ05 ｧ縺08003 ﾀ05 ｧ xyz = = a ﾀﾀ0903 ﾀ05070 ｧ罍陦ｧｧ 0 ｧ閨ﾀ , 0503 ﾀ陦縺 , 0305 ﾀﾀ罍陦ｧﾀ0903 ﾀ050406, 09 ﾀ縺08003 ﾀ05 ｧ x =3 + + y =3 + z =3 = a =3, ﾀｧｧ006 a ｧﾀﾀｧ罍陦ｧﾀﾀﾀ05 ｧｧ0903 ｧ閨06 y = x tg z (a; a; a=4). 703 ﾀ05 縺03 ｧ a 0406 縺08003 ﾀﾀｧ070: z =0;a=4; a=; a ｧﾀｧ06 09 ﾀﾀ0903 ﾀ05 ｧ縺ﾀ05 ｧ az = x + y 縺06 ﾀｧﾀ 04 縺 x = y = z , ﾀﾀ0903 ﾀ縺08003 ﾀ05 ｧ x a + + y b + z c = (x 0; y 0 ; z 0 ) 0304 縺06 閨ﾀｧ060 xx 0 a + yy 0 b + zz 0 c =: ｧﾀｧ縺0 00 ｧ縺08003 ﾀ05 ｧ x + y,,(z, 5) = (4; 3; 0). 703 ﾀ05 縺03 ｧ縺縺08003 ﾀ0504 ｧ 003 縺ｧﾀ 03 ﾀ070 ｧ罍陦ｧ縺0 00 ｧ縺08003 ﾀ05 ｧ z = x, y (; ; 0) ｧﾀﾀｧ罍陦ｧﾀﾀﾀ05 ｧｧ閨06 (a, z )x, a y = (a; a; a).

196 3x. ｮ縺ｧ8 ｧ0 ｧｧｧ縺08003 ﾀ05 ｧ 06 縺 , 0503 ﾀ縺 , 0305 ﾀﾀ陦ｧﾀ0903 ﾀ050406, 09 ﾀ縺08003 ﾀ05 ｧ x = + y = + z = = = a =, ﾀｧｧ006 a ﾀﾀ0903 ﾀ縺08003 ﾀ05 ｧ z = =4, x, y : ) ﾀ0903 ﾀ05 ｧ xoy; ) ﾀ0903 ﾀ05 ｧ x + +y + z =0? 7 04 ｧﾀｧｧ08 09 ﾀﾀ0903 ﾀ x. ｮ縺ｧ8 ｧ0 ｧｧｧ縺08003 ﾀ05 ｧ 06 縺縺03 ｧ 0403 閨ｧｧ. 06 陦ｧ06005 ｰｧ06 u = F (x; y) 0304 縺06 閨 u 閨 (x; y) 縺ﾀ05 ｧ, ﾀﾀ u. 6 5 閨閨0308 ｧ 00 ｧ0 ｧ08 F (x; y) =u, F (x; y) =u, :::,05 閨06 u, u, ::: 0403 ﾀ , ﾀ u 03 ﾀﾀﾀｧﾀ罍ｧ 0406 縺閨ｧ (x; y) ﾀｧ0 ｧｧ 0 陦縺ｧ 00 ｧ0 ｧｧﾀ07 ｧ 0300 ｧ0 ｧ070 ｧ ( ｧ縺0 0 ｧ, ｧｧｧ ) ｧ00 ｧ 00 ｧ0 ｧ070 ｧ 06 縺ｰｧ06 u = F (x; y; z) 0304 縺06 閨ﾀ u 縺0308 ﾀ05 ｧ 05 縺縺縺03 ﾀ05 0 ﾀｷ縺08003 ﾀ05070 ｧ, ｧ00 ｧ縺ﾀ05070 ｧ 06 縺閨0605 F (x; y; z) =u ; F (x; y; z) =u ; ::: 706 ﾀ (x; y; z) 0406 縺ﾀ縺 x = x 0 + l cos, y = = y 0 + l cos, z = z 0 + l cos ﾀ03 ﾀ0903 縺03 ﾀ dl =. ｯ0305 ﾀ dt u = F (x; y; z) 0306 閨0605 ｧﾀ07 ﾀ03 ﾀ0903 縺03 ﾀ v = du dt = du @F cos + @z cos = N l 05 閨06 ; 003 縺ｧ縺08003 ﾀ05 l 0 fcos ; cos ; cos g 06 陦ｧ0 ｧｧ07 l. 7 縺03 ｧ 0403 閨0 07 du @F cos + @z cos = N l ﾀ07 04 縺03 ｧ 0403 閨ｧｧ u = F (x; y; z) ｧｧ l 0 fcos ; cos ; cos g. 06 陦ｧﾀ u = F (x; y; z) ﾀ grad u j k. 06 陦ｧﾀﾀ0903 縺03 ﾀ05 ｧ 0 ｧ0303 ﾀ05 ｧｧ07 ﾀ u ﾀ0504 z =4, x, y. 703 ﾀ05 縺03 ｧｧ0 ｧｧ 06 縺ｧ grad z A(; ).

197 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧﾀ0504 z = arctg y. 703 ﾀ05 縺03 ｧｧ0 ｧｧ 06 縺ｧ grad z: x ) 縺 y = x; ) 縺 y =,x, 04 ﾀ05003 ﾀ05 ｧ (=; =), (; ), ::: 罍ｧｧﾀ05 ｧ 0304 縺06 閨ﾀｧ060 h =0, x 4, y. 703 ﾀ05 縺03 ｧ罍ｧｧ, ﾀﾀｧ h = 00, 9 0, 8 0, 6 0 ｧｧ06 grad h 0304 縺06 閨閨06 ﾀｧ06 00 ｧ0 ｧｧ 0 ｧ縺ﾀ09 05, ｧｧ0 09 縺0605 ｧﾀﾀ05 ｧ. 703 ﾀ05 縺03 ｧ0504 grad h x = ｧ y = ｧ 0 ｧ縺0605 ｧ縺08003 ﾀ05 ｧ z = xy (4; ) ｧ 04 縺03 ｧ 0403 閨ｧｧ u = ln(e x + e y ) ｧｧ, ｧﾀﾀ罍ｧﾀ罍陦ｧｧ 04 縺03 ｧ 0403 閨ｧｧ u = x + y + z (; ; ) ｧｧ lfcos 45 ; cos 60 ; cos 60 g ｧｧ grad u ｧ閨00 ｧﾀ05 縺03 ｧ縺08003 ﾀ05 ｧ 06 縺ﾀ05 縺03 ｧ縺08003 ﾀ05 ｧ 06 縺ﾀ u = x + y,, z ｧｧｧ 0403 ﾀ05 縺03 ｧ0504 grad u 縺06 ﾀｧ07 03 ﾀｧ Ox ﾀ縺08003 ﾀ u = ｧ 04 縺03 ｧ 0403 閨ｧｧ u = x a + y b + z c (a; b; c) ｧｧ陦ｧ06 ﾀｧﾀ0504 z =. 703 ﾀ05 縺03 ｧｧ0 ｧｧ 06 縺ｧ grad x z 04 + y (,; ) ｧｧ jgrad zj ﾀ0504 u = xyz ｧ 04 縺03 ｧ 0403 閨00606 du ｧｧ, dl ﾀ03 ﾀﾀ 03 ﾀ070 ｧ罍陦ｧ , ｧ (; ; ) ﾀ05 縺03 ｧ縺08003 ﾀ05 ｧ 06 縺ﾀ u = x +y, z, 0304 縺06 閨0600 ｧ0504 grad u 縺08003 ﾀ05 ｧ, 04 縺閨縺陦ｧ0 05, ｧ 0403 ﾀ05 縺03 ｧ縺08003 ﾀ05 ｧ, 縺0308 y =0 ｧ z = ﾀ0504 u = x + y + z ｧ grad u ｧ閨00 ｧﾀ05 縺03 ｧ0504 ｧ縺08003 ﾀ05 ｧｧｧ u = z c, x a, y b ｧｧ 04 縺03 ｧ 0403 閨 u (a; b; c) ｧｧ陦ｧ06 ﾀｧ.

198 3x. 709 ﾀ05 縺ｧｧ閨縺 x. 709 ﾀ05 縺ｧｧ閨縺陦ｧﾀｧ07. ｱｧ07 z = F (x; y) ｧﾀ05 縺 , 04 =0 ｧ = ｧﾀ07 09 罍ｧ05 ｧ 06 ﾀ09 ｧ0 ｧﾀﾀｧｧ0 06 縺06 A, B ｧ C ｧ07 04 縺03 ｧ 0403 F 罍ｧ05 ｧ 06 ﾀ (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, ﾀ05 縺 ; 3) B A B C =0, ﾀ05 縺 , ｧ ( ﾀ0300 ｧﾀ ). 3. ｰﾀﾀ05 縺ｪｧ 0509 ﾀ05 縺ｧｧ z = F (x; y) 04 罍ｧ 06 ﾀｧｧ, 0503 x ｧ y ﾀｧ060 '(x; y) = 0, ﾀ03 ﾀ05 04 ｧ0 04 ﾀｧ06 u = F (x; y) +'(x; y) 罍陦ｧﾀ05 縺ｧ (x; y) 閨閨罍ｧｧ070: '(x; y) =0, ｧ縺0308 ｧ閨0705 x ｧ y ｧ 0509 ﾀ05 縺ｧｧ: 030. z = x, xy + y +9x, 6y z = y x, y, x +6y. 03. z = x 3 +8y 3, 6xy z =xy, 4x, y: 034. z = e x= (x + y ) z = sin x + sin y + sin (x + y) 04 罍ｧ 0 6 x 6 = ｧ y 6 = z = x + 04 罍ｧ x + y =. y 037. z = x + y 04 罍ｧ x + y = 縺06 閨0600 ｧ縺03 04 縺縺ﾀﾀ06080, ｧｧ縺08003 ﾀ罍ｧ 06 ﾀｧｧ, V ﾀ05 縺03 ｧｧ04 ﾀ x +4y =4 ｧ 04 縺 x +3y, 6= =0 ｧｧ04 ﾀｧｧ, 0 ｧｧ 0 ｧ縺ｧ0406 縺 x, y = ｧ , 0 ｧｧ (0; ).

199 ｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧ縺06 閨0600 ｧ縺03 09 ｧ00 ｧ0 陦 0 ｧ罍ｧ 06 ﾀｧｧ, 縺08003 ﾀ S = ) ｧ04 ﾀ x +3y =0404 ｧﾀ陦縺縺ｧ09 ﾀ 03 ﾀｧ060, ﾀｧ, 05 09, 閨04 05 縺ｧｧ ) 05 ﾀ04 Ox ﾀｧ0906 閨ﾀ縺ｧ閨縺ｧ 0006 ﾀｧｧ A(0; a) B(c;,b), ｧ縺閨060 ｧﾀﾀｧ07 0 ｧ縺06007 (a >0, b>0, c>0)? a ｰ09 0 ｧｧ 0 ｧ0 ｧｧｧ T = v cos + b v cos 04 罍ｧ 06 ﾀｧｧ a tg +b tg = c, 05 閨06 v ｧ v ﾀ0903 縺03 ﾀ05 ｧﾀ閨ﾀ縺06 08, ｧ閨060 ｧ07 ｧ 04 縺ｧｧ 0509 ﾀ05 縺ｧ08: 043. z =3x +6y, x, xy, y z = x + y, x, 4 xy, y z =x 3, xy +5x + y z =3x, x y + y, 8x z = xy 04 罍ｧ 06 ﾀｧｧ, 0503 x + y = ｧ 0 ｧｧ縺ｧ罍ｧ 06 ﾀｧｧ, 0503 閨00 ｧ陦ｧｧ ) y =4x ｧ , 0 ｧ縺 x, y +4=0. ) ｧ04 ﾀ x a + y =0404 ｧﾀ 0 04 縺ｧ09 0 ｧ b 陦ｧｧ閨縺06 閨0600 ｧ縺ﾀ 0 ｧ罍ｧ 06 ﾀｧｧ, 縺08003 ﾀ S.

200 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 ｧ06 03 陦ｧ縺06009 ｧｧｧ陦ｧ縺06009 ｧｧ060 n 縺07 閨ﾀｧ06 04 ｧ F (x; y; y 0 ;y 00 ; :::;y (n) )=0; () 05 閨06 y = y(x) ｧﾀｧ07. 6ｧｧ07 y = '(x), ｧ06 () 閨06 ﾀ050403, ﾀ07 縺ｧｧ07, ｧｧｧｧ罍ｧｭﾀ00 ｧ縺ｧｧ閨06 (x; y) =0, ﾀ07 ｧｧ07 (). ｱｧ07 y = '(x; C ; :::;C n ), ﾀ03 閨06 縺 n ｧﾀｧ縺03 ｧﾀ , ﾀｧ0 縺ｧｧ07 (), 06 ﾀ00 ｧﾀ縺ｧ罍ｧｧﾀ C, ::: :::, C n. ｭﾀ00 ｧｧﾀｧ閨ｧ060 (x; y; C ; :::;C n ) = 0, ｧﾀｧ0 ｧｧ07 (). 7 罍ｧｧｧ y = '(x; C ; :::;C n ) ｧ00 ｧｧｧ (x; y; C ; :::;C n ) = 縺06 閨ｧﾀ C, :::, C n, ﾀ縺ｧ06 ｧ00 ｧﾀﾀﾀｧｧ07 () , ｧ00607 ﾀﾀ罍ｧ040308, ｧ060 (x; y; C ; :::;C n )=0, ｧｧﾀ縺03 C, :::, C n ｧﾀｧﾀｧ08 d =0; =0; :::; dn n =0; ｧ0, 縺07, 陦ｧ縺06009 ｧｧ06 04 ｧ (), 閨縺 (x; y; C ; :::;C n )= ﾀｧ0 ｧｧ縺06009 ｧｧ縺縺07 閨09 ｧｧ F x; y; dy =0: () ｧｧ06 () ﾀｧ dy, 06 ﾀ00 ｧ , ｧ0 dy = f (x; y): (3)

201 ｧ縺06009 ｧｧ07 ｰｧ06 (3) 0304 縺06 閨 k = tg dy = = f(x; y) ｧ罍ｧ (x; y), 縺06 閨ｧ08 ｧ罍ｧｭﾀ00 ｧ縺ﾀ05 ｧｧ07 f(x; y) 縺06 縺03040 ｧｧｧﾀ縺03 ｧ 0403 閨00606 fy 0 (x; y), ﾀ07, 縺閨縺 (x 0 ; y 0 ) ﾀ05 ｧ 04 縺0308 閨陦ｧ0 ﾀｧ罍ｧﾀ05 ｧｧ06 (3) ｧ縺ｧ06 y = '(x; C), ｧ00 ｧｧ08 ｧ (x; y; C) =0, ｧ縺ｧ 06 陦ｧ0 ﾀﾀ縺ｧ06, ｧ00 ｧ 06 陦ｧ0 ﾀﾀｧ , 06 閨縺07060 ｧﾀｧ070: y = y 0 04 罍ｧ x = x 縺罍ｧ陦ﾀ , ｧ07 y = Cx ﾀ07 縺ｧ060 陦ｧ縺06009 ｧｧ07 3y, xy 0 = ﾀ05 縺03 ｧ0504 ｧ罍ｧ040306, 04 縺閨070 ｧ06 06 縺ｧ: ) (; =3); ) (; ); 3) (;,=3) 縺罍ｧ陦ﾀ , 0503 陦ｧ縺06009 ｧｧ07 ) y 00 +4y =0 ｧ)y 000, 9y 0 =0 ｧﾀﾀｧ06 縺ｧ07: ) y = C cos x+c sin x ｧ)y = C +C e 3x +C 3 e,3x ﾀ05 縺03 ｧ y = Cx 04 罍ｧ C = 0; ; ｧﾀ03- ﾀ05 04 ｧ0504 陦ｧ縺06009 ｧｧ06 ﾀﾀｧﾀ05 縺03 ｧ0504 ｧｧ07 ﾀﾀ0504 : ) 0309 縺ﾀ x + + y =Cx, ) y = x +Cx ｧﾀ03 ﾀ05 04 ｧ0504 ｧ08 陦ｧ縺ｧｧﾀ05 縺03 ｧ0504 ｧｧｧ08, 0304 縺06 閨閨030 ｧｧ08: dy ) dy 3) = y + x. ) dy y = x ; = y, x; ﾀ05 縺03 ｧ0504 ｧｧｧ08, 0304 縺06 閨ｧ060 dy = x + y, ﾀ縺ﾀ050608, 04 閨縺0308 dy ; ; ; 3;::: 7 罍ｧﾀ罍ｧ0300 ｧｧ = 罍ｧ040606, 04 縺閨縺罍陦ｧ0 05. x. 6 7 ｧ縺06009 ｧｧ縺縺07 閨09 ﾀ閨ｧ0 ｧﾀ縺ｧ. 05 縺罍ｧｧ. 6 7 ｧ縺06009 ｧｧ縺縺07 閨09 P+ Qdy =0; () 05 閨06 P ｧ Q ｧｧ x ｧ y, ﾀｧ060 ﾀ閨ｧ0 ｧﾀ縺ｧ, 06 ﾀ00 ｧｧ09 ｧ P ｧ Q 04 罍ｧ陦ｧ縺06009 ｧﾀｧｧ, 04 ｧﾀ070 ｧ x ｧ00 ｧ y, ﾀ00 ｧ

202 3x. 6 7 ｧ縺06009 ｧｧ縺縺07 閨ｧｧ f (x)'(y) + f (x)' (y) dy =0: () 7 閨0600 ｧｧ07 () 0 '(y)f (x), ｧ0 f(x) f (x) + ' (y) dy '(y) =0: (3) ｧ0 ｧｧ07 (3), ﾀ0006 閨 , ｧ () 0306 閨0605 f (x) + ' (y) dy = C: (4) '(y) f (x). 05 縺ｧ罍ｧ070 ｧﾀﾀｧ0 ｧ08 F (x; y; a) = ﾀ07 00 ｧ0 ｧｧ, 0406 縺06 ﾀｧ06 00 ｧ0 ｧｧﾀﾀ縺縺03 陦ｧ縺06009 ｧ縺ｧ06 F (x; y; a) = x ｧｧﾀｧ04 a ｧｧｧ08, ｧ0 陦ｧ縺06009 ｧｧ06 00 ｧ0 ｧﾀﾀ0504 y 0 = f (x; y). ｯ03-05 陦ｧ縺06009 ｧｧ縺罍ｧ閨0605 y 0 =, f(x; y). 6 5 ﾀ0006 閨06060 ｧ08 陦ｧ縺06009 ｧｧ0708: ) ｧｧ08 ｧ ; ) 0403 ﾀ05 縺03 ｧﾀｧ罍ｧ040308; 3) ｧﾀｧﾀｧ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: ｧｧ06 ｧｧ08: 06. x y 0 + y =0: 06. x + xy + y 0 (y + xy) = ' dr +(r, a) d' = st ds =(+t ) dt. 6 5 ﾀ0006 閨06060 ｧｧｧｧ08 ｧﾀｧﾀｧ070: 065. y 0 x = y, y =04 罍ｧ x = y 0 =(y + ) ctg x, y == 04 罍ｧ x = = x y 0 + y =0, y =04 罍ｧ x =, ﾀ05 縺03 ｧ0504 ｧ罍ｧ閨ｧｧ08: ) y 0 (x, 4) = xy; ) y 0 + y tg x = 0,04 縺閨070 ｧ06 06 縺ｧ: ) (0; ); ) (0; =); 3) (0;,=); 4) (0;,) ｧ 09 罍ｧ040606, 04 縺閨縺 (; =3), 06 ﾀ00 ｧｧ09 ｧﾀ罍ｧ縺ｧ09 ｧ06005 陦ｧ06 ﾀｧ 09 ﾀ - 0 ｧ07.

203 ｧ縺06009 ｧｧ罍ｧ縺陦ｧ05 06 縺 A(0; a), MN 04 縺03 ｧ罍陦ｧ罍ｧ縺06 閨0600 ｧ罍ｧｧ 06 ﾀｧ07, 閨04 OAMN = as, 05 閨06 s 閨00 ｧ0 閨0605 ｧ AM ｧ 09 罍ｧ040606, 04 縺閨縺 (a; a), 06 ﾀ00 ｧﾀ閨ﾀ09 ｧﾀﾀｧ 09 - ﾀ 0 ｧｧ 09 罍ｧ040606, 04 縺閨縺 (,;,), 06 ﾀ00 ｧ 0403 閨003 縺閨ｦ縺 , 0 05 縺閨03 00 C, 陦ｧ05 ﾀ07 閨03 5 C ﾀ縺 C, 06 ﾀ00 ｧ閨03 60 C ﾀｧ0? ( ﾀ0903 縺03 ﾀ閨060 ｧ07 04 縺ｧﾀ05 ｧ ) 縺ｧﾀﾀ05 ( ﾀ0. 罍ｧﾀ. 6) 閨陦ｧ0 ｧ0903 閨00 ｧ罍ｧｧ縺縺ﾀ , ｧ縺006, 06 ﾀ00 ｧｧｧ罍ｧ H. ｰ09 0 ｧ閨 OC ^ ( 罍ｧﾀ. 6) 04 縺03 ｧ M. 7 ﾀ OM 0306 閨0605 閨0608 ﾀ罍ｧﾀｧ0003: 0503 罍ｧ H ( ｧ M), 0406 縺05 ｧﾀ x ｧｧﾀｧｧ07 T ( ｧ M) ﾀｧ07 ﾀ縺ｧ08 ﾀｧ00 0 Ox ｧ Oy 閨ﾀ縺06 閨0600 ｧ0504 ｧ 0403 ﾀ05 縺03 ｧ罍ｧ040606, 04 縺閨縺 P (,a; a), 06 ﾀ00 ｧ 0305 縺 AB ﾀ , 閨06 03 ﾀ070 ｧ罍陦ｧ0 05, 閨0600 ｧ05 ﾀﾀ 0 ｧ07 M ｧ03 縺罍ｧｧﾀﾀ ay = = x. 703 ﾀ05 縺03 ｧ0504 ｧｧ 03 縺罍ｧｧﾀﾀｧ0406 縺 xy = c ｧ 03 縺罍ｧｧﾀﾀｧ 06- ﾀ09 ｧ ay = x ｧ 03 縺罍ｧｧﾀﾀｧ04 ﾀ0304 x +4y = a ｧｧ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 ' = y 0 = y ln x; y =04 罍ｧ x = e ( + x )y 0 + y +x = xy; y =04 罍ｧ x = 縺06 閨0600 ｧ罍ｧ040606, 04 縺閨縺 A(,; ), 06 ﾀ00 ｧｧ09 ｧﾀ罍ｧ陦罍陦ｧｧ 09 ﾀ 0 ｧ07.

204 3x ｧ縺06009 ｧｧ縺縺07 閨罍ｧ縺陦ｧ05 06 縺 A(0; a), MN 04 縺03 ｧ罍陦ｧ罍ｧ縺06 閨0600 ｧ罍ｧｧ 06 ﾀｧ07, 閨04 OAMN = a(mn, a) 縺06 閨0600 ｧ0504 ｧ 0403 ﾀ05 縺03 ｧ罍ｧ040606, 04 縺閨縺 (,;,), 閨縺縺 OT, 0305 ﾀﾀｧ Ox 09 ﾀ罍ｧ , 陦ﾀ09 ｧﾀﾀｧ 09 ﾀ 0 ｧｧ03 縺罍ｧｧﾀﾀｧ0406 縺 x,, y = a 縺06 閨0600 ｧ罍ｧ040606, 陦ｧ06 ﾀｧ縺縺縺0 00 ｧ閨06 09 罍ｧｧ 03 ﾀ0406 Ox 縺06 閨0600 ｧｧ0 ｧ06, 06 ﾀ00 ｧ閨04, ｧﾀ070 ｧ罍陦ｧ0 05, ｧ0 ｧ0608 ｧ 04 縺03 ｧ罍陦ｧ , 040 = 陦ｧ 04 縺ｧ09, 0403 ﾀ05 縺罍陦ｧｧ 09 罍ｧ x ｧ縺06009 ｧｧ縺縺07 閨09 : ) 03 閨003 縺03 閨00306, ) 00 ｧ , 3) 706 縺ｧ. 05 閨003 縺03 閨ｧ06. ｰｧ06 P+ Qdy = ﾀ07 03 閨003 縺03 閨0030, 06 ﾀ00 ｧ P ｧ Q 03 閨003 縺03 閨ｧｧ 0305 x ｧ y 03 陦ｧｧ 006 縺060 ｧ罍ｧ0403 陦ｧ05 ﾀｧ閨06 dy y = ' ｧ縺ﾀ07 x 0403 陦ﾀ y = u ｧ00 ｧ y = ux. x. 6ｧ8 ｧｧｧ縺06009 ｧｧﾀ07 00 ｧ , 06 ﾀ00 ｧ縺ﾀｧﾀｧｧﾀｧｧ y ｧ 04 ﾀ縺03 ｧ 0403 閨ｧ8 ｧｧ縺縺07 閨09 ｧｧ y 0 + Py = Q ﾀ0403 陦ｧ05 ﾀ07 09 閨ｧ070 ﾀ閨ｧ0 ｧﾀ縺ｧ 0403 陦ﾀ y = uv. 6 7 縺ﾀ0403 ﾀ0303 縺ｧ07 (04 罍ｧ 09 ｧ07 04 縺03 ｧﾀ ) ﾀ03 ﾀ0503 ｧ , 0503 ﾀ0 00 縺ｧ06 y 0 + Py = 0; y =,Ae R, P. 703 陦ﾀ縺ｧｧ06, ﾀｧ05 07 A ｧ0608 x, ｧ陦ｧ0 A 0 ｧ A. 3. ｰｧ縺ｧ y 0 + Py = Qy n 縺ﾀ , ｧ 00 ｧ , 0403 陦ﾀ y = uv ｧ00 ｧ 04 罍ｧ 09 ｧ縺03 ｧﾀｰｧ縺ｧ 04 罍ｧ0403 陦ｧ05 ﾀｧ陦ﾀ z = y,n ｧ0504 陦ｧ縺06009 ｧｧ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 y 0 cos x, y sin x = sin x.

205 ｧ縺06009 ｧｧ 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 x y y 0 + yx 3 = { ｧﾀｧﾀｧ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 = ｧﾀﾀ罍ｧ040308, 0403 閨09 ﾀ縺ﾀ0504 ﾀ縺06 閨00606 罍ｧｧ 06 ﾀ罍陦ｧｧ 09 ﾀ 0 ｧｧ 03 縺罍ｧｧﾀﾀ縺ﾀ x + y =ax. 0. ｮｧ i ｧﾀﾀ0304 縺0305 ｧｧ060 R, ﾀ 003 ｧ0 閨ｧ0608 L ｧ縺03 閨04 ｧﾀｧ E 06 閨縺陦ｧ縺06009 ｧｧ06 L di + Ri = E: dt ｧｧ06, ﾀｧ05 07 R ｧ L 0403 ﾀｧ, 縺03- 閨04 ｧﾀｧ0006 E 00 ｧﾀ : E = kt ﾀｧ07: i =004 罍ｧ t = ｧ 0703 縺縺09 00, ﾀｧ, 閨07-0 ｧ06 ｧｧ, ｧ06. ｰ09 0 ｧ06. 7 ﾀﾀ0 05 罍ｧﾀﾀｧ06 06 縺09 00, 04 罍ｧ罍陦ｧ0 05, ｧ06 03 ﾀ04 Oy. 07 ﾀｧﾀ罍ｧ M ﾀ OM ｧ 03 ﾀ0406 Oy, ﾀﾀｧOy 0305 縺 ON = OM ｧ0504 陦ｧ縺06009 ｧｧ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

206 3x 4. ｰｧ07 ﾀ陦ｧ縺06009 ｧ 00 0 ｧ 04 縺03 ｧ 0406 閨060 ｧ07 ｧﾀ (, x )y 0, xy = xy ; y =0; 5 04 罍ｧ x = 縺06 閨0600 ｧ罍ｧ040606, 04 縺閨縺 A(a; a), 06 ﾀ00 ｧﾀﾀｧ罍陦ｧ0 05 閨03 09 ﾀ罍ｧﾀ09 ｧﾀﾀｧ. x ｧ縺06009 ｧｧ07, ﾀ03 閨06 縺07 0 ｧ06 陦ｧ縺06009 ｧ縺03 ｧ 0406 閨060 ｧ07 ｧﾀ d(xy) =xdy+ y; d y x = xdy, y x ; d x y = y, xdy y : ｯ 09 ｧｧ07 ｧ縺ﾀ07, 06 ﾀ00 ｧﾀﾀｧ0504 xy = u, y u = x ｧ00 ｧ y = u, y = ux. x ｧ0504 陦ｧ縺06009 ｧｧ07: 4. x dy + xy = : 5. y xdy, y 3 = x dy. ｰ 09 0 ｧ罍ｧ006 縺ｧ06 04 罍ｧ0403 陦ｧ05 ﾀｧ閨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 ｧ縺06009 ｧｧ縺縺07 閨陦ｧ縺06009 ｧｷ罍ｧ縺06060 ｧｧｭﾀ00 ｧ 04 陦ｧ縺06009 ｧｧｧ P+ @y =, 罍ｧ0303 縺ｧ du =0 ｧｧ08 ｧ u = C.. ｭ 6=, 罍ｧ縺ﾀｧ0708 ﾀ06006 ﾀ (x; y) , 0503 P + Q dy = du ｧ07 (x; y) ﾀ07 ｧ罍ｧ縺06060 ｧｧ

207 ｧ縺06009 ｧｧ07 0ｷ罍ｧ縺06060 ｧｧｧ 04 ﾀ : Q =(x), ln =R ) = (y), ln =R (y) dy. P 6 7 ｧ縺06009 ｧｧ07 x ﾀ07 ﾀｧﾀｧｧ08, ﾀﾀ0 05 罍ｧﾀｧ0504 ﾀ0006 閨06060 ｧ06 陦ｧ縺06009 ｧｧ陦ｧ縺06009 ｧ : 35. 4, y y + dy =0. x x 36. 3x e y +(x 3 e y, ) dy = e,y +(, xe,y ) dy = x cos y+(y, x sin y) dy = ｧｧ罍ｧ縺06060 ｧｧｧｧ縺0600 ｧ0504 陦ｧ縺06009 ｧｧ07: 39. (x, y) + xdy = x tg y+(x, sin y) dy =0. 4. (e x, y ) + ydy=0. 4. (+3x sin y), x ctg ydy= , ﾀ05 ｧﾀ0006 閨06060 ｧ08 陦ｧ縺06009 ｧｧ08 ﾀ陦ｧ縺06009 ｧ 0003, ｧ縺0600 ｧｧ07: 43. (3x +y) +(x, 3) dy = (3x y, 4xy ) +(x 3, 4x y +y 3 ) dy = (x cos y +), x sin ydy = ｧｧ罍ｧ縺06060 ｧｧｧｧ縺0600 ｧｧ07: 46. y +(yx, ) dy = (x, 3y ) +xy dy = (sin x + e y ) + cos xdy = (x sin y + y) +(x cos y + x ln x) dy =0.

208 3x 6. ｰｧ07 6ｧｧ縺03 5 x ｧ縺06009 ｧｧ縺縺07 閨09, 006 縺ﾀｧ縺03 ｧ 0403 閨ｰｧ07 6ｧｧ縺03. ｭﾀ00 ｧｧ06 F (x; y; y 0 )= 縺0308 ﾀｧﾀｧ y 0, ｧ閨04 縺ｧﾀｧ y 0 : y 0 = f (x; y) ｧ y 0 = f (x; y), 縺06 縺ﾀｧ x ｧ y 縺ﾀ05 ｧ, 罍ｧ 06 ﾀ09 ｧ 0304 縺06 閨 (x 0 ; y 0 ) ﾀ05 ｧ閨ｧ07 ｧ罍ｧｯ 09 ｧ06 陦ｧ縺06009 ｧｧ07 F (x; y; y 0 )=0,09 縺ｧ (x; y; C) =0 ｧﾀｧ , ｧ00305 ｧﾀｧ , 006 ﾀ03 閨06 縺07 0 ｧ08 04 縺03 ｧﾀｧ縺ｧ08 ﾀ07 ｧｧ 04 罍ｧｧｧ 0403 ﾀﾀｧ , 06 ﾀ00 ｧ 030 ﾀ06006 ﾀ , ｧ0504, ｧﾀｧ04 = y 0 ｧｧ08 F (x; y; ) =0 ｧ F 0 (x; y; ) =0 ｧ00 ｧ 0706 ｧﾀｧ04 C ｧｧ (x; y; C) =0 ｧ 0 C = 罍ｧ 06 ﾀ09 ｧ 03 ﾀｧ縺06 閨ｧﾀﾀ0504 ｧ罍ｧ ).. ｰｧ06 6ｧ閨06 = y 0, ｧ罍ｧ縺ﾀ07 ﾀ0006 閨06060 ｧ縺03 陦ｧ縺06009 ｧ縺 () 0403 x, 0 08 閨060: y = xf() +'(); () = f () + xf 0 () +' 0 () d : ｧ06 00 ｧﾀｧ x ｧｧ , ｧ0: d x = CA() +B(): () ｰｧ07 () ｧ () 0306 閨罍ｧ 06 ﾀ09 ｧ 0304 縺06 閨ｧ08 ｧｷ3 ﾀｧ04 ｧ 0 ｧ (06 ﾀ00 ｧ ), ｧｧ08 ｧ縺006 (x; y; C) =0. 3. ｰｧ縺03 y = x + '() (3) ﾀ07 ﾀﾀｧ07 6ｧｧｧ08 ｧ y = Cx+ '(C) ｧ 03 ﾀ , ｧ08 ﾀ07 ｧﾀｧｧｧ08 y = x + '() ｧ x =,' 0 () ﾀ05 縺03 ｧﾀｧ罍ｧｧ07 y 0 = 4y ｧ06 閨0406 ｧ罍ｧ縺閨縺 M (; 4)? ) ｮ縺06 閨ｧｧﾀ. 00

209 ｧ縺06009 ｧｧﾀ05 縺03 ｧ0504 ｧ罍ｧｧ07 y 0 + y,, = ｧ06 閨0406 ｧ罍ｧ縺閨縺 M (=; = )? , 0503 ｧ罍ｧｧ07 xy 0,, yy 0 +4x =0 ﾀ03 閨06 縺07 05 ﾀ罍ｧ 03 ﾀ05 縺閨06 04 縺ｧ y = x. 703 ﾀ05 縺03 ｧ0504 ｧ罍ｧ040306, ｧﾀ C = ; ; ｧｧｧ07: ) yy 0 + y 0 (x, y), x =0; ) xy 0 +xy 0, y =0 ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ罍ｧｧｧ07, 006 ﾀ03 閨06 縺07 0 ｧ閨00308 ｧ 0406 縺 : ) y =+y 0 ; ) x =y 0, y. 0 ｰ09 0 ｧｧ04 y 0 06 縺06, 04 縺03 陦ｧ縺06009 ｧ縺縺ｧ x, 縺 y ｧｧ06 ｧ 03 ﾀｧｧ08 6ｧ : ) y = xy 0 + y 0 ; ) y =xy 0 + xy0 ; 3) y = y0 y ｧｧ08 ｧ 03 ﾀｧｧ縺03 ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ罍ｧ040306: ) y = xy 0, y 0 ; ) y = xy 0, a +y 0 ; 3) y = xy 0 + y ﾀ05 縺03 ｧ0504 ｧ罍ｧｧ07 y 0 + y = ｧ06 閨0406 ｧ罍ｧ縺閨縺 M (; 3=4)? ｧｧ07, 006 ﾀ03 閨06 縺07 0 ｧ閨00308 ｧ 0406 縺 : ) y = y 0 + y 03 ;)x = ay0 +y ｧｧ06 y =y 0 x + x + y ｧｧ08 ｧ 03 ﾀｧｧ縺03 ｧ 0403 ﾀ05 縺03 ｧ0504 ｧ罍ｧ040306: ) y = y 0 x + y 0 ; ) y = xy0 + y 0 + y ｧ 09 罍ｧ040606, 09 ﾀ縺ﾀ 03 ﾀ070 ｧ罍陦ｧ縺ｧﾀ陦ｧ, a ｧ 09 罍ｧ040606, 09 ﾀ縺ﾀﾀ罍陦ｧ縺06 09 ｧ, ﾀ縺 a.

210 3x ｧ縺06009 ｧｧﾀ00 ｧ縺07 閨 x ｧ縺06009 ｧｧﾀ00 ｧ縺07 閨090304, 閨ﾀｧｧ07060 ｧ縺07 閨09. ｰｧ06 04 ｧ y (n) = f(x) 縺ﾀﾀ0006 閨 n ｧ罍ｧ縺ｧﾀ05 ｧ. 7 罍ｧ閨030 ｧ罍ｧ縺ｧｧﾀ07 03 閨0 04 縺03 ｧﾀ , 縺 n 04 縺03 ｧﾀｰｧ06 F (x; y 0 ;y 00 )=0, 006 ﾀ03 閨06 縺 y 縺006, 0403 陦ﾀ y 0 =, y d 00 = 04 罍ｧ0403 陦ｧ05 ﾀｧ閨06 F x; ; d =0: 3. ｰｧ06 F (y; y 0 ;y 00 )=0, 006 ﾀ03 閨06 縺 x 縺006, 0403 陦ﾀ y 0 =, y d 00 = = d 04 罍ｧ0403 陦ｧ05 ﾀｧ閨06 dy F y; ; d =0: dy ｧｧ07: 63. ) y 000 = 6 x 3 ; ﾀｧ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: 縺06 閨0600 ｧ罍ｧｧ 05 ｧ罍ｧｧ, 03 陦ｧ縺閨0600 0, 0 陦縺閨0608 ﾀﾀ03 ﾀ縺06 閨ﾀｧ00 P (0406 ﾀｧ 04 縺縺06 04 ｧﾀｧｧ 05 ｧ03 0 ﾀ , y 0 ) 縺06 閨0600 ｧ罍ｧ040306, 縺0308 陦ｧ06 ﾀ 09 罍ｧ04 ｧ閨閨00 ｧ縺0 00 ｧ縺06 閨0600 ｧ罍ｧ040306, 縺0308 陦ｧ06 ﾀ 09 罍ｧ04 ｧ閨00 ｧ縺0 00 ｧ縺 [0; ] 0304 縺06 閨0600 ｧ罍ｧ040606, 09 ﾀﾀ07 03 ﾀｧ Ox 罍陦ｧ0 05, 06 ﾀ00 ｧ 09 罍ｧ04 ｧ k = x, 縺003 0 ﾀ閨ﾀｧ Ox (0406 縺閨罍ｧ04 07). 7 罍ｧ罍ｧ0300 ｧ , y 0.

211 ｧ縺06009 ｧｧｧｧ07: 75. y 00 = cos x ; y = ln, y0 =04 罍ｧ x = ( + 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 +: 縺06 閨0600 ｧ罍ｧ040306, 縺0308 陦ｧ06 ﾀ 09 罍ｧ04 ｧ閨00 ｧ縺0 00 ｧｧ00506 縺 (,=; =) 0304 縺06 閨0600 ｧ罍ｧ040606, 09 ﾀﾀ07 03 ﾀｧ Ox 罍陦ｧ0 05, 06 ﾀ00 ｧ 09 罍ｧ04 ｧ k = cos x. x 8. 6ｧ8 ｧ閨003 縺03 閨00306 陦ｧ縺06009 ｧｧ07 ﾀ 0403 ﾀｧｧ09 ｧｧ 05 閨003 縺03 閨ｧ陦ｧ縺06009 ｧｧ06 y (n) + y (n,) + :::+ n y =0; () 05 閨06 i ｧｧ x, ｧ縺ｧ06 04 ｧ y = C y + C y + :::+ C n y n ; () 05 閨06 y, y, :::, y n 00 ｧｧﾀｧ00306 ﾀ縺ｧｧ07 (), C, C, :::, C n 04 縺03 ｧﾀｭﾀ00 ｧｧ09 ｧ ,, :::, n ｧ07 () 0403 ﾀ , 0503 ﾀ縺ｧ07 y, y, :::, y n 閨0705 ﾀ07 ﾀ罍ｧﾀ05 ｧ 06 ﾀｧ07 r n + r n, + :::+ n =0: (3) ) 閨ﾀ縺006 r = a ｧ07 (3) ﾀ05 ｧ m ﾀﾀ m ﾀ縺ｧ08 e ax, xe ax, :::, x m, e ax. ) 閨縺06 00 ｧ縺00608 r = i ﾀ05 ｧ m ﾀﾀ m 04 ﾀ縺ｧ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: ｧｧ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:

212 3x 9. 6ｧ8 ｧ閨003 縺03 閨00306 陦ｧ縺06009 ｧｧ d s dt +ds dt +s =0; 04 罍ｧ t =0 s =, s0 =. 93. y 000, 5y 00 +8y 0, 4y = y IV, 6y =0: 95. y 000, 8y = y ay 00 +3a y 0 + a 3 y = y IV +4y =0: 98. 4y IV, 3y 00, y = 縺06 閨0600 ｧｧｧｧ09, ﾀ03 ﾀｧ 0 ﾀﾀ03 m, 0403 閨ｧ05 ｧ閨00 ｧ00308 l ( ﾀ0304 縺0305 ｧｧ縺縺06 04 ｧｧ0504, 罍ｧｧ07 sin ) 縺06 閨0600 ｧ罍ｧｧ陦ｧ縺閨縺0607 ｧ閨0608 ﾀ0504 ｧ閨縺06 04 縺0607 ｧ0 06 閨00 ｧﾀ07 0 a ﾀ縺06 閨0600 ｧ0504 閨04 ｧ07060 ｧ縺縺06, 06 ﾀ00 ｧ縺縺ﾀ07 ( ﾀ0304 縺0305 ｧｧ縺縺06 04) 縺06 閨0600 ｧ罍ｧｧｧﾀﾀ0304 縺0305 ｧｧ07, 04 縺縺09 ｧﾀ0903 縺03 ﾀ05 ｧ閨04 ｧ07060 ｧｧｧ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: ｧｧ罍ｧｧ07 y 00, y =0,09 - ﾀﾀ (0; 0) 04 縺 y = x. x 9. 6ｧ8 ｧ閨003 縺03 閨00306 陦ｧ縺06009 ｧｧ07 ﾀ 0403 ﾀｧｧ09 ｧｧ. 05 ﾀﾀﾀﾀｧ07: y (n) + y (n,) + :::+ n y = f (x) 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 縺ｧｧ07 (), y ﾀ縺ｧｧ07 (). ｯ縺ｧ06 y ｧ07 () 0306 閨0605 y = u + y : (3)

213 ｧ縺06009 ｧｧ縺06 閨ｧ09 ｧ罍ｧ 0403 ﾀ ,, :::, n ﾀ縺ｧ06 y 陦ｧ05 ﾀ閨縺06 閨ｧ09 ｧﾀ0006 閨06060 ｧ08 ﾀ : ) f(x) ; ) 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 ｧｧｧ08 ﾀﾀ縺ｧ06 y ｧｧ, 0503 ｧ f(x), ｧ 07 ﾀｧ09 ｧｧ. 0ｷ3 ﾀｧ07 ﾀ03 ﾀﾀﾀ0006 ｧ, : ) f(x) , 003 r = 縺罍ｧﾀ05 ｧ 06 ﾀｧﾀ05 ｧ k; ) f(x) =e mx (a cos nx + b sin nx), 003r = m ni 06 ﾀ縺罍ｧﾀ05 ｧ 06 ﾀｧﾀ05 ｧ k ｧ08 03 ﾀﾀ y ｧ 0605 ﾀ f(x) ｧ09 ｧｧ, ｧｧ x k 罍ｧ 09 ｧｧ 04 縺03 ｧﾀｧ0 04 罍ｧ縺ｧ閨003 縺03 閨ｧｧﾀｧ , ｧ00 ｧ罍ｧ 09 ｧｧ 04 縺03 ｧﾀｭﾀ00 ｧ y ｧ y ｧﾀｧ00306 ﾀ縺ｧｧ07 y 00 +y qy =0, 0503 縺ｧｧ07 y 00 +y 0 +qy = f(x) 閨06 6ｧ陦ｧ05 ﾀｧ閨06 y = Ay + By,05 閨06 A ｧ B ｧｧ x, 06 閨縺ｧ06 ﾀｧﾀｧ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 : ｧｧ07: 3. y 00, y 0 + y = e x : 4. y 00, 4y =8x 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 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.

214 3x 0. 7 罍ｧ006 縺03 陦ｧ縺06009 ｧｧｧ y y 00 =6x + e,x : 6. y IV, 8y =7e,3x : x + _x =3t : 8. y y = 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 : ｭ陦ｧ0 ｧ09 0 ﾀﾀ03 閨04 ｧﾀﾀｧ Ox 0403 閨0608 ﾀ0504 ｧﾀﾀｧ0003 a, ﾀｧ, 04 罍ｧﾀ0304 縺0305 ｧｧｧ閨04 ｧｧ06, ｧﾀﾀ0903 縺03 ﾀ05 ｧ閨04 ｧ07060 ｧｧ閨04 ｧｧ07, 06 ﾀ00 ｧ 04 罍ｧ t =0 ｧ x =0 ｧﾀ0903 縺03 ﾀ0504 v = ｧｧ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 = y IV +5y 00 +4y = 3 sin x. 45. y 000, 3y 00 +3y 0, y = e x. ｮ0006 閨06060 ｧｧ07 縺0600 ｧ閨罍ｧ 09 ｧｧ 04 縺03 ｧﾀ : 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 陦ｧ縺06009 ｧｧｧ縺06 閨0600 ｧｧ04 陦ｧ縺06009 ｧｧ07 ｧ縺0600 ｧ : 49. y 0 + y +x = e,x : 50. y 0 + y tg x =tgx. 5. (x, x 3 )y 0 +(x, )y = x 3.

215 3 ｪﾀ縺ｧ06 03 閨003 縺03 閨ｧ07 (04 罍ｧ f (x) =0) ｧｧ閨06 y = x r,05 閨06 r 0403 ﾀｧﾀ閨060 ｧ07 r ﾀ05 04 ｧ0504 y = x r 閨003 縺03 閨00306 陦ｧ縺06009 ｧｧ06 ｧ縺0600 ｧ罍ｧﾀ05 ｧ 06 ﾀｧﾀｧ r. 7 罍ｧ : ) 閨ﾀ縺006 r = a ﾀ05 ｧ m ﾀﾀ m ﾀ縺ｧ08 x a, x a ln x, x a (ln x), :::; ) 閨縺06 00 ｧ縺00608 r = i ﾀ05 ｧ m ﾀﾀ m 04 ﾀ縺ｧ08: x cos ( ln x); x cos ( ln x)lnx; ::: ; x sin ( ln x); x sin ( ln x) ln x; ::: ｧ縺06009 ｧｧ07 5. ( + x )y 0 + y(x, +x )= t ds +ts dt = e t dt: 54. xy 0 =4(y + y). 55. xyy 0 =y + y 4 + x 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 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 閨06 04 罍ｧ 0406 縺05 ｧｧｧ 03 ﾀｧｧ 0406 罍ｧｧ07, 縺ｧ05 ﾀ07, 06 ﾀ00 ｧ 09 ｧ00 ｧ0 陦縺06 ｧ閨06 ｧﾀ05 ｧ0504. ｮ0304 縺0305 ｧｧ06 閨04 ｧ07060 ｧ06 04 罍ｧ罍ｧ0300 ｧﾀ陦ｧ06 ﾀ 0 ｧ縺08003 ﾀ a ｧ a ｧﾀ縺縺縺08003 ﾀ05 ｧ 00 C ｧ 0 縺 C 縺06 閨0600 ｧ縺罍ｧﾀｧﾀﾀｧｧ r (a 6 r 6 a) ｧ 04 罍ｧ r =; 6a. ｰ09 0 ｧ06. ｮ0903 縺03 ﾀ閨060 ｧ縺03 dt 縺閨0 ｧ0906 ﾀ03 dr ﾀ05 09 ｧ030 縺0030 ﾀ04 縺06 閨ｧ縺縺縺09 ｧ陦ｧ縺ﾀｧ07. x. 6ｧ8 ｧ陦ｧ縺06009 ｧｧ n (n) + n, (n,) + + n, 0 + n = ( )

216 3x. ｮｧﾀｧ陦ｧ縺06009 ｧｧ閨003 縺03 閨00306 陦ｧ縺06009 ｧｧ縺ﾀ閨罍ｧ 09 ｧｧ 0403 ﾀｧｧ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. ｮｧﾀｧ陦ｧ縺06009 ｧｧ08 ﾀ 0403 ﾀｧｧ09 ｧｧｧｧ07: 75. dt + y =0,, dy dt dt 76. dt + x, y = et, =3x + y. dy dt, x + y = et. ｰ 09 0 ｧ縺03 陦ｧ縺06009 ｧ縺縺ｧｧ t, ｧﾀｧ 05 縺ｧ08 y ｧ dy dt , 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 ｧｧ07: 79. _x +3x + y =0, _y, x + y =0; x =;y =04 罍ｧ t =0: 80. _x = y, _y = x +sht.

217 ｧ縺06009 ｧｧ07 x 3. 6ｧ8 ｧ陦ｧ縺06009 ｧｧ07 04 ﾀ縺03 ｧ 0403 閨縺縺07 閨09 ( 罍ｧﾀ05 ｧ09) ｧ縺ｧ06 ( ﾀ03 閨06 縺閨縺03 ｧｧｧ) ｧ08: =0; =0; u, =0; =ax y + b. ｰ09 0 ｧ06. = z ｧﾀ縺ｧ z ﾀｧ070: z = y = y 04 罍ｧ x = 縺 u = ｧ 06 ﾀ縺006 ｧｧ縺ｧ縺ｧ = ｧ 06 ﾀ縺006 ｧｧ縺ｧ06. = ﾀ0006 閨06060 ｧ08 陦ｧ縺06009 ｧｧｧｧ06 縺ｧ07, 06 ﾀ00 ｧﾀｧ07, 0503 ｧﾀ縺ｧ07: +4@ =0. @y =0; u = sin = y 04 罍ｧ x =0; u =y = y 04 罍ｧ x

218 388. x 罍ｧﾀ05 ｧ09 5, =0; = x 04 罍ｧ t = ｧﾀ縺ｧ07 陦ｧ縺06009 ｧｧ08: u =0; u =,x, 04 罍ｧ t 90. 4a =0; u = ax 04 罍ｧ t 9. ; u = f = F (x) 04 罍ｧ t

219 ｧ9 ｭ, ｯｧ9 ｭ 0ｷｷｧ80ｷ3 7 ｭ 76ｧ9 ｭ 0ｷ3 7 ｯｭ06 7 縺6ｧ86ｧ9 x ｧﾀ00060 ｧ陦ｧﾀ閨ｧｧ縺06 縺ｧｧ F (x; y), ﾀ04 縺03- ﾀｧﾀ0504 (S) ﾀ0903 ﾀ05 ｧ xoy, ﾀ07 04 縺06 閨0600 ﾀﾀ閨縺00308 ｧﾀ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 ﾀｧ07 i ｧ k, 閨縺ｧ (x i ; y k ) 04 罍ｧ0 閨ﾀ05 ｧ (S) 閨04 S ﾀ05 ｧ (S) 0304 縺06 閨ﾀ縺 S = (S) dy:. ｭﾀ00 ｧﾀ0504 (S) 0304 縺06 閨ﾀﾀｧ a 6 x 6 b, y (x) 6 6 y 6 y (x), 04 罍ｧｧ縺06 縺罍ｧ y = y (x) ｧ y = = y (x) 0406 縺06 ﾀﾀ07 ﾀ 0406 縺05 ｧ x = X (x <X<x ) 閨 , 0503 b y (x) (S) F (x; y) dy = 05 閨06 04 罍ｧ 0403 ｧﾀ00060 ｧｧｧ a y (x) y (x) y (x) F (x; y) dy; F (x; y) dy ｧｧ006 x ﾀｭﾀ00 ｧﾀ0504 (S) 0304 縺06 閨ﾀﾀｧ h 6 y 6 l, x (y) 6 6 x 6 x (y), 04 罍ｧｧ縺06 縺罍ｧ x = x (y) ｧ x = = x (y) 0406 縺06 ﾀﾀ07 ﾀ 0503 罍ｧ y = Y (y <Y <y ) 閨 , 0503 l x (y) (S) F (x; y) dy = h dy x (y) F (x; y) ;

220 3x ｧﾀ00060 ｧ陦ｧﾀ閨ｧ閨06 04 罍ｧ 0403 ｧﾀ00060 ｧｧｧ x (y) x (y) F (x; y) ｧｧ006 y ﾀｭﾀ00 ｧﾀ0504 (S) 0304 縺06 閨縺罍陦ｧﾀｧ ' 6 ' 6 ', r (') 6 r 6 r ('), 閨ﾀ05 ｧ ' r (') S = rdrd'= d' rdr: (S) ' r (') 6ｦ0 04 ｧﾀ 0504 ﾀ閨ｧｧ 0403 ｧﾀ00 ｧ陦ｧ, ｧｧ0 ｧ070 ｧ: 9. xy =4, y = x, x = ) y = x, 4y = x, y =4;)y = x, 4y = x, x =. 94. y =4+x, x +3y = ay = x, ax, y = x. 96. y =lnx, x, y = ｧ y =, ﾀ05 縺03 ｧﾀ05 ｧ, 陦ｧ縺ﾀ07 ｧｧ: ) a x dy; ) a dy a,y ; 3) a,y 0 x 0ｷｧ縺07 閨0309 ｧ罍ｧ縺ｧ07. ｰ09 0 ｧ06. ｪｧｧ07 00 ｧ0 ｧ08, ｧｧｧﾀ0504, 縺06 閨ｧ罍ｧ x, 04 縺06 閨ｧ dy 04 罍ｧ y ﾀ05 縺03 ｧﾀ05 ｧ, 陦ｧ縺ﾀ07 ｧ0-,x ｧ: ) 0 x dy; ), dy y,4 a. a,x dy. 0ｷｧ縺07 閨0309 ｧ罍ｧ縺ｧ07 ｧ 0403 ｧﾀ00 ｧ陦ｧｧﾀ00 ｧ閨04, ｧｧ0 ｧ070 ｧ r = = a(, cos ') ｧ r = a ｧﾀ縺ｧﾀ00 ｧ閨04, ｧ縺 r cos ' = a ｧ 0309 縺ﾀ r =a ｧﾀ00 ｧ陦ｧ, ｧｧ0 ｧ070 ｧ: 30. xy = a, xy =a, y x =, y =x. ｰ09 0 ｧ閨縺ｧ罍陦ｧ xy = u ｧ y = vx, 0403 ﾀ閨縺06 閨ﾀ縺

221 , 05 縺ｧ 09 罍ｧｧ jjj du dv, 05 = ｧﾀｧｧ0504 y = ux, vy = x 3, 縺ｧ縺罍陦ｧ x = r cos 3 ' ｧ y = r sin 3 '. 30. y = ax, y =6ax, ay = x 3, 6ay = x x =3 + y =3 = a = ｧﾀ00 ｧ陦ｧ, ｧｧ0 ｧ070 ｧ: 304. y = x, y = x ax = y, ay ｧ y + x = y = sin x, y = cos x ｧ x = y = a, ax, y = a + x r = 4( + cos '), r cos ' =3( ﾀ縺 ) r = a(, cos '), r = a ｧﾀ罍陦ｧ03 ｧ閨 xy =, xy =8, y = x, y =8x ﾀ05 縺03 ｧﾀ05 ｧ, 陦ｧ縺ﾀ07 ｧｧ: ) b a x a dy; ) a 0 dy a,y ay ; 3) 4 0 8,x x dy. 0ｷｧ縺07 閨0309 ｧ罍ｧ縺ｧ07 ｧ 0403 ｧﾀ00 ｧ陦ｧ. x. ｨC ﾀﾀｧｧ006 縺09 ｧｧ陦ｧﾀ縺003 ﾀ04 縺06 閨ﾀﾀ0308 (04 罍ｧﾀ05 ｧ =) 罍陦ｧﾀﾀ陦ｧ S ﾀ縺003 ﾀ04 縺06 閨ﾀﾀ0308: RR RR xdy ydy x c = ; y c = : () S S ｧ006 縺09 ｧｧ陦ｧ S: J x = y dy; J y = x dy; J 0 = r dy: () (S) (S) (S)

222 3x. ｨC ﾀﾀｧｧ006 縺09 ｧｧ陦ｧ縺06 閨0600 ｧﾀﾀ陦ｧ, ｧｧ0 ｧ070 ｧ: 3. y =0 ｧ 03 閨ﾀｧ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 = 縺06 閨0600 ｧﾀﾀ陦ｧ, ｧﾀ05 縺03 ｧ- 閨0308 x =3 + y =3 = a =3 ｧ03 ﾀ0406 Ox. ｰ09 0 ｧ縺ｧ縺罍陦ｧ x = r cos 3 ' ｧ y = r sin 3 ': 縺06 閨0600 ｧｧ006 縺09 ｧｧ J x, J y ｧ J 陦ｧ 04 縺ｧ09, ｧｧ0 ｧ070 ｧ x = 0, x = a, y = 0 ｧ y = b 縺06 閨0600 ｧｧ006 縺09 ｧｧﾀｧﾀｧ Ox 陦ｧ, ｧｧ0 ｧ070 ｧ y = x=, x = a, y = a 縺06 閨0600 ｧｧ006 縺09 ｧｧﾀｧﾀｧ Oy 陦ｧ 05 縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(0; a), B(a; 0) ｧ C(a; a) { 縺06 閨0600 ｧ縺ｧ006 縺09 ｧｧ陦ｧ, ｧｧ0 ｧ070 ｧ: 30. x + y = a, x =0, y =0: 3. r = a cos ' 縺ﾀ r = a. 33. y = ax, x = a 縺06 閨0600 ｧﾀﾀ: ﾀ y = ax, x = a, y = 0 (04 罍ｧ y>0) ｧ04 ﾀ x a + y =,0305 ﾀﾀ0406 Ox. b 縺06 閨0600 ｧｧ006 縺09 ｧｧﾀｧﾀｧ Oy 陦ｧ, ｧｧ0 ｧ070 ｧ y = a + x, y =x ｧ x =0. a 縺06 閨0600 ｧｧ006 縺09 ｧｧﾀｧﾀｧ Ox 陦ｧ 05 縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(; ), B(; ), C(3; 3) 縺06 閨0600 ｧ縺ｧ006 縺09 ｧｧ陦ｧ, ｧｧ0 ｧ070 ｧ: 38. x a + y b =, x =0, y = y =4, x ｧ y =0: 330. r = a(, cos ').

223 , 05 縺ｧ 09 罍ｧｧｧ x ｧﾀ00060 ｧﾀ閨ｧ , ｧﾀ0406 縺縺08003 ﾀ z = F (x; y), ﾀ0 ｧﾀ0903 ﾀ z = 0 ｧﾀｧ00 ｧ0 陦罍ｧ 06 ﾀ縺08003 ﾀ050406, 0403 縺ﾀ0903 ﾀ05 ｧ xoy ﾀ0504 (S), V = zdy= F (x; y) dy: (S) (S) ｧﾀ00 ｧ , ｧ縺08003 ﾀ05070 ｧ: 33. z = x + y, x + y =4, x =0, y =0, z = z = x + y + a, y = ax, x = a, z =0, y =0(04 罍ｧ y>0) (x + y) + az = a, x =0, y =0, z =0( 縺08003 ﾀﾀ05 縺03 ｧﾀｧ070: x = 0, y = 0, z = 0, z = h 6 a; ﾀ ) x + y = a, x + z = a ( ﾀ ) z = xy, x = a, x =0, y = a, y = az = x, y, z =0, x = a z = xy, x + y = a x + y + z =3a, x + y = a, z =0. ｰ 09 0 ｧ { 縺ｧ縺罍陦ｧ z = mx, x + y = a, z = az = a, x, y, z = x + y + z =4a, x + y = a ( ｧ00 ｧ0 陦 ). 34. x + y + z = a, x + y ax =0( 罍ｧ 09 ｧ00 ｧ0 陦縺0304) 縺ｧｧ0903 ｧ y = x tg z 罍ｧ 09 ｧ00 ｧ0 陦 a x + y = a ｧﾀ0903 ﾀ z = z =ax, x + y = ax z x =, c a, y b, z =0. ｰ 09 0 ｧｧ縺ｧ ( ｧｧ 06 ﾀ09 ｧ0) 縺罍陦ｧ0 05 0: x = ar cos ', y = br sin ' z = ce (,x =a ),(y =b ) ｧ x a + y b = x =3 + y =3 + z =3 = a =3 ( ｧ0504 x = r cos 3 ', y = = r sin 3 ').

224 3x 陦ｧ 09 罍ｧ縺08003 ﾀｧﾀ00 ｧ , ｧ縺08003 ﾀ05070 ｧ: 348. z = a, x, y = ax ｧ z = z = x + y, y = x, y =, z = y + z =4ax, y = ax, x =3a ( ｧ00 ｧ0 陦 ). 35. x a + z b =, x a + y b = ｧ x y + h z = a y 04 罍ｧ 0 6 y 6 h ( ﾀ ) x =3 + z =3 = a =3, x =3 + y =3 = a = z =6, x, y, z =0, x + y =4( ｧ00 ｧ0 陦 ). ｰ 09 0 ｧ { 縺ｧ縺罍陦ｧ z =(x + a), x + y = a z = 4 x + y, z =0, x + y =, x + y = az = x + y, z =0, x + y ax = az = a, x, y, z = 0, x + y ax = 0( 罍ｧ 09 ｧ00 ｧ0 陦縺0304) x a + y b + z c =. ｰ09 0 ｧｧ0504 x = ar cos ', y = br sin '. x 陦ｧ 09 罍ｧ縺08003 ﾀ閨04 ﾀ05 ｧ縺08003 ﾀ05 ｧ F (x; y; z) =0, 04 縺ｧ縺ﾀ0903 ﾀ0504 z =00304 縺06 閨ﾀ0504 ( z ), 040 (@F=@x) +(@F=@y) +(@F=@z) = dy = j@f=@zj ( z) = ( z) N dy: j@f=@zj 縺ｧ罍ｧ 04 縺ｧ縺ｧｧ 0 閨0406 陦縺0605 ｧ罍陦ｧﾀﾀ05 ｧｧ0 = ( y ) N dz; j@f=@yj = ( x) N dy dz: j@f=@xj

225 , 05 縺ｧ 09 罍ｧｧｧｧﾀ00 ｧ閨04: 縺08003 ﾀ05 ｧ 09 ｧ00 ｧ0 陦 z = x,0305 ﾀﾀ0903 ﾀ05070 ｧ y = x=, y =x, x = 縺08003 ﾀ05 ｧﾀ z = xy, 0305 ﾀﾀ0903 ﾀ05070 ｧ x = a ｧ y = a, 04 罍ｧ x > 0 ｧ y > 縺08003 ﾀ05 ｧﾀ y + z = x, ﾀ罍ｧ 09 ｧ00 ｧ0 陦 x + y = a 縺08003 ﾀ05 ｧ az = xy, ﾀ罍ｧ 09 ｧ00 ｧ0 陦 x + y = a 縺08003 ﾀ05 ｧﾀ x + y = z, ﾀ罍ｧ 09 ｧ00 ｧ0 陦 z =x ｧﾀ00 ｧ閨04: 縺08003 ﾀ05 ｧ 09 ｧ00 ｧ0 陦 x + z = a, ﾀ罍ｧ 09 ｧ00 ｧ0 陦 x + y = a 縺08003 ﾀ05 ｧ 00 x + y + z = a, ﾀ罍ｧ 09 ｧ00 ｧ0 陦縺0304 x + y ax = 縺08003 ﾀ05 ｧｧ x + y =az, ﾀ罍ｧ 09 ｧ00 ｧ0 陦 x + y =3a ｮ閨ｧ縺06 閨0600 ｧ閨04 - ﾀ05 ｧ縺08003 ﾀ05 ｧ, ｧ罍ｧ陦ｧ 0 0 ｧ 0 ｧ, 縺030 ｧﾀﾀ00305 縺ﾀﾀ罍ｧ =30, =60. x 5. ｯ縺ｧｧ罍ｧｧ07 ｯ縺ｧｧｧ f(x; y; z), ﾀ04 縺03 ﾀﾀ0504 (V ), ﾀ07 04 縺06 閨0600 ﾀﾀ縺縺00308 ｧﾀ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- ﾀｧ07 j ｧ k, 閨縺ｧ (x i ; y j ; z k ) 04 罍ｧ0 閨ﾀ05 ｧ (V ). ｭﾀ00 ｧﾀ0504 (V ) 0304 縺06 閨ﾀｧ 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:

226 3x 5. ｯ縺ｧｧ罍ｧｧ罍ｧ F (x; y; z) = V 罍陦ｧﾀﾀ 03 閨003- 縺03 閨 V 0403 ｧﾀﾀ縺 : x c = V (V ) xdydz; y c = V (V ) ydydz; z c = V (V ) zdydz: 縺06 閨0600 ｧ , ｧ縺08003 ﾀ05070 ｧ az = x + y, az = a, x, y 縺06 閨0600 ｧ , ｧ縺08003 ﾀ05070 ｧ x + y, z =0, x + y + z = a, 罍ｧﾀ , 縺08003 ﾀﾀ x + y, z =0 閨0600 ｧ x + y + z =az ｧｧ 3: 縺06 閨0600 ｧﾀﾀ06 04 ｧ 0 ｧ閨03, ﾀ0903 ﾀ05070 ｧ x + y + z = a, x =0, y =0, z =0, 06 ﾀ00 ｧﾀ閨ｧ z ｧ縺06 閨0600 ｧﾀﾀ 03 閨003 縺03 閨 , ｧ縺08003 ﾀ05070 ｧ: 373. x + y + z = a, x =0, y =0, z = az = a, x, y, z = 縺06 閨0600 ｧｧ006 縺09 ｧｧﾀｧﾀｧ Oz , ｧ縺08003 ﾀ05070 ｧ ( ﾀ0504 =): 375. x =0, y =0, y = a, z =0, x + z = a x + y + z = a, x + y = a, z = 縺06 閨0600 ｧ , ｧ縺08003 ﾀ050406: ) (x + y + z ) = a 3 x; ) (x + y + z ) = az(x + y ). ｰ09 0 ｧ縺ｧ 09 ﾀ0706 罍ｧ 06 ﾀ09 ｧ罍陦ｧ縺 x = = r sin cos ', y = r sin sin ', z = r cos '; dv = = r sin drd'd 縺06 閨0600 ｧ , ｧ縺08003 ﾀ05070 ｧ: 378. az = x + y, x + y + z =a x + y, z =0, z =6, x, y az = x + y, z = x + y 縺06 閨0600 ｧﾀﾀ , ｧ縺08003 ﾀ05070 ｧ x + y, z = 0 ｧ z = h, 06 ﾀ00 ｧﾀ閨ｧｧ.

227 , 05 縺ｧ 09 罍ｧｧｧ縺06 閨0600 ｧﾀﾀ , ｧ縺08003 ﾀ05070 ｧ x + z =a; x + z = a; y = ax; y =0 (04 罍ｧ y>0), 06 ﾀ00 ｧﾀ閨罍陦ｧ y ｧ縺06 閨0600 ｧﾀﾀ 03 閨003 縺03 閨 x + y + + z = a, z = 縺06 閨0600 ｧｧ006 縺09 ｧｧﾀｧﾀｧ Oz , ｧ縺08003 ﾀ05070 ｧ z =ax, z =0, x + y = ax 縺06 閨0600 ｧ , ｧ縺08003 ﾀ (x + y + z ) = axyz (0406 縺ｧ 09 ﾀ0706 罍ｧ 06 ﾀ09 ｧ罍陦ｧ0 05 0) ( ﾀ ) 縺06 閨0600 ｧﾀﾀ06 ﾀ0706 罍ｧ 06 ﾀﾀ閨縺ﾀ05070 ｧ x + y + z = a ｧ x + y + z = 4a, 06 ﾀ00 ｧﾀ閨縺縺09 ｧﾀﾀｧｧ閨罍陦ｧ0 05 (0406 縺ｧ 09 ﾀ0706 罍ｧ 06 ﾀ09 ｧ罍陦ｧ0 05 0). x 罍ｧｧｧｱ03 縺罍ｧ縺06 閨ｧ06 09 罍ｧｧｧﾀ閨 AB, ^ ﾀ04 縺罍ｧ040308, 0304 縺06 閨縺06 縺ｧ07 P (x;y;z) 閨ﾀ05 ｧｧ 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 ﾀ07 09 罍ｧｧｧ , 閨 AB, ^ ｧﾀ07 R^AB P (x; y; z). 縺ｧ縺06 閨ﾀ07 ｧ R^AB R R Q(x; y; z) dy, R(x; y; z) dz ｧ P+ Qdy + + Rdz ﾀ縺06 閨03 閨060 ｧ08 ｧ , 04 ﾀ05 縺ﾀ罍ｧｧｧｧ ^ 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 : ｧﾀ00060 ｧ06 09 罍ｧｧｧﾀ罍ｧ AB ^ ｧ070 ｧ x = f(t), y = '(t), z = (t), t 04 罍ｧ 0406 縺ｧｧｧ M (t) 0403 閨 AB ^ 閨ｧｧｧﾀ ; ^ AB P (x; y; z) = t B t A P f(t); '(t); (t) f 0 (t) dt;

228 3x 罍ｧｧｧｱ03 縺罍ｧﾀ縺ｧ陦ｧ縺06009 ｧ罍ｧｧｧｧ縺06 03 閨縺 (t) ｧ 0606 陦ｧ縺06009 ｧ 00 (dt) ｧｧ08 09 罍ｧｧ 06 ﾀｧ06 09 罍ｧｧｧｷｧ R^AB P+ Qdy+ Rdz 0304 縺06 閨罍ｧ 0406 縺ｧｧ 06 陦ｧ0 ｧﾀﾀ閨 AB , ﾀｧ FfP ; Q; Rg. 4. ｮ陦ｧ縺06009 ｧ 00. ｭﾀ00 ｧ縺ﾀ05 ｧ (V ) P+ Qdy + Rdz = du, 0503 R^AB P+ Qdy + Rdz = u B, u A, ﾀ05 ｧｧｧｧ u(x; y; z) B ｧ A ｧｧﾀｧｧｧ罍ｧ縺ｧ07 AB, ﾀ05 ｧ (V ). 5. ｱ03 縺罍ｧ0 I (C) P+ Qdy @x, 04 縺罍ｧｧｧ P+ Qdy, (04 縺0305 ｧ04 ﾀﾀ05 縺ｧ) 縺06 (C), 04 閨ｧﾀ05 ｧ (S), ｧｧ縺 , ｧ縺030 (C): S = I (C) xdy, y : ｧ A(; ) ｧ B(; 0) ｧﾀ00 ｧ0504 (C) (x + y) : ) 縺 OA; )0403 閨 OA ^ y = x ; 3) OBA ｧ A(4; ) ｧ B(; 0) ｧﾀ00 ｧ0504 (C) (x + y), xdy : ) 縺 ^ OA; ) OBA ｧ閨0007 ｧ (C) y+ xdy: 閨06 ﾀｧｧ0 ｧｧﾀｧｧｧ罍ｧ縺ｧ07?

229 , 05 縺ｧ 09 罍ｧｧｧｧ A(a; 0;0), B(a; a; 0) ｧ C(a; a; a) ｧ- ﾀ00 ｧ0504 ｧ y+ zdy+ xdz 縺 OC ｧ OABC ﾀｧ FfP ; Qg, 05 閨06 P = x, y, Q = = x. 703 ﾀ05 縺03 ｧ0504 ﾀｧ0006 F 閨縺00 ｧ陦 05 ﾀ03 ﾀ0503 縺030 0 ｧ x = a ｧ y = a ｧ 0403 ｧﾀ00 ｧ罍ｧ 0406 縺ｧｧ 06 陦ｧ0 ｧﾀﾀ縺陦ﾀｧ FfP ; Qg, 05 閨06 P = x + y, Q = = x. 703 ﾀ05 縺03 ｧ0504 ﾀｧ0006 F 閨縺05 ｧ 0309 縺ﾀ05 ｧ x = a cos t, y = a sin t ｧ 0403 ｧﾀ00 ｧ罍ｧ 0406 縺ｧｧ 06 陦ｧ- 0 ｧﾀﾀ縺ﾀ05 ｧｧ罍ｧ 06 ﾀｧｧ P = x + y, Q = x 閨06 ﾀ ? ﾀｧ Ffy; ag 縺06 閨0600 ｧ罍ｧ 0406 縺ｧｧ 0 ﾀﾀ03 m 縺06, ﾀ070 ｧ罍陦ｧ0 05 ｧ 0406 縺縺ｧ04 ﾀ x = a cos t, y = b sin t ﾀｧ Ffx; y; zg ｧﾀ00 ｧ罍ｧ 0406 縺ｧｧ 06 陦ｧ0 ｧﾀﾀ OABCO, ﾀ0306 陦ｧｧ O(0; 0; 0), A(0; a; 0), B(a; a; 0), C(a; a; a) ｧﾀ 0504 ｧ 04 縺罍ｧ縺罍ｧ0 閨0007 I (C) (x + y), xdy 縺06 05 縺ｧ09 ﾀ03 ﾀ0503 縺030 0 ｧ x =0, y =0, x + y = a ｧﾀ00 ｧ0504 ｧ : ) ^ 3) AB ^ AB xy + x dy; ) tg y+ x sec ydy ^ AB cos y, x sin ydy; ｧ0 ｧｧｧ A(; =6) 閨03 B(; =4) 罍ｧ0060 ｧ縺罍ｧ0, 0403 ｧﾀ00 ｧ0504 ｧ I (C) y +(x + y) dy 縺06 4ABC ﾀ 0406 縺00 ｧ0 0 ｧ A(a; 0), B(a; a) ｧ C(0; a).

230 3x 罍ｧｧｧｱ03 縺罍ｧ縺06 閨0600 ｧ罍ｧｧｧ閨ｧ- 04 ﾀ x = a cos t, y = b sin t 縺06 閨0600 ｧ罍ｧｧｧ閨ｧ 09 罍ｧ x 3 + x, y =0( ﾀ0. 罍ｧﾀﾀ. 304). ｰ09 0 ｧ縺ｧ罍ｧ 06 ﾀ09 ｧｧ070, ｧ04 y = = xt 縺06 閨0600 ｧ罍ｧｧｧ閨ｧ閨0609 縺ｧﾀ05 x 3 + y 3, 3axy =0( ﾀｧｧ罍ｧﾀﾀ. 334). 40. ｮﾀｧ罍ｧｧﾀﾀ M, 縺003 ﾀ- 04 縺06 閨縺縺ﾀ05 ｧ x + y = a, 0 ﾀﾀ06 m, ﾀ03 ﾀ縺06 閨罍陦ｧ0 05? ｰ09 0 ｧﾀｧﾀ0504, ds 閨00 ｧ縺ﾀ05 ｧ, 陦ｧ06 ﾀﾀ 03 ﾀ0406 Ox, X ｧ Y 04 縺ｧｧﾀｧ罍ｧｧ07. ｯ0305 X = (C) km cos ds r ; Y = 05 閨06 k ｧ05 09 ｧﾀ (C) km sin ds r ; ｧ A(,a; a) ｧ B(a; a). ｮﾀｧﾀﾀ M, 縺003 ﾀ04 縺06 閨縺 AB, 04 罍ｧｧﾀﾀ06 m, ﾀ03 ﾀ縺06 閨 (0; 0) ｧ A(a; 0), B(0; a) ｧ C(,a; 0). ｮﾀｧﾀﾀ M, 縺003 ﾀ04 縺06 閨 ABC, 04 罍ｧｧﾀﾀ06 m, ﾀ03 ﾀ縺06 閨罍陦ｧｧ A(0; ), B(; 5) ｧ C(0; 5) ｧﾀ00 ｧ0504 (C) (x + y), ydy : ) 縺 AB; ) 0403 閨 AB y = x +; 3) ACB ｧ A(,a; 0) ｧ B(0; a) ｧﾀ00 ｧﾀｧ0003 FfP ; Qg, 05 閨06 P = y ｧ Q = y, x, 04 罍ｧ 0406 縺ｧｧ 06 陦ｧ0 ｧﾀﾀ03: ) 縺 AB; ) AOB; 3)0403 閨 AB y = a, x a , 0503 I (C) y+(x + y) dy

231 , 05 縺ｧ 09 罍ｧｧｧ縺縺罍ｧ0504, 0403 ｧﾀ00 ｧ04 ｧ縺06 07 ｧ0506 縺03, ｧｧ0 ｧ070 ｧ y = x ｧ y = ｧﾀ 0504 ｧ 04 縺罍ｧ縺罍ｧ0 閨0007 ｧ I (C) y, dy x ; 縺06 4ABC ﾀ 0406 縺00 ｧ0 0 ｧ A(; ), B(; ) ｧ C(; ) ｮ罍ｧｧｧ縺06 閨0600 ｧ閨04 07 ｧ0506 縺03, ｧﾀ05 縺03 ｧ閨0308 x = a cos 3 t, y = a sin 3 t ｮ罍ｧｧｧ縺06 閨0600 ｧ閨04, ｧ罍ｧ y + x 4, x =0.( 706 縺ｧ罍ｧ 06 ﾀ09 ｧｧ070, ｧ04 y = xt.) x 縺08003 ﾀｧｱ03 縺ﾀ05 縺0305 陦ﾀ {06 06 ﾀﾀｧｮﾀ縺08003 ﾀｧﾀ0504 F (x; y; z) 縺06- 縺ｧ07 ｧ z = '(x; y) ｧ縺08003 ﾀ05 ｧ S, 04 罍ｧ y) ﾀ縺08003 ﾀｧ縺ｧ04 04 縺06 陦ﾀﾀ縺06 閨0600 ｧﾀ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 閨04 i 縺08003 ﾀ05 ｧ S, (x i ; y i ; z i ) 04 罍ｧ- 0 閨ｧ , 04 罍ｧﾀｧ陦ｧｧ060 ｧ07 ﾀ05 縺060 ｧ05 ﾀｭﾀ00 ｧ 04 縺ｧ縺08003 ﾀ05 ｧ S ﾀﾀ0504 xoy 03 閨 , ﾀ縺070 07, ﾀｧ Oz, 0406 縺06- ﾀ縺08003 ﾀ0504 S 00 ｧ閨 , 0503 ﾀﾀｧ縺ﾀｧ縺ｧｧﾀ縺 (S) F (x; y; z) ds = = () F x; y; '(x; y) s y) ｭﾀ00 ｧ P = P (x; y; z), Q = Q(x; y; z), R = R(x; y; z) 縺06- 縺ｧｧｧ S + ﾀ0503 縺縺08003 ﾀ05 ｧ S, 罍ｧｧ縺0 00 ｧ nfcos ; cos ; cos g,0503 ﾀﾀｧ08

232 3x 縺08003 ﾀｧ縺08003 ﾀｧ縺縺ﾀ07 ﾀ0006 閨06060 ｧ : Pdydz+ Qdz+ Rdy = (P cos + Q cos + R cos ) ds: (S + ) (S) (S). ｱ03 縺ﾀ05 縺0305 陦ﾀ {06 06 ﾀﾀ (P cos + Q cos + R cos ) ds + + (V ) dy dz; 05 閨06, ｧ縺0 00 ｧ縺08003 ﾀ05 ｧ S ｧ 03 ﾀ070 ｧ罍陦ｧ0 05, V , ｧ縺08003 ﾀｧｧﾀｧ閨06 (S z ; 05 閨06 F (x; y; z) = ｧ縺08003 ﾀ05 ｧ, S z 04 縺ｧ07 S ﾀ0903 ﾀ0504 xoy. 3. ｱ03 縺00600 ｮﾀ : I (C) = (S) P+ Qdy+ @x, cos 05 閨06,, , ﾀ070 ｧ罍陦ｧ0 05 ﾀ 003 縺縺08003 ﾀ05 ｧ S, ﾀ0503 縺03006, ﾀ縺 C ﾀﾀ0 05 罍ｧﾀ07 04 縺03 ｧﾀ0803 閨070 ｧ0 04 縺0305 ｧ04 ﾀﾀ05 縺ｧｧﾀ00 ｧ0504 (S) (x cos + y cos + z cos ) ds 縺縺08003 ﾀ05 ｧﾀ0903 ﾀ05 ｧ x + y + z = a, ﾀ縺ｧﾀ00 ｧ0504 (S) x cos (n; i) +y cos (n; j) +z cos (n; k) ds 縺縺08003 ﾀ05 ｧｧ x + y +az = a, ﾀ縺 (05 閨06 x<0, y>0, z>0).

233 , 05 縺ｧ 09 罍ｧｧｧｰ 09 0 ｧ罍ｧ0406 閨07 ｧｧ閨06 (x 3 + y 3 + az dy ) a, (S z) 縺ｧ縺罍陦ｧｰ ' 0306 閨0605 ｧﾀ = 閨ｧﾀ 0504 ｧ 04 縺罍ｧ縺ﾀ05 縺0305 陦ﾀ閨0007 ｧ (S) x cos (n; i) +y cos (n; j) +z cos (n; k) ds; 縺08003 ﾀ05 ｧ 00 x + y + z = a ｧﾀ 0504 ｧ 04 縺罍ｧ縺ﾀ05 縺0305 陦ﾀ {06 06 ﾀﾀ閨0007 ｧ (S) x cos (n; i) +y cos (n; j) +z cos (n; k) ds; 縺縺08003 ﾀ05 ｧ , ｧ縺ﾀ05070 ｧ x + y +az = a, x =0, y =0, z =0, 罍ｧ 0406 縺ｰ 09 0 ｧｧﾀ09 ｧ , ｧ0303, 0 04 罍ｧ006, ﾀ0903 ﾀ05 ｧ z =0 ｧ cos (n; i) =0 ｧ cos (n; j) = 縺ﾀ05 縺0305 陦ﾀ {06 06 ﾀﾀ P = x, Q = y, R = z, ｧ縺閨 : V = [x cos + y cos + z cos ] ds: 3 (S) ｧﾀ00 ｧ縺ｧ04 ﾀ03 ｧ x a + y b + z c =: 縺ﾀ05 縺0305 陦ﾀ {06 06 ﾀﾀ Q ｧ R ( fp ; Q; Rg grad 閨 , 0503 (V ) udydz = (S) du dn ds; 05 閨06 u u ｧ

234 3x 縺08003 ﾀｧ縺罍ｧ縺06 閨03 閨縺閨ｧｧ u = x + y + z 縺08003 ﾀ05 ｧ x + y + z = a ﾀ縺ｮﾀ, 0503 (C) yz + xz dy + xy dz 縺縺罍ｧｧﾀ00060 ｧ060 ｧ縺06 4OAB ﾀ 0406 縺00 ｧ0 0 ｧ O(0; 0; 0), A(; ; 0) ｧ B(; ; ) ｧﾀ 0504 ｧ 04 縺罍ｧ縺ｮﾀ閨0007 ｧ I (C) (z, y) +(x, z) dy +(y, x) dz; 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 ｧｧ縺08003 ﾀ05 ｧ, 04 縺閨縺罍ｧ縺ｧ09 ABC, 0 04 罍ｧﾀ0903 ﾀ05 ｧ x + y + z = a ｧﾀ 0504 ｧ 04 縺罍ｧ縺ﾀ05 縺0305 陦ﾀ {06 06 ﾀﾀ閨0007 ｧ (S) x 3 cos (n; i) +y 3 cos (n; j) +z 3 cos (n; k) ds; 縺08003 ﾀ05 ｧ 00 x + y + z = a. ｰ 09 0 ｧ 06. ｯ縺ｧ縺ﾀ0706 罍ｧ 06 ﾀ09 ｧ陦ｧ I ｧﾀ 0504 ｧ 04 縺罍ｧ縺ｮﾀ閨0007 ｧ (C) x(z, y) + y(x, z) dy + z(y, x) dz 縺06 05 縺ｧ09 ﾀ 0406 縺00 ｧ0 0 ｧ A(a; 0; 0), B(0; a; 0) ｧ C(0; 0; a) ( ﾀｧ ). 4. ｮ縺ﾀ05 縺0305 陦ﾀ {06 06 ﾀﾀ 0403 ｧﾀ00 ｧ0504 (S) x 3 dy dz + y 3 dz + z 3 dy; 縺縺08003 ﾀ05 ｧ 04 ｧ 0 ｧ閨03, ﾀﾀ05070 ｧ x + y + z = a, x =0, y =0, z =0.

235 ｧ9 x. ｪｧﾀ縺07 閨 u + u + u 3 + :::+ u n + ::: ﾀ07 ﾀ0803 閨070 ｧ0 ﾀ07, 06 ﾀ00 ｧﾀ0600 S n n 0406 縺罍ｧ n! ﾀ05 縺060 ｧ05 ﾀ縺06- 閨 S: lim n! S n = S. ｪｧﾀ0003 S ﾀ07 ﾀﾀ0803 閨ﾀ07 縺ﾀ0803 閨070 ｧ08 ﾀ07 縺ﾀ07 ﾀ0803 閨070 ｧ0 ﾀﾀ0803 陦ｧ003 ﾀ05 ｧ縺陦ｧ003 ( 閨03 ﾀ ), u n! 0 04 罍ｧ n!.. 0ｷ罍ｧ 0 09 ﾀ0803 陦ｧ003 ﾀ05 ｧ縺07 ﾀｧｧｧ0 ｧｧ: ｭﾀ00 ｧ u n = f(n), 05 閨06 f(x) ｧ07, ｧ A; 0503 縺07 ﾀ0803 陦ｧ05 ﾀ07, f(x) = ; 0503 縺07 ﾀ0803 陦ｧ05 ﾀ罍ｧﾀ0803 陦ｧ003 ﾀ05 ｧ縺07 ﾀｧｧｧ: 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, =; 縺03 ﾀ 03 ﾀﾀ縺ｮｧ06 縺07 閨0304 ﾀｧｧｧ: 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 () ﾀ 06 縺06 閨06060 ｧ0 ｧﾀｧ 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 ﾀﾀ0803 陦ｧ003 ﾀﾀ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)

236 3x. ｪｧﾀ縺07 閨ﾀ縺07 (3) ﾀ07 03 ﾀﾀ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) ﾀ07 06 ﾀ ( ﾀ ) ﾀ0803 閨070 ｧ0 ﾀﾀ07 00 ｧ陦ｧﾀｧ06 ﾀ0803 陦ｧ003 ﾀ05 ｧ縺07 : ::: ::: ::: 0ｷ3 ﾀﾀ0006 閨ｧ罍ｧﾀ0803 陦ｧ003 ﾀ0504 縺07 : ::: ::: ::: ::: ::: , + 5, + 7, + ::: 43. ln + 3ln 3 + 4ln 4 + ::: 0ｷ3 ﾀﾀ0006 閨罍ｧﾀ0803 陦ｧ003 ﾀ0504 縺07 : ::: ! + 4 3! + 8 4! + ::: ::: ::: ! 4 + 5! ! ::: :::

237 閨03 ｮｧ060 ﾀ 05 縺0030 ｧ 06 ﾀ09 ｧ0 縺07 閨030 ｧ00 ｧﾀ縺0305 縺06 ﾀ- ﾀｧ0608 ｧﾀﾀ0006 閨ﾀ0803 陦ｧ003 ﾀ0504 縺07 : ::: ::: 440. ln + ln 3 + ln 4 + ln 5 + ::: 閨030 ﾀｧ07 縺07 閨 , 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 ｧﾀｧ縺04030 ﾀｧ0504 x, x 4, x 6, ::: 06 陦ｧ0 ｧ09 0 ｧ, 縺030 ﾀ縺03 ﾀｧ陦ｧ0 ｧｧﾀ縺07 : ::: ｰ09 0 ｧｧ0504 u n 縺00306 陦縺0303 ｧ ::: 0ｷ3 ﾀﾀ0006 閨ﾀ0803 陦ｧ003 ﾀ0504 縺07 : 444., + 3, 4 + ::: 445., 3 + 5, 7 + ::: sin, ln 3ln3 +, ::: 4ln4 + sin + sin ::: , 0503 ﾀ0600 S 06 ﾀﾀ0803 閨ﾀ07 縺07, + +, + ::: ｧ05 ﾀ07 04 閨040306, 06 ﾀ00 ｧ 0403 ﾀ閨ｧ縺ﾀ05 ｧ0504 閨ﾀ0006 閨06060 ｧ罍ｧ , ｧｧｧ05 ﾀ , 06 ﾀ00 ｧ 0403 ﾀ閨0308 閨ｧﾀ05 ｧ陦ｧ罍ｧ

238 3x 縺0 07 ﾀ0803 陦ｧ003 ﾀｧ縺ｷ3 ﾀﾀ0006 閨ﾀ0803 陦ｧ003 ﾀ0504 縺07 : ::: ::: ::: ::: ::: ::: ::: ! + 6 5! + ::: 457., 3 + 5, ::: 458., , ::: 459., a + 3a 4, 4a 6 + ::: ｧﾀ縺07 : ::: ::: x 縺0 07 ﾀ0803 陦ｧ003 ﾀｧ縺07. ｮﾀｧ08 x, 04 罍ｧ縺ｧ縺07 u (x)+u (x) +:::+ u n (x) +::: () ﾀ0803 陦ｧ05 ﾀ07, ﾀﾀﾀ0803 陦ｧ003 ﾀ05 ｧ縺07. ｱｧ07 S(x) = lim n! S n(x) ﾀﾀ , 003 ﾀ0504 R n (x) =S(x), S n (x) 03 ﾀ縺07.

239 閨 () ﾀ縺003 ﾀ0803 閨070 ｧ0 ﾀ縺 [a; b], 06 ﾀ00 ｧ閨ﾀ "> ｧ N, 罍ｧ n>n ｧ x 縺 [a; b] 0306 閨ﾀ jr n (x)j <" 罍ｧ縺00308 ﾀ0803 陦ｧ003 ﾀ05 ｧ. 707 () ﾀ0803 陦ｧ05 ﾀ07 03 ﾀｧ縺縺 [a; b], 06 ﾀ00 ｧﾀ06006 ﾀｧﾀﾀ0803 閨070 ｧ08 ﾀ07 縺07 ﾀｧｧｧ c + c + c 3 + :::+ c n + ::: , 0503 ju n (x)j 6 c n 04 罍ｧ a 6 x 6 b 縺06 閨0600 ｧ罍ｧ jxj < ﾀｧ03 ﾀ縺07 +x + + x + x 3 + ::: ｧ , ﾀ0803 陦ｧ05 ﾀ縺縺 [0; =]. 7 罍ｧ n 03 ﾀ jr n (x)j < 0; 00 閨 x 縺 ? , 0503 縺07 x + x(, x) +x(, x) + x(, x) 3 + ::: ﾀ0803 陦ｧ05 ﾀ縺縺 [0; ] ｧ縺縺 [=; ]. 7 罍ｧ n 03 ﾀ jr n (x)j < 0; 0 閨 x 縺 [=; ]? , 0503 縺07 x, x + x3, ::: ﾀ0803 陦ｧ05 ﾀ縺縺 [0; ]. 7 罍ｧｧ08 n ｧ x 縺 jr n (x)j < < 0;? , 0503 縺07 x 3 + x3 +x 3 + x 3 ( + x 3 ) + ::: ﾀ0803 陦ｧ05 ﾀ縺罍ｧ x>0 ｧ縺罍ｧ x >. 7 罍ｧ n 03 ﾀ jr n (x)j < 0; 00 閨 x >? , 0503 縺07 + +x 3 +3x x ::: ﾀ0803 陦ｧ05 ﾀ縺ｧ00506 縺 x<. +7x 7 罍ｧ n ( ｧ罍ｧ x) 03 ﾀ縺07 jr n (x)j < < 0; 0? ｰ 09 0 ｧ 06. ｮ 040 ｧ縺07 ﾀｧﾀﾀ0803 閨070 ｧ0 ﾀ07 縺07 閨 , 0503 縺07 x, + x +4 + x, +9 x +::: +6 ﾀ0803 陦ｧ05 ﾀ縺ﾀ0608 ｧﾀﾀｧ. 7 罍ｧ n ( ｧ x) 03 ﾀ縺07 jr n (x)j < 0; 000?

240 , 0503 縺07 x 3. ｮ縺07 閨03 47 x(x +) + (x + )(x +) + (x + )(x +3) + ::: ﾀ0803 陦ｧ05 ﾀ縺ｧ00506 縺 <x<. 7 罍ｧ n ( ｧ x x>0) 03 ﾀ縺07 jr n (x)j < 0;? , 0503 縺07 +x + +x x x + ::: ﾀ0803 陦ｧ05 ﾀ縺ｧ00506 縺 x<. 7 罍ｧ n 03 ﾀ縺07 jr n (x)j < 0; 0? 706 ﾀﾀ縺07 x 3. ｮ縺07 閨03 a 0 + a x + a x + :::+ a n x n + ::: () ｪｧﾀ0003 R 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 ｧｧ00 ｧｧﾀﾀ0006- 閨ｧﾀﾀ0803 陦ｧ003 ﾀ05 ｧ縺07 () 罍ｧ , ｧ00 ｧ, ﾀ06 a i ｧ , 縺 R = lim. 6 5 ﾀ05003 ﾀ05 ｧ, a n+ n! a n 06 ﾀ00 ｧ縺06 閨 , 0503 縺07 () 03 ﾀﾀ0803 陦ｧ05 ﾀﾀﾀｧ Ox. ｮ縺07 ﾀ0803 陦ｧ05 ﾀﾀ , 003 ｧ縺縺 [a; b], 罍ｧｧ00506 縺04 00 ﾀ0803 陦ｧ003 ﾀ05 ｧ (,R; R) 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 閨ﾀ0803- 陦ｧ003 ﾀｧ09 08 ｧ00506 縺04 00 : x 3 + x x ::: 47., x 5 + x3 5, x ::: x x x ::: P x n P 473. n= : n! 474. (,x) n,. n= n P 3 n x n 475. n= (3n,. ) n

241 閨 ) P n= xn, n!; ) P n= (x +) 477. (x +) x, 3 n!x n (n +) n. (x +)3 (x +) ::: (x, 3) (x, 3)3, +, ::: 縺06 閨0600 ｧ0504 ｧ00506 縺04 00 ﾀ0803 陦ｧ003 ﾀ05 ｧ縺07 ｧｧﾀ060006: x +3x +4x 3 + ::: ｰ09 0 ｧ閨060 ｧ07 ﾀ S ｧﾀ x, x3 3 + x5 5, x7 7 + ::: ｰ 09 0 ｧｧﾀ0 00 ds. x 0 S x +5x +7x 3 + ::: ｰ09 0 ｧｧ04 ﾀ縺07 06 縺06 S, ﾀ03 ﾀ05 04 ｧｧ06 S, Sx ｧ閨06 ﾀ0600 ｧ縺縺 m m(m, ) x + x + ｰ09 0 ｧ , 0503 S0 09 ｧｧ06. m + S0 x m m(m, )(m, ) x 3 + ::: 3 = S, ｧ縺0600 ｧ陦ｧ縺縺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 閨ﾀ0803- 陦ｧ003 ﾀｧ09 08 ｧ00506 縺04 00 : x + 4x x ::: 484., x x 4 x , ::: 4 4 P 485. n= 487. x, + 0 n x n n : 486. (x, ) + P xn, (,)n, n= n,. (x, )3 3 + :::

242 3x 閨03 ｯｧ縺 x + (x +) (x +) ::: 縺06 閨0600 ｧ0504 ｧ00506 縺04 00 ﾀ0803 陦ｧ003 ﾀ05 ｧ縺07 ｧｧﾀ060006: 489., 3x +5x 4, 7x 6 + ::: ｰ09 0 ｧ閨060 ｧ07 ﾀ S ｧﾀ0 00 x S x + x + x3 3 + ::: ｰ 09 0 ｧｧﾀ0 00 ds. 49., 4x +7x, 0x 3 + ::: ｰ09 0 ｧ06. ｮ03 ﾀ05 04 ｧｧ06 S + Sx. 0 x 閨03 ｯｧ縺060. ｱ03 縺縺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 : 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! 閨縺060 ｧｯｭﾀ00 ｧ縺 () ｧ () R n (x)! 0 04 罍ｧ n!,0503 ｧ 0505 ｧ縺ﾀﾀ縺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 罍ｧｧ0708 x, 04 罍ｧ縺0308 lim n! R n(x) =0.

243 閨ｧ06 04 縺07 閨縺ｧ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 ﾀﾀｧｧ 04 罍ｧ 04 ﾀｧ0708 x; ( + x) m =+ m m(m, ) x + x + ::: 03 ｧ0030 ｧ縺07 ; 030 ﾀ0803 陦ｧ05 ﾀｧ ( + 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 ｧ縺ﾀ x ｧｧ: ) cos (x, ); ) sin x; 3) xe x ; 4) sin mx + ｧ 0 04 ｧﾀ 0504 ｧｧﾀﾀ0006 閨ﾀｧﾀ縺罍ｧｧ07 04 縺ｧｧ f (x) = ln ( + e kx ) 縺縺ｧﾀｧ06 04 縺ﾀ x 03 ｧ0030 陦ｧ05 ﾀ07 04 罍ｧ jxj <a. + x a m ｧ , 縺07 ﾀｮｧ0030 ｧ縺 , 罍ｧ jxj < ( + x) 3 =, 3x +6x, 0x 3 n(n +) + :::= (,x) n, : n= 496. ｮｧ0030 ｧ縺ｧｧ06 04 縺ｧｧ +x =, x + 3! x4, ! x6 + ::: 04 罍ｧ jxj < : ｧ縺ﾀ x ｧｧ: ) ln +x, x ; ) ln (, 3x + x ); 3) ln (, x + x ) ｷ罍ｧ縺ｧ縺ｧ- ﾀ 0504 縺07 閨0007 ln (x + +x ). X

244 3x 5. 7 罍ｧｧ07 縺07 閨罍ｧ0300 ｧｧﾀ00060 ｧｧ0504 e x=a 04 縺ﾀ x, a; 0 04 ｧﾀ 0504 ｧｧﾀ- ﾀ0006 閨縺ﾀ縺ｧｧ06 f (x) =x 3, 3x 0403 ﾀ x, ｧ0504 x ﾀ x ｧ縺ﾀ x ｧ06 f (x) = x ｧｧﾀﾀ0006 閨ﾀ0803 陦ｧ003 ﾀ0504 縺罍ｧｧ縺07 閨ｧｧ: ) f (x) = cos x 0403 ﾀ x, ;)f(x) = sin 3x 0403 ﾀ x ｧ縺ﾀ x ｧ06 f (x) = 3 x ｧｧﾀﾀ0006 閨罍ｧﾀ0803 陦ｧ003 ﾀ縺ｧ縺ﾀ x ｧｧ: ) x ; ) cos mx + ｧ 0 04 ｧﾀ 0504 ｧｧﾀﾀ0006 閨縺ﾀｧｧｧ06 f (x) =x 4, 4x 0403 ﾀ x ｧ縺ﾀ x, ｧ06 f (x) = cos x ｧ 0 04 ｧﾀ 0504 ｧｧﾀﾀ0006 閨縺ﾀ縺ｧ縺ﾀ x, ｧ06 f (x) = = sin x ｧ縺ﾀ x, ｧ06 f (x) = x ｧｧﾀﾀ0006 閨罍ｧﾀ0803 陦ｧ003 ﾀ縺ｮｧ0030 ｧ縺 , 0503, x =+ x + 3! x ! x6 + ::: 04 罍ｧ jxj < ｧ罍ｧ縺ｧ060 縺07, , 0 04 ｧﾀ 0504 縺07 閨0007 arcsin x. x 5. 7 罍ｧｧ07 縺07 閨罍ｧ0300 ｧｧﾀ00060 ｧｧﾀｧ0030 ｧ縺07 閨0007 +x ｧ 0403 ｧﾀ00 ｧ0504 ; 004, 0; 99, 90, ｧｧ0400 ｧﾀ04 閨ｧ縺ｧ縺ﾀｧﾀｧ0030 ｧ縺07 閨 x ｧ 0403 ｧﾀ00 ｧ ; 006, 3 0; 99, 3 30, ｧｧ0400 ｧﾀ04 閨ｧ縺ｧ縺ﾀ0504.

245 閨ｧﾀ00 ｧ0504 sin, ｧｧ0400 ｧﾀ04 閨ｧ縺07 閨0007 sin x, ｧｧ縺ﾀ0504. ｰ09 0 ｧ06. x =, 04 陦ｧ 0 08 x = =5 = 0; 縺ｧ縺ﾀ05 ｧ 0304 縺06 閨0600 ｧ0504 ｧ 06 ﾀｧ07 x<0; ｧ060 ｧﾀ00 ｧ陦縺0303 ｧ +x P 0006 ｧｧ06 +x = n= (,)n, x n, ｧ, 04 縺03 ｧ罍ｧ縺縺07, 0 04 ｧﾀｧ06 04 縺07 arctg x. P x = 3 (,) n, x n, ｧｧ arctg x =, n= n, ｧ0504 縺07 閨ｧﾀ00060 ｧ07 : X (,) n, = 3 (n, )3 n= n, : ｧﾀ00 ｧ0504, 縺07 ｧｮ縺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 ｧｧ04 +x =, ｧ x ｧ05.., x sin x 縺06 閨0600 ｧｧ閨06 縺07 閨0304 ｧ x 縺06 閨0600 ｧｧ閨06 縺ｧ06 (x) = ｧ e x x x. e,x ｧｧﾀ00 ｧ0504 (=3), ﾀ , ﾀ閨 , 縺ﾀ , 縺06 閨0600 ｧｧ閨06 縺ｧ06 (x) = 0 ｧ 0403 ｧﾀ00 ｧ0504 (=5), ﾀ , ﾀ閨 , 縺ﾀ , ｧｧ閨06 縺07 縺ｧｧ07 y 00 = x y ﾀｧ 06 ﾀｧ070 ｧ: 04 罍ｧ x =0, y =, y 0 = ｧ 0406 縺縺縺07, 0304 縺06 閨縺ｧｧ07 ( 7 ｧｧ) y 0 =+x,y ﾀｧ 06 ﾀｧ070 ｧ: y =04 罍ｧ x =0. x 3 +x

246 3x 5. 7 罍ｧｧ07 縺07 閨罍ｧ0300 ｧｧﾀ00060 ｧｧﾀｧ閨06 縺07 縺ｧｧﾀﾀ xy y 0 + xy =0 ﾀｧ 06 ﾀｧ070 ｧ: y =, y 0 =004 罍ｧ x = ｧﾀ00 ｧ ; 005, ; 00, 3 0; 993, 0; 997, 0, 3 70, 5 40, ｧｧ0400 ｧﾀ04 閨ｧ 03 ｧ0030 ｧ縺07 ( + x) m m(m, )x =+mx + + :::, ｧｧ縺ﾀ0504.! ｧﾀ00 ｧ0504 cos, ｧｧ0400 ｧﾀ04 閨ｧｧ07 04 縺07 cos x ｧ縺ﾀｧｧ 04 縺07 arcsin x ( 5) x = ==, 0403 ｧﾀ00 ｧ0504, ｧｧ04 07 ﾀ04 05 縺ｧ縺07. ｰ09 0 ｧ06. ｮｧﾀ00 ｧ縺ｧ縺 , ｧ0504 閨06 ﾀ0705 ｧ陦縺閨0308 ｧ 0406 縺縺ﾀ縺ﾀ縺縺ﾀ閨06 ﾀ = arctg + arctg, 0 04 ｧﾀｧ06 閨縺06 ﾀ閨ﾀ縺07 閨 x ==N ｧｧ ln ( + x) 04 縺07, ｧ縺 : ) ln (N +)=lnn + ) lg (N +)=lgn +0; 4343 N, N + 3N 3, ::: ; N, N + 3N 3, ::: ｦ00 07 ln = 0; 693, 0403 ｧﾀ00 ｧ0504 ln 5 ｧ ln 0 ｧ , 閨 M = 0; ln ｧﾀ00 ｧ0504 lg 0 ｧ lg 縺06 閨0600 ｧｧ閨06 縺07 閨00 ｧ006 閨0605 ｧｧ04 ﾀ. 0; ｧﾀ00 ｧ0504 +x 3, ﾀ縺07, ﾀ閨 , 縺ﾀ , 縺06 閨0600 ｧｧ閨06 縺ｧ06 (x) = ｧﾀ00 ｧ0504 ﾀﾀ閨03 0, x cos x 4 ｧ

247 閨ｧﾀ縺罍ｧ縺07, 0304 縺06 閨ｧ06 縺ｧｧ07 y 0 = x + y,06 閨縺ﾀｧ06: y = 004 罍ｧ x = ｧﾀｧ閨06 縺07 縺ｧｧ07 y 00 + xy =0 ﾀｧ 06 ﾀｧ070 ｧ: 04 罍ｧ x =0, y =, y 0 = ｧﾀｧ閨06 縺07 閨ｧ縺閨罍ｧ040308, 04 閨縺罍ｧ04 ｧ 0 k 0 ﾀ縺縺09 ｧ閨00 ｧ006 閨0605 ｧ s. ｰ09 0 ｧ06. 0ｷ3 06 ﾀｧ07 d',05 閨06 C 0403 ﾀ , ｧ ' ｧ ds = s C 縺0600 ｧｧ07 = ds cos ' ｧ dy = ds sin '. x ｯ閨ｧｧ閨縺ｱ03 縺ｯ閨ｧｧ閨縺ｧﾀ縺0608 ﾀ0006 閨06060 ｧ08 04 ｧ 08: F (x + h; y + l) =F (x; y) + F (x; y) +! F (x; y) =F (a; b)+! (x, h + F (x; y) +(y, F (a; + +(y, F (a; b) +::: z = dz! + d z! + :::+ dn z n! x=x0+x y=y 0+y (I) (II) : (III) ｧﾀｧｧｧ F (x +h; y +l) 縺ｯ (I), 06 ﾀ00 ｧ F (x; y) =x + xy + y ｧｧ06 F (x; y) =x 3 +xy 0403 ﾀ x, ｧ y, (0703 縺00600 (II)) ｧｧ06 F (x; y) = ln (x, y) 0403 ﾀ x ｧ y +, 0 04 ｧﾀｧ縺07 閨ｧ 03 ﾀ (0703 縺00600 (II)) ｧｧ06 F (x; y) = sin (mx + ny) 0403 ﾀ x ｧ y, 0 04 ｧﾀ , ｧ縺07 閨ｧ 03 ﾀ (0703 縺00600 (II) 04 罍ｧ a = b =0) ｧﾀ x ｧ y ｧ06 e,x,y (0703 縺00600 (II) 04 罍ｧ a = b =0) 縺06 閨0600 ｧ罍ｧ 0060 ｧ06 z ｧｧ z = x, xy + y (0703 縺00600 (III)) ｧ 0403 ｧﾀ00 ｧ罍ｧ 06 ﾀｧｧ, 0503 x ｧﾀ07 ﾀ閨03,, y ｧﾀ07 ﾀ 3 閨03,8.

248 3x ｱ06 縺ｷｱ06 縺縺06 閨0600 ｧ罍ｧ 0060 ｧ06 z ｧｧ z = cos (ax, by), 0 04 ｧﾀ 04 閨縺 (III) ｧ 03 ﾀｱｧ06 F (x; y) =x y ｧﾀ x, ｧ y +(0703 縺00600 (II)) ｱｧ06 F (x; y) = arctg y ｧﾀ x, x ｧ y, ｧｧ0400 ｧﾀｧｧ縺07 閨ｧｧ06 z = y x 0403 ﾀ x, ｧ y,, 0 04 ｧﾀｧ縺07 閨090304, ｧ 0403 ｧﾀ00 ｧ0504 ; ; 縺06 閨0600 ｧ罍ｧ 0060 ｧ06 z 閨ｧｧ z = x y, y ｧ 0403 ｧﾀ00 ｧﾀﾀ閨03 0, 罍ｧ 06 ﾀｧｧ, 0503 x ｧﾀ閨03,99, y 閨03 5,0. x ｱ06 縺ｷｱ06 縺縺06 閨ｧ06. ｱｧ07 f(x) ﾀ07 06 閨縺ﾀｧｧ罍ｧ縺 [a; b], 06 ﾀ00 ｧ縺 : ) ｧｧﾀ0003 縺 , 04 罍ｧﾀｧ 0406 縺縺03 ; ) ｧｧﾀﾀ05 縺 ; f(x, 0) + f(x +0) 3) f(x) = ﾀ (a; b).. ｱｧ07 f(x), 06 閨縺ﾀｧｧ罍ｧ縺 [,l; l], 縺06 閨ﾀ縺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), f(x) ｧ , 0503b n =0 ｧ f(x) = a 0 + X n= a n cos nx : (3) l ｭﾀ00 ｧ f(x) =,f(,x), f(x) ｧ , 0503a n =0 ｧ f(x) = X n= b n sin nx : (4) l

249 閨03 ｭﾀ00 ｧｧ06 f(x), 0304 縺06 閨縺07 閨030 () 縺 [,l; l], 04 縺03- 閨ｧ罍ｧ03 陦ｧ 06 ﾀﾀ 0406 罍ｧ03 閨030 l, 縺 , f(l, 0) + f(l +0) f(l) =, 閨縺06 閨ﾀ07 縺07 閨030 () ｧ 0 04 ﾀ060 ﾀ縺03 閨ｧｧ. 3. ｭﾀ00 ｧｧ07 f(x) 03 ﾀｧ罍ｧ縺縺 (,; ) jf(x)j ﾀ0803 陦ｧ05 ﾀ07 ｧ 06 閨縺ﾀｧｧ-, 罍ｧﾀ縺 , 縺06 陦ﾀｧｱ06 縺0406: f(x) = + 0 d +, f(t) cos (x, t) dt = = + [a() cos x + b() sin x] d; (5) 05 閨 a() = f(t) cos t dt ｧ b() = f(t) sin t dt: (6),, ｧ縺07 閨03 ｱ06 縺0406 ﾀ0006 閨06060 ｧ罍ｧ03 陦ｧ 06 ﾀ09 ｧｧｧﾀ 0406 罍ｧ03 閨030 : 549. f (x) =04 罍ｧ 0 <x< ｧ f (,x) =,f (x). ｮ縺 , 0503, 3 + 5, 7 + :::= 4 : 550. f (x) =x 04 罍ｧ 0 6 x 6 ｧ f (,x) =f (x). ｮ縺 , :::= : f (x) =x 04 罍ｧ, 6 x 6. ｮ縺 , 0503 ), + 3, 4 + :::= ; ) :::= f (x) = 04 罍ｧ, <x<0;, x 04 罍ｧ 0 6 x 6 :

250 3x ｱ06 縺ｷｱ06 縺ｧ縺07 ｱ06 縺罍ｧ03 陦ｧ 06 ﾀ09 ｧｧｧﾀ 0406 罍ｧ03 閨030 l: 553. f (x) =04 罍ｧ 0 <x<l ｧ f (,x) =,f (x) f (x) =, x 04 罍ｧ 0 6 x 6, f (,x) =f (x), l = 罍ｧ,l <x6 0; 555. f (x) = x 04 罍ｧ 0 6 x<l: 556. f (x) ﾀ05 ｧ (0; ] ｧ09030 ( 罍ｧﾀ. 35) ｧ 04 縺03- 閨 : ) ; ) 罍ｧ03 陦ｧ 06 ﾀﾀ 0406 罍ｧ03 閨030 l = ｧ閨0606 ｧ 0505 ｧｧ08 04 縺07 ｱ06 縺ｧﾀｧﾀﾀ04 縺03 ﾀｧﾀ0506 縺07006 閨00 ｧ00308 l 0304 縺06 閨ﾀ ; 05 閨06 u(x; t) , ｧ 06 ﾀｧ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=; ) ｧ: u = 04 罍ｧ t =0. l, x 04 罍ｧ x>l= 0504 縺06 閨0600 ｧ閨030 ｱ06 縺ｧ06 u(x; t) 縺03 閨ｧ07 ﾀ0506 縺07007 閨00 ｧ00308 l, 縺陦ｧ (04 罍ｧ x =0) 09 縺 , 陦縺 (04 罍ｧ x = l) ﾀ閨060, 0304 縺06- 閨ﾀ u ; 05 閨06 u(x; t) 04 縺03 閨ﾀｧ06, ｧ 06 ﾀｧ070 ) 05 0 ｧ 0030 ｧ: u =004 罍ｧ x =0; =004 罍ｧ x ) ｧ: u = f (x), =004 罍ｧ t 0504 縺06 閨0600 ｧ閨030 ｱ06 縺ｧ06 u(x; t).

251 閨縺ｧ07 ﾀ0506 縺07007 閨00 ｧ00306 l ﾀ 09 縺ｧｧ 0304 縺06 閨ﾀｧ060 ｧ06 ﾀｧ070 u 4 =0 ) 05 0 ｧ 0030 ｧ: u =0 =004 罍ｧ x =0 ｧ x = l; ) ｧ: u = f (x) =004 罍ｧ t 0504 縺06 閨0600 ｧ閨030 ｱ06 縺ｧ06 u(x; t) { ｧﾀ 0504 ｧｱ06 縺0406 閨ｧｧ: 04 罍ｧ 0 <x<; 560. f (x) = ｧ f (,x) =,f (x) 罍ｧ x> 56. f (x) =e,x 04 罍ｧ x > 0 ｧ f (,x) =f (x). 56. f (x), 縺 [,; ] ｧ罍ｧﾀ. 36 ｧ縺ｧ縺07 閨03 ｱ06 縺ｧｧ: 563. f (x) =, x 04 罍ｧ 0 <x6, f (,x) =f (x); f (x +) =f (x): 564. f (x) = j sin xj; ﾀ縺 , :::=. x 04 罍ｧ 0 6 x 6 =; 565. f (x) =, x 04 罍ｧ = 6 x 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) f (x) =e x 04 罍ｧ,l <x<l ｧ f (x +l) =f (x).

252 3x ｱ06 縺ｷｱ06 縺閨030 ｱ06 縺0406 縺0600 ｧ罍ｧ 06 ﾀ ) u =004 罍ｧ x =0, =004 罍ｧ x = ) u = f (x) =004 罍ｧ t ｧﾀ 0504 ｧｱ06 縺0406 閨ｧｧ 04 罍ｧ, <x<; f (x) = 0 04 罍ｧ jxj > :

253 305 ｯ6 5 ｭｯ6ｧ9. AB =9, BC =,6, AC =3, 9,6 = (+ ), 90, (5; 5), (5;,3). 8. B(0; ) ｧ B(0;,4). 9. x = a c, b ; 04 罍ｧ c>jbj 閨ｧ, 04 罍ｧ c = jbj 03 閨0, 04 罍ｧ c<jbj 0 ｧ 03 閨 M (5; 0).,!,!,!. ｨC06005 (;,), R = x AB =,, 04 繦 AB =,4, j ABj = B(5; 8), j,! ABj = B(4;,3). 5.,4,, (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 ﾀﾀﾀ (; ; 5). 9. OC = 5, OD = (3; 3) (; 3), 06 ﾀ00 ｧﾀｧ閨006 ﾀ0503 縺03006, ｧ (5; 7), 06 ﾀ00 ｧﾀ0503 縺 (;,) x = x + x + x 3, y = y + y + y ; 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 縺ﾀ05 ｧ. 43. x, y, =0, D ｧ E ｧ0 ｧｧ. 45. x + y = y = x. 47. x 5 + y =. 48. y = x 4,x y = x. 5. (; 0), (3; 0), (0; 3). 53. y =8(x,). 54. x, y +5 = 0. ｯ03 09 ｧ B ｧ D ｧ0 ｧｧ. 55. x + y = y = x (x +) +(y +), (x, ) +(y, ) = 4 ｧ00 ｧ xy =; 04 罍ｧ x = =; ; ; 4, y = 4; ; ; =; ｧﾀ05 縺03 ｧ罍ｧ ) y = x +3; ) y =,x ) y = x 3, 3; )y =,x 3, y =,; 5x. 63. ) k ==3, b =,; )k =,=3, b = 0;3)k =0, b =,3; 4)k =,3=4, b = k =, b =, y = x ) x 3 + y, =;) x,4=3 + y =. 67. y =0; 4x, 3y =0; y =4; 4x, 3y + = x, y 3 = ｧ00 ｧ, x 4 + y 3 = ,!,!,! OxAB = 8,04 Oy AB = 6, j ABj = A ｧ C 0 04 縺 , B , D 080 ｧ縺ﾀ縺06 閨 : ) 04 ﾀｧ, ｧ縺 y = 3x + ( ﾀ0903 ﾀ0504); ) 04 ﾀｧ, ｧｧ縺 y =3x +; 3) 04 ﾀｧ, ｧ縺 y =4, x ｧ 0 ﾀ縺 ; 4) ｧ,

254 ｧｧ縺 y = 4, x. 73. x, y = a. 74. ｪ06 縺06 t ﾀ罍陦ｧｧ M 0306 閨0605 x = a + mt, y = b + nt. 0ｷ3 ﾀｧ04 x, a t, ｧｧ罍ｧｧ: m = y, b n. 75. ) y = x 3, ; ) y =,x 3,. 76. k =, b = x +y, 4=0, x,y +4 = 0; y =3, y = 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 x +y +4 = 0, 5x +y = 5. ab 88. x, 3y + = 0, 5x, y = 4, 3x + y = , 30 0 ｧ y = 3x ｧ y =, x. 9. x, 5y +6 = 0, 5x + y =, y =x, 6, y =,x (3;,), (3; 3), (,9=5; 3=5); 45, , (5=; 5=). 95. AE: x, 5y =,4, AD: x, y =,; A = 8 6 0, B = , C = x +y, = tg A = 4=3, tg B = tgc = ; S = (;,), (8=3;,). 00. x + y =,4, x, y =,4, x + y = ; 8; 0; ; =. 06. k = 縺 , : 4x, 3y 0 = x, 5y +6=0, 8x, 5y = x, y =0 ｧ x + y, 4= x, y = ｧ x +3y = 4.. x + y = ｧ00 ｧ 4x + y, 8 = 0.. 3x +6y =,. 3. x +3y = x, 4y +0= 0; x =. 6. h = 8= 縺 : x + y = 0 ｧ x, 3y = 0; ﾀﾀｧ07: d =, d =0; 縺 : 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 = , ; S 4 =b =3. 5. a =5. 6. A =36 5 0, B = ( 0+ 5); x,y+6=0, x,4y =4, x,3y+ = y = x+, x,5y =6, y =,x, y = x ｯ03 09 閨04 ｧﾀﾀ0503 縺陦 05, ｧ縺ｧ x,3y = 5, 3x+y = h = h =6= (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 = ) B(; ); ) C(,;,5). 39. y =x+6; = 5; DAB x +y +8x,, 6y =0; A ｧ O 縺ﾀ05 ｧ, B x +y +4x,6y = (0; 0), (,; 5; ; 5). 44. (x,) +(y,) = ｧ00 ｧ (x,5) +(y,5) = = tg =,; 4, = (x +4) +(y +) = x + y, 8y = y = 4x=3 ｧ y = y = x(a, x).

255 = a y, a 3 5. (x, 3) + y =9. 5. x x + y = a. 54. x + y = ax. 55. x + y, 6y, 9= ) (3;,), R =6; ) (,5=; 7=), R =4;3)(0;,7=), R =7=. 57. x +y +4y =0; (0; 0), (;,), (,;,). 58. x + y + ax + ay = y =0, 5x +8y = x + y =3. 6. x + y + ax = (x, ) + y = = 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; a = , e == x 6 + y 4 =, 3 e =, r =4, 3, r = x 64 + y 8 =, r =, r = ; (=7; 4 3=7). 74. (,5=4; 63=4). 75. x 36 + y 4 x = y 3 x =. 78. a + y x = ｧ00 ｧ b b + y a =. 79. x 9 + y x = ｧ00 ｧ y x = 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 = x 9 + y 8 =. 87. e = 5=, r =, r = ) x 6, y 9 = ;)x 0, y 4 =. 90. x, y 4 = ; 3 ｧ x 6, y = a. 93. (0; a ); y + = =. 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 ) ﾀ0.. y = x.

256 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 += y =, 3(x +); 6=3. 3. r =7; 4, d =9; ｧ縺罍ｧﾀ x = 3;, e =; 5, r = 0; 5, d = 8;. x y =. 34. x, y =. 35. ｮ0304 縺陦ｧ y =, x, a = b = ｮ0304 縺陦ｧ y + x =0; ｰｧ06 陦ｧ y = b a x, 閨00 ｧ0 (a + b ). 38. y =; 5x. 39. y = x, 9y +5 = y =x ) x 3y =8; ) x y =;3)x y =,. 45. x, y = y = x x + y = a + b. 49. y = y ｰｧｧ MN: a y 0 x, b x 0 y = c x 0 y ｧ0 y =0, 0 08 閨ﾀ09 ｧﾀﾀｧ N 0406 縺06 ﾀｧ縺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, 縺 MN 閨0600 ｧ05 FF ｧｧ r : r ｧﾀｧﾀﾀ罍ｧﾀ縺 y =x ｧｧ06 y 0 x + y = y 0 ( + x 0 ) ｧ04 y =0, 0 08 閨060 x = + x 0, FM = x, = + x 0 = FM, FMN = FNM. 53. (3; ; ; 4) ｧ縺03 y = x ｧ y =,x=4, y = x= x, y = arctg x + y += ) O (; ); ) tg ' = ) X +4Y = 6;6)Y = 4X; 7) X,4Y =4;8)Y = X =. 6. ) X +4Y =6;)X,4Y = X, Y = ) XY =6;)XY =,6; 3)XY =4;4)XY = =, ｰｧ06 ﾀ05 縺06 ｧ: y = 6(x, x ); y =30 04 罍ｧ x =0; 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 ). 75. ) ｧ04 ﾀ; ) 05 ｧ0406 縺 ) X 5 + Y =, O (3;,); )X, Y =9; 3) Y =X; 4)X =4Y. 77. X +Y =4. ｱﾀ03 04 ﾀ05 縺0308 ﾀｧﾀ (; ) ｧ (,;,). 78. (x +) + y = (x, 3) +(y, 3) = (x, =. 80. x +3y = y ) = 4(x +4) y 6 =. 84. x + y, ax, by =0. a ﾀｧ06 AB =a, 0403 ﾀ0305 OD = a, 閨04 a ｦ罍ｧ O, 閨 AB 5 5

257 ｧｧ AO : OB = m, 03 ﾀ04Ox 04 縺 OB; 0406 ﾀ0504 OB = a, 罍陦ｧ A ｧ B 0306 閨0605: A(,ma; 0), B(a; 0). ｰｧ06 ｧﾀｧ0 ｧｧ: (m,)x +(m,)y =max; 04 罍ｧ m 6= 0309 縺ﾀ0504: x + y = ma x; 04 罍ｧ m =04 縺070 07: x = ｯ O 04 罍ｧ , m, OB 03 ﾀ04Ox. ｰｧ06 ｧﾀｧ0 ｧｧ: (a, b)(x + y )=abx; 04 罍ｧ a 6= b 0309 縺ﾀ0504: x + y = ab x; 04 罍ｧ a = b 04 縺070 07: x = 0. a, b 89. (k x +y )=a (k +); ｧ04 ﾀ 04 罍ｧ k 6=,0309 縺ﾀ0504 x +y = a 04 罍ｧ k =. 90. x +0x + y 5 9 = a arctg (a; a). 94. A( 6; 0), B(;,), C(, ; ); S 4 ABC = = ; y = x, x, +y = = ax, by + jabj a + b =0; d = ｧ a + b 0060 ｧ , ｧ0 4(y, x) =(y + x)(y, x), 0305 ﾀ06 : ) y = x; ) x +y =4; ﾀ0006 閨 , ｧ 0406 縺06 ﾀｧ縺 y = x ｧ00 ｧ 0 04 縺 x + y =4; 0 08 閨060 x =, x =,6; 閨00 ｧ罍閨 x + y (x, ) = a(x + y) y =( ｧ04 ﾀﾀ縺030 (; 0)) xy = y = x, 6x X, Y =4; 8 (x, ; 5) O (;,3). 307., y = (05 ｧ0406 縺ﾀ縺030 (; 5; 0)). ; ﾀ0504 M (x; y) 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 ﾀ縺ﾀ : X +Y =4a cos ' = +tg ' = = 5, sin ' = 5 ; ｧ06 X, Y = x +8xy,, 3y =0; 縺ﾀ ' = arctg (=) 04 罍ｧ0403 陦ｧ05 ﾀｧ閨06 X, Y =4( ﾀ0. 309). 3. y =x +(e, )x. 33. ) 7 04 縺 y = x; ) (0; 0); 3) 00 ｧ縺ﾀ0504; 4) (3; 4); 5) 縺 x = 0, y =,x; 6) 縺 y = 4; 7) 縺 y = x ｧ y = x. 34. ) (;,), X = ; ) (; ), 6 + Y 4 X, Y = 9; 3) X +5XY +Y = ) X 4 + Y 4 ) X 4, Y X =. 36. ) Y X =;) 4 = ; 8, Y 4 =. 37. ) Y = = 5X; ) 縺 x, y = ) 3y =x, 7 (x, );

258 ) (;,); 3) 4y =,x, X, Y = 8; (; 0); tg ' =,= (x, ) +(y, ) = 縺ﾀｧ 0,45, ｧ0 Y = X a + a. ｰｧ06 x + y = a 0304 縺06- 閨閨 AB ( 罍ｧﾀ. 87), 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 x y =0;) (,; ); 3) 縺 y = x, x +6y = ) X + Y 4 =;)X 0, Y 5 =. 35. ) Y =4 X; ) 04 縺 x + y =. 36. ) y = x, ; ) 3y = x, 5 (x +). 37. ) 7x, xy +7y, 48x, 48y + 44 = 0; ) x +4xy + y +6x +6y, 8 = (x, y), a(x + y) +a =0; Y = a X. 39. x, 4xy, y, 4x +8y, = 0; X, Y =3; ) r = a a sin ;) r = r = a sin (, ) sin (, ') r = cos ' sin ' =acos ' ) 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, ) 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) ) 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 r = a sin ' b. a( sin ') 344. r = OB AB = ｧ00 ｧ 04 閨0609 縺罍陦ｧ y = cos ' x(x, a) = a, x 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 r = a( + cos '); (x + y, ax) = a (x + y ) ﾀ0504 C 閨04 ｧ縺0605, C ﾀ縺0605 ｧM ('; r) ｯ OCC = MC C = ' ｧ CO = C M = a, 0503 OM k CC. ｮ04 縺ｧ縺 COMC 0 CC, ｧ0 a cos ' + r + a cos ' = a ﾀ06 r = a(, cos ') ) r max =504 罍ｧ ' =0, 80 ; r min =04 罍ｧ ' =90, 70 ;)r max =4

259 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 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 c =a r = b + a cos ' ｷ3 4OAM: r = OM = OA cos ', 003 ｧ 4OAB: OA =a sin ', r = a sin ' ﾀ A 0 03 ﾀｧ Ox, 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, x =3 + y =3 = a = y = x + x. 36. (3y + x ) = 4x (a, y ) 縺罍陦ｧ r = OM = AB = BD sin ' = a tg ' sin '; 04 閨0609 縺 y = x3 a, x ( 罍ｧﾀ. 85) ｧ04 06 縺06 t OA ﾀ Ox, 閨060 x = a ctg t, y = a sin 8a 3 t. 0ｷ3 ﾀｧ04 t, ｧ0 y = x +4a x = a(t, sin t), y = a(, cos t) x = a(cos t + t sin t), y = = a(sin t,t cos t) 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 x =(R+r) cos t,r cos, r (R + r)t, 05 閨06 t 縺ｧ0 ｧｧ縺0304. r t, y = (R, r) sin t, r sin R, r t X = P X i = 8; Y = P Y i =,; OM = 64+4 = ) c = a + b ;)a =c, b c = (a, b) 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 X = X + X + X 3 = =, 3, Y = Y i =6, OM = 9+36 = ) a =3(c, b); ) c =b, a OM = r =5 ; cos =0; 5, cos =,0; 3, cos = 0; r = 7; cos = = ｧ00 ｧ M (3 ;3;,3), r =3( i + j, k) u =i, 6j +3k, u =7. 39.,! OC = i, j + k, OC,! = 6; AB = k, 4j, i, AB = B(4;,; 5) ｧ00 ｧ B (4;,;,7), cos ==7; cos =,3=7; cos = 6= a = b, 0; 8c u = 3 5, cos =, cos = cos = cos == ｧ00 ｧ D(4; 0; 6) c = b, a B = C = cos ' = r

260 == 0=0; 36; ' = cos ' == 5=0; 894; ' arccos 0; b a = 4 = ) + 3;) (a + b) = a + b +ab cos ' ( 縺ﾀｧ- 006 ﾀ0304); (a + b) +(a, b) =a +b ( ﾀﾀ陦ｧ ) R = (a + b + c + d) =0 4+ 5; ｧ cos ( [a; m) = (m, n)m (m, n) = 5 ; cos ( da; n) = 7 =, = =6.,,! 45. OM =(i + j +k),,,! ON =(i +j + k); cos = 5= cos ' = = cos ' = 0; 6 0; ' ,! AB,! CD AB 48. D(,; ; ); ' = a b = =, OM = = (n + m) = 7, ON = (3m,,! OM,,! ON + n) = 3; cos ' = OM ON = = 7 9 = 7 9; 08 0; 89, ' = , cos = a cos ' =,= a b 04003: ),6j; ),k; 3) 6i, 4j +6k 閨 : ) 6; ); 3) ; , h = 4;. 49. ) (k, i); ) a c; 3) a c; 4) 閨 , 0403 ﾀ05 縺陦ｧ , 04 閨陦ｧ ; , S 4 =3 7= S 4 =7 5, BD = = ja + bj = ja, bj = 5, S = ; V =5, V =4, H =7 3= c =5a + b = V =4, H = c = a+b V = j(a+b)[(b+c)(a+c)]j =jabcj (m n) = jm nj cos = sin cos =0; 5 sin cos = 7, cos = 3 7, cos = x+4y,z = x+y = x,y +z = y,3z +7 = y +z = x+y = x a + z c = x + y + z = x 4 + y 3 + z =. 46. cos = 3, cos =, 3, cos = 3 ; =48 0, = , = x,, y, 3z +4 = x,4z = x +y = x + y 4 + z 4 = ) 45 ;) x, y, 3z = x +3y +4z = x+y+z = a. 47. x,y+z =. 47. x,y+z = x,y = 0 ｧ x +3y = ) x, y +z =, x,y +z =,; )x+y,z =0 ｧ x+y +z = ) x,8y +9z =; ) x, y +z =0 ｧ x, y, z = (;,; ) x, 4y + z =. 48. y, 5z +0 = ｰｧﾀ0903 ﾀ05 ｧ x + y, z =0;

261 ﾀﾀ0903 ﾀ z =0: cos ' = 6=3 0; 865, ' = jaj 3. abc 484. y = z x +y + z =0 ｧx +y + a b + a c + b c + z+4 = x+4y+4= ) (5; 4; 0) ｧ (7; 0; );)(0;,4; 0) x, 3 ｧ (; 0; ) x =,z +3, y =,z +5; = y, 5 = z, x, 4, = y, 3 = z. 49. Pf0; 0; g. 49. ) P = i; )P = i + k; 3) P = j+k. x = y, = z, 3 3 4,5 ; cos =0; 3, cos =0; 4, cos =,0; x =, z = ｪ06 縺06 t ﾀ罍陦ｧｧ M 0306 閨0605 x = 4+t, y =,3+3t, z =+t; x, 4 = y +3 = z, ) x =,+t, y =,t, z =,+3t; )x =+t, y =,t, z =+t ) x, a = y, b = z, c, ｧ05 x = a, y = b; )z = c ｧ 0 0 x, a m = y, b n cos ' = cos ' = ｧ P = NN = i+3j+5k. ｰｧ07 04 縺 : x +4 = y, 3 = z x +y = 0, z = ; = (4; ; 0), (3; 0; ), (0;,6; 8) x =6, 3z, y =,z +4; x, 6,3 = y, 4, = z ; ﾀ0006 閨03: (6; 4; 0), (0; 0; ) Pf; ; g; = = arccos 5. 7 罍ｧ0406 閨ｧｧ 06 ﾀ縺006: ｧ x = y, 4 3 ﾀｧ縺ｧ閨06 x, x = y +4 = z Pf0; ; 0g y =,3, x, z = 0. 6 x = y +7 = z, 5 = z 0 ; cos ' = 6 0; 95, ' = ｧ- = y = z, =3, ｧ0 x ｧ06 ｧﾀ縺 : = y, = z A(0; ; 0),,,! AMf3;,; 4g, Pf; ; g, d = sin == ｧ08 04 縺 Am + Bn + C =+(,)+(,) 3=0, (,;,; 3) ｧﾀ0903 ﾀ05 ｧ, 縺0308 (,;,;,3) ｧﾀ0903 ﾀ05 ｧ. 56. y + z + = 0 04 ｧﾀｧ閨06 x, 0 = y, = z ｧ07 04 縺 x, y + z +5 = 0.

262 x, 5y + z, = x +y, z =. 50. x 3 = y = z ; (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 = x, 3,9 = y, = (; ; ); (,; ; ); (6; ; 0). 53. (3;,; ). 53. x,y,z = (,; 3; ) x, = y 5,4 = z ｯ03 09 ｧ, 0 04 縺 : O(0; 0; 0) ｧ A(; ; 0); ｧ縺03 04 縺 :,! OAPP Pf0; 0; g ｧ P f;,; g, d = jp P j = ) C(; 5;,; 5; ), 5 R =; 5 ;)C(0; 0; a), R = a (x, ) +(y +) +(z, ) = x + y + z =8x x + y + z, a(x + y + z) = y = =ax,x. 54. x +y =ax, x +z =ax, y +z = a (; 7; ), R = (3Y, ) = (3X,) ) y =0, x = a,az ( ); ) x =0, y = a,az ( ); 3) z = h, x+y = a(a, h) 04 縺070 07, x +y = a ( ﾀ0. 罍ｧﾀﾀ. 30). ｨC ｧ00 ｧ0 陦罍ｧ 06 ﾀ縺08003 ﾀ0504 x +(y, z +) =8, 0703 縺ｧ x (y +) + = ｧ04 ﾀ x, y +3z, 7 = x +(y +4) + z = 4. (x, ) (y +4) = (x, z) +(y, z) = 4(x, z) x =4, z y = x + y a = z c h x =z[h(y + a), az] (0; a; 0), 縺ﾀ0504 z = a, x +(y, a) = a 縺00 ｧ0 (0; 0; 0), z = h, x = hy 罍ｧ z = 0 x = a; 04 罍ｧ y = h x + y = a ; 04 罍ｧ x = c 04 縺07- a, c z = y, 縺08003 ﾀ閨04 ｧ07060 ｧ縺07- h 00308, ﾀ0903 ﾀ05 ｧ yoz ｧ 0406 縺06 ﾀ縺ﾀ0504 ABC ( ﾀ0. 罍ｧﾀﾀ. 3) ｧ 03 ﾀ04 Ox ) z = x + y ;03) y + z = = x. 56. ) z = e,(x +y ) 4 ;)z = x + y (x + z )=6y x + z = z(y + a) ) x + z = y ; 03) z = x + y 縺ﾀｧOx ｧ Oy 縺ﾀｧOz 0 45, ｧｧ縺08003 ﾀ05 ｧｧﾀ0903 ﾀ05 ｧｧ閨06 = X, Y, X = a ﾀ06 ﾀｧ06: X = a Y, a + a = ｧ04 ﾀﾀﾀ070 ｧ a ｧ a x + y a + z 45 = ) 3; 84; 03) c ) x + y, z a c = (03 閨ﾀｧ0406 縺ｧ ); 03) x a, y + z c =( 閨ﾀｧ-

263 縺ｧ ) 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 + ), 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( ﾀ). 57. x + y = az. x 574. x + y =4, x, y = z; x + y =z, x, y = a + y + z =. a 576. x +y,z =,a ( 閨ﾀｧ0406 縺ｧ ) x =, z + y 4a x = 3z y = 3z ) ｮ0706 ﾀ縺030 (0; 0; a) ｧ陦ｧ06 ﾀ030 R = a; ) ｧｧ縺0605 Oz; 3) 09 ｧ00 ｧ0 陦; 4) 05 ｧ縺ｧ 06 ﾀ09 ｧｧ ; 5) ﾀ; 6) ｧ 06 ﾀ09 ｧ08 09 ｧ00 ｧ0 陦; 7) ﾀ; 8) ｧｧ07; 9) ﾀ; 0) 09 ｧ00 ｧ0 陦. 58. x + y =+z, x, y =, z; x + y = 3(z, ), 3(y, x) =z x + y = = az z = a, x + y y = 3z x +4y = 4, a 3x, 4y =z; z =0, 3x =4y , a sin ( + ) sin (, ). 59., a. 594.,b. 595.,x. 596.,4a (x, y)(y, z)(x, z) sin (, ) ｧ縺 y = x ) x y x y x y = 0; ) x y 3, 5 = amn a(x, z)(y, z)(y, x) 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 ; 6; ,; 0; k, 8k, 3k k,,k,,7k. 69. x = y = z = ﾀﾀ縺06 閨 : x = +5z, y = 5, 7z ﾀﾀ ; 3 3,;, ;,;. 66. k, k,,4k. 67. x = y = z =0. 68.,k, 3k, 5k 縺06 閨 : y =7, 3x, z =8, 7x ) +5i; ) a +b ;3)5,i; 4),+i; 5)i; 6)+i ) cos i sin 3 ; 4 ) sin cos + i sin ) 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..

264 cos k 3 +i sin k 3 ; k =0; ; :::; ),, i 3 ;),i, i 3 ; 3) i, 3 i ;4)+i,,; ; 365i, 0; 365, ; 36i ) +i ; ) 6 (cos ' + i sin '), ' =45, 65, 85 ;3)( 3+i), (,+i 3) ),, i 3;) i ) 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 sin (x=) sin (nx=) cos [(n +)=]x ) 7, 4i ; ) b(3a, b )i. sin (x=) ) 4 e i=4 ; ) e i=3 ; 3) e,i= ) 5(cos 0 + i sin 0); ) e,i= ;3)e,3i= ｯ03 09 ｧ罍ｧ 09 縺0605 ﾀ縺030 C(z 0 ) ｧ r = ) 8i; )5(, i 3); 3), ) i ;)cos ' + i sin ', 05 閨06 ' =0, 7, 44, 6, ),,i 3;)i, 3i; 3)3, sin nx, i 5 3i ),,, 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 = =, ) < 0, ' =60, z = 4 cos 0, z ; 3 = 4 cos (0 0 ). f() f() k k 665., ; 7,0; 3 ; 85 <x<; 86: ; 7 ; 87,3; 0; ; 4,0; ; 5; 0; 54;,; ) ; 305; )4 ｧ 0; 30; 3),0; 68l; 4)x = =; 494, x =,0; 798 x 0 08 閨縺 x = 4 x +, x 縺 x = x4 +3x, ),6,, i ; ),; ; ) = 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) ｧ04 x = z,, ｧ0 z 3, 3z +=0; =0; z =,, z = z 3 =; x =,4, x = x 3 =, ; 76 ｧ,; ) ; 7; )3; ; x<. 68. x = 0, x = ) x >,; ),3 6 x 6 3; 3) 0 6 x ),4 6 x 6 0; ), 6 x ) x > 0; )x 6 4.

265 ) k 6 x 6 (k +); ),4 6 x ) f(0)=, f()=, f(,) = 3, f() = 3, f(a +) = a + a ) b + a; ) ah. b + a b 690. F (4; 3) = 9, F (3; 4) =, ) ｪ ; + ab + a ) ; 3) ; 4) ; 5) ; 6) ｧ , ｧ 006 f(x )+f(x ) > f x + x log a x a x <x ) 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, n> 3 3 lg (=") ｧ00 ｧ n> =0; jj <", n> x =; lg 0; 3 lg 3 ; 5 ; 6 7 ; ; :::! ; jx, j < 0; 0, n > 50; jx, j <", n>, " x =4; 3; ; 3; 0; :::! 3+0; x =; ; 9; " ; 99; :::! 3, 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; 罍ｧ jxj > 500; 罍ｧ jxj > 7; lim n! x 縺罍ｧ006 縺 , 縺030,, 縺 , , ﾀ , 縺ﾀ06006 ﾀ 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 = ) 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 = +;

266 375. x =5; 4; ; 4; 0; 4; 00; :::! 4+0; x =3; 3; 9; 3; 99; 3; 999; :::! 4, 0; x =,0; 5;,; 4;,; 49;,; 499; :::!,; 5+0; x =,; 5;,; 6;,; 5;,; 50; :::!,; 5, ｯ縺ﾀ0006 閨ﾀ0504 ｧ縺06 閨0600: lim n! x = ﾀ罍ｧ lim x 006 ﾀ06006 ﾀ ) 0; ); 3); n! 4) 0; 5); 6)0; 7)0 04 罍ｧ a>, = 04 罍ｧ a =, a 04 罍ｧ 0 <a< ),0; 6; ) = =. 739.,= =3. 74.,= 04 罍ｧ a>0 ｧ 04 罍ｧ a< = m= ,= ) =3; ),; ,. 750.,3=. 75. =. 75. = = ,. 755.,. j sin xj 756. lim x!+0 sin x, cos x =,= ; , ,= , = = cos x ) ; ),=. 77. =. 77. =. j sin xj 773. = lim =, m = x!,0 x 779. = ), sin x; ),= ; = = = , =. 789.,. 790.,= =. 794.,=. 795., ) =0; ) ) 3=4; ) [ ｧ罍ｧ006 縺06 ) x = t, 罍ｧ006 縺06 ) +x = t 4 ]. 798.,a ),; ),0; ) 3; )3=. 80. ) ; ),=. 80. ),; ),0; ),; 5; ); ), ; ), ) ; ) ) 4; ) ; 3) 罍ｧ! 0 ( + ) 3, ) ; 5; )a=b; 3); ｧ ) ; )3; 3). 85. ) 7 罍ｧ x =0;)04 罍ｧx = n, ; 3)04 罍ｧx = 罍ｧ x = 縺罍ｧ 06 ﾀｧ07 ｧ縺 ( (, 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 =, ｧｧｧ縺0304 I 縺03 ( 縺ﾀｧ縺06 縺ﾀ05 ｧ) 罍ｧ x = 縺ﾀｧ06 ( 罍ｧﾀ. 37) 縺罍ｧ x = 0, lim y =, lim y = 0, x!+0 x!,0 lim y =( 罍ｧﾀ. 38) 縺罍ｧ x =. 8. ) 7 縺0304 I 縺03 x! 04 罍ｧ x =0, 04 罍ｧ lim y =0, x!+0 lim y =, x!,0 x!+ lim y =, x!, lim y =

267 = ( 罍ｧﾀ. 39); ) 縺0304 I 縺03 04 罍ｧ x = a, 04 罍ｧ 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 罍ｧ罍ｧ x> ｧ,x lim y =, x!,0, 04 罍ｧ x< ; lim y = x!+0. 7 ｧﾀｧﾀｧﾀｰｧ06 x, y =00304 縺06 閨 y ﾀｧﾀﾀ閨ｧ08 x. 0ｷ3 0 ｧ08 閨0406: y = x ｧ y =,x 縺06 縺ﾀ ( 縺 ) 0 03 閨0 ｧ08 06 ﾀﾀｧ Ox 0304 縺06 閨ﾀｧ060 y = x, 0 陦縺0605 ｧｧ060 y =,x. ｪﾀ縺ｧ 04 罍ｧ x = ; ; 3; ::: 縺06 閨0600 ｧ : (,jxj 04 罍ｧ n, <x<n; y = +jxj 04 罍ｧ n < x < n +; : (,x 04 罍ｧ n, <x<n; y = +x 04 罍ｧ n < x < n +; 05 閨06 n =0; ; ; 3; ::: 縺0304 II 縺03 04 罍ｧ x =,; lim y = +, lim y =,, x!,,0 x!,+0 lim y x! = 罍ｧ x = 縺ﾀｧ06

268 縺06 縺ﾀ05 ｧ; 04 罍ｧ x = ｧ 05 縺ｯ03 09 ｧ縺0304 : ) x = =0;)x =;3)x =0;4)x =0;5)x = ｧ x = ﾀｧﾀﾀｷ3 0 ｧ08: ) 縺06 縺 y = 4, x ｧ y =, 4, x ; ) ｧﾀ縺 : (, 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 ) x =,= ｧ y =,; ) y = x; 3) y =,x. 83. ) y = x; ) x + y =,a; 3)y = x ; 4)y =,= ) y =0;)y = x; 3) x =0 ｧ y = x : ) y = x 3 =3; )y = x ) x =0 ｧ y =;)x =0 ｧ y =,x ) x = =,, y ==;)x = ｧy =, x + ; 3) x =, x =,, y = ( 罍ｧﾀ. 40); 4) x =, x =, ｧ y =,x =e ) e,=3 ;)e ) e ;)e, ) e, ;)e, ) 3; )e = e. 84. ) ; ),; 3)lna ｧ ) e 6 ; ) e e ) =e ; ), = e ) =x; ), ) 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 ) (x, +x ) ;) b a ) (x, ) ; r a 30 +x ), x x 4 ;),x. x ), ; ) x ) x 3, x ), x x x ;) 4 x ), tg x(sin x, x) x ) x( cos x,x sin x); ) sin x + cos x ; ) x 3 3, x 4 ; ) x 3 3x 3, x ) sin x x ; ), x sin x x 3 + x (x +) ) (, 4x) ; ) 4x, sin x 4x x cos x.

269 ), sin x ;) 866. ) x, x 6 ; ) x x( x +). 86. ) gt; ) a t sin. 86. ; 3 3 x 3 +. x 0; ; , ),6bx(a, bx ) ;) x, 3 x ) 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 ) (x +) 4x ;). x 4 (x 87. ), +) x 3 + x 3 ; x ), + sin x ( + sin x). 87.,= ,,,=9,,= ) 6 cos 6x; cos x, sin x ; ), sin x 3. ) b sin(a, bx) ) 876. ),0(, 5x) 3 ; ) 3),tg4x cos 4x x ) 0x (, x ) ;), x ; 6, x sin x x, sin x sin 3 x cos x ) sin x; ), sin x; 3) tgxsec 3 x. 88. sin x sin x, tg 4 x. 4, sin x cos x ( + cos x) 3 x (, sin x + + sin x); 罍ｧ cos x >0, ,09 04 罍ｧ cos x <0, 04 罍ｧ cos x =0 y ﾀ06006 ﾀ ( lim y 0 =, x!=4,0 lim y 0 =, ) x!=4+0 0 sin 4x ( + cos 4x) , ctg (x=3). sin (x=3) 888. sin x( + sec x) x, x,. 890., x x x,. 89., sin t dr a sin '. 89. ) =, ;) dr a d' cos ' d' = sin ' ' + cos ('. + =4) 893. f , = ab a + b, f0 () = 0, f cos x 4x + sin 4x ab =, a + b. x(, 3x ). 897., sin 4x., x sin 6x 3( + cos 6x) ) sec6 x;)3x sin x cos x (, sin x) ds dt = sin (t=4) t=,. sin (t=) 90. dr d' = (3x +) cos '. 903., x 3 4x. + r 904., k = tg = y = 8, 4x, x, 4y =. 6

270 y = x y = 0, y = (3x, ) y =,x y =, x ｧ arctg ) 5,,, 5; ) 3, 3 3 3, 3,. 95. y = x, 3x b 陦ｧ05 ﾀ07 ｧ 06 ﾀｧ07 y 0 =x + b =4+b =, c ｧ 06 ﾀｧ07, 0503 (; ) ﾀ 0 ｧ y =,4x +8, y =, 5 x, ; ' = arctg y =4x, y =,4x x 4y = y = (3x +8) ｧ y = = ｧ00 ｧ (,;,4). 93. (=; 7=4). 94. ; ; ; ｧ y, 0 =,, y0 + =. 97. y0, =, =; y 0, + ==. 98. y = x ｧ y =,x. 99. y = x ; x = x =. 93. x = x = y, = x, x =, y = 4x; ) ln x+;), ln x 0; 4343 (x +) ;3) ) ; x x x 3 (x +) ) x(x +) ), tg x ;)ctg x cos x x + x a x a 4, x x(, x ) cos x. 944., 4x a + x x ), ctg x sin x ; ) y = x, ﾀ 0605 ﾀ07 04 x, ax ( e;=) ) 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= x e x e x (, e x ) a ex=a cos x a, sin x a 956. ),e,x sin x; ), x (x, ) x 959., ln a ) x x (ln x +);)x sin x x a(eax, e,ax ).. cos x ln x + sin x x 963., tg x sin x. 964., x, x. 965., x +x cos x. + sin x 967. x(, x ) ctg x ctg x, sin x tg x + cos x. x 97.,. 97.,x ax + x a e,x=a (ex=a,e,x=a 4 ). 974., (e x, e,x ). e x e 4x e 4x +., ln x 977. x=x. x y = =, x r, x +x. 98. x +x. 98., x, 4x a jaj a, x.

271 a 984. a + x , x, x +x ), x ; 3e 3x )., e 6x 988., x x x, arctg x a. 99. x, x. 99. x 6x, ) x jxj, x ; ) x + x ex, e x arccos x e x, e 8x r 4 t, r x, (, 4)= ) sh x; ) th x; 3) ch x ; ) th x; ),4= sh x ) cth x;)= sh x ) = ch x; )4sh4x., x 005. x +; 75y =; 85a y =3; 76x +3; ) x x, ; 4x, ) tg 3 x = et (e t, ) x dt e t x x, 4x. 0. ds dt =tg5 t. 03. a. 04. ) x + a x(x, a ;)cos (ln x). 05. ) , 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 ) ,, (, 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 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 + ) ) 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 ,, 6, ) 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) 縺ｧｧ09 f (n) (x) =x e,x=a, a n + + n xe,x=a, n, n(n, ) + e,x=a, n, ﾀ06 f a a (n) (0) =

272 n(n, ) = a n, (,) n, n(n, ) = a n, (,) n. 04. f 0 (x)=,xe,x =,xf(x) 縺ｧｧ09 f (n) (x)=,xf(x) (n,) ｧ ), x y ; r ) y ;3) b x x + y x, y y. a 045. ), ;) y x +y x, y ), 3 x ;) e,x + y e y + x. 047., ex sin y + e,y sin x e x cos y + e,y cos x = ),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) ) xx 0 a + yy 0 b = ; ) yy 0 = = (x + x 0 ) x + y = a arctg ), b x a y ; ) x, ay ),a ax, y y ;), R 3 (y, ) ;3), ( + y ) ;4), 6a 3 y 5 (x +y) y =, x, 3 ｧ y = x x +y =4. 06., e. 06. e(e, ) ) dy = nx n, ; )dy =3(x, ) ) dy = x +x ;)ds = gt dt ) dr = 4 sin 'd';) = =, dt t ) sin tdt;)sin udu. 068.), a3 x (a + x ) +)d ;)( ; 3), sin ' d'; 4), x ) 0; 04; )0; ) dv = x, =3x =0; 75; dv 3bds =0; x ｧ00 ｧ 0; 6%;)d =. 07. ) 6 0; 8f 6 5x < 0; 005; ) 陦ｧ06 ﾀｧ 006 罍ｧ0504 ﾀ縺ﾀ x 3 % ) S = R, S ds = R dr; )V = 4 3 R3, V dv =4R (, x) dr ) ;)b x 3 sin (a, b') d'; 3), tdt., t du 075. ), tg x;) u 4u, ;3),e,t dt ) x ;)tg d; 3) b( + e,bt ) dt ) 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; x ) 4y = x 3 ;)y = x 3, x 079. ) a + y b =;)x=3 + + y =3 = a = ) x, y 3at =;)y =. 08. x = +x +t, 3

273 y = 3at (4, )a y = x x+y = a ), +t3 a sin 3 t ; ) t + 4t ;3), 3 4a. sin 4 (t=) 086. ) y =,x, x; )(y +) 3 = x x + y = a ) 3t, 4t 3 ;3) t = a g, x = a g 095. v = dt ; dv dt =av, w = dv d x dt = y = x, a 3 gt x = at, 4et ; ), 4 sin 3 t ; dt = a, gt; d x =,g; 06 縺06 dt (0403 ﾀ ). 09. dt = t, 4t +3; t =, t =3. = w; 0406 縺ｧ dt = a. =,g ｧ0403 ﾀ dt a h(r, h) = a r 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 dh dt = d(! ) =!d!, d(! ) d! d! dt =! =! d' d' dt d' =!" =" 縺0 ｧｧｧ ; 3.! 0703 縺縺03 ｧ 0403 閨00308 f 0 (x) =x, ; < < 罍ｧｧ0, ｧ罍ｧ x = 縺03 ｧ 0403 閨 , x = ( 閨ﾀ ) 罍閨03 (AB): k = 9, 3+ =; f 0 (x) =x =, x =; x =09 ﾀ罍閨 f(b) = b, f(a) = a, f 0 (c) = c; 0403 陦ﾀ05 04 ｧ縺ｧ b, a =(b, a) c; c = b + a. 06. c = 閨ﾀ罍ｧ x =, 縺ｧｧ縺03- ｧ 0403 閨ｱｧ縺06 縺03040 ｧｧ縺03 ｧ 0403 閨罍ｧ縺06 09 [0; ], 003 縺ﾀ0504 s = f(t) ｧ06 閨04 ｧ07060 ｧ07, t ｧ t ｧ閨04 ｧｧ縺ｧ閨06 t ｧ t 0 08 閨0605 ﾀ07 t 3, 04 罍ｧ縺030 f(t ), f(t ) = f 0 (t 3 ), = f 0 (t 3 )= ds t, t dt t (x) = f 0 (x) 0 b f(b). ｯ (b) =(a) =0 ｧ 04 ｧ00506 縺 a f(a) (a; b) ｧﾀ07 04 縺03 ｧ 0403 閨 (x), 縺閨06 a ｧ b 0 08 閨0605 ﾀ07

274 3x = c, 04 罍ｧ縺030 0 (c) = 0, f 0 (c) 0 b f(b) = 0, f(b), a f(a), f(a) =(b, a)f 0 (c). ｱｧ07 (x) 06 ﾀ閨閨04 4AMB, 05 閨06 M 閨 AB.. b3, a 3 b, a = 3c c, c = = (a + ab + b ). 3. ｰｧ09 ｧﾀ dy 3(a + b) = f0 (t) ' 0 (t), t = c k = f0 (c) ' 0 (c). ｰｧ09 ｧ06005 ﾀ k = y, y = x, x f(b), f(a) = ; 縺ｧ閨06 a ｧ b 0 08 閨0605 ﾀ07 t = c, 04 罍ｧ縺030 '(b), '(a) k = k, ﾀ罍閨06. 7 罍ｧ ' 0 (t) 6= 0, 0503 '(a) <'(c) <'(b) ( ｧ00 ｧ縺0305), ｧﾀ 0 ｧ陦ｧ05 ﾀ罍ｧ r r r a + ab + b 4 閨0605 ｧ. 7. c =. 8. ) 3, ; ), 4 ;3) s ln. 9. ) 5 4 ;) 3 ; ｱｧ07 y = jx, j 006 ｧ縺03 ｧ 0403 閨罍ｧ x = x =,= =. 4. na. n, a =b. 7. =. 8. = ) ; ) e 縺07 閨 a, b. 45. = = ln a b. 48. = ,=3. 5.,. 53. =e. 54. = e 罍ｧ x =, y min = 罍ｧ x =, y min =,6=3; 04 罍ｧ x = y max = +6=3; ｧ 0406 縺06 ﾀｧ07 ﾀ Ox: x =0, x ; 3 = 3 3; 罍ｧ x =, y max = 5=3, 04 罍ｧ x =3 y min =,9; ｧ 0406 縺06 ﾀｧ07 ﾀ Ox: x =0, x ; 3 ; 5 3; 罍ｧ x = y max =5, 04 罍ｧ x =0 y min =, 04 罍ｧ y =0 x ; 罍ｧ x =0 y = 縺0605 ｧ03; 04 罍ｧ x =3 y min = =,7= 罍ｧ x =, y max =,, 04 罍ｧ x = y min =; ﾀｧ x =0 ｧ y = x= 罍ｧ x =0 y min =, ( ) ｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ0406 Ox: x = 罍ｧ x =0 y max =, 04 罍ｧ x! y! 0, y =0 ﾀｧ罍ｧ04 07 ﾀｧ罍ｧﾀｧﾀｧ Oy ( ?) 罍ｧ x = y max =,4, 04 罍ｧ x =5 y min =4; ﾀｧ x =3 ｧ y = x, 罍ｧ x =0 y min =0, 04 罍ｧ x ==3 y max =4= 罍ｧ x =4 y max =, 04 罍ｧ y =0 x =3 ｧ00 ｧ x =5, 04 罍ｧ

275 y =,3 x =,4 ｧ00 ｧ x = 罍ｧ x =0 y max =; ﾀｧ y =0. ｮｧ罍ｧﾀｧ Oy 罍ｧ x = y max = + 3 ;, 04 罍ｧ x = 5 y min 0; 罍ｧ x = 3 y max = 4 3, 3 ; 45, 04 罍ｧ x =, 3 y min = 3, 4 3,; 45. 罍ﾀｧ x = 罍ｧ x = y max =, 04 罍ｧ x! 0 y!,; 04 罍ｧ x! y! 0. 罍ﾀｧ x =0 ｧ y =0. ｯ縺06 ﾀｧ07 ﾀ 03 ﾀ0406 Ox: +lnx =0, ln x =,, x = e, 0; 罍ｧ x = y min =, 4,0; 8, 04 罍ｧ x =, y max 0; 8. 罍ﾀｧ y = x. 76. ) 7 罍ｧ x = y max ==e. 罍ﾀｧ y =0. ) 7 罍ｧ x ==e y min =,=e; lim y = x!+0 = ; 04 罍ｧ x = y = ) 7 罍ｧ x =0 y min =0 r 4n + ( ), 04 罍ｧ x = y max =;)04 罍ｧx =0 y min =0 ( ). 78. y min == 04 罍ｧ x = =3; 3=4; 5=4; :::; y max = 04 罍ｧ x =0;=; ; 3=; ::: ﾀ0504 ﾀｧ07 09 罍ｧ x 6 ; y max = 04 罍ｧ x = ; y =004 罍ｧ x =0 ｧ x = 罍ｧ x = y max = ; ﾀ0504 ﾀｧ07 09 罍ｧ x>0. 8. 罍ﾀｧ x = ｧ x = 4( 縺030403) y min =,=9 04 罍ｧ x =,, y max =, 04 罍ｧ x = 罍ｧ x = y min =; 罍ｧ04 07 ﾀｧｧ 06 ﾀ09 ｧ 04 罍ｧ0300 ｧﾀ y = x = ｧ0903 ﾀｧOy 罍ｧ x =0 ｧ x = y min = 3 4 ; 6, 04 罍ｧ x = y max =( ｧ0 ｧｧ ) 罍ｧ x =0 y0406 縺0605 =0, 04 罍ｧ x = y max =0;, 04 罍ｧ x =3 y min =,5; 罍ｧ x =, y max =0, 04 罍ｧ x =,; y min,;, 04 罍ｧ x =0 y0406 縺0605 = 罍ｧ x = y max =, 04 罍ｧ y =0 x =; ﾀｧﾀｧ罍陦ｧ罍ｧ 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 y = x, x = 罍ｧ x = 4 +k y max =, 04 罍ｧ x = +k 縺罍ｧ 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( ﾀｧ k = ) 罍ｧ x =0 y min =0; 04 罍ｧ x = y max =4=e =; ﾀｧ y = 罍ｧ x =, y min =, 04 罍ｧ x =0 y max =3, 04 罍ｧ y =0 x 罍ｧ x = y max =4; 04 罍ｧ y =0 x =0, x =4.

276 罍ｧ x =, y min =,4; 04 罍ｧ y =0 x =, x =, 罍ｧ x =0 y min =0, 04 罍ｧ x =, y max =4=3; 04 罍ｧ y =0 x =0, x =, 罍ｧ x =, y min =,4, 04 罍ｧ x =,3 y max = 罍ｧ x =0 y max = 0, 04 罍ｧ x = y =, 04 罍ｧ x =4 y min =8; ﾀｧ x = ｧ y = x + ( 罍ｧﾀ. 4) 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) 罍ｧ x = y min =,4, 04 罍ｧ x =0 y max =0; 04 罍ｧ y =0 x =0, x ; 3 = 8 ; 罍ｧ x = 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 y = 罍ｧ x =, y min =,= e,0; 6, 04 罍ｧ x = y max 0; 6; 03 ﾀ04Ox ﾀｧ罍ｧ x = y min = (, ln ) 0; 6; 03 ﾀ04Oy ﾀｧ罍ｧ x = y =; 04 罍ｧ x = e 7; 4 y 3; 罍ｧ x = ｧﾀ. 43 y max =0, 04 罍ｧ x = y min =,3 3 4,4; 8, 04 罍ｧ x =5 y = ｧ閨ｧ罍ｧﾀ罍ｧ x =+ 6 y max 0; 34, 04 罍ｧ x =, 6 y min,0; 34, 04 罍ｧ x = y = = ; 57.

277 罍ｧ x = 4 y min = + ; 57, 04 罍ｧ x = 3 4 y max = +3; 7; ﾀｧ x = 0 ｧ x = 罍ｧ x =, y max =, ; 85, 04 罍ｧ x = y min ; 8, 04 罍ｧ x = 0 y = =. 罍ﾀｧ y = x 罍ｧ x = y min =, 04 罍ｧ x = 0y =, 04 罍ｧ x = y = 罍ｧ x = =6 ｧ x = 5=6 y max = ; 5, 04 罍ｧ x = = y min = 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 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 ﾀｧ07 ﾀ 03 ﾀ070 ｧ: x =0, y =; 5; y =0, x = 3 ; 7. 罍ﾀｧ x =, ｧ y =, x 罍ｧ x = y min =, 04 罍ｧ x =, y max =,, 04 罍ｧ x =0 縺0304. 罍ﾀｧ y = x ｧ x =0. 4. ) 7 罍ｧ x =0 y = a. ｯ03 09 ｧ 0406 縺06 ﾀｧ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 ﾀｧ罍ｧｧｧｧ08; ｧﾀ罍ｧ y = ae,x, 縺0308 ｧ閨0705 ﾀｧ 0509 ﾀ05 縺ﾀ05 縺03060 ｧﾀ09 罍ｧ y = ae,x. 05 ﾀ04 Ox ﾀｧ ) 7 罍ｧ x =, y max =, 04 罍ｧ x = 縺0605 ｧ03, 04 罍ｧ x = y min =,; 04 罍ｧ y =0x =0, x ; 3 ; 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; 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 ﾀｧ罍ｧ x = y max =; 04 罍ｧ y =0 x = = 0; 7. 罍ﾀｧﾀｧOx ｧ Oy 罍ｧ x =0 y max =, 04 罍ｧ x = y min =0; 04 罍ｧ y =0 x = 罍ｧ x =, y min ==3, 04 罍ｧ x = y max =3, 04 罍ｧ x =0 y =; ﾀｧ y = 罍ｧ x =, y max =; 04 罍ｧ y =0 x =0, x =,4; ﾀ0504 ﾀｧ07 09 罍ｧ x ) 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 x =, ｧ y = x +5;)y min =004 罍ｧ x =n, y max = 04 罍ｧ x =(n +) ｧ0 ｧ0060 y ﾀ06006 ﾀ ( ｧ) ｧ ah=4. 5. a= ﾀ ; cos = (03 閨罍ｧ 06 ﾀ0003m 04 ｧｧ, 0503 a m 6,05 閨06 a 04 縺ｧ07 AB ｧ AB

278 閨03 縺0305 ｧ) ﾀｧｧﾀｧ09 ﾀｪ06 縺06 a ﾀｧﾀﾀｧ閨 a= x = D=, v y = D 3= ; l 5; 6 0; 0304 縺06 閨ﾀﾀｧｧｧ l = ; 4 sin + ; 6 cos. 36. v max = 8 閨罍ｧﾀ x = 閨 S max = R 04 罍ｧ 0403 ﾀ x = R. 38. (; ). 39. ab 罍ｧ x = ﾀ0 ｧ 3 ; 7 ﾀ0. 4. x =; ｮｧ陦 05 ﾀ03 ﾀ0503 縺 D= 罍ｧ = =3 陦ｧ P. 45. F = cos + sin ; tg = =0; 5, ) y = x, y 00 => 0, 09 罍ｧﾀ06 閨ｧ 09; ) y = x 3, y 00 = 6x, 09 罍ｧｧ罍ｧ x > 0 ｧ縺罍ｧ x < 0, x = 縺0605 ｧ03 ; 3) y = e x, y 00 = e x > 0, 09 罍ｧﾀ06 閨ｧ 09, (0; ) 縺06 ﾀｧ07 ﾀ Oy; 4) y = lnx (x > 0), y 00 =, < 0, 09 罍ｧﾀ06 閨縺0809, x (; 0) 縺06- ﾀｧ07 ﾀ Ox; 5) (0; 0) 縺0605 ｧｯ03 09 ｧ 0406 縺0605 ｧ03 09 罍ｧ : ) (;,8=3); )(= ; e,= );3)( 3; 3=) ｧ (0; 0); 4)04 罍ｧ x =, ln,0; ﾀ0504 ﾀｧ07 x>,. ｯ03 09 ｧ縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ (,; 0) ｧ (0; ln ). y 04 ﾀ06 閨06 04 ﾀ , 09 罍ｧ縺0809. 罍ﾀｧ x =,. 53. y > 0, y = 0 ﾀｧ ) ｮｧ罍ｧﾀｧ Ox ﾀ0504 ﾀｧ07 x > 縺ｧ 09, 0 ｧ縺ｧ 09 ﾀ 0605 ﾀ07 Ox (0; 0). 07 罍ｧﾀｧ 06 ﾀ ( ﾀ0506 ﾀ 03 ﾀ0406 Oy ); ) , 縺06 閨03 閨罍ｧ04 07, 003 ﾀ閨04 ｧ陦ｧ0 ｧ ) 7 罍ｧ x =0 y max =,, ﾀｧ x =,, x = ｧ y = 0(05 罍ｧｧ); ) 04 罍ｧ x = y max =, 04 罍ｧ x =, y min =,, 0406 縺06 ﾀﾀ07 ﾀ Ox 04 罍ｧ x = 3, 0406 縺0605 ｧ03 04 罍ｧ x =, ﾀｧﾀｧOx ｧ Oy. 56. ) ﾀ0504 ﾀｧ07 x>0, 04 罍ｧ y =0 x =, ﾀｧﾀｧ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 , 04 罍ｧ x = 0y = ) 7 罍ｧ x = 0 y min =, ﾀｧ x =, ｧ x, y =0; ) ﾀｧ罍ｧﾀｧ Oy, 04 罍ｧ y =0 x = = 0; 7, 04 罍ｧ x = y min =,, ﾀｧﾀ04 Oy. 58. ) ﾀ0504 ﾀｧ07 x> 0, 04 罍ｧ x = y min =, ｧ 09; ﾀｧﾀ04Oy; ) Oy 03 ﾀ04 ﾀｧ罍ｧｧ, 04 罍ｧ x =0

279 y min = a, 04 ﾀ06 閨ｧ 09; 09 罍ｧﾀｧ0 ｧ ) 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 x = ｧ y = x; ) 04 罍ｧ x =, y min =,3, 04 罍ｧ y = 0x =, 3 0; 5,0; 6, ﾀｧﾀｧ Ox ｧ Oy. 60. ) ｮｧ罍ｧﾀｧ Ox ｧ Oy, ﾀ0504 ﾀｧ07 jxj <, 04 罍ｧ x = y 05 =, 04 罍ｧ y = 0 x = 0 ｧ00 ｧ x = ;) ｧ y = x + y min = 304 罍ｧ x =, x y = x, x 0406 縺06 ﾀ Ox 04 罍ｧ x = 3 4 ; 6, ｧｧﾀｧ y = x ｧ x = 罍ｧ x =, y min =, 3 6,; 5, 04 罍ｧ x = y max ; 5 ( ｧ ), 03 ﾀ04 Ox ﾀｧ , 8x ｧ0303 y =! 0, x!. (x +) 4=3 +(x, 4) =3 +(x, ) 4=3 6. ｮｧ罍ｧﾀｧ Ox, ﾀ0504 ﾀｧ07 x > 0, ﾀｧﾀ04Ox ( lim x! y =0), 04 罍ｧ x =0509 ﾀ05 縺 y 05 = =e 0; ) 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; ) (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 sin 3x + C. 80., cos x + C. 8., 3 e,3x + C tg 5x + C. 83. (ex=, e,x= )+C (4x, )3= + C.

280 (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 ln j+sinxj + C. 98. ln j +lnxj + C. 99. sin3 x + C , cos4 x + C. 30., 4 3 sin 3 x + C. 30. cos x + C. 303., cos x sin x +C C. 305.,e cos x +C sin x 3 ex3 +C. 307., e,x + C e x + C (x +) 3 + C (x3, 8) 4 + C. 3. 0; 5 3 ( + x 3 ) + C. 3.,, x + + C. 33., + cos x + C. 34. ( + ln x) 3 + C ( 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 x + C. sin x, 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 ) 0; ln x, 5 x C; ) 3 arctg x 3 + C. 33. ) arcsin x + C; )ln (x + x +5)+C. 33. ) ln jx + x, 4j+C; ) arctg x +C ) arcsin x +C; ab ln bx, a ) x3 arctg 6 + C ) x arcsin + C; ) 3 bx + a + C ) x arcsin + C; ) 3 4 ln (x4 + x 8, ) + C ) ; 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 x, arctg x + + C x3 3 +3x ln x, 3 x C arctg (x +)+C.

281 arctg x, 3 + C. 34. ln (x ++ x +x +3) + C arcsin x + + C arcsin x, x +3 + C arctg C arcsin 4x, 3 +C ln j3x, + 9x, 6x +3j C arctg +ln 3 x, 3! x + C arcsin + 3 x + +ln(x + +x ) + C ln(x +5), 5 arctg x + C x + ln x, x + + C. 35. x 3 3, x + arctg x + C arcsin (e x )+C arctg (x )+C ; arctg x + + C arctg x, +C arcsin x + +C ln (x + x +),, x + arctg + C ln (x ++ 4x +4x, 3) + C x ln jxj,x + C. 36. x ln jx, j, x + x +lnjx, j + C. 36. ex x, + C x + arctg x, x + C x sin x + +x cos x, sin x + C ex (sin x, cos x) +C x (ln jxj,, ) ln jxj C. 368.,x ctg x +lnjsin xj + C. 369., + C. x xarcsin x +4, x + C. 37. x arcsin x +, x + C. 37.,e,x (x 3 +3x +6x +6)+C x ln (x +), x + arctg x + + C x (cos ln x + sin ln x) +C x 3 ln jxj, 3 + C. 376.,e,x= (x +4x +8)+C x arctg x, ln ( + x )+C. x sin x + ctg x + C x tg x+lnj cos xj+c ; 5e x (sin x+cos x)+c x,,, x arcsin x + C. 38., 38. x x, arctg x,, + C x + 4 sin x + sin x + C. sin 4x + cos x, +C x sin x sin 4x C x 8, x sin 4x, 3 +C x 8 sin 4x sin 8x, C x sin 4x, sin 3 x + 48

282 3+ C. 390., cos x + 3 cos3 x, cos5 x C. 39. sin3 x 3, sin5 x 5 + C sin4 x, 6 sin6 x + C sin x, sin 3 x sin5 x, 7 sin7 x + C x + 4 sin x + 3 sin x, 8 sin3 x + C. 395.,, sin x + C. 3 sin x 396. cos x + cos x + C ln j tg xj + C ) tg ln x + x + C; )lntg + h 4 + C lntg x x +lntg + i + C 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 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 ) 5 sin 5 6 x, cos x x + 5 sin3 x + 5 sin x + C cos x 408. ) sin x + tg ln x sin x + C; ) cos x + tg x ln x + 3 sin x sin 4x + C x, 4 sin x + sin 4x + C. 3 x sin 4x 4.,, sin3 x + C. 4. sin x, sin3 x + sin5 x + C cos5 x, cos3 x + C x, 4 cos x, 3 sin x + 8 cos3 x + C ln j tg xj,x + C ( sin x, sin 4x) +C. 47. cos x + + cos x +tgx + C. 48., 4 cos x + 4 x3 + x + C. 49. ) 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 x + 4x +ln(x, )8 jxj + C. 44. x +ln x, x x + C. + C.

283 a ln x, a x + x, a ax + C. 46. ln Cx(x, ) + x,. 47. ln x, x +, x + + C ln (x +x, 0), arctg x + + C ln(x, 0; x +0; 7), 5 arctg 0x, + C ln jx +j x, x x C ln + arctg x, + jxj (x +) + C. 43. ln 4 x, x arctg x, x +C ln, 3 jx +j, x + + arctg x + C ) b 3 arctg x b + bx x + b + C; x(5b +3x ) ) 8b 4 (x + b ) + 3 b arctg x x +9 + C ), b 8(x +x +5), 6 arctg x + + C; ) (x, 3)(3x, 8x + 3) 8 (x, 6x + 0) + 3 arctg (x, 3) + C. x ln + x, jx +j x + + C x, 4(x +) + 8 arctg x + C a ln x 44. x + a +C ln x, 3 x + 3, + C a, b ln x + b x + a +C 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 ln,, ) lnc(x, x + x + arctg (x, ) + C jx +j ln x arctg x + C. +C. + C(x, )3 + C ln. x, C(x, ) 3, ). (x +) lnC(x, (x, ) x + x, ln jxj x, x + + a ln x + a (x, ) 45. ln 4 x +x +4, x + arctg + C , x + x +x + + arctg (x +) + C 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.

284 (x + )(x, ) = 4 ln x, x + x +, (x, ) x +, (x, ) (x + )(x, ) = ln x,, arctg x + C. x +, arctg x + C. 3 x x (3x +) + + C ( x +, 3) + C. 5 x 3 x , + 6 x, ln ( + 6 x) + C (3x, ax, a ) a, x + C. " (x 4 +), 3 x 4 ++ln( 3 x 4 ++) 4 # + C. (x, 4) x C arcsin + C (, 04 罍ｧ x > 0 ｧ 3 x r Cx + 04 罍ｧ x< 0) ln x ++ x +x , a, x + a x C(x h +) + C ln + x +x x a, x + a arcsin x i +C. a x x + C arcsin x, x 4 (, x ) 4, x + C. 47. x 3 3a (a + x ) 3 +C , (x, ) 縺陦ﾀ x, = sin t, , 4 sin t cos tdt= arcsin x, x, x, arcsin x + C., (x, ) 3+x, x + C (x +5) x +x +, 3; 5ln(x ++ x +x +)+C. 475., 3, x, x, arcsin x + + C x, a ax, x + a arcsin x, a + C. a ln +x 3, 4 +x arctg 4 +x 3 +C. 479., 3 (, x 3 ) 4x +C.

285 m +,+ + = + 3 n ﾀｧﾀ0006; ｧ04 x, + = = t, ｧ0 x, x,3 (x, +) 3= =, t, dt t =, +x x +x + C. 48. m + = ﾀｧﾀ0006; ｧ04 a, bx = t, n ｧ0 t, a a, bx (x, ) x, dt b t = b + C C. a, bx 3 (3x +)= (3x +) =3 +lnj(3x +) =3, j +C x, x + + ln ( x +)+C. 485.,0; 3(x 3 +3a) (a, x) + C. r 486. x, x, + arctg + C (x +) 3! (x 3 +) x C ln ( + +x )+ + +x + C x + 3 (4, x ) 3 + C; 罍ｧ006 縺閨003 ﾀﾀ陦ｧ0504 ﾀ07 r0305 ｧ罍 09 ｧﾀ05 ｧ x 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 arcsin, x, x +C x 4x + x, lnjx ++ 4x + x j + C. 495., x x, x + 7 arcsin x, + C. 3 +x 496., x + +x ln + + C. jxj +x 497., + C ｧ04, x 3 = t, 0 08 閨060 x x x 3, x = dt 3 3 t, =, x3 3 ln,, x C ｧ04 x ==t, 0 08 閨060 dt, =, dt 3, t, t 4, (t +) = arccos x + x + C.

286 ln (ex +), arctg (e x )+C tg3 x, tg x + x + C. 50. ex, ex + 4 ln (e x +)+C lntg x + C arctg tg x 506., ctg3 x 3 + C ln tg(x=)+ tg (x=),, ctg x + C arctg tg x + C. + C e x +lnje x, j + C tg4 x, tg x, ln j cos xj + C. 50. e x ln e x, e x + + C. 5. tg (x=) arctg + C. 5. tg3 x 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 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 x x + C. 56., x ch 3x + C. 57., ch3x + 5 x, sh 4x + C. 58., x C. 59. sh5 x + C x, cth x + C ch x, +C (0403 ｧｧ0504 ﾀ0 00 ｧﾀ00 ｧｧ ch x, ). 53. sh x, ch x +C ln jx + x, 3j + + x x, 3+C ln jx x + x +3j, +3 +C x ++ x + ln x +, +x (arctg x) x + C C a ln x + a x,, x ax + C tg, + C arcsin x + C ( ｧ b x = sin t) ab arctg a tg x + C. x + x sin x + cos x r, x +x = + C. 54. ln C(e x +), x, e,x., x, x = arcsin x +, x + C.

287 C x tg x +lnjcos xj, x + C. tg 546. ln x + + cos x + C. 547., cos x arctg + C x =3, x =6 + b b +4 ln (x =6 b, 3ax +)+C C ( ｧ0504 ax + b = t). 6a(ax + b) , ctg3 x , x + arctg x + C. 55., tg x + ( 閨0600 ｧ0504 ｧﾀ00 ｧｧ cos x ｧｧ0504 tg x = t). 55. b a + b ln x + C b(n, )(a, bx 3 ) n, + C 04 罍ｧ n 6= ｧ, 3b ln ja, bx3 j + C 04 罍ｧ n = 閨0600 ｧ縺陦 05, ｧ0504 x + + = sin t ( ｧ00 ｧ 0706 縺0600 ｧ閨縺06 閨ｧ09 ｧ ); x +, x, x + arcsin x + + C. 555., x + ( x +) + C ln x +x, arctg x +C x arctg ex, x+ 4 ln (4 + ex )+C ln C x + + x 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 ﾀｧ罍 09 ｧﾀ05 ｧ ; 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 ln C(x, ). 564., x 3 3= x + + C arctg x 3, +C ( ｧ0504 x 3, =t ). x x + x + x ｧ0504 x =. t ; 5[x +lnjsin x + cos xj]+c x arcsin x +, x + C 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 tg4 x + C ( ｧ0504 tg x = t) ln jxj, x + ln jx +j + C. x

288 3574., sin x = cos x + sin x = + sin x + C (+ 04 罍ｧ cos x>0 ｧ, 04 罍ｧ cos x<0) 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 x arctg x,, ln j+xj + C tg x + C ( ｧ0504 tg x = t) ln jxj,, x + x ln (x +) + C. 58. ln a arctg (ax )+C. 58. ( x + (x +7) j x +, j + cos x)+c x ++ ln + C ( x ｧ0504 x + = t ) x, x,, x arcsin x + C x ｧ0504 x =. 586., 3x +3x + t 3(x +) 3 + C ( ｧ0504 x + = = t) ax + x, a ln jx + a + ax + x j + C ( ﾀ. 53, ) ln (x, ) jx + xj + C. 589., + cos x + sin x sin x + C ln C(x +x +) + x, x + 8 arctg x ( ﾀ07 0, x ｧｧ 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; a (e, ) ln ( + ) ｧ04 x = t ｧｧ 0060 ｧ04 ﾀﾀ縺06 閨060003, ｧ0 3 tdt t, =, t + ln (t, ) 3, 3 = ( + ln ) ,, ln , e a(, ) ln ( ｧ0504 x = e + 4 = a sin t) a +ln(+ ) ln, , ln ) ;) 3 4 ;3)

289 , a , ln; 5, arctg e, 4 0; , 4. 6.,. 63., ln. 64. ) ;) 3 4 ;3) ab 縺03 ｧ 0406 閨060 ｧ07 03 ﾀｧ07 ( h) ﾀ h ln ; ; a ( ﾀ0. 罍ｧﾀﾀ. 309). (4, )a ; 8 ( ﾀ0. 罍ｧﾀﾀ. 307) ; ｧ0504 x =asin t ( 罍ｧﾀ. 84, ﾀ. 335) a sh = a (e, e, ) ; 35a a a a a r max = 404 罍ｧ ' = n, 罍ｧ ' = n =45, 5 ; r min = 04 罍ｧ ' = =, n, 罍ｧ' =, n = 35, 35. ｮﾀ05 縺陦ｧ06 ﾀ縺03 04 罍ｧ 45 ｧ 35. 0ｷ3 ﾀ閨 =4 =4 (3 + sin ') d' = r = a(sin ' + cos ') =a cos ', 4 a a ; r max = a 04 罍ｧ ', 4 = = 0, ' = 4 ; r min = 004 罍ｧ ', 4 =, ' =, 4 ｧ閨04 4 3=4 S = (a ) cos ', d' = a ﾀ07 04 縺03006, 06 ﾀ00 ｧ 4,= 縺ｧ 09 閨0609 縺罍陦ｧ0 05 0: x + y = a(x + y) 0309 縺ﾀ a (0 +7 3)a a ; 5, 6ln ( ﾀ0. 罍ｧﾀ. 5 0 ﾀ. 307) ,,x x + = 8 ( ﾀ0. 罍ｧﾀﾀ. 307) r max =4, ' = n, ' = n =90 ｧ00 ｧ 70 ; r min =, ' = n, ' = 80 n =0 ｧ00 ｧ閨04

290 3 = S = (3 + cos ') d' = a a a (e, e, ) = a sh ab arctg b a a h 8a b ; a b. 3 sh 674. a a ! a ( +) a a b ; a V = = a a a ｯ03 09 ｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ 04 罍ｧ t = 0 ｧ t = 4 8, 4 8 s = t 4 +t 3 dt = ln ( + 3) a sh 0 ; 35a 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 ; x = t ; s = x 3=5 5=4 t dt t, = =; 35 + ln ; ｯ03 09 ｧ 0406 縺06 ﾀｧ07 ﾀ cos x cos x = =3 0 d(sin x), sin x =ln(+ 3) ; ) 4 3;)0; 5ln(ch) ; ) 8a; )a a ln ( + +4 ). 3a ln ln3, ln(+ ) ; a (sh +) ln ( + ) a , 6 ab ; 4a. 7. 9; ; ; 0 0 ｧｧ006 ﾀｧｧ07 ; ah R ;

291 J x = ab3 3, J y = a3 b J x = ab3, J y = a3 b ; M x = M y = a3 6 ; x c = y c = a a M y x = y=0; ab, a a M y = xy = 4 ba ; S = 0 = 0, y c = ) ; R 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. ) ; ; R+h R mgr x = mgrh R + h R H ; t = = 00 ﾀ 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 閨ｧﾀｧﾀ00 ｧ04, ｧ0 t 4 ﾀ ah = = 0 h. 74. ; 閨0606 ﾀ I x = a 4 sin t cos tdt= a x c = 0, y c = H R 0 R 0 " V0 a 0 y y y xdy = = R 000 (H, H x) x , 0V 0,, V 0 r t = 4R R 5 s 0; 8 g = ﾀ ) ; 3 ｧ ) 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 ) ; ) =; 3) =4; 4) ; 5) ln ; 6) ) 6 ;) 4 + ln ;3),. 75. ) 6 3 ;) ﾀ0803 陦ｧ05 ﾀ07; 3) 6. 8 #

292 ) ﾀ0803 陦ｧ05 ﾀ07, ｧ0303 < +x 3 +x 3 x, 3= 0 ( ﾀ ); ) 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 x3, > x, e,x x e,x ﾀ0803 陦ｧ05 ﾀ07 ( ﾀ ); 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 ); 5) x x x 4 + x 04 罍ｧ x>, 4 e,x ﾀ0803 陦ｧ05 ﾀ ) b a 03 ﾀﾀ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 a a ln ( + ) ) ;) 3 ;3); 4) ﾀ0803 陦ｧ05 ﾀ ) ln ( + ); ) ; 3), ) 3ln ;) ; + ab + b 3) e, ;4)a ;5) ) "(h) =0;)j"(h)j 6 < 0; ; 8 閨 ln = 0; 693; j"(h)j 6 0,4 < 0; ; 罍ｧ0300 ｧ ; R = R = 縺00 ｧ006 (; 0) R = ; 縺00 ｧ006 (0; ) R = R = =4a. 78. y max = 04 罍ｧ x =; R = e (4; 4) (3;,). e

293 (0; ) X +8Y 3 = (X) =3 + Y =3 = 3 = X =3,Y =3 =(a) = X = a cos t, Y = a sin t ｧ00 ｧ X +Y = = a k = e x ( + e x ),3= ; k max = ln x =, 3 3, 0; ) R = a r3 ar; ) ;3) 3 3r a (,; 3) ;, , 3 ; X = 4, 3,0; 7; Y =,,; X 3, 7Y = X =,t + t, Y = 4t + t ; 閨ﾀ05 縺03060 ｧ07 09 罍ｧｧﾀ03 ﾀ05 04 ｧｧｧ08 x, y, X, Y 閨0007 t =0;; 3= (X +Y ) =3,(X,Y ) =3 = (X +Y ) =3 +(X,Y ) =3 =a =3 ; 04 罍ｧ縺ﾀｧ06 04 罍ｧｧ x =3 +y =3 =(a) =3, ﾀ05 縺03 ｧ閨03 06 ﾀﾀ05 縺03 ｧﾀｧｧ 04 閨ｧｧ縺 t ; sh ; ln 83. y = 4 3 x, x 9 ; dr dt w = 4jt, j 4t, 6t +5, w n = w n = ; x, t 88. x, x a = y, t t = y, x +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, _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, ｧ縺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 ﾀﾀﾀ0903 ﾀ05 ｧ: 3x, 3y + z =. 8. N =3(i + j), B =,i + j +k. ｰｧ縺0 00 ｧ: x = y, z =0; 03 ｧ003 縺0 00 ｧ: x,, = y, = = z. 8. 0ｷ3 ﾀｧ04 t, ｧ0 x + y = z ｧｧ 06- ﾀ縺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 : x = z ｧ y = 0; 縺0 0004: 03 ﾀ04 Oy; 03 ｧ-

294 縺0 0004: x + z = 0 ｧ y = 罍ｧ t = x,a = z, b= ; b a b 4 4ab y = a. 84. cos =, cos =, cos = ; a + b a + b a + b ｧﾀｧｧ閨ｧ 09 罍ｧｰｧ07 04 ｧｧ0 ｧｧ: x = sin t, y =, cos t, z = t, 05 閨06 t 縺0305 ( 罍ｧﾀ. 44). ｭ陦ｧ0 ｧｧ003 縺 C (04 罍ｧ t = =): = i + j + k 罍ｧ t = v = 7 ｧﾀ. 44 = a(i + j), w = ai. 87. x, = y, = z, x, = = y, ｧ z = 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. ｰｧ縺0 00 ｧ: x =, z = y +; 03 ｧ003 縺0 00 ｧ: x =, y + z = 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, , 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,

295 w = _v = 7, w n = v 5 R = v = 5 =; 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 +) =( (t +) ) ｮ0 00 ﾀ03 ﾀ05 04 ｧ罍ｧ罍陦ｧ縺0304 r _r r... r _r r t t t 3 t 3t 0 6t t,6t 6ｦ閨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 +) = R = (x + y) = 4, =, R = 3, = ｧｧ0 ｧｧ: = b a + b, : =, 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, =, R = 3, =, ) 6 5 ﾀﾀ0903 ﾀ0504, 09 縺ｧ (0; 0); 4)x + y 6 a ;5)xy > 0 (0406 縺ｧ 05 縺0605 ｧ陦 00503); 6) x + y < ; 7) 04 ﾀﾀ0903 ﾀ0504, 09 縺縺 y = x. ｰｧ07 ) ｧ ) 0304 縺06 閨ｧ閨ｧ07; 3) 縺08003 ﾀｧ縺ﾀｧOz 09 罍ｧ z = 4 x ｧ y = 0 ( 罍ｧﾀ. 45); 4) ﾀ0706 縺06; 5) ﾀ, 閨0007 ｧｧ縺ﾀｧ07: x = a, z = ay ｧ y = b, z = bx ( 罍ｧﾀ. 46); 6) 縺08003 ﾀｧ07 09 罍ｧ z =, y = 縺0605 Oz; 7) ﾀ, x ﾀｧ0 ｧ y = kx, z = kx ｧｧ0 ｧﾀ0503 縺030k, 0 ｧ0 ｧ 05 ｧ0406 縺ｧ y = h, (x, h)(z + h) =,h, ｧｧ0 ｧ 0406 縺00 ｧﾀｧ Oy ｧ 03 閨006 ｧﾀｧﾀ0903 ﾀ05 ｧ y = x (x = h, y = h); ｧｧ0406 縺ﾀ07 04 ﾀｧ0708 x = h ｧ00 ｧ z = h ( 罍ｧﾀ. 47). :

296 s = (, x)(, y)(x + y, ) ﾀ0504 ﾀ06006 ﾀｧｧｧ: 0 <x<, 0 <y< ｧ x + y>, ﾀ罍ｧ 05 縺ｧ09, ｧｧ0 ｧ070 ｧ x =, y = ｧ x + y =. 7 ｧﾀｧﾀ x z =(x, y +x)x =0;, y z =(y, x +y)y =,0; 9, z = x z + y z, xy =0; 縺06 縺ｧ 03 閨ｧﾀﾀ05 ｧ jyj 6 jxj ｧｧ z = + x, y ｧ z =, x, y ｧﾀ縺ｧ 0 ｧ縺08003 ﾀ05070 ｧ 09 縺ﾀ ( ﾀ 03 ﾀ0406 Ox). 7 罍ｧ006 縺030 縺ｧｧ, 0304 縺06 閨ｧ060 z =

297 = x, y, ﾀｧｧ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 ｧ縺 x =, x = ｧｧ0 ｧｧ縺ｷｧ閨縺06 閨06060 ｧ06 ﾀﾀ縺ｧ 0 ｧ縺08003 ﾀﾀﾀ縺06 閨ｧｧｧ jyj 6 jxj, ﾀ罍ｧ 03 ﾀ05 縺閨06 04 縺ｧ y = x ｧｧ08 04 縺 ) 6 5 ﾀﾀ0903 ﾀ0504, 09 縺縺 y =,x; 3) ｧ罍ｧｧ04- ﾀ x a + y = ｧｧ04 ﾀ06; 4) 04 ﾀﾀ0903 ﾀ0504; 5) ｧ罍ｧ b jyj 6 jxj ｧﾀ0503 縺030 08; 6) 0904 陦ﾀ0903 ﾀ05 ｧ x > 0 ｧ y > 縺08003 ﾀ0504 ) 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 07 ﾀｧ0 ｧ z = h, x + y =4=h ｧ z =4=x, y =0( 罍ｧﾀ. 48) 縺08003 ﾀ05 ｧ 5) { 6) ｧ 06 ﾀ09 ｧ06; 7 ｧﾀ縺08003 ﾀ0504 4) ｧ x(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) x( 3 x, 3, t) 3 3t( 3 t, = a, b = b, a ab sin =. = e,xy =,x = 5t =

298 r =, (x = t x, x 874., =,a sin (ax, by), = b sin (ax, 875. =, yjxj x = jxj x x, = 3y =, 3x (3y, = ctg (x, =, ctg (x, t). = cos (x = sin x cos (x +y) ) 0; 075; ),0; e, 0; ,0; ; 閨 ; 3 ﾀ ) dz = =, y x + y + x + x y dy; )ds =lnt+ xdt. 89. z = t =0; 043, dz =0; ; ,30 ﾀ dz dt =,(et + + e,t )=,cht dz = ey + xe y = x, x, =,x 4+ x y @v, @x @u. @u cos ' + sin cos ' r. = = (Ax+By) cos t,(bx+cy) sin t =(A,C) sin t+b cos t; ) dz dt = = et e 4t +. y 906. = @y x r =, x y y ), 3 x ; ) yex, e y xe y, e. x ,. 9. ) (,; 3) ｧ (,;,); ) (; ) ｧ (,3; ). = 3, =,y = = x = = b c. 98. dy = x 4y. 99., y x. 90. x + xy + y. 9. xy =, = y x, z ; ; 0; , 6y x 4 ; x 3 ; 0; xy (x + y ), y, x (x + y ),,xy (x + y ) ) x 4 (3y, 4xy dy +x dy );), y(, xdy) xy.

299 @y z z z = + z ; @v : ｦ0 04 ｧﾀ , ｧ , ｧ0 z, + z x + x ; z @v z ;,y z,4y + y 8x (, y) ; : =, = 4x (, = = 9t 3 t ;, 8x 7t 3 3 t d u =, y x + + x dy, d3 u = y x 3 3, 3 x dy z. z,v + + x u = + x ln y, cos y + C. 96. u = x z + x +lny,, arctg z + C u = xy, x + 3y + C u = x sin y + + y ln cos x + y + C u = xy + sin y + y + C u = x x ( + t +)+C u = x ln y, x cos z + yz + C u = x, 3y + C y = x +x; ﾀ0504 ﾀｧ07: +x > 0; z x >,. ｯ03 09 ｧ 0406 縺06 ﾀｧ07 ﾀ Ox: y =0, x =0 ｧ00 ｧ x =,. 05 ﾀ O(0; 0) ﾀ05 縺 y 04 罍ｧ x =, 3, y 05 = 3 3 ( 罍ｧﾀ. 49) y = (x +) x +; x >, ﾀ0504 ﾀｧﾀ : (,; 0) ｯ03 09 ｧ 0406 縺06 ﾀｧ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, ﾀ0504

300 ﾀｧ07 x >, x =0, y =0 03 ﾀｧ 0300 ｧ縺罍ｧ x = y =0, 04 罍ｧ x = y =. ｯ縺0605 ｧ03 : x = 4 3, y = ( 罍ｧﾀ. 5). 97. y = x, x ; ﾀ0504 ﾀｧ07 jxj 6, ｧ00 ｧ 7 ｧﾀｧﾀｧﾀ. 5, 6 x 6. ｯ03 09 ｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ: 04 罍ｧ y =0 x =0, x =, x 3 = =,. 05 ﾀ O(0; 0) ﾀ05 縺罍ｧ x = 0; 7 y 05 = ( 罍ｧﾀ. 5) y = x x x ﾀ0504 ﾀｧ07 x > > 0; ｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ070 ｧ: 04 罍ｧ y = 0 x = 0 ｧ00 ｧ x = ; 03 ﾀ O(0; 0) 縺03 ﾀ 09 ﾀ y = x. 709 ﾀ- 7 ｧﾀ. 5 7 ｧﾀ縺ｧｧ07 y = x, x x; 04 罍ｧ x = 4 9 y max = 4 ( 罍ｧﾀ. 53) y = (x, ) x; ﾀ0504 ﾀｧ07 x > 0; 04 罍ｧ y =0 x =0 ｧ00 ｧ

301 x =; 03 ﾀ (; 0) 罍ｧ04 07 ｧｧ, ｧ 0 罍ｧﾀ. 5, 003 ﾀ閨04 ｧ y = (x + a), x +a ;09 罍ｧ04 07 x ﾀﾀ05 ｧ, 05 閨06 x ｧ x +a ｧｧ, 罍ｧ,a 6 x<0. 05 ﾀ (,a; 0) ; x =0 ﾀｧ罍ｧｧﾀﾀ03 ｧ r, , 罍ｧﾀ. 85, 003 ﾀ a x 3, y y = ; ﾀ0504 ﾀｧ07 y 6 x. ｯ03 09 ｧ 0406 縺06-3 ﾀｧ07 ﾀ 03 ﾀ070 ｧ: 04 罍ｧ x =0 y =0 ｧ00 ｧ y =,3. 05 ﾀ (0; 0) 閨060 ﾀｧｧ y = kx + b. 7 閨0600 ｧ y 3 y ｧ07 0 x 3 : =, 3 = 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 y = x,. + xy + y 709 ﾀ05 縺ｧｧ x = '(y) = 3 y 3 +3y : 04 罍ｧ y =, x 05 = 3 4 ; 6; 04 罍ｧ x =0 y =, 縺0605 ｧ03 ( 罍ｧﾀ. 54) x 3 + y 3, 3axy =0 閨0609 縺ｧﾀ05 ( ﾀ ). 05 ﾀ O(0; 0) ﾀ09 ﾀｧ y = 0 ｧx = 閨060 ﾀｧ y = kx + b. 7 罍ｧ0406 閨060 y 3 y y ｧｧ閨06 +, 3a =0; k = lim =,, x x x x! x r b = lim (y + x) = lim x! x! 7 ｧﾀｧﾀ. 55 3axy x =,a. 0ｷ305 09, y =,x, a ﾀｧ004-, xy + y x ( 罍ｧﾀ. 79) y =. ｮｧ罍ｧﾀｧ Ox x, a

302 ｧ Oy ﾀ0504 ﾀｧ07 jxj > a ｧ jyj > jxj. O(0; 0) 03 ﾀｧ 0300 ｧ縺罍ｧ x = a 0509 ﾀ05 縺 y = a. 罍ﾀｧ x = a ｧ y = x ( 罍ｧﾀ. 55) y = x, x; ﾀ0504 ﾀｧ07 x 6. ｯ03 09 ｧ 0406 縺06 ﾀｧ07 ﾀ 03 ﾀ0406 Ox: 04 罍ｧ y =0 x =0, x =. 05 ﾀ (0; 0) ﾀ05 縺 y: 04 罍ｧ x = 4 3 y 05 = = = ; 08. (07 罍ｧ04 07 ｧ縺006, 罍ｧﾀ. 49.) 980. y = = x a a, (x, a) ; ﾀ0504 ﾀｧ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 ( ). 7 罍ｧ y 0 x(a, =0, ax, x x) + = ax, x =0, x = 3a, y 05 = a 5 4 a ( 罍ｧﾀ. 56). 98. y = (x +) x ﾀ0504 ﾀｧ07 x > 0 ｧｧ 0300 ｧ縺 (,; 0). ｯ縺0605 ｧ03 04 罍ｧ x ==3. 07 罍ｧ , 罍ｧﾀ. 5, 003 ﾀｧﾀｧﾀﾀ05 ｧﾀｧ07: ) x > 0; ) x<,a. 罍ﾀｧ : y = x + 3a, y =,x, 3a ｧ x =0. ｯ (,a; 0). 709 ﾀ05 縺 y 04 罍ｧ x = a y 05 = 3 3a ; 6a y = x x +5; ﾀ0504 ﾀｧ07 x >,5. 05 ﾀ (0; 0) ﾀ罍ｧ0903 ﾀｧﾀ05 縺 y: 04 罍ｧ x =,4 jyj max =8, 04 罍ｧ x =0 jyj min =0( 罍ｧﾀ. 57).

303 y = x x, ﾀ05 ｧﾀｧ07 jxj > ﾀｧ 0300 ｧ縺 O(0; 0) ｧ , ｧ 0 罍ｧﾀ. 5, ﾀ閨ｧ閨03 ﾀｧ罍ｧｧ 09 罍ｧﾀ罍ｧ y =0 x =0 ｧ x =,4; 04 罍ｧ x =0 y =0, y =,. 05 ﾀ (0; 0) ﾀﾀ k =. 7 罍ｧ x =,8=3 y max =; 8 ｧ 04 罍ｧ x =0 y min =,. 罍ﾀｧ y = x+. 07 罍ｧ縺06 ﾀﾀｧ罍ｧ x =,0; 4 ｧ r ｧﾀ x , 04 縺0308 閨07 06 縺06 (0; 0) ｧ (0;,) ) y = (x,a) a, x ;09 罍ｧﾀ , 05 閨06 x ｧ a, x ｧ陦ｧｧ, 罍ｧ 0 6 x 6 a. ｯ03 09 (a; 0)03 ﾀﾀﾀ k =. ax 罍ﾀｧ x =a ( 罍ｧﾀ. 84). ) y = ; ﾀ0504 ﾀｧ07 x, a jxj >a ｧ jyj >a ﾀｧ 0300 ｧ縺 (0; 0). 罍ﾀｧ x = a ｧ y = a 閨閨縺ｧ08 ﾀｧ罍ｧ040308, 09 縺ﾀ , 00605, ｧ0303 jxj >a ｧjyj >a. 07 罍ｧ04 07 ﾀ03 ﾀ0503 ｧ05 ｧ縺0608 ﾀｧ罍ｧ , 04 罍ｧ0300 ｧｧ08 ﾀ07 09 ﾀｧ x = a ｧ y = a ) y = r a, x = x ; ﾀ0504 ﾀｧ07,a <x6a. ｯ03 09 ｧ 0406 縺06 ﾀｧ07 ﾀ x + a 03 ﾀ0406 Ox: y =0, x =0, x = a. 05 ﾀ (0; 0) 罍ﾀｧ x =,a. 07 罍ｧ04 07 ﾀ05 縺ｧｧﾀ縺0605 ｧ03 0 ｧ060 罍ｧﾀﾀｧ Oy ｧﾀｧﾀｧ Oy a. ) ﾀ05 ｧﾀｧ07: x > a; x<,a ｧ x =0. ｯ03 09 (0; 0) ｧ 0300 ｧ縺罍ﾀｧ x =,a, y = a, x ｧ y = x, a. 7 罍ｧ x =, a( 5+),; 6a y 05 3;3a ) y =,x =4; )y = x ) y = R; )y =0 ｧ y =,x ) y =;)y = 罍ｧ 06 ﾀﾀ , ｧ ; 3) y = ｧ罍ｧ 06 ﾀﾀｧ 0305 ｧ ; 4) y = x, 4= ｧ , y = x 罍ｧ 06 ﾀﾀ x =3 +y =3 = a = y =, x3 x (x +y ) = = 4a xy ｮﾀ罍ｧ08 y = x tg, gx b cos. 0ｷｧ ( ﾀ003 ﾀ05 ｧ09) y = b g, gx b ) x + y = = ;)y =4x; 3)y = y =4(x +) x =3 + y =3 = = l = y =,4x = x +4y, z = xy 0 + yx 0 = =zz xy 0 z 0 + yx 0 z 0 + zx 0 y 0 =3a xx 0 a + yy 0 b, zz 0 c = x + y, z = x, 3 = y, 4 = z, 5 ; (0; 0; 0). 3 4,5

304 cos =, cos = cos = y = 0, x + z + = 0; 縺08003 ﾀ0504 ｧ罍ｧﾀ. 45, ﾀﾀﾀﾀ0504 x, y +z = a. ｭ06 ﾀﾀｧ a ｧｧ縺08003 ﾀｧ縺ｧ0 ｧｧﾀ07 04 ﾀｧ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, 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 cos = 3 ; cos =, 3 ; cos =, ﾀ0903 ﾀ0504 z + y, x = a, = a ) z =4;)x +y +z = 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; = 4 dl. 0. du dl =+ ; grad u =i +j + +k, jgrad uj = grad u = 6 = 4i. 04. a + b + c. 05. grad z =0; 3i,0; 64j, jgrad zj = =0; du yz + xz + xy = = dl 3 7 ｧﾀ. 58 = grad u =(xi + yj, zk), jgrad uj =z. 08. grad u = xi + yj + zk 3 =, jgrad uj = , u a + b + c z min =, 04 罍ｧ x =,4, y =. 03. z max = 04 罍ｧ x = = y = z min = 0 04 罍ｧ x =, y =, ﾀ05 縺 z min =, e 04 罍ｧ x =,, y = z max = 罍ｧ x = y = z min = 04 罍ｧ x = y = z max =,4 04 罍ｧ x = y =, ｧ z min =404 罍ｧ x = y = x = y = 3 V, z =0; 5 3 V (8=5; 3=5), (,8=5;,3=5) ｧ 0 ｧ0 ｧｧｧ

305 z = d = x +(y, ) 04 罍ｧ 06 ﾀｧｧ x, y, 4=0. 0ｷ3 ﾀ ( 5; ). 04. R =, H =. 04. ) 縺00 ｧ003 (3;,) ｧ (0; ); ) 0006 閨縺ｧ 05 09, sin : sin = v : v, ｧ 04 縺03 ｧﾀ陦ｧ罍ｧ縺03 閨 z min =904 罍ｧ x =0 ｧ y = z min =004 罍ｧ x = y = z min =004 罍ｧ x =0 ｧ y = z min =004 罍ｧ x =, y = z max =04 罍ｧ x = y =, z min =, 04 罍ｧ x =,y = V = ) ｧ 0 ｧ0 ｧ0060 d = x, y +4 ｧ00 ｧ 0 ｧ0 ｧ z = x r, y +404 罍ｧ 06 ﾀｧｧ 4x, y =0; ｧﾀ (; ); )ab. S 050. R = x ｰｧ07 ｧ罍ｧ040308: ) y = 3 3 ; ) y = x 3 ;3)y =, x xy0 = y ) y, x = xyy 0 ; ) x + y = xy y = Cx, y =,x xy = C, xy =, x + y = C, x + y = y = Ce x, y =4e x y = = Ce =x. 06. x+y =lnc(x+)(y+) r = Ce =' +a s = = t, +Ct y = Ce x, y = e x, y = C sin x, ; t y = sin x, x + = C; y =,x ｧ06 ｧ : y ) y = C(x, 4); )y = C cos x. 6 5 ﾀ06 ｧ罍ｧ縺ｧ縺06 ﾀﾀ04Ox 04 罍ｧ x =, 縺罍ｧ x =(n, ) x x (03 ﾀｧ) y = x y= a +y 0, r y y = a +y 0, y 0 =, ; ｧ0 y = a ch u, a sh u a u 0 = sh u 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 ｧ0 ｧ07, ｧ00 ｧ y = a 04 縺 y = ax. 07. y =4(x +) ｦｧｧﾀ縺06 t ﾀ閨0605 T ; dt dt =,k(t, 0 C), 05 閨06 k ｧ 0406 ﾀｧ09 ｧ縺縺09 ｧﾀ05 ｧ; ln (T, 0 C) =,kt + C; 04 罍ｧ t =0 T = 00 C, 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 , ｧﾀｧ0 006 ｧ 0406 ﾀ05kt k: k 0 0 = ln 6 ln, t = 400 ｧ P X i =,H + T cos = 0, 0

306 P Yi =,x + T sin = 0, tg = dy = x H, y = H x + C ( ) ｰｧ06 09 ﾀ Y, y = y 0 (X, x) ｧ04 Y =0, 0 08 閨ﾀ09 ｧﾀﾀｧ A 0406 縺06 ﾀｧ07 09 ﾀﾀ 03 ﾀ0406 Ox: X A = x, y y ﾀｧ06 X A =x, x =, y y0 ; 縺0600 ｧ陦ｧ縺06009 ｧｧ06, 0 08 閨060 ｧﾀ罍ｧ xy =,a (05 ｧ0406 縺 ) x +y = C y, x = C x +3y = 3a y = Cx y = Ce,=x. 08. y = Cx,. 08. y = ( + x) = C(x + x + a ) y = C, x r = C cos ', r =, cos '. +Cx 085. y = x ln x, x + C, y = x ln x, x y = C +x x + +x, +x y = x + +x xy =, y = aex=a y = x, x x y = C 陦ｧ06 ﾀ OM = x + y,0305 縺ｧ MN = y cos = y +tg = y +y 0. 0ｷ3 ﾀ罍ｧ04 07 ｧ00 ｧ x +y = C (0309 縺ﾀ0504), ｧ00 ｧ x,y = C (05 ｧ0406 縺 ). 09. y = Cx y, x = Ce x=(y,x) x, y = Cx s = t ln C t y = Cx 3, x 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 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,) ｧ04 X = ｧｧ 09 ﾀ Y, y = y 0 (X, x), 0 08 閨060 Y 0 =,ON = y, xy 0, ON = xy 0, y = OM = x + y ﾀ06 y = x, C. 6ｦ006 縺閨ｧ閨ｧ07. C. y = Cxe,y=x. 3. y = ln C(x + a + x ) 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

307 = +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= x + 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 = = 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 M (; 4) 04 縺 M ;! 閨罍ｧ y =(+x) ｧ y =(3,x). 5. y = sin (C x). ｪ06 縺縺閨罍ｧ y = sin x, ｧ y = sin 4 3 4, x. 5. y = Cx + ; 03 ﾀｧ y = x. 53. ) y = x + C C ｧ x + y = C ;)x r + y x = C ｧ00 ｧ (y, C) =4Cx. 05 ﾀｧ x =0 ｧ y =,x ﾀ0504 ﾀｧ : 04 罍ｧ x > 0 y >,x, 04 罍ｧ x < 0 y <,x ﾀ 0605 ﾀ07 03 ﾀｧ Oy ｧ (x, C) 04 縺 y =,x. 54. ) y = + ; 03 ﾀｧ y =; 4 ) x =,, y =, + C. 55. ) y = (C + x +) ;03 ﾀｧ y = 0; ) x = Ct, t 3, y = Ct, 3t, 05 閨06 t = ; 3) Cy =(x, C) ; 03 ﾀｧ y =0 ｧ y =,4x. 56. ) y = = Cx, C ; 03 ﾀｧ y = x 4 ;)y = Cx, a +C ; 03 ﾀｧ x + y = a ;3)y = Cx + C ; 03 ﾀｧ y =; 5x=3.

308 3(x + C) 57. y =, ; 06 縺06 M ; 縺0308 閨0605 閨罍ｧ040306: y =, x ｧ y = x, x ) x =+ 3 +C, y = + 3 ;)x +(y+c) = a. 59. y =, x 4 + Cx+ C ; y =, x. 60. ) y = Cx+ ; 03 ﾀｧ C (x y =4x; )y = C(x +)+C +) ; y =, 縺06 09 ｧ 09 ﾀ Y,y = y 0 (X,x) 0 03 ﾀ罍陦ｧ0 05: X A = x, y y 0, Y B = y,xy ﾀｧ06 X A Y B =a ; (y, xy 0 ) =,4a y 0, y = xy 0,4a y ｧ縺03. 6ｧ縺ﾀﾀ0504 y =,Cx a C, 罍ｧ04 07, 0304 縺06 閨ﾀｧ xy = a, 0605 縺ｧ06 ｧ (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, x ｧ06 09 罍ｧｧ 05 ｧ C y = EI 3 = (C x + C ) (x, b) y = a ch = a, e (x,b)=a + e,(x,b)=a. 4 a 74. y = x y = C x + C, ln cos x; ﾀｧ 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 (C y, ) = (C x + C ). 8. y = C, C cos x,, x. 8. ｮ 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 =

309 = 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 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 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 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, x 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 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 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 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).

310 y = C e x +C e 3x + 6 (5 cos 3x,sin 3x). 38. y =(C x + C )e,x 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, ) ) y = e mx (C + C x)+ 40. y = C e x= + C e,x=, x 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 y = C 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 ) 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 g dr = =, k 4r ; T = k 8r + C; k ｧ C 陦ｧ0 ｧ 06 ﾀｧ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+).

311 ) 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 ) 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. ｪｧ06 = F 04 罍ｧﾀ05 ｧｧ 06 ﾀｧ閨06, 縺0600 ｧ罍ｧﾀ05 ｧ 06 ﾀｧ06 Ady, Bdy+ C =0;04 閨ｧ '(x; y) = ｧ (x; y) = 04 縺03 ｧﾀｧ 04 罍ｧ縺ｧ 04 縺ｧ縺ｧ06 ( ﾀ0. ｧ 94 ｧ 94) 罍ｧ006 縺縺0600 ｧｧ06 +4 dy +3dy =0, dy + =0, dy +3 =0, y + x =, y +3x = 縺ｧ06 04 罍ｧ u = ﾀ06 u = '() + () ='(y + x) + (y +3x) ｦ罍ｧﾀ05 ｧ 06- ﾀｧ06 x dy, xy dy + y =0, ｧ00 ｧ (xdy, y) =0, y y ｧ00 ｧ d = 0; = ｧ ; 04 罍ｧ0 ｧ0 060 y. 0ｷ305 09, x x y 罍ｧﾀ05 ｧ09 ｧ: = ｧ y =. ｰｧ06 04 罍ｧｧ ( ﾀ0. ｧ x 944 ｧ u =0; = '() + (), 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 ) u = y ln x +y u = xt' x t + (xt); ﾀ縺ｧ06 u = x ( + t 3 ). 89. u = e,x '(x, t) + (x); ﾀ05t 縺ｧ06 u = (x, t)e,t, x. 90. ｪﾀ縺ｧ06 u = xat + + f(x, 3 a3 t 3 at) +f(x + at). 9. u = + a, 4ln 3; ) 3 3 x+at x,at F (z) dz. 9. 6, 5 9a ;) , e.

312 3 a x a a 97. ) dy = dy ) a 0 dy 0 0 a,y a,y = 0 a 0 y = 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 閨ﾀ , 4+x 0 dy = 6 4 3, a. a ; 457a a ln a a ; a , a , b x a ln. 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 ,y + dy = ; a. 33. (3; 4; 8) ; a ; 4a ; 56a a4 a4 a a4 a4 88a4 3a ; 3a ; 4b a ab(a + b ) 4 0 8,x ; y =4 35a a ｮｧﾀ0903 ﾀ z = h, x + y = = a(a, h) 縺 , 縺08003 ﾀｧ00 ｧ0 陦罍ｧ-

313 ﾀ09 07 ( 罍ｧﾀ. 59). 0ｷ3 ﾀ V = ( 罍ｧﾀ. 60) ( ｮ0. 罍ｧﾀ. 46, ﾀ. 303.) a 0 a,x 0 zdy= a a3 8 9 a a3 a ｧﾀｧﾀ a V =4 = 340. a a a3 0 m cos 'd' a abc abc, e a 0 r dr = 4m3 3 ( 罍ｧﾀ. 6). 9 (3, 4) ( 罍ｧﾀ. 6) a 3 ( 罍ｧﾀ. 58) a a3. 7 ｧﾀ. 6 7 ｧﾀ. 6

314 3349. V = 350. V =4 0 3a x ax 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 ｧﾀｧﾀ a 0 b (b=a) a,x a, x a h 0 0 dy = 6ab. 3 ya h, x dy = a h 陦ｧ 03 ﾀｧｧﾀｧ, ｧﾀ ( 罍ｧﾀ. 65) a a ln ( ﾀ0. 罍ｧﾀ. 45, ﾀ. 303) a a3 6.

315 abc a. 36. a a 3 (, ) a a 4 (, ) a. x y + z = dy = z 80 R sin a; 04 罍ｧ =60 ｧ =30 (S) = R a 陦ｧ06 ﾀﾀｧ07 r = 3 a a3 6 3 (, ). 37. a a,x dy a,x,y a a zdz= a a 4 ; a 4 ; a ) a ; 0; a 3 a3 a3 ; ) (8, 7) a h a 4 0; ; 3a. 8 a 3 3 a ka,05 閨06 k ｧ09 ｧ縺ｧﾀ05 ｧ縺 OA; >< (x + y) = 0403 閨 OA; 3 >: OBA: 388. ) 8; ) (xdy+ y)= ｧ08 ﾀ , @y ) ; 5a ; ) a a. 39. a mab ) ; ), ; 3), a ab = kmm kmm a Y = a. 40. X =0, Y = kmm. a 40. Y =. a 404. ),6; ), 5 3. ;3), ) 3a ; ) a a ;3) a a3 a ｧﾀ縺 a ｧﾀ縺 a ｧﾀ縺 a ; 5a ﾀ0803 陦ｧ05 ﾀ07. x ﾀ0803 陦ｧ05 ﾀｮ0803 陦ｧ05 ﾀ07, ｧ0303 (x +) 3 = ｮ0803 陦ｧ05 ﾀ07, 8

316 3 ｧ0303 陦ｧ05 ﾀ07, ｧ0303 +x = ﾀ0803 陦ｧ05 ﾀ07, ｧ (x +), = 4 ln x x x = ｮ x = ln. 43. ｮ0803 陦ｧ05 ﾀ07. 4 u n+ 43. ｮ0803 陦ｧ05 ﾀｮ0803 陦ｧ05 ﾀｮ0803 陦ｧ05 ﾀ07, ｧ0303 lim = n! u n < < ﾀ0803 陦ｧ05 ﾀﾀ0803 陦ｧ05 ﾀｮ0803 陦ｧ05 ﾀﾀ0803- 陦ｧ05 ﾀｮ0803 陦ｧ05 ﾀﾀ0803 陦ｧ05 ﾀｮ陦ｧ05 ﾀﾀｮ0803 陦ｧ05 ﾀ07 03 ﾀｮ0803 陦ｧ05 ﾀﾀｮ0803 陦ｧ05 ﾀ07 03 ﾀﾀ縺縺06 ﾀｧｧ00060 縺ｧ閨06,, 4 + 3,, ,, +::: ｧ04 閨0608 ﾀ0504 ｧ07 04 ﾀ , ｧ0 縺07, 縺閨縺ﾀ縺縺06 ﾀｧ縺 n 縺 : 4n, 3 + 4n,, n = = 4n, 3, 4n, + 4n,, 4n + 4n,, ; 04 罍ｧ n =; ; 3; ::: 4n 0406 縺縺縺07 ﾀﾀ S, 0403 ﾀ0006 閨0 ｧ06 閨04 縺07 ﾀﾀ S ｮ0803 陦ｧ05 ﾀﾀ0803 陦ｧ05 ﾀ07, ｧ x, 99 =. 陦ｧ05 ﾀ07, ｧｮ0803 陦ｧ05 ﾀ07, ｧ0303 x +x 4 = ﾀ x, x = ｮ0803 陦ｧ05 ﾀｮ0803 陦ｧ05 ﾀ07, ｧ0303 u n+ lim = < ｮ0803 陦ｧ05 ﾀ07, ｧ0303 lim n! u n n! u n+ u n 0n + = lim n! 3(0n +) = = < ｮ0803 陦ｧ05 ﾀｮ0803 陦ｧ05 ﾀﾀｮ0803 陦ｧ05 ﾀﾀ罍ｧ a> ﾀ0803 陦ｧ05 ﾀ07 03 ﾀ , 04 罍ｧ a = ﾀ0803 陦ｧ05 ﾀﾀ , 04 罍ｧ a< ﾀ0803 陦ｧ05 ﾀｮ縺07 S(x) =, x 04 罍ｧ x<, 03 ﾀ R n = S, S n = xn, x 縺 [0; =] jr n j < lg 000 < 0; 00, n, > n, lg ; n >.

317 ｧ ( x ﾀ S =, (, x) = 04 罍ｧ 0 <x6 ; 0 04 罍ｧ x =0 ( (, x) n 04 罍ｧ 0 <x6 ; ｧ03 ﾀ R n = 0 04 罍ｧ x =0: 7 罍ｧ n 03 ﾀ R n 0306 閨 , 0 04 罍ｧ006, 0;9, x<,, n 0;9, 縺 [0; ] 縺07 ﾀ0803 陦ｧ05 ﾀ縺縺 [=; ] 030 ﾀ0803 陦ｧ05 ﾀ縺003, ｧ罍ｧ xjr n j < n <", n>, lg " lg ; 04 ﾀ05003 ﾀ05 ｧ, jr nj < 0; 0 04 罍ｧ n > ﾀ縺06 閨ﾀ07 縺閨縺縺縺 [0; ] jr n (x)j < xn+ n + < 6 0;, n n +> 0 ｧ00 ｧ n > ｧﾀ S = ｧ03 ﾀ 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 n 03 ﾀ R n 0306 閨 , 0 04 罍ｧ006, 0;, x 3 < < n, 0,, 罍ｧ x > 0 縺07 ﾀ0803 陦ｧ05 ﾀ縺罍ｧ x > 030 ﾀ0803 陦ｧ05 ﾀ縺003, ｧ罍ｧ x > jr n j 6 6, lg " <", n, > n, lg ; 04 ﾀ05003 ﾀ05 ｧ, jr nj < 0;00 04 罍ｧ n > > 罍ｧ罍ｧ x 縺 ( ｧ00 ｧ 04003) ｧﾀﾀ0803 閨ﾀ07 縺 ::: ｮ0006 閨 , 縺07 ﾀ0803 陦ｧ05 ﾀ縺003 閨ﾀ0608 x > 0, R n (x) ﾀｧﾀ縺07, R n (x) < (=3)n, =3 = < 0;0, n, n, > 50, ｧ00 ｧ n > 5, 04 罍ｧ x > jr n (x)j < n 6 6 0;000, n > 00, 04 罍ｧ x u n =, x + n S n = x, x + n ; S = n! lim S n = x x + n,, 04 罍ｧ x 6= 0.

318 ﾀ05003 ﾀ05 ｧ, 04 罍ｧ x>0 R n (x) = x + n < 6 0;, n > 0. n 罍ｧ罍ｧ x 縺 ( ｧ00 ｧ 04003) ｧﾀﾀ0803 閨ﾀ07 縺 ::: 縺07 8 ﾀ0803 陦ｧ05 ﾀ縺003 閨ﾀ0608 x > 0, R n (x) < (=)n, = = < 0; 0, n, n, > 00, ｧ00 ｧ n > ,3 6 x<3. 47., x , 6 x 縺03 ﾀﾀ0803 陦ｧ05 ﾀﾀ0608 ｧﾀﾀｧ. 474., <x , 3 <x< ) R =0; 3 ) R = e. 477.,5 6 x< <x 罍ｧ jxj <. (, x) +x 480. arctg x 04 罍ｧ jxj (, x) 04 罍ｧ jxj <. 48. ( + x)m , <x , 3 6 x ,0; 6 x<0;., x 486., 6 x , 6 x< , 6 x< ( + x ), x 04 罍ｧ jxj <. 490., ln (, x) 04 罍ｧ, 6 x< 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 ln ( + e kx )=ln+ kx + k x!, k4 x ) 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 + ::: :

319 ) 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 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,) x 3, 3x =,+3(x, ) +(x, ) x 4 =, 4(x +)+6(x +), 4(x +) 3 +(x +) x =, (, (x +)=) = =, + x ) cos x = (x +) (x +) ::: (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 ( ﾀ0. ﾀ003 ﾀﾀ ); ) sin 3x = 504. X n= n (3x + )n, (,). (n, )! 3 x =, 3, (x +)=, + x + (x +) + 3! 3 +! 5(x +) :::=, + x + 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!

320 3 ) cos mx + =, mx 4! X = n=, m x! + ::: = (mx) n, (n, )! cos (n, ) x 4, 4x =(x +) 4, 8(x +) 3 + 0(x +), 6(x +). ( ! = ): 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!, ::: (x, =3) 6 6! 508. sin x 3 = n (x, ) n 3 n sin ( ! = ). n! n= x = + x, 4 (x, 4) 3(x, 4)3, 3! 6 +,! 9 3! 5. arcsin x = x + x ! x ! x7, ::: 3 5(x, 4)4, + ::: 4! r : : 7 +::: 5. 0; 99 = =, 0; 008, 0; 004 = 0; 996; 90 = = = 9; r = = ; 99 = 3, 0; 009 0; 997; = = arctg x = x,, x3 3 + x5 5, ::: 57. = 3, , sin x = ; ; ) x = 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 ﾀ縺ﾀ050406

321 < x. 5. (x) = ! x7 7, :::; 5 < 0 3 +x = x + 3 x3 3, 3! x ; 008 ﾀ縺ﾀ < 0; 縺03 陦ｧ縺06009 ｧ縺ｧ06 n ｧﾀ05 04 ｧ04 x =0, ｧ0 y (n+) 0 = n(n, )y (n,) ﾀ06 y 00 0 = y000 0 =0, y IV 0 =, y V 0 =3, yvi 0 =0 ｧ陦ﾀ05 04 ｧｧｧ縺縺060 y = y 0 + y0 0! x + y00 0! x + :::, 陦ｧ0 y = + x + x x x 8 + ::: 53. y =+x , x3 3 + x4, ::: ｧﾀｧﾀﾀ縺07 閨09 09: I 0 (x) =, x + + x4 4, x ::: ;005 ;005; ;00 ;0004; 0;993 0;9965; 3 0;997 0;999; 0 = =0;5; 57. =6 58. = 53. s = =4; 5; =; ! :::, = ( + 0; ; 0047) 3; 4: 3 + 5, ::: +, , ::: = X = 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, = : e6 5, ::: # ; 05 閨06 e 0509 ﾀ罍ｧﾀｧｧ04 ﾀ, a ﾀ04 ( ﾀ

322 3 ｧ ) ;5 0 +x 3 = x + x4 4, x7!7 + ::: 0;5 0 = , ::: 0; 508 ﾀ縺ﾀ < (x) = = x, x 5! x 9 4! 4 4 9, :::; =, + ::: 0; x3 ﾀ縺ﾀ < y = 3 + x x ::: 縺03 陦ｧ縺06009 ｧ縺ｧ06 n ｧ 0403 陦ﾀ05 04 ｧ04 x =0, 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 閨ﾀ C = RL, R 陦ｧ06 ﾀ 09 縺罍ｧｧ L 閨00 ｧ縺閨罍ｧ罍ｧﾀｧ閨0308 ( 罍ｧﾀ. 88, ﾀ. 336) F (x + h; y + l) =x + + xy + y + h(x + y) +l(y + x) +h + hl + l x 3 +xy = = 9+(x,)+8(y,)+3(x,) +8(x,)(y,)+(y,) +(x,) 3 + +(x,)(y,) 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) =0;, dy =,0;, 3! 4! z =(x, y) +(y, x) dy +, dy + dy =,0; 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) x y =,,, (x,)+(y+),(x,) +(x,)(y+) arctg y x = y,(x,)y+::: 547. y x = + (y, )+(x, )(y, ) + + 0; +0; 0; + 0; (y, ) + :::; ; ; =; =,0; 0, dy =0; 0; z =yx +(x, y) dy + y + xdy, dy + 3 dy,0; 407.

323 X 4 X n= sin (n, )x n,, 4 X n= cos (n, )x. (n, ) n cos nx 3 sin x +4 (,). n 55. 4, sin x sin 3x, +, ::: + 3 n= cos x cos 3x cos 5x ::: sin x + 3x sin + l 3 l + 5 sin 5x l + ::: cos x cos 3x ::: l 4, l cos x + 3x cos + ::: + l l 3 l 556. ) cos x, x cos + 3x cos 33 + ) sin x + x sin + 3x sin 3 + ::: 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 x 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

324 3 56. f(x) = 4 (, cos ) sin sin x d cos 3x cos 5x cos x ::: j sin xj =, 4 cos x cos 4x cos x, sin 3x sin 5x 3 +, ::: l, 4l cos (x=l) cos (3x=l) , cos x cos 3x ::: cos (x=l) 568. sh l l, l cos (x=l), + l X cos 6x :::. 3 + :::., sin x cos x + + l + ::: + sin (x=l) +, + l + ::: : sin (x=l) + l + ::: : n + n u = a n cos t sin x, n=0 05 閨06 a n = n + f() sin d f(x) = sin cos x d. 0

325 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ｧｷ300 縺6 5 ｯｯ 7 ｭｯ04 ｭ03 ｰ 0ｷ36ｦ06 7 繖 70ｷ縺ｧｧ縺0 00 ｧ05 ｧ 06 ﾀ罍ｧﾀ0903 ﾀ05 ｧ... 9 x 罍陦ｧｧ 0 04 縺ｧﾀ0903 ﾀ05 ｧ. 7 ﾀﾀｧ閨06 閨ｧ... 9 x ｧ縺ｧｧ閨04 05 縺ｧ09 ｧｧ09... x 3. ｰｧ06 00 ｧ0 ｧｧ罍ｧ 06 ﾀﾀ x 4. ｰｧ06 04 縺 : ) ﾀｧ09 ｧ , ) , 3) 縺ﾀ x 5. ｰ閨06 04 縺ｧ. ｰｧ縺 , 04 縺閨070 ｧ08 06 縺ｰｧ06 04 縺 , 04 縺閨縺06 閨ｧ. ｯ縺06 ﾀｧ07 閨縺 x 縺ｧ06 04 縺ﾀﾀｧｧ閨03 04 縺ｰｧ07 03 ｧﾀﾀ罍ｧﾀ. ｰｧ縺 , 04 縺閨070 ｧ08 06 縺縺06 ﾀｧ07 閨縺 x 7. ｮｧ 0 04 縺 x 縺ﾀ x ｧ04 ﾀ... 4 x ｧ0406 縺 x x. 6 7 ｧ縺罍ｧﾀ03, 陦ｧ縺03 ｧ 09 ﾀ罍ｧ縺縺07 閨 x 3. 7 縺ｧ06 閨0609 縺罍陦ｧ = = 08 + b08 + c ｧ x = ay + by + c. 06 ｧ0406 縺 xy = k x 4. ｮｧ 0 09 罍ｧ縺縺07 閨 x ｧ06 00 ｧ0 ｧｧ縺縺07 閨 x 縺罍陦ｧ x 7. 縺ｧ 06 ﾀ09 ｧ06 09 罍ｧ縺ｧ 0403 ﾀ00 ｧ縺07 閨 x 8. ｯ 0 ﾀ09060 閨罍ｧ縺 x. ｮｧ縺0304. ｰｧﾀ x. 7 縺罍陦ｧｧｧ縺03 ﾀ05 0 ﾀ x 3. ｮ縺縺03 ｧ 0406 閨060 ｧ06 閨縺 x 縺縺03 ｧ 0406 閨060 ｧ06 閨縺 x 5. ｮ縺03 ｧ 0406 閨060 ｧ06 05 縺縺

326 ｧ縺0 00 ｧ05 ｧ 06 ﾀ罍ｧ縺03 ﾀ05 0 ﾀ x. ｰｧﾀ0903 ﾀ05 ｧ... 6 x. 05 ﾀｧﾀ0903 ﾀ x 3. ｰｧ07 04 縺 x 4. 7 縺ｧﾀ0903 ﾀ x 5. ｮ0706 罍ｧ 06 ﾀ09 ｧ06 ｧ 09 ｧ00 ｧ0 陦罍ｧ 06 ﾀ09 ｧ縺08003 ﾀ05 ｧ x ｧ 06 ﾀ09 ｧ縺08003 ﾀ05 ｧｧ縺08003 ﾀ05 ｧｧ x ｧ04 ﾀ03 ｧ, 05 ｧ0406 縺ｧ閨03 ｧｧ閨ﾀ x 縺06 閨0600 ｧｧ x. ｮｧﾀｧｧ x ﾀ00306 ｧﾀ x 4. ｰｧﾀ00 ｧ08 ﾀｧ 04 罍ｧ0300 ｧ縺ｧｧ閨060 ｧｧ x. 706 縺ｧｧ003 ｧｧｧ x. 7 縺06 閨ﾀ0006 閨ﾀ05 ｧｧｧｧ. 706 ﾀｧ 0306 ﾀｧ x 3. ｮﾀ縺06 閨ﾀ09 縺0305 ｧ縺06 閨ﾀｧ 0 ｧ x 4. 7 縺06 閨ｧ07 sin 04 罍ｧ! x 縺06 閨ﾀ05 ｧ 04 ｧ, ｧ x 6. ｮ罍ｧ006 縺ｧﾀ00060 ｧ06 04 縺06 閨 x 7. ｮｧﾀ x 縺06 縺ﾀｧｧ... 0 x 9. 罍ﾀｧ x 0. ｪｧﾀ0003 e 縺03 ｧ 0403 閨0 07 ｧ陦ｧ縺06009 ｧ x. 7 縺03 ｧ 0403 閨ｧ 06 ﾀ09 ｧ08 ｧ 05 罍ｧ罍ｧ 06 ﾀ09 ｧｧ x. 7 縺03 ｧ 0403 閨0 07 ﾀｧｧ x ﾀｧ 003 縺ﾀ罍ｧ x 4. ｮ0006 ｧ 006 陦ｧ縺06009 ｧ縺ﾀ05 ｧ縺06 縺ｧｧ.. 3 x 5. 7 縺03 ｧ 0403 閨罍ｧ070 ｧ 06 ﾀ09 ｧ08 ｧｧ08 4 x 6. 7 縺03 ｧ 0403 閨罍ｧ罍ｧ 06 ﾀ09 ｧｧ x 7. 7 縺03 ｧ 0403 閨ｧ0406 縺ｧ 06 ﾀ09 ｧｧ x 8. ｮ罍ｧ006 縺03 ｧｧ 0 陦ｧ縺06009 ｧ縺ｧ06. 8 x 9. 7 縺03 ｧ 0403 閨ﾀ00 ｧ縺07 閨 x 0. 7 縺03 ｧ 0403 閨ｧｧ....

327 ｧ06 5 x. 6 7 ｧ縺06009 ｧｧｧ... 3 x 罍ｧ 06 ﾀ09 ｧｧ07 09 罍ｧ罍ｧｧ07 04 縺03 ｧ 0403 閨 x. ｮ0903 縺03 ﾀ0504 ｧ06 ﾀ0903 縺060 ｧ x. ｯ0603 縺ﾀ縺06 閨 x 3. 7 ﾀ09 縺0305 ｧ縺06 閨ﾀｧ0003 6ｧ80304 ｧ x ﾀ05 0 ｧ06 ｧｧｧｧﾀｧ0060 ｧ 0 ｧ0 ｧ x 5. 6ｦ0 ｧ 03 0 ｧｧ08 ｧ 0 ｧｧｧｧｧ0 36 x ｧﾀ05 ｧｧｧ 0406 縺0605 ｧ03 09 罍ｧﾀ05 縺03060 ｧ06 09 罍ｧ縺06 閨ｧ x 縺06 閨ｧｷ罍ｧ縺ｧｧ x. 0ｷ罍ｧ縺ｧ陦ﾀｧﾀ縺06 陦ﾀ x 3. 0ｷｧ x, a a,, x x + k ｧ 09 0 ｧ0 04 罍ｧ0403 閨070 ｧ06 ﾀ x 4. 0ｷ罍ｧ縺ｧﾀ x 5. 0ｷ罍ｧ縺ｧ06 05 罍ｧ罍ｧ 06 ﾀ09 ｧｧ x 6. 0ｷ罍ｧ縺ｧ06 09 ｧｧ 06 ﾀ09 ｧｧ08 50 x 7. 0ｷ罍ｧ縺ｧ縺0308 ｧ罍 09 ｧｧ 06- ﾀ09 ｧｧ x 8. 0ｷ罍ｧ縺ｧ縺ﾀ09060 閨ｧ08 55 x 9. 0ｷ罍ｧ縺ｧ06 05 ｧ0406 縺ｧ 06 ﾀ09 ｧｧｧ0406 縺ｧ- 06 ﾀ09 ｧ陦ﾀｧ x 0. ｮ罍ｧ006 縺03 0 ｧ罍ｧ縺ｧ縺06 閨ｧ x ｧﾀ00060 ｧ縺06 閨ｧ x ｧﾀ00060 ｧ閨 x ｧ x ｧ0 閨0605 ｧﾀ罍ｧ x 閨縺08003 ﾀ05 ｧｧ x 6. 6ｦ0 ｧｧ 07 ｧｧ09 ｧ x ﾀ0303 ﾀｧ x 8. ｮ縺06 閨ｧｧｧ x 9. ｱ03 縺ｧ08 ｧ 0703 縺00600 ｮｧ004 ﾀ罍ｧ04 ｧﾀｧ 04 縺03 ﾀ05 0 ﾀ罍ｧ x. 07 罍ｧ04 ｧﾀ罍ｧｨC06005 ｧ陦ｧ06 ﾀ 09 罍ｧ04 ｧ x ｧ0 閨0605 ｧ 09 罍ｧ縺03 ﾀ05 0 ﾀ

328 ｧ06 x 3. 7 縺03 ｧ 0403 閨ｧｧ 0403 ﾀ縺06 ｧｧ 06- ﾀｧ罍ｧ 06 ﾀｧ06. ｭﾀ0506 ﾀ縺ｧ09 09 罍ｧ x 罍ｧ04 ｧ 0 ｧ 09 縺ｧ06 04 縺03 ﾀ05 0 ﾀ罍ｧｪﾀ縺03 ｧ 0403 閨00306, 陦ｧ縺06009 ｧ 0003 ｧｧ08 04 罍ｧｧ x. ｱｧｧ閨縺ｧｧ罍ｧ 06 ﾀｧｧ x. ｪﾀ縺03 ｧ 0403 閨縺縺07 閨 x 陦ｧ縺06009 ｧ縺縺07 閨 x 4. 7 縺03 ｧ 0403 閨00306 ﾀｧ x 5. 7 縺03 ｧ 0403 閨ｧ x 6. ｪﾀ縺03 ｧ 0403 閨00306 ｧ陦ｧ縺06009 ｧﾀ00 ｧ縺07 閨 x 7. 0ｷ罍ｧ縺ｧ陦ｧ縺06009 ｧ x ﾀｧﾀ罍ｧ x ｧﾀﾀﾀ09 ｧ08 09 罍ｧ x ﾀﾀ0903 ﾀ0504 ｧ 003 縺縺08003 ﾀ05 ｧ... 0 x. ｮ縺ｧ8 ｧ0 ｧｧｧ縺08003 ﾀ05 ｧ 06 縺縺03 ｧ 0403 閨ｧｧ. 06 陦ｧ x. 709 ﾀ05 縺ｧｧ閨縺ｧ縺06009 ｧｧ x ｧ06 03 陦ｧ縺06009 ｧｧｧ x. 6 7 ｧ縺06009 ｧｧ縺縺07 閨09 ﾀ閨ｧ0 ｧﾀ縺ｧ. 05 縺罍ｧｧ x ｧ縺06009 ｧｧ縺縺07 閨09 : ) 03 閨003 縺03 閨00306, ) 00 ｧ , 3) 706 縺ｧ... x ｧ縺06009 ｧｧ07, ﾀ03 閨06 縺07 0 ｧ06 陦ｧ縺ｧ縺03 ｧ 0406 閨060 ｧ07 ｧﾀ x ｧ縺06009 ｧｧ縺縺07 閨陦ｧ縺06009 ｧｷ罍ｧ縺06060 ｧｧ x ｧ縺06009 ｧｧ縺縺07 閨09, 縺ﾀｧ縺03 ｧ 0403 閨ｰｧ07 6ｧｧ縺 x ｧ縺06009 ｧｧﾀ00 ｧ縺07 閨090304, 閨ﾀｧｧ07060 ｧ縺07 閨 x 8. 6ｧ8 ｧ閨003 縺03 閨00306 陦ｧ縺06009 ｧｧ07 ﾀ 0403 ﾀｧｧ09 ｧｧ... 8 x 9. 6ｧ8 ｧ閨003 縺03 閨00306 陦ｧ縺06009 ｧｧ07 ﾀ 0403 ﾀｧｧ09 ｧｧ... 9 x 0. 7 罍ｧ006 縺03 陦ｧ縺06009 ｧｧｧ x. 6ｧ8 ｧ陦ｧ縺06009 ｧｧ x n y (n) + + ax n, y (n,) + :::+ an,xy 0 + any = f(x)...

329 ｧ06 7 x. ｮｧﾀｧ陦ｧ縺06009 ｧｧ08 ﾀﾀｧｧ09 ｧｧ... 3 x 3. 6ｧ8 ｧ陦ｧ縺06009 ｧｧ07 04 ﾀ縺03- ｧ 0403 閨縺縺07 閨09 ( 罍ｧﾀ05 ｧ09) , 05 縺ｧ 09 罍ｧｧｧ x ｧﾀ00060 ｧ陦ｧﾀ閨ｧ x. ｨC ﾀﾀｧｧ006 縺09 ｧｧ陦ｧﾀ縺003 ﾀ- 04 縺06 閨ﾀﾀ0308 (04 罍ｧﾀ05 ｧ =)... 8 x ｧﾀ00060 ｧﾀ閨ｧ x 陦ｧ 09 罍ｧ縺08003 ﾀ x 5. ｯ縺ｧｧ罍ｧｧ x 罍ｧｧｧｱ03 縺罍ｧ x 縺08003 ﾀｧｱ03 縺ﾀ05 縺0305 陦ﾀ {06 06 ﾀﾀｧｮﾀ閨 x. ｪｧﾀ縺07 閨 x 縺0 07 ﾀ0803 陦ｧ003 ﾀｧ縺 x 3. ｮ縺07 閨 x 閨03 ｯｧ縺 x 5. 7 罍ｧｧ07 縺07 閨罍ｧ0300 ｧｧﾀ00060 ｧ x ｯ閨ｧｧ閨縺 x ｱ06 縺ｷｱ06 縺罍ｧｧ縺罍ｧ ( 閨0007 ﾀ )... 33

330 縺罍ｧ ( 閨0007 ﾀ ) 7 罍ｧｧ06

331 3 7 罍ｧｧ06 333

332 罍ｧｧ06

333 3 7 罍ｧｧ06 335

334 罍ｧｧ06

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