di-problem.dvi
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- しなつ はぎにわ
- 5 years ago
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1 III 005/06/6 by. : : : : : : : : : : : : : : : : : : : : :. : : : : : : : : : : : : : : : : : : : : : : : : : : 3 3. : : : : : : : : : : : : : : 4 4. : : : : : : : : : : : : : : : : : : : : : : 5 5. : : : : : : : : : : : : : : : : : : : : : : 6 6. : : : : : : : : : : : : : : : : : 7 7. : : : : : : : : : : : : : : : : : : : : : 8 8. : : : : : : : 9 9. : : : : : : : : : : : : : : : : : : : : 0 0. : : : : : : : : : : :. sin : : : : : : : : : : : : : :. cos : : : : : : : : : : : : : : 3 3. log ( + ) : : : : : : : : : : : 4 4. : : : : : : : : : : : : : 5 5. e : : : : : : : : : : : : : : : : 6 6. : : : : : : : : : : : : : : 7 7. : : : : : : : : : : : : : : 8. (A): (B): (C):
2 . f () = a y = f () (a; f (a)) y f (a) =f 0 (a)( a) a f 0 (a) y = jj [ ; ] y = a (» a» ) y = jj jj p y = jj y =0 ( ( : ) y = 0 ( : ) (?) ( ) ( ) " " " " " y = p jj =0 y = =3 =0. (A) y = A + B + C (A 6= 0) a; b a + b
3 . dy d = lim y!!0 (Leibniz:646 76) dy d 50 (Cauchy: ) y = f () y dy = df () dy = f 0 () dy ( ) y = d = y = f () dy = f 0 ()d f 0 () dy dy = d dy d d dy dy; d d ; y y y y = f ( + ) f() z = g(y) dz = g 0 (y)dy w = f ()+g(y) dw = f 0 ()d + g 0 (y)dy y = f () y = f () dy = df () =df () y = c (c ) ) dc =(c) 0 d =0d =0 () y = cf () (c ) ) d(cf )=(cf ()) 0 d = cf 0 () d = cdf y! 0 y dy = f( + ) f() f 0 ()d! dy y (3) y = f () +g() ) d(f + g) =(f () +g()) 0 d =(f 0 () +g 0 ()) d = df + dg (4) y = f ()g() ) d(fg)=(f ()g()) 0 d =(f 0 ()g() +f ()g 0 ()) d = gdf + fdg (5) y f = () f f 0 () ) d = d f 0 ()g() f ()g 0 () gdf fdg = d g() g g() g() = g. y = dy = dy d d =(3 +6 ) d.. (A) y = + 3
4 3.. 3 y 5 + y 6 + =0 d( 3 y 5 )+d(y 6 )+d() d() = d(0) d( 3 )y d(y 5 )+6y 5 dy +d 0 d =0d (3 d)y (5y 4 dy) +6y 5 dy +d =0 (3 y 5 +)d +(5 3 y 4 +6y 5 ) dy =0 dy d = 3 y y 4 +6y 5 :. 3 y 5 + y 6 + =0 3 y (5y 4 dy dy )+6y5 d d +=0 dy d. a + y = a d+ydy =0: = 0, y = y 0, d = 0, dy = y y 0 0 ( 0 )+y 0 (y y 0 )=0: 0 + y 0 y = 0 + y 0 = a ( 0 ;y 0 ). 0 = a, y 0 = 0 a = a. = a (a; 0) y y 0 = f 0 ( 0 )( 0 ). y = 3p y 3 = 3y dy = d: = 0, y = y 0, d = 0, dy = y y 0 3y 0 (y y 0) = 0: 3y 0 y = 0: 0 =0,y 0 =0 =0 y y 0 = f 0 ( 0 )( 0 ) 3. (C) y = + 3 y = t, y t t = t 0 t =0,t = ± 4
5 4. s `. ` T T =ß g % s ` T (`) =ß g ` T = T (` + `) T (`): T T (` + `) = = T (`) T (`) r r` + ` = ` + ` ` : ` =0:0` T p T (`) = :0 =:00995 ο 0:0 % dt ß = p d` = g p` T (`)d` ` : d` = ` =0:0` dt =0:0T (`). % 4. (A) r % 5
6 5. f () = a f (a) = a f () <f(a) ( 6= a) f (a). I f () M () I f ()» M () f ( 0 )=M 0 I M = ma f () I f () I m () m =minf () I. f () = a = a 9 f 0 () < 0 ( <a) >= f 0 (a) =0 f () = a f 0 >; () > 0 ( >a) " " f 0 () > 0 f 0 (a) =0 f 0 () < 0 9 ( <a) >= f () = a ( >; >a). f () = a f 0 (a) =0 f 00 (a) < 0 ) f () = a f 00 (a) > 0 ) f () = a f 00 (a) =0 ) 5. (A) I (0; ), (0; ], [0; ), [0; ] 4 f () = ma f () min f () I I p 6. (A), 3, 4, cosh, jj 6
7 6. f () I (a; f (a)), (b; f (b)) f () f () I. f () I f 00 () < 0 (f 00 () > 0) I convave ( ) conve ( ).. y = f () C `: y = a + b lim (f () (a + b)) = 0! C!`. f (), f 0 (), f 00 (), f () 7. (A) e ; 3 ; log ; log ; n lim n log =0;!+0 log : log lim!+ n =0 8. (B) f () = 3 ( 3 + )log ( >0) 9. (C) f () = sin cos () f 0 () > 0(0<<ß). (0 <<ß) () lim f () =0, lim f () =.!+0!ß 0 p p 0. (C) F (t) = t + ( t) + ( + t) (t ) + (0» t» ) () t (0; ) F 00 (t )=0F 00 (t) =0 t = t t = t () t (t ; ) F 0 (t ) = 0 F 0 (+0) = lim F 0 (t) = 0 F 0 (t) = 0 t = 0 t!+0 t = t (3) F (0), F () F (t) (0» t» ) 7
8 7. Rolle. f () [a; b] (a; b) f (a) =f (b) f 0 (c) =0 c (a; b). f () [a; b] (a; b) f 0 f (b) f (a) (c) = c (a; b) b a f (b) f (a). g() = ( a) +f (a) f () b a. f (b) =f (a) +f 0 (c)(b a); a<c<b: f (b) =f (a) +f 0 (( )a + b)(b a); 0 < <: f (a + h) =f (a) +f 0 (a + h)h; b = a + h: f ( + ) =f () +f 0 ( + ) ; 0 < <: c c f(a) = f(b). f () I f () I f 0 () > 0 f 0 () < 0 f 0 () =0 ) f () I ) f () I ) f () k k Cauchy. f (), g() [a; b] (a; b) g 0 () 6= 0 ( (a; b)) f (b) f (a) g(b) g(a) = f 0 (c) g 0 (c) c (a; b). = g(t), y = f (t) (a» t» b) t = a (g(a);f(a)) f (b) f (a) t = b (g(b);f(b)) g(b) g(a). dy fi fi fi f 0 (c) dfi = =c g 0 (c) (g(t);f(t)) (a» t» b) f (b) f (a). F () =f () g() g(b) g(a). (B) p p + () lim! ( + )= lim p p =0k =3; 4; 5;:::! ++ lim (kp + kp )! () sin <( >0) c f(a) = f(b) 8
9 8. 0 0, de l'h^opital. >a lim 0 f 0 () lim!a+0 g 0 () = ` ) lim f ()!a+0 g() = `: lim f!+ () =!+ lim g() =0 f 0 () lim!+ g 0 () = ` )!+ lim f () g() = `:!a+0 f () = lim g() =!a+0. f 0 (e) g 0 (e) f () f (a) = g() g(a) = f () g() (a <e <):. lim, lim!a 0!,. n e () lim! n =! lim () lim! log n = lim! (3) lim!+0 n log = e = lim nn! = = lim nn lim!+0! log = lim n!+0 e e = = lim n(n )n! n! = : (e ) n n =0: = n (n+) = (log ) lim!+0 n n = (4) lim (sin ) log y =(sin) log!+0 lim log y = lim log(sin ) cos = sin = lim =: lim!+0!+0 log!+0 = y = e.!+0 (5) lim! ß (tan ) sec y = (tan ) sec 4 lim! ß log y = lim! ß 4 4 log(tan ) cos = lim! ß 4. (AοB) sec = tan sin 0 : () lim ( +) log! () lim 0 : (3) lim log (4) lim!+0 : (5) lim! 0 0 : (7) lim!0 = lim! ß 4 lim n!+0 n =0: sin = : lim y! ß = e. 4! ß! arctan ß arctan a (6) lim cos (a 6= 0)! a b (a; b > 0) (8) lim! ß sec sin ß a + a + + n a n (a ;:::;a n > 0) n ß arctan ± : (9) lim (0) lim!!0 0 0 : () lim () lim!+0! 0 : (3) lim!+0 (4) lim! log(tan) : (5) lim!+0 log(tan ) 9 p k p + k (k =; 3;:::) a (6) lim (a; b > )! b 3
10 9. f 0 (c) = f (b) f (a) b a (a <c<b) f (b) =f (a) +f 0 (c)(b a) (a <c<b) Taylor. f () [a; b] n (a; b) n c (a; b) f (b) =f (a) +f 0 (a)(b a) + f 00 (a)! (b a) f 000 (a) + (b a) 3 f (n ) (a) + + 3! (n (b )! a)n + R n ; R n = f (n) (c) (b a) n n!. n =. R n. g() = f (b)+f()+f 0 ()(b )+ f 00 () (b ) + f 000 () (b ) 3 +! 3! + f (n ) () (n (b )! )n + K(b ) n g(b) =0 g(a) =0 K g() [a; b]. a<c<b c = a + (b a) (0< <) b = f () =f (a) +f 0 (a)( a) + f 00 (a) f () = a a =0 f () =f (0) + f 0 (0) + f 00 (0)!! ( a) f 000 (a) + ( a) 3 f (n ) (a) + + 3! (n ( )! a)n + R n ; R n () = f (n) (a + ( a)) ( a) n (0 < <); n! f 000 (0) + 3 f (n ) (0) + + 3! (n )! n + R n ; R f (n) ( ) n () = n (0 < <); n! f () Maclaurin 3. (A) 0
11 0. 0 < < (5) ff () e =+! +! + + n (n )! + R n(); R n () = e n! n : () sin = 3 3! + 5 m 5! + +( )m (m )! + R m+(); R m+ () = ( ) m cos( ) m+ : (m +)! (3) cos =! + 4 m + +( )m 4! (m)! + R m+(); R m+ () = ( ) m+ cos( ) m+ : (m +)! (4) log( + ) = + 3 n + +( )n 3 n + R n(); R n () = ( ) n n( + ) n n : ψ! ψ! ψ! ψ! (5) ( + ) ff ff ff =+ + ff + + n ff + R n (); R n () = ( + ) ff n n : n n. ψ! ff ff(ff ) (ff (k )) = ff = n ( ) k k! ψ! n n(n ) (n (k )) = = k k! n! (n k)!k! = nc k R n () = n (5). (5) ff = + = ( ) n n + R n (); R n () =( ) n n : ( + ) n+ (4) = = n + R n (); R n () = n ( ) n+ :. sin sin cos cos 4. (A) 5. (A) 0 < < () sinh = + 3 3! + 5 n + + 5! (n )! + R n+(); R n+ () = () cosh = + 4 4! + 6 n + + 6! (n )! + R n(); R n () = cosh (n +)! n+ : cosh n : (n)!
12 . sin sin = 3 3! ! 7! ( ) n n R n+ : (n )! n cos R n+ =( ) (n + )! n+ (0 < <): () = () = 3 3! (3) = 3 3! + 5 5! (4) = 3 3! ! 7! sin () (4) () (4) sin (4).5 y () () (3) sin
13 . cos cos =! ! 6! + 8 8! ( ) n n R n+ : (n)! R n+ =( ) n+ cos (n +)! n+ (0 < <): () =! () =! + 4 4! (3) =! ! 6! (4) =! ! 6! + 8 8! cos () (4) () (4) cos (4) ().5 y cos - (3) () -.5 3
14 3. log ( + ) log( + ) = R n = ( )n+ n n( + ) n (0 < <): () = () = (3) = (4) = ( ) n n R n : n log( + ) () (4) () (4) log( + ) y (3) () log( + ) (4) () 4
15 4. = n + R n : R n = () = n ( ) n+ (0 < <): () = + (3) = (4) = () (4) () (4) y () ().5 (4) (3)
16 5. e e =+ +! + 3 3! + 4 n + + 4! (n )! + R n: R n = e n n! (0 < <): () = + () = + +! (3) = + +! + 3 3! (4) = + +! + 3 3! + 4 4! e () (4) () (4) e 4 y () () (4) e (3) -4 6
17 6.. lim f () =0, lim g() =0!+0!+0 f () () lim =0f()=o(g()) (! +0)!+0 g() fi fi fi () >0 0 fi f () fi fi fi g() fi» C ( ) f()=o(g()) (! +0) o O Landau.! 0 = O( )=o() = O( )=o() =o() n = O( n )=o( n ) f () = f (0) + f 0 (0) + f 00 (0) = f (0) + f 0 (0) + f 00 (0) + O( 3 )!! = f (0) + f 0 (0) + O( ) f ( + ) = f () +f 0 () + O(( ) ) (! 0) + + f (n ) (0) (n )! n + O( n ) f () =a 0 + a + a + + a n n + O( n+ ) (! +0) f () n. e 3 O( n+ )=o( n ) e =+ +! + 3 3! + O(4 ) (! 0): 6. (A)! 0 ( ) () + + ( 4 ) () log cos ( 4 ) (3) tan ( 3 ) (4) arctan ( 3 ) (5) a ( ) (6) e sin ( 3 ) 7
18 7.! 0 sin = 3 sin sin.! 0 3! + O(5 ) = 3! + O(4 ):! (! 0). =+ + O( ); sin = 3 tan = + 3 tan = O(5 ) + O(4 ) 7. (A)! 0 = 6 + O(5 ); cos = 3 + O(5 )(! 0) O(5 ) + = + O( ); sinh = + 3 tanh = O(4 ) + O(4 ) 6 + O(5 ); cosh =+ 3 + O(5 )(! 0) = O(4 ) 3 + O(5 ):. 0 0 e e sin lim!0 sin (AοB) sin cos () lim = () lim!0!0 = sinh!0 log( + a) (4) lim = a (5) lim!0 (7) lim!0 sin 3 = 6 sin!0 (8) lim cos 9. (AοB) sin () lim!0 3 () lim!0 (3) lim!0 sin + e + log( ) (5) lim (6) lim!0 arctan (7) lim e +! p (9) lim 3 +! tan (3) lim =!0 e a = a!0 = (6) lim = (9) lim!0 p (4) lim!0 log( + ) (8) lim!0 (0) lim! 8 sin cos!0 log( + ) + e e sin sin = p ( + a)( + b) p ( a)( b)
Chap10.dvi
=0. f = 2 +3 { 2 +3 0 2 f = 1 =0 { sin 0 3 f = 1 =0 2 sin 1 0 4 f = 0 =0 { 1 0 5 f = 0 =0 f 3 2 lim = lim 0 0 0 =0 =0. f 0 = 0. 2 =0. 3 4 f 1 lim 0 0 = lim 0 sin 2 cos 1 = lim 0 2 sin = lim =0 0 2 =0.
(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)
2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x
grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si
I 8 No. : No. : No. : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No. : I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin.
5 36 5................................................... 36 5................................................... 36 5.3..............................
9 8 3............................................. 3.......................................... 4.3............................................ 4 5 3 6 3..................................................
( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED)
rational number p, p, (q ) q ratio 3.14 = 3 + 1 10 + 4 100 ( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED) ( a) ( b) a > b > 0 a < nb n A A B B A A, B B A =
I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n
sin cos No. sine, cosine : trigonometric function π : π = 3.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even.
08 No. : No. : No.3 : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No.0 : No. : sin cos No. sine, cosine : trigonometric function π : π = 3.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin
I No. sin cos sine, cosine : trigonometric function π : π =.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even.
I 0 No. : No. : No. : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No.0 : I No. sin cos sine, cosine : trigonometric function π : π =.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd
I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10
1 2007.4.13. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 0. 1. 1. 2. 3. 2. ɛ-δ 1. ɛ-n
[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
![[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s](/thumbs/94/118579915.jpg "[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s")
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1 1. x 1 (1) x 2 + 2x + 5 dx d dx (x2 + 2x + 5) = 2(x + 1) x 1 x 2 + 2x + 5 = x + 1 x 2 + 2x x 2 + 2x + 5 y = x 2 + 2x + 5 dy = 2(x + 1)dx x + 1
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di-problem.dvi
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mugensho.dvi
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difgeo1.dvi
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
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sim0004.dvi
4 : 1 f(x) Z b a dxf(x) (1) ( Double Exponential method=de ) 1 DE N = n T n h h =(b a)=n T n = b a f(a) +f(b) n f + f(a + j b a n )g n j=1 = b a f(a) +f(b) n f + f(a +j b a )g; n n+1 j=1 T n+1 = b a f(a)