CALCULUS II (Hiroshi SUZUKI ) f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b)
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1 CALCULUS II (Hiroshi SUZUKI ) f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b) lim f(x, y) = lim f(x, y) = lim f(x, y) = c. x a, y b (x,y) (a,b) P A. f(x, y) A(a, b) (a) f(a, b) (b) lim (x,y) (a,b) f(x, y) (c) lim (x,y) (a,b) f(x, y) = f(a, b) f(x, y) f(x, y) = { xy x +y (x, y) (0, 0) 0 (x, y) = (0, 0) hsuzuki@icu.ac.jp 1
2 x- y-0 y = mx x 0 lim f(x, mx) = lim x 0 x 0 mx x + m x = m f(0, 0). 1 + m m = 0 x (0, 0) 0 m = 1 y = x (0, 0) 1/ (0, 0) 1. f(p + h, q) f(p, q) f(p, q + h) f(p, q) lim, lim h 0 h h 0 h f(x, y) (p, q) x y (partial derivative) x (p, q) = f x(p, q) = D x f(p, q), y (p, q) = f y(p, q) = D y f(p, q). f(x, y) x = f x = D x f, y = f y = D y f f x f(x, y) y x y f(x, y) = x y + e x f x = xy + e x f y = x (p, q) f(x, y) a, b ɛ(x, y) = f(x, y) f(p.q) a(x p) b(y q) lim (x,y) (p,q) ɛ(x, y) (x p) + (y q) = 0 f(x, y) (p, q) (a, b) (p, q)
3 . z f(p, q) = a(x p) + b(y q) (p, q, f(p, q)) 3. f(x, y) f(x, y) df = dx + x y dy 1.1 (1) f(x, y) (p, q) (a, b) = ( (p, q), (p, q)). x x () (p, q) f(x, y) f(x, y) (p, q) f(x, y) (p, q) 1.4 p 1 x 1 p x P =, X =.. p n x n f(x) f(x) P lim f(x) = f(p ). X P f(x) P A = (a 1,..., a n ) lim X P ɛ(x) = 0, ɛ(x) = f(x) f(p ) A (X P ). X P z f(p ) = (gradf)(p ) (X P ), gradf(p ) = ( x 1 (P ),..., x n (P )). 3
4 .1.1 f(x, y) x y t x = x(t) y = y(t) t f(x(t), y(t)) t df dt = dx x dt + dy y dt. t = p x(t) = x(p) + x (p)(t p) + ɛ(t) y(t) = y(p) + y (p)(t p) + ɛ (t) x(t) y(t) t t p ɛ(t) (t p) 0, ɛ (t) (t p) 0 f(x(t), y(t)) f(x(p), y(p)) = (x(p), y(p))(x(t) x(p)) + (x(p), y(p))(y(t) y(p)) + ɛ(x(t), y(t)) x y f(x, y) (x(t), y(t)) (x(p), y(p)) ɛ(x(t), y(t)) (x(t) x(p)) + (y(t) y(p)) 0 f(x(t), y(t)) f(x(p), y(p)) t p = x (x(p), y(p))(x (p) + ɛ(t) t p ) + y (x(p), y(p))(y (p) + ɛ (t) ɛ(x(t), y(t)) ) + t p t p ɛ(x(t), y(t)) lim t p t p ɛ(x(t), y(t)) = lim (x (p) + ɛ(t) t p (x(t) x(p)) + (y(t) y(p)) t p ) + (y (p) + ɛ (t) t p ) = 0 df (x(p), y(p)) = dt x (x(p), y(p))x (p) + y (x(p), y(p))y (p) 4
5 . f(x, y) x = x(u, v) y = y(u, v) u, v u = x x u + y y u, v = x x v + y y v. v x y f u u.1.3 f(x 1,..., x n ) x i = x(u 1,..., u m ) (i = 1,..., n) u j (j = 1,..., m) u j = n i=1 x i x i u j. x 1 x (,..., ) = (,..., u u m ) u 1 u m x 1 x n x n x u 1... n u m (x 1,..., x n ) (u 1,..., u m ) (gradient) grad(f) = ( x 1,..., x n ).1 1. f(x, y) = x 8 + x 5 y 9 x(t) = 3t 4t y(t) = 5t 4 F (t) = f(x(t), y(t)) t = 1 x(1) = 1 y(1) = 1 df dt = dx x dt + dy y dt = (8x 7 + 5x 4 y 9 )(6t 4) + 9x 5 y 8 5 = ( 8 + 5) 45 = 51. f(x, y) = x + y x(u, v) = u + 3v y(u, v) = uv (u, v) = ( 1, 1) u, v ( 1, 1 ( 3 ) z = f(x, y), x = ρ cos θ, y = ρ sin θ ) = ( 1, ) ( x ) + ( y ) = ( ρ ) + 1 ρ ( θ ). ρ = x x ρ + y y ρ = x = x θ x θ + y y θ = x 5 cos θ + y sin θ ( ρ sin θ) + y ρ cos θ
6 ( ρ ) + 1 ρ ( θ ) = ( cos θ + x = ( x ) + ( y ). y sin θ) + 1 ρ ( x ( ρ sin θ) + ρ cos θ) y. f(x, y) y ( x f) = f y x = f x,y, x ( x f) = f x = f x,x..4 [Schwartz] (p, q) f x, f y, f xy f xy f yx f x,y (p, q) = f y,x (p, q). f(x, y) = log x + y f = f x + f y = f(x, y) f(a + h, b + k) = f(a, b) + hf x (a + hθ, b + kθ) + kf y (a + hθ, b + kθ) 0 < θ < 1 a, b, h, k F (t) = f(a + ht, b + kt) F (1) F (0) = F (θ) 0 < θ < 1 x = x(t) = a + ht y = y(t) = b + kt df (t) dt = dx x dt + dy y dt = hf x (a + ht, b + kt) + kf y (a + ht, b + kt) f(x, y) F (1) F (0) f(a + h, b + k) f(a, b) = hf x (a + hθ, b + kθ) + kf y (a + hθ, b + kθ). 6
7 3. f(x, y) n n + 1 f(a + h, b + k) = f(a, b) + (h x + k )f(a, b) + y + 1 n! (h x + k y )n f(a, b) + R n R n = 0 < θ < 1 1 (n + 1)! (h x + k y )n+1 f(a + θh, b + θk) 3. y = f(x) x y y x (explicit function) F (x, y) = 0 y x (implicit function) F (x 1,..., x n, z) = 0 z x 1,..., x n F (x 1,..., x n, y) = 0 y = f(x 1,..., x n ) x 1 x,..., x n x 1 0 F + F F = 0 x 1 y x 1 y = F x 1 (x 1,..., x n, y) x 1 F y (x 1,..., x n, y) y = f(x 1,..., x n ) 3.3 [ ] F (x, y) (p, q) F x (x, y) F y (x, y) F (p, q) = 0 F y (p, q) 0 x = p y = f(x) (1) F (x, f(x)) = 0 f(p) = q () F F (x, f(x)) + x y (x, f(x))f (x) = [ ] F (x, y, z) (p, q, r) F x (x, y, z) F y (x, y, z) F z (x, y, z) F (p, q, r) = 0 F z (p, q, r) 0 (p.q) z = f(x, y) 7
8 (1) F (x, y, f(x, y)) = 0 f(p, q) = r () x = F x F z, y = F y F z n 3.3 z = f(x, y) P (p, q) P Q(x, y) f(p, q) < f(x, y) z = f(x, y) P (minimum) P f(p, q) f(p, q) > f(x, y) z = f(x, y) P (maximum) P f(p, q) (extremum point) x y P (p, q) z = f(x, y) f x (p, q) = f y (p, q) = 0 f x (p, q) = f y (p, q) = 0 f(x, y) (stationary point) 3.5 f(x, y) (p, q) A = f xx (p, q), B = f xy (p, q), C = f yy (p, q) (1) B AC < 0 (p, q) (a) A > 0 f(p, q) (b) A < 0 f(p, q) () B AC > 0 (p, q) 8
9 3. n = 1 f(p + h, q + k) f(p, q) = hf x (p, q) + kf y (p, q) + 1 ( ) h f xx (p + θh, q + θk) + hkf xy (p + θh, q + θk) + k f yy (p + θh, q + θk) f x (p, q) = f y (p, q) = 0 = 1 ( h k k f xx(p + θh, q + θk) + h ) k f xy(p + θh, q + θk) + f yy (p + θh, q + θk) h/k h, k B AC B AC < 0 0 A (a) A > 0 f(p + h, q + k) f(p, q) > 0 f(p, q) (b) A < 0 f(p + h, q + k) f(p, q) < 0 f(p, q) B AC > 0 h/k f(p + h, q + k) f(p, q) (p, q) B AC = f(x, y) = 4xy y x 4 f x = 4y 4x 3, f y = 4x 4y, f xx = 1x, f xy = 4, f yy = 4 f x (x, y) = f y (x, y) = 0 y = x x = 1, 0, 1 D = 16 48x = 16(1 3x ) (1, 1), ( 1, 1) (0, 0) ( 3. n = 1 ) f(x, y) f x, f y (C ) a, b h, k θ 0 < θ < 1 f(a + h, b + k) = f(a, b) + hf x (a, b) + kf y (a, b) 1 (h f x,x (a + θh, b + θh) + hkf x,y (a + θh, b + θk) + k f y,y (a + θh, b + θk)) 9
10 3.7 f(x), g(x) [a, b] (a, b) g (x) 0 a < u < b f(b) f(a) g(b) g(a) = f (u) g (u). (1) F (x) F (x) = f(x) f(a) f(b) f(a) (g(x) g(a)) g(b) g(a) F (a) = F (b) = 0 Roll F (u) = 0 (1) 3.8 F (x) 0 t 1 < θ < 1 F (t) = F (0) + F (0)t + 1 F (θt)t. () f(x) = F (x) F (0) F (0)x, g(x) = x f(0) = g(0) = t u f 1 (x) = F (x) F (0), g 1 (x) = x 3.7 v 0 u v 0 t v = θt 0 < θ < 1 F (t) F (0) F (0)t t = f (u) g (u) = F (u) F (0) = F (v) u = F (θt. θ t > 0 t < 0 F (t) = f(a + ht, b + kt) Chain Rule F (t) = f x (a + ht, b + kt)h + f y (a + ht, b + kt)k. f x (a + ht, b + kt), f y (a + ht, b + kt) Chain Rule F (t) = f x,x (a + ht, b + kt)h + f x,y (a + ht, b + kt)hk f y,x (a + ht, b + kt) + f y,y (a + ht, b + kt)k Schwartz f x,y = f y,x 3.8 t = 1 0 < θ < 1 f(a + h, b + k) = F (1) = F (0) + F (0)t + 1 F (θt)t = f(h, k) + hf x (a + h, b + k) + kf y (a + h, b + k) + 1 (h f x,x (a + θh, b + θh) + hkf x,y (a + θh, b + θk) + k f y,y (a + θh, b + θk)). 10
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)
5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h 0 f(a + h, b) f(a, b) h............................................................... ( ) f(x, y) (a, b) x A (a, b) x
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A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................
5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
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No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y
No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ
III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2 lim. (x,y) (1,0) x 2 + y 2 lim (x,y) (0,0) lim (x,y) (0,0) lim (x,y) (0,0) 5x 2 y x 2 + y 2. xy x2 + y
III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2. (x,y) (1,0) x 2 + y 2 5x 2 y x 2 + y 2. xy x2 + y 2. 2x + y 3 x 2 + y 2 + 5. sin(x 2 + y 2 ). x 2 + y 2 sin(x 2 y + xy 2 ). xy (i) (ii) (iii) 2xy x 2 +
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
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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.
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. ttp://www.morikita.co.jp/books/mid/00571 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. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π
4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan
f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a
3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (
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1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
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9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin
4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P = 90, = ( ) = X
4 4. 4.. 5 5 0 A P P P X X X X +45 45 0 45 60 70 X 60 X 0 P P 4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P 0 0 + 60 = 90, 0 + 60 = 750 0 + 60 ( ) = 0 90 750 0 90 0
II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1
II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2
I y = f(x) a I a x I x = a + x 1 f(x) f(a) x a = f(a + x) f(a) x (11.1) x a x 0 f(x) f(a) f(a + x) f(a) lim = lim x a x a x 0 x (11.2) f(x) x
11 11.1 I y = a I a x I x = a + 1 f(a) x a = f(a +) f(a) (11.1) x a 0 f(a) f(a +) f(a) = x a x a 0 (11.) x = a a f (a) d df f(a) (a) I dx dx I I I f (x) d df dx dx (x) [a, b] x a ( 0) x a (a, b) () [a,
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
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名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト
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Chap9.dvi
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1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
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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
, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,
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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.
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5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3)
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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.
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