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) 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.. 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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