.1.1 Y K L Y = K 1 3 L 3 L K K (K + ) 1 1 3 L 3 K 3 L 3 K 0 (K + K) 1 3 L 3 K 1 3 L 3 lim K 0 K = L (K + K) 1 3 K 1 3 3 lim K 0 K = 1 3 K 3 L 3 z = f(x, y) x y z x-y-z.1 z = e x +xy y 3 x-y ( ) z 0 f(x, y) = z 0 (x, y) ( ) y = f(x, y) x-y 1. z = e x +xy y.1 z = x +y z = 1 z = f(x, y) 4 1 19
.1 z = e x +xy y 1 0.8 0.6 z 0.4 0. 0 1 0 y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0, 0, 0) (1, 0, ) (0, 1, 1) n f(x 1, x,..., x n ) n x 1, x,..., x n n n 3 () 3 n (hyperplane) a 1, a,..., a n, b a 1 x 1 + a x + + a n x n = b (.) 0
. z = e x +xy y 1.5 1 0.5 y 0 0.5 1 1.5 1.5 1 0.5 0 0.5 1 1.5 x 1 1 f(x) df(x) dx = lim f(x + h) f(x) h 0 h f(x, y) (x 0, y 0 ) (λ, µ) t f(x, y) g(t) g(t) f(x 0 + λt, y 0 + µt) g(t) t = 0 g(0 + h) g(0) f(x 0 + λh, y 0 + µh) f(x 0, y 0 ) lim = lim h 0 h h 0 h f(x, y) (x 0, y 0 ) (λ, µ) (x 0, y 0 ) (λ, µ) ( ).1 f(x, y) = x + y (, ) (1, 1) ( + h) + ( + h) ( + ) h + 8h lim = lim h 0 h h 0 h = lim h + 8 = 8 h 0..1 (1, 1) (1, 1) 1
(1, 0) (x 0, y 0 ) f(x 0 + h, y 0 ) f(x 0, y 0 ) lim h 0 h f(x, y) (x 0, y 0 ) x x f(x, y) x f(x, y) x z = f(x, y) x f(x, y) x z f(x, y) x x f x (x, y) (0, 1) y.1 F K F L Y K F K(K, L) Y L F L(K, L).3 f(x, y) = x y f(x, y) x y.3 f x = 4xy f y = x i) x + 3y + 4 ii) (x + y) 3 iii) e x +xy+y iv) x y
1 n 1 f(x) x = x 0 f(x 0 + h) f(x 0 ) lim = c h 0 h c f (x 0 ) = c 9 h 0 o(h) h o(h) f(x 0 + h) f(x 0 ) ch (.3) 0 c f(x) x = x 0 f(x) = f(x 0 ) + f (x 0 )(x x 0 ) + 1 f (x 0 )(x 0 x) + x = x 0 + h f(x 0 + h) = f(x 0 ) + f (x 0 )h + 1 f (x 0 )h + (.3) o(h) h h 0 o(h) h 0 f(x, y) (x 0, y 0 ) (u, v) a = (h, k) o(h, k) o(h, k) f(x 0 + h, y 0 + k) f(x 0, y 0 ) (uh + vk) (.4) o(h, k) a 0 0 a a = h + k f(x 0 + h, y 0 + k) f(x 0, y 0 ) (uh + vk) lim = 0 (.5) a 0 a a = (h, k) k = 0 ( u = f x (x 0, y 0 ) f ) x (x 0, y 0 ) v = f y (x 0, y 0 ) ( f ) y (x 0, y 0 ) 3
f(x, y) (x 0, y 0 ) (x 0, y 0 ) f(x, y) z = f(x, y) (x 0, y 0 ) z = ux + vy + α = f x (x 0, y 0 )x + f y (x 0, y 0 )y + α α = f(x 0, y 0 ) f x (x 0, y 0 )x 0 f y (x 0, y 0 )y 0 (.6) (.6) z = f(x, y) (x 0, y 0 ).4 f(x, y) = 0.5x + y 1 (, 1) f x (, 1) = 0.5 f y (1, 1) = 1 (.6) α = 1 0.5 1 1 = 1 z = 0.5x + y 1.5 f(x, y) = 1 8 x + 1 y z = f(x, y) (, 1) f x (, 1) = 1 8 = 0.5 f y (, 1) = 1 1 = 1 α = 1 8 + 1 1 0.5 1 1 = 1 z = 0.5x + y 1.4.5 (1, 1).1 f(x, y) i) (λ, µ) f(x, y) (λ, µ) ii) f(x, y) x y f(x 0 + λh, y 0 + µh) f(x 0, y 0 ) lim = λu + µv h 0 h u = x f(x 0, y 0 ) v = y f(x 0, y 0 ) 4
.3 z y 1 x z y 1 x 5
.1 f(x, y) x y.6 f(x, y) = { xy, x +y (x, y) (0, 0) 0, (x, y) = (0, 0) f(x, 0) = 0 f(x, y) x 0 f x (x, 0) = 0 f y (0, y) = 0 f x (0, 0) = f y (0, 0) = 0 (0, 0) (λ, µ) ( ) λµh f(0 + λh, 0 + µh) f(0, 0) h = 1 h (λ + µ f(0, 0) )h λµ = 1 h (λ + µ ) λ µ 0 h 0 h = x k = y f = f(x 0 + h, y 0 + k) f(x 0, y 0 ) (.4) f = u x + v y + o(h, k) = f x (x 0, y 0 ) x + f y (x 0, y 0 ) y + o(h, k) (.7) (.7) x x y y f f x y 1 a = (h, k) a 0 o(h,k) a 0 df = f x (x 0, y 0 )dx + f y (x 0, y 0 )dy (x 0, y 0 ) f.7.1 df = f x (x, y)dx + f y (x, y)dy (.8) Y K = 1 3 K 3 L 3 dy = 1 3 K 3 L 3 dk + 3 K 1 3 L 1 3 dl Y L = 3 K 1 3 L 1 3 6
.5 1.4 c 1 c U(c 1, c ) = 1 γ cγ 1 + α 1 γ cγ f(x, y) x f(x,y) x x y f(x, y) x x f(x,y) f x xx f(x, y) x x ( ) f(x, y) f(x, y) x y f(x,y) x y ( ) f(x, y) f(x,y) y x f(x, y) x y y x x f yx, f(x,y) f y yy.8 f(x, y) = x e y 1 f(x, y) x = xe y f(x, y) y = x e y f(x, y) x = e y f(x, y) x y = xe y f(x, y) y = x e y f(x, y) = xe y y x f xy. f(x, y) f(x,y) y f(x,y) x y f(x,y) x y f(x,y) y x f(x, y) y x = f(x, y) x y..6 f(x, y) = x 4 y 3 + xy f xy = f yx.7 f(x, y) = x e x y 7
n f(x 1, x,..., x n ) (.9) (a 1, a,..., a n ) x i lim h 0 f(a 1, a,..., a i + h,..., a n ) f(a 1, a,..., a i,..., a n ), i = 1,..., n (.9) h y = f(x 1,..., x n ) x i f(x 1,..., x n ) x i, x i f(x 1,..., x n ), y x i, f xi (x 1,..., x n ), f i (x 1,..., x n ), (.10) y = f(x 1, x,..., x n ) (a 1, a,..., a n ) y = f 1 (a 1,..., a n )x 1 + f (a 1,..., a n )x + + f n (a 1,..., a n )x n + c c = f(a 1,..., a n ) f 1 (a 1,..., a n )a 1 f (a 1,..., a n )a f n (a 1,..., a n )a n (.11) f j (a 1,..., a n ) = f x j (a 1,..., a n ) df = f 1 (x 1,..., x n )dx 1 + f (x 1,..., x n )dx + + f n (x 1,..., x n )dx n (.1) f(x 1,..., x n ) x i x j i j f(x 1,..., x n ) ( ) f(x1,..., x n ) x i x j x j x i i = j f(x 1,..., x n ) x i f xi x j k k f(x 1,...,x n) x k 1 1 xk xk n n ( ) f(x1,..., x n ) x i x i k 1 + k + + k n = k.8 f(x, y, z) = x + xy + xz + yz 8
. z x y z = f(x, y) x y t x = g(t) y = h(t) z t f(x, y) g(t) h(t) t f(g(t), h(t)) t dz dt = f dg x dt + f dh y dt (.13).9 z = x + xy y = e x dz dx z x y = e x z = x + xe x dz dx = x + e x + ( 1)xe x = x + ( x + 1)e x f(x, y) = x + xy, h(x) = e x dz dx = f x + f dh y dx = x + y + x( e x ) = x + ( x + 1)e x.10 G 1 G p 1 p G 1 p 1 p I c(p 1, p, I) I M(p 1, p, u) p 1 h(p 1, p, u) = c ( p 1, p, M(p 1, p, u) ) h(p 1, p, u) p 1 h p 1 h p 1 = c p 1 + c I M p 1.9 f(x, y) = (x 4 + x y + xy) 3 x = t + 1 y = t + 4 df dt 9
. K L F(K, L) F(K, L) = 5K 0.4 L 0.6 (.14) L L + L F(K + K, L + L) = F(K, L) K K L Y = F(K, L) K L dk dl dy = F K dk + F L dl dy = 0 dk dl = F L F K F K (K, L) = K 0.6 L 0.6, F L (K, L) = 3K 0.4 L 0.4 dk dl = F L(K, L) F K (K, L) = 1.5K L L ɛ K 1.5 K L ɛ x y f y = f(x) ( ) dy dx x y F(x, y) = 0 x + y 1 = 0 (.15) x + y e xy = 0 (.15') y = f(x) (.15) (.15') F(x, y) dy dx 30
.4 0.5 1 8 7 F(K,L)=10 6 5 K 4 3 F(K,L)=.5 1 F(K,L)=5 0 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 L.3 ( ) F(x, y) (x 0, y 0 ) F(x 0, y 0 ) = 0 F y (x 0, y 0 ) 0 x = x 0 i) F(x, f(x)) = 0 ii) y 0 = f(x 0 ) f(x) y = f(x) dy dx = F x(x, y) F y (x, y).11 F(x, y) = x + y e xy dy dx (1, 0) F(1, 0) = 0 F x (x, y) = 1 ye xy F y (x, y) = 1 xe xy dy yexy = 1 dx 1 xe xy 31
.10 F(x, y) = 3x + 4xy + y 9 F(x, y) = 0 y = f(x) dy dx.3 y = e x t + t 1 x t t f(x, y, t) t f(x, y, t) = 0 (x, y) x{y t.1 f(x, y, t) = x (t 1)x y+t t 0, 1,, 3, 4 f(x, y, t) = 0.5.5 x (t 1)x y + t = 0 16 14 1 10 y 8 t=4 6 t=3 4 t= t=1 0 t=0 1 0 1 3 4 5 6 x t C C 3
C C.6.6 x (t 1)x y + t = 0 C 14 1 10 8 6 y 4 0 C 1 0 1 3 4 5 6 x t t t (x(t), y(t)) (.16) t f(x(t), y(t), t) = 0 (.16) d dt f(x(t), y(t), t) = f x(x(t), y(t), t) dx(t) + f y (x(t), y(t), t) dy(t) + f t (x(t), y(t), t) (.17) dt dt (x(t), y(t)) C dy(t) dt / dx(t) dt dy dx = lim y x 0 x = lim y(t + t) y(t) t 0 x(t + t) x(t) y(t + t) y(t) t = lim t 0 t x(t + t) x(t) = dy(t) dt /dx(t) dt (.18) f(x, y, t) = 0 x y (x(t), y(t)) dy dx = f x(x(t), y(t), t) f y (x(t), y(t), t) (.19) 33
C C (.18) (.19) f x (x(t), y(t), t) dx(t) dt (.17) (.0) + f y (x(t), y(t), t) dy(t) dt (.16) (.1) = 0 (.0) f t (x(t), y(t), t) = 0 (.1).3 f(x, y, t) = e x t + t y (.16) (.1) e x(t) t + t y(t) 1 = 0 e x(t) t + 1 = 0 t y = x.7.11.6 C