f (x) f (x) f (x) f (x) f (x) 2 f (x) f (x) f (x) f (x) 2 n f (x) n f (n) (x) dn f f (x) dx n dn dx n D n f (x) n C n C f (x) x = a 1 f (x) x = a x >

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1 x = a f (x) a x h f (a + h) f (a) h (5.1) h 0 f (x) x = a f +(a) f (a + h) f (a) = lim h +0 h (5.2) x h h 0 f (a) f (a + h) f (a) f (a h) f (a) = lim = lim h 0 h h 0 h (5.3) f (x) x = a f (a) = lim h 0 f (a + h) f (a) h (5.4) f (x) x = a f (x) x = a (a, f (a)) P (a + h, f (a + h)) Q (5.1) PQ ( 5.1) h 0 PQ α (5.4) α x = a f (x) PQ P 2. y Q P x a a + h Fig.5.1 x (5.4) f (a) a x f (x) f (x) d f f (x) d D f (x) dx dx (x ) ( x ) 5-1

2 f (x) f (x) f (x) f (x) f (x) 2 f (x) f (x) f (x) f (x) 2 n f (x) n f (n) (x) dn f f (x) dx n dn dx n D n f (x) n C n C f (x) x = a 1 f (x) x = a x > a x f (x) f (a) x a > 0 x a f (x) f (a) lim > x a+0 x a = 0 (5.5) f (x) f (a) lim < x a 0 x a = 0 (5.6) (5.5) (5.6) x = a 0 f (a) = 0 f (x) x = a f (a) = ( ) f (x) x = a 1 f (x) x = a f (a) = 0 f (x) f (x) = x x = 0 f (x) = x 3 x = 0 f (0) = 3x 2 = 0 f (a) = 0 a f (x) 2 f (x) f (a) = 0 x = a f (a) > 0 f (x) f (a) < 0 f (x) f (a) = 0 5-2

3 3. (5.4) f (x) g(x) 5.1 {c f (x)} = c f (x) (5.7) 5.2 { f (x) ± g(x)} = f (x) ± g (x) (5.8) (5.7) (5.8) (5.4) 5.3 { f (x)g(x)} = f (x)g(x) + f (x)g (x) (5.9) (5.4) f (x + h) f (x) = h f (x) + h f (h) (5.10) f (h) lim h 0 f (h) = 0 ( ) f (0) = 0 g(x) g(x + h) g(x) = h g (x) + h g (h) (5.11) f (x + h)g(x + h) f (x)g(x) = { f (x) + h f (x) + h f (h)}{g(x) + h g (x) + h g (h)} = h{ f (x)g(x) + f (x)g (x)} + h{ f (x) g (h) + g(x) f (h)} + h 2 { f (x)g (x) + } h h 0 2 { } 0 3 { } f (x + h)g(x + h) f (x)g(x) lim = f (x)g(x) + f (x)g (x) h 0 h (5.9) 5-3

4 5.4 x g(x) 0 { } f (x) = f (x)g(x) f (x)g (x) g(x) {g(x) 2 } (5.12) { } 1 = g (x) g(x) {g(x) 2 (5.11) } 1 { 1 h g(x + h) 1 } = 1 g(x + h) g(x) g (x) + g (h) = g(x) h g(x + h)g(x) g(x) { g(x) + h g (x) + h g (h) } { } 1 h 0 = g (x) g(x) {g(x) 2 } f (x) g(x) = f (x) 1 (5.9) g(x) { } f (x) = f { } (x) 1 g(x) g(x) + f (x) = f (x) g(x) g(x) + f (x) g (x) {g(x) 2 } = f (x)g(x) f (x)g (x) {g(x) 2 } 5.5 { f (g(x))} = f (g(x))g (x) (5.13) (5.10) g(x + h) g(x) = h g (x) + h g (h) (5.14) y = g(x) f (y + k) f (y) = k f (y) + k f (k) (5.15) h k k = h g (x) + h g (h) (5.16) h 0 k 0 f (g(x + h)) f (g(x)) = k f (g(x)) + k f (k) = h f (g(x)) g (x) + h g (h) f (g(x)) + h f (k) g (x) + h g (h) f (k) (5.17) h f (g(x + h)) f (g(x)) h = f (g(x)) g (x) + g (h) f (g(x)) + f (k) g (x) + g (h) f (k) h

5 z = f (x,y) x y (x,y) z x,y 2 z 1 (x,y) z = f (x,y) x y 1 (a,b) f (x,y) (5.4) lim h 0 f (a + h,b) f (a,b) h (5.18) f (x,y) x = a,y = b x y lim k 0 f (a,b + k) f (a,b) k (5.19) f (x,y) x = a,y = b y 2. x,y (5.19) a,b x,y x f (x,y) f x (x,y) f x x f (x,y) (??) y f y (x,y) f y y f (x,y) f (x,y) f x (x,y) y = x y x 1 x f y (x,y) x y 1 5-5

6 f x (x,y) x 1 x 2 f xx (x,y) = 2 f x 2 y x,y 2 f yx (x,y) = 2 f y x y f yy (x,y) = 2 f y 2 f xy(x,y) = 2 f x y x y f yx (x,y) = f xy (x,y) 2 f y x = 2 f x y n C n C 3. (x,y) ds f (x,y) ds x dx y dy ds x dx f x y dy f y f x = f x dx f y = f y dy 5.6 ( ) d f = f x dx + f y dy 4. (chain rule) x,y ds φ ψ f φ = f x f ψ = f x x ψ + f y x φ + f y y φ y ψ chain rule 5-6

7 ( ) ( ) f (x) ( ) f (x) ( ) ( ) ( ) ( ) 6-1

8 6 6.2 ( ) 1. f (x) f (x) ( f (x) ) f (x) ( ) f (x) f (x) ( f (x) ) f (x) f (x) = x 2 (x > = 0) x = 0 f (x) = x 2 (x < 0) x = 0 2. f (x) = 0 x = c f (c) f (x) 6.1 f (x) x [a,b] x = c f (c) = 0 f (x) f (c) > 0 x = c f (c) < 0 x = c x = c f (c) = 0 f (c) > 0 c f (c + c) > 0 (c,c + c) x ( f (x) > f (c) c f (c c) < 0 (c c,c) x ( f (x) > f (c) f (c) x = c (c c,c + c) f (c) < 0 x = c f (x) = x 4 x = 0 f (0) = 0 f (x) = 0 ( f (x) = x 3 ) ( ) 3. f (x) f (x) = ± 6.2 f (x) x [a,b] f (x) a b f (x) = 0 ( ) 6-2

9 6 c f (x) > 0 c f (c + c) > f (c) x = c f (c c) < f (c) x = c f (x) < 0 ± f (x) = 0 f (x) = x x = 0 x = 0 ( ) 4. (1) f (x) = 0 f (x) < = 0 ( ) (2) 6.3 f (x,y) 1. f (x) f x = 0 f (x,y) 2 f x 2 2 f y 2 2 f x y 2. f (x,y) f x = 0 f y = 0 (c x,c y ) f (c x,c y ) f (x,y) 6.3 f (x,y) x,y D (x,y) = (c x,c y ) f x = 0 f y = 0 f (x,y) 2 f x 2 > 0 2 f y 2 > 0 (c x,c y ) 2 f x 2 < 0 2 f y 2 < 0 (c x,c y ) 0 ( ) f x 2 2 f y 2 6-3

10 6 2 f x 2 > f 0 2 y 2 < 0 2 f x 2 < f 0 2 > 0 y f (x,y) x,y D f (x,y) f x = 0 f = 0 ( ) y f x f y 0 4. (1) f x = 0 f y = 0 f 2 x 2 < 0 2 f < 0 ( ) y2 (2) (1) (2) ( ) ( ) y = ax + b f (x,y) ( ) ( ) f (x,y) ( ) a f (x,y) ( ) g(x,y) = C (6.1) ( ) 2. (6.1) y = y(x) x = x(y) f (x,y) f (x,y(x)) f (x(y),y) x y 6-4

11 6 3. x y t x = φ(t) y = ϕ(t) f (x,y) s f (φ(t),ϕ(t)) t 4. () (6.1) g(x,y) C = 0 λ f (x,y) x,y,λ F(x,y,λ) F(x,y,λ) = f (x,y) + λ(g(x,y) C) (6.2) λ () 0 F(x,y,λ) f (x,y) (6.2) x,y,λ F x = 0 F y = 0 F λ = 0 x y λ ( ) 6-1 y = 1 x f (x,y) = x 2 + y 2 x + y 1 = 0 F(x,y,λ) = x 2 + y 2 + λ(x + y 1) (6.3) 2x + λ = 0 2y + λ = 0 x + y 1 = 0 λ = 1 x = y = 1 2 ( 1 2, 1 2 ) (6.3) 1 2 y x x 6-5

12 6 6-1 (1) d dx (xx ) ( y = x x ) (2) 2 ( ) y log(xy) x y (3) ( log(x 2 y 2 ) ) + log(x x y( 2 y 2 ) ) 6-2 () x 2 + y 2 = 1 f (x,y) = xy 6-3 (1) f (x) = 1 x = 0 Taylor ( Maclaurin ) ( ) x + 1 (2) (1) ( x ) 1 (3) x f (x) = x = 0 Taylor (x + 1) 2 6-6

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),

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