1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim = lim(t 1) = 0 (t 1) 3 (t 1) 2 t 1 (t 1) 2 t 1 (t 1) 3 = o((t 1) 2 ) (t 1) 1.3 f (x) x = a lim f (x) f (a) f (a)(x a) lim x a x a x a = 0 f (x) = f (a)(x a) + f (a) + o(x a) (x a) f (x) f (a) x a = f (a) f (x) f (a)(x a)+ f (a) f (x) { f (a)(x a)+ f (a)} x a y = f (a)(x a) + f (a) x = a y = f (x) x a y = f (x)
2 1.4 lim h 0 f (x) x = a lim f (a + h) f (a) f (a)h h = 0 h 0 f (a + h) f (a) h = f (a) h 0 f (a + h) f (a) f (a)h h f (a + h) = f (a) + f (a)h + o(h) (h 0) f (a + h) f (a) Ah = o(h) (h 0) h A 0 = lim h 0 f (a + h) f (a) Ah h A = f (a) f (x) x = a f (a + h) = f (a) + Ah + o(h) (h 0) ( f (a + h) f (a) = lim h 0 h ) A A A = f (a) y = f (x) x x y f (x + x) f (x) y y = f (x + ) f (x) ( 1 2)(3) a x h x f (x + x) = f (x) + f (x) x + o( x) ( x 0) y = f (x) x + o( x) ( x 0) y f (x) x y f (x) x x 1 y f (x) x f (x) (differential) dy, d f, d f (x) dy = d f = d f (x) = f (x) x f (x) = x f (x) = 1 dx = d f (x) = 1 x = x dx = x dy = d f = d f (x) = f (x)dx 1 x
3 y = f (x) dy = f (x)dx y f (x) x y f (x) x x lim x 0 1.5 y f (x) x x = 0 f (x) = x 2 (differential) d f (x) = 2xdx 2 2 z = F(x, y) 1.4 F(x, y) (a, b) h, k A, B F(a + h, b + k) = F(a, b) + Ah + Bk + o( h 2 + k 2 ) ((h, k) (0, 0)) 1 A 2 A, B k = 0 F(a + h, b) = F(a, b) + Ah + o(h) (h 0) f (x) = F(x, b) f (a + h) = f (a) + Ah + o(h) (h 0) A = f (a) A F(x, y) y y = b F(x, b) x x = a (a, b) F(x, y) x F (a, b), F x(a, b) F(x, y) x x = a F(a, y) y y = b B (a, b) F(x, y) y F (a, b), F y(a, b) A = F (a, b), B = F (a, b) a x b y h x k y z = F(x + x, y + y) F(x, y) z = F x + F y + o(s) (s 0)
4 s = ( x) 2 + ( y) 2 F F = F (x, y), F = F (x, y) F(x, y) y x F(x, y) x x y F 2.1 y F(x, y) = x 2 y + 3x y F = 2xy + 3, F 1 z F dz, df, df(x, y) dz = df = df(x, y) = F F x + y = x2 1 x + F y z = F(x, y) 2 F(x, y) = x F = 1, F = 0 dx = df(x, y) = 1 x + 0 y = x dy = y dz = df = df(x, y) = F F dx + dy 2.2 z = F(x, y) = x 2 y dz = 2xydx + x 2 dy F(x, y) 3 dz = F F dx+ dy z F 3 2 x+ F y 3 F(x, y) (a, b) F x (x, y), F y (x, y) (2 ) (a, b) F(x, y)
5 u = F(x, y, z) du = df = df(x, y, z) = F F F dx + dy + z dz 2.1 x = ϕ(t) z = F(x, y) y = y(t) dz = z dx + z dy 3 u = F(x, y, z) du = u dx + u dy + u z z = F(ϕ(t), y(t)) t t x, y, z x, y, z s = ( x) 2 + ( y) 2 z = f (x + x, y + y) f (x, y) = f x (x, y) x + f y (x, y) y + o( s) t 0 x 0, y 0 s 0 o( s) t dz = o( s) s s t ( = o( s) ) 2 ( ) 2 ( ) 2 ( x y dx dy + 0 + s t t = 0 ( t 0) z = f x (x, y) x + f y (x, y) y t t t f dx + f dy + o( s) t ) 2 t 0 dz =
6 3 x = x(t) y = y(t) xy C : F(x, y) = 0 1 x (t 0 ) y 0 P C (t 0 ) a = x(t 0 ), b = y(t 0 ) P(a, b) P P P C df = F dx + F dy 1 t F x (a, b)x (t 0 ) + F y (a, b)y (t 0 ) = 0 F x(a, b) (a, b) F y (a, b) F x(a, b) F y (a, b) F x (a, b)(x a) + F y (a, b)(y b) = 0 C : y = f (x) F(x, y) = y f (x) F x (x, y) = f (x), F y (x, y) = 1 f (a)(x a) + 1 (y b) = 0 y b = f (a)(x a) 3.1 F(x, y) = x 2 + 2xy + 2y 2 5 F x (x, y) = 2x + 2y, F y (x, y) = 2x + 4y C : x 2 + 2xy + 2y 2 5 = 0 (x 0, y 0 ) C (2x 0 + 2y 0 )(x x 0 ) + (2x 0 + 4y 0 )(y y 0 ) = 0 (x 0 + y 0 )(x x 0 ) + (x 0 + 2y 0 )(y y 0 ) = 0
7 xyz S : F(x, y, z) = 0 2 F(x, y, z) x x(t) P(a, b, c) S P S ( ) l : y = y(t) z z(t) a = x(t 0 ), b = y(t 0 ), c = z(t 0 ) df = F dx + F dy + F z 2 t dz F x (a, b, c)x (t 0 ) + F y (a, b, c)y (t 0 ) + F z (a, b, c)z (t 0 ) = 0 F x (a, b, c) x (t 0 ) x (t 0 ) F y (a, b, c) y (t 0 ) y (t 0 ) P l F z (a, b, c) z (t 0 ) z (t 0 ) F x (a, b, c) F y (a, b, c) P π l F z (a, b, c) l P S π P S P π P S F x (a, b, c)(x a) + F y (a, b, c)(y b) + F z (a, b, c)(z c) = 0 S : z = f (x, y) F(x, y, z) = z f (x, y) F x (x, y, z) = f x (x, y), F y (x, y, z) = f y (x, y), F z (x, y, z) = 1 f x (a, b)(x a) f y (a, b)(y b) + 1 (z c) = 0 z c = f x (a, b)(x a) + f y (a, b)(y b) 3.2 K : z = x 2 + y 2 A(x 0, y 0, z 0 ) K f (x, y) = x 2 + y 2 f x (x, y) = 2x, f y (x, y) = 2y z z 0 = 2x 0 (x x 0 ) + 2y 0 (y y 0 )
8 z = f (x, y) dz = f x (x, y)dx + f y (x, y)dy z c = f x (a, b)(x a) + f y (a, b)(y b) + o( (x a) 2 + (y b) 2 ) ((x, y) (a, b) c = f (a, b) (x, y) (a, b) z f x (a, b)(x a) + f y (a, b)(y b) + f (a, b) z = f x (a, b)(x a) + f y (a, b)(y b) + f (a, b) (a, b, f (a, b)) (x, y, z) (a, b, f (a, b)) z = f (x, y) 4 xy C t : f (x, y, t) = 0 E {C t } (i) E C t (ii) E C t 2 4.1 E C t : f (x, y, t) = 0 f (x, y, t) = 0 1 E (x, y) f t (x, y, t) = 0 2 E C t P(x, y) C t 1 x = x(t), y = y(t) t P E E 1 f (x(t), y(t), t) = 0 t f x (x(t), y(t), t) x (t) + f y (x(t), y(t), t) y (t) + f t (x(t), y(t), t) 1 = 0 3 x = x(t) E P E y = y(t) x (t) C y t f (x, y, t) = 0 (t ) (t)
9 f x (x(t), y(t), t) (x x(t)) + f y (x(t), y(t), t) (y y(t)) = 0 f x(x(t), y(t), t) f y (x(t), y(t), t) x (t) f x(x(t), y(t), t) y (t) f y (x(t), y(t), t) f x (x(t), y(t), t) x (t) + f y (x(t), y(t), t) y (t) = 0 4 4 3 f t (x(t), y(t), t) = 0 2 {C t } 1 2 1 2 t t 1 2 C : f (x, y) = 0 (a, b) f x (a, b) = f y (a, b) = 0 (a, b) C 4.2 f (x, y, t) = 0 1 C t : f (x, y, t) = 0 (x, y) x(t), y(t) f t (x, y, t) = 0 2 x (t) 2 + y (t) 2 0 x = x(t), y = y(t) 3 x = x(t) C : {C t } C t y = y(t) C t C 3 1 f (x(t), y(t), t) = 0 4 C P(x(t), y(t)) C t 4 f x (x(t), y(t), t)x (t) + f y (x(t), y(t), t)y (t) + f t (x(t), y(t), t) = 0 5 3 2 f t (x(t), y(t), t) = 0 6
10 6 5 f x (x(t), y(t), t)x (t) + f y (x(t), y(t), t)y (t) = 0 f x(x(t), y(t), t) x (t) f y (x(t), y(t), t) y 7 (t) x (t) 0 C P y (t) f x(x(t), y(t), t) = 0 P C t f y (x(t), y(t), t) f x(x(t), y(t), t) 0 C t P f y (x(t), y(t), t) 7 C C t P C C t 4.1 l θ : (cos θ)x + (sin θ)y = 1 (0 < = θ < = π) 1 C 1 θ (sin θ)x + (cos θ)y = 0 2 1 cos θ 2 sin θ x = cos θ 1 sin θ + 2 cos θ y = sin θ l θ x = cos θ C : (0 < y = sin θ = θ < = π)
11 4.2 C t : x 4 x 2 +(y t) 2 = 0 f (x, y, t) = x 4 x 2 +(y t) 2 f t (x, y, t) = 2(y t) f (x, y, t) = 0 f t (x, y, t) = 0 x(t) = 0 x(t) = 1 x(t) = 1 t y(t) = t y(t) = t y(t) = t x = 0, x = 1, x = 1 x = 1, x = 1 f x (x, y, t) = 4x 3 2x f y (x, y, t) = 2(y t) f x (0, t, t) = 0 f y (0, t, t) = 0 (0, t) C t x = 0 2.5 2 1.5 1 f (x, y, t) = 0 1 f t (x, y, t) = 0 2 (x(t), y(t)) C x (t) = 0 y (t) = 0 t ( ) 4.3 C t : y = t(x 1) f (x, y, t) = t(x 1) y f t (x, y, t) = x(t) = 1 x 1 f (x, y, t) = 0 f t (x, y, t) = 0 y(t) = 0 1 (1, 0) C t 0.5 0-1 -0.5 0 0.5 1-0.5