009 IA 5 I, 3, 4, 5, 6, 7 6 3. () Arcsin ( (4) Arccos ) 3 () Arcsin( ) (3) Arccos (5) Arctan (6) Arctan ( 3 ) 3. n () tan x (nπ π/, nπ + π/) f n (x) f n (x) fn (x) Arctan x () sin x [nπ π/, nπ +π/] g n (x) cos x [nπ, (n+)π] h n (x) g n (x) gn (x) h n (x) h n (x) Arcsin x 3. () cos (Arcsin x) () tan (Arcsin x) ( < x < ) (3) sin (Arctan x) (4) sin ( Arctan x) x 4. f(x) = x 4 5 f D 0 3x + D D x f 0 f(x) D x f 0 f(x) D 5. log( + x) 0 x < x 6. x < x x x log ( + ) = x n=0 a n x n {a n} n=0 7. x y () x y 3 () sin(x y 3 ) (3) x y (4) x + y 8. f(x, y) g(x, y) (a, b) g(x, y) = ( x y )f(x, y) g g (a, b) = x y (a, b) = 0, a + b <
9. R x 0 {(x, 0) x 0} D D f(x, y)? () f y (x, y) = 0 f y x f(x, y) = g(x) x g () f x (x, y) = f y (x, y) = 0 f 0. f(x, y) { ( x + y sin f(x, y) = ) x +y (x, y) (0, 0) 0 (x, y) = (0, 0) f(x, y) (0, 0) (0, 0) x y. f(x, y) f(x, y) = xy x + y (x, y) (0, 0) 0 (x, y) = (0, 0) f(x, y) (0, 0) x y 0 u (0, 0) u (0, 0) () f(x, y) (0, 0) x y () f(x, y) 0 u (0, 0) u (3) f(x, y) (0, 0). f(x, y) f(x, y) = x y x 4 + y (x, y) (0, 0) 0 (x, y) = (0, 0) f(x, y) (0, 0) (0, 0) 3. f(x, y) f(x, y) = () f(x, y) (0, 0) (x y) 3 x + y (x, y) (0, 0) 0 (x, y) = (0, 0) () f(x, y) (0, 0) x y (3) f(x, y) (0, 0) 4. f f(x, y) = { ( x + y ) sin ( ) x +y (x, y) (0, 0) 0 (x, y) = (0, 0) f (0, 0) x f x y f y (0, 0)
009 IA 5 I, 3, 4, 5, 6, 7 6 3 () π 6, () π, (3) π 6, (4) 3 4 π, (5) π 6, (6) π 3 () Arcsin sin θ = π θ π θ Arcsin = π 6 () () y 0 tan x = y 0 x tan x 0 = y 0 x 0 x = x 0 + mπ m Arctan y 0 tan x = y 0 x π/ < x < π/ tan x = y 0 x Arctan y 0 x = Arctan y 0 + mπ m f n (x) (nπ π/, nπ + π/) nπ π < f n (y 0 ) < nπ + π fn (y 0 ) = Arctan y 0 + nπ y 0 fn (x) = Arctan x + nπ
5 () y 0 y 0 sin x = y 0 x sin x 0 = y 0, π/ x 0 π/ x 0 x = mπ + ( ) m x 0 m cos x = sin (x + π/) ( cos x = y 0 sin x + π ) = y 0 cos x = y 0 x x 0 x = mπ + ( ) m x 0 π = m π + ( ) m x 0 m sin x = y 0 x π/ x π/ Arcsin y 0 x 0 Arcsin y 0 x x x = mπ + ( ) m Arcsin y 0, x = m Arcsin y 0 + ( ) m Arcsin y 0 m nπ π g n (y 0 ) nπ + π, nπ h n (y 0 ) (n + )π n gn (y 0 ) x m = n h n (y 0 ) x m = n + gn (y 0 ) = nπ + ( ) n Arcsin y 0, h n (y 0 ) = n + π + ( ) n+ Arcsin y 0 y 0 [, ] gn (x) = ( ) n Arcsin x + nπ, h n (x) = ( ) n+ Arcsin x + n + π h 0 (x) = Arccos x Arccos x = Arcsin x + π y = x
5 3 y f f π Arctan x = f 0 f O x f Arcsin x = g 0 g g g y O g x h h y O h 0 = Arccos x x h 3 : f n, gn, h n () sin(arcsin x) = x cos θ sin θ cos θ π/ θ π/ 0 cos θ = sin θ π θ π π/ Arcsin x π θ Arcsin x cos (Arcsin x) = sin (Arcsin x) = x () cos(arcsin x) (cos(arcsin x)) = cos (Arcsin x) Arcsin x = sin(arcsin x) cos(arcsin x) = x + C = x x x
5 4 C x 0 C = 0 cos(arcsin x) = x () () tan(arcsin x) = sin(arcsin x) cos(arcsin x) = x x (3) tan (Arctan x) = x sin θ tan θ π/ θ π/ cos θ 0 cos θ = cos θ cos θ = sin θ + cos θ cos = tan θ + θ sin θ = tan θ cos θ = θ = Arctan x sin (Arctan x) = tan θ tan θ + tan (Arctan x) tan (Arctan x) + = x x + (4) sin ( Arctan x) = sin (Arctan x) cos (Arctan x) sin (Arctan x) (3) cos (Arctan x) (3) cos θ = θ = Arctan x + tan θ cos (Arctan x) = π < θ < π x + x sin ( Arctan x) = x + = x + x x +
5 5 () () () 4 4 5 4 5 f(x) = x x x = + x + x + + x n + xn x x = + x + ( x ) + + ( x ) n + ( x ) n x f 0 n R n (x) = xn x ( x ) n x ( ) = x n x n ( x) x < 0 0 D x < x 5 5 log( + x) ( ) log( + x) = = ( + x) + x ( ) log( + x) = ( + x) ( log( + x) ) = ( + x) 3 ( log( + x) ) = 3 ( + x) 4. ( log( + x) ) (n) = ( ) n (n )!( + x) n
5 6 log( + x) 0 n p n (x) p n (x) = log( + 0) + ( + 0) x + = x x + x3 xn + ( )n 3 n ( + 0) x + a ( + 0) 3 x 3 3! lim p n(a) = log( + a) < a n + + ( )n (n )!( + 0) n x n n! x x a a > p n (a) n log( + a) a > a n lim n n = 4 0 n=0 a n n a n 0 p n (a) n log( + a) < a p n (a) log( + a) R n+ (x) = log( + x) p n (x) < a lim R n+(a) = 0 n R n+ (x) R n+(x) = + x + x x + ( ) n x n = ( + x) ( x + x + ( ) n x n ) + x = ( ) n xn + x R n+ (0) = 0 R n+ (a) = R n+ (a) R n+ (0) = R n+ (c)a = ( ) n + c a c 0 a < a < c < n c n 0 R n+ (a) 0 cn 0 e x sin x, cos x S S N = N n=0 a n a N = S N S N+ N S S = 0
5 7 0 e x e x x x x3 3! xn n! = ec x n+ (n + )! c 0 x e c x n+ (n + )! e x x n+ (n + )! n 0 x e x 0 x e x 4 log( + x) 0 x < x < 0 4 4 4 f f = g f (k) = g (k ) f(x) 0 n f(x) = f(0) + f (0)x + f (0) x + f (0) x 3 + + f (n) (0) + R n+ (x) 3! n! g(x) = f (0) + f (0)x + f (0) x + + f (n) (0) (n )! xn + R n+(x) = g(0) + g (0)x + g (0) x + + g(n ) (0) (n )! xn + R n+(x) f(x) n f n f n f n log( + x) /( + x) ( x) n = ( + x)( x + x + ( ) n x n ) + x = x + x + ( ) n x n + ( ) n xn + x () x /( + x) 0 n lim x 0 x n x n + x = lim x 0 x + x = 0 0
5 8 () 0 x x log( + x) = x x + x3 xn + ( )n 3 n + t n 0 + t dt log( + x) 0 n R n+ (x) = x 0 t n + t dt n 0 0 Arctan x 0 x Arctan x () x x + x = x + x 4 + ( ) n x n + ( ) n xn + x n n 0 0 x Arctan x = x x3 3 + x5 xn + ( )n 5 n + ( )n x 0 t n + t dt n = n ) t n + t tn x 0 t n + t dt x 0 t n dt = x n+ n + x n 0 x Arctan x 0 Arctan x Arctan x [, ] 0 x > 5 a > 6 < X log( + X) = X X + X3 3 X4 Xn + + ( )n 4 n +
5 9 5 X = /x < /x x < x ( log + ) = x x x + 3x 3 4x 4 + + ( )n nx n + ( x x log + ) = x x x x + x x x 3x 3 + x 4x 4 ( )n x nx n = 3x + 4x + ( )n (n + )x n + a n = ( )n n + R n U R m f : U R m n m f n n n = n n 3 n = m = 0 () : f : U R U R (a, b) U A B f(a + u, b + v) f(a, b) Au Bv lim = 0 (u,v) (0,0) u + v (A B) (Df) (a,b) f (a, b). T (v) ( u ) v u Au + Bv Au + Bv = (A B) T (v) v T (A B) (A B) (Df) (a,b) ( ) u () u = R v d f(a + ut, b + vt) f(a + ut, b + vt) f(a, b) dt = lim t 0 t=0 t f (a, b) u f (a, b) (a, b) u (Df) (a,b) ( u) = Au + Bv A, B ()
5 0 (3) (a, b) (a, b) (4) f(x, y) (a, b) x f f(a + t, b) f(a, b) (a, b) := lim x t 0 t f x (a, b) f (a, b) y f f(a, b + t) f(a, b) (a, b) := lim y t 0 t f y (a, b) U (a, b) () f (a, b) (Df) (a,b) f (a, b) (a, b) f Df. i j x (, ) y (, ) (i, j) ( ) e j i x 0 ( ) 0 y (5) (a, b) x y A = f (a, b), x B = f (a, b) y A, B (). f : U R m U R n f m n f i (x,..., x n ) f p = (a,..., a n ) U (Df) p () T (Df) p ( v) = f x (p) f x (p). f m x (p) f f x (p) x n (p) f f x (p) x n (p)..... f m x (p) f m x n (p) v v. v n v = ( ) f f (a, b) (a, b) x y () (A B) (6) f f C v v. v n
5 7 7 () x x y 3 = xy 3 y x3 y 3 = 3x y () x sin(x y 3 ) = xy 3 cos(x y 3 ) y sin(x y 3 ) = 3x y cos(x y 3 ) (3) y xy = yx y y xy = y ey log x = (log x)e y log x = x y log x (4) x + y x = x (x + y ) = (x + y ) x x = x + y x y y x + y = y x +y (4) x y y f(x, y) 7 8 f(x, y) g(x, y) 5 (8) g(x, y) x + y g(x, y) x + y = (x, y) g(x, y) = ( x y )f(x, y) = 0 f(x, y) = 0 x +y < g(x, y) > 0 (x, y) x + y < g(x, y) < 0 (x, y) x + y < g(x, y) 0 x + y = a + b < (a, b) g(x, y) x + b < φ(x) = g(x, b) x + b <
5 x = a φ(x) φ (a) = 0 g(x, y) x y f(x, y) φ(x) φ (x) = g x (x, b) (a, b) g x (a, b) = 0 y ψ(y) ψ(y) = g(a, y) a + y < ψ(y) y = b ψ (b) = 0 ψ (y) = g y (a, y) g y (a, b) = 0 g g (a, b) = x y (a, b) = 0 a + b < 0 f(x, y) (a, b) f f (a, b) = (a, b) = 0 x y 9 () f(x, y) = { x x > 0 y > 0 0 f y (x, y) = 0 f(, ) = f(, ) = 0 f y () f(x, y) (x 0, y 0 ), (x, y ) f(x 0, y 0 ) = f(x, y )
5 3 (x 0, y 0 ) (, y 0 ) D f(x, y 0 ) x f(x 0, y 0 ) f(, y 0 ) = f x (c, y 0) c x 0 f x 0 0 f(x 0, y 0 ) = f(, y 0 ) (, y 0 ) (, y ) D f(, y) y f(, y 0 ) f(, y ) = f y (, c ) c y 0 y f y 0 0 f(, y 0 ) = f(, y ) (, y ) (x, y ) D f(, y ) = f(x, y ) f(x 0, y 0 ) = f(, y 0 ) = f(, y ) = f(x, y ) 0 ()
5 4 C = = = 0 0 (0, 0) C (0, 0) 0 (x, y) ( ) sin x + y x + y < ε = f(x, y) < ε f(x, y) (0, 0) x y f(x, y) x y x ( ) f(x, 0) f(0, 0) x + 0 sin ( ) x lim = lim +0 x = lim x 0 x x 0 x x 0 x sin x (0, 0) x z = x sin x x = 0 z = 0 x 0 z () f(x, y) x y x f(x, 0) f(0, 0) lim = lim x 0 x x 0 x 0 x +0 0 = lim 0 = 0 x x 0
5 5 x y f f (0, 0) = (0, 0) = 0 x y ( ) u () (0, 0) u = φ(t) := f(ut, vt) t v φ(t) t = 0 f φ(t) t 0 φ(t) = (ut)(vt) (ut) + (vt) = uv u + v φ(0) = 0 t 0 uv/(u + v ) t = 0 0 u v 0 uv 0 φ(t) 0 t = 0 f (0, 0) u (3) (0, 0) lim f(x, y) = f(0, 0) (x,y) (0,0) (x, y) (0, 0) f(0, 0) (x, y) (0, 0) f(0, 0) y = x (x, y) (0, 0) xx lim f(x, x) = lim x 0 x 0 x + x = 0 = f(0, 0) f(x, y) = c c (0, 0) ( ) u u = f (0, 0) u v ()
5 6 u φ(t) := f(ut, vt) t φ(t) t = 0 t 0 φ(t) = (ut) (vt) (ut) 4 + (vt) = u vt u 4 t + v φ(0) = 0 φ (0) v 0 φ(t) φ(0) u v lim = lim t 0 t t 0 u 4 t + v = u v v = 0 φ(t) 0 φ (0) = 0 (0, 0) (3) (x, y) (0, 0) f(x, y) f(0, 0) y = x x 0 f(x, x ) = x x x 4 + (x ) = x 0 f(0, 0) = 0 f(x, y) f(x, y) = c 3 () f(0, 0) = 0 ε x + y < δ = f(x, y) < ε δ f(x, y) = (x y) 3 ( x + y )3 x + y x + y x x + y y x + y x y x + y ( x + y ) 3 f(x, y) x + y = 8 x + y
5 7 δ = ε 8 x + y < δ f(x, y) 8 x + y < 8δ = ε (0, 0) () f x (0, 0), f y (0, 0) f(x, y) f f(x, 0) f(0, 0) (0, 0) = lim = lim x x 0 x x 0 x x = lim = x 0 x 3 f f(0, y) f(0, 0) ( y) 3 (0, 0) = lim = lim y y 0 y y 0 y y = lim( ) = y 0 (3) f(x, y) (0, 0) P (x, y) P (x, y) = (x 0) + ( )(y 0) + f(0, 0) = x y (x y)3 f(x, y) P (x, y) = x + y (x y) = (x y)((x y) (x + y )) x + y xy(x y) = x + y y = x, x > 0 (x, y) (0, 0) f(x, y) P (x, y) lim = lim (x,y) (0,0) x + y x +0 y= x x>0 4x 3 x x = lim x +0 4x 3 x 3 = 0 f(x, y) (0, 0)? No g(x, y) = y(3x y ) x + y (x, y) (0, 0) 0 (x, y) = (0, 0) (0, 0)
5 8 4 (0, 0) (0, 0) (0, 0) x y f(x, y) (0, 0) ( ) f f(x, 0) f(0, 0) (0, 0) = lim = lim x sin x x 0 x x 0 x θ sin θ ( ) 0 x sin x x ( ) lim x sin x 0 x = 0 f (0, 0) = 0 x f(x, y) x y f (0, 0) = 0 y f(0, 0) = 0 0 f(x, y) (0, 0) lim (x,y) (0,0) f(x, y) x + y = 0 f(x, y) lim x + y sin (x,y) (0,0) ( ) x + y 0 f(x, y) (0, 0) f x (x, y) (0, 0) (x, y) (0, 0) f x (x, y) ( ) f (x, y) = x sin x x + y x ( ) x + y cos x + y (0, 0) ε δ a + b < δ f x (a, b) f x (0, 0) ε (a, b)
5 9 ( f x (a, b) f x (0, 0) = a sin a + b ) a ( ) a + b cos a + b (a, b) a = 0 0 b = 0 b = 0 ( ) f x (a, 0) f x (0, 0) = a sin a ( ) a cos a a a n = nπ a = a nπ a > 0 f x ( ), 0 nπ f x (0, 0) = nπ n a δ n /(δ π) a = / nπ a + 0 < δ f x (a, 0) f x (0, 0) = nπ f x (x, y) (0, 0) f(x, y) x y f y (x, y) (0, 0) C 0 z = x sin x x = 0 z = 0 x 0 z C
5 0 C :