(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

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1 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n m n m n 0 2 = π =

2 (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 = 0, (10) ( ) ( b) = b b + x = b x b b x = b x P (x) x α α P (x) = n ( + b) n = n + n n 1 b + 2 n(n 1) n 2 b 2 n! k!(n k)! k b n k + + b n 1-9. A A x x A minimum A 2

3 A x x A mximum A c A upper bound A x x c A A A A A sup A 2-1. f y = f(x) x y y = f(x) c, A x c f(x) A f(x) A x c lim f(x) = A x c A x c f(x) limit x c+ x c c x c x c c lim x c+ f(x), lim x c f(x) x = c f(x) x = c lim f(x) = f(c) x c f(x) x = c lim f(x) = f(c) x c+ f(x) x = c lim x c f(x) = f(c) f(x) [, b] < c < b c x = x = b 3

4 [, b] f(x) f() f(b) M f(c) = M c [, b] 2-5. x x x x y = x+y ( x ) y = xy 2-6. x x > 1 0 < < > 1 0 < < 1 x = y (, ) (0, ) x y = x y 1 y log x x log x + log y = log xy, log x y = y log x 2-8. y = log x (0, ) (, ) log x > 1 0 < < (1 + 1 n )n n e = lim n (1 + 1 n )n e x 1 lim = 1 x 0 x 4

5 x e xi x = i e 0i = 1 x > i x e xi x < i x e xi x e xi e xi e yi = e (x+y)i, e xi 1 lim x 0 x = i sin x cos x e xi e xi = cos x + i sin x 3-1. y = f(x) f (x) y dy dx (f(x)) 3-2. y = f(x) f (x) x = f () f () = lim x f(x) f() x f () = lim h 0 f( + h) f() h f(x) f () f(x) x = f () f(x) x = f(x) x = x = f(x) 5

6 f (x) f (x) f(x) 3-3. (x ) = x 1 2 (cf(x) + dg(x)) = cf (x) + dg (x), (f(x)g(x)) = f (x)g(x) + f(x)g (x), ( g(x) f(x) ) = f(x)g (x) f (x)g(x) f(x) 2 (f(x) 0 x 3-4. y = f(t) t = g(x) y = f(g(x)) y = f(t) t = g(x) (f(g(x)) = f (g(x))g (x), dy dx = dy dt dt dx 3-5. (e x ) = e x, (log x) = 1 x, (sin x) = cos x, (cos x) = sin x e x 1 lim = 1 x 0 x e ix 1 lim = i x 0 x (e xi ) = ie xi 3-6. f(x) x = f () x = x = +h f( + h) f() x f(x) ( + h) 6

7 h 0 f(x) x = f () > 0 x = f () < 0 x = 3-7. y x x = f(y) x x = f(y) y 1 x = f(y) 1 y y = g(x) x y y = g(x) x = f(y) x = g(y) g (y) > y = f(x) y = f(x) y = 1 g (f(x)), dy dx = 1 dx dy 3-8. x = sin y [ π 2, π ] [ 1, 1] 2 dx dy = cos y > 0 ( π 2 < y < π 2 ) y = sin 1 x x = cos y [0, π] [ 1, 1] dx dy = sin y < 0 (0 < y < π) y = cos 1 x x = tn y = sin x cos x ( π 2, π 2 ) (, ) dx dy = 1 cos 2 y > 0 y = tn 1 x (sin 1 x) = 1, 1 x 2 (cos 1 x) = 1, 1 x 2 (tn 1 x) = x 2 7

8 3-9. y = f(x) f (x), (f(x)), y dy, dx f (x), (f(x)), y d 2 y, dx 2 f () > 0 y = f(x) x = f () < 0 y = f(x) x = n y = f(x) n f (n) (x), y (n) d n y, dx n 4-1. [, b] f(x) (, b) lim f(x) = x + f() x = lim f(x) = f(b) x b f(b) f() x = b = f (ξ) b b ξ x = f(b) f() x = b f (ξ) x = ξ b ξ b 4-2. (, b) f(x) f (x) > 0 ( < x < b) f (x) < 0 ( < x < b) f(x) f (x) C 1 C 1 f(x), x f(x) = f() + f (ξ)(x ) ξ x f(x) x = C 1 f(x) = f() + f ()(x ) + o( x ) o( x ) R(x) lim x x = 0 1 f(x) f() + f ()(x ) y = f() + f ()(x ) y = f(x) 8

9 (, f()) 4-3. f(x) f (x) C 2 C 2 f(x), x f(x) = f() + f ()(x ) f (ξ)(x ) 2 ξ x f(x) x = C 2 f(x) = f() + f ()(x ) + f () (x ) 2 + o((x ) 2 ) 2 o( x 2 ) f(x) x = f(x) f() + f ()(x ) + f () (x ) f(x) n f (n) (x) C n C n f(x), x f(x) = f() + 1 1! f ()(x ) + 1 2! f ()(x ) ! f (3) ()(x ) (n 1)! f (n 1) ()(x ) n n! f (n) (ξ)(x ) n ξ x 4-5. f(x) g(x) f() = g() = 0 x = f(x), g(x) f (x) lim x g (x) = A lim f(x) x g(x) = A lim x f(x) =, lim g(x) = x f (x) f(x), g(x) lim x g (x) = A f(x) lim x g(x) = A 9

o( x 2 ) f(x) x = f(x) f() + f ()(x ) + f () (x ) 2 2 4-4. f(x) n f (n) (x) C n C n f(x), x f(x) = f() + 1 1! f ()(x ) + 1 2!

10 5-1. F (x) = f(x) F (x) f(x) f(x) 1 F (x) f(x) { F (x)+c c } f(x) f(x)dx f(x) 5-2. f(x) f (x) f (x)dx = f(x) + c x dx = x+1 ( 1 ) 1 dx = log x, e x dx = e x x cos xdx = sin x, sin xdx = cos x, (cf(x) + dg(x))dx = c f(x)dx + d g(x)dx, f (x)g(x)dx = f(x)g(x) f(x)g (x)dx, f(x)dx = f(g(t))g (t)dt x = g(t) 1 dx = tn x, cos 2 x y x 5-6. y = f( y x ) u = y x 10

11 5-7. y + P (x)y = Q(x) 1 1 y = e P (x)dx ( e P (x)dx Q(x)dx + c) 6-1. b f(x)dx f(x) x = x = b F (x) f(x) b b f(x)dx, f(x)dx = F (b) F () [, b] C 1 f(x), g(x) b f (x)g(x)dx = [ f(x)g(x) ] b b f(x)g (x)dx C 1 x = g(t) g(α) =, g(β) = b b f(x)dx = β α f(g(t))g (t)dt 6-2. [, b] f(x) [, b] n : = x 0 < x 1 < x 2 < < x n 1 < x n = b i [x i 1, x i ] f(x) m i M i x i x i 1 m i n i s( ) x i x i 1 M i n i S( ) s( ) s S( ) S s S s = S f(x) [, b] [, b] f(x) f(x) [, b] b f(x)dx [, b] 6-3. [, b] f(x) G(t) = 11 t f(x)dx G (t) =

12 f(t) G(x) f(x) b f(x)dx = G(b) f(x) 6-4. [α, β] x = f(t), y = g(t) (x, y) (f(t), g(t)) t β f (t) 2 + g (t) 2 dt α 7-1. (x, y) z = f(x, y) 2 z = f(x, y) y x 1 z z x x x y 1 z y z y z x, z y f x(x, y), f y (x, y) z = f(x, y) f x (x, y) f y (x, y) (x, y) = (.b) f( + h, b) f(, b) f(, b + k) f(, b) f x (, b) = lim, f y (, b) = lim h 0 h k 0 k f x (, b) f y (, b) z = f(x, y) (x, y) = (, b) f x (, b) f y (, b) (x, y) = (, b) x y f x (, b) (, b) x f y (, b) (, b) y 7-2. z = f(x, y) x = h(t), y = k(t) z = f(h(t), k(t)) z = f(x, y) x = h(t) y = k(t) z = f(h(t), k(t)) 12

b) f( + h, b) f(, b) f(, b + k) f(, b) f x (, b) = lim, f y (, b) = lim h 0 h k 0 k f x (, b) f y (, b) z = f(x, y) (x, y) = (, b) f x (, b)

13 ϵ(x, y) ϵ(x, y) (x ) 2 + (y b) 2 0 (x )2 + (y b) 2 0 (x ) 2 + (y b) 2 ϵ(x, y) = o( (x ) 2 + (y b) 2 ) 2 f(x, y) f(x, y) = f(, b) + A(x ) + B(y b) + o( (x ) 2 + (y b) 2 ) A B (, b) f(x, y) (, b) (, b) f(x, y) = f(, b) + f x (, b)(x ) + f y (, b)(y b) + o( (x ) 2 + (y b) 2 ) z = f(x, y) z x, z y C 1 f(x, y) (, b) C 1 (, b) z = f(x, y) C 1 x = g(t), y = h(t) z = f(g(t), h(t)) dz dt = z dx x dt + z dy dy dt z = f(x, y) C 1 x = g(u, v), y = h(u, v) z = f(g(u, v), h(u, v)) z u = z x x u + z y y u, z v = z x x v + z y y v 7-3. z = f(x, y) x z x f x(x, y) x 2 z x 2 f xx(x, y) y 2 z y x f xy(x, y) z = f(x, y) y z y f y(x, y) x 2 z x y f yx (x, y) y 2 z y f yy(x, y) 2 13

C 1 x = g(t), y = h(t) z = f(g(t), h(t)) dz dt = z dx x dt + z dy dy dt z = f(x, y) C 1 x = g(u, v), y = h(u, v) z = f(g(u, v), h(u, v)) z u = z x x u + z y y u, z v =

14 2 z x, 2 z 2 y x, 2 z x y, 2 z y 2 f xx(x, y), f xy (x, y), f yx (x, y), f yy (x, y) z = f(x, y) z = f(x, y) C 2 z = f(x, y) (, b) C 2 f yx (, b) = f xy (, b) 7-4. (, b) C 2 f(x, y) f(x, y) f(, b) + f x (, b)(x ) + f y (, b)(y b) f xx(, b)(x ) 2 + f xy (, b)(x )(y b) f yy(, b)(y b) C 2 f(x, y) (x, y) = (, b) f x (, b) = f y (, b) = 0 f xx (, b)f yy (, b) f xy (, b) 2 > 0 f xx (, b) > 0 f(x, y) (, b) f xx (, b)f yy (, b) f xy (, b) 2 > 0 f xx (, b) < 0 f(x, y) (, b) 7-6. (, b) C 1 F (x, y) F (, b) = 0 F y (, b) 0 x = y = f(x) f() = b F (x, f(x)) = 0 C 1 F (x, y), G(x, y) F (x, y) Λ = { (x, y) G(x, y) = 0 } (, b) G x (, b) 2 + G y (, b) 2 0 H(x, y, λ) = F (x, y) λg(x, y) H x (, b, λ) = H y (, b, λ) = 0 λ G(x, y) = 0 F (x, y) H(x, y, λ) 8-1. [, b] g(x), h(x) h(x) g(x) ( x b) D = { (x, y) h(x) y g(x), x b } f(x, y) f(x, y) dxdy f(x, y) D D 14

C 2 f(x, y) (x, y) = (, b) f x (, b) = f y (, b) = 0 f xx (, b)f yy (, b) f xy (, b) 2 > 0 f xx (, b) > 0 f(x, y) (, b) f xx (, b)f yy (, b) f xy (, b) 2 > 0 f xx (, b) < 0 f(x, y) (, b) 7-6.

15 2 D f(x, y) dxdy = b { g(x) h(x) f(x, y) dy}dx f(x, y) 0 z = f(x, y) (x, y) D [, b] [c, d] = { (x, y) x b, c y d } 2 f(x, y) [, b] = x 0 < x 1 < x 2 < < x n = b [c, d] c = y 0 < y 1 < y 2 < < y m = d [, b] [c, d] : = x 0 < x 1 < x 2 < < x n = b, c = y 0 < y 1 < y 2 < < y m = d (i, j) [x i 1, x i ] [y j 1, y j ] = { (x, y) x i 1 x x i, y j 1 y y j } f(x, y) M ij m i,j (x i x i 1 )(y j y j 1 ) M ij S( ) m ij s( ) S( ) S s( ) s s S s = S f(x.y) [, b] [c, d] f(x.y) [, b] [c, d] f(x, y) dxdy [,b] [c,d] D D 1 D 0 1 D (x, y) D [, b] g(x), h(x) D f(x, y) f(x, y)1 D (x, y) f(x, y)1 D (x, y)dxdy f(x, y) dxdy [,b] [c,d] D C 1 u = g(x, y), v = h(x, y) (x, y) D (u, v) E 1 1 E f(u, v) f(g(x, y), h(x, y)) E f(u, v) dudv = D (u, v) f(g(x, y), h(x, y)) (x, y) dxdy 2 15

< y m = d (i, j) [x i 1, x i ] [y j 1, y j ] = { (x, y) x i 1 x x i, y j 1 y y j } f(x, y) M ij m i,j (x i x i 1 )(y j y j 1 ) M ij S( ) m ij s( ) S( ) S s( ) s s S s = S f(x.

16 (u, v) (x, y) = (x, y) u v (x, y) x u v (x, y) x (x, y) x (x, y) y = u v x y u v y x u = g(x, y), v = h(x, y) 8-4. (u, v) D = { (u, v) u b, c y d } x = f(u, v), y = g(u, v), z = h(u, v) C 1 S : x = f(u, v), y = g(u, v), z = h(u, v) ((u, v) D) (y, z) (z, x) (x, y) ( (u, v) )2 + ( (u, v) )2 + ( (u, v) )2 dudv D 8-5. n f(x 1, x 2,, x n ) n D f(x 1, x 2,, x n )dx 1 dx 2 dx n n n n n 9-2. lim x f(x) = A ϵ-δ ϵ > 0 0 < x < δ x f(x) A < ϵ δ > lim f(x) = A lim g(x) = B (1) lim(f(x)+g(x)) = x x x A + B (2) lim cf(x) = ca (c ) (3) lim f(x)g(x) = AB (4) A 0 x x g(x) lim x f(x) = B A 9-5. lim n n = ϵ-n ϵ > 0 n N n < ϵ N 16

v) )2 + ( (u, v) )2 + ( (u, v) )2 dudv D 8-5. n f(x 1, x 2,, x n ) n D f(x 1, x 2,, x n )dx 1 dx 2 dx n n n n n 9-2.

17 9-6. lim n n = lim n b n = b (1) lim n ( n + b n ) = + b (2) lim c n n = c (3) lim n b n n = b (4) 0 lim b n n n = b lim n n = lim = n n n ϵ N ϵ-δ 9-7. lim f(x) = A lim n =, n (n = x n 1, 2, 3, ) n lim f( n ) = A n (x, y) (, b) (x ) 2 + (y b) 2 0 ( n, b n ), n = 1, 2, 3, ( 0, b 0 ) lim (n 0 ) 2 + (b n b 0 ) 2 = 0 n lim f(x, y) = A ϵ δ ϵ > 0 0 < (x,y) (,b) (x )2 + (y b) 2 < δ (x, y) f(x, y) A < ϵ δ , 2,, n, ϵ n, m N n m < ϵ N 17

18 10-4. [, b] x x δ(x) U δ(x) (x) = (x δ(x), x+δ(x)) [, b] U δ(x1 )(x 1 ), U δ(x2 )(x 2 ),, U δ(xk )(x k ) (1),(2),(3),(4) R R + (1) R R (2) R + R + (3) R R + (R R + = ) (4) R R + (R R + = R) R, R + R R [, b] f(x) (1) (2)[, b] (3)f() < M < f(b) f() > M > f(b) f(c) = M c ( c b) (4) ϵ x x < δ, x, x b f(x) f(x ) < ϵ ϵ δ ( D D D D (x, y) (x, y) δ(x, y) U δ(x,y) (x, y) U δ(x1,y 1 )(x 1, y 1 ), U δ(x2,y 2 )(x 2, y 2 ),, U δ(xk,y k )(x k, y k ) D D f(x, y) (1) (2) D (3) ϵ (x x ) 2 + (y y ) 2 < δ, (x, y) D, (x, y ) D f(x, y) f(x, y ) < ϵ ϵ δ 18

[, b] f(x) (1) (2)[, b] (3)f() < M < f(b) f() > M > f(b) f(c) = M c ( c b) (4) ϵ x x < δ, x, x b f(x) f(x ) < ϵ ϵ δ ( 10-7.

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0. () 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >