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1 c /(21)

2 n n α { n } α α { n } lim n = α, n α n n ε n > N n α < ε N {1, 1, 1,, ( 1) n 1, }, {1, 2, 3,, n, } n { n } lim n =, n n n n { n } { n } n n+1 [ n n+1 ] n N { n } ( ) ( ) 1. n = (1 + 1/n) n n N n < n+1 n < 3 { n } e { n } { n1, n2, n3,, ni, } n 1 < n 2 < n 3 < < n i < { n } { n } { n } 2. n = 1/n + ( 1) n n 1 < n < 2 { n } 2n 1 1 n 2n 1 n 3. { n } ε N c /(21)

3 m n < ε m, n > N 2 α n α { n } { n} { n } + { n } lim n [ lim n ] n n lim n = lim sup{ k k n} lim n = lim inf{ k k n} n n n n R D D D x y y x y = f (x) y = g(x) D y f (D) = { f (x) x D} y = f (x) z = g(y) f (x) g(y) z = g( f (x)) f g z = g f x y y f (x) I x 1 x 2 x 1 < x 2 f (x 1 ) f (x 2 ) [ f (x 1 ) f (x 2 )] f (x) I f (x) y = f (x) I x 1, x 2 x 1 x 2 f (x 1 ) f (x 2 ) f (x) 1 1 f (I) y y = f (x) x I f (I) I x = f 1 (y) f (x) x y y = f (x) y = f 1 (x) y = x 2 ε δ c /(21)

4 f (x) A < ε 0 < x x 0 < δ A x x 0, lim f (x) = A, f (x) A x x 0 x x0 x = x 0 f (x 0 ) lim f (x) x x 0 K δ f (x) > K [ f (x) < K] 0 < x x 0 < δ x x 0 f (x) ( ) ( ) lim f (x) = [ ], f (x) [ ] x x 0 x x 0 x x 0 x 0 x 0 lim f (x) [ lim f (x)] x x 0+0 x x 0 0 ε K f (x) A < ε x > K [ f (x) A < ε x < K ] lim f (x) [ lim f (x)] = A x x L K f (x) > L x > K [ f (x) < L (x > K)] lim f (x) = [ ] lim f (x) x x f (x) x = x 0 lim f (x) = f (x 0 ) f (x) x = x 0 x x 0 f (x 0 ) x x 0 f (x 0 ) lim f (x) [ lim f (x)] = f (x 0) f (x) x 0 x x 0+0 x x 0 0 f (x) I f (x) I I b x = [x = b] 4. f (x) [, b] f () < k < f (b) f () > k > f (b) f (c) = k < c < b c 5. f (x) [, b] f (x) [, b] 6. f (x) [, b] ε c /(21)

5 δ x, x x x < δ, x, x [, b] f (x) f (x ) < ε c /(21)

6 x = f (x) f ( + h) f () lim h 0 h (1 1) f (x) x = x = f (x) f () h 0 0 (1 1) f () f +() = lim h +0 f ( + h) f (), f f ( + h) f () h () = lim h 0 h (1 2) f + () f () f (x) I f (x) I I f () f (x) d(x)/dx f f 7. f (x) x = f ( + h) = f () + Ah + o(h) A h 7 f (x) x = f (x) x = y = f (x) I g(y) J f (x) J g( f (x)) I dg( f (x)) dx = g ( f (x)) f (x) y = f (x) x = f 1 (y) f (x) 0 y dx dy = 1 dy/dx = 1 f (x) f (x) 0 x = ϕ(t) y = ψ(t) ϕ(t) ϕ (t) 0 y x c /(21)

7 dy dx = ψ (t) ϕ (t) f (x) [, b] (, b) f () = f (b) f (c) = 0 < c < b c : x = b f (x) 9. f (x) [, b] (, b) f (b) f () b = f (c) < c < b c 10. f (x) g(x) [, b] (, b) g() g(b) (, b) f (x) g (x) 0 f (b) f () g(b) g() = f (c) g (c) < c < b c f (x) n f (x) n f (n) (x) d n f (x)/dx n n f (x) n C n C 11. f (x) g(x) x = g (x) 0 f (x) g(x) 0 x lim f (x)/g (x) ± lim f (x)/g(x) x x f (x) lim x g(x) = lim f (x) x g (x) f (x) g(x) ± x = ± 12. f (x) [, b] n 1 (, b) n c < c < b c /(21)

8 f (b) = f () + f () 1! (b ) + f () (b ) f (n 1) () 2! (n 1)! (b )n 1 + R n R n = f (n) (c) n! (b ) n R n = f (n) (c) n! (b c) n 1 (b ) : n = 1 f (x) x = I C R n 0 n f (x) I f (x) = f () + f ()(x ) + f () 2! (x ) f n () (x ) n + n! x = = f (x) c (c ε c + ε) ε f (x) < f (c) x c [ f (x) > f (c) x c ] f (x) x = c f (c) 13. (i) f (x) x = c f (c) = 0 (ii) f (x) x = c x = c ε f (x) > 0 [ f (x) < 0] c ε < x < c f (x) < 0 [ f (x) > 0] c < x < c+ε f (x) x = c c /(21)

9 f (x) F (x) = f (x) F(x) f (x) F(x) = f (x)dx f (x) F(x) f (x)dx f (x)dx = F(x) + C C : C : 1 1 x 1 1 x α dx = xα+1 α + 1 α 1 sin xdx = cos x dx 2 x = x 2 sin 1 > 0 dx x = log x cos xdx = sin x dx x2 + A = log x + x 2 + A A 0 e αx dx = eαx α α 0 dx cos 2 x = tn x dx x 2 + = 1 x 2 tn x 2 dx = 1 2 (x 2 x sin 1 x ) > 0 x2 + Adx = 1 2 (x x 2 + A + A log x + x 2 + A ) A 0 ϕ(t) C 1 f (x)dx = f (ϕ(t))ϕ (t)dt (x = ϕ(t)) f (x) g(x) C 1 f (x)g (x)dx = f (x)g(x) f (x)g(x)dt c /(21)

10 2 R(X) R(X Y) X Y A. R(sin x) cos xdx = R(t)dt sin x = t R(cos x) sin xdx = R(t)dt cos x = t B. R(cos x sin x)dx = R( 1 t2 1 + t 2t t ) 2 dt 2 tn x 1 + t2 2 = t C. R(cos x sin x) = R( cos x sin x) tn x = t R(cos x sin x)dx 3 R(X Y) X Y x + b n A. R(x n )dx d bc 0 n = 2, 3, (x + b)/(cs + d) = t cx + d B. R(x x2 + bx + c)dx (i) > 0 x2 + bx + c = t x (ii) < 0 x2 + bx + c = (x α)(β x) = (β x) (x α)/(β x) (x α)/(β x) = t α < β f (x) I = [, b] I x 0, x 1, x 2,, x n = x 0 < x 1 < x 2 < < x n = b I k = [x k 1 x k ], x k = x k x k 1 k = 1, 2, 3,, n, = mx x k k I k ξ k R[ f {ξ k }] = n f (ξ k ) x k k=1 0, {ξ k } R[ f {ξ k }] J f (x) [, b] J f (x) [, b] J b f (x)dx c /(21)

11 2 I k f (x) M k m k S = M k x k, s = m k x k S R[ f,, {ξ k }] s S s S = inf S s = sup s S = s [0, 1] f (x) = 1 x : 0 x : [0 1] M k = 1 m k = 0 S = 1 s = f (x) [, b] f (x) [, b] 15. f (x) [, b] f (x) F(x) = x f (t)dt [, b] F (x) = f (x) F(x) f (x) f (x) G(x) b f (x)dx = G(b) G() 3 f (x) (, b) b b b ε lim f (x)dx, lim ε +0 +ε ε +0 b ε f (x)dx, lim f (x)dx ε ε +0 +ε b f (x)dx (, b] [, ) (, ) b b b f (x)dx = lim f (x)dx, f (x)dx = lim f (x)dx, b f (x)dx = lim,b b f (x)dx 16. f (x) [, b) b= f (x)dx x2 f (x)dx x1 f (x)dx = x2 x 1 f (x)dx 0 x 1, x 2 b 0 b f (x) dx b b f (x)dx f (x)dx b c /(21)

12 R 2 D D 2 D P(x, y) z z = f (P), z = f (x, y) D x y z P(x, y) P 0 (, b) P 0 f (x, y) c P P 0 f (x, y) c lim f (x, y) = c, lim f (P) = c, f (x, y) c (x, y) (, b) (x,y) (,b) P P 0 f (, b) lim f (P) P P 0 1 f (, b) lim f (x, y) = f (, b) (x,y) (,b) f (x, y) (, b) D f (x, y) D f ( + h b) f (, b) lim h 0 h (1 3) f (x, y) (, b) x (1 3) (, b) f x (, b), f x(, b) (1 4) f (x, y) D x f x (x, y) D f (x, y) x y f y (, b), f y (, b) f y(x, y), f y (x, y) f x (x, y), f y (x, y) x y f xx (x, y), f xy (x, y), f yx (x, y), f yy (x, y) c /(21)

13 f (x, y) n f (x, y) C n n f (x, y) C n f (x, y) C f xy f yx f xy = f yx 3 f xxy = f xyx = f yxx. 2 f (x, y) (, b) f = f ( + h b + k) f (, b) = f x (, b)h + f y (, b)k + o( h 2 + k 2 ) (1 5) f (x, y) (, b) f (x, y) (, b) d f = f x (, b)dx + f y (, b)dy f (x, y) (, b) (, b) f x f y (, b) f (x, y) (, b) C 1 f (x, y) x = x(t), y = y(t) f (x, y) t d f dt dx = f x dt + f dy y dt 17. f (x, y) D C n {( + ht b + kt) ( t b)} D n 1 1 f ( + h b + k) = k! (h x + k y )k f (, b) + 1 n! (h x + k y )n f ( + θh b + θk) (0 < θ < 1) k=0 n = x y F(x, y) = 0 y y y = f (x) x y = f (x) F(x, y) = 0 f (x) F(x, y) (, b) C 1 F(, b) = 0 F y (, b) 0 x = c /(21)

14 b = f (), F(x f (x)) = 0 C 1 y = f (x) f (x) f (x) = f x (x, y)/ f y (x, y) D 2 f (P) D P 0 u ε (P 0 ) D P 0 P f (P 0 ) > f (P) [ f (P 0 ) < f (P)] u ε (P 0 ) f (P) P 0 f (P 0 ) P 0 f (x, y) (, b) f x (, b) = f y (, b) = f (x, y) (, b) C 2 f x (, b) = f y (, b) = 0 (, b) = f xx (, b) f yy (, b) f xy (, b) 2 (i) (, b) > 0 f xx (, b) > 0 [< 0] f (, b) (ii) (, b) < 0 f (, b) 20. f (x, y) g(x, y) C 1 g(x, y) = 0 f (x, y) (, b) (, b) g(x, y) (g x (, b) 2 + g y (, b) 2 0) f x (, b) λg x (, b) = 0, f y (, b) λg y (, b) = 0 λ c /(21)

15 f (x, y) D D K = {(x, y) x b c y d} f (x, y) = 0 ( x, y) K D K [, b], [c d] = x 0 < x 1 < x 2 < < x m = b c = y 0 < y 1 < y 2 < < y n = d m, n = {k i j }, k i j = {(x, y) x i 1 x x i y j 1 y y j }, 1 i m 1 j n x i = x i x i 1 y j = y j y j 1 = mx{ x i y j } K i j (ξ i η j ), {(ξ i η j )} f (x, y) R[ f {(ξ i, η j )}] = f (ξ i η j ) x i y j 1 0, (ξ i η j ) J f (x, y) D J f (x, y)dxdy D D 3 f (x, y, z) f (ξi η j ζ k ) x i y j z k (ξ i η j ζ k ) f (x, y, z) D 3 f (x, y, z)dxdydz D 1 K i j f (x, y) M i j m i j S s S = inf M i j x i y j, s = sup m i j x i y j S = s K = {(x, y) x b c y d} f (x, y) K K f (x, y)dxdy = d b c f (x, y)dxdy = b d c f (x, y)dydx 1 2 : f (x, y) f (x, y)dxdy K ϕ 1 (x) ϕ 2 (x) [, b] D = {(x, y) x b ϕ 1 (x) y ϕ 2 (x)} f (x, y) D c /(21)

16 f (x, y)dxdy = b ϕ2(x) D ϕ 1(x) f (x, y)dydx uv E C 1 x = x(u, v) y = y(u, v) xy D 1 1 f (x, y)dxdy = f (x, y) J(u, v) dudv D E J(u, v) (x, y) J(u, v) = (u, v) = x/ u x/ v y/ u y/ v x = r cos θ y = r sin θ J(r θ) = r dxdy = rdrdθ uvw E C 1 x = x(u, v, w) y= y(u, v, w) z = z(u, v, w) xyz D 1 1 f (x, y, z)dxdydz = f (x, y, z) J(u, v, w) dudvdw D E (x, y, z) x/ u x/ v x/ w J(u, v, w) = = y/ u y/ v y/ w (u, v, w) z/ u z/ v z/ w x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ dxdydz = r 2 sin θdrdθdϕ D f (x, y) D 1 D 2 D 3 D D D n D {D n } J = lim f (x, y)dxdy J {D n D n } f (x, y) D n J f (x, y)dxdy D f (x, y) D {D n } f (x, y) D {D n } f (x, y) D f (x, y) D f (x, y) D 1 1 : (sin x/x)dxdy = π/2 sin x/x dxdy = (sin x/x)dxdy D = {(x, y) x 0 1 y 0} D c /(21)

17 x y = f (x) y, y,, y (n) x F(x, y, y,, y (n) ) = 0 F y (n) n n n dy dx = f (x)g(y) dy/dx x y g(y) 0 1/g(y) dy/dx = f (x) dy g(y) = f (x)dx + C C : g(y) = 0 y 2 dy dx = f ( y x ) dy/dx y/x u = y/x xdu/dx = f (u) u 3 1 dy + P(x)y = Q(x) dx 1 e P(x)dx x y = e p(x)dx ( P(x) Q(x)e dx + C) C : 4 P(x, y)dx + Q(x, y)dy = 0 (1 6) F(x, y) df = 0 c /(21)

18 F(x, y) = C C : (1 6) P/ y = Q/ x Pdx + (Q Pdx)dy = C C : y (1 6) M(x, y) (1 6) M(x, y) D = d/dx (D n + p n 1 D n p 0 )y = Q(x) p 0, p 1,, p n 1 : (1 7) n Q(x) = 0 (D n + p n 1 D n p 0 )y = 0 (1 8) (1 7) 21. (1 7) y p (x) (1 8) y c (x) (i) (1 7) y(x) y(x) = y c (x) + y p (x) (ii) F(λ) = λ n + p n 1 λ n p 0 = 0 k λ 1, λ 2,, λ k m 1, m 2,, m k m i = ny c(x) y c (x) = c m1 (x)e λ1 x + C m2 (x)e λ2 x + + c mk (x)e λk x C mi (x) i = 1, 2,, k m i 1 λ = α ± jβ β 0 m e αx {( x + + m 1 x m 1 ) cos βx + (b 0 + b 1 x + + b m 1 x m 1 ) sin βx} c /(21)

19 { n } n=1 { n} n=1 n = n + (1 9) n=1 n n { k } 1 n n n S n {S n } n=1 {S n } S (1 9) S {S n } (1 9) 22. n = n + ε N n > m > N m+1 + m n < ε n n 0 n : n = 1/n n 0 n n 2 n = n + n 0 n= 1, 2, n 23. n n {S n } 24. n, b n K n Kb n n= 1, 2, (i) b n n (ii) n bn 25. n lim n n = r n (i) 1 > r 0 n (ii) r > 1 n 26. n lim n+1/ n = r n (i) 1 > r 0 n (ii) r > 1 n 27. f (x) > 0 [1 ) c /(21)

20 f (n) = f (1) + f (2) + + f (n) + f (x)dx 1 3 n n n n n n n 28. n bn n = b n : 1 1/2 + 1/3 1/4 + = log /3 1/2 + 1/5 + 1/7 1/4 + = 3 2 log n > 0 n 0 n ( 1) n 1 n = ( 1) n 1 n { n } n f n (x) { f n (x)} { f n (x)} fn (x) x = x 0 { f n (x)}, f n (x) x = x 0 I x f n (x) f (x) n, n k=1 f k(x) F(x) n { f n (x)} f n (x) I f (x) F(x) I f n (x) { f n (x)} ε f n (x) f (x) < ε n> N x N = N(ε) { f n (x)} I f (x) f n (x) I { f n (x)} I f (x) f (x) I n=0 n x n = x + 2 x n x n + (1 10) R(> 0) (1 10) x < R x > R R x R = x = 0 x R = 0 0 n x n c /(21)

21 n (i) r = lim n R = 1/r n (ii) r = lim n+1 / n 0 r R = 1/r n n x n R(> 0) ( R R) x : x n=0 0 n t n dt = x n=0 0 nt n dt : ( n=0 n x n ) = n=1 n n x n 1 c /(21)

Chap11.dvi

Chap11.dvi . () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x