( z = x 3 y + y ( z = cos(x y ( 8 ( s8.7 y = xe x ( 8 ( s83.8 ( ( + xdx ( cos 3 xdx t = sin x ( 8 ( s84 ( 8 ( s85. C : y = x + 4, l : y = x + a,

Transcription

1 [ ] 8 IC. y d y dx = ( dy dx ( p = dy p y dx ( ( ( 8 ( s8. 3 A A = ( A ( A (3 A P A P AP.3 π y(x = { ( 8 ( s8 x ( π < x x ( < x π y(x π π O π x ( 8 ( s83.4 f (x, y, z grad(f ( ( ( f f f grad(f = i + j + k x y z i, j, k x, y, z ( f, h (x, y, z h ( f h grad(f f grad(h grad = h h ( x + y + z ( ( (,, grad x + y z +.5 ( y = x 3 sin x ( y = log(x +.6 z x, z y ( 8 ( s84 ( 8 ( s8

2 ( z = x 3 y + y ( z = cos(x y ( 8 ( s8.7 y = xe x ( 8 ( s83.8 ( ( + xdx ( cos 3 xdx t = sin x ( 8 ( s84 ( 8 ( s85. C : y = x + 4, l : y = x + a, x + 4 = x + a,., a. ( a =, C l. (, a. (3, a. (4, a. (5 a =,. ( 8 ( s A =, B =, C = ,. ( A + B A B AB. ( B C. (3 A A. ( 8 ( s83. (x y + dx (x y + 5dy =,..3 ( x y + =, x y + 5 =. ( ( (X, Y,. (3 (. (4 (3, (x, y. x e x y dydx (.,. ( 8 ( s833

3 (. (3. (4,. ( 8 ( s834.4 (A + B 3, (A + I A, B n I n ( 8 ( s84.5 (,,, (,,, (,, R 3 A = A ( 8 ( s84.6 A a, b A, x = a + b, x = Ax, x 3 = Ax,, x n = Ax n x n a, b ( 8 ( s843.7 f(x, y = e x+y Taylor f(x, y = ( + (.8 (cos x r sin x dx, r ( 8 ( s844 ( 8 ( s845.9 x + y = ax x + y + z = a, x, y, z R, ( R ( x = r cos θ, y = r sin θ J = (3 R V (x, y (r, θ ( 8 ( s85. x, f(x f(x = x sin x, ( f(x f (x f (x ( f (x = x f (x = x (3 y = f(x π x π,,,,, (4 x x sin x. A b A = ( Ax = b x ( 8 ( s85 3, b = 3

4 ( A (3 A n, n ( 8 ( s853. ( a, b, c, p, q, r, D = p q r, N = a b c k, (D + N k = D k + M, M =, α, β, γ α β γ ( A n A n = O, A 3 = O, O.3 V W ( 8 ( s854 V = {(x, x, x 3, x 4 x + x + 4x 4 =, x + 5x 5x 3 + 6x 4 = } W = {(x, x, x 3, x 4 x + 5x 4 =, x + 4x 5x 3 + 7x 4 = } ( V W ( V W (3 V + W = {v + w v V, w W } ( 8 ( s855.4 D = {(x, y 4x + y < 4, x >, y > }, xy dxdy D ( 8 ( s856.5 ( {f(xg(x} (n = ( x > h(x = x 4 log x lim h(x x + (3 < m < 4 m, ( h(x m h (m (x, lim x + h(m (x (4 lim x + h(4 (x ( 8 ( s857.6 f(x = x a x, g(x = x e x a > f(x a x a ( y = f(x ( y = g(x 4

5 (3 (4 a a f(xdx a f(xdx = g(xdx a ( 8 ( s87.7 p, q ( p q ( A = Ax = λx λ x p q { a n+ = ( pa n + qb n ( {a n } {b n } b n+ = pa n + ( qb n < p <, < q < {a n } {b n }, a, b lim n a n lim n b n p, q, a, b ( 8 ( s87.8 d y + 4y = f(x dx (a x = y = dy dx = ( f(x = ( f(x = sin x (a y p = x(a cos x + B sin x A, B (3 f(x = N= sin Nx y S x y S x x ( 8 ( s873.9 z = x + yi (x, y ( z i = (3 z i + z = 4 (4 ( z i = z z i z = (4 ( 8 ( s874.3 A p n A X X n B(n, p B(3,.5 ( A r P (r P (r n, r, p n ( P (X = X= 5

6 (3 X P (X B(3,.5 B(8,.5 (4 B(n, p µ n, p ( 8 ( s875.3 f(x, y R {} C x = r cos θ, y = r sin θ (x, y R {} ( f x + f y = f r + f r r + f r θ ( f(x, y r = x + y θ f f x + f =, r = f =, r = f = y f.3 ( 8 ( s88 ( R f p ( lim n b.33 A A = a sin nx dx = (b a π.34 a, b, c B B = x+p x f(tdt = (b > a. p f(tdt (x R. ( 8 ( s88 P AP P b a 4 c ( 8 ( s883 B ( 8 ( s P (,, 3 π : x + y + z = H HP e x + 3y + z = x + y + 3z = 3x + y + az = x log x dx ( 8 ( s89 x = y = z = a ( 8 ( s89 ( 8 ( s x = t sin t, y = cos t ( < t < π y = y(x dy dx d y dx t ( 8 ( s894 6

7 .39 z = x 3 3xy + y 3 S S P (,, S.4 dx e y3 dy x.4 y = y x + y = y(x y( = e π ( 8 ( s895 ( 8 ( s896 ( 8 ( s897.4 f(x x [a, b] f(x m x m f(x a < x < x m x m < x < b [a, b] a < x < x < b x, x f(x f(x x 3 [x, b] [x, x 3 ] x 4 x m f(x f(x f(x O a x x mx x 3 b x [a, b] f(x ( [a, b] x, x u : v = v : w v/u w v a x x b x u v [a, b] f(x f(x f(x f(x > f(x x < x m < b x < x 3 < b x 3 f(x 3 f(x f(x < f(x 3 x < x m < x 3 x < x 4 < x x 4 f(x > f(x 3 x 3 < x m < b x 3 < x 4 < b x 4 x m ( C - -3 ( v/u 7

8 double f(double x;/ f(x / double searchmin(double a,double b / f(x [a,b] xm / { double x, x, fx, fx, t, s, upper, lower; double tolerance =.e-5; / / upper=b; lower=a; t= - (- ; x= - +t; x= -3 -t; fx=f(x; fx=f(x; while({ if(fx>fx{ / f(x>f(x / lower=x; x=x; fx=fx; t= - (- ; x= - -t; if(x-x tolerancereturn x; fx=f(x; }eles{ / f(x<f(x / upper=x; x=x; fx=fx; t= - (- ; x= -3 +t; if(x-x tolerancereturn x; fx=f(x; } } } ( 8 ( s A B = {a + b a A, b B} ( {,, } {, 4, 7} {, } ( A (B C = (A B (A C (3 A, X A = A X = A X (4 A, Y A = A Y = Y Y (5,,, ( x.44 f(x = Sin x + ( 8 ( s899 ( lim x f(x, lim f(x, x lim f(x x x ( f (x (3 f(x ( 8 ( s8 8

9 .45 4(x y ( dxdy, D : y, x y, x + y. D + (x + y ( xydxdydz, E : x, y, x + y + z. E ( 8 ( s E A = 3 λ W (λ = {x R 4 Ax = λx} ( det(λe A ( W (λ {} λ (3 ( λ W (λ.47 A u, v x 6 8 A =, u =, v = 4 ( u, Au, A u ( v, Av, A v ( 8 ( s83, x = (3 x Ax, A x, A 3 x z i = x x x 3. ( 8 ( s84.48 ( f(z = z + z + ( ( f(z (a f(zdz. (b f(zdz. z i = 3 ( 8 ( s85.49 X X X x x X x x = ( x + X x 3 l x x < l x X x x ( X ( X = 7 X l =. x 3 (3 [, 7] f(x = A x + B g(x = ax + b 7 g(x = 7 x f(x = A x + B A, B 9

10 ( 8 ( s86.5 ( {,, 3} 3 ( n = 3 {,, 3} {a, a,, a k }, k 3 a a a k {a,a,,a k } (3 n(n {a,a,,a k } a a a k = n {,,, n} {a, a,, a k }, k n ( 8 ( s87.5 π/4 R(> Im Γ R Γ R Γ ( R, Γ( R, Γ(3 R, Γ (3 R Γ ( R O π 4 Γ ( R R Re ( Γ R dz = Γ R e z G R, P R, C R, S R G R = R e x dx, P R = Γ ( R e z dz, C R = R cos r dr, S R = R ( P R P R π/4 e R cos θ Rdθ sin r dr, θ π 4 4 θ cos θ lim π P R = R (3 (, ( cos x dx sin x dx π e x dx = ( 8 ( s88.5 x 3 ( cos x ( x ( 8 ( s89.53 f(x = ( x + + x f (x f(x x

11 ( f ( ( f ( f( (3 f (f ( ( 8 ( s8.54 X { X = X Y p ( X P (X = i (i =,, X E(X, X V (X p ( (X = i Y P (Y = j X = i (i, j (3 (X = i Y E(Y X = i (i =, (4 Y P (Y = j (j =,,, Y E(Y p.55 ( d y dx + dy dx 6y = 6x3 + 3x 4x + 3 ( 8 ( s8 ( dy dx + xy = x y ( 8 ( s8.56 ( A A = x x y 3 3 x y z 4 x ( B B n B = x n =,, 3,, x.57 f(x = e x sin x ( 8 ( s8 ( f (x x π f (x = ( y = f(x (3 f(xdx ( 8 ( s83.58 (x m dx dt = mx ( ( ( t = x = x

12 ( ( (x (K x dx dt = mx(k x ( K ( t = x = x x x ( 8 ( s83.59 φ = φ = r r = x + y + z, x + y + z ] (x a.6 f(x = exp [ ( f(x f (x f (x = x ( f(x f (x f (x = x ( 8 ( s833 ( 8 ( s834.6 f(x, y = x.6 y.4 x + y = 4, x, y x.6 y D.63 ( 8 ( s835 (4 x ydxdy, D = {(x, y x + y, x, y },, ( R 3 a b a b ( 8 ( s836 a b ( R 3 a b 3 (3 ( 5 ( 8 ( s p p < p < ( X P (X = k, k =,,, 3 ( X E[X] [ (X ] (3 X E E[X] (4 n Y (5 Y ( 8 ( s838

13 .65 z = f(x, y x = r cos θ, y = r sin θ ( z + x ( z = y ( z + r r ( z θ ( 8 ( s x y a D xydxdy D ( 8 ( s83.67 n A A k (k =,, k A A k = AA A }{{} k ( A = A A ( A k(=,, A k A k (A k A A (A k = (A k (A k A k k A ( 8 ( s83.68 x = cos t, y = sin t t f(x, y x, y ( cos x + cosh y ( f(x, y ( 8 ( s83.69 M tr(m n A, B tr(ab = tr(ba.7 f(x = e x sin 3 x ( 8 ( s833 ( n f (n (x ( f(x (3 x y = f(x x ( 8 ( s834 x.7 P = p(x p(x = x x, a, b, c, d f(p(x = p(x a b c d f : P P ( P 3 p(x x ( f (3 {x 3, x, x, } f ( 8 ( s835 3

14 .7 ( f(x = ln( + x, ( < x x n (n =,, 3, ln x x ( + x ( g(x = ln, ( x < x n x (3 ln.73 ( (x a + y = r (a > r > ( 8 ( s836 ( y V ( 8 ( s837.74,, 4,, 3 3 4, 3 A ( A, ( A 3 ( 8 ( s ( x, y x y.76 A = l : y = ax P P ( , B = ( , O = ( ( 8 ( s839 ( AB, B T A B T B ( X O, AX = XA = O X ( 8 ( s83.77 y = x (t, t C(t x = y > m ( C(t ( C(t m t (3 C(t, m x S(t S(t t (4 S(t 4

15 ( 8 ( s83.78 ( lim x (3 lim x x + 4x + 3 x { log(x + + log( 3x + 3x sin 3x ( lim x sin 5x }.79 ( π π ( I(a = x sin xdx π π (x a sin x + dx a.8 ( a = (,,, b = (,, ( 8 ( s83 ( 8 ( s833 ( x = (,, 3, y = (,, (3 A(,, 3, B(3,,, C(, 4, 4 ( 8 ( s834.8 ( 3 ( (3 A 5 B 3 (4 ( ( y = x + x + ( y = + cos x x (3 y = x e e x (x > (4 y = log ( x + x + ( 8 ( s835 ( 8 ( s84.83 ( t = tan x dt sin x, cos x t dx π dx ( + cos x ( 8 ( s84.84 ( x, x, x 3, y, y, y 3, z, z x = 3y + y + y 3, x = y + y 3, y = 3z z, y = az, y 3 = z + 4z x, x z, z z, z x = kx a k k 5

16 ( ( 8 ( s f = x + x + x x 3 + x x 3 M T M ( f x = (x, x, x 3 T A f = x T Ax A ( A p, p, p 3 (3 (x, x, x 3 T = y p + y p + y 3 p 3 = (p, p, p 3 (y, y, y 3 T f y, y, y 3 (p, p, p p, p p 3.86 ( x(x y dy dx + y = ( ( 8 ( s844 dy dx xy = x (3 d y dx + y = sin x ( 8 ( s845 x (.87 T : R 3 R x + 4y + 3z y x + y + z z R 3 a =, a =, a 3 = ( ( R b =, b = ( {a, a, a 3 } R 3 ( (T (a, T (a, T (a 3 = (b, b A 3 A.88 A = ( 8 ( s846 ( A A ( n A n.89 n I n = π sin n x dx ( 8 ( s847 ( I ( n 3 I n = n n I n ( 8 ( s848 6

17 .9 f(x, y = x x sin y cos y ( f x, f y, f xx, f xy, f yx, f yy ( f x (x, y = f y (x, y = (x, y (3 f.9 ( x + x = a x + a x ( x x x + >.9 f(x = (ax + a x, g(x = (ax a x ( 8 ( s849 ( 8 ( s85 ( a x, a x f(x, g(x ( f(x + y f(x, g(x, f(y, g(y ( 8 ( s85.93 (x, y (x, y xy a, b, c, d x = ax + by y = cx + dy ( A = ( k k x y = ( a c b d ( f ( f k ( ( xy x y ( 8 ( s ( 7 ( A = ( 8 ( s854 ( A A ( A A (3 AX = X ( 8 ( s87.96 t > xy D(t : t x + y 4t, x, y xy F (t = + x dxdy + y D(t 7

18 ( x = r cos θ, y = r sin θ F (t rθ ( F (t ( 8 ( s87.97 ( + y y ( dy x + ( y dx y = x.98 ( π ( e iθ dθ ( π e iθ dθ.99 {a n } a = a n+ = 3a n + 4 (n =,, a n + 3 ( 8 ( s873 ( 8 ( s874 ( a n < (n =,, ( {a n } (3 {a n } ( 8 ( s875. ( V W, W W W, W +W V ( 4 R 4 4 a = (,,,, a = (,,,, a 3 = (,,,, a 4 = (,,, 3 a a W a 3 a 4 W W W, W + W ( 8 ( s876. ( R. A = ( ( (3 (4 (a ( ( (c ( ( (e (3 closure (3 ( Z (b ( ( (d ( ( (f (4 interior (4 (3 R [, ] [, ] [, 3] [a, b] = {x R a x b} 3 4. ( 8 ( s877 ( 8 ( s88 8

19 ( ( ( a d e.3 A = B = AB =, b c f. ( ( b = c =, CA = C. ( ( D DA =. (3 A. (4 A..4 xe x dx. ( 8 ( s88 ( 8 ( s883.5 ( D = {(x, y x, y 3} xy. ( xydxdy. D.6,..7 Cramer. ( 8 ( s y + z =, z + x =..9 x, y, z x + 3y + 6z = ( 8 ( s89 x + 5y + x = 6 3x y + z = x + 3y + z = ( 8 ( s89 ( 8 ( s893 f(x, y, z = e x+y+z., x, y, z. n= ( 8 ( s894. n ( ( n(n + n + n=. f(x = x sin x. g(x, y = log(x + y g x (x, y, g y (x, y ( 8 ( s8 ( 8 ( s8 ( 8 ( s83 9

20 .3 ( a A = 3 a A a A A ( ( b, c B = 5 b c 4 ( (, 3 b c B.4 n [, ] f n (x g n (x f n (x = e nx n k= ( 8 ( s84 x k k!, g n(x = e (n x e nx ( x x f n (x g n (x ( x x g n (x n (3 f n (x dx n ( 8 ( s85 x.5 R 3 = x = y : x, y, z R z ( A 3 R 3 T T (x = Ax (x R 3 p R 3 v W T (W Ap t (A v t (A A A ( a, b, c x a + y b + z = C C c.6 ( (4 y = x 3 6x + 9x x y z C ( 8 ( s86 ( ( (3 (4.7 ( (4 ( xdx ( cos xdx (3 dx (4 x ( 8 ( s87 4 (x + 3x + dx ( 8 ( s88

21 .8 8 m/s t ν m/s ν(t = 8 at ( (3 a ( ( ν(t P (t dp (t ν(t = a dt (3 35 m a.9 ( (x +(x dx (. ( x x dx (q > (3 + q ( 8 ( s89 e x cos(axdx (a > ( 8 ( s8 ( A [ ] n A = sgn a p a p a npn n p p p n S p p p n n S n n sgn a. = a B B (n. (R 3 ( b =, b =, b3 = ( 8 ( s8 { b, b, b3 } R 3 ( R 3 { e, e, e 3 } f : R 3 ( e, e, e 3 R 3 ( e, e, e 3 / / 5/ f f : R 3 ( b, b, b3 R 3 ( e, e, e 3 3/ 3/ 7/ R 3 ( e, e, e 3 { e, e, e 3 } R 3. 4l ( 8 ( s8

22 ( x, y V x, y, l ( V x V x y V y (3 y x ( 8 ( s83.3 < x f(x = x + C x = p ( < p xn C n ( lim f(x ( lim f(x x x + (3 f(x f (x (4 C p n (5 p n (6 ( 8 ( s84.4 A, B, C n a. n A n + A B b. n B n + 3 B 3 C c. C C A B, C ( B P ( C P (3 C P 3 (4 B P 4 ( 8 ( s8.5 xyz 4 O(,,, A(,,, B(, 5,, C(,, ( AB yz ( 3 A, B, C x (3 OBC (4 OABC.6 y = x y = x + ( A, B ( ( 8 ( s8 (3 P A B P AB ( 8 ( s83

23 .7 d y dx + dy + ay = (a a > dx ( ( y( =, y ( = (3 y(π = a.8 t[s] f(t F (ω = ( 8 ( s84 f(te jωt dt j = ω [rad/s] a {, < t < a ( f(t = F (ω f(t F (ω, t <, t > a F (ω F (ω { exp( at, t > ( f(t = F (ω f(t F (ω, t < (3 f(t a F (ωe jωa (4 f(at a > ( ω a F a.9 A = ( k 3 k 3 (k, < k <,. ( A. ( A. ( (3 A n = a n c n b n d n ( 8 ( s85 (n =,,, lim n (a n + d n..3 ( e x. (, m, ( < e! 3! + (m!. (3 m 3, ( < e! 3! + (m!.3 f(x, y =. x + y ((x, y (,,. ( 8 ( s8 < (m! < 5 ( 8 ( s8 ( f x + f y. 3

24 f ( < ε <. I(ε = ε x +y x + f y dxdy. (3 lim ε + (sin εi(ε. ( 8 ( s83.3 ( d dx tan (x ( d dx xe x ( 8 ( s x y log(x + y ( 8 ( s83 ( dx ( x π e x sin x dx ( 8 ( s O xyz A, A B, B, l,, a = AA, b = BB, r = OA, s = OB ( a b n ( n A, B P, Q OP, OQ (3 P Q, l, l r, s, a, b (4 A(,,, A (,, B(,,, B (,, 3 (3 ( 8 ( s Z = ai + bj ( (, I = J =, a b, a b ( Z T, a, b, I, J T ( Z + Z T Z Z T a, b, I, J (3 ZZ T a, b, I, J (4 Z, a, b, I, J (5 J Z ( 8 ( s y = sin x + cos x ( x π ( y, y, 4

25 ( y, y x (3 x 3π 4, y x ( 8 ( s836.38,,, (3, y =.39 y, y, dy/dx, d y/dx ( y + 6y + 9y = ( xy + y = 4 (3 y = sin θ cos θ = cot θ x y + cos x x 4y sin y ( 8 ( s837 cos x = sin x ( 8 ( s84.4 x = a(t sin t, y = a( cos t t = t (x, y.4 a x dx, a >, x a cos x = cos x sin x = sin x cos x ( 8 ( s84 ( 8 ( s843.4 x + (y b = a (b > a > x ( 8 ( s x, y, z, w a, b, c x + y + z + 3w = a x + 3y + 5z + w = b 3x + 4y + z + 7w = c ( 8 ( s B x B x + y = 4 ( ( [ ] B =, x = x y.45 ( 8 ( s846 ( dx dt = x t x( = ( x dy dx = y y( = (3 x dy dx = y y( = 5

26 ( 8 ( s ( y = x y ( y = x y (.48 A A = ( 8 ( s848 ( 8 ( s849 4 ( A ( A P A.49 i ( 8 ( s84 ( ( i cos θ + i sin θ 3 i 3 ( 8 ( s84.5 ( dy dx = y ( x(x y dy dx + y = (3 x dy + y = x (4 d y dx dx + y = cos x ( 8 ( s84.5 ( (x 3 y 3 x y 4 3 ( (3 4 log + log 3 + log 4 3 (4 5 log 5 7 (5 log 3 log 7 5 log y = (log x log x + 3 x.5 x 8 ( s = log x s ( y s (3 y x.53 y = a sin x + b cos x ( y = A sin(x + θ ( a =, b = 3 A θ ( 8 ( s843 ( 8 ( s844 (3 x π x ( 8 ( s845 6

27 .54 ( y = x ( y = x + (3 y = cos x sin x (4 y = e x (5 y = x 3x ( 8 ( s ( e x ( x 3 e x (3 x log x ( 8 ( s k m x t k m d x dt + kx = m x k ( ω = x = A cos ωt + B sin ωt m ( m =.6(kg, k = 4(kg/sec (3 dx t = x = (cm, = (cm/sec A, B 3 dt ( 8 ( s P (x, y t x = a cos(πt, y = a sin(πt ( a = e t t t a x ( t P (3 dx dt, dy dt (4 P t ( 8 ( s C 4 C 4 C t 4 C y a dy = ay dt ( 4 C y ( y 6, a (3 4 C y =.5y ( 8 ( s84 7

28 .59 ( ( 3 4 ( ( 3 ( T 4 cos π sin π cos π sin π ( 6 6 (3 5 (4 3 6 sin π cos π 3 3 sin π cos π A λ x ( A, λ, x ( A = (3 ( ( 8 ( s ( 8 ( s84.6 ( x 8 x 7 3x log(cos x.63 lim x x ( 8 ( s843 ( 8 ( s844 ( 8 ( s85.64 f(x, y = log x + y ( 8 ( s85.65 f(x = x 5 e x ( 8 ( s853 π.66 ( 4 e x sin x dx ( dx x(x + ( 8 ( s a > D(a = {(x, y x a, y 3} y I(a = ( + x + y dxdy D(a ( D(a I(a ( lim I(a a A A = ( A A 4 6 ( 8 ( s855 8

29 ( 5x + y = 5 x 3y z = 5 x + y + z = 5 ( 8 ( s f : R n R m R l l ( f ( f (a f( = (b r r a, a, a r k, k, k r R R n f(k a + k a + + k r a r = k f(a + k f(a + + k r f(a r ( 8 ( s857.7 ( (3 dy dx = ky + cos ωx ( dy dx + y x = dy dx = y x + x y 3 u = y. ( 8 ( s858.7 (x, y A A y x + y = cx (c.7 I = e x sin x dx ( 8 ( s859 ( 8 ( s85.73 dy dx = (x + y + ( 8 ( s85.74 A = ( A A ( A.75 A = x a x x a a x a a x ( 8 ( s85 ( A ( A = x ( 8 ( s86.76 i i a + bi a, b ( 8 ( s86 9

30 .77 x dy dx x + xex =.78 9x + 6 ( x 3 dx ( + x + sin 7x cos xdx ( 8 ( s863 ( 8 ( s f(x, y = (x + y e y f x + f y ( 8 ( s865.8 y = x + 9 x D (x + ydxdy D ( 8 ( s866.8 A = (a,, B = (b, 3 A, B 3 6 b > a > a, b A 3 B ( 8 ( s867.8 x 3 + y 4 = ABCD AB = a S D y C S a A O a B x (x < a x a.83 p(x p(x = + ( a x < a x a + ( x a a (a < x a > ( xp(xdx ( ( 8 ( s868 x p(xdx ( 8 ( s A A =

31 ( A A ( A A 3x + y + 5z = (3 A A 7x + 3y + 4z = 3 x + y + z = i4 i =.86 x = y = dy dx = y x 3 ( 8 ( s86 ( 8 ( s86 ( 8 ( s86.87 A = ( 3, B = ( det A, det B, det C 3 3, C = ( A (3 C (rank C ( 8 ( s x, y f(x, y D f x + f y = f(x, y D D = R ( f(x, y = x 3 axy a ( f(x, y g(x, y = y f x + x f y g(x, y f(x, y ( 8 ( s ( cos(πxdx ( cos (πxdx.9 y = x + sin x (, ( 8 ( s865.9 ( 8 ( s866 x x 4 + ( 8 ( s867.9 a = (,,, b = (,, α α < 8 ( 8 ( s868 3

32 ( 5.93 y = mx y = mx m ( 8 ( s d y dx + 6 dy + 9y = 5ex dx ( 8 ( s ( 8 ( s86.96 (x, y r r x θ x = y = n θ = π/n θ θ =, x = r, y = θ i+ = θ i + θ (i =,,,, n x i+ = y i+ = (x, y, (x, y,, (x n, y n r, θ i, θ i+ ( 8 ( s86.97 z = 3 + i, z = + 3i Re(z z z z Re ( z z z.98 y = x A(, ( P (x p, x p ( 8 ( s87 ( A x p P A.99 (. a = dy dx + y = x ( d y dx 4 dy + 3y = ex dx ( 3 b = ( ( 8 ( s87 ( 8 ( s873 3

33 ( ( x y ( (a bx = X = z w ( a, b ( 5 (3 c = a b 3 (4 R a, a a b. A = ( 4 3 ( 8 ( s874 λ = λ = 5 ( A λ = λ = 5 ( λ W. f(x = xe x ( f(x ( 8 ( s875 ( f(x y = f(x (3 f(x x x = q (q > S(q lim q S(q ( 8 ( s ( 5 4 ( 5 4 (3 4 (4 (5 ( 8 ( s877.4 ( f(t F (s F (s = f(te st dt (Re[s] > { t = (a f(t = δ(t = t { t (b f(t = u(t = t < δ(tdt = 33

34 { t t (c f(t = t u(t = t < { (d f(t = e αt e αt t (α > u(t = t < ( RC (a ν i (t ν o (t i(t ν i (t t = ν o (t = R i(t C ν o (t (b ν i (t, ν o (t i(t V i (s, V o (s I(s (a G(s (c ν i (t G(s = V o(s V i (s ν i (t = u(t = ν o (t V o (s { t t < (d V o (s ν o (t (e G(s g(t (f g(t (c ν o (t ( 8 ( s878.5 N ( N 4 (a N = 8 4 (b N = 4 4 ( 3 N = (a N = 5 (b N = (3 3 (a N = 6 A E A. 34

35 B. C. 3 D. 3 E. 3 (b N = 3 A E A. B. C. 3 D. 3 E. 3 ( 8 ( s879.6 xy (x i, y i, i =,, N ( G ( x = ( ȳ xy G θ uv ( x i y i (x i, y i (u i, v i θ ( u i = v i = ( x i y i ( x ȳ ( u N J(θ = vi (4 i= J(θ θ (3 v i θ u i v i = (5 θ (4 θ J(θ N = u i v i = (6 θ i= (6 (3 N N x iy i = (x i y i (7 i= = N i= (8 N x iy i i= (x i i= x i y i ( (3 y i (8 35

36 (8 = = (9 N i= N x iy i i= (x i y i ( ( (4 θ (4 θ ˆθ G y = tan ˆθ ( x + ( ( 4 (,, (3,, (4, 3, (6, (a 4 G N N (b 4 x iy i N, y i i= i= x i (c 4 θ cos 7 4 π sin 3 i=.7 A = [ ] ( 8 ( s87 ( A α, β (α > β [ ] z z ( P AP P P AP (3 γ n = α n β n A n γ n, γ n, γ n+.8 ( x( xyy = y( + xy ( y + y + 5y = exp x y 6x + x.9 ( x 3 + 3x dx x 3 { π } ( π(x 3 + x cos 4 (x + dx ( 8 ( s88 ( 8 ( s88 ( 8 ( s883. N N N N N k ( k N P k ( N = 3 P k (k =,,, 3, 4, 5, 6 ( x y x y (3 P k 36

37 . R 4 a = 3 4 ( 8 ( s884, b = k 4, c = p 6 3 q i a b π 3 ii a, b, c ( k ( p, q ( 8 ( s89 ( (. ( R O π 4 F r e =, e = ( ( A = 5 F t : R R F t F r F C : x + y = F F (C.3 ( (i tan x ( 8 ( s89 (ii sin x + x ( ( f(x = sin x + x + tan x ( 8 ( s893.4 (x + y 6 (x y 8 dxdy, D = {(x, y : x, y, x + y } D.5 ( n= n + 4n + 3 ( 8 ( s894 ( log( + t = t t + t3 3 + ( n tn +, < t n f(x = ( + x log x 3 x ( 8 ( s895.6 D = {(x, y x, y } V = e max(x,y dxdy max(x, y x, y D x > y x y.7 A = D ( 8 ( s896 37

38 ( A 3 = A + A I I 3 ( ( n 4 A n = A n + A I (3 ( A 5 ( 8 ( s897.8 (x + yy (x + y = ( ( y( = ( 8 ( s898.9 ( (x + x x + 5x dx ( (3 {sin(ωt + a + cos(ωt + b} dt sin 3 θ dθ ω, a, b. (r, θ r = + ε cos θ ( 8 ( s83 ε ( (x, y ( ε (a ε = (b < ε < (c ε = (d ε > a a. b b c c λ. 3 λ ( 8 ( s83 ( 8 ( s833 = ( 8 ( s834.3 i, j, k OA = ai + bj + ck OB = bi + cj + ak A, B, C OC = ci + aj + bk ( AB ( AB AC (3 ABC ( 8 ( s835.4 xy y = f(x = a log(x + A(,, B(3, 3 a ( (3 a log 38

39 y ( A (, f( B (3, f(3 3 g(a g(a a A B ( g(a a (3 g(a f(x O 3 x (.5 A = ( 8 ( s836 m m + 5 m m m ( A A ( A = A m ( 8 ( s837.6 cm 5cm.5cm 3 ( t y(cm x(cm S(cm V (cm 3 x, y x y ( x(cm t (3 S(cm t S ( 8 ( s838.7 x( f(x = 5 e 5 x x >, e ( (5. y. f(x = 5 e 5 x O 5 5 x ( f(xdx ( (3 (4 (5 lim x x e ax =, (a > 39

40 ( 8 ( s839.8 m = (, 3, n = (3,, ( m n θ cos θ ( m n (3 A = (, 4, m n ( 8 ( s83.9 ( f(x = n n + x ( f(x = + x (3 ( g(x = log( + x.3 ( a = (,, b = (, 6, θ ( ( a b ( 8 ( s83 (3 ( a b r a b ( 8 ( s83 ( 3.3 A A =.3 N 8 n ( 8 ( s833 ( N n = 8 N 63 n ( ( lim n.33 log N n x ( 8 ( s834 x 5x + dx ( 8 ( s ( (x y 8 x 6 y ( x + y + z = 8 (x, y, z.35 f(x = { λe λx (x (x < ( 8 ( s836 ( ( (3 ( 8 ( s837 ( a b + ci.36 ( λ, λ a, b, c, d b ci d c i = 4

41 ( λ, λ x, x (x x =.37 ( I = x ij x i j (i, j =, α α e αx dx = ( x e αx dx.38 3 A = a =, b = 3 4 π α > α 4, c = 3 4, d = ( A ( A 4 x j x j j= ( 8 ( s838 ( 8 ( s839 ( d a, d, c d = xa + yb + zc (x, y, z ( 8 ( s83 3x.39 ( lim x x + ( e x sin(x lim x x ( 8 ( s θ A(θ = ( cos θ sin θ sin θ cos θ x x + dx ( 8 ( s833 E ( t A(θA(θ = E t A(θ A(θ ( A( θ A(θ (3 θ, θ A(θA(θ = A(θ + θ ( 8 ( s834.4 R 3 x, y, z x + y + z = f(x, y, z = 3x + xy + xz + 4yz (x, y, z ( 8 ( s f(x, g(x [a, b] g(x > b f(xg(xdx = f(ξ b a a g(xdx ξ [a, b] ( 8 ( s833 4

42 .44 p n k b k n, p q = p n (px + q n = nc k p k q n k x k n C k = ( b k k= n! k!(n k! n b k = k= ( µ σ (3 r r P ( r µ aσ a a p.45 I(a = exp[ x ] cos ax dx ( 8 ( s8333 ( (x, y (r, θ x + y dxdy π ( I( I( (3 I(a I(a a.46 a < a < z w = z ai + aiz i = ( 8 ( s8334 ( z C = {z z = } w ( z C = {z z = } w.47 C C = {z z = } ( 8 ( s8335 ( D = {z z < } f(z f(z f( I = dz C z ( g(z = z φ(z = g(ζ C ζ z dζ D = {z z > } φ(z.48 ( w ( 8 ( s w = ( w n, n= ( w < 4

43 w = w = z Log( + z z =, Log( + z π <Im log( + z π log( + z ( z ( (3 lim + z n = e z n n.49 z y z x y x lim ( nlog + z = z n n ( 8 ( s8337 ( 8 ( s834.5 lim x + x x ( 8 ( s834.5 x sin x + ( x π sin sin.5 x( x dx.53 y y 3y = 3x + x ( x (.54 lim x x ( + x + x.55 dx x( + x.56 a, a, b, b ( a a x, y Q(x, y = (x + a y b + (x + a y b ( 8 ( s8343 ( 8 ( s8344 ( 8 ( s8345 ( 8 ( s8346 ( 8 ( s8347 Q x = Q y = x, y a, a, b, b ( 8 ( s D = {(x, y : x, y x x } dxdy + y D ( 8 ( s

44 x + z =.58 x + y + z w = 3 k x y + z + w = k 3 A rank A.59 A = 3 3 A n (n.6 A = ( 8 ( s834 ( A ( 8 ( s834 ( A x = 3y = 3z (3 A 3 ( n+ n + 3 n n= ( N S N = N ( n+ n + 3 n n= ( 8 ( s835 ( ( N S N ( + a N = a N N C j a j + Na + j= N(N a N ( 8 ( s835.6 x y xy + y = f(x (A x > ( f(x = y = x (A ( u y = uy w w = u (A w x (3 f(x = (A (4 f(x = x x (A ( 8 ( s (y a = a(a + x, (y a = a(a x a > 44

45 ( ( x y (3 y (4 x ( 8 ( s y(x xy (x + y + y = x e x (A ( (A xy (x + y + y = y = e px (p ( y = e px u (p ( u x (A u (3 ( (A ( 8 ( s A(, A (a, B(, B (a + b, a f a, b f F ( F a, b ( F a, b (3 f n f n (x, y f n (x n, y n a, b, x, y ( 8 ( s ( t x p x + p t + t =,,, X t t =,,, x =,,, P (X t = x p p p p p p p p p p p p x : ( t y / y + / y t + t =,,, Y t t =,,, y =,,,,,, P (Y t = y 45

46 / / / / / /.67 / / / / / / y : ( 8 ( s8357 x (t + a(tx (t + b(tx(t =. ( a(t, b(t ( x, x ( J [ x (t J(t := det x (t x (t x (t ] J : J (t = a(tj(t. ( ( x, x ( [ x ( x ( J( = det x ( x ( ] t (x (t, x (t, (x (t, x (t (3 x, x, x 3 ( t x (t x (t x 3 (t det x (t x (t x 3 (t = x (t x (t x 3 (t (c, c, c 3 (,, t c x (t + c x (t + c 3 x 3 (t = ( 8 ( s a f(z g(z f(a ( a g(z f(z a f, g a g(z a ( a g(z f(z a f g a 3 g(z a z n (3 n dz (z n z = ( 8 ( s [ π, π] f(t. 46

47 π ( {a k } a k := f(t sin kt dt, k =,,. π lim a k = a k < k ( ( {a k } π lim N π N f(t a k sin kt dt = k= k= π π f(t dt = k= a k ( 8 ( s835.7 X Y { λe λx, x > f(x = λ >., x ( X < Y P (X < Y ( min{x, Y } min{x, y} x y (3 a < b < P (a < X Y < b.7 ( 8 ( s835 x + y + 3z = a x + 3y + 4z = b 3x + 4y + 5z = c ( (rank ( (3 ( 8 ( s836.7 dy ( y dx = f x ( ( ( dy dx = x + y xy.73 ( 5 ( a.74 A = c ( b d ( 8 ( s836 dx (x + (x + 9(x + 4 ( 8 ( s8363 ad bc = A a + d t t = a + d A = f(ta + g(ti f(t, g(t I A ±I t = t = A 4 = I A 4 = I ( 8 ( s838 47

48 .75 g a,b,c,d (t g a,b,c,d (t = at + b ct + d g = g a,b,c,d (t ( g ( g = ( g g = d/dt a, b, c, d, â, ˆb, ĉ, ˆd (ad bc, â ˆd ˆbĉ a, b, c, d ad bc ( g = g a,b,c,d (gâ,ˆb,ĉ, ˆd(t = g a,b,c,d gâ,ˆb,ĉ, ˆd(t.76 y x x y dxdy ( 8 ( s838 ( 8 ( s r : [a, b] R + (x, y x = r(θ cos θ, y = r(θ sin θ, a θ b α a b π/ α A, α L A = b a f(rdθ, L = b a g(r, r dθ f(r g(r, r r(θ = / cos θ A L π/3 dθ cos θ = 3 ( 8 ( s ( f(x, y f/ x, f/ y sin sin (i f(x, y = e 3x cos y, (ii f(x, y = sin x y ( f(y, y f(ax + bx, ax + bx (a, b f f x x x x ( 8 ( s ( 5 3 A = 3 7 P.8 y = f(x f (x f y (x = lim x, ( y(x = u(xv(x : y (x = u (xv(x + u(xv (x ( 8 ( s8386 x = lim x f(x + x f(x x ( y(x = sin(x : y sin x (x = cos(x, lim x x = ( 8 ( s839.8 ( Z = f(x, y = sin(x cos(y, x = e t, y = log e (t, dz/dt t 48

49 ( xy (x, y φ(x, y = x + y =, Z = f(x, y = x + y ( I = a x b dx, a >, b > ( I =.83 I n = (x, y (x + a dx (a >, n ( 8 ( s839 ( I { } x ( n, I n = (n a (x + a n + (n 3I n I n, I n (3, I 3.84 m t v(t,.85 A =.86 A = m dv(t dt = mg cv(t (m, g, c e ax dx, a > ( 8 ( s8393 ( 8 ( s8394 v(t,, t = v = X, Y 7 3, B =, x = , ( 8 ( s8395 X + Y = A, X Y = B ( 8 ( s8396, t xax t x x ( 8 ( s A t A = A A A, A = O ( 8 ( s8398 x y + z =.88 4x y + z = 3 x + y + 3z =.89 ( 8 ( s8399 a + b + c c b c a + b + c a b a a + b + c ( 8 ( s839 49

50 .9 ( ( 3 ( 8 ( s839.9 A A = X + Y X Y M M, M = M, M = M [ ] A = 3.9 x y z = ( 8 ( s839 ( 8 ( s8393 [ ].93 A (, ( A = ( A x ( A, f(x, y ( x f(x, y = (x y A (x, y y f(x, y f(x, y = a(x + y + b(x y a, b 3.94 ( ( 3 x y z = a, b, c 3 ( 8 ( s8394 a b c ( 8 ( s84.95 n =, C n : r n = cos nθ ( C xy x ( C ( θ π/4 l l = dx x 4 ( 8 ( s84.96 c, c y ( x y xy =, y( = c, y ( = c 5

51 ( n y (n ( ( y ( 8 ( s843 x p sin.97 p > R f f(x = x (x > (x ( f R ( f R p (3 f R p ( 8 ( s84.98 R n m {a,, a m } ( {a,, a m } {a,, a m } m ( c,, c m R c j a j = c = c = = c m = j= ( ( ( {a, a, a 3 } a =, a =, a 3 = (3 A λ, λ, λ 3 3 e j (j =,, 3 λ j {e, e, e 3 } R 3.99 log ( n n k= n k < log n < k k= 3 3 ( 8 ( s84 ( x > log( + x x x + x3 3 n (3 n γ n = k log n γ n > {γ n } k=.3 A = 4 a ( 8 ( s843 a ( A ( π : x y + z = π : x + y 4z = π 3 : x + y + az = (3 a = 5 A 5

52 ( 8 ( s844.3 a, b a ( e ax sin bx dx, e ax cos bx dx ( dy + ay = cos bx dx ( (3 y( x ( y(x a b.3 x ( F (x = ( x ln(x ( F (x = x 3 + x.33 cos(axe x dx ( a ( 8 ( s845 ( 8 ( s846 ( 8 ( s x y y = x (, (a, a ( 8 ( s848 ( 8 ( s ( n.37 f(x, y = ( S { n(n n 3 = S =! + 3! + 3 4! + + } n (n +! + ( 8 ( s84 + x y (, x, y x, y, xy ( 8 ( s84.38 l /4 x y x y 5

53 y l O l x ( 8 ( s n A, B n E n A k A k A E ( A k = E k A A k ( a AB = ae AB = BA.3 A = ( 5 5 ( 8 ( s844 ( ( ( u, u u, u P = (u, u (3 ( P P t AP P t P ( 8 ( s845.3 y = x ( x x V y V.3 dx dt = 3t x.33 f(x = x + 3x + 9 ( 8 ( s843 ( 8 ( s843 ( 8 ( s8433 ( y = tan x ( 8 ( s8434 ( π < x < π x = arctan y tan y x = arctan y ( 8 ( s8435 e x.36 e x + dx ( 8 ( s

54 .37 A A e φ A = A A φ A e A e θ A ( 8 ( s F, F P F P F P P F + P F = a F, F (ae,, ( ae, e ( x, y ( P F x θ P F r F y O P θ F x (3 F P (.39 A = ( 8 ( s8438 ( ( (3 P t P AP, t P P.3 f(x = x x x +.3 y = y(x ( y + y = ( y + y + y = x ( 8 ( s844 ( 8 ( s844 ( 8 ( s f(x R R f (x > ( a < b f(a = f(b = f (a < < f (b g(x a < b a, b g(a = g(b a < ξ < b ξ g (ξ = ( a < b < c f(a = f(b = f(c = 54

55 (3 a < b f(a < < f(b f(x = x a b.33 n { f n (x = max 3 4n 3 x + 3 } n x, ( 8 ( s845 max{a, b} a b ( f n (xdx ( x [, lim f n(x n (3 lim n f n (xdx lim f n(xdx n ( 8 ( s I n n n a R n n x R A(x f(x A(x = I n + x( t aa, f(x = det A(x t a a det A(x A(x ( f( = ( aa( = f( = (3 f(x x m m n (4 x, y R f(xf(y = f(x + y + xy (5 f(x = ( + x m (6 m = f(x = + x ( 8 ( s ( V V R f f a b V M = c d f (M = ad bc f (M = a + d f f V R R V.36 x = log t, y = t + t d y dx ( 8 ( s8454 ydx t ( 8 ( s ( (a lim x( x + x x (b lim x cos x x 55

56 ( (a x sin x (b log x x + (c a x log x a a.38 ( e x cos x dx ( ( 8 ( s846 t + 3t dt ( 8 ( s a f(x, y = x 3 + y 3 3axy f(x, y ( 8 ( s D = {(x, y ; x + y log x 4} + y dxdy x + y.33 x, y, z 4x + y + z = ax 5y + z = 8 x 4y z = 9 3x y + z = 5.33 f(x = x tan x ( f(x (, ( ( x 3 f x dx.333 x + y z = x + 4y z + 6w = 7 x + 3y + 4w = 6 3x + 8y z + 8w = ( D ( 8 ( s8464 a ( 8 ( s8465 ( 8 ( s8466 ( a, b, c x = a, y = 7, z = b, w = c a, b, c.334 a > x n = ( n a, n =,, + a ( {x n } ( x n (3 nx n n= n= ( 8 ( s8467 ( 8 ( s A A A, B AB = BA A B.336 n f(x = x+ x ( 8 ( s8469 t n t n dt + 56

57 ( f(x x = n ( ( n lim x f(x, lim x f(x ( 8 ( s { } ( lim (x + a(x + b (x a(x b x a, b ( lim x x 3 x tan(t dt ( 8 ( s ( a, b, c R 3 a b, b c, c a ( W, U R 3 x x W = y R 3 x + y z =, U = y R 3 x + y =, z = z z.339 ( 8 ( s846 y y e x dxdy ( 8 ( s E 3 3 A = 4 4 B = E + A, 3 ( B O B 3 = O ( 3 u B u (3 ( u u, Bu, B u R 3 (4 f : R 3 R 3 x Ax (x R 3 (3 {u, Bu, B u} ( 8 ( s8464 {.34 C z = f(x, y x = r cos θ y = r sin θ r θ z = f(r cos θ, r sin θ ( ( ( ( z z z + = + ( z x y r r θ ( z x + z z r θ r θ y ( 8 ( s

58 c c.34 A = c c c c ( A ( A c (a c = (b c = (3 ( (a,(b A (4 ( (a,(b A.343 ( d y ay = dx ( [, l ] d y dx + a y = { y( = y(l = ( 8 ( s847 a (3 d y dx + a y = f(x f(x d y dx + a y = y = c sin ax + c cos ax c c x y = { } sin ax f(x cos axdx cos ax f(x sin axdx + c sin ax + c cos ax a ( 8 ( s a C C ( z ( z = a (a z > a (b z < a dz n C (z z n ( ( f(z = log(z b+log (z a df log z b b > a dz b C dz.345 ( m, n (a (c π π π π sin mx cos nxdx = sin mx sin nxdx = { ; m n π ; m = n (b π π ( 8 ( s8473 cos mx cos nxdx = { ; m n π ; m = n ( π f(x f(x = a + (a k cos kx+b k sin kx (a π π f(xdx (b π π f(x cos pxdx (c π π k= f(x sin pxdx ( p =,, 58

59 (3 π f(x = x ; π < x π (4 (3 (a (k (b k= k= k (c ( k k k= ( 8 ( s8474 π.346 e ax dx = (a > a a π 4a = d ( ( e ax dx = 3/ da a e ax dx = x e ax dx, x e ax dx = π 4a 3/ ( ( ( h > e hx h ( x 4 e ax dx < (3 (, ( (.347 f(x, y = x 3 x + y, (x, y (,, (x, y = (, ( f(x, y (, + x hx4, x R ( 8 ( s8475 ( f(x, y (, x y (3 f f (x, y (x, y (, x y.348 x, x,, x n n A = [a ij ] n ( 8 ( s8476 ( x, x,, x n n ( x, x,, x n y j = a ij x i (j =,,, n y, y,, y n A (3 x, x, x 3, x 4 a A = a a a ( y, y, y 3, y 4 a i= ( 8 ( s

60 .349 P A : P P (Af(x = P = {f f(x = ax 3 + bx + cx + d, a, b, c, d R}, x f(ydy f ( 4 x4 ( A f n (x = x n, n =,,, 3, (Af n (x ( W f, f, f, f 3 A (3 A 4 (4 A.35 {a n } n=, {b n } n= f : R R α, β, x R ( 8 ( s8478 ( lim n = α M n a n M n ( lim n = α, lim n = β lim nb n = αβ n n n (3 lim f(x α ε > lim n = x {x n } n= x x n f(x n α ε (n =,, (4 (a lim f(x = α. x x (b {x n } n= lim x n = x lim f(x n = α. n n.35 a, b, c 3 3 A A = a + b + c a c a + b c a c a + c a + c a + b c a + c a + b + c ( 8 ( s8479 ( abc A = A A ( a = b = c = A ( 8 ( s O = (,,, A = (,,, B = (,,, C = (,, O, A l, 3 O, A, B π ( B l P OA e = OA e P B = P B OA, P B OA P B ( C π Q OQ = αe + βe α β Q 6

61 ( 8 ( s x π 3 f(x = log(cos x, ( cos x g(x = log + sin x cos x sin x ( g(x g (x cos x ( y = f(x ( x π 3 ( 8 ( s e f(x = e x n (n f (n (x ( f (n (x = p n (xe x p (x, p (x ( k lim x + xk f(x = (3 I = ( (log x dx ( o (log x dx = lim ε + p (xp (xf(xdx ε (log x dx ( 8 ( s8473 ( 8 ( s R 4 4 a =, a = 3, a 3 = 5, a 4 = R 4 V ( V ( 6, V (3 R 4 R 4 (4 a, a R 4 a 3, a 4 R f(x, g(x g(x D = {(x, y ; x, y g(x} D ( 8 ( s849 d g(x dx d f(y dy f ( =, f(g( = 3, f(g( = dxdy ( 8 ( s8493 6

62 a b c.358 A = b c a c a b ( A a, b, c ( a, b, c A A (3 a = + i, b = i, c = A i (4 a =, b = + 3 i, c = 3 i A A i ( 8 ( s ( tan x ( x x.36 y = x + x ( 8 ( s8495 ( 8 ( s lim x x sin x x 3 ( 8 ( s x x 3 e x + log( x e D x log x dx log(x + y dxdy ( D : x + y ( 8 ( s8498 ( 8 ( s8499 ( 8 ( s849 x + y + z =.365 y x + y + 4z = 3x + y + z = A = ( 8 ( s ( 8 ( s x = cos 4 t, y = sin 4 t dy dx, d y t dx ( 8 ( s x cos 3x n g(x = cos ax (a > g (n (x = a n cos (ax + n π ( 8 ( s8494 6

63 .369 π cos x + cos x dx ( 8 ( s z = tan (u + v, u = x y, v = x y z x, z y x, y ( 8 ( s g(x, y = f(x, y g(x, y = x + y, f(x, y = x + y.37 D.373 A =.374 A =.375 A = x y dxdy, , B = D = {(x, y y < x } [ 3 4 ] ( 8 ( s8497 ( 8 ( s8498 AB, BA rank(a det A A ( 8 ( s8499 ( 8 ( s849 ( 8 ( s a, a,..., a r (r n R n a, a,..., a r.377 A = ( 8 ( s849 ( A ( P A ( 8 ( s dy/dx ( y = 3x + 4x ( y = /x (3 y = x ln x (4 y = sin(3x (5 y = x/e x ( 8 ( s T P V (,( a, b, R (P + a/v (V b = RT ( ( P/ T V ( ( P/ V T ( 8 ( s

64 .38 exp exp(x = e x ( x + 4x + ( cos(x (3 4x exp(x (4 /(x 9 (5 x ln x.38 ( (dy/dx + xy = ( (dy/dx + y = x ( 8 ( s8496 ( 8 ( s x a t a = π cos πt t x t = ( 8 ( s8498 [ ] i.383 i i.384 ( f(x x = x f(x = e x ( g(x x = x g(x = x (3 lim x f(x g(x.385 y = dy dx, y = d y dx.386 A = ( y 4y y = ( y 4y y = x 8 (3 (, B = ( A B ( t A t B ( 3 4 (3 A A (4 t B det( t B t A t B A B ( 8 ( s8499 ( 8 ( s8493 dy dx + y = y (cos x sin x ( 8 ( s (.388 A =, x + x + x 3 = x + px + x 3 = px + x = ( 8 ( s8493 p ( 8 ( s84933 ( A λ, λ ( λ, λ (3 A 64

65 ( 8 ( s ( lim x 3 x + 5 x 3 4x + x ( lim x + 5x 4x + 7 ( 8 ( s ( y = (x 3x + 3 ( y = (sin 4x log(x 3 ( 8 ( s C ( (3x + sin x dx ( sin x cos 3 x dx (t = sin x ( 8 ( s I y = x 3 x +, I = [, 3 ] ( 8 ( s lim x x 3 x ( 8 ( s D x + y dxdy,.395 ( dy dx = x + y x y ( d y dx + 4 dy dx + 3y = D : x + y cm, x cm ( x 9 x 9 cm ( 8 ( s8494 ( 8 ( s cm x cm x cm ( 8 ( s x + y + z 9 z = z.398 9x + 4y + 3z = 5x + y + z = 7x + 3y + 4z = ( 8 ( s84943 ( 8 ( s

66 .399 P x y (, x y 75 x y P x y 75 = x.4 x = r cos θ, y = r sin θ J = r y r r.4 dr r.4 xy D = { (x, y } x + y 4 ( 8 ( s84945 x θ y θ ( 8 ( s85 ( 8 ( s85 dxdy D x y 4 ( 8 ( s µ λ > ( A =, B = ( λ µ, P = ( x y AP = P B λ, µ, x, y.44 A = ( ( 8 ( s854 ( A λ λ ( λ λ x x ( 8 ( s x >, y > u(x, y = x y, v(x, y = x + y u u J = x v y v J x y.46 ( y = x ( f(s, t = s + t + ( 8 ( s853 ( 8 ( s D D = {(x, y x y x +, x } ( D xy ( (x + y dxdy.48 ( (x + dy dx + y = ( 8 ( s8534 D 66

67 ( (x + dy + y = (x + sin x dx ( 8 ( s ( xe x dx ( (4 (x + cdx (x + a(x + b dx a x ( a < x < a, a > cos x (a b (3 cos xdx ( 8 ( s854.4 ( (a xdx dy = x(xdy ydx (b (y + cos xdx + (xy sin ydy = ( y + 4y + 4y = xe x ( 8 ( s854.4 a = (a, a, a 3, b = (b, b, b 3 a b ( a a 3 a b = b b 3, a 3 a b 3 b, a a b b ( a = (a, a, a 3, b = (b, b, b 3, c = (c, c, c 3 ( a a a 3 (a b c = b b b 3 c c c 3 ( ( a b a b.4 o xyz ( ( ( 8 ( s8543 ax + by + cz + d =, (a, b, c ( ( n = (a, b, c ( ( ( ± a + b + c lx + my + nz = p, (p ( p O (3 Q(x, y, z ( h h = lx + my + nz p ( 8 ( s f(x = x 3 + 6x ( 8 ( s

68 .44 y = (x y = x + 4x S ( 8 ( s a = (,, 3, b = (,, 6 c.46 A = [ 3 4 ], B = [ 3 ] ( 8 ( s8547 T ( X = (AB T, ( Y = (BA T, (3 X, (4 Y ( 8 ( s h r l V S ( 8 ( s (x, y (x, y T x = y, y = x ( 8 ( s ( y = (8 x + 3x ( y = log x e x (3 y = sin x ( + cos x ( 8 ( s854.4 ( x dx ( dx x( + x ( 8 ( s854.4 ( y = sin x cos x ( y = sin x ( π < y < π.4 ( π sin xdx ( π x cos xdx ( 8 ( s8543 ( 8 ( s A, A, A 3 A = cos θ sin θ, A = sin θ cos θ, A 3 = ( 8 ( s dy dx = y + x + ( 8 ( s

69 .45 3x + y + 5z = 5 x + y + 3z = x y z = 3.46 ( + xy + ( + yx dy dx =.47 A, B AA t BB t A t A B t B ( 3 A = B = ( 8 ( s855 ( 8 ( s855 ( 8 ( s C C = 3.49 f(x =.43 sin x cos x sin x + cos x 3 x(ax + n dx ( a, n.43 f(x, y = x y + y cos x + y 3 ( 8 ( s8554 df dx ( 8 ( s8555 ( 8 ( s8556 f x f x y ( 8 ( s e iθ = cos θ + i sin θ ( (cos θ + i sin θ n = cos nθ + i sin nθ ( e i(θ+ϕ = e iθ e iϕ cos(θ + ϕ = cos θ cos ϕ sin θ sin ϕ sin(θ + ϕ = sin θ cos ϕ + cos θ sin ϕ ( 8 ( s π f(t = π t ; π t π f(t + π = f(t ( π t π ( 69

70 .434 d [ ( ] tan dx 3x ( 8 ( s8559 ( 8 ( s x x 4 + dx ( 8 ( s A A = ( ( A.437 c b c a b a b + c ab ca = A ab c + a bc ca bc a + b ( 8 ( s855 ( (ax + b n ( x (3 log(ax + b (4 e x.438 ( x n ( xdx ( b a ( 8 ( s8553 (x a(b xdx ( 8 ( s ( 8 ( s8555 ( ( 3.44 A =, B = 6 4 ( A + 3B ( AB, BA.44 ( ( π lim x tan x (3 log( + x dx ( 8 ( s8556 ( f(x = e sin x.44 a b b b b a b b b b a b b b b a ( 8 ( s856 a, b ( 8 ( s856 7

71 .443 x 3 3xy + y 3 = y = f(x ( 8 ( s dy e x dx y ( 8 ( s I n = (log x n dx (n log ( I, I I ( ( n I n I n ( 8 ( s x + 5y 4z = 7 3x + y 3z = 6 5x + 4y z = ( ( A = ( ( A a 3 A 3 ( (3 ( ( 8 ( s z = sin xy ( z z ( x y (3 z x (4 z x y.448 u = x y z x y z ( u ( (4 ( 8 ( s858 u x (3 u y u z (5 u x + u y + u z ( 8 ( s ( ( ( 8 ( s8583 7

72 .45 a, b, c a a b b c c ( 8 ( s R 4 v,, v k v,, v k ( v,, v k R 4 ( v v,, v k v,, v k, v = v,, v k (3 v = 3, v = 7 3, v 3 =, v 4 = 5, v 5 = v, v, v 3, v 4, v 5.45 f(x = e x (sin x + cos x ( 8 ( s8585 ( 8 ( s {α n }, {β n } n > N α n β n N lim α n lim β n n n.454 f x G(x = x a ( 8 ( s8587 (x u{f (u + f(u}du ( 8 ( s ( dx ( x dx (3 x + x dx ( 8 ( s f(x, y C θ x = u cos θ v sin θ, y = u sin θ + v cos θ (.457 A = f f u v ( f u + f v f x f y 3 (a (e ( 8 ( s858 (a A T, (b AA, (c A, (d A, (e A A T A A A.458 x [ ( 8 ( s858 ] [ ] [ ] x x =, x = x ( 8 ( s858 7

73 .459 A = i.46 ( f(x = Ae kx d d f(x f(x dx dx ( ( f(x = Ae kx.46 ( φ(x, y, z = x + xy z φ = [ i i ] ( 8 ( s8583 d dx 4k x f(x = Cf(x C ( x, y, ( A(x, y, z = (A x, A y, A z = ( y 3, x 3, A = ( Az y z ( 8 ( s8584 φ A ( y Ax i + z z A ( z Ay j + x x A x k y i, j, k x, y, z.46 e x cos x ( 8 ( s8585 ( 8 ( s