[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ), x = u + v, y = u v z u, z v.4 = { (x, y); y x y, y } ( ) ( s46) ( ) ( s463) () () x ydxdy.5 A k 3 4 5 6 () A ( ) ( s464) () A k (.6 tan x π, π ) () y = tan x. () x >, tan x > x 3 x3..7. () dx x( + x ) () π ( ) ( s465) 6 ( s646) sin x 4 cos x dx 6 ( s646).8 f(x) F (x) = x f(t) dt
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x) ( 6) ( s6463) 6 ( s6464) ( ) s β. α, β, γ f(x) g(s, t) = γ + (t α)f t α g(s, t) g(s, t) = γ + (t α) g (s, t) + (s β) g t s (s, t) ( 6) ( s6465). = {(x, y) ; x + y π },. sin x + y dxdy. ( 6) ( s6466) xy dxdy : x + y, y x +, x ( 6) ( s6467).3 A t A A () A + t A A t A () A = t A A = t A 6 3.4 () 5 4 3 () x + 4y + z = x + 6y + 3z = 5x + 4y + 3z = x + y + z = ( 6) ( s6468) ( 6) ( s6469).5 () A n λ n u, v u, A u = λ u, A v = u + λ v (*) u, v
() a, b, c, d x (a + ( d)x + ad) bc = a λ a b A = () (*) c d u, v ( ) ( ) ( ) u v u v (3) () u, v u =, v = P = u v u v P AP.6 (). (a) log( + x 4 ) (b) sin x () α, β, tan x (x > ) f(x) = β (x = ) αx α + β (x < ) (. tan x π, π (a) f(x) x = β. (b).7 f(x) = x + x x 3 + x ( 6) ( s646) ).. lim f (x) = lim f (x) α. x + x,. () f(x) = A x + Bx + C x A, B, C. + () f(x)dx. ( 7) ( s746) (3) 3 f(x)dx. ( 7) ( s746).8 () f(x, y) = x x 3 xy. () = {(x, y) : x, y, x + y 3},. x ydxdy.9 A = a () a. () A. ( 7) ( s7463)., a. (3). ( 7) ( s7464) 3
. a, b a > b >, { x = a cos t y = b sin t ( t π). (). () t = π, (). 4 (3) y. ( 7) ( s7465). A, B, O, E. (), (), (3)?, A, B. () (A + B)(A B) = A B. () AB = O A = O B = O. (3) A + A E = O A. ( 7) ( s7466).. (). 3 () 3 3 () 3 a a 3 b b 3 c c 3 ( 7) ( s7467).3 f(x) (, ). < a < b, f(b) = f(a) + f (a)(b a) + K (b a) K. () F (x) = f(b) {f(x) + f (x)(b x) + K (b x) }, F (x). () K = f (a + θ(b a)) < θ <. (3) x > f (x) δ δ >,. f(x) lim x x = ( 7) ( s7468).4 a >, x >, f(x) x a f(t)dt = + log x., f(x) a. ( 7) ( s7469) 4
.5 I. I = π/3 π/4 dx sin x ( 7) ( s746).6 () (a) lim x( x + x ) x () (a) x sin x x (b) log x + (b) lim x cos x x (c) a x log x a a.7 () e x cos x dx () ( 8) ( s846) t + 3t dt ( 8) ( s846).8 a f(x, y) = x 3 + y 3 3axy f(x, y) ( 8) ( s8463).9 = {(x, y) ; x + y log x 4} + y dxdy x + y.3 x, y, z 4x + y + z = ax 5y + z = 8 x 4y z = 9 3x y + z = 5.3 f(x) = x tan x () f(x) (, ) ( ) () x 3 f x dx.3 x + y z = x + 4y z + 6w = 7 x + 3y + 4w = 6 3x + 8y z + 8w = () ( 8) ( s8464) a ( 8) ( s8465) ( 8) ( s8466) () a, b, c x = a, y = 7, z = b, w = c a, b, c.33 a > x n = ( ) n a, n =,, + a () {x n } () x n (3) nx n n= n= ( 8) ( s8467) 5
( 8) ( s8468).34 A A A, B AB = BA A B.35 n f(x) = x+ () f(x) x = n () () n lim x f(x), x ( 8) ( s8469) t n t n dt + lim x f(x) ( 8) ( s846).36 { } () lim (x + a)(x + b) (x a)(x b) x a, b () lim x x 3 x tan(t )dt ( 8) ( s846).37 () 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.38 ( 8) ( s846) y y e x dxdy ( 8) ( s8463).39 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) {.4 C z = f(x, y) x = r cos θ y = r sin θ r θ z = f(r cos θ, r sin θ) 6
() () ( ) z + x ( ) z = y ( ) z + r r ( ) z θ z x + z z r θ r θ y ( 8) ( s8465).4 (). x a (a) lim x a x a (). (a) e x cos x (b) x + sin 3x (b) lim x x sin x (c) x x (x > ).4 () π x sin xdx. ( 9) ( s946) () dx. x x3 ( 9) ( s946).43 f(x, y) = y x e t dt. f x, f y, f y x. ( 9) ( s9463).44 a, = {(x, y) : a x a, a y a},. e x y dxdy.45 () λ λ λ λ ( 9) ( s9464) = λ. () () λ λ, λ x λ x λ x 3 = λ (x, x, x 3, x 4 ). x 4.46 () log xdx. ( 9) ( s9465) () < a <, f(a) = (a) f(a). e log x a dx. (b) a < a <, f(a) a. ( 9) ( s9466).47 a, b, c, [ a A = a ], B = 3 b b [, C = c 3 c ], 7
[ = 5 3 ], F = [ 3 4 7 5 ]. () AB. () C t., t. (3) AB = C t + F, a, b, c..48 x, f(x) = x t 3 + t dt. + ( 9) ( s9467) () f(x). () x > f(x) >. f(x) (3) lim x x..49 () 4 x x x x () f(x) =. ( 9) ( s9468) f(x). f(x)..5 x > f(x) = ( + x) 3,. () f (x), f (x), f (x). ( 9) ( s9469) () f(x) x = x., 3 f (x). (3) (), (8.) 3 3..5 (x, y) (, ) f(x, y) = () f x, f y..5 a. A = () A A. 3 8 3 a x + y. () f x + f y.. ( 9) ( s946) ( 9) ( s946) () A A a., a A. (3) A = 4, a., A A. ( 9) ( s946) 8
.53 x y : y x, z = xe y z =. ( 9) ( s9463).54, dy/dx. y = x x ( 9) ( s9464).55. log x dx (x > ) ( 9) ( s9465).56. π/.57. sin 3 θ dθ ( 9) ( s9466) dy dx = xy ( 9) ( s9467).58 P, P (x, y ), (x, y ). P, P O P OP = θ, cos θ. ( 9) ( s9468).59 a, b, c, A, B, C a + b A =, B = a b, C = c. () AB + C. () AB + C, a, b, c. (3) ( t B)C = ( t C)B, a, b, c., t B B, t C C..6 a, A b A =, b, Ab, A b 3 B. B = ( b, Ab, A b ).,. () Ab, A b. () B B. (3) B B, a..6 f(x) = ex e x, g(x) = ex + e x.. () f(x). () f(x). (3) g(x) f(x). (4) a >. a + x dx ( ) ( s46) a 4, b =.. ( ) ( s46) 9
( ) ( s463).6 f(x, y) = x + xy + y x + y. () f(x, y). () f(x, y). (3) f(x, y). ( ) ( s464).63, 3.,. 3 () 3 4 () f 3 4 5 a b c, 3 d e..64 () x 3. () x 4. ( ) ( s465) (3) f(x) = O. x (4) g(x) = O 3. + x ( ) ( s466).65 ()., x > n. x n log x dx (). π x sin x dx (3) 3 A = ( x, x, ), B = (, x, x 3 ), C = (x,, x 3 ). n (4)., a. lim n n + k + a.66 () (a). lim (b) x n lim x e x =. (). x x cos x, n. k= (a) log x + (b) sin x (c) (sin x) sin x.67 (). (). π 4 dx (x + ) tan x dx ( ) ( s467) ( ) ( s468) (x + )(x ) ( ) ( s469)
.68 () ϕ(u, v), ψ(u, v), J(u, v) = ϕ ψ u v ϕ ψ. ϕ, ψ v u J(u, v). (a) ϕ(u, v) = e u cos v, ψ(u, v) = e u sin v (b) ϕ(u, v) = u + v, ψ(u, v) = tan v u () = {(x, y) ; x, y, x + y },. x y dxdy.69 A =, C = 3 4 6. ( ) ( s46) () AB = C B. () C. ( ) ( s46).7 (). (a) x tan x (b) x + 9x () f(x) = ex e x, g(x) = ex + e x,. (a) {g(x)} {f(x)} =. (b) y = f(x). (c) y = f(x) h(x), h (x). ( ) ( s46).7 (). y = log x 4 x ( x e) () a >, a,. e x(log x) a dx.7 () f(x, y) = x 4 + y 4 (x y). ( ) ( s46) () = {(x, y) π x π, y },. x sin(xy) dxdy.73 A = a a a a., a > ( ) ( s463)
() A. () A a. (3) A. (4) A. ( ) ( s464).74 () (a, b). (a) y = x log x, (a, b) = (e, e) (b) y = ex e x e 3x, (a, b) = (, ) + e x (). 3 x + (a) lim x 4 x (b) lim x tan x x x 3 (, tan x π, π ). (3) f(x) f (x)., f(x) = c f (x) =. ( ) ( s46) (.75 () x = sin t π < t < π ),., x. dx x () S(x) = x log t dt.. (a) S (e) (b) S(e) (c) e e S(x) log x dx (d) e S(x) x x dx ( ) ( s46).76 z = f(x, y). x = r cos θ, y = r sin θ,. (a) x z x y z z z = r cos θ sin θ y r θ (b) ( ) z + x ( ) z = y ( ) z + r r ( ) z θ.77..78 A c b A = c a. b a () A. ( y ) e x dx dy ( ) ( s463) ( ) ( s464) () a, b, c. b + c ab ca ab c + a bc ca bc a + b ( ) ( s465)
.79 3 A = 3 7 9 9 3 3. () A. () A. (3) A. (4) P A. ( 3) ( s346).8 c. 4 A = c c c. () A x y z w = 3 c.. () A x y z w = c.. ( 3) ( s346).8 y = (x + )e x n y (n). ( 3) ( s3463).8 f(x, y) = xy( x y) f x = f y = (x, y). f. ( 3) ( s3464).83. (). π ( ) cos y x dx dy y π 3
() = {(x, y) : x, y, x + y },. x + y + dxdy ( 3) ( s3465).84 a, b, < a <. xy, C : y = log x, l : y = ax + b,. () b a. () b > a. (3) a (), C, 3 l, x =, y =, a. ( 3) ( s3466).85 () f(x) = tan x + tan ( x (x > )., tan x π, π ). (a) f(x) f (x). (). e 3x e 4x (a) lim x sin 5x (b) cos x lim x log x (b) f(). ( (c) lim x log x ) x ( 3) ( s3467).86 (). e x cos x dx () t = tan x,. π + cos x dx ( 3) ( s3468).87 f(x, y), (x, y) = (a, b) f f (a, b) = (a, b) = x y., g(z). F (x, y) = g (f(x, y)), (), (), (3). () () F f (x, y) x x (x, y) g (f(x, y)). F F (a, b) = (a, b) =. x y (3), f x (a, b) + f y (a, b) =, F x (a, b) + F (a, b) =. y ( 3) ( s3469).88 = {(x, y) ; x + y, x y },. x dxdy ( 3) ( s346) 4
.89 x, y, z x + y 3z = x + y + z = x 3y + z = a, a. ( 3) ( s346).9 (). (a) x sin x lim x x 3 (b) lim x x x () x ex e x e x. + e x (3) f(x) = cos (sin x)., cos x [, π]. (a) f( π ). (b) f(x). (c) lim f (x) lim f (x). x π + x π ( 5) ( s546).9 (). e (log x) dx (). x = t3 3 t, y = t ( t ).9 f(x, y) = x y. () f f x () y. ( 3, 3, ( 5) ( s546) 3 ) z = f(x, y). (3) Ω = { } (x, y, z) : x + y a z f(x, y)., < a <. 56 4.93 () 6 54. 8 6 4 () 6 4..94 f(x) = 8 4 x.. (log x) ( 5) ( s5463) ( 5) ( s5464) 5
() lim x + f(x). () (, ) y = f(x),. (3) = {(x, y) x >, y f(x)}. x + y. ( 6) ( s646).95 (). (). x 3 x x dx + e x dx e x ( 6) ( s646).96 f(x, y) = x + txy + y + x + y t.,. ( 6) ( s6463).97. ( ) + y3 dy dx x.98 A = 4 6 4. ( 6) ( s6464) () A. () () A..99 (). ( x log( + x) (a) lim x x (b) lim cos 4 ) x x x () x x x + sin x. (3)., cos x [, π].. (). ()., (a) xe x dx (b) cos x + cos ( x) = π x(x ) dx e x dx = x e x dx ( 6) ( s6465) ( 7) ( s746) π. (c) x e x x dx ( 7) ( s746) 6
. f(x, y) = tan y f. y x x x f y f x f y.. = { (x, y) x + y 4, y x },..3 A =, E =. xydxdy,, n. ( 7) ( s7463) ( 7) ( s7464) (i) E (ii) A (iii) A n ( 7) ( s7465).4 () C z = f(x, y) x = r cos θ, y = r sin θ z = f(r cos θ, r sin θ) z x + z y = z r + r z θ. () f(x, y) = x 4 8xy 3 + y 4 + 5. f(x, y) = y. ( 7) ( s7466).5. x y + y dydx ( 7) ( s7467).6 a, C : y = x e x, C : y = ae x S. () y = x e x,,. () S a. (3) S lim x + a. ( 7) ( s7468).7 x. 3 R 3 3 v =, v =,. () v, v., v 3 = () v, v, v 3, x. (3) x = 5, v, v, v 3 R 3. x ( 8) ( s846) 7
3.8 () A A = 3 4. 3 (a) A. (b) P, A. () B, B., B,..9 () f(x, y) (, ). xy f(x, y) = x + y (x, y) (, ) (x, y) = (, ) () C f(x, y) x f y y f x = ( 8) ( s846)., z = f(x, y) x = r cos θ, y = r sin θ z = f(r cos θ, r sin θ) r. (3) f(x),. x dy y. a >, f(x) = (e x )(e x a). dz z f(t)dt = x (x t) f(t)dt ( 8) ( s8463) () y = f(x),, lim f(x),. x ± () y = f(x) x S. S (3) S (), lim a + (a ) 3.. (). (a) lim x log sin x (b) lim (sin x)x x + x + (). (a) x + x (b) tan (x + 3) ( 8) ( s8464). (). x 4x dx ( 9) ( s946) (). (3) y = sin x sin x dx ( x ), y = π y y V. [, sin x π, π ]. ( 9) ( s946) 8
.3 a, f(x, y) = log(x a + y a ). () () f (x, y), x f (x, y), x f x (x, y) + f (x, y). y f x (x, y) + f (x, y) = x >, y >, a. y ( 9) ( s9463).4 = {(x, y) x, x y }..5 A = (). (). log( + y )dxdy 3 3 3. () A. () A. (3) () A..6 R 3, v =, v = 3 ( 9) ( s9464) ( 9) ( s9465), v 3 = 6 4 V = {c v + c v + c 3 v 3 c, c, c 3 R}.. () v, v, v 3. () v, v V. x (3) V = y R 3 αx + βy + γz = α, β, γ. z a (4) a + V a. a + ( 9) ( s9466).7 (). A = 3 3 9
(a) A,. (b) P, A. () a, + a a B = a a 3. B,..8 x y = (, ). ( 9) ( s9467) ( 9) ( s9468).9 z = f(t) C. t = x + y, yz x xz y =.. x + y = 4 x + z = 4 S, ( 9) ( s9469) ( 9) ( s946). a, b, a >. f(x) = bx + x + a x =,. () a, b. () f(x), lim f(x),. x ± (3) y = f(x) x, y S. ( 9) ( s946). P, Q, R, S. [ 6 P =, Q = 3 9. ], R =, S = 3 3 6 4 6 3 3 5. (),. (i) S RQ (ii) t P RS (iii) R t Q + t (Q t R) (iv) SP P () P, Q, R, S,. x (3) x. T = S + x., T, x. x.3 a, A.. A = a a a a ( 9) ( s946)
() A rank A. x () a =, x, y, z A y x (3) x, y, z A y = = z a a.,,. 7 3. z 3 a a, ( 9) ( s9463).4 () tan x = sin 3 5 x., tan x sin x. () y = (x + )e 3x n. (3) n I n = sin n xdx.. I n = n sinn x cos x + n n I n (n 3) ( 9) ( s9464)