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[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v = v grad u 3 l ux, y, z = c c : v l = xi + yj + zk + v.3 z e z e z = e z z z 9 s9 3 e nz n 4 d dz enz = ne nz 9 s93.4 π fx = x π < x < π x = π π π = 4 3 + 5

y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s9.7. 3 x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a = 3 y = fx, y x y y x y,. : e =.78, log e = fx = a. 9 s94 3.9 A det A A. A =.... 3 6. 3 9 s95. 9 s94

x + y 3z = 9 x y + z = 3x + y + z = 34 x =, y =, z =. 9 s94.,. tan 3x x 3 lim = lim x x x e x = 4 lim x cos x log cos x = log x 5 lim = x log cos x.3,. log x 3 lim x log sin x = 9 s943 π 4 x cos xdx = π + π 3 sin x cos xdx = 3 5 e x 4 x dx = 3 x log xdx = 4 e 3 + 6 x x dx = 5 xe x dx = + e 7 x e x dx = e 9 s944.4. y = 3 + x x x 3, 3 log, e. π..5 A =..6 B = 3, Im f Ker f..7 fx =. 9 s945 A, t AA, A., t A A 9 s95 fv = Bv f : R 4 R 3 9 s95 x =. + sin x 9 s953.8 = {x, y x, y } dxdy 4 + x + y 3. 9 s954 3

.9 x, y, < x < π, < y < π, fx, y.,. fx, y = sin x + sin y sinx + y fx, y f x, f y, f x, f y, f x y. f x = f = x, y. y 3 fx, y,. 4 z = fx, y x = π, y = π.. y = yx y, z = zx.,. 9 s955 z = y 4 dz dy, y dx dx. z = y 4, dy dx + yp x = y5 Qx z. 3 dy dx + xy = xy5.. x y, fx, y fx, y = 4 x + y. 9 s956 x + y, z fx, y R,. R. R. 3 xy x + y. z = fx, y.. fx = e x. fx x. 9 s957 fx. 3 y = fx..3. x log xdx, log x..4. 9 s96 x dx 9 s96 f, f, f Kerf Imf,. x x x + y x + y + 3 a f y =, b f y = x 3y y + z 4 z z, V W F : V W, KerF = {v V F v = }, ImF = {F v W v V }.,. 9 s963 4

.5 A,,,. 3 6 4 A = 3 3 9 s964.6. xy dy dx = x + y d y dx + n dy x dx + a y =,., n, a. a n =. b n =. 9 s97.7 x i, y i, x i+, y i+,, x i+ y i+ x y = = A x i y i,., i, A. x y A. = 4 3, x 3 y 3 = 8 3,. A,., A. 3, A n., n, 4 x n, y n., n, 9 s97.8 n, z n = n n z, z,, z n., n z i arg z i, arg z < arg z < < arg z n < π.,, z, z,, z n P, P,, P n, O.,. 3 P, P, P 3., P P P 3 S 3., 3 P, P, P 3 O, OP P, OP P 3, OP 3 P G, G, G 3. G G G 3 A 3. 3, n P, P,, P n O n G, G,, G n. n P P P n S n n G G G n A n r n = A n S n. 4 3 r n n lim n r n. 9 s973.9 x, y, z, xy x a + y = a > y 4a. 5

x.y, z = x, y, z., z >. a 3, z P, P,, a, a 3, a 3 x, y. P, P,. 9 s974.3 n., n 3.., n P., n. k. 3, k. 4,. 9 s975.3 x, y {x >, y > } fx, y = x log y. f. f. 9 s98.3., a >, b >. ax + by dxdy.33. y x y x x y x y x +y 9 s98 9 s983.34 R 3 = {x, y, z x, y, z R 3 } H = {x + y + z = }. f : R 3 R 3, v R 3, H v. f A. 3 A..35 6 d y dx + dy dx..36. fx, y = y 9 s984 y = y = yx y = 3 lim yx = x + 9 s99 + xy 3y x3 3 + x x 9 s99.37 xy y = log x x x =, y. x dxdy 9 s993 6

.38 A. 3 5 4 6 8 4 5 5 A. E, t te A =..39 A, B A, B, A B. a + b c + d b A = 3, B = a + b c + d a b a b c d 3a b a + b c + d a b.4 A, E. A =, E = detxe A = x λ x λ. λ, λ. 9 s994 9 s9 V = {x R 3 : A λ Ex = }, V = {x R 3 : A λ Ex = }. V v V v. 3 A λ Ev 3 = v v 3. 4 v, v, v 3 R 3. T : R 3 R 3 T x = Ax, v, v, v 3 T..4 fx = x cos x x 3. gx = log + x x 3. 9 s9 3. x cos x log + x x log + x lim, lim x x sin x x e x 9 s93.4 y = fx C x, y = sin t, t cos t, t π/. t = π/4 C P x, y.. f x, P C. f x. 3 C x. 9 s94.43 z + i = 3 C,., i. z dz + C C z 4 dz 9 s95.44 A = [ 4 3 ]. 7

.. 3..45,. x + y + 4x + y 3 dy dx = d y dx dy dx 6y = 4ex : x =, y =,.46 fx = 9 s9 : x =, y = dy dx = 9 s9 + e x x. 9 s9 log x.47. lim x x.48. 9 s9 x + y + z = x + y + 3z = 3x + y + z = 5 9 s93.49 6., 4,..,,. **,,,. 9 s94.5, 5,. T, F. p q p q p q p q p q q p T T F F T T T T F F T F T F F T T F F T T F F T T F T,, 7 5 = 9.,, 7 5 = 9. 3 7 5 = 35,. 4 7 5 =,. 5,, 7 5 =. 9 s95 8

.5. lim x sin x x.5 R 3 A = 3 3 3 lim x sin x x x.. 9 s96 A T = A + AT, A T = A AT., A T A. T. 3 T {x R 3 T x = }..53. = {x, y, z x, y, z, x + y + z π/}.., a. x sina + xdx 9 s97 3, 3. z sinx + y + zdxdydz 9 s98.54. dy dx = ay + b, a, b, y = y y b. dy dx = kyp y, k, p, y = y < y < p. 9 s99 sin x cos x.55 lim x x x x 3 cos x sin x x x dx x < x < x.56 3x + 4xy + 5y = f = x + y.57 a b x y z = ax by z = x y 4az = 4b 9 s93 9 s93 9 s933 9

.58 X, Y fx, y = π e x +y / Z = X + Y < x <, < y <.59 A A = A... 9 s934 9 s935.6 V C n. V f : V V f f = f, V = Im f Ker f., Im f Ker f, V. Im f = {fv v V }, Ker f = {v V, fv = } 9 s936.6 n θ, θ,, θ n, Θ i, j θ i j n. p k = θ k + θ k + + θ n k k.. t ΘΘ., t Θ Θ. p p p... p n p p p 3... p n A = p p 3 p 4... p n+........... p n p n p n+... p n Vandermonde det Θ = nn /. det t ΘΘ = i<j n i<j n θ i θ j θ i θ j 3 θ,, θ n n A tr A k = p k k., tr..6 gx n n d xgx = xg n x + ng n x. dx R fx x u n x = x y n fydy n! n d u n x = fx, n =,,. dx 9 s937 9 s938

.63 a, b a + b =, = {x, y R x + y }, ax + by dxdy. ax + by..64 lim x + x + x. x { u = ax + by v = bx + ay fx = sin x. 9 s939 9 s93.65 fx [a, b]., fxdx = fc b a a x = c a, b. fx, gx [a, b]., [a, b] gx > b, b a gxdx fxgxdx = fc a x = c a, b. b 9 s93.66 x xy + y =,.,,. x xy + y = x, π π g, y f f g,. A f g : x x A y y, A A. 3. e x xy+y dxdy {x, y < x <, < y < }.67 A, B : A B = {x x A B x A B}.,, A A. 9 s93 A B,., U. A U B A B, A, B, A, B,,. 3 A, B, C, A, B, C,.

U A B C,,,,,. 9 s933.68 fx = x ln x x >,. f x, f x. fx. 3 x + x + fx. 4 fx. y = x ln x x x = a < a < Sa. 5 Sa. 6 Sa lim Sa. a + : e =.7..69 f : R 3 R 3 f x x 9 s934 ax + ax + x 3 = ax + x + ax 3 a R,. x 3 x + ax + ax 3 f F. f. 3 f, f..7. 9 s935 x +y +z dxdydz., = {x, y, z : x +y +z } 9 s936.7 d y dx 5 dy dx + 6y =. 9 s937.7 fx, y = x 3 + y 3 + 3xy +. f f f x y x y. fx, y. 9 s938

.73.. x + y z = y + z = x + y z = x + y z = y + z = x + y z = 9 s939.74 A. t xax = x 4xy + 5y [ ] [ ] x [ ] A =, x =, t x = x y. 5 y A. P t P t P AP. 3 x. 9 s93.75 x >. log. log + x > x x 9 s93.76 A, B v, w. a, b. A, B A, B a, b, v, w. 9 s93.77 ABC, tan A, tan B, tan C,. 9 s933.78 G. T.,,..,., G. A T, A G. T 99%. 9 s934.79! = 3 9. 5! = 3 3 5. 9 s935.8 x 3 sin x cos x = A sinx α, A, α. A >, α < 36..8 4 cos x cos x + 9. 9 s936 9 s937 3

.8 3 3. 9 s938.83 i z = z a + bi. i =, a, b. + i 9 s939.84 A, B, C,. A, B, C,,,. A : B, C. B : A, C. 3 C : A. 9 s933.85 4, 3,, log.3,. 8 8, 3 4 3, 4 log 3 + 3 log, log 3.48. log 7, log 7. 9 s933.86 C : x + y + z 6x 4y + 4z 8 =. C yz. C y. 3 C A6,,. 9 s933.87, 5,, 5, 5 3, 5..,. 3, 3, 3,. 3 3,. 9 s9333.88 y = fx = x 4 8x + ax + b x = α, β α < β x. a, b, α, β. y = fx x. 9 s9334.89 k= kk + = 3 + 4 + 3 5 +. 4

.9 A = k = + 3 + 5 +. k=, k = + + 3 + = π π. 6 3 k=. 9 s9335 A. A n n. 9 s9336.9 Ax = b, A = 3 α b = β x = x x x 3 x 4 α, β. A ranka.. 3. 9 s9337.9 fx = + x α,. + x α = + αc i x i i=, αc i. αc i = αα α i + i! i gx x 3. gx =.93. 4 x dx.94 fx. fx = dx x 3 + + x + x 9 s94 9 s94 x 5 x + 4 5 6 x 6 fx. fx, x. 9 s943.95 A R S. 5 3 A = R + S,, A = 7 5 3, R S. 8 6 9 s944 5

.96 a, b, c. x a = b = 3 c = 3 a b., x. 9 s945.97. x dy dx = x + y dy dx + y tan x + cot x = 9 s946.98. dx x + y = et dt 3x + dy dt y = 9 s947.99 a, b, A = a a a a, B = b b b b. A. AB a, b.. A = 4 4 3 3. 9 s948 A. P AP P. 9 s949. + t = t + t t 3 + + n t n + n tn + t t,. log + x = x x + x3 3 x4 xn + + n 4 n + R nx x >, R n x = n x t n dt. + t 3 x R n x n. 4 < x < R n x n. 9 s94 6

. y = x y = x + xy., ydxdy..3 A = 3 a.,. A, a ±. a = 4, A..4,. 9 s94 9 s95 3 + 3 4 + + nn + n + = nn + n + n + 3 n =,, 3, 4 9 s95.5 x, y, fx, y = x y + ax + by 9,. fx, y =, 3, 4. a b. fx, y., x y..6 tan θ = x,. sin θ = x x. cos θ = + x + x..7,.. 9 s953 9 s954.,,.,,, 5., 5. 9 s955.8 y = sin x π x π.., y = Arcsin x fx = Arcsin x,., fx = Arcsin x f x. 3 f x,.9 a =, b =, c = f xdx. 9 s97,. 7

a, b, c. 3 d = a, b, c. 5., y y = e x, y =, y =. z n = lim n z n. n i n =,,,,. z n. 3. x, y, r, θ.. n= 9 s97 9 s973 nz n. n= 9 s974 ρ >. fx, y = r sin θ A = {x, y < x < ρ, < y < ρ} Iρ = fx, ydxdy B = {r cos θ, r sin θ < r < ρ, < θ < π/} Jρ = fx, ydxdy A B. Iρ Jρ ρ,..3 x, y, r, θ.. fx, y p u = a, b, p u, f p. u 9 s975 fp + tu fp lim t t fx, y = r sin 3θ o u = cos φ, sin φ f o., u f x o, f o. y fx, y p,, f f p x y f p = a f p + b f u x y p. f o..4 A = / / / / / /,. 9 s976 A λ < λ < λ 3 v, v, v 3. 8

ε u n u n = A n + ε, n =,,,., ε = + ε, A., u n v, v, v 3. 3 x, x = x x. u n,. u n+ b lim n u n a lim u n n d u n n. u n c lim n u n.5 {a, b, c}. 9 s977 n n X. 3, f : X Y,. a fa B = fa fb c f C = f C f b fa B = fa fb d f C = f C f, A, B X, C, Y., X X, Y Y, fx = {fx Y x X }, f Y = {x X fx Y }. x + x + x 3 = 3.6 x + x x 3 = x + 5x 4x 3 = 3.7 A =.8 bc ca ab b + c c + a a + b. k sin x sin x.9 lim x x 3.. 9 s978 9 s98, A a, b, c k. 9 s98 9 s983 9 s984 β. sinλx + µdx, α, β, λ, µ., λ. α 9 s985 {. = x, y x, y π }, e x sin y dxdy. 9 s986 9

. log =.3, log 3 =.477. 4,. 4 3,..3 A = 4 3. 9 s987 A. A. 9 s988.4 n, n+ + 3,..5 A = 4. 9 s989 A. A. 3 A..6. x + y + z = a x + 3y z = 3x + ay z =.7 = {x, y x + y π}.. 9 s99 9 s99 x sinx + y dxdy 9 s993.8 P t xt = ft, gt., ft, gt [, ], f t, g t., P t = O, t = A,., dx dt t = f t, g t OA. 9 s994.9 r = 3,, F =,,.. xy, r e. N = r F. 3 r F. 9 s9

.3 C, R, L, qt, a = R L, b =, qt CL,. d q dt + adq dt + bq = a 4b >,. 9 s9.3 e ix = cos x + i sin x, i =,.,. cosα + β = cos α cos β sin α sin β sin 3α = 3 sin α 4 sin 3 α.3 5.. 9 s93 5...33 θ Rθ = cos θ sin θ sin θ cos θ, K = 9 s9.. RθK = KR θ. tan θ Aθ, Aθ = RθK. 7π π 3 A Aθ = R Aθ. 9 s9.34 A3,, B, 4. P x + y =, ABP. π.35 sin xdx. k= b 9 s93 [a, b] fx S = fxdx. [a, b] b a a n n, a = a < a < a < < a n < a n = b. n fa k + fa k S n = b a S. n [., π. ] 3, π sin xdx S 3 9 s94.36 y y + y = y =, y = y = yx. y. z = log y, z = zx. y.

.37 A x y z, A 9 s95 x y y = z 3 3. z x A. A 3. 3 A. 9 s9.38 fx f x = fx, f =.,. e x fx =., fx. fx. 3 lim log n! 3! + 4! + n n!.39 ε = {x, y x + y ε, y x} < ε < Iε = dxdy.,. π x y ε. 9 s9 log Iε ε. Iε. 3 lim. ε + ε.4 A = 4.. 9 s93 A. n, 3 n A n. 9 s94.4 R 4 V. V = {x, y, z, w R 4 x + y + z + w =, x + y + z + 3w = } V. V..4. < a < b < π, < c <, sin d d sin b b < sin a a. 9 s95 = c d, π..43 = {x, y R x, y }.. e x +y dxdy 9 s96 e x +xy+4y dxdy 9 s97

.44 a, a,, a n M n log + x a k x k Mx n+ x, a, a,, a n. lim n k= k= n k+ = log. k 9 s98.45. { x x fx = x = { x m x > fx = x. fx.. m 3, f x. 9 s99.46 n, I n. I n = I. x n e x dx,. n,. I n = e + n I n 3. I n = I n 4. lim n n!.47 ux, y, z u =., v = x u x + y u y + z u z n! e n k= k! 9 s9 u = u x + u y + u z, v. 9 s9.48 n, n = {x, y R x + y n, x, y } e x y y dxdy. n e x R n = {x, y R x n, y n}, y dxdy y dxdy =. lim n R n e x 3..49 { e x n e x dx x + x + 3x 3 + 4x 4 = 4x + 3x + x 3 + x 4 = 9 s9 x x x 3 x 4 R4 V. V R 4. V. 3 R 4, V. 9 s93 3

.5 4 4 A b R 4,. 3 A = 3 5, b = 3 8 8 a x = x x x 3 x 4 R4 Ax = b a. a, Ax = b. 3 ImA = {Ax x R 4 }..5 A = 5 4 4 6 5 6.. 9 s94 A. P AP P. 9 s95.5 M. A = M, F : M M 3 4 F X = AX XA X M, M KerF = {X M F X = O}, ImF = {F X X M}., O. 9 s96.53. d dx cos x < x < d dx xex 3 d dx log e x x x > e 9 s93.54. x sin xdx x + x x + 5 dx 9 s93.55 / β L = β/ β β/ β / β, < β <,. L x x = L y y, x y = x y. 4

L L L/ β + I = O., I, O. 3 L. / γ 4 L M = γ/ γ γ/ γ / γ x x = ML y y, x y = x y., < γ <. 5 L. 9 s933.56 A, B, C S. a = a i, b = b j, c = c k., i, j, k., n = a b + b c + c a S. S P x, y, z r, S r a b + b c + c a = a b c. S x, y, z, a, b, c. 3 a = 3, b =, c =, O S d. z C c n A i a S k o j r P b B y x.57 f θt = + sin θt + 3 cos θt +, f θ t, θ t = 5 + cos θ t + sin θ t + θ t,.58, 4 xydxdy 9 s934 df dt. df dt. 9 s935 ; x y x 9 s936.59., 4 y =., y, y, dy dx, d y dx. 5

y + y = y = y + 3y 3 xy + xy = 4 yy + xdy + y x dx =.6. sin x lim x e x = d = dx tan x.6 fx =,. x f x, f x f x. f n. 3, fx x =. 9 s937 9 s94 9 s94.6 { x, y, r, θ. x = r cos θ, r, θ < π,. y = r sin θ r θ, x y. r θ x x. 3, log r x..63. a 9 s943 b a x a dx a > 9 s944.64 y = x, x = y. 9 s945.65 x, y x + y a a >, y.66.. Ax = 3 x, A = 3 6, x =, =. 5 6 9 x x 3 9 s946 A. A. 3. 9 s947 6

.67 a, b, c,. a = 3 5, b = 3 6, c = p, q, r,. p = 3, q =.68 regular matrix., r = A B, AB. 6 9 3 9 s948 A, A + B, E + A B. E..69,. x dy dx = y 9 9 s949. y =..7,. x dy dx = y + 9 9 s94. y =..7 R 3, a =, b =, c =.. a, b, c. 9 s94, a, b, c,,. 3, a, b, c., p a.7 dfx dx, q a, b, r a, b, c p, q, r. = fx,. 9 s94. f =. 9 s943.73 gx = cos x, x = π Taylor. 9 s944 7

x 4.74 lim x x 3. 9 s95.75 fx, y = x 4 xy x + y. 9 s95.76 fz = ux, y + ivx, y vx, y = y x, ux, y + y, fz z., z = x + iy, i. 9 s953.77 sin xdx. 9 s954.78 x + ydxdy., = {x, y x, x + y 3}. 9 s955.79 dy + xy =. dx dy dx + xy = x + x3, y =. 9 s956.8 3 P,,, P, 3, 5, P 3, 5, 7,. P 3 P P l. P l M. 3 l M..8. dy dy = y = y y dx dx 3 dy dx = y + sin x 4 d y dx 3 dy dx y = 9 s957 9 s958.8 d x dt + ω x =., ω. d x dt + ω x = cosωt., ω,, : a ω ω ; b ω = ω. 9 s959.83 fx, y = loge x y + e xy.,. f f x,. f x y 9 s96 8

.84 3 4 5 3 5 7 9 7 6 5 4 3 3 5 7 9 x y z u v =. 9 s96.85 n A A 3 = O., O. A =., A A. a b n =. A =, A a + da = O. c d 3 n = A = O. 9 s963.86., xy,. x + y dxdy.87 dy = yy dx,. y = a., a. 9 s964 yx, lim x yx. 9 s965.88 3 A,,, B 3,,, C,, 3. e 3x.89 lim. x x [ : lim x x log e + x =.].9 fx = x + 5x + x +, x 3.9, dz dt. fxdx. z = sinx cosy, x = e t, y = log e 3t 3.9 3. 4,. 9 s966 9 s967 9 s968 9 s969 9 s96 9

.93 4. 8 6. 4 + 8, 6. 4 8, 6...94 A. A = 3 9 s96 A. A P,, P AP P. d x t x t 3. = A, x =, x = dt x t x t.95 log x x = a, Taylor., a. 9 s96 9 s963.96 lim x sin x x =. 9 s964.97 logsin t + t..98 9 s965 e x e x + e x dx. 9 s966.99,. x dy + y =, x = y =. dx 9 s967. F, P Q, W, W = F P Q. F = 4i 3j + k, P 3,,, Q,, 4 W., i =,,, j =,,, k =,,. 9 s968. e x, x =,., e 3,. 9 s969. φx, y, z, Ax, y, z = A x i + A y j + A z k,.., i, j, k 3 x, y, z 3

rot grad φ = :. div rot A = 9 s96.3., C. sin x sin 3xdx sin x sin 3x, cos,...4. 9 s96 9 s96.5. 3 x x + dx.6 y dy dx = ex.7 A = 3 a a e log xdx 9 s963 y =...8,. A = 3 a a, B = 3 6 A = 6, a. a =, A t A B t A B. 3 B B. 4 B. 9 s964 9 s965 5 4.,. 9 s97.9,..,., 3 8. 3

3,.. y = fx = x + x. fx < x < 3. fx t, ft h. 9 s97 3 h x =, x =, y = St., t t. 4 St t =. 5 t =, y = fx h, x =, x =. 9 s973. S : x + y + z + x 6y 4z + 5 = P. : x = y = z +, a,, α. 3 S P α. α, a. 4 α S, S P. a, P h. 9 s974. A, B, 9. A, 9 9. B, 9 m m 9m.. A,. B,. 3 A, B, m = 3 m. 4 A B n n 9. A B,, m n. 9 s975.3,,,,. t,, t +,.75,.5. t,, t +,.5,.5,.5. t,, t +,.75,.5. t,,, X t, X t, X 3 t, t +,,, X t +, X t +, X 3 t +.. x t + x t x t + = A x t A. x 3 t + x 3 t 3

A. 3 A. 4 A., 3..4 fx = logx + x + 9 s98,., x >. fx x. e fxdx., e. 3 fx, y = fx. 9 s98.5 3,... n, P n. 3 n. 4 n 5% n., log =.3, log 3 =.477, log 5 =.699. 5 P n, N,, 6 5 N. N np n. n=.6 x > fx = log x x., π e e π..7. x x+y x y zdzdydx.8 a. 9 s983 9 s99 π π x sin y y dydx 9 s99 a a a a a a 9 s993.9 A, P AP P 3. A = 9 s994 33

. a, b, c,.. a =, 3, 5, b =,,. a =, 3, 5, b =,,, c =,,,, a b c. x x 3y z + 4y + 6z z z = λ x x, y, z x : y : z., x, y, z.. x + dx t = x + x +..3. dy dx = y x.4. y z 9 s93 λ, λ 9 s93 9 s933 d y dx + 4 dy dx + 3y = 9 s934 xy x + y dxdy : x + y, x >, y > 9 s935.5 k, k A rank, k, k. k A = k Av = v = x y, v x, y, z z, A rank. 9 s936.6 x. a x xa x 3 x 3 x 4 x x cos t sin tdt 9 s937.7. e log e xdx π e x sin xdx 9 s938.8 n, n! n,. 9 s939.9. y x = xy y x < xy 9 s93 34

.3 E =, O =,. 4 A =, A 5A + 6E A 5. 3 a b A = A A 8E = O, a + d, ad bc. c d 9 s93.3 R 3, 3 e = t,,, e = t,,, e 3 = t,,.., t. λ e + λ e + λ 3 e 3 = λ, λ, λ 3., e, e, e 3. 3 R 3 a = t a, a, a 3 e, e, e 3. 9 s93.3 u, u, u 3, 3 P, Q, R p, q, r,,. p = u u, q = u + u 3, r = u + u 3 q p Q P, u, u, u 3. αp + βq + r, p q, α, β. 9 s933.33 x 3 3x x. y = x 3 3x. 3 x 3 3x y =.,3,,,,,, x, y.,,,,,, x, y,,,.,,,.,,,. 9 s934.34 6,. 6.. 3 4. 9 s935 35

c cos x.35 fx = x π x > π,. fx c.. 3. 4. 9 s936 a c.36 A = a, b, c T t T T = T t T = I c b [ ], t T T, t α T AT = β. n A n α n = T T. β n a + b = α + β, ab c = αβ. 3 a = b =, c =, α, β., T. 9 s937.37 xy = {x, y : x + y <, < x, < y} fx, y = xy x y. fx, y x, y. I = fx, ydxdy..38 A =, a = A, A 3. Aa, A a, A 3 a. c 9 s938 c. 3 a Aa, A a c..39 fx = x x + 4. lim fx lim x x fx. fx..4 a. xy C : y = x + a < x <, C : y = log x x > 9 s93 9 s93... a. 36

y = a C. 3 C, C y = a. 9 s933.4.,. 9 s934 a b.4. b a.43 x, y,, r, θ., 9 s935 x = r cos θ y = r sin θ dx dy dr dθ. x, y r, θ t, dt dt dt dt. 9 s936.44 I. I = x cos x dx.45. 9 s937., a. d y. dx + a y = Ae ibx, A, a, b, i. a y = Be ibx. b. d y dx + a y = 9 s938.46 r = xi + yj + zk., i, j, k, x, y, z.. r, r., = i x + j y + k. r, div r. z w = ck, v = w r, v., c. v, rot v..47 A = 9 s939,. xe A., x, E 4. A. 9 s934 37

e x x.48. lim x + x.49. 9 s934 e x + e x + dx 9 s9343.5 fx, y = 3x + y 3 6xy. fx, y. fx, y..5 d y dx + 3 dy + y =. dx 9 s9344 d y dx + 3 dy dx + y =, y =, y =. 9 s9345.5 xy x xy + y =. x, y x xy + y =, x + y,..53 y = y + y.. 9 s935 y = 3,...54 A = 5 4. 9 s935 A,. P, A. 3 A n..55 ABCEF, Q.. AP, AQ a, b. CQ F P R, 9 s9353 AB = a, AF = b, C P, E AR a, b. 3 AR CF S, CS : SF..56 N S N = n= n= N u n., n= 9 s9354 lim N S N = S S. u n = nn+n+3 = 4 + 3 5 + 3 4 6 + 4 5 7 + + nn+n+3 + 38

u n = n + r n = 3r + 4r + 5r 3 + 6r 4 + + n + r n + n= n=, r <.,.57 x, y, z r, θ, ϕ : lim n nrn =. 9 s9355 x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ r >, θ π, ϕ < π,. e x +y +z Iα = x + y + z dx dy dz α, α = I. 3 Iα α. 4 Jα, β = B x + y + z α logx + y + z dx dy dz β α, β. {, B = x, y, z : x + y + z < }. 4.58 A = a a a a a 9 s9356,., a. A. a, a, a. 3 A, a. a,. 4 A, A n., a <. 9 s9357.59, n,, n., Q, n. n. i M i. M i n. i, i,, i m n m. M i, M i, M im n m. 3 E P E. A A P A A, A A P A A. P A A = P A + P A P A A. A, A,, A N P A A A N N P A A A N = l S l l= N i 39

S l = k <k < <k l P A k A k A kl ii., ii, N,,, N l k, k,, k l. l = N S = P A j. i. j= 4 Q, n = n j j! j=. 5 lim Q, n. n 9 s9358.6 x t + axt = sin t., a. x t + axt =.. 3 a >. xt x = α, x = β., α, β. 9 s9359.6. sin x + x dx 9 s935.6 fx = x 4 π x π fx = a + a n cos nx + b n sin nx, a n n =,,,, b n n =,, 3,. n+ n = π. n= 9 s935 n=.63 X, Y N,, W, W Z., W = X + Y, Z = W. N, fx, Φt Φt = t fxdx W..,. z, P Z < z Φ. 3 Z. 9 s935.64 z, e z = 3i z., i. a >,..65. x x 4 + a 4 dx 9 s936 dy dx = a x + y a > d y dx 4 dy + 4y = 6ex dx 4

9 s936.66,. x + 3x + 3x 3 x x x x 3 A, A. T, T AT T., T T. 3. 9 s9363.67, u, u, u 3.. v = u + u 3, v = u 3 + u, v 3 = u + u, v, v, v 3,,. w = u u 3, w = u 3 u, w 3 = u u, w, w, w 3,,..68. a b + c b c + a c a + b 9 s938 9 s938.69 xy,,,.. e x y e x +y 4 3 x + ye x y.7. xy dydx = {x, y R x, y } x + y dydx = {x, y R x + y } 3 x + y e x +y dydx = {x, y R x + y } xy 4 x + y dydx = {x, y R y x, x + y } 9 s9383 9 s9384.7 fx x. fx >,. f + f + + fn F x = lim n x ftdt., f + f + + fn F n n fxdx f + + fn + fn. lim n =. fn =., F n 9 s9385 4

.7. y y = y + y = 9 s9386.73 {a n } lim a an n. a n+ = n a = a = 4, a n = a n =,, 3,. a <, lim n =. n 3 a > 4, lim n =. n 4 4 > a >, lim n =. n.74. xy dxdy, : y x x y dxdy, : x + y 3 x ye x+y dxdy, : x + y, x y 9 s9387 9 s9388.75 y = yx y + 5 x y =.,. y = ze x /, z = zx. z, z..76 R 3 e, e, e 3 e = T R 3 R 3,, e = 9 s9389, e 3 =. T e = e + e, T e = e + e 3, T e 3 = e + 3e e 3.,. T. KerT, ImT..77 A = [ 3 3 5 ]. 9 s938 A. A. 3 A., P AP = B P B. 4 n, A n. 9 s938 4

.78. : e. y = x 3 y = log sin x 3 y = x x.79. lim x sin bx sin ax 9 s939 a, b sin x lim x x 9 s939.8 r,. θ.,..8. : e. e.8 z = x a + y b.83 = log x dx x. { x, y π x sin xdx 9 s9393 9 s9394 a, b > x, y, z, x, y x a + y b } a, b > 9 s9395, x dxdy. 9 s9396.84 A =,, 5 B =, 3,,., a, b, c a, b, c x, y, z. A B J = A B. xy, A, x C. 3 = B AA B/ A A. 9 s9397.85 u, v. u = a b c v = a b c a b c a 3 b 3 c 3 9 s9398.86 A λ x., x A λ x, Ax = λx A. [ ] A =.87. 3 3 3 9 s9399 9 s939 43

.88 A. A =.89. [ 3 3x + y + 3dx + x + 3y + dy = y 4y + 3y = x.9,.. ] 9 s939 9 s939 OA = 3 i + k, OB = 3 i + 4 j + k, OC = j + 3 k., i, j k x, y z, O OA OB. OA OC, OA OC θ cos θ. 3 OA OC OAC. 4 OA, OB OC..9 Rθ = [ cos θ sin θ sin θ cos θ ],. 9 s9393 RαRβ = RβRα = Rα + β R α = Rα.9 3 4.93 [ 3 9 6 3 6 4 3 4 ]..94..95. 3 4 5 9 6 5 3 5 8 9 7 9 3..96 m. 9 s9394 9 s9395 9 s9396 3x + 6y + z = 4 4x + 9y + z = 3 x + 3y + 4z = 9 9 s9397 9 s9398 m 3, 5, 3. 4 3 9 s9399 7 44

.97 [ 3 4 ]. 9 s939.98 7. {7, 76, 7, 68, 78, 73, 74} cm.. 9 s939.99 N P N = n = λn n! e λ, λ >, n =,,,., N : e λ λ n = n! 9 s939 3 4.3 A, ranka. A = 4 3 n= 3 4 m, n, m n A, A m n A. ranka ranka. 3 m, n, m n A, A m n A. A ranka ranka. 9 s94.3 fx [a, b], a, b] F x F x = x ftdt.. x a.3 a F x x F x fx F x. a, b] fx F x. 3 a, b] F x. 4, F x = ex. x dy y y. p dp dy = y y3 y <, p = p C [,. { y y + y 3 =, y <, x R, 3 x sin x.33 lim x x cos x.. lim yx =, x x x e x n., n. 9 s94 9 s943 3 z = fx, y, x = gt, y = ht C d z, dt. 9 s94 45

.34 I n x,., n. I n x = x sin n t dt I x, I x, I x, I 3 x. sin n t = sin n t sin t, n, I n x. 9 s94.35, P Q. a b a b P =, Q = a b c d c d c d.36 l = 3 6 9 s943 7 l. a = 8 l b, l c, a = b + c, b c..37 A,. A =.38. 9 s944 3 3 3 3 3 5 5 P, P AP., P. 3 n, A n. π π π π cos nx dx, π π cos mx cos nx dx. 3 π, fx =. sin nx dx { π π π/4 n. sin mx sin nx dx 9 s945 π π π < x < π/4 < x < π sin mx cos nx dx n, m 9 s943.39,,,,..4..3...,. 46

3., log = 3.3, log 3 =.58,..3 a. A = a a a a,. 9 s943 a = 3, A. A, a. 3 a, A..3 x, fx = sinx x. 9 s944 lim x fx. f x. 3 lim x f x. 9 s944.3 = {x, y : x, y }, S = {x, y : x + y 4},. dxdy S dxdy..33 ux. xydxdy. 3 S xydxdy. 9 s9443 u + u 3u + = u =, u = 3 v 3v + v = vx. ux = log yx,, ux..34 fx, y = x 3 6xy + y 3,. fx, y. fx, y..35 fx x = x 9 s9444 9 s945 ε, δ x x < δ fx fx < ε.,. fx = x,. x =, ε = δ. x =, ε > δ, fx = x x =. 47

fx = m x, m > n m x =, x, 3 5. 3. a f 3 b f c f 4 8 4 M. x < x x m M m > M. 5 fx x =..36 α >, ft =. f. t >, lim x e αx x t =. 3 t >, αft + = tft. 4 n, fn +. 9 s945 e αx x t dx t >., 9 s9453.37 M 3 R 3, A M 3 R., L A = {B M 3 R B O, AB = O} R A = {C M 3 R C O, CA = O}., O 3., L A.,. A. 4 6 a, A = a + 7 8 a, 4, a. 3 P L A 3 P. 4 R A = { t Q Q L A }., t Q Q. 9 s9454.38 M R,,,.,. { },,, M R. a b tr : M R R tr = a + d, tr. c d 48

3 Ker tr = {A M R tra = } M R. 4 Ker tr. 9 s9455.39. 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 >.3 π x sin xdx. 9 s946 dx. x x3 9 s946.3 fx, y = y x e t dt. f x, f y, f y x. 9 s9463.3 a, = {x, y : a x a, a y a},. e x y dxdy.33 λ λ λ λ 9 s9464 = λ. λ λ, λ x λ x λ x 3 = λ x, x, x 3, x 4. x 4.34 log xdx. 9 s9465 < a <, fa = a fa. e log x a dx. b a < a <, fa a. 9 s9466.35 a, b, c, [ a A = a ], B = 3 b b [, C = c 3 c ], 49

[ = 5 3 ], F = [ 3 4 7 5 ]. AB. C t., t. 3 AB = C t + F, a, b, c..36 x, fx = x t 3 + t dt. + 9 s9467 fx. x > fx >. fx 3 lim x x..37 4 x x x x fx =. 9 s9468 fx. fx..38 x > fx = + x 3,. f x, f x, f x. 9 s9469 fx x = x., 3 f x. 3, 8. 3 3..39 x, y, fx, y = f x, f y..33 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 5

.33 x y : y x, z = xe y z =. 9 s9463.33, dy/dx. y = x x 9 s9464.333. log x dx x > 9 s9465.334. π/.335. sin 3 θ dθ 9 s9466 dy dx = xy 9 s9467.336 P, P x, y, x, y. P, P O P OP = θ, cos θ..337 x,. x d y dx x dy dx 3y = 9 s9468 x = e z dy dy d y x., dx dz dx d y dz, dy dz, x. x = e z x i z. 3 x i..338 z = x y. i 9 s947 z = x y xy z =,. z = x y, x + y = xy z = : z x y, x + y. 3 z = x y, x + y = : z = x y, x + y. 9 s947.339 φ = x + y + z, φ =., u, v rx, y, z = u, v, u v., r 3.,. P =,, u r u. v r v 5

r u, r P. v x, y, z. 9 s9473.34 X. X a P X a, P X a.. a < T P X a = + πa sin T a T T T a X X X, X < X. X B. 3 X px. 4 X. 5 X..34 p >, f n p = 9 s9474 dx, n =,,. + nxp. lim n fn n < ε < f n p [, ε ] [ ε, ]. < f n p < ε + ε + nε p 3 lim n f np =. 9 s9475.34 ω, yx, < x < y + y + y = sin ωx. ω =,. ω,. 3 ω, x yx. 9 s9476.343 3 S : xu, v, yu, v, zu, v, u, v., u, v., S µs x y y z z x µs = u u x y + u u y z + u u z x dudv v v v v v v a b.,.. c d 5

S z = fx, y, x, y µs = + f x x, y + f y x, y dxdy. x + y + z = 4π. 9 s9477.344 A = a ij i, j =,. A T A. x = x x y = y y x, y = x y + x y. x, Ax, Ax = x, x, A T A = AA T = E., E. A. A. 3 A, θ cos θ sin θ A =. sin θ cos θ.345 p n A n. i = j {,,, n } p i = j = n A n i,j = i j =, A n i,j A n i, j., A = p. p = 3 A 3. p = A 3. 9 s9478 3 A n A n a n. {a n } n, a n, n =,,,..346 a, e, fx = e ax.. n, fx n n f n x. fx x. x n 3 N,. n N! n=n 9 s9479 9 s947.347 n =,,,. x n x + dx n,,.,. x n x + dx 9 s947 53

.348 4 A. A = 4 4 4 4 4 4 9 s947.349 A n,,. α, C n x. c : x, Ax = αx n =, α. n 3, α. 9 s9473.35 dy/dx. y = x 3 + 3x + 5 y = sin x 3 y = x + 3 4 y = x log x 5 y = expx/x 9 s949.35 fx = expx, f x f x. fx = expx, f, f, f. 3. fx = f + f x + f x /! + + f n x n /n! + f n θx x n /n! < θ < fx = expx 4., expx = e x. 4 4, exp. 9 s949.35. x 3/7 dx 4 x log x dx 5 cos3x dx 3 4x dx 9 s9493 tan x dx.353 y = x y = x +..354.,..355 x x =, x 3. y x + 9yx =. 9 s9494 9 s9495 54

3 fx = x π x π fx = a + [ a n cosnx + b n sinnx ], a, a n, b n. n= 9 s9496.356. x log x dx. C. 4 sin x e 3x dx 3 x dx x { } y dxdy, = x, y x, x y x 9 s9497.357 3 Ax = A x x, y, z, A y x, y, z, A z x, y, z. Ax Ax = Ax Ax Ax.358., log. x sin x lim x x 3 lim log + x x.359 fx, y = xy3 x y..36 fx = e x sin x,. a >, I a =.36 3 A = fx dx. 5 3 8 4 3 a 9 s9498 x fx dx. A A. 9 s9499 9 s949 9 s949 9 s949.36 3 a = 3,,, b = 4,,, c =, 3, R 3,. a, b, c. a, b, c W = a, b, c dim W..363 3 B = 9 s9493,. 9 s9494 55

.364. x lim x x lim x x sin x 9 s9495.365 R,. 9 s9496.366. x x + x + e x cos x 9 s9497.367. x dy dx + y = x3 y 3 y y + y = 3 y y + y = cos x + sin x 9 s9498 [ ] [ ].368 e = e =. V V [ ] [ ] [ ] y = fx fe = 5 x fe =, x = 3 fx..369 A =, x 9 s9499 A + A 8E = O., A A., E, O, A,,, A.., A = A 9 s949.37 α,. x + x + x 3 = 5 x x 4 = 7 x + x + x 4 = 4 x 3x x 3 x 4 = α.37 x,., log e. 9 s949 y = sin 3x cos 3x y = x sin x 3 3 y = log x 4 y = x.37 n y n. y = sin ax y = e ax sin bx 9 s949 9 s9493 56

.373., C. x dx x + dx x + a 9 s9494.374 x a + y b = x V., a, b,. 9 s9495 4x + 3x 7.375 lim x x 5x + 3. y = log + x x. 3 x e x dx. 4 fx, y = x 5xy + 3y, x, y. f f., 3, x y 9 s9496.376. dy = x y dx.377. d y dt dy 3t = 5 sin t dt 9 s9497 x + y z = 3x + y + z = 7 x y + z = 9 s9498.378., x y z z, i, j, k., i + j + k =. i, j, k m y i, j, k.,,, i >, j >, k <. m 5m x 9 s9499.379 a, b A,. A = a b a 4 A, b > a. A b., a., a. 9 s95 57

.38 R.. θ = θ = π/. BC. θ, x y R θ. 3 θ = π/, AB. π/ dx AB = + dθ dy dθ dθ AB. 4 A { π/ A = R θ sin θ cos θ + θ sin θ } dθ π 4. I = π/ C R O θ sin θ cos θ dθ. y θ θ = π/ B x A θ = 9 s95.38 x + y + z = r, fx, y, z = r = x + y + z f x x, y, z, f y x, y, z, f z x, y, z. f xx x, y, z + f yy x, y, z + f zz x, y, z.., f x x, y, z =.38 π/ fx, y, z x cos 3 x sin x dx.,, f xx x, y, z = fx, y, z x,. x 3 e x dx. 3 x y dxdy, = {x, y x y }. { 4 x = e t sin t y = e t cos t, 9 s953 t π, L. 9 s954 [ ] a.383 A. A = b,. [ ] A B = BA. a b. A, a b. [ ] 3 3 A, a b. 58

.384,, y = dy dx. y + y = y + y = x + 9 s955 3 y = x = y = y + y = x +..385 z/ x z/ x y. z = x 3 5xy + 9 s956.386. A = 9 s957 5 8 9 s958.387 x. I = x log x dx 9 s959.388. y = tan x cot y xx yy + y =.389. sin x cos 3 x dx 9 s95., a, b. a + y b = 9 s95.39 fx = 9x 3 + 6x 9x +, fx =. x.39 x y = a c b d x y 9 s95,.,,, 3, 7, 3, 5. a, b, c, d., x + y =., a, b, c, d..39 fx, y = e x3y f xy x, y. 9 s95 9 s95.393 = {x, y R x, y, x + y a }, x y dxdy., a >. 9 s95 4.394. 9 s953 59

.395 f : R 3 R 3 fx =. x x R 3, f. f..396 W = a + b a b 9 s954 a, b P,. P W. W P Q..397 fx, y = log + x + y,. fx, y f x, f y. 9 s953 x = r cos θ, y = r sin θ r >, θ < π, f r, θ f r, f θ x, y. 9 s953.398 fx, y = x + y 3 xy.,,,..399, = {x, y y, y x y}... 9 s9533 xy dxdy 9 s9534.4. dx + x + sin t = 9 s9535 dt.4. xe x dx dx 4 a + x a >.4. x + x dx 3 + x + 3 sin x cos x dx 9 s954 a x + xy dx + xydy = b xy + sin x dx + x y + cos y dy =. y + 4y + 3y = 3e x 9 s954 6

.43 ABC EF GH, E EC H E θ., a = F G AB, b = A, c = AE, c, l. b A C a B 9 s9543.44.. A =. A, A 3, A 4, A 5, A n..45 A = 3 4, B = 3 4 5, C = 9 s9544, AC, BC, BA. 9 s9545.46 y = log x x = a, log x = kx k. 9 s9546.47 x a + y =.,. b 9 s9547.48 xyz 4 A,, 6, B,,, C 3,,, x,, 3. a = s AB + t AC s, t, a x, y, z. 4 A, B, C,, x..49. y = cosθ sinθ y = tanx logx 9 s9548 9 s9549.4. dx 3x + 3 sin θ dθ 9 s954.4. 5 9 3 6 7 3 6 9 s954 6

.4.. { π θ π}.. y = 3 sin θ + π/3 9 s954.43 f f3x fsin x = a lim. 9 s9543 x x.44 dx 4x. 9 s9544 + 6x + 3.45 P,,, P 3,,. x, y, z,, 3, 5, 4. 3 O,,, P, P,,, 3, 4,,. OP OP. [ ] [ ] 4 A =, B =, A BA + B = A B. [ ] 5..46 A = a a a a >. 9 s9545 9 s955.47 d u dx du dx u = x+ 5 u =, u =, u = ux x.., x u = ux 9 s955 3.48 A =, B =, BA. 4 5 6 k A =, AX = X. 4 3 3 4.49 a = 4, b = a b x. a + b a b x. x 3 9 s9553,..4 A = 3 4 9 s9554 λ, λ u, u. 9 s9555 6

.4 cos θ sin θ sin θ cos θ..4 fx, fx = n= 9 s9556 f n x n., f n x = n! fx n. x, e x sin x x 3..43 yz + zx + xy = zx, y. 9 s9557 9 s9558.44, a + jb a, b., j =. 8 5 j 3 + j + j3 + j + j.45 x dy 3y + x =,. dx 9 s9559 ux = yx, xdu = u. x dx x = y =,. 9 s955.46 cos θ sinθ cos θ sin3θ = A sin4θ, A. cos 5 θ = B cos θ + C cos3θ + cos5θ, B, C,. [ ] [ a a.47 A I, A =, I = b b + a b. 9 s955 ], A 5A I = 9 s955 e x + e x.48 lim. x cos x.49 x dx. x + fx = log sin x. 9 s956.43 a b c a b c. 9 s956 9 s9563 xy.43 lim x,y, x + y. 9 s9564 63

.43. x y dxdy = {x, y : x + y, x, y }.433 x d y dy + ax + by = a, b dx dx,., x >. 9 s9565 x x = e t dy dy., x dt dx. d y dy, x dt dx d y dx. 3,, x x = e t. 4 3, x d y dy + 3x dx dx + y =. 9 s957.434 A, B, C, P n n n, x, y, z, x, y n., x.,.,. y = Ax, z = By., z = P x P A B. y = Ax, z = Bx., z = P y P A B. 3 y = Ax, y = Bx, y = Cy, x = Cx, B A C..435 A = x y z 3 4 3 4 4 3., 9 s957 A. x = y = z =, A. 4 9 s958.436 3 R 3 a =, a =, a 3 =, a4 = 3., a, a, a 3. a 4 a, a, a 3. 9 s958.437 4 R 4 W : x W = y z R4 : x y 3z + w =, x y + z + w =. w 64

W R 4. W..438 fx = + sin x sin x..439 log x x. n lim n + 3. n n= log n n 9 s9583., fx x π log x dx. x,. 9 s9584, 4 lim a n = a n b n = a n + a n+ + + a n, lim b n = n n?...44 z = x y x + y, x = eu+v, y = e u v, z z u v..44 u = 3x + y, v = x 3y, xdxdy, = {x, y : 3x + y, x 3y }..44 n fn = x n e x dx,. 9 s9585 9 s9586 9 s9587 f. fn + = nfn. 3 n, fn + = n!. 4 f = e x dx. 5 f3f 5 f 3., f = π..443 a A = A. 3 5 4 4 a,. A., A A. 3 A ranka. 9 s9588 4 Ax = b, x.,, Ax = b, x., b =. 5 Ax =, x., a x., x, x =. 9 s9589 65

.444 u u x + u y + u z =. u = x y z x y z u = x + y + z.445 u = ux, t = πt e x π 4t, e x dx =., t >..446 A =,. 9 s958 ux, tdx = 9 s958 A,. A. 3 A λ i x i i =,, 3. 4 A..447 y = x e x. y y y. y x. 9 s958 3 y x x min y min,, y x x max y max. 4,, y. 9 s9583.448, fx π, [ π, π],. fx. fx = a + a n cos nx + b n sin nx a = π n= π π fxdx, a n = π π fx { x π x fx = x < x π π π fx cos nxdx, b = π π π, fx fx sin nxdx n =,, 3, 9 s9584.449,. dxt = Axt dt [ ] [ ] [ ] x t, xt =, A =, x =. x t 66

A λ, λ. A ν, ν. 3 T = [ν, ν ] [ xt ] = T yt, yt y t., yt =. y t 4 3 yt. 5 4 yt, xt. 9 s9585.45 x, y, z a =,, 3, b =, 3,,. a b. a b. 3 a b, 3, 4, 7. 4 3 x + 3y z 6 =..45 C C = 3,. C. = P CP P. 3 C n. 9 s959 9 s959.45 x., < x <. 9 s9593.453. x log x x x + x x 9 s9594.454 t = m, v rv. g, mg rv = m dv dt. t v z..455 y = e xx = expx x. fx = tan x f x = tan x. 9 s9595 9 s96.456 y = yx y y 8y =. 67

y y 8y = 5 cos 3x,.457 A =.,. 5 4 5 3 9 s96,.458 x + y x. dxdy, x + y.459. x + x dx dx 3 x qx q.46 fx = x e x,. 9 s963 9 s964 π cos x sin x + dx 9 s96 fx., g x = gx. fx x x.,. 3 fx., e 3,.3..46 y = x 3 x x +. 9 s96 x, x V..46 x = a, x =., a. a = x, x, x 3. x, x, x 3 a. a 9 s963, x 3 = a.463 A =,. 9 s964 ] A λ i, u i i =,, 3.,. A u i i =,, 3, R 3. 68

.464 n.,. + + + + > n 3 n 9 s965 9 s966.465 y = ax 3 x + y + y =., a. 9 s967.466 log x 4 log x =..467 9 s968 cos x sin x cos x dx. 9 s969.468 X, X. X 4 6 8 P.4, 3.. 9 s96.469 3cm. 6cm r dr/dt., t. 9 s96.47 n, + + 3 + + n = nn + n +. 6 9 s96.47 [, ], dx + x. 9 s963.47 N N. t, k N t., t = N = N. t = N = N. t = 8 N..473 A = p q p q,. 9 s964 A. p = q =, An n =, 3, 4, 9 s963 a.474 R 3,R 3 V = a a, b R b. 69

u, v V u + v V. V R 3. 3 V. 9 s963.475. lim x + x x log x 3 lim x x.476 I = a I = I = a x ax x dxa >,. a a a t + a a t dt. a a t dt. 9 s9633 3 I. 9 s9634.477 R 3 V = xy, yz, zx, divv rotv., divv V, rotv V. 9 s9635.478 {x, y, z R 3 : z = x + xy + y },,..479 y y = e x cos x..48 y 3y + y = e x sin x. 9 s9636 9 s9637,.. 5 3 4 8.48 8 4 7 6 9 9. 6 8 8 6 7 4.48 4 4 5. 9 8 9 s9638 9 s964 9 s964.483 7..484. 9 s9643 7

3x + x x 3 + x dx x π 3 π 6 dx cos x sin x tan x = t 9 s9644.485 fx, y = x 3 3x y + 3xy 3x..486. xe y dxdy, = {x, y x y, y }.487 x + y dxdy, = {x, y x + y } 7 5 5 7. 9 s9645 9 s9646 9 s965 x x.488 =, y x + y =, y. 9 s965 x y + z =.489 x + y + z =, x = y = z = k, 3x + y + kz =,. 9 s9653 sin x.49 lim x e x e x. 9 s9654.49 z = x + y x, y =, 3..49. 9 s9655 xe y dxdy = {x, y x, y x } 9 s9656.493. d y dx = dy y + x dx = dy 3 dx + y = x 4 d y dx 5 dy dx = 9 s9657.494 π fx = π x π x < π., + 3 4 + = π. 9 s9658.495. i. lim z i z 3 + z + i lim z i z + z i 9 s9659 7

.496, z = ± z = ±i, C.,. sin πz dz z C C z z 4 dz 9 s965.497,, 3., X, Y.,. P X = P Y =., X Y,..498 X fx =, P X a = 9 { x x x > a 9 s965 9 s965.499, 5. fx,,. fx = x ia, a a >, i i =, x, < x <. fx,,. ux = x a x + a vx = x + a ux, vx x. d ux dx d vx dx 3 ux, vx. 4 ux, vx < x < a. a o uxdx a vxdx 5 x = a fa. fa. 9 s966 7