() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

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0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A () Q = P AP P, Q () A k 0.4 e x 0.5 () x ( x + ) α lim x e x = α : xα ( 5) ( s5590) ( 5) ( s55904) () ( cos x ) α sin x ( 5) ( s55905) 0.6

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) = λ n + a n λ n + + a λ + a 0 a n,, a 0 I λ A f A (A) = A n + a n A n + a n A n + + a A + a 0 I = 0 ( ) A =,. () A () A ( 6) ( s6590) 0.0 f(x) = x log e e x () f (x) () f(x) () 0 f(x)dx ( 6) ( s65904) 0. (x, y) (x + y ) = 4x y

() (r, θ) (r = x + y, tan θ = y/x). () () ( 6) ( s65905) 4 5 0. () A = 6 4 0 0 4. 0 0 ( ) () A = AB = I B. 4 I. ( 7) ( s7590) ( ) ( ) ( ) 0 0. V, A = B = C = 0 0 0 A, B, C V,. 0.4 A = ( ) () A.. (). ( 7) ( s7590) () Φ = x T Ax Φ = aξ + bξ b a. x = (x, x ) T ξ = (ξ, ξ ) T U x = Uξ. ( 7) ( s7590) 0.5, y = dy dx. (x + y 4)y = x y + ( 7) ( s75904) 0.6 y = dy dx. y y y = x + x ( 7) ( s75905) 0.7. a x b x 0.8. lim x 0 x sin x ( 7) ( s75906) ( 7) ( s75907) 0.9 (x, y, z) a = (,, ), b = (,, 0),.

() a b. () a b. () (a b), (, 4, 7). (4) () x + y z 6 = 0. 0.0 C C = 0 0,. () C. () D = P CP P. () C n. ( 9) ( s9590) ( 9) ( s9590) 0. x., < x <. ( 9) ( s9590) 0.. () x log x () x x + (x )(x ) ( 9) ( s95904) 0. t = 0 m, v rv. g, mg rv = m dv dt. t v z. (λ 4)x + y 5z = 0 0.4 () x + (λ )y 0z = 0 4x y 5(λ 5)z = 0 ( 9) ( s95905), λ. () λ,. ( 0) ( s0590) 0.5 (x, y, z) a = (a, a, a ), b = (b, b, b ),. () (a, b), a b. () a b. () a b, a b. 0.6 f(x) = x x = 0 x4. ( 0) ( s0590) ( 0) ( s0590) 4

0.7. dy y = cos x sin x ( 0) ( s05904) dx 0.8. x dx ( 0) ( s05905) x (x + y ) dxdy 0.9. ( 0) ( s05906) x+y x,y 0 0.0 (x, y, z) a = (,, ), b = (,, ),. () a b θ. () a b n. ( ) ( s590) 0. a, b, c.. x + y + z = a x + y + 4z = b x + 4y + 5z = c 0. A A = 4,. () A. () A n. n. 0.. ( ) ( s590) ( ) ( s590) () y = x sin x () y = x ( ) ( s5904) 0.4. x dy dx + (y + 5) = 0 ( ) ( s5905) 0.5 f(x) = cos x., π x π. () f(x). () f(x) f(x). ( ) ( s5906) 0.6. () sin x sin 4x dx () x e x dx ( ) ( s5907) 5

0.7 a,. x y + z = x + y z = x + ay + z = x y + (a + )z = a 5 ( ) ( s590) 0.8. () x + y z 4 = 0 x y + z 4 = 0. () () (0,, 0). 0.9 A = 4 ( ) ( s590) P, A. ( ) ( s590) 0.40. () f(x) = xe x () f(x) = sin x cos x ( ) ( s5904) 0.4. dy dx = x y + y x ( ) ( s5905) 0.4 f(x) = e x. () x = 0 x. () () e 0.0. 0.4. () dx () e x sin x dx x log x 0.44 A A = () A. () A. () t A. 0.45 a =.,. k ( ) ( s5906) ( ) ( s5907) ( 4) ( s4590) 6 b = 4, 6

() a b k. () k a a. ( 4) ( s4590) 0 0.46 A A =,. 0 () A. () A. () P AP P P AP. 0.47. () f(x) = (x x ) 6 () f(x) = sin x ( 4) ( s4590) ( 4) ( s45904) 0.48. dy dx = y x + y 0.49 f(x) = x. ( 4) ( s45905) () f(x) x. () () f(0.0). ( 4) ( s45906) 0.50. ( ) log x () x cos x dx () dx x ( 4) ( s45907) cos θ sin θ 0.5 A A =,. sin θ cos θ () A A. () A A. () θ = π, x =, x x. x x y = Ax = y. ( 5) ( s5590) 0.5 (x, y, z) a = (,, ) b = (,, ),. () a b θ, cos θ. () a b. () (0, 0, 0), (,, ), (,, ). 7

0.5 A = 7 5 0 B = 5 4 ( 5) ( s5590),. () AB BA. () A. () A, B, AB. 0.54. () f(x) = x (x + ) () f(x) = sin 4 x cos x ( 5) ( s5590) ( 5) ( s55904) 0.55. ( ) ( x + xy dx + x y ) dy = 0 ( 5) ( s55905) 0.56 f(x) = e x sin x,. () f(x) x. () () f(0.0). ( 5) ( s55906) 0.57. () sin x dx () x 5x + 6 dx ( 5) ( s55907) 0.58 a = 0.59 A =, b =, c = k,. () a (a b)., a b, a b. () a, b, c, k. () (0, 0, 0), a, b. 0 0, x = x x x,. ( 6) ( s6590) () x T Ax = x T Qx Q., x T x. () A A. () A A. 0.60 A = 4,. ( 6) ( s6590) 8

() A,. () A P, A. () A n n., n. 0.6. () f(x) = (x ) e x () f(x) = log sin x (x nπ, n ) () f(x) = x x (x > 0) 0.6,. (x y + )dx + (y x + )dy = 0 0.6 5. () ( 6) ( s6590) ( 6) ( s65904) ( 6) ( s65905) cos x x4, + x + x 4. () tan x x 5, x + x + 4 x 5. () lim x 0 x tan x x, 5. ( 6) ( s65906) 0.64. () dx () sin x x (log x) dx ( 6) ( s65907) 0.65 A (),(). 4 A = 0 0 0 () A A. () A. ( 7) ( s7590) 0.66 a, b, c (),(). 7 6 + k a = 6 k, b =, c = 0 () a b k. 0 () a, b, c k. ( 7) ( s7590) 9

0.67 A (),(). 0 A = 0 () A. () A. ( 7) ( s7590) 0.68. () f(x) = ex x () f(x) = sin (x + ) ( 7) ( s75904) 0.69,. 0.70 f(x) = (y x + )dx + (x y + y)dy = 0 x + x. ( 7) ( s75905) ( 7) ( s75906) x + 4 0.7 lim. x 0 x ( 7) ( s75907) 0.7. () x log x dx () dx sin x cos x ( 7) ( s75908) 0.7 x, y, z. x y z = 4 0 ( 8) ( s8590) 0.74 A (),(),(). A = 4 (). () P AP P P AP. () A 5. 0.75 a, b, a =, b =. (), (), ()., u. ( 8) ( s8590) u 0 0

() a b a, b. () a b θ cos θ. () a b u. 0.76. ( 8) ( s8590) () f(x) = x () f(x) = x e x () f(x) = tan ( x) 0.77,. ( ) x + log y dx + x y dy = 0 ( 8) ( s85904) 0.78. ( 8) ( s85905) log( x) x, x + x + x. ( 8) ( s85906) x sin x 0.79 lim x 0 cos x. ( 8) ( s85907) 0.80. ( ) 0dx () 8x 4x () + sin x + cos x dx ( 8) ( s85908) 0.8 4 A,. a b c d A = b a d c c d a b d c b a () t A A, t AA. () (), A A. ( 8) ( s85909) 0.8 () A.,. A = () P, a, b, c. a 6 P = b 6 c 6 ( 8) ( s8590)

0.8 t(0 < t < π) x = a(t sin t), y = a( cos t),., a 0. () dy dx. () (), d y dx. ( 8) ( s859) 0.84 x + xy + y = x y x.. () () dy x y. dx dy = 0 (x, y). dx () () d y, y x. dx ( 8) ( s859) 0.85 a = y i x j C : r = (cos t) i + (sin t) ( j 0 t π ),. () d r. () a d r. () (), C I = C a d r. ( 8) ( s859) 0.86.. () f(x) = xe x () f(x) = x x + () f(x) = cos (log x) ( 9) ( s9590) 0.87. dy dx = y x y x ( 9) ( s9590) 0.88 f(x) = e x cos x x. ( 9) ( s9590) 0.89 x( + x ) dx. ( 9) ( s95904) 0.90. f(x) = cos x dx ( 9) ( s95905)

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