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0  . . ˜. W˜: A =   . a2n

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. †‚ l1 , l2 : 1. y²: ü†‚É¡; 2. ¦ü†‚ úR‚. n. •þ α1 , α2 ‚5Ã', α3 , α4 ‚5Ã', … α1 o. A Ú B ´ü‡ØÓ Ê. α1 , · · · , αn ´ V 8. 3

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A

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u α3 , α4 , y²: ùo‡•þ‚5Ã'.

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(α, αi ) = ci .

A Š©O• λ1 = λ2 = 9, λ3 = −9 Ú λ1 † λ2 ©OéA A A ±9¦ A é z

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•þ• α1 = (1, 0, −1), α2 =

P.

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˜. W˜K(z˜ 5 ©, 1.

60 ©)

g-‚ x2 − 4xy + y 2 + 10x − 10y + 21 = 0

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-¡a.´

•þ, K•þ| α1 + α2 , α2 + α3 , α3 + α4 , α4 + α1

.

5. 3 3 ‘¢•þ˜m R3 ¥,

α1 = (−1, 1, 1)T , α2 = (1, −1, 0)T , α3 = (1, 0, −1)T , β = (−4, 3, 4)T . K β

3Ä {α1 , α2 , α3 } e ‹I´ .  a1 + b1 a2 + b1   a1 + b2 a2 + b2  6. n > 2, K det  .. ..  . . 

n > 1, Ý

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λ2 + 2λ

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2

λ

an + b1

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an + b2 .. .

     

0

λ2 − 1

λ

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···

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 8. λ− Ý

. XJ-‚•§´ x2 −y 2 −1 =

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7.

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u

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A

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é¡V‚5¼ê• f (X, Y ) = tr(X T Y ), X, Y ∈ V,

¢‚5˜m V þ

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Ï):   x1 + x2 + x3 + x4 + x5 = 1      3x1 + 2x2 + x3 + x4 − 3x5 = −2   x2 + 2x3 + 2x4 + 6x5 = 5     5x + 4x + 3x + 3x − x = 0 1 2 3 4 5

5

g

x−1 y z ˜mþk†‚ l1 : = = Ú l2 : (x, y, z) = (3 + 2t, t, 3t − 3). 3 1 0 √ … π † l1 ål´ 91, ¦ 𠕧.

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‚5˜m U

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50 ©)

1. ü‡²¡ z = x + 2y Ú z = −2x − y 2. : (0, 2, 1)

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(α1 , α1 )

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A ´Ã•‘‚5˜m V

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50 ©)

1. 3 R3 ¥, †‚ x = y = z †²¡ z = x − y 3

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Y

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‹I©O• A(1, 0, 1, 0), B(0, 1, 0, 1), C(1, 1, 1, 1), K 4ABC

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u

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gê•

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.

5. ¦‚5•§|    a2 x1 + x2 + x3 = 1   x1 + ax2 + x3 = a    x + x + x = a2 1

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A

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1

0

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λI − A

.

Ð Ïf|•

{λ, λ + 1, λ2 , λ2 , (λ − 1)2 , (λ − 1)3 }, KA

• õ‘ª dA (λ) =

, rank(A) =

n > 2, K¢ g. Q(x1 , · · · , xn ) =

8.

n X

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i=1

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úR‚•§.

ü‡f˜m. ¦y:

dim(W1 ∩ W2 ) + dim(W1 + W2 ) = dim W1 + dim W2 . 11. ( 20 ©)

A ´ê• F þ n ‘‚5˜m V

‚5C†. ®• A

A õ‘ª ϕA (λ) = f (λ) · g(λ), Ù

¥ f (λ) † g(λ) ´ê• F þ ü‡pƒ õ‘ª. ¦y: (1) Imf (A ) = Kerg(A ); (2) V = Imf (A ) ⊕ Img(A ). 12. ( 15 ©)

Q(x) ´ n

¦y: V ´ Rn 13. ( 20 ©)

R

2×2

¢

g., V = {x ∈ Rn |Q(x) = 0}.

f˜m ⇔ Q(x) ´Œ ½½ŒK½ . þ ‚5C† A (X) = AX − XA, Ù¥ A =

(1) ¦y: f (X, Y ) = tr(X T AY ) ´ R2×2 þ SÈ; 9

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1

1

2

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(2) ¦ ImA 3 f e ˜|IO 14. ( 15 ©)

n > 2, ¦Xe n

¢•

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A = (aij )n×n 

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1 ..

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..

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..

1 .

    ,  

O

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10

i + j ∈ {n, n + 1}; i+j ∈ / {n, n + 1}.

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1. ü†‚ 1 − x = 2y = 3z † x = y + 2 = 2z + 4 ¢ê a, b,  1   1 3. ¢•   0  0 2.

c ÷v 1

0

1

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0

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3

2

1, x, x , x

x − 1, (x − 1) , (x − 1) e Ý  λ  1   Ð Ïf|• . 1   1

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A, B ´ n

A † B Ãú 11. ( 20 ©) Ú f2 (x)

E•

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¿‡^‡´ g(A ) Œ_.

11

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ª. ¦y: Kerh(A ) = Kerf1 (A ) ⊕ Kerf2 (A )

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1.

: †‚ x + 1 = y + 2 = z + 3

2.

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3. ý

ål•

l%Ç•

. •§•

.

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f (x) = x2 − 2ax + 2 Ø g(x) = x4 + 3x2 + ax + 2, K~ê a = .   0 1 0 1    −1 0 1 0  −1  A= , A Jordan IO/•  0 −1 0 1  , K A =   −1 0 −1 0

4.

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Z

a

7. ( 15 ©)

b

f (x)dx = f (ξ)

g(x)dx. ξ

lim an = a ∈ R, y²:

n→∞

(1) ˜?ê

∞ X

an xn

ÂñŒ» R > 1.

n=0

(2)

f (x) =

∞ X

an xn , @o lim (1 − x)f (x) = a. x→1−

n=0

Z (3)

lim (1 − x)

x→1−

8. ( 15 ©)

0

x

f (t) dt = a. 1−t

z = z(x, y) 3 R2 þkëY

ê, …÷v•§ 14

Á(½ λ

∂2z ∂2z ∂2z +5 − 2 = 0. 2 ∂x ∂x∂y ∂y

Š, ¦ 3C† ξ = x + λy, η = x − 2y, (λ 6= 2) 23

(1)

e, •§ z{• ∂2z = 0. ∂ξ∂η ¿dd¦Ñ ‡©•§ (1)

).

9. ( 15 ©) → − F =



1 y x bx cxy a − + , + 2,− 2 y z z y z



Ù¥ a, b, c ´n‡~ê. → − (1) ¯ a, b, c ÛŠž, F •k³|. → − (2) F •k³|ž, ¦Ñ§ ³¼ê. 10. ( 15 ©)

u(x, y) 3 R2 þkëY

ê, …ð u

Š. y²: u ÷v•§

∂2u ∂u ∂u = ∂x∂y ∂x ∂y

¿©7‡^‡• u(x, y) = f (x)g(y).

24

¥I‰ÆEâŒÆ 2015 cïÄ)\Æ•ÁÁKêÆ©Û

15.

o ‡&ú¯Ò: sxkyliyang 1. ( 15 ©) ¦4•  lim

x→+∞

2. ( 15 ©) ¦ 3. ( 15 ©)

¼ê F (x, y) = √ a, b ´

1 sin x

x

Z

| sin t|dt. 0

x y 34«• x > 0, y > 0, x + y 6 1 þ •ŒŠ. +p 2 1+x 1 + y2

ê. OŽ -È© ZZ

(x2 + y 2 )dxdy, D

Ù¥ D ´ý 4. ( 15 ©)

y2 x2 + 6 1. a2 b2

R > 0. OŽ-¡È© ZZ  S

 1 3 xy + x dydz + yz 2 dzdx + R3 dxdy, 3 2

Ù¥ S ´þŒ¥¡ x2 + y 2 + z 2 = R2 ( z > 0) ••Šþ. 5. ( 15 ©) OŽ2ÂÈ© Z 0

6. ( 15 ©)

n > 0. ¦yØ

+∞

1 dx ( n > 1). 1 + xn

ª 1 < 2n + 2

7. ( 15 ©)

Z

π 4

tann xdx <

0

1 . 2n

α α ∈ (0, 1), {an } ´ î‚4Oê , … {an+1 − an } k.. ¦4• lim (aα n+1 − an ). n→∞

8. ( 15 ©) ?Ø?ê

∞ X √ √ ( n + 1 − n)α cos n

^‡Âñ5ÚýéÂñ5.

n=1

9. ( 15 ©)

f (x) ´«m [0, 1] þ

ëY¼ê¿÷v 0 6 f (x) 6 x. ¦y: Z

1

Z

2

x f (x)dx > 0

¿¦¦þª¤• 10. ( 15 ©)

2

1

f (x)dx

.

0

ª ¤këY¼ê f (x).

f (x) 3 [a, +∞) þkëY

¼ê, …

lim sup |f (x) + f 0 (x)| 6 M < +∞.

x→+∞

¦y: lim sup |f (x)| 6 M. x→+∞

25

16.

¥I‰ÆEâŒÆ 2016 cïÄ)\Æ•ÁÁKêÆ©Û o ‡&ú¯Ò: sxkyliyang

1. ( 15 ©) ¦4• 1 lim x→+∞ x4 + | sin x| 2. ( 15 ©) ê

Z

x2

0

t3 dt. 1 + t2

{an } ÷v 1 , n > 1. 1 + an

a1 = 3, an+1 = y²: ê

{an } Âñ, ¿¦Ù4•. Z +∞ sin x −xu 3. ( 15 ©) y²: e dx 3 [0, +∞) ˜—Âñ. x 0 4. ( 15 ©) OŽÈ© ZZ

z(x2 + y 2 )3

p 1 − (x2 + y 2 )dS,

S

Ù¥ S ´þŒ¥¡ x2 + y 2 + z 2 = 1, z > 0. 5. ( 15 ©)

¼ê f (x) ± 2π •±Ï, f (x) = 1 − x, x ∈ [−π, π).

¦ f (x)

Fourier ?ê, `²Ù Fourier ?ê´Ä˜—Âñ.

6. ( 15 ©) y²: e¼ê f (x) 7. ( 15 ©)

ê f 0 (x) 3«m (0, 1) Sk., K¼ê f (x) 3«m (0, 1) Sk..

f (x) 3«m [0, 3] þëY, 3 (0, 3) SŒ , …÷v f (0) + f (1) + f (2) = 3, f (3) = 1.

y²: 3«m (0, 3) S•3˜: ξ, ¦ x2 y2 + =1 a2 b2

f 0 (ξ) = 0.

8. ( 15 ©)

S ´dý

ƒ‚† 2 ‡‹I¶Œ¤ «• ¡È, ¦ S

• Š.

9. ( 15 ©)

¼ê f (x) 3«m (0, 1) þ•à¼ê, =?‰ (0, 1) ¥ ü: x1 , x2 , ±9?¿ t ∈ (0, 1) k f ((1 − t)x1 + tx2 ) 6 (1 − t)f (x1 ) + tf (x2 ),

y²: ¼ê f (x) 3«m (0, 1) þëY. 10. ( 15 ©) 00 F (x, y) ∈ C ∞ (R2 ), F (0, 0) = 0, Fx0 (0, 0) = 0, Fy0 (0, 0)Fxx (0, 0) > 0.

y²: d F (x, y) = 0 (½

Û¼ê y = f (x) 3 x = 0 NC÷v f (x) 6 f (0) −

26

00 1 Fxx (0, 0) 2 x . 4 Fy0 (0, 0)

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17.

o ‡&ú¯Ò: sxkyliyang 1. ( 15 ©) ¦4• R x2 0

lim

x→0

2. ( 15 ©) ¦1

sin tdt . tan x4

.-¡È© ZZ

x3 dydz + y 3 dzdx + (z 3 + 1)dxdy, S

Ù¥, S ´þŒ¥¡ x2 + y 2 + z 2 = 1, ••÷¥¡ {•þ• . 3. ( 15 ©) y²: 2 π

Z

+∞

0

 1 − x, sin2 u cos(2ux)du = 0, u2

x ∈ [0, 1] x>1

. 4. ( 15 ©)

α > 0, {an } ´4Oªu á Z ak+1 ak+1 − ak 1 (1) 6 dx. α+1 α+1 x ak+1 ak

(2)

∞ X ak+1 − ak k=1

ak+1 aα k

ê

. ¦y:

Âñ.

f (x, y) ∈ C 1 (R2 ).

5. ( 15 ©)

(1) y²: éu?¿

a, b, c, d ∈ R, Ñ•3 (ξ, η) ∈ R2 , ¦ f (a, b) − f (c, d) = (a − c)

(2) e

∂f ∂f = , … f (x, 0) > 0 é?¿ ∂x ∂y

∂f ∂f (ξ, η) + (b − d) (ξ, η). ∂x ∂y

x ∈ R Ѥá, y²: éu?¿

(x, y) ∈ R2 , Ñk

f (x, y) > 0. 6. ( 15 ©) y²: Z

+∞

sin(x2 )dx

0

^‡Âñ. 7. ( 15 ©)

D ´1wµ4-‚ L ¤Œ «•, ¼ê f (x, y) 3 D þk

ëY

ê, …÷v

∂2f ∂2f + = 0. ∂x2 ∂y 2

(1) ¦y: I −f (x, y) L

∂f ∂f dx + f (x, y) dy > 0. ∂y ∂x

(2) e f 3 L þð•~ê c, ¦y: f 3 D þ•ð•~ê c. 8. ( 15 ©)

f : [0, +∞) → [0, +∞) ´˜—ëY , α ∈ (0, 1]. ¦y: ¼ê g(x) = f α (x) •3 [0, +∞) þ˜—ë

Y. 9. ( 15 ©)

 z z x − ,y − = 0. y²: y x   ∂z ∂z (xFu + yFv ) xy + z − x −y = 0. ∂x ∂y

F (u, v) ∈ C 1 (R)2 , … F



27

10. ( 15 ©)

«m I = [0, 1], fn (x) † f (x) þ´ I þ ëY¼ê, n, 1, 2, · · · . … fn (x) > fn+1 (x), ∀x ∈ [0, 1].

lim fn (x) = f (x).

n→∞

¦y: (1) éu?¿

ε > 0, I=

∞ [

{x|x ∈ I, fn (x) − f (x) < ε}.

n=1

(2) fn (x) ˜—Âñu f (x).

28

18.

¥I‰ÆEâŒÆ 2018 cïÄ)\Æ•ÁÁKêÆ©Û o ‡&ú¯Ò: sxkyliyang

1. (1) ¦4• π

lim |x|arctan x+ 2 ;

x→−∞

(2) ®• ak • êê

∞ X sin(a x) k , … 6 | tan x|. x ∈ (−1, 1). y²: k2 k=1

ak = o(k 2 ), k → +∞. 2.

Φ(x) •±Ï• 1 (1) ¦ Φ(x)

iù¼ê.

ëY:Úmä:

a.;

(2) OŽÈ© Z

1

Φ(x)dx. 0

− − 3. ®• Ω • R3 ¥ k.•, → n •ü •þ. ¦y: •3± → n •{•þ 4. ®• f (x) •±Ï u 2π 5.

Û¼ê,

²¡²© Ω

x ∈ (0, π) ž, f (x) = −1. Á|^ f

NÈ.

Fourier ?êOŽ

∞ X

1 . (2n − 1)2 n=1

ϕ(x) •k³| F (x, y, z) = (x2 − y, y 2 − x, −z 2 ) e ³¼ê, ¦n-È© ZZZ ϕ(x, y, z)dxdydz, Ω

Ù¥ Ω • x2 + y 2 + z 2 6 1. 6. ®• f ∈ C 2 [0, 1], f (0) = f (1) = 0, … f (x) 3 x0 ? (1) ¦ f (x) 3 x = x0 ?

Lagrange {‘

• Š -1,

Taylor Ðmª;

f 00 (ξ) = 8.

(2) y²: •3 ξ ∈ (0, 1) ¦

7. ®• Dt = {(x, y) ∈ R2 |(x − t)2 + (y − t)2 6 1, y > t}, f (t) =

ZZ

p x2 + y 2 dxdy, OŽ f 0 (0).

Dt 2

00

8. ®• u(x) ∈ C[0, 1], u(x) ∈ C (0, 1), u (x) > 0, - v(x) = u(x) + εx2 , ε > 0. (1) y²: v(x) • (0, 1) þ î‚à¼ê; (2) y²: u(x)

•ŒŠuà:?

.

9. ®• Br = {(x, y) ∈ R2 |x2 + y 2 6 r2 }, B = B1 , u(x, y) ∈ C(B) ∩ C 2 (B), 4u =

∂2u ∂2u + 2 ∂x2 ∂y

4u > 0, ∀(x, y) ∈ B, y²: u(x, y) 3 B þ •ŒŠu>. ∂B þˆ ;   Z d 1 (2) 4u = 0, ∀(x, y) ∈ B, y²: u(x, y)dS = 0, ∀r ∈ (0, 1). dr 2πr ∂Br Z 1 (3) y²: u(0) = u(x, y)dS. 2πr ∂Br Z 2−t
10. ®• u(x, t) äk ëY , …÷v utt (x, t) = uxx (x, t), P F (t) = u2t (x, t) + u2x (x, t) dx, y (1)

t

dF (t) ²: 6 0. dt

29