Методические указания по выполнению контрольной работы № 2 по математике для студентов инженерно-технических специальностей заочной формы обучения

yc

z; ycc

dz

dz

dy

dz

z;

yccc

d 2 z

z2

§ dz ·

2

ɢ ɬ. ɞ.

¨

¸

z

dx

dy

dx

dy

dy2

¨

¸

©dy ¹

ɉɨɞɫɬɚɜɢɜ ɷɬɢ

ɜɵɪɚɠɟɧɢɹ

ɜ

ɭɪɚɜɧɟɧɢɟ

ɜɦɟɫɬɨ

c cc

(n)

, ɩɨɥɭɱɢɦ

y , y ,..., y

yy

cc

y

4

c

2

,

c

0 .

( y )

y(0) 1, y (0)

Ɋɟɲɟɧɢɟ.

Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ

ɧɟ

ɫɨɞɟɪɠɢɬ ɧɟɡɚɜɢɫɢɦɭɸ ɩɟɪɟɦɟɧɧɭɸ,

ɩɨɷɬɨɦɭ ɩɨɥɚɝɚɟɦ y

c

z( y) . Ɍɨɝɞɚ y

cc

z

dz

dy , ɢ ɭɪɚɜɧɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ

dz

z

y3

.

dy

y

z

Ɋɟɲɚɹ ɟɝɨ, ɧɚɯɨɞɢɦ z

ry

y2 C1 . ɂɡ ɭɫɥɨɜɢɹ yc

z 0

ɩɪɢ y

1 ɢɦɟɟɦ

C

1 ,

ɫɥɟɞɨɜɚɬɟɥɶɧɨ,

z

ry

y2 1

ɢɥɢ

dy

ry

y2 1 .

ɂɧɬɟɝɪɢɪɭɹ ɷɬɨ

1

dx

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ

ɫ

ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ

ɩɟɪɟɦɟɧɧɵɦɢ, ɢɦɟɟɦ

arccos

1

r x

C2 . ɉɨɥɚɝɚɹ

y

1

ɢ x

0 ,

ɩɨɥɭɱɢɦ C2

0

, ɨɬɤɭɞɚ

1

cos x ɢɥɢ

y

y

y

sec x .

eDx [(C C

2

x ... C

r

xr 1 ) cosEx (C

r 1

C

r 2

x ... C

2n

xr 1 )sinEx] .

1

ɉɪɢɦɟɪ

5.1. Ɋɟɲɢɬɶ

ɭɪɚɜɧɟɧɢɟ

y IV 5ycc 4 y 0 .

ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ

ɭɪɚɜɧɟɧɢɟ k 4 5k 2 4

0

ɢɦɟɟɬ ɤɨɪɧɢ

k

r1, k

3,4

r2 . Ɉɛɳɟɟ ɪɟɲɟɧɢɟ

1,2

ɉɪɢɦɟɪ

5.2.

Ɋɟɲɢɬɶ

ɭɪɚɜɧɟɧɢɟ ycc 2 yc 5y

0 .

ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ

ɭɪɚɜɧɟɧɢɟ k 2 2k 5

0 ɢɦɟɟɬ ɤɨɪɧɢ k

1 r 2i . Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ

1,2

y

ex (C cos 2x C

2

sin 2x) .

1

ɉɪɢɦɟɪ

5.3.

Ɋɟɲɢɬɶ

ɭɪɚɜɧɟɧɢɟ

ycc 2 yc y

0 .

ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ

ɭɪɚɜɧɟɧɢɟ

k 2 2k 1 0

ɢɦɟɟɬ 2–ɤɪɚɬɧɵɣ

ɤɨɪɟɧɶ

k

1, ɩɨɷɬɨɦɭ ɨɛɳɟɟ

1,2

ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ

y

e x (C

C

2

x) .

1

ɉɪɢɦɟɪ 5.4. Ɋɟɲɢɬɶ

ɭɪɚɜɧɟɧɢɟ y IV

8 yccc 16 yc

0 . ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ

ɭɪɚɜɧɟɧɢɟ

k 5 8k 3 16k

0

ɢɦɟɟɬ ɤɨɪɧɢ

k1

0 , k2,3

2i,

k4,5 2i . Ɉɛɳɟɟ

ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɬɚɤɨɜɨ

y

C1 C2 cos 2x C3 sin 2x C4 x cos 2x C5 x sin 2x .

y(n)

a1 y(n 1) ... an 1 yc an y f ( x) ,

(5.4)

ɝɞɟ ai R,

f ( x) – ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ.

ɉɭɫɬɶ

y C1 y1 C2 y2 ... Cn yn –

(5.5)

­

C1c(x) y1 C2c (x) y2

Cnc (x) yn

0;

°

C1c(x) y1c C2c (x) y2c

Cnc (x) ync

0;

°

®

°

(n 1)

(n 1)

(n 1)

°

f (x),

¯C1c

(x) y1

C2c (x) y2

... Cnc (x) yn

ɝɞɟ

Cic

dCi (x)

– ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɣ Ci

( x) . Ⱦɥɹ ɭɪɚɜɧɟɧɢɹ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ

dx

ycc p x yc q x y

f (x)

ɞɚɧɧɚɹ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɜɢɞ

­Cc

(x) y

Cc

(x) y

2

0;

®

1

1

2

¯C1c(x) y1c C2c (x) y2c

f (x).

ɉɪɢɦɟɪ 5.5. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ ycc yc

1

.

1 e x

³(e

x

1)dx ³

dx

e

x

x ³

exdx

e

x

1

1

e

x

e x x ln | ex 1| C2 ,

~

,

~

– ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ.

ɝɞɟ C1

C2

Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɡɚɩɢɲɟɬɫɹ ɬɚɤ:

y

ln(e

x

~

x

( e

x

x ln(1 e

x

~

1) C1 e

) C2 ) .

L( y) { y(n)

a y(n 1)

... a

n

y

f (x) ,

(6.1)

1

ɝɞɟ ai R, f ( x)

– ɧɟɩɪɟɪɵɜɧɚɹ

ɮɭɧɤɰɢɹ.

ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɨɞɧɨɪɨɞɧɵɦ

ɭɪɚɜɧɟɧɢɟɦ ɛɭɞɟɬ

y(n) a y(n 1)

... a

n

y 0 .

(6.2)

1

ɉɭɫɬɶ

k n a k n 1

... a

n

0

(6.3)

1

y

y y *.

1)

ȿɫɥɢ

ɩɪɚɜɚɹ

ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (6.1)

ɢɦɟɟɬ ɜɢɞ:

f (x) P ( x)eDx ,

n

ɝɞɟ Pn (x)

–ɦɧɨɝɨɱɥɟɧ ɫɬɟɩɟɧɢ n, ɬɨ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (6.1) ɦɨɠɟɬ ɛɵɬɶ

ɧɚɣɞɟɧɨ ɜ ɜɢɞɟ:

y* xr eDxQ(x) ,

ɝɞɟ Q(x) A xn A xn 1 ... A

– ɧɟɤɨɬɨɪɵɣ

ɦɧɨɝɨɱɥɟɧ

ɫɬɟɩɟɧɢ n ɫ

0

1

n

ɉɪɢɦɟɪ 6.1. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ycc y

xe2 x .

Ɋɟɲɟɧɢɟ. ɋɨɫɬɚɜɥɹɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ

k 2 1

0

ɞɥɹ

ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ. ȿɝɨ ɤɨɪɧɢ

k1 1, k2

1.

Ɍɚɤ

ɤɚɤ

ɱɢɫɥɨ D 2 ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɟ ɹɜɥɹɟɬɫɹ, ɬɨ r 0

. ɋɬɟɩɟɧɶ

ɇɚɯɨɞɢɦ

yc (2ax 2b a)e2 x ,

ycc (4ax 4b 4a)e2 x

ɢ, ɩɨɞɫɬɚɜɥɹɹ ycc,

ycɢ y ɜ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɢɦ (ɩɨɫɥɟ ɫɨɤɪɚɳɟɧɢɹ ɧɚ e2 x )

4a 4ax 4b ax b x .

Ɉɬɤɭɞɚ ɧɚɯɨɞɢɦ:

x

3a

1,

a 1/ 3

x0

4a 3b 0, b 4 / 9

ɂɫɤɨɦɨɟ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ:

y*

1

(3x 4)e2 x ,

9

ɚ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ

y C ex C

2

e x

1

(3x 4)e2 x .

1

9

2) ȿɫɥɢ ɩɪɚɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (6.1) ɢɦɟɟɬ ɜɢɞ

f (x)

eDx (P (x) cosEx Q

m

( x) sinEx) ,

(6.4)

n

ɝɞɟ Pn (x) ɢ Qm (x) – ɦɧɨɝɨɱɥɟɧɵ n–ɣ ɢ m–ɣ ɫɬɟɩɟɧɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɬɨɝɞɚ:

ɚ) ɟɫɥɢ ɱɢɫɥɚ DriE ɧɟ ɹɜɥɹɸɬɫɹ ɤɨɪɧɹɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ

(6.3), ɬɨ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (6.1) ɢɳɟɬɫɹ ɜ ɜɢɞɟ

y*

eDx (uS ( x) cosEx vS ( x)sinEx) ,

(6.5)

ɝɞɟ

us ɢ

vs

–

ɦɧɨɝɨɱɥɟɧɵ ɫɬɟɩɟɧɢ s

ɫ ɧɟɨɩɪɟɞɟɥɟɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɢ

s

max{n,m};

ɛ) ɟɫɥɢ ɱɢɫɥɚ DriE ɹɜɥɹɸɬɫɹ ɤɨɪɧɹɦɢ ɤɪɚɬɧɨɫɬɢ r

ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ

ɭɪɚɜɧɟɧɢɹ (6.3), ɬɨ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (6.1) ɢɳɟɬɫɹ ɜ ɜɢɞɟ

y*

xreDx (us (x)cosEx vs (x)sinEx) ,

(6.6)

ɝɞɟ

us ɢ

vs

–

ɦɧɨɝɨɱɥɟɧɵ ɫɬɟɩɟɧɢ s

ɫ ɧɟɨɩɪɟɞɟɥɟɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɢ

s

max{n,m}.

1.

ȿɫɥɢ ɜ (6.4)

P (x) { 0

ɢɥɢ

Q

m

(x) { 0 ,

ɬɨ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ

y ɬɚɤɠɟ

n

ɢɳɟɬɫɹ ɜ ɜɢɞɟ (6.5), (6.6), ɝɞɟ s

m (ɢɥɢ s

n ).

2. ȿɫɥɢ ɭɪɚɜɧɟɧɢɟ (6.1) ɢɦɟɟɬ ɜɢɞ L( y)

f1 (x) f2 (x) , ɬɨ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ

y ɬɚɤɨɝɨ

ɭɪɚɜɧɟɧɢɹ

ɦɨɠɧɨ

ɢɫɤɚɬɶ

ɜ

ɜɢɞɟ

y

y y

,

ɝɞɟ

y

– ɱɚɫɬɧɨɟ

1

2

1

ɪɟɲɟɧɢɟ

ɭɪɚɜɧɟɧɢɹ

L( y)

f1 (x) ,

ɚ

y2

–

ɱɚɫɬɧɨɟ

ɪɟɲɟɧɢɟ

ɭɪɚɜɧɟɧɢɹ

L( y)

f2 ( x) .

ɉɪɢɦɟɪ 6.2. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ

ycc yc ex e2 x x .

Ɋɟɲɟɧɢɟ. ɋɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɨɞɧɨɪɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɜɢɞ

ycc yc

0 ,

ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ

k 2 k

0

ɢɦɟɟɬ

ɤɨɪɧɢ

k1

0,

k2

1. Ɉɛɳɟɟ

ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ:

y

C C

2

e x .

1

ɉɪɚɜɚɹ ɱɚɫɬɶ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɟɫɬɶ ɫɭɦɦɚ

f ( x)

f1 (x) f2 ( x) f3 (x)

e x e2 x x .

ycc yc ex ; ycc yc e2x ;

ycc yc x .

ɑɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɩɟɪɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɳɟɦ ɜ ɜɢɞɟ

y

Axex , ɬɚɤ ɤɚɤ D

1

1

ɹɜɥɹɟɬɫɹ

ɨɞɧɨɤɪɚɬɧɵɦ

ɤɨɪɧɟɦ

ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ

ɢ Pn ( x) 1

–

ɦɧɨɝɨɱɥɟɧ ɧɭɥɟɜɨɣ ɫɬɟɩɟɧɢ. ɉɨɫɤɨɥɶɤɭ

y c

Aex Axex ;

y s

Aex Aex Axex

2Aex Axex ,

1

1

ɬɨ, ɩɨɞɫɬɚɜɥɹɹ ɷɬɢ ɜɵɪɚɠɟɧɢɹ ɜ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ, ɢɦɟɟɦ

2 Ae x Axe x Ae x Axe x

e x ɢɥɢ Aex ex

A

1 ɢ

y

xex .

1

ɑɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɦ ɧɚɯɨɞɢɬɶ ɜ ɜɢɞɟ

y2

Ae2x , ɬɚɤ

ɤɚɤ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɜɬɨɪɨɝɨ

ɭɪɚɜɧɟɧɢɹ

D

2

ɧɟ

ɹɜɥɹɟɬɫɹ

ɤɨɪɧɟɦ

ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ Pn (x)

1 – ɦɧɨɝɨɱɥɟɧ ɧɭɥɟɜɨɣ ɫɬɟɩɟɧɢ.

Ɉɩɪɟɞɟɥɹɹ,

ɤɚɤ

ɢ ɜɵɲɟ,

ɩɨɫɬɨɹɧɧɭɸ A,

ɩɨɥɭɱɢɦ

y2

1

e

2x

.

ɑɚɫɬɧɨɟ

2

ɪɟɲɟɧɢɟ ɬɪɟɬɶɟɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɦ ɧɚɯɨɞɢɬɶ ɜ ɜɢɞɟ

y3*

x(Ax B) ,

ɬɚɤ ɤɚɤ ɜ

ɩɪɚɜɨɣ ɱɚɫɬɢ ɬɪɟɬɶɟɝɨ ɭɪɚɜɧɟɧɢɹ D 0 ɹɜɥɹɟɬɫɹ ɨɞɧɨɤɪɚɬɧɵɦ ɤɨɪɧɟɦ

ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ

ɭɪɚɜɧɟɧɢɹ

ɢ

Pn ( x) x

– ɦɧɨɝɨɱɥɟɧ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ.

ɉɨɫɤɨɥɶɤɭ y3 c

2Ax B,

y3 s

2A, ɬɨ,

ɩɨɞɫɬɚɜɥɹɹ

ɷɬɢ

ɜɵɪɚɠɟɧɢɹ

ɜ ɬɪɟɬɶɟ

ɭɪɚɜɧɟɧɢɟ, ɢɦɟɟɦ

2 A 2 Ax B B

x . ɉɪɢɪɚɜɧɢɜɚɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ x ɢ

ɫɜɨɛɨɞɧɵɟ

ɱɥɟɧɵ

ɜ

ɥɟɜɨɣ ɢ

ɩɪɚɜɨɣ

ɱɚɫɬɹɯ

ɪɚɜɟɧɫɬɜɚ, ɩɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ –

2 A 1,

BA B

0 , ɨɬɤɭɞɚ ɧɚɯɨɞɢɦ A

1

,

B

1.

2

ɋɥɟɞɨɜɚɬɟɥɶɧɨ,

y3

§

1

·

x¨

x 1¸.

©2

¹

ɋɭɦɦɢɪɭɹ

ɱɚɫɬɧɵɟ

ɪɟɲɟɧɢɹ,

ɩɨɥɭɱɚɟɦ

ɱɚɫɬɧɨɟ

ɪɟɲɟɧɢɟ

y

ɢɫɯɨɞɧɨɝɨ

ɭɪɚɜɧɟɧɢɹ

y

y

y

xex

1

§1

x

·

3

e2x x¨

1¸. Ɍɨɝɞɚ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɚɧɧɨɝɨ

1

2

©2

¹

ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ

y y y C C

ex xex

1

e2x

§1

·

2

x¨

x

1¸

1

2

©2

¹

C (C

2

x)ex

1

e2x

1

x2 x.

1

2

2

ɉɪɢɦɟɪ

6.3.

ɇɚɣɬɢ

ɱɚɫɬɧɨɟ

ɪɟɲɟɧɢɟ

ɭɪɚɜɧɟɧɢɹ

ycc y

4x cos x ,

ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ y(0)

0,

c

1.

y (0)

Ɋɟɲɟɧɢɟ. ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ

ɭɪɚɜɧɟɧɢɟ

k 2 1 0

ɢɦɟɟɬ

ɤɨɪɧɢ

k1 i, k2

i .

ɉɨɷɬɨɦɭ

ɨɛɳɢɦ

ɪɟɲɟɧɢɟɦ

ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ

ɨɞɧɨɪɨɞɧɨɝɨ

ɭɪɚɜɧɟɧɢɹ

ycc y

0

ɛɭɞɟɬ y C1 cos x C2 sin x .

Ⱦɥɹ

ɩɟɪɜɨɣ ɱɚɫɬɢ

ɞɚɧɧɨɝɨ

ɭɪɚɜɧɟɧɢɹ

D

0,

E

1,

Pn ( x) 4x – ɦɧɨɝɨɱɥɟɧ

ɩɟɪɜɨɣ

ɫɬɟɩɟɧɢ;

(n

1),

Qm ( x) 0

– ɦɧɨɝɨɱɥɟɧ

ɧɭɥɟɜɨɣ

ɫɬɟɩɟɧɢ

(m

0) ;

s

max{1,0}

1, DiE

i

ɞɚɧɧɨɝɨ

ɭɪɚɜɧɟɧɢɹ ɧɚɯɨɞɢɦ ɜ ɜɢɞɟ y x(( Ax B)cos x (Cx D)sin x) ɢɥɢ

y ( Ax2 Bx) cos x (Cx2 Dx)sin x .

ɇɚɯɨɞɢɦ

y c

(2Ax B)cos x (2Cx D)sin x

( Ax2 Bx)sin x (Cx2 Dx)cos x

(2Ax B C 2 Dx)cos x (2Cx D Ax2 Bx)sin x;

y cc

(2A 2Cx D)cos x (2Ax B Cx2 Dx)sin x

usin x ( Ax2 Bx) cos x (Cx2 Dx)sin x

4x cos x.

ɉɪɢɪɚɜɧɢɜɚɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ cos x, sin x, x cos x, x sin x ɜ ɨɛɟɢɯ ɱɚɫɬɹɯ

ɪɚɜɟɧɫɬɜɚ, ɩɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ

cos x

2A 2D

0;

sin 0 x

2C 2B

0;

x cos x

4C B B

4;

x sin x

4A D D

0.

Ɋɟɲɚɹ ɷɬɭ ɫɢɫɬɟɦɭ, ɧɚɯɨɞɢɦ A 0, B 1, C 1, D

0 . Ɍɨɝɞɚ