Сборник задач по высшей математике 2 том

O,

z ~ O,z > 4,

6.13.19. fx+Y(z) = { ~z, 1

°< Z ~ 2,

Z2

6.13.20. fx+y(z) = _1_ e-T;

1 -

4z,

2 < z ~ 4.

2.Ji

M(Z) = 0; D(Z) = 2; X + Y = Z '" N(O; V2).

0,

z ~ 0,

Z2

°< z

O,

z ~ 0, z ~ 2,

2'

~ 1,

{

z,

°< z < 1,

6.13.22. Fz(z) = 1 (2 _ Z)2

fz(z) =

1-

2

' 1 < z ~ 2,

2 - z,

1 ~ z < 2.

1,

2 < Z;

O,

z ~ 0,

6.13.23. fz(z) = { 0,5· (1- e-O,3z),

°< z < 2,

0,5· (eO,6 - 1)e-O,3z,

2 ~ z.

Zi

-8

2

7

1

2

9

; B) M(Z) = 2,35;

6) Pi

0,19

0,36

0,45

;

Pi

0,1

04

0,4

01

D(Z) ::::! 30,13;

M(W) = 2,1;

D(W) = 5,69 . 6.13.25. M(X) ::::! 1,42; D(Y) ::::! 0,78.

1

2'

Y E (-1; 1),

1

6.13.26. a) g(y) = { 7r.J1=Y2

.

. 6) M(Y) = 0; D(Y) = 2'

0,

y't (-1,1),

6.13.27. a) Fy(y) =

1

~ arctg (1 -

y); Jy(y) =

1

2 ;

-2 -

71'(1

+ (1 - y) )

6) Fy(y) =

~ + ~arctg W; Jy(y) =

/

4; (Fx(x) =

~ + ~arctgx).

37r(y3 + y3)

6.13.28.

a

)

9

()

= {

61,

Y E [3 -

371'; 3 + 371'],

y

71'

6) g(y) = {*,

0,

B OCTa.JIbHbIX CJIy'laHx;

y E

~o;.;1!:j'. .

B) g(y) =

7rYv'-lny

7r2

{1,

Y E(e- 42 ,1),

0,

y't

°'2'

0,

y't(e-T,1).

6.13.29. a) g(y) = f( -y), Y E lR; 6) g(y) =

fey -

10), Y E R;

B) g(y) = 2~ . (f(v'Y) + f(-v'Y», y > 0;

r) g(y) = {seC2 y. f(tgy),

YE (=~~~),

6.13.30. a) Fy(y) = Fx (y; 3);

0,

y 't (

2' 2) .

6.13.31. a) g(y) =

1

1

-- 2

fiLe

2y , Y # 0;

y2y 271'

()

= {

I,

Z E [0,1]'

d] CJIe,ll;OBaTe.JIhHO,

fz Z

0,

Z 'F [0,1 ;

Z

1 {O,

Z

~ 0,

~ 1,

Fz(z) = [ ! f(x) dx = z,

0

< z

- 00

1,

1

< z.

t ~ 0,

0< t ~ 1,

1 < t.

0< z ~ 1,

1

1 < z ~ 7r,

6.13.35. /z{z} = 7r'

1

7r<z~7r+l,

1i'(7r-z+l),

0,

B OCTaJIhHhIX CJIy'laHx.

!e-~z

z > 0

Fz(z) =

{O,

_~

z ~ 0,

6.13.36. fz(z) = { 2

'

,

Z > O.

0,

z ~ 0;

1 -

e 2,

-t, z~-I,

{~'

z~-I,

6.13.37. Fz(,) ~{2 i';

-1 <, o. fz(,) ~

1;

-1 <,,, 2,

1 -

3z'

2 < z;

3z2 '

2 < z.

6

hIJIa MeHhme z.

F

z

()

= {

Z2,

Z E [0,1],

f

()

=

{2Z,

z E [0,1],

HHX

z

0,

d

[

]

Z

Z

0,

d [

]

Z 'F

0, 1 ;

Z 'F

0, 1 .

0,

z ~ 2,

Z2 -

2z

2 < z ~ 3,

- 8 -

6.13.40. fz(z) =

2z -3

3 < z

~ 4,

8

5 + 4z -

Z2

4 < z ~ 5,

8

'

0,

5 < z.

YICa3axue. f(z) = j~X' fz(z -

x)dx, {Z _ ~~:~!-1 .

o

6.13.41. h(z) = {

O'

(_>. z

->. Z)

z ~ 0,

AlA2

e

z > 0.

--- e

2

-

1

'

Al - A2

z

_,

_, ( _ )

{o~x~z

YICa3aXUe. J(Z) = j Ale"lZ. A2e

"2 Z

'" dx,

:: ;;:: 0 '.

o

o12. e-O,lz . z

z> 0,

6.13.42. a) M(Z) = 20. YICa3axue. J(z) ='{

,

0,

z

~ OJ

6) M(Z) = 1. YICa3axue. J(z)

= {

Z'

°< z ~ I,

2 -

z,

1 < z ~ 2,

0,

B OCT8JIbHbIX CJly'laHx.

6.13.43. M(Y) = ka + bj u(Y) = Iklu. 6.13.44. Jy(y) = {

~'

°~ y ~ Ij

!

1 < Y ~ 2.

3'

y < 0,

'" ,

0,

y> 2.

6.13.45. Fy(y) = {FX ( VY4+ 9) -Fx (- VY4+ 9),

y> -9,

0,

y ~ -9j

Fz(z) =

I - Fx (! In!)

z > °

{

2

2'

y;;:: 8,

{

3

z'

'6.13.46. Jy(y) =

(y -

6)

y < 8j

0,

z ~ 0.

0,

-vz

z >0,

_e_,

Jz(z) =

{ 2VZ

6.13.47. a) HeTj

6) ,ll;a.

0,

z ~ 0.

O'

z ~0,

{O'

z

~ 0,

=

~

°~ z ~ 7r,

6.13.48. Fz (z)

{ 2~' 7r

°

7r,

Jz (z) =

2~

,

< z

1 --

7r < z·

~

'

7r

< z.

2z'

,

2z

2

Z2

6.13.49. Jz(z) = {

z· e- 2

,

z > 0,

(pacnpe,ll;eJIeHHe P3JIez)j

0,

Z ~ 0,

z2

O'

z ~ -2,z > 2,

Fz(z)= { l-e- 2

,

z>O,

z+2

-2

< z ~ 0,

6.13.50. J(z) = { -4-'

0,

z ~ 0.

2-z

°< z ~ 2j

-4-'

0,

z ~ 0,

z> 11,

z

0< z

~ 3,

6.13.51. fz(z) =

24'

1

3

< z ~ 8,

8'

11- z

8

< z ~ 11.

----U-'

§ 14. np~enbHble TeopeMbl TeopMM BepoSiTHocTeiii

1

0,33. 6.14.5. a) p ;;:: 0; 6)

8

6.14.2. P ;;:: 3". 6.14.3. 0,7; p ;;::

p ;;:: 9 ~ 0,89;

u) p ;;:: ~~ ~ 0,99. 6.14.6. a) p

= 1 - e-6 ; 6) p;;:: ~~. 6.14.7. p;;:: ~~6~ ~ 0,97.

6.14.21. Jy(y) ~

1

_ (y - 60)2

; P(A) ~ 0,971. 6.14.22. a) p ~ 0,499;

tn=e

8

2· v 27r

6)

p ~ 0,841. 6.14.23. (-0,0251; 0,0251). 6.14.24. a)

2

P{X ;;:: 30} ~ 3";

6)

P{IX - 201 < 8} ;;:: 0,74.

6.14.25. p;;:: 0,75; P{IX -

501 < lO} ~ 0,954.

7

35

15

7

2

6.14.26. p ;;:: 9; p = 36· 6.14.27. p;;:: 16· 6.14.28. P(A) ;;:: 16; P(B) ;;::

3".

6.14.29. a) p ;;:: 0,936; 6) p ;;:: :~. 6.14.30. p ;;:: ~. 6.14.31. a) n;;:: 250;

6)

2

;;:: 300. 6.14.34. )J,a.

n;;:: 1250. 6.14.32. P{O < X < 6} ;;:: 3". 6.14.33. n

6.14.38. n ~ 761.

8

~ 0,982; 6) 1;

6.14.39. p ~ 0,034.6.14.40. p;;:: 9; a)

7

P{IX - 3501

8

u) ~ 0,997. 6.14.41. a) P{X ;;:: 400} ~ 8; 6)

< 25} ;;:: 15·

6.14.42. n ~ 200.

6.14.47. HeorpaH1PleHHo B03pacTaeT.

y2, V = -2xy.

7.1.5. u = x 2 - y2 -

2x, v = 2xy - 2y + 1. 7.1.6. u = x 3 -

3xy 2, V = 3x2y _ y3.

7.1.7. u = x, v = y. 7.1.8. u = 2x, v = 0.7.1.9. u = X

x+y-1

2 + (y -1)

2'

x-y+1

x 2 _y2

-2xy

v = x 2 + (y _ 1)2 . 7.1.10. u = (x 2 + y2? , v =

(x2 + y2)2 .

7.1.11. u = x - ~, v = y +~. 7.1.12. u = e'" cosy, v = -e'" siny.

x +y

x +y

7.1.13. u = eY cos x, v = -eY sinx. 7.1.14. u = shx cos y, v = chxsiny.

arctg ~

rrpll x > 0,

7.1.17. u = - arg z = (-1)· { sign(y)· 11' + arctg ~

rrpll x < 0,

sign(y) . ~

rrpll x = 0,

v = !In(x2 + y2). 7.1.18. u = In(x2 + y2), V = 2argz (CM. OTBeT

rrpe,!l;bI,IQ'rn;eil:

Arg J(z) = cp + 211'k,

k E Z. 7.1.22. IJ(z)1

= 1, Arg J(z) = cp + 211'k, k E Z.

7.1.23. IJ(z)1 = r3,

Arg J(z) = 3cp + 211'k,

k E Z. 7.1.24. IJ(z)1 = rn,

Arg J(z) = ncp + 211'k, k E Z. 7.1.25. IJ(z)1

= \, Arg J(z)

= -5cp + 211'k, k E Z.

r

7.1.28. J(z) = iz. 7.1.29. J(z) = Z2.

7.1.30. J(z) = tz

7.1.36. J(zI) = 0,5 - 2,5i,

J(Z2) = O.

7 1 37

•

J()

i

~

11'. .

11'

=

..,fi

. ..,fi

••

Zl

= e

4

= cos 4' + %sm

4'

2

+ %2'

.11'

= cos ~ + isin

~ = i.

J(Z2) = e'2'

7 1

38

•

J()

-i~

(11')'

. (

11')

=

1

.v'3

••

Zl

= e

3 = cos -3 +%sm

-3

2'+ %2'

J(Z2) = 2e- i ¥ = 2 (cos i - i sin i) = ..,fi + i..,fi.

7.1.39. J(zI) = e . cos 1 - i . e· sin 1, J(Z2) = 2.

7.1.40. J(Zl) = 0, J(Z2) = i~.

7.1.43. u = -2xy, v = x 2 -

y2. 7.1.44. u = x - 3xy2 -'1, v = -3x2y + y3 + 2.

2

2

x(x + 1) + y(y - 1)

7.1.45. u = x -

y

, v = -2xy. 7.1.46. u =

X

2 + (y -1)

2'

x - y + 1

_ _I(

)2

2

_ _I(

)2

2

V =

x 2 + (y -1)

2'7.1.47.

u - Xv

x - I

+ y

,

v - Yv

x - I

+ y .

7.1.48. u = - sinxsh(y + 1), v = cos x ch(y + 1).

7.1.49. u = - sh(2xy) . cos(x2 - y2), V = ch(2xy) sin(x 2 _

y2).

x

Y

x

Y

- 2 -- 2

- 2 -- 2 .

7.1.50. u = eX

+ Y cos -- , v

= -ex

+ Y

sm -- .

x2

+ y2

x2

+ y2

sin 2x

sh2y

7.1.51. u =

ch2y+cos 2x '

v = ch2y+cos 2x .7.1.52. IJ(z)1 = ar,

ArgJ(z) = cp + 211'k, k E Z. 7.1.53. IJ(z)1

= pr, ArgJ(z) = cp + (J

+ 211'k, k E Z.

7.1.54. IJ(z)1 =

~, Arg J(z) = -ncp + 211'k, k E Z. 7.1.55. J(z) = -\.

r

z

fez) = eiz -

z + iC, C E lR. 7.2.39. b.u = 0, vex, y) E R2 \ (CO, O)};

v(x,y) =

2 x 2 + 2xy + C; fez) = 2i

+ Z2 + iC, C E lR. 7.2.40. b.u = 0,

2(x

+ y ) .

z

V(x,y) E R2

\

(CO, O)}; vex, y) =

2 2xy 2

2

+ C; fez) = --\ + iC, C E R.

(x

+ y

)

z

7.3.14. -~.

7.3.15. -2 + i. 7.3.16. - ;~ - 3i. 7.3.17. - j

+ ~i. 7.3.18. - ~ +

~i.

7.3.19. -2 - i. 7.3.20. -3 + 3i.

7.3.21. cos 1 + sin 1 - e . sin 1 + i(sin 1 - cos 1

+ e . cos 1). 7.3.22. 1 - ~ + i.

7.3.23.

i(1 - sh 1). 7.3.24. -

sh 71'. 7.3.27.

6:. 7.3.28. O. 7.3.29. ~(3 + 2i).

7.3.30.

O. 7.3.31. ~(v'3+ i).

7.3.32. ~ +

~i. 7.3.33. ~ + 3i. 7.3.34. -971'.

HJIH He cyrn;ecTByIOT o.n;HOBpeMeHHO jf(z) dz = -

jf(z) dz. 7.3.50. 3Ha'leHHe

/1

/2

1

Z

Z3

-

... =

~ (-1 t

2n - 1

,

0 <

I I

< +00

(

7.4.5. a ) Z -

-2' +

-4'

L.J

-(-)-, z

z

BeCb pfi.n; eCTb

..

n=O

2n.

)6)

1

z

Z3

... =

~(-I)n2n_l

, 0 <

II

+00

(

rJlaBHM 'IacTb j

Z -

-2' + -4' -

L.J

-(-)-, z

z <

BeCb pfi.n;

..

n=O

2n.

eCTb npaBHJIbHM 'IacTb). 7.4.6. a) Z2 -

4

6

+00

(_I)n+l

z3'

+

;

- ... =

E

(

z2n,

.

5.

n=l

2n -

I)!

o< Izl < +00 (BeCb PM eCTb npaBHJlbHM 'IaCTb)j

4

6

... =

+00

(-It+ 1

6) Z2 - z3' +

z5' -

E

(

2n

_

)' z2n, 0 < Izl < +00 (Becb pfi.n; eCTb maBHM

..

n=l

1.

'IacTb).

1

1

1

+00

1

·(

1

)n=

0

1

7.4.7. a) 1+--1+-2"

(z

+ 1)

2+···=E r

+ 1

E

- (),(z+l)n,

z +

.

n=O n.

z

n=-oo

-n.

o< Iz + 11 < +00 (1 -

npaBHJlbHafi 'IacTb,

~1 +

-2\ .

1

2 + ... -

rJlaBHafi

z+

.

(z+l)

'IacTb)j

+ ... = E 1..

t

6) 1+_1_+.1.

1

1

1

(+I)n

z + 1

2!

(z + 1)2

n=O n!

(z + l)n =

n=-oo (-n)!

z

,

o < Iz + 11

< +00 (BeCb pH,ll; eCTb rrpaBllJIbHaH 'IaCTb).

7.4.8. a) e

3

e3

e3

1

e3

1

t

+ ... =

+00 e3

1

t

+ - +1 + -2' .

2 + ... + ,. . (

+

E ,.. (

+

z

.

(z + 1)

n.

z

1

n=O n.

z

1

o

e3

Iz + 11 <

+00 (Becb pH,ll; eCTb rrpaBllJIbHaH qaCTb);

=

nfoo (-n)! (z + l)n, 0 <

3

e3

e3

1

e3

1

+ ...

+00 e3

1

6)

e + z + 1 + 2!

. (z + 1)2 + ... + n! . (z + It

= n~o

n!

. (z + It

o

e3

Iz + 11 <

+00 (Becb pH,ll; eCTb rrpaBllJIbHaH qacTb).

=

nfoo (-n)! (z + It, 0 <

7.4.9. a )

.

sin 2

2

cos 2 3

+ ... +

( -1 t

sin 2

2n

sm 2 + cos 2· z -

- 2' z

-

- 3' z

(

)'

. z

+

..

2n.

+

(-It cos 2

2n+l

+ ... =

+00

k

(_I)n sin 2

(

),. z

E CkZ

, (C2n =

()'

' n = 0, 1, 2, ... j

2n + 1 .

k=O

2n .

C2n+1 =

(-It cos 2

0, 1, 2, ... ), 0 < Izl

< +00 (Becb pH,ll; eCTb rrpaBllJIbHaH

(2n+ 1)'.

' n =

qaCTb) j 6 )

.

+ cos 2· z -

sin 2

2

cos 2 3

+ ... +

( -1 t

sin 2

. z

2n

sm 2

- 2' z -

- , - z

()'

+

.

3.

2n.

+

(-It cos 2

2n+l

_

+00

k

_

(-It sin 2

_

.

(

),. z

+ ... -

E CkZ , (C2n -

()'

' n -

0, 1, 2, ... ,

2n + 1 .

k=O

2n .

(_I)n cos 2

0,1,2, ... ), 0 < Izl

< +00 (Becb pH,ll; eCTb rrpaBllJIbHaH

C2n+l =

(2n+ 1)'.

' n =

qaCTb). 7.4.10. cos 3

.

3

1

cos 3

1

2 -

sin 3

1

3

+ ...

+ sm

. -- 3 -

- 2'·

(z +

3)

- 3'·

(z

+

z

+

.

.

3)

(-It cos 3

1

(-It sin 3

1

+00

1

... +

(2n)!

. (z + 3)2n

+

(2n + I)! . (z + 3)2n+l

+ ... = k~OCk . (z + 3)k

(_l)n cos 3

(-It sin 3

,n=0,1,2, ... ),

(C2n=

()'

,n=0,1,2, ... jC2n+l=

(

2n

+

)'