РГР (2 семестр)

- I2.

.

7·'

. tg

(4/7') 13.·

I'

X:?:·arcsht6x.

I4

I'

7_.;._sx_-_I

I1111

.

.

lm

arctg3 9x

. lffi

,__..,

x->O

x-+o ln(l + tg9x)

15 •

.

4x 2 -1

I1111

.-.1.2 arcsin(l- 2x)

H~pnanT 29.

I.

..

8x3 +1

.

5-~

.

ln 3 x+lnx-2

I1111 .

2•

1nn

3.

11m

In

6

x-1

·,_.-112J6x2 +lOx+!

x-+23x-2../15-3x

x-->r

4.

}~~~ x( .J4x2 + 7 +2x)

5. lim'

tg{6+ 2x).,. tg(6·-·2x)

·

·~~o sin(~+ 2x)- sin(6 -2x)

6.

lim .

sin 9.t

'

7

r

..fi-2cosx

8.

. .

_,__.o

,.f2x + 7-

·1-f

. ,l~~4 ...

4

lim(cos7xt'ss.isms.,

'>J

I

,.

-

X

·' -+0

9.

lim

4

') . 3

)1/Sx

10.

lim(tg6x+tg9x)"K2·'

+~~Ill

~

, ....o( 4- sm3x

.....o

II.

lim Jcos4x-cos5x ·ctg6x

12.

limtg(7Jrx)·ctg(5Jrx)

t-+0.

x-+1

·

13•

.

ln[tg(3x+Jr/4)]

14.

.

ln(cos6x)

.

arcsin8x

·

,

IJill

sin 13x

11111

ln(cos9x)

15 . 1IITI-;=====

, ...o

<-+o

x-+o

~tg2x·sin 3

5x

'•

nap11311T 30.

. .

2x3 + 5x2 + 5x + 2

.

I - ~

I .

11111

.

2•

1

,/

IITI

,_._ 1

5 + 4x- x 2

x-+ 2 2- <!9- .J2x- 3

3.

lim

Ssin

x+l

4.. 1i.m(2x+~I0-8x3 )

5. lim-co_s__

_x_-_I

3

4

6

l·-+-n'66sin 2 x+5sinx+l

x-+->.

x-+OI-cos3 7x"

.

,.

tg2x·sin5x

7

J'

t

g

2

6

1lffi

·.

. ~

•

X

8

J' ( .r.-:----1

x-tO]

lm

r::

J

·

-..;cos8x

HO

+ cosx

1m

-.JI+x-x)''x

-.Jl.-

1

.t->0

2

9.

Iim[tg(Jr/4-x)tK'

10.

lim(l--:-tg3x+tg9x}1151" 7 '

X-+0

.t--+0

'

11.

lim.J2x+3x

2

·ctg(..[S;) 12.

lim( .

2

1

.

--.--)

x-+0

.t-+0 s·II12XSII1 X

Sill 2 X

13

_ lim

arctgi9x

14

_ lim I-ctg1rx

15.

.

</1 + x -I- sin x

x.....o.JI + 7x + 9x2

ln(tg1rx)

Inn------

-I

X--+ 114

<-+0

ln(l +x)

l. f(x)=sitJ3x+~2.f(x)=(l-2x)2'x+

I

,/(0)=7

2x

x2

- 4

x3 + 2x2 + 2x + I

3.

f(x) = O,x < -;r;sin x,-Jr < x :c 0;1r~x ~ 0 4.

f(x) = arctg--

2

,

x-3

5.

f(x) =2xl(l-x)

BapnanT 2.

r.-:-

x+3

x+6

- -.Jt+x-1 +

2. f(x)=(l+5x)J'' +-·J-,/(0)=0

l.f(x)-

2

- 16

.

x

-8

-

X

X

3.

IX+ 51

5

21/x + 31/x

5. f(x) =e

2

X+llx

f(x) = ---- 4.

f(x) = ··

x+S

X

211 x-3llx

BapnanT3.

1_ /(x)=sin(x-2)+_1_

2

+2x,/(l)=-2

2_ /(x)=(cosx)"<-Hl _x

x 2 '- 4

x2 +x

cos I

x 2

-

4

262

3. .f(x)=x+l,x<O;(x+l)2 ,0<x:s;2; 4-x, x>2

4./(x)=--1

5. .f(x) = x~e- 11 '

HapnanT 4.

I. ((x)= ~-1+

x+3

•

X

X 2 +X;- 6

2. .f(x)=sin7x +

x+5

,.f(O)=-I

3

.

.f(x)Jx+41_~

· x

x 3 - x 2 - x + I

x + 4 x

.

I

4. .f(x) = arcctg--

5. .f(x) =sin 211cx-JJ

x-6

BapuanT 5.

I.

/(x)=arcsin3x+ ~+4

2. .f(x)=(l- 3x)41r+

l,~~·.f{O)=I

X

X

-8

·

X 4 -

3.f(x)=-x,x<O;x\O:s;x<2;3,x>2 4 . .f(x)=____.;.

I

5. .f(x) = x. 5"'

BapnanT 6.

I.

.f(x) =

~-2- x+3

2. .f(x)=(s~nx)+-f-,f(l)=l

x

x4 -I

sm I

x

-4

. 3.

lx+3l·

·

6

5. .f(x) = x 2 • 2-li(J-xJ

f(x) = ---X

2 4. .f(x) = arcctg

.

x+3

(x-1)·3

Bapnaan 7.

l..f(x) = arctg7 x +

2x _ 1

x

2

25

X -

sin5x

x+3

263

2. f(x) =

r;:: +

r-;:

. ,f(O) =-I

I

vx + 2 - v2 x 2 + 2x + I

3. f(x) =-2,x < -1Z" 12; 2sin x, -1Z" 12 :s;

.X:<: 1Z" 12; l,x ~ 1Z" I 2

:31/(x-2)

_ 2

5. .f(x) =COS 7' 1x

4. f(x) = l!(x- 2)

+

2

i''

3

BapuanT 8.

rr

I. f(x)= ~l-x-x

2

-.1 + 2x+3

.

((x)=

s,in3x

+ x +6 ,f(O)=l .:

'

X

X 3 -1_25

2

.

):• -

2x

X 4 -J

.

Ix

2

+5xI

1

.

,

· 3. f(x) =

x+5

- x

4. .f(x) = arcctg--

5. .f(x) = e'114 -·'

>

3-x

Bapuan'f 9.

I. f(x) = l-cos7x +--'

x 2

2x2 -3x +I

2.

f(x) =(I+ 2sin 5x)31 x +

x + 5

,f(O) = -2

x 2 +2x.+l

lx + 21

r;--;

I

3.

x :s;

1

f(x) =--,x < -2;v4

-x2

,-2 :s;

2;--x > 2

x+2

·

x-2

4.

f(x) =_I_.

5.

f(x) =2x-llx

I+ 51/x

BapuanT 10.

2

x + 3

) =sin 3 2x

x + I

l

I

f( ) _ J1 - x- 2x

-I

f(

O) _

·X-

+2

I

2'X

1+4

,(-5

X

X -

Jx-

X

-16

x+2

4.f(x)=

3

,

I

I

11 ·'

3.f(x)=x+--

,

5.f(x)=x+le-

1X+ 21

I

+

2(x -1)/(x

-4)

I.

.f(x)= sinx-siJ17x + .~+3

2. .f(x)=(cosSx)3'·'' +-_1_,/(0)=0

X

X -25

·

X3-I

3.

f(x) =

2lx-II

I

·

4. .f(x) = arctg- 5. f(x) = ln(l + 2311 <+ 2>)

x2 - x-1

1''·'

BapnanT 12.

I . .f(x) = (x /3)2'(,~3>- (x +Ire,

2. .f(x) = sin(3x22x) + _I_,.f(O)= I

x

x-3 .

3. .f(x) Jxl ,x < o;/J7,o :5; X :5; I;I/(x -l),x >I

4. .f(x) = arctg(211 x)

X

5.

I

((x) = -2-

.

X .

X

X -

,

IX+ II

2

+X

4. .f(x) = arcctg(5

211

3

9

'

3. .f(x) = --x

x- )) 5. .f(x) == 2x'<

-x >

x+I

BapnanT 14.

l.f(.;)=(2-cos8x)2'''+(x-5r8

3

.f(O) = _2

2. .f(x) = I- cos

3x _

I

x2

(x-4)''

lx + 31

' ,-;:--2;

I

3. .f(X) =--,X< -3; v9- X

,-3 :5; X :5; 3;--x> 3

.x+3

x-3

265

BapnanT 15.

3

l..f(x)=sinx.-sini +

I

2 ./(x)=(cosx)"<x- >+--f--,/(3)=-2

x-I .

2x2

- 3x

cos 3

x -I

_

lxl

.

r.--:2

. l

3. .f(x) ---,x <0,v4-x- ,0:5; x :5; 2,--x > 2

x

.

·

x-2

4. .f(x) = arcctg(l/311 x). 5. f(x) = lxle-llx

BapnanT 16.

L f(x)=(2-cos5x)2tx' +

~-3

2.

f(x)= arcsin7x +

. x 2 ,((0)=5

X

-9

.

X

(x-2)

3.

f(x)=l-x 3 ,x<O;(x-Ii,0:5;x:5;2;4-x,x<2

4.

3

X

f(x) = ar.cctg--

5. f(x) = 11x

x-4

3

BapnanT 17.

3

I.

f(x) =cos x- cos 2 + ~ 2. f(x) ~ sin

4x __x_,.f(O) = 0

x-2

x

x3

x-I

3.

f(x)=

lx2- 41

-x

4. f(x)=

51/x + 2.1/x

.

x-2

llx

II.T

5. f(x)=arcsin(l/(x-4)]

·

5

-2

BapnanT 18.

l. j(x) = sin(x + 2) +_I

x2 -4

(x2 +x)2

2. f(x) =(I- 2sin 7x)11 x +~,.f(O) = -2

x+2

3. f(x)=2X,-l:5;x:5;l;l,x=l,x-l,l<x:5;4

4.

f(x)= 1 _ 5 ~1( 3 -x)

I. .f(x) =xsin(2/ x) + 1/(x -1)6

2. .f(x)=arctg3l;+

x:5·

,.f(O)=-l

x

x 3 -5x -5x+l

3.

jx+4j

.

·

·

I

f(x) =

-

x 4.

f(x) = arcctg-·- 5.. f(x) =sin 51114 -'1

x

2 -16

x+5

BapuanT 20. ·

I.

.f(x)=(s~nx)2 '<-•-J> +~ 2.

f(x)=

~x _ -1/x3 ,/(0)=1

sm3

x -8

3.

f(x) = cos(1Z'X h),jxj ~ l,jx -lj,jxj >I

4.

f(x) = _

:,

_,

1

7

13

1

1

I.

/(x)=sin(3-x)

1_ 2. /(x)=(2x+l) r. +(x-1)-s,.f(O)=I

x3 - 27

x4 -I

. 3x +I

3.

f(x) = lx+41 - x2

4.

f(x) = arcctg 2 .

5.

f(x) = x 2 • T 1114"-xl

x 2 -16

(x-3)

3

BapuanT 22. I. f(x) =

arctg4x

.x-1

.

+-·-

x

X 3 -125

2.f(x)= arcsin3x

+

x+3

,.f(O)=-l

Jx+3 .:...fj

x2 +4x+4

3.

f(x) = 2-Jx:o::; x Sl,4- 2x,l < x < 2.5,2x.:. 7,x 2:!

2.5

51/(x-J) -1

1

f(x) = 51 ,1x_31 +I

4.

5. f(x) = cos3 'l•-ll

f(x) = </1-

2

2. f(x) = arctg2x + -;-,f(O) = 1

1.

2x- x -1 + x + 3

X

X 3 - 8

x2 + 3x X -I

3.

f(x)=

lx2 -

6xl

4.

1

5.

1 9

2

,

-3x

/(x)=arcctg--

f(x)=6x < -x l

x-6

4-2x

BapuanT 24.

l.f(x) = arcsin(x -I)+ __

x 2 -1

(x + 2)4

2.f(x)=(l-sin9x)51 '+

1

,/(0)=-4

x 2 -2x+ I

3.

f(x) =1- x

2

,x $

O;x,O < x < l;l,x 2:! 1 4.

.

+

I

f(x) =

_,_

31

1

7211

5.

f(x) =32x-Jix

BapuanT 25.

1.

f(x) =(cos7x)11

,2

+

x+2

·

sin 4 x

x

,/(0) = 5

2: f(x) = -

-

+

x 4

-16

(x 2 -1)2

'

3x

4

3.

·

_

2

2x + 4

4.

_

'

3

5.

-1

I -l/(n2)

f(x)- x

+-

-

21

f(x)-

J+3''

2

2

f(x)- x- 2

e

1X+

-1}/(x

-9)

BapuanT 26.

1.

f(x)= cosx-cos5 +__!_

2. f(x)= ex -e3'

-~,.f(O)=O

x2 - 25

x3

X

X- 1 .

_12x-4j

_

_I_

3.

f(x)-

x3 _

+ x +I

4.

f(x)- arctg 2''<>+4

8

5.

f(x) =In(!+ 5611 .... 41 )

3.

lx3 1

0;~,0 ~ x ~ 1;$,x >I

f(x) = -·-,x <

X

4.

f(x) = arctg(31'<x+J))

1

5. f(x) = - -

x2 ·1

BapuanT 28.

1. f (X) = COS 9X -

COS 2x

1

·

2

---

X

•

x6 -I

2.

f(x) = ~4

)1/.i

+ (x3 -1r2 ,f(O) =1

(

1+3x

3. f(x) = jx + 21x 2 + 3x

4. f(x) = arcctg(241<5-'1) 5.

f(x) ~ J''< 25 -'21

x+2

'·'·

BapuanT 29.

I.

1

f(x) = (2-:- cos6xt'

+ (x + 1)-9

2.

.sinxsin7x _

I

,/(0)=-2

f(x) =

x2

(x + 6)J

~·

1

f(x)=O.Sx2,jxj<2_;2.5,jxj=2;3,jxj>2 4. f(x)=-

5.

f(x) = 9''(4x-2)

BapuanT 30.

I. f(x)=sinx-sin9 ++

1

91

2. f(x)=(cosx) '<'-+-!-J(9)=0

x -

9

2x - I

cos 9

x -1

jx3 -1j

2

1)

...

11

3. f(x)=--+x+l

5. f(x)=jxJ·S-

4. /(x)=arcctg(l/i'< -'

'

' \ X2 -1

arctg(2x2 + 3x)

()

. ·

)7.•

Y3 = ~

;

y4 = x+sm 4x

;

4x +.Jx2 +6x

\

tg39x·arccos6 x 7

. 1

•

Ys =

~

; y6 = sm

(arcsm(1 I x)).

x

13 · 4 3x - 2ctgx

2.

x = e1 sint;y = e1 cost; arctg(yl x) = ln~x2 + y 2 •

,

2

r

,

3.

y1 =cos· x

;y2 = arctgvx. 4.

y = x· - 3x + 2;x0 = 2.

5.

y1 =~ln 2

x-4;y2 =sin(xllnx).,Q. J24i,;

tg48.

7.

y, = x 1 In x, [l ;e]; y

= ~0...:. x 2 )(l+ 2x2 ),

[ -1; 1].

2

I.

y1 = 3tg6 6x + ~x2 + ln(x + x2 ) ; y2 = ctgif7 · 2arc 1116 x;

y3

=~3~3x + x2 + 4x.

=(cos 6x)txx.

ln(2x+31nx) '

y4

'

Ys =

ctg 5 6x ·~arcsin Fx

; .Y6

q ,.---:-'" s

~

= arc9tg -varcsm x .

c6sx3 • 6 4x- tgx

2.

x = ln(1 + t2 );y =I- arctgt;

tg(y + x) = xY.

3.

y

= xJI;;2;y

2

= 5';" 3 '. 4.

y = X 3 + 2x2 - 4x -3;x0 = -2.

1

5.

y1 = tg 2 x;y2 = x f(x +In x).

6. Jl5.9; sin63.

1.

·../X

r.::;;:

8

y3

=

arcsin 5 (x2 + 4x)

y1 = 2"" x + 3ctg(l I x); y2

= vlg1x · arctgx

;

4J;

. ;

+ 2cos5x

y

1

2

x+ll; Ys

= ~x

8

·ln

5

x · arccos

9

4x ·(3x

8

-

5x);

4

= x "<

y

6

=cos9 (arctg.Jsin5x)

2.

X= 31og2 ctgt; y =5/~( 6/+Sl;,

sin(xy) =X+ In y.

3.

Yl =~ctgx2; y2 =70/csinSx. 4.

y =In x; Xo =1.

5.

y 1 =arccos(l/x); y2 =xsin5x. 6. 4,83 ; arcsin0,52.

I

-1'insx+~2x 2

+3X'y·

-arcsin r;sin(l/x)··

y _tg (x

+ 2Fx).

3

2

'

YI-

·

'

2-

'>/X

"

3-

.

•

ln(3x+sm7x)

Y

= (lnx)J2X+7;