*** . .

9

1.1 )

,,,

R,

+i=+i

+i=0

;)

i

i2003;)

z=+i;

- .) ;)

C

z2+z+=0

;

z2+2z+2=0

;)

z=+i

;

z=68i

;) z, z, z, z

;) .) z , = z ;)

zw

;)0 1 2z z z z z z , = =

>0

z1z2

;)1 2z z ;) 1 2z z z z 2, + =

>0

1 20 z z < < ;1.2 )

,,,

R,

: +i=+i +i=010 )

z=+i

,

R, : i2=i4+=,

=0,1,2,3 z z = = )

z,w

C.

:( )zz w , zw , zw = = = =

zzw , , z , z zw= = = = =2z =) z , = z=) 0z z , =

>0,

z0

R=)1 2z z z z , = z1z2,

12

,

12

z1z2.) z 2 5i 3 + = ,-

R=

) z 2 i z 6 3i + = + :)

z24z+8=0

:z1=z2=)

zw

:1.3 ) 5 10iz ,2 i+=+:11 .z=97i. z=4+3i. z=6+8i .z=76i. z=35i)

=i1821+i2003: .i. 2i. -1 .2. 0)

=(1+i)2004(1i)2004: .1. i. 0.-i. -1) z 3, = z : .3z.2z.1z.9z.9z ) : .2z zz = . z z = .22z z = .4z 2 zz= = . z w z w + )

z 2 5i 7 + + = : . . . .. ) z, z 5i 2 = - : .z=7i. z=3i. z=5+2i .z=25i . z=2i) 2z 1 z 2 , = : . z 4 = . z=1 . z=-1.z=i . z.)

z 1 i z 3 5i = +

: .x+y+2=0 . 2xy+3=0 . xy+3=0 .2x+y4=0 . xy+4=012 )

z 3 7i 5 + =

: .(x3)2+(y+7)2=25. 3x7y5=0 .2 2x y13 7+ = .2 2x y17 3 = . (x+3)2+(y7)2=251.4 )

,

C

2+2=0,

==0. )

,

C

+i=0,

=0

=0. )

22z 2iz 1w ,z 3 +=+

22z 2iz 1w .z 3+ +=+ ) z z, =

z

R,

z z, = z- . )

z2+9=0

C. ) zz

z

C. )

z=+i,

,

C, 2 2z . = + )

z z z z = = =

z

C. ) z , = 2z .z= ) z z i z z = =

N*. )22z z, =

z

,22z z, = - z. )

z w z w . + = + ) 0z z , + =

(>0)

1 2z z z z , = (z1z2)

.

13

-. )

z=+i

C,

,

R,

: z i = z z 2 2Re(z) z z 2i 2Im(z)i + = = = = ( ) ( )z z = z z z =

z

, z z = .)

z=+i,

,

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: 22 2z z zz = + = z zzw z w , w 0w w = = z z , =

Z*(z0)) : z =

>0,

2zz= :222 2z z z z zz= = = = 14 . ) z w -zw.:( ) z w d A(z), B(w) AB = = )

>0,

0z z = 0

z0

.,

z0=x0+y0i,

:(C):(xx0)2+(yy0)2=2

) 1 2z z z z , =

z1z2,

-

12,

1

2

z1z2.) 1 2z z z z , + =

>0

1 2z z , <

1

2

z1z2.

2.1

z,

:C:x2+y2=4:)4z = ,z

) 4iw = z +z :C1:x2+y2=8)

C:x2+y2=4,

z 2, =

(0,0)

=2.

:2 4z 2 z 4 zz 4 z (1)z= = = = 15)

w=x+yi

z=+i.

:(1) 4 1 zz (2)z z 4== = (2)4i 1 zw z z 4i z 4i z iz i i( i) ( )iz z 4 = + = + == + = + = + + + = + +

:w=+(+)ix+yi=+(+)ix y x2(3) y y x2+ = = + = =

z=+i

C:x2+y2=4,

:2 2(3)2 2x y y x 4 42 2+ + = + = (x2+y2+2xy)+(y2+x22xy)=162(x2+y2)=16x2+y2=8

w=x+yi

z=+i

:(2)4i 1w z z 4i z iz (1 i)zz z = + = + == + = +

:wz1 i=+

wwz 2 2 2 w 2 1 1 w 2 21 i 1 i= = = = + =+ +

,wR 2 2, = :C1:x2+y2=816 ().

z=+i

w=x+yi.

:2z 2 z 4 = = 24i 4iz 4iz 4i( i)w z z z ( i)z zz 4z+= + = + = + = + + =

=(+i)+i(+i)=()+(+)i:x yx 2()y y x2+ == = + =

2+2=4,

z=+i

x2+y2=4.,():2 22 2 2 2x y y x4 x 2xy y y 2yx x 162 2+ + = + + + + =

x2+y2=8

zw

w=f

(z).

z

w.. z,w,:

z=x+yi

.

w=f

(z)

+i,

w=+i

,

R.

=0,

w

=0,

- w.

=0

=0

.- z,- .17. zC1C2,

w,:

z=+i

w=x+yi. :x A(, )x yi A(, ) B(, )i ()y B(, )= + = + =

C1(),xy.-C2.,

w=f

(z)

z,

z=f-1(w).:z=f1(w)+i=(x,y)+B(x,y)i A(x, y)(1) B(x, y)= =

C1,

(1)

C2.C2wC1z.

w=+i

z=x+yi,

C1xy.

2.2 z( )24z = z . )

z = 0 z = 1. )

z0,

1z = .z

)

z0,

z6=1.18 )

z

4 2z = z . ) z

,

z0;) 4 2z z . = z z, = :( )4 2 4 2 2 2z z z z z z 1 0 = = = ( ) ( )2z 0 z 1 z 0 z 1 = = = = )

z0

z 0, z 1. = :21z 1 zz 1 zz= = = ) 1zz= :4 2 4 621z z z z 1z= = = )

z 0 z 1. = = : z 0 z 0, = = z 1, = :z6=1(z3)21=0(z31)(z3+1)=0(z1)(z2+z+1)(z+1)(z2z+1)=0(z=1

z=-1

z2+z+1=0

z2z+1=0)

=14=-3 0

z0,z1,z2

C,

z1z2;[4]. 1 2 1 2zz z z =

z1,z2

C.[8]. ()-().)

,

C

2+2=0,

==0. ) 22z z =

z

C. )

z=+i,

,

C,

2 2z . = + ) z 1, =

z=1

z=-1. ) z , =

>0,

2z .z= 24 )( ) z z z z . = = )

=4+

,,

N*,

i=i. ) 22z z = z. ) : i) z z, = z, ii) z z, = z. ) z z , =

,

, 1z z R, =

R>0

. [10]

zw:2004 20042004 2004z (z) 2002wz (z) 2002+ +=+ +

) 1w .w= [13]) ,w,z-C.[12]

,:=x+ix,=y+iy,=+ix+y+=x+y+=0:)1 1 1 , , = = = [8]25) ++=0[9]) 2+2+2=0[8]

z

w

:2 2z, w 0, w z z w = = .:) z w 1 = = [5])1 1z wz w= = [4]) w=-

z[4]) w3=-1z3=1[4]. )

z

w.[4])

z

:z2004=1[4]

26

3.1 )

f

g

;)

f;g

) g

ffg; ;)

f:AR

,, ;)

f:AR

x0;)

f:AR

11;) f -1

f:AR;

f-1;f-1;) ;3.2 )

(,)

Cff, )

f

g

,

x

)

f:AR.

f

:f

(A)=27)

g

f

=

g

f

)

f:AR

,:x10}) - : .

f

(x)=x3 . f

(x)=ex

.

f

(x)=x .f

(x)=lnx . f (x) ln x = )

x,y>0

00

x0,

>0 . f

(x)1 . 00

x

R) i) f

x=,

:[ ](1)h 0 h 01limf ( h) f () lim f ()f (h) f ()2 + = = f () 0h 0 h 0f () limf (h) 2f () limf (h) 2 = = h 0limf (h) f (0) (3) = f0.

f

(0)=0,

f

(x)=0

x

R.

f

()0,

f

(0)=2.46 - ii)

R.

f

x=,

-:h 0limf ( h) f () (4)+ = :(1) (3)h 0 h 0 h 01 1limf ( h) lim f ()f (h) f () limf (h)2 2 + == = ==

1 1f ()f (0) f () 2 f ()2 2= = = ,

f

R.6.3

f:RR

:f2(x)+2f

(x)+2x0

x

R

f

x0=0.:f2(x)+2f

(x)+2x0f2(x)+2f

(x)+112x[f

(x)+1]22x[ ] 220 f (x) 1 x 0 f (x) 1 x + + x f (x) 1 x 1 x f (x) 1 x + + :( ) ( )x 0 x 0lim 1 1 lim 1 x 1 x = + = x 0limf (x) 1.=

x=0

:f2(0)+2f

(0)+10[f

(0)+1]20f

(0)=-1

[f

(0)+1]20.47:x 0limf (x) 1 f (0)= = f

x0=0.6.4

f:RR

:f3(x)+3f2(x)f

(-x)=-2x3

x

R.: ) f, ) f

(x)=x

.. ) ( )2x 1x - 2A = lim .f(x) - 1

) 2x +2xB = lim .f (x)

.) :f3(x)+3f2(x)f

(-

x)=-

2x3(1)(1)

x

x

:f3(-

x)+3f2(-

x)f

(x)=2x3(2)(1)(2):f3(x)+3f2(x)f

(-

x)+f3(-

x)+3f2(-

x)f

(x)=0[f

(x)+f

(-

x)]3=0f

(x)+f

(-

x)=0f

(-

x)=-

f

(x)

x

R.

f.)

f

(-

x)=-

f

(x),

(1):f3(x)+3f2(x)(-

f

(x))=-2x3-2f3(x)=-2x3f3(x)=x3f

(x)=x

x

R.B. ) :( )2 2 2x 1 x 1 x 1x 2 x 2 1A lim lim lim (x 2)(x 1) (x 1)f (x) 1 = = = =

48 - :2x 1 x 11lim lim(x 2) 1 0(x 1) = + = 0

g()>1. h()=f

()g()=f

()=[f

()1]0

f

()0.(1)

x2:2 22 22 2 3x f (x) 3xx x (3)x x x + :2222x 0 x 0 3x 3xlim lim 3 9x x = = =

(3)x 0f (x)lim 9.x=

f

(0)=9.

.fx0.,()Cf( )0 0A x , f (x ) :

=f

(x0)

():yf

(x0)=f

(x0)(xx0)()xx,:==f

(x0). fgx0

(x0,y0)(): f

(x0)=g(x0)(1) f

(x0)=g(x0)(2)(1)

(x0,y0)

CfCg.(2)CfCgx0.64 . CfCg,,x1x2,:yf

(x1)=f

(x1)(xx1)yg(x2)=g(x2)(xx2),:f

(x1)=g(x2)f

(x1)x1f

(x1)=g(x2)x2g(x2),CfCg:2 11 22 1g(x ) f (x )f (x ) g(x )x x= =

.Cf(,)

:

(x0,y0)

. ()Cf:():yf

(x0)=f

(x0)(xx0) ()

(,)

f

(x0)=f

(x0)(-x0). x0,.

8.2 CfCg

f

(x)=lnx+x+2

g(x)=x3x+3

.) .) Cf

Cg.)

(,)

Cf

Cg.

=f

()

=g(),:f

()=g()(1)

>0

Cf

Cg

,:f

()=g()(2)65(1):ln++2=3+3(3):1f (x) (lnx x 2) 1x= + + = +

g(x)=(x3x+3)=3x2-1(2):2 3 211 3 1 3 2 1 0 ( 1)(3 3 1) 0 1 + = = + + = =

32+3+10

R

(0,) 21 1 2f (x) + f =x x x

x>0,) f

(x)=lnx,

x>0.) :xf (xy) f 2f (x) (1)y + =

(1)

x=1

y=x

:1 1f (x) f 2f (1) f (x) f 0 (2)x x + = + =

) (1)x(y):x 1f (xy) y f 2f (x)y y + =

x=1

y=x

:21 1 1 1 2f (x) x f 2f (1) f (x) f (3)x x x x x + = + =

f

(1)=1

x0

(

x>0).84 ROLLE...... ) (2):21 1 1 1f (x) f 0 f (x) f 0 (4)x x x x + = =

(3)(4):2 12f (x) f (x) f (x) lnx c, x 0x x= = = + >

f

(1)=0,

0+c=0c=0.

:f

(x)=lnx

x>0

f

(x)=0

x

(,)

f

=[,],

f-..;

f

(x)=0

x

(,)

(,).

-:12c , x (, )f (x)c , x (, ) =

f,:12c , x (, )f (x) c, x c , x (, ) = =

f,:- +1 2x x lim f (x) lim f (x) f () c c c = = = =

f

(x)=c,

x

(,),

f(,).

85

. ) Rolle-.[3]) f

f

(x)=f

(x)

x

R.

f;[3]. ) f.f

f

(x)=0,

x,f-.) fg

f

(x)=g(x)

-x,c,f

(x)=g(x)+c

x

.[14]. ()-().) f [,]

f

()=f

(),

f- Rolle. ) f

[,]

(, ),

Cf

(,f

())

(,f

()). )

f:R

f

(x)=0

x

A, f. )

f

(x)=g(x)

x

,

f

(x)=g(x)+c. )

f

(x)=f

(x)

x

R,

f

(x)=cex. [5]86 ROLLE......

f:RR

:f

(x+y)=f

(x)+f

(y)+3xy(x+y)x,y

R. ) f

x0=0,

fR.[6]) f

x=,

.[7]. f

x0=0

f

(0)=0.) fR.)

Cf

Cf

,.[12]

f:(0,+)R

:f

(1)=0,f

(1)=-1f

(x)=e2f(x)

x>0:) f,[5]) f

(x)=[f

(x)]2,[5]) :g(x)=-

f

(x)e-f(x),x>0,[5]87) f

(x)=-

ef(x)

x>0,[5]) f

(x)=-

lnx,

x>0.[5]

f

[,]

(,)

f

()=f

()=0.

(,)

f

()>0,

(,),

f

()0. )

x0

(,)

f

(x0)=0,

f

(x0)- x0. ) f

x0

(,)

f

x0,f

(x0). )

f

(x)0

(,)

f- ,f

()f

(,). ) f

=(,) ,f . 91)f

x0

[,]

x0,

f

(x0)=0. ) f

=[,],

f . ) f

[,],

- f

[,]

f- .

92 -

. ,-,.:(x)B(x)A(x)B(x)(x)

x

,

.)

(x)B(x)

:1() :f

(x)=A(x)B(x),x

f. f

(x)

f

(x0)=0

x0

.:f

(x)f

(x0).2(...)(...).-

[,x]

[x,]

..,x

.

.)

A(x)B(x)(x)

,().1....932:A(x) B(x)A(x) B(x) (x) B(x) (x)

.. ,

ee,

...-(

f

(x)=xlnx

-,

>e).. , 0 g(x) = xe - (1 + e )ln(1 + e ) :) g,) g(x)0,) f

(0,+).) :x x x x x x x x xx1g(x) [xe (1 e )ln(1 e )] (e xe ) e ln(1 e ) (1 e ) e1 e= + + = + + + =+

x x x x x x x xe xe e ln(1 e ) e xe e ln(1 e ) = + + = + = xx x x x x xxee [x ln(1 e )] e [lne ln(1 e )] e ln 01 e= + = + = 0. f

(1)[f2(1)+f

(1)+]=-10,

f

(1)0,

:2 22 ln < -

,>0

>.

x,

=x

:2 2 2 2x x2xln x 2xln x 0 (1) < + < x2xln ,-.

xln x.x(1):2x 2ln x 0 x + < :2x f (x) 2ln x , x x= + :2 2 2 2 22 2 2 2 1 2 x 2x (x )f (x) 2 1 1x x x x x x+ = = = = f

[,+)

f

(x)

:f

() + < < < .98 - 2 2xg(x) 2xln x ,= +

x

((1)),:

g()=0x 1 xg(x) 2ln 2x 2x 2ln 2 2x x = + = + g()=0 1 2( x)g(x) 2 2x x= =

g(x),

g

[,+).

:x>g(x)0.[7]) :x3x2+

lnx

x>0

=1.[9]106 -

:f

(x)=2x33x212x+8) f.[4]) f.[4]) f.[5]) f.[5]) :2x3+8=3x(x+4)[7]

107

13.1 A.)

f

;) f ;)

(x0,f

(x0))

Cf;) f

(x0,f

(x0))

-

Cf

x0

,

f

(x0);) ;)fx0

f

(x0)=0,

(x0,f

(x0))-

Cf

;)

f

(x)=x46x2+3.. ) DeLHospital0.0 ) xx 0e 1lim ;x

) DeLHospital . ) 22x2x lnx 1lim .x lnx 2++ ++ +

)

x=x0

Cf;)2xf (x) ;x 1=

108 DELHOSPITAL)

y=

Cf+-;) 22x x 2f (x) .x x 3+ += +

)

y=x+

Cf; ;) 2xf (x) 3x 5x 1= ++. .)

y=x+

Cf,;)Cf,2x 3x 3f (x) .x 1+ +=+

13.2 ) =x0

Cf, ,) f ) =

Cf+(-), ().) 222x 1f(x)x 2x 1+= + ) Cf+,

0.

:==)2x 2f (x) 2x 1x 3+= + ++ 109) DeLHospital: i)x 1x 1e xlimx 1=ii)2xxxlime+= iii)xlnxlimx+= iv)x 0xlimln(x 1)=+

13.3 ) f ,f. ) 1f (x)x=

(-,0)

(0,+). ) ff

,f. )

f

(x)>0

x, f. ) f,

f

(x)0f

(x)>f

(0)f

(x)>0117f

(-,0]

[0,+)

-R.f

: x0,f

(-,0]

[0, +),

R.f

f

(0)=0.)

f

fR.

f

(x)

f,:f

(0)=0f

(-,0]

[0,+).

f

(-,0]

[0,+);

.f

.)

f

(x)0,

f

[0,+),

x0=0.. )

f

: xf

(0)f

(x)>0 x>0f

(x)>f

(0)f

(x)>0

f

(x)>0

x0,

f

(x)0

x0.

f

(0)=0,

f

(x)=0

x=0.) :ex+e-xx2+2ex+e-xx220f

(x)0f0,:f

(x)0

x

R .118 DELHOSPITAL x=x0 y=y0 y=x+,00x xlim f (x)= 0x xlim f (x)+= 0xlim f (x) y+=0xlim f (x) y= [ ]xlimf (x) (x ) 0+ + = [ ]xlimf (x) (x ) 0 + = x0- Df

-.+-+-:0x xlim f (x)

0x xlim f (x)+

:xlim f (x)+

xlim f (x)

:xf (x) limx= [ ]x limf (x) x=

14.4

f

R

:f(0)=f

(0)=0f

(0)=2) x 0f (x)A = lim .x

) 2x 0f(x)B = lim .x

) 2x 0 x + f(x) = lim .f (x)ln(x + 1)

) :x 0 x 0f (x) f (0) f (x)f (0) 2 lim 2 lim 2 (1)x 0 x = = =

=2.119)

f

x0=0,

,:x 0limf (x) f (0) 0= = 00 DeLHospital:2x 0 x 0 x 0f (x) f (x) 1 f (x) 1B lim lim lim 2 12x 2 x 2 x = = = = =: 00x 0 x 01 f (x) 1 f (x) 1 1B = lim == lim = f (0) = 2 = 12 x 2 1 2 2

f

x0=0,

f

(x)

x0=0

x 0lim f (x) = f (0) = 2.) 2x 0f ( x )lim ,xx2.:( )222x 0 x 0f (x) xxxf (x) ln(x1)x x x f (x) lim limf (x)ln(x 1) +++= =+

:00x 0 x 0 x 01x 1 ln(x 1) 1lim lim lim 1x 1 x 1 + +== = =+

:1 1 12 1+= =

14.5 f

x0=0,

f

(0)=1

f(0)=0,:) x2x 0x - xeA = lim ,x

120 DELHOSPITAL ) x 0f(x)B = lim ,x

) xx 0xf(x)lim = -1.x - xe

) DeLHospital:0 0x x x x x x0 02x 0 x 0 x 0x xe x e xe x e e xeA lim lim lim 12x 2 x = == == = )

f

(0)=1

f

(0)=0,

:x 0 x 0f (x) f (0) f (x)f (0) 1 lim 1 lim 1x 0 x = = =

:00x 0 x 0 x 0f (x) f (x)B lim 1 lim limf (x) f (0) 1x (x) = = == = = = f

,:x 0limf (x) f (0)= ) :2x 0x x x xx 0 x 0 x 02 2 2x 0xf (x) f (x)f (x)xx xx xe x xe x xex x xlimxf (x) B 1lim lim lim 1A 1 x xelim = = = = = =

14.6 32x - 4f(x) = .x

)

f

.)

f

.)

f.)

Cf.)

Cf

f.121)

R

-

x3x24=0.)

f.) f

=R*.

:3 4 3 3 3 32 4 3 3x 4 3x 2x(x 4) 3x 2x 8 x 8f (x)x x x x + += = = =

f

(x)=0x3+8=0x3=-

8x=-

2,f

(-,-2]

(0,+)

[-2,0).8 4f( 2) 34 = = f.) :3 5 3 2 3 33 6 4 4x 8 3x (x 8) 3x 3x 3x 24 24f (x)x x x x + + = = = =

x0.

,f

(-,0)

(0,+).

Cf-.) ()().) :332 2x 0 x 0 x 0x 4 1limf (x) lim lim (x 4)x x = = =

:32x 0 x 01lim(x 4) 4 0 limx = < = +,

x=0

Cf.122 DELHOSPITAL :33x xf (x) x 4lim lim 1x x = = [ ]32 2x x xx 4 4limf (x) x lim x lim 0x x = = =

,

y=x

Cf+-.) -f-,.)

x3x24=0Cf.

x=0.

x0

:33 22x 4x 4 x f (x) x = = = Cf

y=,

R.

,yy:

0.

>-3

,

>0,

y=

Cf.) Cfyy.( )*f . = +x + x 0lim f(x) = - , lim f(x) = + f

=(0,+),

f()=(-,+).

,f,f(),R.12314.7 --

f:RR

.:1g(x) =f(x) ) g.) : + x xlim g(x) lim g(x) ) Cg.)

g.)

g.)

g.)

g.) :f

(x)=0(x=-

2

x=0

x=2)g:=R{-

2,0,2}) x xlim f (x) lim f (x) += = + :x xlim g(x) 0 lim g(x) 0 += = ) x 2 x 0 x 2lim f (x) 0, limf (x) 0 limf (x) 0. = = = f-

2,

0

2

:

x 2 x 2lim g(x) lim g(x) + = = +

x 0 x 0lim g(x) lim g(x) + = + =

x 2 x 2lim g(x) lim g(x) + = = +124 DELHOSPITAL ,

x=-

2,

x=0

x=2

Cg.x xlim g(x) 0 lim g(x) += =

y=0

-Cg(-)(+).) f

(-,-2],

(-,-2)

1g(x)f (x)= .:

x1B e dx, x 1x 1 = > +

2 21 1 dx, x 0,2 x x = +

B. ) .) I xxdx; =

) A x2xdx. =

) ;) lnxA dx B xdx,x= =

x

(0,).)22x 1A dx, x 3.x 5x 6+= > +

12915.2 .) f (x)g(x) dx =.) I lnxdx, =

=. ) ( )A f g(x) g(x)dx, =

=

,

u=

du=.) 32 x 1I 3x e dx,=

=.) 2 99A x(x 4) dx, = =+c.)22x 3A dx.x 3x 2+=+ +:22x 3 A cx 3x 2+=+ =++ +

B. ) : f (x) dx = f (x) dx =

)

f

(x)0

x

[,],

f (x) dx

) cdx =

) : f (x) dx , =

R [ ]f (x) g(x) dx , + =

,

R.) f

[,],

f

(x)0

[,]

f

[,],

f (x) dx

)f

,,

.

:

f (x) dx =+

130 . ) f

[,]

: i)( )xdf (t)dtdx=ii)( )tdf (x)dxdt=

iii)( )df (y)dydx=

) g,( )g(x)f (t)dt =

) 32xt 1xf (x) e dt,+=

f

(x)=) Gf

[,],

: f (x)dx =

) : i)0 xdx = ii)120(3x 2x 1) dx + =

iii)2 1x02xe dx =iv)e1 lnxdx =

15.3 .) : . f (x) dx f (x) c = +.

F

(x)=f

(x),

Ff .f (x) f (x)e f (x) dx e c = + .(x)dx ln (x) c(x)= +

.

f (x) dxf

(x)) 2x 1f (x) ,x+=

x>0,

: .21 1f (x) dx cx x= + + .1f (x) dx lnx cx= +

.1f (x) dx ln( x) cx= + +.2 31 2f (x) dxx x=

.

131) 2f (x) ,x 1=+

x>-1,

f (x) dx: .2(x+1)+c.

2ln(x+1) . 2ln(-1x)+c .2ln(x+1)+c . 2 x 1 c + + )

f

(x)=xx,

f

(0)=f

(0)=1,

: .f

(x)=x+x+1. f

(x)=xx+2 . f

(x)=2x3x+1.f

(x)=xx+2 . f

(x)=-

xx+2) 13f (x) x = (1,2),

: .33 5F(x) x x4 4= + . F(x) x 1 = +. F(x)=x2+1 . F(x) x x 1 = +. F(x) x x = + . ) 220A (3x 2x 1) dx, = +: .3. 5. -7 .-4. 6) 32 12 21 2x 7x x dx 2 dx,x 5 x 5+= ++ + : .12= . =3 .32= .=5. =2) I 2 f (x)f (x) dx =: .f

()f

(). f

()+f

() . f2()+f2() .f2()f2() . f

()f

()f

()f

())

Cf

(0,0)

(1,1),

- 10I f (x) dx =: .2. 1. 3.5 . 0132 ) x21f (x) 1 t dt, = : . f (x) x x = . f

(x)=2x . f

(x)=-

2x . f (x) x = . f

(x)=2x)fRt6 40 xf (x) dx t t , = +: .f

(1)=7. f

(1)=5. f

(1)=3.f

(1)=8. f

(1)=10) 51I x 2 dx,= : .5. -3 . 9.7. 10) x 12txg(x) e dt+=: . . . .. ) 2 h22 h 01L lim 5 t dth+= +: .2. 3. 5 .1 . 7) [ ]0f (x) f (x) xdx 2 + =

f

()=1,

f

(0): .1. 2. 3 .5 . 4. ) - : .32f (x) dx

.1 32 1f (x) dx f (x) dx+

.1 32 1f (x) dx f (x) dx

.0 32 0f (x) dx f (x) dx+

.

133) CfCg

x=

x= : . [ ] f (x) g(x) dx

. [ ] [ ] f (x) g(x) dx f (x) g(x) dx

. [ ] [ ] f (x) g(x) dx f (x) g(x) dx +

. g(x)dx f (x)dx +

. [ ] [ ] f (x) g(x) dx f (x) g(x) dx

) f

(x)=x2

y=-

x+2

: .3. 5. 7 .92. 6)

f

(x)=x3

g(x)=x

x=-

2

x=1

: .114.73.85 .137.112 )

f

(x)=x

g(x)=x

x=0

x=2 : . 3. 5. 2 2 . 3 5. 4 2 ) : .13.57.2 3ln3 2+ .3 5ln2 3+ .

3ln22ln3134 15.4 .)

F

f,

G

f

F+c,

c

R. )

F

(x)=f

(x),

x

,

f (x) dx F(x) c. = + ) f (x) dx f (x). = ) 1dx ln(x 2) c,x 2= + ++(x>0). ) xdx x c. = + )x x dx c. = + ) 1dx 2 x c.x= + . ) ( )xf (t)dt f (x), =fR. ) 2x40f (x) 1 t dt, = +8f (x) 2x1 x . = + ) 14 3 20(5x 4x 3x 2x 1) dx 5. + + + + = ) 0 xdx 0. = ) 2 20 0f (x) dx f (x) dx. =

135

..: 11f(x)f (x)dx f (x) c, 1 1+= + +

21f (x)f (x)dx f(x) c2= +

f (x)dx ln f (x) cf (x)= +

1f (x) 1dx f (x) c, 1 1 f (x) += + +

f (x) f (x)e f (x)dx e c = +

f (x)dx 2 f (x) cf (x)= +

. I A(x)dx =(x)f

(x),.. 1I f (x)dx=

x=f

(y),

dx=f

(y)dy.

f

()=

f

()=,

:( ) [ ] 1 I f f (y) f (y)dy yf (y)dy yf (y) f (y)dy= = =

,Cf1fC

y=x,

Cf.

136 16.1

f:RR*:x1 1 2+ =f(x) f (x) e

x

R

f

(0)=1,

:) :( )xI = f(x) + f (x) e dx )

f

(x)=ex,

x

R

.) .:( )x x xI f (x) f (x) e dx f (x)e dx f (x)e dx = + = + =

x x x xf (x)e dx f (x)e f (x)e dx f (x)e c = + = +

) :x x1 1 2 f (x) f (x) 2f (x) f (x) f (x)f (x) e e++ = = ( )()xf (x) f (x) e 2f (x)f (x) + = ( )x 2 x 2f (x)e f(x) f (x)e f(x) c (1) = = +

(1)

x=0

:f

(0)=f2(0)+cc=0:f (x) 0x 2 xf (x)e f(x) f (x) e , x= = 16.2 fg

[,],

FfGg[,]. f(x) dx = g(x) dx, :) F()G()=F()G(),)

(,)

,

f

()=g().137) FGfg

[,],

: f (x) dx F() F() =

g(x) dx G() G() =

: f (x) dx g(x) dx F() F() G() G() = =

F()G()=F()G())

f

()=g()

h(x)=f

(x)g(x).

h

H=FG,

:( ) H(x) F(x) G(x) f (x) g(x) = =

x

[,] H

[,]

[,]. H

[,],

(,).:H()=F()G()H()=F()G()

H()=H(),

.-Rolle

(,)

,

H()=0.

:H()=0f

()g()=0f

()=g()16.3 2xt t4f(x) = e dt ,

x

R

.)

f

(x).)

f

.) 20I = 3f(x) dx. ) :( )22 2x xx x 2 f (x) e x 2xe , x = = 138 ) :f

(x)=0x=0f

(-,0]

-

[0,+).) :[ ]() 2 2 2200 0 0I 3f (x) dx 3 (x)f (x) dx 3 xf (x) 3 xf (x)dx = = = ==

2 2 2 2x x x x2 20 06f (2) 0 3 2x e dx 0 2 3x e dx = =

x

[0,2],

x x. = :( )3 3 3 2 2 22 x x 3 x0 0 0I 2 3x e dx 2 e (x )dx 2 e dx = = = =

[ ]32x 8 802 e 2(e 1) 2(1 e ) = = = 3 22 x0A = x e dx ( )3x = y.

I f (x)dx =-,: x=+-y dx=-

dy: I f ( y)( dy) f ( x)dx J = + = + =

2=+J,

1I (I J).2= + -

+J

.139:-I f (x)dx =

x=-y.

16.4 x-xxI = dx.e + 1:)xx-xe xI = dxe + 1 ) =)

x=-

y,

dx=-

dy.: x y- xx ( y)( y)I dx ( dy)e 1 e 1 = = =+ +

x y x x- - -yy xx xe xdy dx dxe 1 e 1 1 e = = =+ + +

) :x x x- xx xe xI dx I dxe 1 e 1= =+ +

.:x x x x x- -xx xe x xx xe x2I dx dxe 1 e 1 e 1 += + = = + + +

x x- -xx(1 e )dx xx dxe 1+= = =+

[ ] -- -x( x)dx xx xdx = = + =

[ ]-( ) () x 2 = + =

2=2=.140 ()..fh(x)I g(t)f (t)dt, =g(t),: - ,:( ) ( ) I g h(x) f h(x) f (x) = ,,, .. : I f (x t)dt I f (xt)dt.. = =

xt=u

xt=u,

... xx tI e f (t)dt.=-: xx tI e e f (t)dt,=ex( t). e-x, e-xI..f

(),x..f

()--f.14116.5

f:RR

:yy xxx, y f(t)dt = e (y - 1) - e (x - 1) )

f(x)=xex.)

Cf,

xx

x=1.) :x xy x x yy yf (t) dt e (y 1) e (x 1) f (t) dt e (x 1) e (y 1) (1) = =

f,xyf ( t ) dtx.y(1)x.:( )xx yyf (t) dt e (x 1) e (y 1) =

f

(x)=ex(x1)+exf

(x)=xexex+exf

(x)=xex

f

(x)=xex(1).) ,

x=1,

-

f

(x)=0,

x=0.

f

(x)0

x

[0,1]

1 1 1x x0 0 0E f (x) dx xe dx x(e )dx = = = =

1 1 1x x x0 0 0xe e dx e e e e 1 1 = = = +=

1..().

142

. ) f;) f

;[4]. :) 0dx dx = =

)1dx x dx , 1x= =

) xdx xdx = =

)2 21 1dx dx x x= =

)x x dx e dx = =

[5]. fFf,:)

G(x)=F(x)+c,

c

R,

f,[3])

G

f

G(x)=F(x)+c,

c

R.[6].()-().)

f

. 143) F

f

- . )

F

G

: f:AR,

G(x)=F(x)+c. ) f (x)dx f (x) c. = + ) Ff, F(x)dx f (x) c. = + ) f (x)g(x)dx f (x)g(x) f (x)g(x)dx. = ) ( )f g(x) g(x)dx f (u)du, = :u=g(x)du=g(x)dx [7]

f,g:RR

f

(0)=1

g(0)=1.

fggf.

h(x)=f

(x)+g(x)

(x)=f

(x)g(x),

:) h(x)=h(x)(x)=-

(x),[8])

H(x)=e-xh(x)

(x)=ex(x)

,[8]) f

(x)=ex

g(x)=ex,

x

R.[9]

f:RR

:(x2+1)f

(x)2x

x

R)

f

(0).[5]144 ) 22xf (x) .x 1=+

[5])

f.[9]) Cf,xx

x=1.[6]

:1 2xx 2 41 xf (x) g(x) f (t)dt x 0(1 x ) 1 x= = + +

)

g(x)=0

x

R*.[13]) 1I f (x)dx, =

R*.[12]

145

.:) f (x) dx f (x) dx f (x) dx. = =

)

f

(x)0

x

[,],

f (x) dx

) f (x) dx = [ ] f (x) g(x) dx + =

)

f

,,

,

: f (x) dx =+

) f

[,],

f

(x)0

x

[,]

f ,)f,

xF(x) f ( t )dt =, :

F(x)=

x

, ( )g(x)f (t)dt , =g-

f

g.) : f (x)g(x) dx =

( ) f g(x) g(x)dx , =fg-

,

u=g(x),

du=

u1=g()

u2=g().146 ) fg

x=

x=

(

0

-

[,],

[,]

,:f () = 152 17.30 fg

[0,1]

:f

(0)=g(1),f

(1)=g(0)f

(0)f

(1))

(0,1)-,

f

()=g().) ;17.31

f:RR -

xf

(x)2x2+x

x

R,

f

(0).17.32 :2xf (x)4 x=

)

f.) :x 2 x 2M limf (x) N lim f (x) = = ) f.17.33 f:h 0limf (x h) f (x)+ = ( )( )2 2f (x) x f (x) x 0 + =

x

R.

f.17.34 f

[1,4]

:f

(1)+f

(2)=f

(3)+f

(4),f

(1)f

(2)f

(3)f

(4):)

(1,2)

,:f (1) f (2)f ()2+= ) f.17.35

f:RR -

[f

(x)x2][f

(x)+x2]=2x2+1

x

R.)

f

(x)=0.) f,:f

(2000)=4

106+117.36

f:RR

f

(xy)+f

(x)+f

(y)+3=x+y+xy

x,y

R.)

f

(1)=0.)

f.) :2xx2xlimf(x) 1++

17.37

f:RR

f2(x)=1+2ex[1f

(x)]

x

R.

f

(2004)

) f,:2=-1) ,f-

x0=1.17.40 :22x , x 0f (x)x0, x 0= =

:) f

x0=0,) Cfxx.17.41 f:x+2f

(x)x2+x+2

x

R.

:) f

x0=0,) f

x0=0.17.42 fx0=2

:h 0f (2 h) 3lim 5h+ = :) f

(2),) f

x0=2.17.43 f-

x0=0

f

(0)=1,

:x 0f (3x) f (2x) f (x) 3f (0)A limx+ + = 17.44

f,g:RR -

x0=,

:)2 2x f (x) x f ()A limx =

)x g()f (x) g(x)f ()B limx =

17.45 f-

f

()0,

f .17.46

y = f

(x)

x

[,].

(,)

Cf

(,f

())

-,

(,f

())

(,f

()),

: 2+= 17.47 :f

(x)=x2+x+2) Cf

:y=3x+5.154 ) Cf

(2,7).17.48

f

(x)=x3.

Cf,:) ::y=3x+5) :7 , 93 17.49

f

(x)=lnx

(-1,-2)

(2,1).) .) -Cf.17.50 -

f,g:RR

g(x)g(x)0

-

x

R.

:f (x)h(x) ,g(x)=

x

RCh

(,h())

-,:f ()h()g()= 17.51 :f

(x)=x2+x+21g(x) 1x= ,Cf

Cg

:x=1,

-.17.52 :2xx 1f (x)e+= ) f

(x)f

(x).) f

f

.) f Cxx.17.53

f:RR

.-:) f

,) f

f

(0)=0.17.54

f,g:RR

g(x)=f

(x2x+2)

x

R,fR.Cf(2,3):y=4x+6Cg

(1,g(1)).17.55 :2x x 1, x 1f (x)x , x 1 + 0.15717.75 :2xf (x)x 1=

:) f,) f.17.76

f:RR

,:2f

(x)=[1+f

(x)]x

x

R

f

(0)=f

(0)=1

f

(1)=3:) f

,)

f.17.77

f,g:RR-

[f

(x) g(x)][f

(x) g(x)] = 0,

x

R.

f

(0)=1

g(0)=0,

:) [f

(x)g(x)]2=1,) f

(x)=g(x)+1

x

R

f

(x)=g(x)1

x

R.17.78 :21f (x) 2(x 35) 750x = + ) :23200f (x) 2 , x 0x= ) f-.) f.17.79 :f

(x)=x+(1)x2+1) f-.)

f

(x)=0.)

f

(x)0.)

0e.)

exxe

x>0.)

+1 > ( + 1)

e.)

xx

x>0,

-

=e.17.81 :f:RR

f

(x)>0

:(f

(x)ex+1)(f

(x)+ex1)=0

x

R.

f

(x).17.82 :f

(x)=(2x28x)lnxx2+8x+2158 ) f-.)

f.17.83

f:RR

:2f3(x)+3f2(x)+6f

(x)==2x3+3x2+6x+5

x

R.

f-R,f-.17.84

,>0,

x+x2

x

R.

=1.17.85 :f:RR

f 3(x) + 3f

(x) = x3 + 3x

x

R.) f-.) f

(x).17.86 :f

(x)=3(21)x4+4x3+12x2+12x+2,

R,f

(-1)f.17.87 :f:RR:f

(1+x)f

(1x)2x

x

R:) :g(x)=f

(1+x)f

(1x)2x,) f

(1)=1.17.88 :f

(x)=-

x3+12x4) f.) -

f

(x)=0.) f.) f.17.89

N*

2,

x0

:2(x+1)>2+2x+(1)x2

17.90 -270cm2.-2,5cm3cm.) x-,(x)x.) ,.17.91 159,.-,-.

-30.) --.) -.) -.17.92 :f

(x)=x39x2+12x(lnx1)+22) f

(x).) f.) ,,

Cf.17.93 :4 3 2x x xf (x) x 12 6 2= + + +

,,

R.

,f.17.94 :f

(x)=x3(++3)x2+(+4)x+1,f

(1)f(2,f

(2))

Cf.17.95 :f

(x)=6x2(lnx1)2x3+3x2+2002) :g(x)=lnx+1-x) f.17.96

f:RR

.:[f

(x)]3+3f

(x)=x33x+2

x

R,

:)

f,)

Cf,)

f.17.97 :2x x 3f (x)x + +=+

,,-

Cf

:x=2y=3x+5160 17.98 :2( 1)x 2x 3f (x)3x 2+ +=

():2xy+=0.

,()

Cf

+.17.99 f

(x) +

():y=3x2,

:22xf (x) xf (x) 3x x 3A lim2f (x)(x 1) 6x 5++ + +=+ +

17.100

f:RR + ():y=3x+4, R ,:2xf (x) 6xlim 1xf (x) 3x 5x 2++= + +

17.101 fR

R.

,-DeLHospital,:00x f (x) f ()f () limx = ==

x x [f (x) f ()]lim limf (x)(x ) = =

:x limf (x) f ()= f

x=.-f

.-;17.102

f:RR

:exf

(x)+x=1+(x+1)f

(x))

ex=x+1.)

f

(0).)

f

(x).17.103 fx0=0,

f

(0)=0

f

(0)=1.

-:)x2x 0x xeA limx= )xx 0xf (x)B limx xe=

17.104

f:RR

-:ef(x)+f

(x)=xx+1

x

R.)

f

(0)=0.) f

(x)f

(x)f

(x).) :4x 0f (x)A limx= 17.105

f:RR

-:f3(x)+3f

(x)=3ex3x-3

x

R.) x 0limf (x).

161) f-.) :x 0f (x)A limx= )

f

(x)=0.17.106 :3 222x 3x 6x 1f (x) 6lnxx+ += ) f

(x).)

f

-.) :6x2lnx=2x3+3x26x+1) f

(x).17.107 :2x x 2f (x)x 1+ +=

)

f.) f.17.108 :22 3xf (x) xx= + ) f.) -:x3x23x+2=0

R.17.109 :22x 1f (x)x 1=+

) :3322x 3xf (x)(x 1)+=+

) f-.) :22 2 21 xf (x) 3(x 1) x 1=+ +

) f

Cf.) f.)

Cf.) f.17.110 :f

(x)=x+1+ln(x2+1)e-x

)

f

(x).) f-.) :ex[x+1+ln(x2+1)]=1) xA lim f (x).+= 17.111 :f

(x)=ex1ln(1+x))

f.162 ) f -.)

f.) x 0limf (x)-f.) :1+ln(1+x)=ex

)

x>-1

-:ex1+ln(1+x)17.112 f:f

(x)=6x+4

x

R.

Cf

(1,5)

::x+7y+10=0

f.17.113 -fxx 2x 1++

x0.f,

Cf

-

(0,2).

17.114 :)2 A xdx, x ,2 2 =

)2B xdx, x (0, ) =

17.115

f:RR*,

-R.-Ff,CF

:y=2004.17.116 :*f:+

f

(x)=2f

(x)

f

x0=0

2.-

f.17.117 f:RR

:f

(0)=2(1)f

(x)f

(x)=ex(ex+1)

x

R.) :f2(x)=(ex+1)2

) f.17.118 :)2A x lnxdx =

) B ln(x 2)dx = +

)lnx dx, x 02 x= >

16317.119 f,R.-

g(x)=(f

(x)+f

(x))ex,

x

R.

:( )xg(x)dx f (x) f (x) e dx = + =

x xf (x)e dx f (x)e dx = + =

x x xf (x)e dx f (x)e f (x)(e )dx = + =

x x x xf (x)e dx f (x)e f (x)e dx f (x)e = + =

:xg(x)dx f (x)e (1) =

g(x)dx-g,:xg(x)dx f (x)e c = +

(1)-c;17.120 :)2xA dx, x (0, )1 2x=

) B x ln(2 x)dx = +

17.121 :)2lnxA dxx=

)2x B dx, x ,1 2x 2 2 = +

17.122 I ln xdx, =:+-1=xlnx17.123 :) A x ln(1 x)dx, x (0, ) = +

) B (x 1) xdx = +

)x e dx =

17.124 :)122x 3I dx, x 1x 3x 2+= > + +

)3 222x 4x 8x 6I dx, x 2x 3x 2+ + += < + +

17.125 :)2x x2x x3e 4eA dx, x 1e 3e 2= > +

)x xB e 1 edx = +

17.126

f:RR

-xxxe-2x

x

R.

f

(0)=-1,

f.17.127 :xg(x) (x t)f (t)dt =

f

f

(x)>0

x

R.) g(x).) g.17.128 2x4 31tf (t)dt x 2x 3 = + -

x

R,

f

(1).164 17.129 :f:(0,+)Rx1xf (x) lnx 1 f (t)dt = + +

x>0.17.130 :)420xA dx x=

)30B x x 1 = +

17.131 :)2e1lnxA dxx=

)440B xdx =

17.132 :4 3 22312x x 16x 9A dxx 9x =

17.133 2 xt1f (x) 2 e dt, =:10I f (x)dx =

17.134 : *40I xdx, = ) : 21I I 1=

>2.)

5.17.135 :214 8C: y x x9 3= + 225 10C : y x x9 3= + )

xx.) -

xx.) -.-C1C2..17.136 f.) : 0 0xf (x)dx f (x)dx2=

) :20xxI dx3 x=+

17.137 f-,

f

(0)=0

f

(0)=1,

:x0x 0xf (t)dtA limx x=

16517.138 fR:x 11 xtf (t)dt f (x) f (t)dt =

x

R

Cf-

(2,1),

f.17.139 :f

(x)=-

x2+4x

(3,f

(3)).

Cf.:) .) -,

Cf

xx.17.140 :x2+y2=8:21y x2= 17.141 :*f: (0, )+ + :22004f (x) x f (x) f (1)e= = ) f:1xf (x) 2004e= ) --:2f (x)g(x)x=

xx

x=1

x=2.

166

18.1.1 :z2+z+=0,

,

R1,:) 1, =) 2, ) 2z z 0 + + = 1.18.1.2 :f,g:RR :) f

(xy)+x+y=xy+f

(x)+f

(y)

-

x,y

R, ) g(xy)+g(x)+g(y)=xy+x+y+3

x,y

R.18.1.3

f:RR

:f2(x)+2x2xf

(x)+1

x

R.)

f

(1).) fx0=1.18.1.4 fx0=0

:23x 01xf (x) x lim 7x= ) :2x 01limx 0x =

) x 0limf (x).

)

Cf

.)

f

(0)=0.18.1.5

f:RR

-

x1=2.

:3f (x 1), x 1g(x)f (2x), x 1 + 1) f -.)

f.) :2x 2x x 2 2 x 3x 2 + ;) -

x=-

x.18.2.9 :)21 x xA dx, x ( , )1 x+ += +

)2(x x)x B dx, x ,2 2 x+ =

)2 x dx, x 0,2x 2x 2 = +

)22x dx1 x=+

18.2.10 :f:RR:)xx t x0x e f (t)dt xe f (x)+ =

x

R)x3 216xf (x) 6 f (t)dt 4x 3x 11 = + +

18.3.1 z,:5z z = )

z . ) :5z z = ) -.18.3.2

f,g:RR :170 (f

g)(x)=x21

x

R.) f g ,

f

g

.) fg

11

-

fg

11.) fg.18.3.3

f:RR -

x0=0

x

R

:f3(x)xf2(x)x2f

(-

x)=x2x)

f

(0).) x 0f (x)lim .x )

f

(0)=1.18.3.4 :f

(x)=x33x2+2x+1-Cff,:) ::x+11y+17=0) ::x+y2004=0)

(3,7).18.3.5

f:RR

:f

(x2)x23x+2f

(x3)+2x-4

x

R.)

f

(x)=x2+x.) ()-1M , 0 .2 -:i) ()Cf-

R.ii) Cf-.iii)-,R,:1y2= 18.3.6 :)1 x 1 1lnx 1 x x+< 0)1 e e < <

,>0

0.) 2x 2 x

x>0,-

=e.18.6.7

f:RR

-

f

(0)=f

(0)=0

f

(0)=2.) x 0f (x)lim .x

) x 0f (x)lim 2.x= ) :2x 0 x f (x)lim 1f (x)ln(x 1)+=+

18.6.8 f--

:y=2x+3.) xf (x)A lim .x= 176 ) xB lim [f (x) 2x].= ) ,:2 22 3xx 3 f (x) 2x x 3lim 4x f (x) 2x 2x 1+ + + += +

18.6.9

f:RR

.-:x0g(x) (x t)f (t)dt, x = )

g(x).)

g

.) Cg.18.6.10

f:RR

-

f

(0)=0

f

(0)=2.) x 0f (x)A lim .x= ) :( )x20dg(x) x f (t)dtdx=

) :x20x 0 x 0x f (t)dtx f (x)B lim limx x x x = =

) :x220 x 01lim x f (t)dt 12x 2x 2 =+

18.7.1

f:RR

(f

f

)(x)=2x

x

R.)

f

(1)=1.) f11.-

f-1;) f-R.) :f

(x2+x)=f

(2x+2)18.7.2

00,

N*.

-:) :xxg(x) =

(0,+),) =e.18.7.3 :f

(x)=3x4+4x3+44

,

R.) f -.)

f.) :34+43+440

,,

R.17718.7.4 -

y=f

(x),f

[0,5]

(0,5).

) f.) f.)

Cf.18.7.5

f:RR

-,

f

(0)=2

x0=0

-0.:f (x)x, x 0g(x)0, x 0= =

) x 0f (x)A lim .x= ) 2x 0f (x)lim 1.x= ) g-

x0=0.) g.) g(x).18.7.6 F-

f

(x)

:2F(x) x 1f (x)e 2xe+=

x

R.

F(0)=1,

:)

F(x)=x2+1,) f,) CfCF,) CfCF.18.7.7 fR.:) :[ ]20f 3 f (x) f (x) xdx 22 = + =

f

(0)=1,) :f

()=1[ ]04f (x) f (x) 2xdx 2 + =

f

(0)=2.18.7.8 :) A xln(x 1)dx, x 1 = + >

)x2xeB dx, x 1(1 x)= > +

18.7.9 :)30x 2A dxx 1+=+

)3x 10B e dx+=

)2xxe dx1 e=+

178 )83x 1 dxx x 1+=+

18.7.10 :2lnxf (x)x= ) Cf

xx.) :x 0 xlimf (x) lim f (x) +

) -Cf,xx:x=1x=e

18.8.1 zw:3 7z w wz 1 = = ) zw.) zw.18.8.2 :22x 2x 1f (x)x 1+ +=+

) f-.)

f.)

f.)

Cf.18.8.3

f:[0,+)R

-

f

(0)=0

f

(x)>0

x>0.)

x>0,

-,

f

(x)=xf

().)

f

.) :f (x)g(x)x= .18.8.4 :) 2x+3x+4x=9x ) xx=2x+4,x>018.8.5 :f:RR :3 x 21f (x) f (x) 2 e x x 12+ + = +

x

R.) f

(0).)

exx1=0.) f.) f-.18.8.6 :( )2 2f (x) xln x x 1 x 1 1 = + + + +) f

=R.) :179( )221f (x) ln x x 1 f (x)x 1= + + =+

) f-.) f.) :( )2 2xlnx x 1 x 1 1 + + = + 18.8.7 :x21e x, x 02f (x)ln(x 1) x, x 0x+ = + >

) f,:=1) f,.18.8.8

f:RR

-:2 21f (x ) f(x)4

x

R.)

f

(0)

f

(1).) f.) f.18.8.9

f:RR

-

f

(0)=1:( )x x y0 0 02x tf(t)dt f(t)dt dy xx 2 + = +

x

R.) :x0xf (x) f (t)dt x xx = +

)

x0

f-.)

f

(x)=x

x0.) f.18.8.10

f:RR*

f

(x)=2xf 2(x)

x

R.

f

(0)=-1:) :21f (x)x 1= +

) :10I f (x)dx =

x=y.18.8.11 -:)22f (x) ln xx= + )2 4g(x) lnx 2 xxx= +

180

18.9.1

=3+4i

z z z 50. + = ) z-().) z0,.z0;)

w

C

w 2, z w 3. 18.9.2

f:RR

:f

(0)=f

(0)=f

(0)=2yf (x y) f (x) yf x2 + = +

x,y

R.

:)y y yf (x y) f x f x2 2 2 + = + + +

x,y

R, ) f

(x)=2+2x

x

R, ) f

(x)=x2+2x+2.18.9.3 :f

(x)=x4+2x3x+1) f.)

Cf.) Cf

.18.9.4 :)42x 0xlim 12x 2x 2=+

)2x 0xx xlim 1xe 1 x2=

)2 xx 0 x elim 1x x =

)2x 01xx lim 1x= 18.9.5 :2x x 2f (x)x 1+ +=

) f-.) f .) f.)

Cf.)

f.)f.18.9.6 :)

ex-1=xlnx+1)

ex-11xlnx18.9.7 F-

f:RR

:f

(0)=1f

(x)F(-x)=1

x

R181:) F(x)F(-x)=1

x

R, ) F(x)F(-x)=1

x

R, )

g(x)=e-xF(x)

-R, )

f

f

(x)=ex.18.9.8 fR:8 64 3f (3x)dx f (4x)dx =

:)2412 f (x)dx 0, =

) :x12g(x) f (t)dt =

...-

[12,24],)

(12,24)

,:f

()=018.9.9

(2,-3)

-

x2=4y.:)

,) -.18.9.10 :t 4xtxe 5tf (x) dt, x1 e+= +)

f.)

f

(x)=x5+x.18.9.11 :f (x) 1 x = +

x

[0,].) f-.)

f.) -

Cf,xx

x=0

x=.18.9.12 :f

(x)=x33x2+5x62+23

) f.)

Cf.) -R,

Cf-.

182

19.1

z=x+yi,

x,y

R

.)

+i

:z + 8iw =z +6 )

x

y,

Im(w)=0.)

x

y,

Re(w)=0.) ().) .).)

z-

6(x,y)(-

6,0).

:[ ][ ]2 2x (y 8)i (x 6) yi z 8i x yi 8i x (y 8)iwz 6 x yi 6 (x 6) yi (x 6) y+ + + + + + + += = = = =+ + + + + + +

2 22 2x 6x y 8y ( xy xy 6y 8x 48)i(x 6) y+ + + + + + + += =+ +

2 22 2 2 2x y 6x 8y 8x 6y 48i(x 6) y (x 6) y+ + + + += ++ + + +

) :Im(w)=08x+6y+48=04x+3y+24=0,

(-

6,0).183) :2 22 22 2x y 6x 8yRe(w) 0 0 x y 6x 8y 0 (1)(x 6) y+ + += = + + + =+ +

) :x2+y2+6x+8y=0(x2+6x)+(y2+8y)=0(x2+2

3x+9)+(y2+2

4y+16)=9+16(x+3)2+(y+4)2=52

(1)

(-3,-

4)

=5.

(-

6,0)

-

z,

Re(w)=0.) (1)

(0,0).

C.)

C:(x+3)2+(y+4)2=52,

(-

3,-

4)

=5,

z 3 4i 5. + + = 19.2 z - 1 = z - 3i ,

z

C .) z

x3y+4=0.)

(1,0)

(0,3);)

z z .)

z=x+yi.

:z 1 z 3i (x 1) yi x (y 3)i = + = + 2 2 2 2(x 1) y x (y 3) + = + x22x+1+y2=x2+y26y+92x6y+8=0x3y+4=0().184 ) z 1 z 3i =

(1,0)

(0,3).

.) z,z 1 z 3i , = -.

.

:1 3= =

:y=-3x

y=-3x

x3y+4=0

:2x 3(3x) 4 0 10x 4 0 x5 + = + = = :2 6y 3x 35 5 = = =

:2z (1 3i)5= + 19.3

z

3z + z = 0. ) z. ) 3z + z = 0. ) z1z2,31z 32z . ) 2004 20041 2A = z + z . ) 3 3z z 0 z z . + = = :( )( )3 23z z z z z z 1 0 z 0 z 1 = = = = = 185) z 0. = :3 3z z 0 z 0 z 0 + = = = z 1. = :3 3 2 2z z 0 z 1 0 (z 1)(z z 1) 0 (z 1 z z 1 0) + = + = + + = = + =

=14=-30.

>0,.186 19.5 z7 3z z = 1. ) z = 1. ) z zz

.) 7 3z z = 1. ) 7 3z z 1. = z z , = :7 3 7 3 107 3zz 1 z z 1 z z 1 z 1 z 1 = = = = = z 0

z

C.) :2 1z 1 z 1 zz 1 zz= = = = ) 1z ,z= :37 3 7 4 2 21z z 1 z 1 z 1 (z 1)(z 1) 0z = = = + =

(z=1

z=-1

z=i

z=-

i)..19.6

z z - 2 - 2i = 3 2. ) z

.) : i) ii) ) :z 2 2i 3 2 = 187 z (2 2i) 32, + = z

(2,2)

32. = ) C

(2,2)

32 = :C:(x2)2+(y2)2=18(1)

C

.i) -z.:

:y=x(2)(2)2 2(x 2) (y 2) 18 + =

(x2)2+(x2)2=18

(x2)2=9x2=3(x=5

x=-1)

x=5

y=5.

x=-1

y=-1.

z=5+5i.ii)

z=-1i.

19.7 fR:(

f

f

)(x)f

(x)=x

x

R)

f

(x).)

f

(0).) f-1f

11.

:f

(f

(x))=x+f

(x)(1)

x1,x2

R

f

(x1)=f

(x2).188 :( ) ( )(1)1 2 1 1 2 2 1 2f f (x ) f f (x ) x f (x ) x f (x ) x x = + = + = f

11,

f-1:f

(R)R.) (1)

x=0

:( ) 11f f (0) f (0) f (0) 0= = 19.8

f:R(0,+)

g:R(-,0).

,

R

f

()=

g()=,

:) 0

x

R,

=f

()>0.

g(x)x

xP(x)..) g(x)0f

(x)>f

(0)

f

(x)f

(0)

x

R.

f

(0)=0

.212 ) :ex+x2=x+1ex+x2x1=0f

(x)=0

f

(0)=0

f

(x)>0

x0.

x=0

f

(x)=0.)

f

(0)=0

,:f

(x)0ex+x2x10ex1x(1x)

x

R.19.30

f(x)=xx,

00.)

+1>(+1)

>e.)

2x=x2

x>0.) f

=(0,+).

:2lnx 1 lnxf (x)x x = =

f

(x)=0lnx=1x=ef

-f

(0,e]

[e,+).)

>e

f

[e,+).

ln 1f () f (e) eln e< < < lne0.

:e x e xlnx 1f (x) f (e) elnx x lnx lne x ex e

exxe

x>0.)

e + > + +

ln+1>ln(+1)+1>(+1)

)

x>0

:x 2 x 2ln2 lnx2 x ln2 lnx xln2 2lnx f (2) f (x)2 x= = = = =

(0,e]

f,

11.

f

(x)=f

(2)x=2215

[e,+)

f,

11.

:f

(2)=f

(4)4

(e,+):f

(x)=f

(2)f

(x)=f

(4)x=4

x=2

x=4.19.33

f(x)=ex-1lnx1.)

f

.)

ex-1=lnx+1.)

x+elnx

x>0,

=e.) f

=(0,+).

:x 1 x 11f (x) (e lnx 1) ex = =

x

x 121f (x) e 0x= + >

x

f

(1)=0f

,: x0f

[1,+).

f

(1)=0

f.) :ex-1=lnx+1ex-1lnx1=0f

(x)=0: f

(1)=0 xf

(1)f

(x)>0 f

(0,1].216 x>1f

(x)>f

(1)f

(x)>0 f

[1,+).

f

(x)>0

x1,

x=1

f

(x)=0.)

g(x)=xelnx,

x>0.

g(x)0=g(1)

x>0.

-Fermat

g(1)=0.

:xeg(x) lnx= eg(1) 0 ln e 0 ln 0 e= = = = eh(x) lnxx=

h(e)=0.

=e

:exe+elnxex-11+lnxf

(x)0,0f.19.34 :)

xx=ee

=(0,+)

x=e,)

x+elnx

x>0,

=e.) :x e x ee ex e lnx lne xlnx e lnx lnx 0x x= = = = = :eg(x) lnx ,x=

x>0:eg(e) lne 1 1 0,e= = =

x=e

g(x)=0,

.21721 eg(x) 0x x= + >

x>0.g(x),

x=e

-.)

f

(x)=xelnx,

x

=(0,+).

-

f

(x)0

x>0.

f

(1)==0,

:f

(x)f

(1)(1)

x>0.

(1)

x0=1

(-)f

(x).,

x0=1

-

=(0,+)

f

x0=1

(-).Fermat

f

(1)=0.:xef (x) lnx=

f

(1)=0lne=0ln=eln=e=ee(2),

xx=ee

(0,+)

x=e.

(2)

=e.19.35 :x + x f(x) = ln - lnx - ln2 x + x +

>0.)

f

.)

00f2(x)+1>0

x

R f.) (1)

x=0

:f3(0)+3f

(0)=0f

(0)[f2(0)+3]=0f

(0)=0

x=0

f

(x)=0.

f-(),-.) f

=R,

:

x0f

(-,0)

(0,+).f.) f

=R,

f,.19.39 :f(x)=2+ex-1g(x)=2+e1-x

)

f

g.) CfCgfg.)

x=

CfCg,-

Cf

Cg

.223) fg

=R.

x

R

: f

(x)=ex-1>0,

f. g(x)=-e1-x0,

g(x)=xexex+1.)

g(x).)

f

=(0,+).224 )

x

R

: g(x)=(xexex+1)=ex+xexex=xex

g(x)=0x=0g

(-,0]

[0,+).: xg(0)g(x)>0 x>0g(x)>g(0)g(x)>0

g(x)>0

x0.)

x>0

:x x x2 2e 1 xe e 1 g(x)f (x) 0x x x += = = >

g(x)>0

x>0.

f.19.41

f:R*R,

:1f (x) =x

x0f(1)=f(-1)=2)

f.)

Cf

.)

f.) xx

Cf

.

Cf

,.) 1f (x)x=

x0.

( )1ln xx= R*-,:ln x , x 0f (x)ln x , x 0 +

225: f (1) 2 ln 1 2 2 = + = =

f

(1)=2ln1+=2=2 f (x) ln x 2, = +

x

=R*.) :f ( x) ln x 2 ln x 2 f (x) = + = + =

x0.

f,Cfyy.) f

=R*.

:( )x 0 x 0limf (x) lim ln x 2 = + = ( )x xlim f (x) limln x 2+ += + = +f

f

()=R.)

y=

Cf-

(,f

())

(,f

()).

:f

()=f

()==-yy.Cf,:f

()f

()=-1f

()f

(-)=-121 11 1 ( 1 1) ( ) = = = =

(1,2)

(-1,2).19.42

f,g:RR ,

:f(x)g(x)=x4x

R

y=3x7

Cf

+

.226 ) : i) x +g(x)limxii)2x +g(x) + 5x + 2xlimxf(x) - 3x + 1

)

y=2x3

Cg

+

.) i)

y=3x7

Cf+,: [ ]x xf (x)lim 3 limf (x) 3x 7 (1)x+ += =

f

(x)g(x)=x4g(x)=f

(x)x+4(1)x x xg(x) f (x) x 4 f (x) 4lim lim lim 1 3 1 2x x x x+ + + + = = + == =

ii)2 2x xg(x) 5x 2x f (x) x 4 5x 2xlim limxf (x) 3x 1 xf (x) 3x 1+ ++ + + + += = + +

( )2x xf (x) 2x4x x x1x4f (x) 4x 4 2x 3 4 0 0lim lim 17 0 xf (x) 3x 1f (x) 3x+ ++ + ++ + + + + += = = = + + +

)

y=2x3

Cg+,-:[ ]x xg(x)lim 2 limg(x) 2x 3x+ += = xg(x)lim 2,x+= ()(i).:[ ] [ ] [ ]x x xlimg(x) 2x limf (x) x 4 2x limf (x) 3x 4+ + + = + = + =[ ](1)xlimf (x) 3x 4 7 4 3+= + == + = ,,

y=2x3

Cg+.19.43

f(x)=2+ln(x1)

g(x)=2ln(x1),

x>1,

Cf

Cg

.227.:)

Cf

Cg,)

f

g..

Cf

Cg

..) fg

=(1,+).

CfCg

f

(x)=g(x).

:f

(x)=g(x)2+ln(x1)=2-ln(x1)2ln(x1)=0x1=1x=2CfCg

(2,2).) :1f (x) 0,x 1= >f.x 1 xlimf (x) lim f (x) . += = + f,

f

()=R.: [ ]1 g(x) 2 ln(x 1) 0x 1= = 0.

-,-:(-2,-1),(0,1)(2,+))

x

Df

:3 32 2( x) 4( x) x 4xf ( x) f (x)( x) 1 x 1 = = =

f.,Cf.) :3 2 2 32 2 2x 4x (3x 4)(x 1) 2x(x 4x)f (x)x 1 (x 1) = = =

4 2 2 4 2 4 22 2 2 23x 3x 4x 4 2x 8x x x 4(x 1) (x 1) + + + += =

f

(x)>0

x

Df,

f:1=(-,-1),2=(-1,1)3=(1,+))f1,23.:3 32 2x x x xx 4x xlim f(x) lim lim lim xx 1 x = = = =

32x 1 x 11lim f (x) lim (x 4x)x 1 = = +

:f

(1)=(-,+)=R230 3 32 2x 1 x 1 x 1 x 11 1lim f (x) lim (x 4x) lim f (x) lim (x 4x)x 1 x 1+ + = = = = +

:f

(2)=(-,+)=R332 2x x x 1 x 11 xlim f (x) lim (x 4x) lim f (x) limx 1 x+ ++ + = = = = +

:f

(3)=R,f:f

(Df)=f

(1)

f

(2)

f

(3)=R) :x3x24x+=0x34x=x2x34x=(x21)

x=1

x=-1

.,

x1

:32x 4x f (x) x 1= =

f

(1),

f

(2)

f

(3)

R

f1,23f

(x)=

1,23.:1

(-,-1),2

(-1,1)3

(1,+)19.45 fR

f2(x)xf(x)+x23=0

x

R .)

x=

f,

=-1

=1.)

Cf

.)

x=

ffR,

f

()=0.

:f2(x)xf

(x)+x2-3=0(1)231:2f

(x)f

(x)f

(x)xf

(x)+2x=0(2)

x=,

:2f

()f

()f

()f

()+2=0-f

()+2=0f

()=2

x=

(1):f2()f

()+23=0(2)2(2)+23=0323=02=1(=-1=1)) f

x=.

f-,

f

()=0.

(2):2[f

(x)]2+2f

(x)f

(x)f

(x)f

(x)xf

(x)+2=02[f

(x)]2+2f

(x)f

(x)2f

(x)xf

(x)+2=0

x=

:2[f

()]2+2f

()f

()2f

()f

()+2=0[f

()]2f

()+1=0,

=14=-3-1.)

f(x)

.)

x1+ln(x+1)

x>-1,

=e.)

x+1>0x>-1.

=(-1,+).

Cfyy

f

(0)=11ln1=0,

(0,0).)

x

:x x1f (x) [e 1 ln(x 1)] ex 1= + = +

f

(0)=11=0.x21f (x) e 0(x 1)= + >+

x

.f

,x=0-

f

(x)=0.f

(x): x0 f

2=[0,+)

f

2=[0,+).) f

f

(0)=0

()f

(x).) Cfxx,

f

(x)=0.234 : f

(0)=0,

x=0

. Hf

(x)

(-1,0],

:xf

(0)f

(x)>0 Hf

(x)

[0,+),

:x>0f

(x)>f

(0)f

(x)>0

f

(x)>0

x0,

x=0

f

(x)=0.)

f

(0)=0

f

(x).

x>-1:f

(x)f

(0)ex1ln(x+1)01+ln(x+1)ex

)

1=(-1,0]

:x 1 x 0lim f (x) limf (x) f (0) 0 = + = =

f1

:f

(1)=[0,+)

f

(x)0

x

=(-1,+),

f:f

()=[0,+)f

(2),

2=[0,+)

,

[0,),

R

{+}.

f

(1)=[0,+)

:f

()=f

(1)

f

(2)=[0,+)) :g(x)=x1ln(x+1),x>-1: g(x)0=g(0)

x>-1

()g(0)()g(x).235

x0=0

=(-1,+)

g x0=0.Fermat

g(0)=0.

:x1g(x) lnx 1= +

g(0)=0ln1=0ln=1=e19.48

f:RR

f

(0)=f

(0)=0

f

(x)0f

(x)0g(x)>g(0)g(x)>0 x0f

(x)>f

(0)f

(x)>0)

x>0

xf

(x) > > > > < 0.

h(x)=f

(x)x,

x0.

:22 21 f(x)h(x) f (x) 1 1 01 f(x) 1 f(x)= = = 0h(x)0

f

(x)>0.

g

[0,+),

:x>0g(x) > >

xf

(x)0.) Cf

y=x+1.

:23x 1 1f (x) x 1 x 1 x 1 x3 x= + + + = + = :[ ]11 11 1213 333 1 1E f (x) (x 1) dx dx 3lnxx x x |= + = = + = | \ .

1 13ln1 3ln 3 3ln3 21 3 = + + =

19.55 3 2xxf(x) = e .x + 2

)

f.)

f

.) : x + x -A = lim f(x) B = lim f(x) ) ,, x -2lim f(x). ) x 0lim f(x) .) :x+20x0x-2

x0:Df=R{0,-2}252 ) :3 3 3 3 3 2 2 2 2x x x x x2 2 2x 2x(x 2) x x 3 x 4x 3f (x) e e e e ex 2 x 2 x 2 (x 2) x (x 2) + + = = + = = + + + + +

3 3 2 2x x2 2x 4x 3(x 2) x x 6e e(x 2) (x 2)+ + + = =+ +

x

Df.

f

(x)=0x2+x6=0(x=-3

x=2)f:(-,-3][2,+):[-3,-2),(-2,0)(0,2]) 3 2xx xxlim lim e 1,x 2+ += + =+

=+. 3 2xx xxlim lim e 1,x 2 = =+

=-.)

x0.

:32xx 2 x 21lim f (x) lim x ex 2+ + = = + +

x 2lim f (x).) :x 03lim ,x= 3xx 0lim e 0.= x 0lim f (x) 0 0 0.= = x 03lim ,x+= + 3xx 0lim e .+= + :253( )33 3x3x x22xx 0 x 0 x 0 x 02 33x1 2 1xx xee 3 elim x e lim lim lim2+ + + + = == = ==

( )( )3x32xx 0 x 023x1xe3 9lim lim e2 2+ + = = =+

x 0 x 0lim f (x) lim f (x), + x 0limf (x).19.56 ,,:

P(x)=x3+x2+x+.

,,-.P(x).;:.-,.:P(x)=3x2+2x+,

P(x)0

x

R.

:

=1.

,:P(x)=x3+x2+x+

=2,

:=4212=42122=-820,

P(x)-.P(x).f

=R,

x xlim P(x) lim P(x) += = +(

>0),P(x)

P(A)=R.254

0

P(A),

P(x).-,P(x).19.57

f:RR

:f(x+y)=f(x)ef(y)-y+f(y)x,y

R) f(0).)

R*,

f()=0,

f(x)=0

x

R .) f.) .) :f

(x+y)=f

(x)ef(y)-y+f

(y)(1)

x=y=0

:f

(0)=f

(0)ef(0)+f

(0)f

(0)ef(0)=0f

(0)=0)

f

()=0,

0.

(1)

x=

:f

(+y)=f

()ef(y)-y+f

(y)f

(+y)=f

(y)

y=x

:f

(+x)=f

(x)(2)(1)

y=

:f

(x+)=f

(x)ef()-+f

()f

(x+)=f

(x)e-(3)(2)(3):( )f (x) f (x)e f (x) 0 f (x) 01 e= = =

x

R,

0

1e-0.)

f

(x)=0

(1).()f

x=0,

f-R.255

f

(x)0

x0.

(1)

y=-

x

:f (0) 0f ( x) x f ( x) xf (0) f (x)e f ( x) f (x)e f ( x) (4)= + += + = (4)x

x:f

(-

x)ef(x)-x=-f

(x)(5)(4)(5):f

(x)f

(-x)ef(-x)+f(x)=f

(x)f

(-x)ef(-x)+f(x)=1f

(-x)+f

(x)=0f

(-x)=-f

(x)(6)

x0,

f

(x)f

(-

x)0

x0.

f

(0)=0,

(6)

x=0.

f.) (1):f

(x)=0

f

(x)0

x0.

f,(5):( )f (x) x f (x) xf (x)e f (x) f (x) 1 e 0 x = =

x0

:1ef(x)-x=0ef(x)-x=1f

(x)x=0f

(x)=x

f

(0)=0,

f

(x)=x

x=0.

f

(x)=x

-(1).,,

f

(x)=0

f

(x)=x.

19.58 F

f:RR ,

:F2(x)F(x)F(x)

x

R ,

0:) F(0)=F(),)

f(x)=0

R .256 ) :F2(x)F(x)F(x)(1)(1)

x=0

:F2(0)F(0)F()(1)

x=

:F2()F()F(0):F2(0)+F2()2F(0)F()[F(0)F()]20F(0)F()=0F(0)=F())

00,

(0,+).. f

x0=1,

f

(1)=2:) f

(0,+),) f,) I = f(x)dx. .) :f

(xy)=f

(x)+f

(y)+xyxy(1)

x=y=1:f

(1)=f

(1)+f

(1)+111f

(1)=1) f

x0=1,

:x 1limf (x) f (1) 1 (2)= =

x0>0.

:[ ]00 0 0 0x x h 1 h 1lim f (x) limf (x h) lim f (x ) f (h) x h x h = = + + = (2)0 0 0 0 0 0 0h 1 h 1f (x ) limf (h) lim(x h x h) f (x ) 1 (x x 1) f (x ) = + + == + + = 258 f

(0,+).

0xhx= :

x=x0hh1,00x x h 1lim f (x) limf (x h). = ) f

>0,

:x limf (x) f ()= [ ]h 1 h 1limf (h) f () lim f () f (h) h h f () = + + =

g(h)=f

()+f

(h)+hh,:h 1limg(h) f ()=

f

(h)=g(h)f

()h++h [ ]h 1 h 1limf (h) lim g(h) f () h h f () f () 1 1 = + + = + + = :h 1limf (h) 1 f (1)= = f

x0=1,

-f

(0,+).. ) 0xh,x=

x=x0h

h1

xx0.

-

x0>0.:00 0 0x x h 10 0 0f (x) f (x ) f (x h) f (x )lim limx x x h x = =

0 0 0 0h 10f (x ) f (h) x h x h f (x ) 1limx h 1+ + = =

( )0 0h 10f (h) 1 (1 x h x h)1limx h 1 + + = =

2590h 10x (h 1) (h 1) 1 f (h) f (1)limx h 1 h 1 = + =

[ ]0 00 0 01 1 1f (1) x 1 (2 x 1) 1x x x= + = + = +1f (x) 1x= +

x>0.) :1f (x) 1 f (x) (lnx x) f (x) lnx x cx= + = + = + +

x>0.

:f

(1)=1ln1+1+c=1c=0

f

(x)=lnx+x.) :I f (x)dx (lnx x)dx (x)lnxdx xdx = = + = + =

2 21 x xxlnx x dx xlnx x cx 2 2= + = + +

19.61 f

[0,],

0 f(x)dx = 2 Ff.)

F(0)F().)

(0,)

,

f()=.) Ff,:0 f (x)dx F() F(0) =

:0 f (x)dx 2 =

F()F(0)=2,

F(0)F()=-2.260 )

f

()=

g(x)=f

(x)x,

x

[0,].g

G(x)=F(x)+x,

:G(x)=(F(x)+x)=F(x)x=f

(x)-xG

[0,]

(0,).

: G(0)=F(0)+1 G()=F()1:G(0)=G()F(0)+1=F()1F(0)F()=-2().RolleG

[0,]

(0,)

,:G()=0f

()=0f

()=19.62 f

f(1)=f

(1)=0.) : 1 120 01f(x)dx = xf (x)dx2

) 36f (x) = ,x + 1: 10I = f(x)dx ) .:[ ]1 1 1100 0 0f (x)dx xf (x)dx xf (x) xf (x)dx = = =

12 2 21 10 00x x xf (1) f (x)dx 0 f (x) f (x)dx2 2 2 |= = + = |\ .

1 12 20 01 1 1f (1) x f (x)dx x f (x)dx2 2 2= + =

261) ():2 21 1 1 123 30 0 0 01 1 6x 3xI f (x)dx x f (x)dx dx dx2 2 x 1 x 1= = = = =+ +

31 1330 0(x 1)dx ln(x 1) ln2x 1+ = = + = +

:21303xI dxx 1=+

x3+1=y,

3x2dx=dy.

:2 21 11I dy ln y ln2y = = =

19.63

f:RR

xt - f(t)0f(x) = e dt

x

R .)

f

(x)ef(x)=ex

x

R .)

f.) :xt f (t )0f (x) e dt (1)=

(1)x,-f.(1):f

(x)=ex-f(x)f

(x)ef(x)=ex(2)) (2):( )f (x) x f (x) x f (x) xf (x)e e e (e ) e e c (3) = = = + (1)

x=0

f

(0)=0,

(3)

x=0

:ef(0)=1+cc=0

ef(x)=exf

(x)=x

x

R.262 19.64

f(x)=(x1)lnx,

x>0.) f.) f,.) f.) Cf.) :I = f(x)dx )Cf,xx

x=e.) f

=(0,+).

:x 1f (x) lnxx= +

x>021 1f (x) 0x x= + >

x>0f

.: x0,f

(0,1]

-

[1,+).)

xf

(1).

x>1

f

(x)>f

(1).

f

(x)f

(1)=0

x>0.

f

(1)=0

f.) xlim f (x) ,+= + f0

=(0,+),:f

()=[0,+)263) :x 0limf (x) ,= +

x=0

Cf.xlim f (x) ,+= + Cf.) :2 2 2x x x 1I (x 1)lnxdx x lnxdx xlnx x dx2 2 2 x = = = =

2 2 2x x x xx lnx 1dx x lnx x c2 2 2 4 = = + +

)

f

(1)=0

f

(x)>0

x1.:

x

(0,1)

f

(x)>f

(1)f

(x)>0,

f-

(0,1],

x

(1,+)

f

(x)>f

(1)f

(x)>0,

f-

[1,+).

f

(x)>0

x1.

f

(x)=0

x=1.

-

[1,e]

f

(x)0,

:e2 2e11x xE f (x)dx xlnx x2 4 |= = + = |\ .

2 2 2e e 1 e 3e e 12 4 4 4 | | | |= + + = | |\ .\ .

19.65

y=f(x)

R: xt x02 e f(t)dt = (e + 1)f(x) - 4

x

R)

f

.264 ) f(0).) f:xxef (x) = f(x)1+ e

x

R)

f.) :( )xtx01f (x) 2 e f (t)dt 4 (1)e 1= ++

.f.)

x=0

((1))

f

(0)=2.) :2exf

(x)=exf

(x)+(ex+1)f

(x)xx xxee f (x) (e 1)f (x) f (x) f (x)1 e = + =+

x

R.) :xx xxef (x) f (x) f (x)(1 e ) e f (x)1 e= + = +

f

(x)(1+ex)(1+ex)f

(x)=0x xx 2 xf (x)(1 e ) (1 e )f (x) f (x)0 0(1 e ) 1 e+ + = = + +

xxf (x)c f (x) c(1 e )1 e = = ++

:f

(0)=22c=2c=1

f

(x)=1+ex

x

R.26519.66

f:RR

: g(x) = f(x - t)dt ,x , < ) g.)

g()=g().)

(,)

,

f()=f().) :g(x) f (x t)dt =

xt=y.

dt=-

dy

: x x 0 x x x x 0g(x) f (x t)dt f (y)dy f (y)dy f (y)dy f (y)dy = = = = + =

x x 0 0f (y)dy f (y)dy =

( )x x 0 0g(x) f (y)dy f (y)dy f (x ) f (x ) = =

x

R)

g()=g(),

: 0 0 0 0f (y)dy f (y)dy f (y)dy f (y)dy =

0 0 0 0f (y)dy f (y)dy f (t)dt f (t)dt 0 (1) = + =

t=-

y,

dt=-

dy.: 0 0 0f (t)dt f ( y)( dy) f ( y)dy = =

f,

f

(-y)=f

(y)

y

R.

: 0 0 0 0f (t)dt f ( y)dy f (y)dy f (t)dt = = =

266 : 0 0f (t)dt f (t)dt 0 + =

(1),

g()=g().) : g()=g(), g

[,], g

(,).Rolle,

(,)

,

g()=0.:f

()f

()=0f

()=f

()19.67 e21lnxA = dx.(1 + lnx)

) 2x-121x - 1A = e dx.x

) .)

1+lnx=y.

lnx=y1

x=ey-1.

dx=ey-1dy

:e 2 2y1 x12 2 21 1 1lnx y 1 x 1A dx e dy e dx(1 lnx) y x = = =+

) ():2 2x 1 x 121 1x 1 1A e dx (x 1)e dxx x = = =

2x 12 2x 1 x 1 x 11 11(x 1)e 1 e 1e (x 1)e dx xe dxx x 2 x = + + = + =

[ ]2 2x 1 x 111e e e ee dx e e 1 12 2 2 2 = + = + = + =

26719.68 :2x-322x-31(x - 1)eI = dx(x - 1)e + 2 - x

)

x=3y

: 22x-31(2 - x)I = dx(x - 1)e + 2 - x

) .)

x=3y,

:dx=-

dy:2(3y) 3 32y1 22(3y) 3 32y2 1(3 y 1)e (2 y)eI ( dy) dy(3 y 1)e 2 (3 y) (2 y)e y 1 = = = + +

3 2x2 23 2x 2x 31 1(2 x)e (2 x)dx dx(2 x)e x 1 (x 1)e 2 x = = + +

e2x-3.) -.:2x 32 22x 3 2x 31 1(x 1)e 2 xI I dx dx(x 1)e 2 x (x 1)e 2 x + = + = + +

2x 32 22x 31 1(x 1)e 2 xdx 1dx 1(x 1)e 2 x + = = = +

12I 1 I .2= = 19.69

f:(0,+)R

: x1- f(t)dtf(x) = e

x>0268 .:)

f

(x)=-

f2(x)

x>0)1f(x) =x

x>0. Cf-.:) =,) ,..) :x1 f (t )dtf (x) e (1)=

x>0.

f,x1 f (t)dt-.,(1),f,:( )x x1 1f (t )dt f (t )dt (1) x21f (x) e f (t)dt e f (x) f (x)f (x) f(x) = = == =

:f

(x)=-

f2(x)(2)

x

R.) (1)

f

(x)>0

x>0.

(2):2 f (x) 1 1 11 ( x) (x) x c (3)f (x) f (x) f (x) f(x) = = = = +

(1)

x=1

f

(1)=e0=1

(3)

x=1:11 c 1 1 c c 0f (1) = + = + = :1 1x f (x) , x 0f (x) x= = > 269. ) :21f (x)x=

x>0.

(,f

())

Cf.:yf

()=f

()(x)21 1y (x ) =

y=0

x=0

:21 1(x ) x x 2, = = =

(2,0),21 1 2y (0 ) y , = = 2B0, . :A B A B20x x y y 2 0 1, , ,2 2 2 2 ++ + + = =

.) ,:1 1 2E OA OB 2 2 ..2 2 = = = ,-.19.70

f(x)=x2(e-x2)

g(x)=x2(x2ex).)

f(x)g(x)

x

R .)

f(x)=g(x).) CfCg

x=1.) fgR.:f

(x)g(x)x2(e-x-2)x2(x2ex)x2(e-x2x2+ex)0e-x+exx220(1)270

h(x)=ex+e-xx22.

: h(x)=exe-x2x,

h(0)=0 h(x)=ex+e-x2,

h(0)=0 h(x)=exe-x

h(x)=0ex=e-xx=0h(x)h

(-,0]

[0,+).

: xh(0)=0 x>0h(x)>h(0)=0

h(x)>0

x0.

hR,h.:

xh(0)=0.h

(-,0]

[0,+).h(0)h,:h(x)h(0)ex+e-xx220(1),

f

(x)g(x)

x

R.) h

(-,0]

[0,+).

: xh(0)h(x)>0 x>0h(x)>h(0)h(x)>0

h(x)>0,

h(x)0

x0.

h(0)=0,

x=0

h.:f

(x)=g(x)=0x2h(x)=0x=0271) CfCg

(0,-1).

f

(x)g(x)

x

[0,1],

:[ ]( ) ( ) [ ]1 12 x 2 2 x0 0E f (x) g(x) dx x e 2 x x e dx= = =

( ) ( ) ( )15 31 1 12 x x 4 2 2 x x0 0 00x 2xx e e dx x 2x dx x e e dx5 3 = + + = + + =

( ) [ ] ( )1 12 x x x x001 2x e e 2x e e dx5 3 = + + + =

( ) [ ] ( )1 1x x x x001 13e 2 xe e 2 e e dxe 15 = + + + =

[ ]1x x01 1 13e 2 e 2 e ee e 15 = + + + =

1 2 2 13 5 13e 2e 2e ee e e 15 e 15= + = 19.71 fgR:f

(x)g(x)=(x2+2x1)ex

x

R)

h(x)=f(x)g(x),

-

(0,-1).)

Cf

Cg.) :f

(x)g(x)=(x2+2x1)ex(1):h(x)=(x2+2x1)ex

h(0)=-1,

Ch

(0,-1).

-:2 x 2 xh(x) (x 2x 1)e dx (x 2x 1)(e )dx = + = + =

272 x 2 x x 2 xe (x 2x 1) (2x 2)e dx e (x 2x 1) (2x 2)(e )dx = + + = + + =

x 2 x xe (x 2x 1) (2x 2)e 2e dx = + + + =

2 x 2 x(x 2x 1 2x 2 2)e c (x 1)e c = + + + = + :h(0)=-1-1+c=-1c=0

h(x)=(x21)ex.) :f

(x)=g(x)f

(x)g(x)=0h(x)=0(x21)ex=0(x=-1

x=1)

h(0)=-1 0x

)

f

.)

Cf.)

f.) f.)

f.)Cf,

xx

x=1

x=e.) lnx 1f (x)x+=

=(0,+).

:2 21x x 1 (lnx 1)lnxf (x) , x 0x x += = >

f

(x)=0lnx=0x=1f

(0,1]

[1,+).

f

(1)=1

()-f.) :x 0 x 01limf (x) lim (lnx 1)x = + =

x 0 x 01lim lim(lnx 1) .x+ = + + =

x=0

Cf.x x x xlnx 1 (lnx 1) 1lim f (x) lim lim lim 0x (x) x+ + + ++ += == = =

y=0

Cf+.277)

1=(0,1]

2=[1,+),

:

f

1

2,x 0 x 1limf (x) lim f (x) f (1) 1, = = =

f

(1)=(-,1],x x 1lim f (x) f (1) 1 lim f (x) 0,++ = = =

f

(2)=(0,1].f:f

()=f

(1)

f

(2)=(-,1]

(0,1]=(-,1]) :2 4 3lnx x 2xlnx 1 2lnx f (x)x x x = = =

x>0121f (x) 0 1 2lnx 0 lnx x e x e2= = = = = f (0, e :) e,+

( )( ) M e, f e , :3 eM e,2e

Cf.) f.

f.278 )

f

(x)>0

x

[1,e],

:e e e1 1 11 lnxE f (x)dx dx (1 lnx)(1 lnx)dxx+= = = + + =

e2 2 211 1 1 1 3(1 lnx) (1 1) (1 0) 2 ..2 2 2 2 2 = + = + + = =

e1 f(x)dx

1+lnx=y.

1dx = y dyx: 22211y 4 1 3E = y dy = = - = ..2 2 2 2

19.75 : x12xt + ln tf(x) = dt ,t + 1

x0) f(x) = ln x

x0.) Cf,xx

y=1.) x12xt ln tf (x) dt,t 1+=+

x0.

:11 x xx12 2 2 21 1 1xt ln t t ln t t ln t t ln tf (x) dt dt dt dtt 1 t 1 t 1 t 1+ + + += + = + + + +

( )2 2 2 2 21 1x x1xlnx ln x 1 x ln x 1 xln xf (x)x 1 x x 1 x(1 x )1++ + = = + = + + + +

22x xln x 1 xln x 1x x(1 x )+ + = =+

x0.279:ln x , x 0f (x)ln x , x 0 +

:121t ln tf (1) dt 0 f (1) 0t 1+= = =+

=0

=0.

f (x) ln x =

x0.) f,:f

(-x)=f

(x)

x

R*.

Cfyy.Cf

x>0,

f

(x)=lnx,Cf.:f (x) 1 ln x 1 x e x e = = = =

=21,

1-Cf,

y=1.

:[ ]e ee111 1E e 1 lnxdx e (x)lnxdx e xlnx x = = = =

=e[(ee)(01)]=e1

=2e2.19.76 f(x) = 2x + x ,

x0.)

f.)

f.) f-1.) 2 -10I = x f (x) dx. 280 )

x>0

:1f (x) (2 x) 02 2x x= + >+

2+x>0.

f

=[0,+),

f

[0,+).)

f

(0)=0

xlim f (x) .+= + :xf (x) 2x x x2x = + = + :x 1 1 x 1x x x x x x1lim 0,x+= :xxlim 0x+= :xxlim2 2 0x+ + = >

:x xxlimx2 lim f (x)x+ + | |+ = + = + | \ .

f

f

()=[0,+).) f,

11.

f-1

f,

[0,+).)

x=f

(y).

:

dx=f

(y)dy

0=f

(y)y=0 2 f (y) 2y y 2 y 2 = + = = 281:( )2 2 21 10 0 0I xf (x)dx f (y)f f (y) f (y)dy f (y)yf (y)dy = = = =

22 22 2 20 001 1 1 y f(y) dy yf(y) f(y)dy2 2 2 |= = = | \ .

2 22 2001 1 1 12f(2) (2y y)dy 2 4 y y2 2 2 2 = + = =

2 2 2 2 21 14 (4 1) (0 1) 4 2 22 2= + = = 19.77 31 - xf(x) = .(1 + x)

)

f

.)

f

.)

Cf.) Cf.) f

=R{-1}.:3 23 6 4 1 x (1 x) (1 x) 3(1 x) (1 x) 3(1 x)f (x)(1 x) (1 x) (1 x) + + + = = = = + + +

4 41 x 3 3x 2x 4(1 x) (1 x) + = =+ +

x

.

f

(x)=0x=2f(-,-1)

(-1,2]

[2,+).

1f (2)27= .282 )

x-1

:4 34 8 5 x 2 (x 1) 4(x 2)(x 1) x 1 4(x 2)f (x) 2 2 2(x 1) (x 1) (x 1) + + + = = = = + + +

5 59 3x 6(3 x)2(x 1) (x 1) = =+ +

f

(x): f

(-,-1)

[3,+),

(-1,3], 1M3,32 Cf.) :x 1 x 1lim f (x) lim f (x) + = = +

x=-1

Cf.:x xlim f (x) 0 lim f (x) += =

y=0

Cf..)

Cf

:1 130 01 xE f (x)dx dx(1 x)= =+

1+x=y

:dx=dy28322 23 3 2 21 112 y 2 1 1 1E dy dyy y y y y | |= = = + = |\ .

1 1 1(1 1)4 2 4 = + + =

,-.

19.78

R=1.-.-,:) ,) ,,.) ,(-

(0,0)).:C1:(x1)2+y2=1

(1,0)

R=1.:C2:x2+y2=2

(0,2)284 C1C2:2 22 2 2(x 1) y 1x y + =+ =

2x ,2= :2 2 22 2 2 (4 ) 4 y x y4 2 = = = 2 2 4 , (0, )2 2

:222 4 224 2y (x 0) y x0 = =

y=0

:2M2x2 4 =

22(AM) , (0, 2)2 4 =

) :22AM x()2 4 = =

( )( )( )2 2 22 0 0 0 2 2 2 4 limx() lim lim2 4 2 4 2 4 + = = = +

( )( )2 222 0 0 2 4 lim lim 2 4 44 (4 ) + = = + =

4.28519.79 (2917.)-6.6.66,.66,,.f(t)g(t),h(t)-,

t

[6,12].

:) h(6)h(12)

290 ) :22 2100 x 100K(x) 20 1 20x x = + =

K(x)=0x=10 (x) -

x=10.) :(10)=20(51+10+10)=20

71=1420) :100 10010 x 10= = .19.83 1600..-425.2.-20.40,-.) (x): 16K(x) = 40 x + + 160 x

x.) ,;) .291) 1600,-32002,:3200

2=6400 :3200 32100x x=

100. :32 64020 x x = 40x,x 40.:640 16K(x) 6400 40x K(x) 40x 160x x = + + = + +

) :22 216 x 16K(x) 40 1 40x x = =

(x)=0x216=0x=4 -

x=4.

-,4.) :16K(4) 404 160 40 168 6720 4 = + + = =

292 :32 328 x 4= = 19.84 2000,5000.-500,-1000..:) 2000,) ,,) .) 2000x..,.:=

=1000

=500

:1000=500=2

=2.

x2000,

=x,:xx 2 2= = :x(x) 5000 , x 02= > 293:x(x) 0 5000 0 x 100002> > <

x

(0,10000). 500 1000 ; x =

:500x x1000 2= = ) (x):xE(x) (2000 x)(x) (2000 x)50002 = + = + :x 1E(x) 5000 (2000 x) 4000 x2 2= + =

(x)=0x=4000,

4000+2000=6000.) :4000E(4000) (2000 4000)50002 = + =

=6000

3000=1819.85 (t)-,t.2000201800.:294 ) (t),t,) 40.) (t)(t)--,:(t)=(t)

(t)>0,

:[ ](t)(t) (t) ln(t) (t)(t)= = = 1t c t1ln(t) t c (t) e (t) ce+ = + = = 1cc e . = :

(0)=2000c=2000,

(t)=2000et(1),20 209 1 9(20) 1800 2000e 1800 e ln .10 20 10= = = = 1 9ln t20 10(t) 2000e , t 0. = (()):(t)=(t)(t)(t)=0(t)e-t(t)e-t=0( )t t t(t)e 0 (t)e c (t) ce = = = .:t1 9ln t2020 109(t) 2000e 200010 = =

(t)=(t)

.295) 40:( )21 99 9ln 402ln ln20 1010 10(40) 2000e 2000e 2000 e = = = =29 812000 2000 2081 1620 10 100 = = = =

19.86 6..149m3.-

te4-tm3/h,

t-6..,25m3/h.612,:) Q(t),)

Q(t)

t,) ,) .(

e454.)) ,:Q(t)=-

(te4-t+25)m3/h) :4 t 4 t 4 tQ(t) (te 25)dt t( e )dt 25dt t(e )dt 25t = + = = =

4 t 4 t 4 t 4 tte e dt 25t te e 25t c, t 0 = = + +

(t=0

6..).:Q(0)=149e4+c=149c=149e4

:Q(t)=(t+1)e4-t25t+149e4

Q(t)=(t+1)e4-t25t+95,t0296 )

f

(t)=Q(t).

: f

(t)=Q(t)=-(e4-tte4-t) f

(t)=0e4-t(1t)=0t=1-,

t=1,

-

6+1=7..) t

Q(t)=0.

-:Q(4)=5100+95=0

Q(t)0

x

R.

f.) f:( )x xx x xlim f (x) lim (x e 1) lim e 0 = + = = xx xlim f (x) lim (x e 1)+ += + = +f

f

()=R.(f-f

().,,.)) f,,

11.

:f-1:f

(R)R

f

(R)=R

f-1:RR.)

f-1(x)=x

f

(x)=x.

:f

(x)=xx+ex1=xex=1x=0) .:2 2x x 2 2 2 x x 2e e x 2 x x e 1 (x 2) e 1+ + + + + + 2f (x ) f (x 2) (1) + f,(1):x2x+2x2x20x

[-1,2])

f

(0)=0,

f

f

(x)x

x

[0,+),

f-1(x)0

x0.

:e10E f (x)dx=

298

x=f

(y),

dx=f

(y)dy,

:( ) [ ]1 1 11100 0 0E f f (y) f (y)dy yf (y)dy yf (y) f (y)dy= = = =

12y0y 1 3f (1) e y e e 1 (0 1 0)2 2 2 | |= + = + + + = |\ .

y=x:[ ]121 1x x0 00xE e f (x) dx (e x e 1)dx ex e x2 = = + = + =

1 1 3e e 1 (1) 22 2 2 = + = =

)

x=f

(y),

dx=f

(y)dy.:1 1 1 12 2 200 0 01 1 1I f (y)yf (y)dy y f(y) dy yf(y) f(y)dy2 2 2 = = = =

12 2 2 201 1 1 1 10 10 2e f(x)dx e e 2e 2e (3e 5)2 2 2 2 3 3 3 = = + = =

299

20.1

f:CC

( )f (z) f z z z + = +

z

C.:)( )f z z , = )

f

(z)=z

z

C.20.2 ,,.:) z z z, = ) ,,: + + = + + .20.3 u,zw:22z z 1wz z 1+ += +

z

C:x2+y2=1.) u u u. = ) 1z .z= ) w.

.20.4 :z=x+yi

x,y

R:2 2z 1 z 3 2i 6 + =

(x,y).) C,.) C,,-.20.5 ,-,.: 1 w + + = =

:) w w, = ) w,) ,.20.6 ,,300

(, ),

.) .) : z + +=+ +

,,.) z--.20.7 z1:z z1z z+ = ) z.) z

z3=1

z3=-1.) :2004 2004 20041 2 8A z z z = + ++

z1,z2,,z8

z().20.8 ,,-.,:++1,:)1 1 1 , , = = = )

(++)(++)=,)

(+)(+)(+)=0,)

.20.9 ,,,:()4+()4+()4=0:x=,y=,=x y = = :) 1, = = ) x y , = = ) ,.20.10

f: R R

(x+y)f

(x)=f

(x)+f

(y).

-:) f

(0)=0, ) f

(1)=0,) f.20.11

f:RR:f

(x)x+11+f

(x+y)f

(x)+f

(y)

x,y

R.

:) f

(0)=1) f

(x)+f

(-x)2

x

R) f

(x)=x+1,

x

R30120.12

f:RR

:f

(xy)+f

(x)f

(x)f

()1

x,y,

R.

:) f

(0)=1

f

(1)=1) f

(x)=1

x

R20.13

f:RR

:f

(x+y)+f

(x-y)=f

(3x)

x

R.

:) f

(0)=0,) f

(2x)=2f

(x)

x

R,) f

(3x)=f

(x)

x

R,) f

(x)=0

x

R.20.14

f:RR

:f

(0)=0:( )2f xf (x) f (y) f(x) y + = +

x,y

R.

:) ( ) f f (x) x =

x

R,) f2(x)=x2

x

R,) f

(x):f

(x)=xf

(x)=-

x20.15 :f:(0,+)R :xyln(xy)yf

(x)+xf

(y)f

(xy)

x,y>0.

:)

f

(1)=0,)21 1f f (x)x x =

x>0,)

f

(x)=xlnx,x>0.20.16

f:RR

:f

(1)=1,f

(x+y)=f

(x)+f

(y)21 1f f (x)x x =

x,y

R.

:)

f

(0)=0,)

f,)

f

(xy)=f

(x)f

(y),)21 1 f (x)f1 x (1 x) =

21 1 2x f (x)f1 x (1 x) + =

x0

x1,)

f

(x)=x

x

R.20.17

f: R R

:1f (x) xfx =

x0,

f

(x+y)=f

(x)+f

(y)1

y-

x.) :2f (x) xf 1x =

x0.) :2 2f f (x) 1, x 0x x =

302 ) f

(0).)

f.20.18 -

f:RR

-:( ) ( ) f f (x) x f 1 f (x) 1 x = + =

x

R.20.19 :f:RR :f2(x)2f

(x)x=4x+2x+1

x

R

:f 32 =

) f,.) f

(0).) :g(x)=f

(x)x,x

R,.)

f.) :xf (x)A limx+= 20.20 -

f

(-x)=3x2

x

R.) f

(0).) Cf-

(0,f

(0)).) f.20.21 .,-150m2..2m3m.) .) ,;) -;20.22 :f

(x)=x3+x2+x+3

x0=1

-,

4.)

=0.)

.20.23 :xf (x) ln (x)ln(x)2 =

x 0, .2 = 303) f.) f.20.24 :xx exe

x>0,

:) :xx exf (x) e =

,) =e.20.25 :f

(x)=x48x3+22x224x+

R) f -.) f.) :xA lim f (x), B f (1), f (2)= = = x f (3) E lim f (x)+= =

R.)

f

(x)=0,

R.20.26

f:RR*:f

(x+y)=f

(x)ef(y)-1

x,y

R:) f

(0)=1,) f

(x)=ef(x)-1

x

R,)

y=ey-1,)

f

.20.27 :xx1f (x) , 0 1= + < ) f -.)

f.)

f.) :xx1 1 + = + ) f-.20.28

f,g:RR

f

(x)>0

x

R,xlim f (x)= xlim f (x) .+= + )

f.) -

f

(x)=0.)

(f

g)(x) x (g

f

)(x)

x

R,

g=f-1.20.29 :2f (x) 5 144 (20 x) 3x 108 = + + ) f -.) f.304 ) :25 144 (20 x) 3(36 x) + = 20.30

f:RR -

f

(1)=1

f

(x)0.) f -.) :22lnx 1ln2x ln(2x 2)2 1 x = ++

20.32 :f:RR:21f (x)1 x=+

x

R.

f

(0)=0,

-:)

f-1,) f

(x)=x

x , ,2 2 ) f-1(x)=x,

x .2< 20.33 f-

,

,

)

(e+2)+(e-+2)>2

3

N*

R*.20.34

f:(0,+)R

-

f

(1)=0,

-:( ) f f (x) f (x) 0 x 0 + = > :) f

(1)=1,) (f

f

)(x)=x

x>0,) xf

(x)+f

(x)=0

x>0,) f

(x)=lnx

x>0.20.35 :32(1 x)f (x)x= ) f-.305) f ,,-.)

Cf.)

f.) f.)-

x3+(3)x2+3x1=0.20.36 --:K(x)=6

10-6x26

10-3x+4/x.100.) 500.) 200-.20.37

f,g:RR

-xf (x) f (t)g(t) dt =

x

R.) f

().) :h(x)=f

(x)e-G(x)

(

R),

G-g.) f.20.38

f:RR

:10f (x) x f (xt) dt 2 x = + ) f-.)

f

(x)=f

(x)

-

x

R.) :g(x)=e-xf

(x),x

R.)

f.20.39 :)101A dx, x 0x 10x= >+

)61B dx, x 0x x= >+

20.40

f:RR

-

[0,]:f

(0)=-1,f

()=1f

(x)=f

(x)

x

R.) :g:RR,g(x)=f

(x)+f

(x)) 0 f (x) dx 0. =

) :20 (x x)f (x) dx 0 =

) :0 xf (x) dx =

306 20.41

f:(0,+)R

-:x1f (t)f (x) 2004(x 1) dt x 0t= + >

.) :xf

(x)f

(x)=2004x) :f (x)g(x) , x 0x= > )

f

(x)=2004xlnx.. -Cf,xx

x=e.20.42 :120ln(x 1)I dxx 1+=+

:)120ln2 1I dx,2 x 1=+1 yxy 1=+

)ln2I8= 20.43 :f:(0,+)R:x1 f (t )dtf (x) e=

x>0.:) f,) f

(x)=-

f2(x)

x>0,)1f (x)x=

x>0..

Cf

.:) =.) ,-.20.44 :0122ln(x 2)I dxx 1+=

) 1 2yx ,y 2+= +:0122ln3 1I dx2 x 1=

) 2ln 3I .4= 20.45

f:RR

-

f

(0)=f

(0)=2.

10g(x) f (xt) dt, =

x

R.:)2x 0f(x) 4lim 8,x= ) [ ]1g(x) f (x) g(x)x=

x0,)

g

x0=0,)

g(0)=1.20.46 :2xf (x) e , g(x) lnx = = 3071 e0 1I f (x) dx g(x) dx = +

=e.20.47 :f:RR* :14 3 30f (x) 1 2x t f (xt) dt =

x

R.

:)x3 30f (x) 1 2 t f (t) dt =

x

R, ) i) :421g(x) x ,xf (x)= ,ii)241f(x) ,x 1=+

x

R, )41f (x)x 1=+

x

R, ) [ ]xlimxf (x)x 0.+= 20.48 f g f g

g(x)+g(-x)=1

x

R.) :0 0f (x)g(x) dx f (x)g( x) dx=

) : 0f (x)g(x) dx f (x) dx=

) :22x-2xA dx1 e=+

20.49

f:(0,+)R

-

f

(1)=3.

Ff:1 3f (x)Fx x =

x>0,

:) 1h(x) F(x)Fx =

(0,+),)1F(x)F 1x =

x>0,) :3F(x)(x)x=

(0,+),) f

(x)=3x2

x>0.20.50 :f

(x)=2(x)+2(x)) :20 0f (x) dx 2 f (x) dx =

) :20A f (x) dx =

) :20xf (x) dx2=

308 20.51 :f

(x)=x33x2+3x) f.)

Cf

::y=x) Cf

1fC .

20.52 :120ln(x 1)A dxln(2 x x )+=+

)

x=1y,

-:120ln(2 x)A dxln(2 x x )=+

) 1A .2=

30920

20.53

f:RR

:( ) f x f (x y) f (2x) y + + = +

x,y

R.:) f

(0)=0) (f

f

)(x)=x

x

R) f

11) fR) f

(x)=x

x

R20.54 :3 23 22x 6x 5x 2, 2 x 1f (x) x, 1 x 12x 6x 5x 2, 1 x 2 + + + = < < +

)

f

(x)

f

(x).) f

.) Cf.20.55 :22x 3xf (x)x 2+=+

) f-.) -

f.

.) f .) Cf.) Cf

xx.20.56 :23 xf (x) x e = ) f-.)

Cf,xx

x=1.20.57 :2 xt0f (x) e dt =

) f -.) f.)

Cf,.20.58 fg-

[,],

-

(,)

,: g() f (t)dt f () g(t)dt =

310 20.59 :xx t x1 e lnt dt e lnx + +

x>0,

:=e20.60 f[,]

: f (x)dx 0 =

(,)

,:f (x)dx f ()( ) =

20.61 ,: 0 2 = = + = :) 2=) ==

311

.fx0.fx0-,

f

(x0)=0.[14]. :) fx0,f .) fx0,) fx0x0 Cf,)

f

x0,) f

[,],

(,)

,

x0

(,)

,[5]. ()-().) f

x=

f

(x)=0

x

R{}, f. ) fR,, [,]

Rolle. 312 ) f,- f. )

f

(x)>0

x

,

f . ) FGf,

F=G+c. ) f f

() ,. [6]

f:A=(-,)R

:f2(x)2f

(x)+2x=0

x

(-,)f

(-,0)

(0,),

:)

f

x0=0,[8]) f

(x),[9]) f

(x),f.[8]

f:RR

f

(0)=0

:f

(xy)=f

(x)f

(y)+x

y

x,y

R.

:) f

(0)=1[10]) f

(x)=x[15]313

f,g:(0,+)R

f

(1)=g(1)=0,

f

(x)=-

eg(x)g(x)=-

ef(x)

x>0.

:) fg,[5]) f=g,[8])

h(x)=e-f(x)x

(0,+),[5]) f

(x)=-

lnx,

x>0.[7]

. )

z=+i

,

R

;)

f:AR

11;)

f:AR

,;[3]. 1 2 1 2zz z z =

z1,z2

C.[7]. () -().) z , = 2z ,z=

>0

z

C. )

,

C

2+2=0==0. 314 ) f

(,),

f m. )

f:AR

11,

f

(x)=y

y

R. )

f

(x)=0

x

,

f. [5].

(x)=-

x

x

R.[10]

f:RR

:f

(0)=f

(0)=1f2(x)=f

(x)f

(x)

x

R :) f

(x)=f

(x)

x

R,[15]) f

(x)=ex,

x

R.[10]

F

f:RR

f

(0)=1

f

(x)F(-

x)=1

x

R.)

g(x)=F(x)F(-

x)

.[8])

F(x)F(-

x)=1

x

R.[8]) f.[9]

:t 4 2xtxe 5t 3tf (x) dt, x1 e+ += +315) f

.[15]) :x 4 2x-e 5x 3xI dx, 1 e+ += +[10]

. )

f:AR;[1])

f

(x)0

x

[,]

f

[,],

f (x) dx ;

[1]) f

,x

,

xF(x) f (t) dt; =

[4]. f

[,]

Gf[,],

f (x) dx G() G(). =

[5]. :) ( )g(x)df (t) dtdx=

) fgR,

(f

g)(x)= ) 0x xlim f (x) 0=

f

(x)0,

z

C 2.1 2z z z z , =

z1z2

3. z , =

>04.

z5.

z.2zz= . z z = .

. z z = .zz= .

:1 2 3 4 5

[5]. . 1. f2. f

(x0)=0

x0

(,)3. Bolzano4. f5. f. f

[,]

f

()f

()0.h 0limf ( h) f ()+ = . f

(,)324 :1 2 3 4 5

[5]. fF,:)

G(x)=F(x)+c

f,[3]) Gf

G(x)=F(x)+c,

c

R.[7].:) Ff,:f (x) dx f (x) dx = =

) xdx , =

xdx =

) f (x)g(x) dx =

) ( ) f g(x) g(x) dx , =

u =

du = ) f

[,],

f

(x)0

x

[,]

f , f (x) dx

[5]

f:R*R

11

:(f

f

)(x)

f

(x)=

x

R*

0.)

(f

f

)(x)=x

x0.[6]325) f.[5])

1

Cf

Cg

.) :i)

Cf

Cg,[8]ii)

f

g.[8]) CfCg.[9]326

. ) x0;) Bolzano.) -f

=(,);[6]. f

[,],

f

()f

()

f

()f

(),

x0

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,

f

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f:AR,

f:

{ }y / y f (x) x A = )

x=x0

C , - . )

f

(x)f

(x0)

x

,

f:AR . ) f (x) ln (x) , = :(x)f (x)(x)= (x). 327)

f:AR

, f

(). [5].

N*-{1},

:(x)=x-1

[6]

f:RR

f

(x+y)=f

(x)f

(y)x

y

x,y

R.)

f

(0)=1.[8]) f

x0=0,

R.[9]) f

x=

f

()0,

fR.[8]

:x f (x) 2lnx 2ln , 0 x= + > ) f.[10]) 2 2 ln 2

.[8]) 2 2x2xln x .= [7]328

f:RR

:23 xxf (x) 3f (x) e x 12+ = +

x

R.)

g(x)=ex+x1,

g(x)=0

g.[8])

f.[6])

f.[6]) f.[5]

. ) fx0,f ( )0 0A x , f (x ) ; [2]) :21(x) x= [7]. fx0,f-.[10]329. () -().) 0x xf (x)lim 1,g(x)= f g x0

0 0x x x xlim f (x) lim g(x) 0, = = 0x xf (x)lim 1.g(x)= ) x0f,-

x=x0

- f. ) fx0f- x0, ( )0 0A x , f (x )

Cf. )

f

(x)=x++g(x)

xlim g(x) 0,+=

y=x+ Cf. ) f f

(x)0

x

,

f . )2 2x limf(x) ,=x lim f (x) .= [6]

f,g:RR

g(0)=0

:2f

(x)+f

(1y)+g(x)g(y)=3(x+1)26y

x,y

R.

:) 2f

(x)+f

(1x)=3(x+1)26x

x

R,[9]) f

(x)=x2+2x

x

R,[9])

f

g

.[6]330

f

g:RR

:f

(x)g(x)=(x2+2x1)ex

x

R)

h(x)=f

(x)g(x),

Ch

(0,-1).[15])

Cf

Cg.[10]

100 .,-100.100.,,10.) .[10]) ,.[10]) ;[5]

331

.fg

f

(x)=g(x)

x,c,:f

(x)=g(x)+c[7].

(x)=x

x

R.[8]. :) 0z z , =

>0

z

C,

)

z

C,

z z z z + = =) f,f) 0x xlim f (x)= +-,

0x x1limf (x)=) f

[,],

f .)

>1,

x xx xlim lim + = =) 0x xlim f (x) 0,> x0.) 0 0x x x xlim f (x) lim f (x), + 0x xlim f (x).

) 0h 0limf (x h) ,+ =0x xlim f (x)=) f (x) dx f (x) dx , = = Gf

[,].[10]332

f,g:RR

:f2(x)+g2(x)+2x2xf

(x)+2g(x)x

x

R.

:) f

(0)=0g(0)=1,[7]) f (x) x x g(x) x x , [9]) fg

x0=0.[9]

,

++0.

2+2+2=0

,,:)2 1 1 1 2 + + = + + [9])1 1 1 , = = = [7]) 2 + + = [9]

f:(-1,+)R x>-1:( )x t x2 2 20 0 0x 2 f (u)du dt (x 2x)f (x) (t 2t)f (t) dt + = + +

333) :x0x f (u) du (x 1)f (x) + = +

x>-1.[13]) f.[12]

.fgx0,-

f+g

x0

(f+g)(x0)=f

(x0)+g(x0).[4]. :) 1(x ) x ,=

RZ [5])( )x x ln, =

>0[5])( )1ln xx=

x

R*

[5]. () -().) 00 00x xx x x xx xlim f (x)f (x)lim lim g(x) 0.g(x) lim g(x) = ) 0x xlim f (x) ,= 0 0x x x xlim f (x) lim f (x) . = = 334 )

f

(x)=g(x)

x

R

f,g- R,

f

(x)=g(x). ) dy dy du dx.dt du dx dt= ) ( )( )g(t )df (x) dx f g(t) g(t),dt=fg- R. )

f:RR

x limf (x) 0,<

f

(x) +

) f.22.3

f:RR

f3(x)+f

(x)=x

x

R.f:) ,) ,) .22.4 -,.:) ,,,) 2+2+2=0,) ,2,22-,) 4,44

.22.5 :f:(0,+)R:[ ]xy1xyf (xy) f (y) xy 1 f (y) f (yt)dt 1 = +

x,y>0.

f

(1)=0,

-:f

(x)=lnx,

x>022.6

f:RR

:f

(x)=f

(1x)

x

R339

f

(0)=1

f

(1)=2,

:10I f (x)dx =

22.7 :120xI dx1 x=+

2,: 21I I 1=

22.8

f:RR

-,:( ) ( )2x x t0 0 02 dt f (t)dt f (t)f (u)du =

22.9 f-R,:1 x 10 0 xf (t)dt tf (t)dt (t 1)f (t)dt f (x) + + =

x

R.22.10 :2xx xf (x)1 1 x x+=+ + +

) f

(x).) :f

(x)+f

(-

x)=x+xx) :2-2f (x)dx 2 =

22.11 :f:RR10x f (xt)dt f (x) 1 =

x

R.22.12

f,g:RR

-:1 10 0f (x)dx 1 g(x)dx = +

(0,1)

,

f

()=g()+32.22.13 :f:(-1,+)R

f

(1)=f

(1)=0

36f (x) ,x 1=+

:10I f (x)dx =

22.14 :txtxe tf (x) dt, xe 1+= + )

f

(x)=x+x.) f-.) :10I f (x)dx=

22.15 f:RR

: f (x)dx f () f () =

0

f

(1)=1

f

(x)>0

x>0,

f

(x)=x

x

R.22.19

f:RR :f

(1x)+f

(1+x)=f

(x)

x

R:)0 11 0f (x)dx f (x)dx=

)2004 20030 2002f (x)dx f (x)dx =

22.20 :x, x 0f (x)x1, x 0= =

:x0xx 0 t 20f (t)dt xA limte dt x 1 =+

22.21 :2 3xx xf (x) e 1 x2 6= + + + + :310xI dxf (x)=

22.22 :22 2xf (x) , x Ax (x 2)= =+ ) f-.) :22202 2x (2 x)I dxx (x 2)= +

22.23 :xf (x) ln 1 , x [0, )2 = + =

.) :1f (x) , x 1 x x= + +

341) f -Cf.. : 220x x xI dx1 x x+ +=+ +

) x y2= -:20 1I 1 dx4 1 x x = + + +

) .22.24

f:RR :( )2f (x)f yf (x) 1 x f (y) f (x) =

x,y

R.

:) f

(0)=0,)

f

(x)=0

-

x=0,) f

(1)=1

f

(x1)=f

(x)1,

x

R,) ( )2f (x)f f (x) x , = ) ( ) f (x) f f (x) 2x, + = )f

(x)=x

x

R.22.25

f:RR -,:( )f (x)I 1 xf (x) e dx = +

22.26 ,,

++=.

,.22.27 :4xlnx, x 0f (x)0, x 0> = =

) :x 0lim(xlnx) 0= ) f.) :I xlnxdx =

) -Cfxx1.22.28 f

[,]

(,)

:f

()=6+f

()

1,2,3(,),:1 2 31 2 3 f ( ) f ( ) f ( )+ + = 22.29 f R:x20 (t 1)f (t)dt + =

0 1x 02 tf (t)dt 4 xtf (x)dt =

x,

f

(0)=0

f

(0)=2.342 ) 22xf (x) .x 1=+

) () Cf,xxx=0,

x=

.) 10cm/sec,3-

ln10.

343

23.1

f:RR

(f

f

)(x)=2x

x

R.:) f

(1)=1,)

f

R,)

f

.23.2

f:(0,+)R

:xf (x) f (y) fy =

x,y>0.

f

(x)=0

,:) f

11,) :f

(x2+3)+f

(x)=f

(x2+1)+f

(x+1))

f

(x)>0

x>1,

f.23.3

f:RR-

(3,2)

(5,9).) f.)

f-1.) :( )1 2f 2 f (x x) 9,+ + =

x

R) :( )1 2f f (x 8x) 2 2 < 344 23.4 f

f

(x)=x3+x+1.) f.) f.)

f-1.)

f-1(x)=x.23.5

f:RR

:2x 2lim f (x) 2x 3x 3 4 + + =

) x 2limf (x).

) :22 x 2f(x) 9limf(x) 7 4+

23.6 :22x 1x ( 3)x 2 A limx 4x 3 + + = +

) ,

=3.)

=2,

.23.7 f

[0,4]

f

(0)=f

(4).

h(x)=f

(x)f

(x+2)

x

[0,2].

:) h,)

f

(x)=f

(x+2)

[0,2].23.8 f,g

h:RR:( )22g (x) g(x) 1 h(x) f (x)g(x) + = =

x

R.

gCf,Ch

(,),

f

()0,

:) g()=1, ) g()=0, ) g()=0,)

Cf

Ch

.34523.9

f

(x)=3xg(x)=-

x2+3x+1

(0,1)

(1,3).23.10 fR. ( ) M , f () Cf.)

f

(x)=f

()(x)+f

()

.) Cf-Cf-.23.11

f:[0,+)R

Ff.F(0)=0

f

(x)=e-F(x)

x0,

:)

f

(x)=-

f2(x)

x0,) f.23.12 2 xt1f (x) e dt, =:10I f (x)dx =

346

..,.1.

f:AR.

f-1,f

11.

- f

11

: i)

f

(x1)=f

(x2)

x1,x2

x1=x2. ii) f. iii)

y=f

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y

R (

y

f

()).

f-1

f.

f-1(x)

y=f

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yx.2.

f:AR.

f

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y

R,

y=f

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x,

x

.ii)

1,2,,

ff.:f

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f

(2)

f

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f

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3. . x.

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11.348 - ,: i) , ii) , iii)Rolle...7.

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