1

2 2001

:

: (4)

1o

A.1. z1, z2 . : z1 z2 = z1 z2.

7,5

.2. , .

z :

. z z z 2 =

. zz 22 = . z - z = . z z = . z z i =

5

.1. i, 3 - 1 z i 4 3 z 21 =+= , .

1

2

1. z z 21 . 4 2. z 21 . 2

3. 2

2z . 25

4. 1z . 5 5. z i 2 . 2

. 5

. 10 7,5

.2. 1, z z =

z

1 z = .

5

2 f :

3 x ,

3x

e-1

3 x ,x f(x) 3-x

2

>

=

. f , = 1/9. 9

. Cf f (4, f(4)).

7

2

3

. f, xx x=1 x=2.

9 3 f, R, :

f3(x) + f2(x) + f(x) = x3 2x2 + 6x 1 x R,

, 2 < 3 . . f .

10 . f .

8 . f(x) = 0

(0,1). 7

4 f, R, o :

i) f(x) 0, x R ii) f(x) = , x R. dt(xt)ftx 2 - 1 1

0

22 g

x - f(x)

1 g(x) 2= , x R.

3

4

. (x)2xf - )x(f 2= 10

. g . 4

. f :

x1

1 f(x)

2+= . 4

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

7

( ) 1.

( , , ). . , .

2. . .

, .

3. . 4. . 5. : (3)

. 6. : (1)

.

K

4

1

30 2002

: : (4)

1o

A. f ' [, ]. G f [, ],

.

)(G)(Gdt )t(f =

12

.1. f(x) = x. f R

f(x) = x .

8

.2. , . . f [,]

(,], f [,] .

1

. , 1-1 , .

1

1

2

. f x0 , 0f(x)

x xlim

0

= . 0 f(x)

x xlim

0

= 1

. f R ,

. dx)x(xf)x(xfdx)x(f = 1

. f(x) > 0 x , 0 f(x)x x

lim

0

> 0 . 1

2 z f() = i z, IN*.

. f(3) + f(8) + f(13) + f(18) = 0 . 7

. z= Arg(z) = ,

f(13) =

++

+ 2

i2

.

8

. z= 2 Arg(z) = 3

,

0, z f(13).

10

2

3

3 f, g R . fog 1-1.

. g 1-1. 7

. : g(f(x) + x3 - x) = g(f(x) + 2x -1) .

18

4 . h, g [, ]. h(x) > g(x) x [, ],

.

dx)x(gdx)x(h >

2 . R f,

:

x R f(0) = 0 . ,1xe)x(f )x(f =

) f f. 5

) , f(x)x f(x) 2

x 0.

12

) f, x = 0, x = 1 xx,

)1(f 2

1 E

4

1

4

( )

1. ( , , ). . , .

2. . .

, .

3. . 4. . 5. : (3)

. 6. : (1 1/2)

.

K

4

1

29 2003 :

:

1o

A. , f x0, .

8

.

;

7

. , .

. z _z ,

zzz == . 2

. f .

f(x)>0 x , f .

2

1

2

. f, ,

, c IR . c)x(fdx)x(f += 2

. f , f .

2

. f x0 . f x0 f(x0)=0, f x0.

2

2 z=+i, ,IR w=3z

_zi +4,

_z z .

. Re(w)=3+4

m(w)=3. 6

. , w y=x12, z y=x2 .

9

2

3

. z, y=x2, .

10

3 f(x) = x5+x3+x .

. f f .

6

. f(ex)f(1+x) xIR. 6

. f (0,0) f f 1.

5

. f 1, x x=3 .

8

4 f [,] (,). f() = f() = 0 (,), (,), f()f()

4

. f(x)=0 (,).

8

. 1, 2 (,) f(1)0.

9

. f .

8

( )

1. ( , , ). .

2. .

.

. 3. . 4.

. 5. : (3)

. 6. : 10.30 .

K

4

1

8 2003 :

: (4)

1o

A. f . F f , :

.

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

. G f

G(x) = c)x(F + , c R . 10

. , . . z1, z2 ,

z z z z z z 212121 ++ . 2

. f ' (, ), x0, f .

f (x) > 0 (, x0) f (x) < 0 (x0, ), f (x0) f .

2

1

2

. f : R 11 , x1, x2 A :

x1 = x2, f(x1) = f(x2) .

2

. f, g , :

= dx g(x) (x) f g(x) f(x) dx (x) g f(x) . 2

. x = x0 f ;

7

2 . ()

z :

2 z = m (z) 0 . 12

. , z (),

z

4 z

2

1 w

+=

xx . 13

2

3

3

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

x=+ .

5

. f, x .

6

. 0 f(x) 1 x (x) f 2 =++ . 6

. ( ) 1 2 ln dx 1x

1

1

0 2 +=+ . 8

4 f IR , :

f(x) = )x2(f f (x) 0 x IR . . f .

8

. f(x) = 0 .

8

. (x) f

f(x) g(x) = .

g xx, 45 .

9

3

4

( )

1. (, , ). .

2. .

.

.

3. .

4. .

5. : (3) .

6. : 10.00 .

K

4

1

1

27 2004 :

: TE (4)

1o

A. f ' x0 . f x0 , f(x0)=0

10

. f x0 =

5

. .

. .

2

. ` ? f(x)lim0xx , ` ?? -/ f(x)limf(x)lim

00xxxx

2

. f, g x0, fg x0 :

(fg)(x0) = f(x0) g(x0) 2

2

2

. f, . f(x)@0 x , f .

2

. f [,]. G f [,],

/? )(G)(Gdt)t(f 2

2

f f(x)=x2 lnx .

. f, .

10

. f .

8

. f. 7

3 g(x)=exf(x), f

IR f(0)=f(23 )=0 .

. (0,23 )

f()=/f(). 8

3

3

. f(x)=2x2/3x, I()= dx)x(g

0

, IR 8

. )(I lim -

9

4 f: IR IR f(1)=1. x IR ,

g(x)= )1x(z

1z3dt)t(fz

3x

1 /-/ 0,

z=+iC, , IR *, :. g

IR g.

5

. N z

1zz -?

8

.

Re(z2) =2

1/ 6

. A f(2)=>0, f(3)= >, x0 (2,3) f(x0)=0.

6

4

4

( )

1. (, , ). . .

2. , . .

, .

3. .

4. .

5. : (3) .

6. : 10:30 .

K

1

1

5 2004 :

: (4)

1o

A. f .

f f(x) = 0 x , f .

9

. , .. f x0

, .

2

. .

2

. f, g IR fog gof, .

2

2

2

. C C f f1 y = x xOy xOy.

2

. f x0,

f(x) lim )x(flim k0x x

k

0x x = , f(x) 0 x0, k k 2.

2

. f (, ) [, ].

6

2 f: IR IR f(x) = 2x + mx 4x 5x, m IR , m > 0.. m f(x) 0 x IR .

13

. m = 10, f, xx x = 0 x = 1.

12

3 f: [, ] IR [, ] f(x) 0 x [, ] z Re(z) 0, m(z) 0 Re(z) >Im(z) .

3

3

z

1 z + = f()

z

1 z

22 + = f2(), :

. z= 1 11

. f2() < f2()

5

. x3f() + f() = 0 (1, 1).

9

4 f [0, +) IR ,

+= 210 2 dt 2xf(2xt) 2x f(x) .. f (0, +).

7

. f(x) = ex (x + 1). 7

. f(x) [0, +). 5

. f(x)lim x + f(x)lim x .

6

4

4

( )

1. (, , ). .

2. , . .

, .

3. .

4. .

5. : (3) .

6. : 10:00.

K

1

1

31 2005 :

: (4)

1o

A.1 f, [, ].

f [, ] f() f() f() f() , x0 (, ) ,

f(x0) = .

9

.2 y = x + f +;

4

B. , . . f [, ] f() < 0

(, ) f() = 0, f() > 0.

2

. ( )g(x)f(x)lim0

xx+ ,

f(x)lim x x 0

g(x)lim x x 0

.

2

2

2

. f f1 f y = x, f1 .

2

. f(x) > 0 x0 f(x)lim0x x

= 0, += f(x)

1 lim

x x 0

.

2

. f ,

( ) ) f(- f(x) dt)t(fx

= x . 2

. f , x x , .

2

2

z1, z2, z3 z1=z2=z3= 3. . :

z

9 z

11 = .

7

. z

z

z

z

1

2

2

1 + . 9

. : z1 + z2 + z3= 3

1 z1 z2 + z2 z3 + z3 z1.

9

3

3

3

f f(x) = ex, > 0.

. f .

3

. f, , y = ex.

.

7

. () , f, yy,

() = 2

2 - e .

8

. 2

() lim

2

+

+ .

7

4

f IR ,

2 f(x) = ex f(x) x IR f(0) = 0. . :

2

e 1 ln f(x)

x

+= .

6

. N : x

dt t) - f(x lim

x

0

0 x

.

6

4

4

. :

h(x) = dt)t(f t x

x 2005 g(x) =

2007

x

2007

.

h(x) = g(x) x IR . 7

. 2008

1 dt)t(f t

x

x 2005 =

(0 , 1).

6

( )

1. ( , , ). .

2. , . .

, .

3. . 4. . 5. : (3)

. 6. : 10:30 .

K

1

1

6 2005

:

: (4)

1o

A.1 f x)x(f = . f (0,+) :

x2

1)x(f = .

9

.2 f:A IR 1-1; 4

B. , . . ,

f 0, f .

2

. f (,) xo. f (,xo) (xo,) , ( ) )x(f,x oo f.

2

2

2

. .

2

. f,g fog gof, fog gof.

2

. xx.

z,z 2

. f IR *, :

= dx)x(fdx)x(f . 2

2

. z1, z2

z1+z2=4+4i , i55zz2 21 += z1 , z2 .

10

. A z,w z 1 3i 2 w 3 i 2 : i.

z, w , z=w 10

ii. z w. 5

3

3

3

f, IR f(x)0 x IR . . f 1-1.

7

. Cf f (1,2005) (-2,1),

( ) 2)8x(f2004f 21 =+ . 9

. C f, Cf

(): 2005x668

1y += .

9

4

f: IR IR , 2005

x

x)x(flim

20x= .

. :

i. f(0)=0

4 ii. f(0)=1.

4

4

4

. IR , : ( )( ) .3)x(fx2 )x(fxlim 2222

0x=+

+

7

. f IR f(x)>f(x) x IR , :

i. xf(x)>0 x0. 6

ii. .

1

27 2006

:

: (4)

1o

A.1 f, .

:

f(x)>0 x , f .

f(x) 0)x(f >

x0.

2

1

2

. H f() f .

2

. , x IR . 1 -xx 3 x)3( = 2

.

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

2 2

f(x) =2+(x-2)2 x2.

. f 1-1.

6

. f -1 f .

8

. i. f f -1 y=x.

4

ii. f f-1.

7

2

3

3

1 z,z,z 321321 zzz === . z z 0z 321 =++

. : i. 321321 zzzzzz == .

9

ii. 4zz2

21 Re . 1)zz( 21 8

. z1 ,z2,z3 , .

8 4

f(x)=1x

1x

+ lnx.

. f.

8

. N f(x)=0 2 .

5 .

g(x)=lnx (,ln) >0 h(x)=ex (,e) IR , f(x)=0.

9 .

g h .

3

3

4

( )

1. ( , , ). . .

2. , . .

.

3. . 4.

. 5. : (3)

. 6. : 10.30 .

K

4

1

5 2006

:

: (4)

1o

A.1 : (x)=x, xIR . 10

.2 f . f ;

5

B. , . . z1, z2 , :

2121 z z z z + . 2

. f, g xo

g(xo)0, g

f

xo :

[ ] 2)g(x )g(x )(x f)(x g )f(x )(x gf o ooooo =

. 2

. x0 [ ]x

1 xn =l .

2

1

2

. f: IR 11, y f(x)=y x .

2

. f [,]. G f [,], . G() G() f(t)dt =

2

2

1x

x

e1

e 1 f(x) ++

+= , xIR . . f

IR . 9

. dxf(x)

1 . 9

. x

3

. () f x=2 yy yo=3,

i. ().

9

ii. f, (), xx

5

3x = .

9

4

f(x) = nx1)(x1)n(xx ll ++ x>0. . i. : 0x ,

x

1 nx 1)n(x >

4

( )

1. ( , , ). . .

2. , . .

.

3. . 4. . 5. : (3)

. 6. : 10.30 .

K

4

1

24 2007 :

: (5)

1o

A.1 z1 , z2 , :

2121 z zz z = . 8

.2 f, g ;

4

.3 y = f +; `

3

B. , , , , , . . f [,]

x[ , ] f(x) 0 . > 0 dx f(x) 2

. f x . f f(x) > 0 x .

2

1

2

. f x0 g x0 , gof x0 .

2

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

2

. > 1 . 0 lim x x

= 2

2

2ii 2

z ++= IR .

. z (0,0) =1.

9

. z1, z2

2ii 2

z ++=

= 0 = 2 .

i. z1 z2 .

8

2

3

ii. :

)z( )(z 22

1 = .

8

3

:

f(x) = x3 3x 22

IR + 2

, Z .

. f , .

7

. f(x) = 0 .

8

. x1 , x2 x3 f, (x1 , f(x1)) , B(x2 , f(x2)) (x3, f(x3)) y = 2x 22 .

3

. f y = 2x 22 .

7

3

4

4

f

[0, 1] f(0) > 0.

g [0, 1]

g(x) > 0 x [0, 1]. :

F(x) = , x [0, 1], x0 dt g(t) f(t) G(x) = , x [0, 1]. x0 dt g(t)

. F(x) > 0 x (0, 1].

8

. N : f(x) G(x) > F(x)

x (0, 1].

6

. N :

G(1)

F(1)

G(x)

F(x)

x (0, 1].

4

. :

x dt g(t)

dt t dt g(t) f(t) lim

5 x

0

x

02 x

0

0 x

2

+ .

7

4

5

( )

1. ( , , ). .

2. , . .

.

3. . 4.

. , .

5. .

6. : (3) .

7. : 10.30 .

K

5

1

3 2007

:

: (4) 1o

A.1 f x0, .

10

.2 Rolle ;

5

B. , , , , , . . f()

f . 2

. f, g, g [,],

=dx)x(g)x(f

dx)x(f dx)x(g . 2

. f ,

=

x

dt)t(f f(x) x. 2

1

2

. f (,), (,) = = . )x(flim

x + )x(flimx 2

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

2

2

++

3

3

f(x) = ex e lnx, x > 0.

. f(x) (1, +).

10

. f(x) e x > 0. 7

.

dt)t(fdt)t(f dt)t(f4

2

2x

3x

2x

1x

2

2

2

2 += ++++

(0, +). 8

4

z1 = +i

1

12

z2

z2z

+= ,

, IR 0. z2 z1 IR .

. z2 z1 = 1. 9

. z1 .

6

. >0,

z

2

1z

1

. 0 )i1z()i1z( 20120

1 =+++ 10

3

4

( ) 1. (,

, ). .

2. , . .

.

3. . 4.

. , .

5. . 6. : (3)

. 7. : 10.00 .

K

4

1

24 2008 :

: (5)

1o

A.1 f(x) = xln , x* * :

( )x

1xln =

10

.2 f [,];

5

B. ,

, , , , . . f:A 11,

f1 : )A(fy y,))y(f(f A xx,))x(f(f 11 ==

2

. f f .

2

1 5

2

. z2+z+=0 ,, 0 , .

2

. f ,

f( x ) > 0 x.

2

. A f ,,

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

2

z w

3i)(3wi)(1w 6z)22i( ==+ :

. z .

6

. w .

7

. w

6

. wz 6

2 5

3

3

=>=

0x , 0

0x,lnx x f(x)

. f 0. 3

. f .

9

.

x

ex = . 6

.

f(x+1)>f(x+1)f(x) , x > 0 .

7 4

f

+= 203 45f(t)dt 3x)10x(f(x) .

f(x)=20x3+6x45 8

3 5

4

. g

.

h

h)(xg(x)glim(x)g

0h

= 4

. f () g ()

45f(x)h

h)g(x2g(x)h)g(xlim

20h+=++

g(0)=g(0)=1,

i. g(x)=x5+x3+x+1 10

ii. g 11 3

( )

1. ( , , ). .

2. , . .

.

3. .

4 5

5

4. . , .

5. .

6. : (3) .

7. : 10.30 .

K

5 5

1

3 2008 :

: (4)

1o

A. [, ]. G f [, ],

= )(G)(Gdt)t(f 10

. ;

5

. , , , , . . 11,

. 2

. f , f , .

2

.

dx)x(f

1 4

2

xx xx.

2

. , , :

+i=0 =0 =0 2

. (, x)(x, ) . :

`

0==

)(f(x)limf(x)lim

oo xxxx

``

2

2

2

3i1z1

+= z2+z+=0, .

. =1 =1.

9

. . 1z31 = 8

. w, :

1zzw 1= 8

2 4

3

3

.x,xln x f(x) 2 02 >=. : f(x)1 x>0.

6

. f.

6

.

0x

0x

,k

,)x(f

xln

)x(g

=

>

=

i. k g .

6

ii. 2

1k = , g ,

, (0, e).

7

4 f [0, +) f(x) > 0 x 0. :

F(x) = , x [0, +), x0 dt f(t) = x dt)t(ft

)x(F)x(h

0

, x (0, +).

3 4

4

. =+10 1 1)(Fdt)]t(F)t(f[e t 6

. h (0, +).

8

. h(1)=2, :

i. < 20 tf(t)dt 2 dt f(t) 20 6

ii. )(Fdt)t(F 12

11

0 = 5

1. ( , , ). .

2. , . .

.

3. . 4.

. , .

5. . 6. : (3)

. 7. : 10.00 .

K

4 4

1

( ) 20 2009

:

: (5)

1o

. f . f x 0)x(f = , f .

10

. f x0 ;

5

. , , , , . . z1, z2 ,

2121 zzzz = 2

. f () x0A, f(x)f(x0) xA

2

1 5

2

. 1x

1xlim

0x=

2

. f .

2

. f [, ] f(x)

3

. w

02

z12ww =+

. 0z

8

3

,1x ),1xln( (x)f x >+= 10 >

A. 1)x(f ,1x > =e 8

. =e,

. f .

5

. f ]0,1( ),0[ +

6

. , ),0()0,1( + , 0

2x

1)(f

1x

1)(f =+

(1, 2) 6

3 5

4

4

f [0, 2]

( ) 0dt)t(f2t20

=

= x0 ],2,0[x,dt)t(ft)x(H

=+

=

0 x,

t

tlim

],(x,dt)t(fx

)x(H

)x(G

t

x

2

2

0

0

116

203

. G [0, 2].

5

. G (0, 2)

2x0,x

)x(H)x(G

2

5

1. ( , , ). .

2. , . .

.

3. . 4.

. 5.

. 6. : (3)

. 7. : 10.00 . .

K

5 5

1

9 2009 :

: (5)

1o

A. f(x) = x . f (0 , + ) :

x2

1)x(f =

9

B. f xo . f xo ;

6

. ,

, , , . . z

( ) z)z( = 2

. f 1-1, f .

2

1 5

2

. f(x) = 0 f(x) < 0 xoxx

lim o

oxxlim )x(f

1 = +

2

. f(x) = x. H f 1= }{ 0xx =

= )x(f -x

12

2

. f, ,

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

2

2

z :

( ) ( ) 08zi2zi2 =++ . N

z = x+yi .

10

2 5

3

. N .

1z

2z

8

.

21 z,z

40zzzz 2121 =++ 22

lim

7

3

= f(x) ln[(+1)x2+x+1] - ln(x+2), x > -1 -1 . ,

f(x) . +x 5

. = -1 . f

. 10

. f

6

. f(x) + 2 = 0 0

4

3 5

4

4

f: [ ]2,0

x2exk)x(f4)x(f4)x(f =+ , 0 2x )0(f2)0(f = , f (2) = 2 f(2)+12 e4, f(1) = e2

k .

.

g(x) = 3x2- x2e

)x(f2)x(f , 0 2x

Rolle [0,2].

4 . (0,2) ,

)(f4)(f + = 6 e2 + 4 )(f 6

. k = 6 g(x) = 0 x [0,2].

6

. 0,ex)x(f x23= 2x 5

.

dxx

)x(f2

1 2 4

4 5

5

1. ( , , ). .

2. , . .

.

3. . .

4. . , .

5. .

6. : (3) .

7. : 10.00 . .

K

5 5

1

1 4

( ) 19 2010

:

: (4)

A1. f . F f , :

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

f

G f

G(x)=F(x)+c, c 6

A2. x=x0 f ;

4 A3. f

. f ;

5 4. ,

, , , .

) +i +i .

2

2 4

) f . f , .

) f (,), (,),

)x(flimB)x(flimAxx + ==

) (x)=x, x ) 0)x(flim

0xx

3

3 4

f(x)=2x+ln(x2+1), x 1. f.

5 2. :

( ) + +=+ 1x 1)2x3(ln2x3x2 4 22 7

3. f f .

6 4.

=1

1

dx)x(xfI

7

f: x :

f(x)x f(x)x =3+ x

0

dtt)t(f

t

1. f

f(x)=x)x(f

)x(f

, x 5

2. g(x)= ( )2)x(f 2xf(x), x, .

7

4

4 4

3.

f(x)=x+ 9x2 + , x 6

4.

+++ < 2x

1x

1x

x

dt)t(f dt)t(f , x

7

1. ( , , ). .

2. , . .

.

3. . 4.

.

5. .

6. . 7. : (3)

. 8. : 10.00 . .

K

1

7 2010 :

: (5)

A1. f(x) = x, x, )x( = x

8

A2. f

[,] ;

4

A3. f

x0A () , f(x0); 3

4. , , , , .

) f(x) = x, > 0, ( ) 1= xx x ) fog gof,

fog = gof

) += )x(flimxx0

, 010

= )x(flimxx

1 5

2

) f

[,] f(x) x[,],

0

dx)x(f 0

) zC zzz =2 10

z1, z2

z1 +z2 = 2 z1 z2 = 5 B1. z1, z2

5

B2. w

221

22

21 zzzwzw =+

w

(x+1)2 + y2 = 4

8

B3. w 2

2 Re(w) + Im(w) = 0 6

2 5

3

B4. w1, w2 w

2 421 = ww , 221 =+ ww

6

f(x) = (x2)lnx + x 3, x > 0

1.

f

5

2. f

(0,1] [1, + ) 5

3. f(x) = 0

.

6

4. x1, x2 3 x1 < x2,

(x1, x2) , f() f() = 0

f ( ))(f, .

9

3 5

4

f: f(0) = 1 f(0) = 0

1. f(x) 1 x 4

2. +=+

x

xdt)xt(fx

limx 3

1

0

3

0

6

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

3.

f(x) = x2xe 2, x

8

4.

h(x) = +2xx

dt)t(f , x 0

5

1. ( , , ). .

2. , . .

.

3. . 4.

.

5. .

6. . 7. : (3)

. 8. : 09.30 . .

K

5 5

1

1 4

( ) 16 2011

:

: (4)

A1. f x0 . f x0 , : f (x0) = 0

10

A2. f . y=x+ f + ;

5 A3. ,

, , , .

) z 0 z0=1 ) f:A 1-1,

Ax,x 21 : x1x2, f(x1) f(x2)

) x1={x |x=0} : x

1)x(

2 =

) : 1x

xlim

x=+

2

2 4

) C C f f1 y=x xOy xOy.

10

z w iz 3 , :

2i3zi3z =++ i3z

1i3zw +=

B1. z

7

B2. i3z

1i3z =+

4

B3. w 22 w

8

B4. : zwz = 6

f : , , ( ) 0)0(f0f == , :

( ) )x(fx)x(f1)x(f)x(fex +=+ x.

3

3 4

),xeln()x( x =1. : f x 8

2. f .

3 3. f

.

7 4. = x

)xeln( x

2

,0

7

f, g : , x :

i) f(x)>0 g(x)>0

ii) += x0

t2

x2dt

)tx(g

e

e

)x(f1

iii) += x tx dt)tx(f ee )x(g0

2

2

1

1. f g

f(x) = g(x) x. 9

2. :

f(x) = ex, x 4

4

4 4

3. :

x

f

)x(flnlim

x 10

5

4.

= x dt)t(f)x(F1

2

xx yy x=1.

7

( ) 1. ( ,

) . .

2. . . .

3. . 4.

. , .

5. . 6. . 7. : (3)

. 8. : 10.00 . .

1

1 5

6 2011

:

: (5)

A1. f(x)=x x

= x )( x 10

A2. f, . f .

5

3. , , , , .

) z=+i, , z z=2

) f x0A () f(x0), f(x) f(x 0) xA

) f

, 1-1 .

2

2 5

) 0 f(x)>0 x= )x(f0xxlim 0,

+= )x(f1

lim0xx

) f x0 .

10

z, w,

:

iz =1+Im(z) (1) w( w +3i)=i(3 w +i) (2)

B1. z

y=4

1x2

7

B2. w (0,3) =2 2 .

7

B3. , z, w z =w.

5

B4. N , , u ,

3

3 5

,,, .

6

y= x , x . 0 (0,1) xy , .

t, t 0 m/min16)t(x = 1. ,

t, t : 0 x(t)=16t

5

2. (4,2) , , .

6

(0,1)

(4,2) y= x

y

O x

4

4 5

3. .

6

4. t0 (0,4

1),

d=() .

8

xy.

f: , 3 , :

i) )0(f1x

)x(flim

0x+=

ii) f(0) < f(1)-f(0)

iii) 0)x(f x 1.

f x0=0.

3

2. f . 5

g(x)=f(x)x, x , : 3. g

)x(xg

x

0xlim

6

5

5 5

4 . >2 20

dx)x(f

5

5. g, xx x=0 x=1 ()=e

2

5,

10

dx)x(f

(1,2) ,

0

dt)t(f =2

6 ( )

1. ( , ) . .

2. . . .

3. . 4.

. , .

5. . 6. . 7. : (3)

. 8. : 18.00

K

1

1 4

( ) 28 2012

:

: (4)

A1. f . x , f

0(x)f >

7

A2. f [, ];

4 A3. f .

f x0A ; 4

A4. , , , , .

)

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

) = + , f(x)

2

2 4

) ,x

1)(x

2= x{x |x=0}

) ,(x)g(x)dxf[f(x)g(x)](x)dxgf(x)

+= g,f

[,] 10

z w :

4=+z +1z 22

1_ (1)

12= w 5_w (2)

B1. z = 1

6

B2. z1, z2 z 2=zz 21

_ , .zz 21 + 7

B3. w

14

y

9

x

22 =+ w

6

B4. z,w (1) (2) :

1 wz 4 6

3

3 4

f(x)=(x1) nl x1, x>0 1. f

1=(0,1] 2=[1,+). f

6

2. x>0 .

,ex 20131-x = 6

3. x1, x2 x10, x=e xx

7

f : (0,+) , x>0 : f(x) 0

e

xx f(t)dt

21xx

1

2 + xx = nl f(x)edt

f(t)

tnt

x

1

+ l

1. f .

10

4

4 4

f(x) = ex( nl xx), x>0, : 2. : ( )( ) ( ) ( ) + xfxf

1xflim 2

0 x

5

3. nl xx1, x>0,

( ) dt, f(t)xF x = x>0,

>0, ( 2). :

F(x) + F(3x) > 2F(2x), x>0 ( 4).

6

4. >0. (,2) :

F() + F(3) = 2F()

4 ( )

1. ( , ) . .

2. . . .

3. . 4. .

, .

5. . 6. . 7. : (3)

. 8. : 10 .30 . .

1

1 4

14 2012

:

: (4)

A1. f (, ), x0, f . f(x)>0 (, x0) f(x)

2

2 4

) f [, ]. G f [, ],

= )(G)(Gdt)t(f 10

z, z1,

w=1z

1z

+

.

:

B1. 1z = 7

B2. 4

z

1z

.

6

B3.

+21 z

1

z

1 (z1+z2)4, z1, z2

z

6

B4. u,

uui=w

iw, w0, x2y2=1

6

f: , :

xf(x)+1=ex , x .

3

3 4

1. f(x)=

=

0x,1

0x,x

1ex

6

2. o f1 .

6

3. f ( ))0(f,0 . , f ,

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

8

4. ( )[ ])x(fn)x(x llnlim0 x+

5

f:A =(0,+ ), :

f(A)= ( ]0, f (0,+ ), 2f(x)+ ,2dt

t

1tfee

x

1x

x

1

)t(f)x(f)t( +

+=

+ x>0

F(x)= dtfx

1)t( , x>0

1. f(x)=

+1xx2

n2

l , x>0

8

4

4 4

2. F ( ))x(F,x 00 , x0>0, . (x0, ) >x0, F M( ))(F,

: F() x(1)y+2012 (1)=0 6

3. >1,

[ ]0

3x

)1x()1(

1x

x)(f)1()(F 35 =++

+

, x, (1,3) 5

4 .

0x,dt)t(ftdtx

tf

x

1

x

x

2

>

6

( ) 1. ( ,

) . . 2.

. . .

3. . 4. .

, .

5. . 6. . 7. : (3)

. 8. : 18 .30

K

2003

.... 2003

-

1

. ) ( )f x x= . f

R : ( )'f x x= . 8

) ,f g :

( ) ( )' 'f x g x= x .

c : ( ) ( )f x g x c= +

x 5

) f . :

f Ax 0

. 3

. .

) RAf : 1f , f

.

) f 0x ( )0 0f x > , ( ) 0f x >

x 0x .

) f ],[ ,

( )0 ,x ( )0' 0f x > .

) f

. ( )'' 0f x > x .

) f [ ],a ( ) 0f x ( ) 0a

f x dx

> ,

[ ]0 ,x ( )0 0f x > .

) f ],[

( ) 0a

f x dx

= , f .

9

2003

2

( )( )ln

, 0ax

f x ax

= > .

. f (1, ( )1f )

0x y = , .

5

. 1a = :

) f .

5

) .

7

) : ( ) ( )1

1

+

> + 8

8

3

f [ ], 0 < <

z a i= + )()( fifw += ( ) 0f .

. :

) ( )

1

1

1

i zz

f i w

+ =

+

( )f a a=

5

) z iw= ,z w 0

, .

5

. 2 2 2

z iw z iw = + . :

) ( ) ( ) 0a f f =

4

) ,z w 0 .

3

) ( )0 ,x a ,

f ( )( )0 0,M x f x 0(0,0).

8

2003

4

f ( )''f x R :

( ) ( ) ( ) ( )0 1

2

0 0

1 '' 2 ' 4

x

x

t f t dt t f t dt x t f x dt+ = Rx ,

( )0 0f = ( )' 0 2f = .

) Rxx

xxf

+

= ,1

2)(

2

10

) ( )E a

f , xx 0x = 0x a= > .

a 10

/ sec3cm ,

( )E a , 3a cm= .

5

) g :

( ) ( )2g x x f x+ Rx .

(i) 2y x= +

g x+

5

(ii)

g , + 0x =

2x = , : ln 5E

5

2004

.... 2004

-

1

. :

f , [ , ] .

f [ , ]

f ( ) f ( )

, f ( ) f ( ) ,

x 0 ( , ) ,

f ( x 0 ) =

5

. f , ,

.

0 1 2 3

1

y

x x

y

y = f (x)

, I 1 , I 2 , I 3 . 3

10

I f (x) dx= 2

=3

02 dx (x)' f I 2

3

30

I f ''(x) dx= 3

2004

. .

1 . x

x lim

0x

2 .

x

1xlim

0x

3 . lnx lim0x +

4 . x

x e

1 lim

.

. 0

. 1

. +

8

. f ( x ) = x, IN -{0, 1}. , f

IR f ( x ) = x - 1

. 5

2

.

f , g IR

f ( x ) g ( x ) = 1 , f ( x ) 1 x IR...

2xf(x)

2g(x)L

x

lim

+=

+

,

0

0

.

. i ) L . 6

i i ) f g

+ . 6

. g IR . 6

. : f ( x ) g ( x ) = x + 4 x IR . 7

2004

3

x IR. +

=

x

0 t

dt e

2g(x) , > 0

z = g ( x ) + x i z + i | z 1 | .

. , i ) g i i ) z

g- 1

. 4

. , :

. R e ( z ) I m ( z ) , x IR 7

. = 1 . 7

.

12

2 t t 0 0

1 1 1 1 dt dt

1 e e e 1 e<

1 2,x x :

1 2x x= ( ) ( )1 2f x f x=

2. ( ) ( )0 0

lim limx x x x

f x g x

< ( ) ( )f x g x< 0x .

3. f [ ],a ( )0 ,x a

( )0 0f x = , ( ) ( ) 0f a f < .

4. f [ ],a ,

( )0 ,x a ( )0' 0f x < .

5. ( ) 0a

f x dx

= f

[ ],a , ( ) 0f x [ ],x a . 10

2

( ) ( )2 ln 2 , 0f x x x x= >

) : ( )ln

' , 0x

f x xx

= > .

4

) ( )0

lim 'x

f x+

.

2005

OEE

EMATA 2005

2005

2

3

) f . 8

)

( )ln x

g x

x

= , xx 1

x

e

=

2

x e= .

10

3

z = ex

+ (x 1) i, x .

) : ( ) ( )Re Imz z> x .. 8

) ( )0 0,1x 2

2w z z i= + + .

8 ) z .

9

4

f ( )1

02

f =

( ) ( ) ( )' 'xe f x f x x f x+ + = x .

) f xex

xf+

=1

)( , x

( ) ( )f x f x x+ = x .. 7

) ( )limx

f x+

.

6

) ( )2

2

I f x dx

= .

6

) : ( )2

0

04

f x dx

.

6

2006

2006

1

1

. ) f 0x

. f

0x ,

: ( )0' 0f x = . 11

) 0

x x=

f ;

4 .

.

) :f 1 2,x x

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

) ( ) ( )( )0

limx x

f x g x

( )0

limx x

f x

( )0

limx x

g x

.

) ( )0

limx x

f x

= + ( ) 0f x x

0x .

) f

,

( )'' 0f x x .

) ( ) 0a

f x dx

= a < ( ) 0f x =

[ ],x a . 10

2006

2006

2

2

z 1

z iw

i z

+=

+

z i .

) : w i

zw i

=

+

5

) 1z = w ,

xx.

6 ) : w z .

7

) f [ ],a ( ) 1f a >

( )z f a i= ( )w f i= . ( ) 0f x = (,).

7

3

( ) 1xf x e ax= 1a > . )

f ( )( )0, 0f . 4

) f .

8

) ( )a

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

i) : ( )2

12

aa

a e a = .

7

ii) ( )lima

a

+

.

6

2006

2006

3

4

f ( ) 0f x >

( ) ( )1

0

, ,g x t f xt dt t x= .

:

) ( ) ( )2

0

1x

g x t f t dtx

= 0x .

6

) g 0

0x = .

6

) ( ) ( )0

x

x g x f t dt < 0x > .

7

) ( ) ( )2 1

1 0

3t f t dt t f t dt = ( )1,2

: ( ) ( )2g f = . 6

OE

E

EM

ATA 2

007

2007

1

1

&

1 . ) Fermat.

4

) f, ( )' 0f x > .

f .

9 .

.

1. Af : ( )f x

x , f .

2. ( )0

lim 0x x

f x

= , ( )f x 0x

( )0

lim 0x x

f x

= .

3. f ( ) ( ) 0f a f < ( ) 0f x

( ),x a , f [ ],a .

4. ,f g

( ) ( )' 'f x g x= x , ( ) ( )f x g x=

x .

5. f ,

: ( ) ( )f x d x f x d x = .

6. ,f g [ ],a

( ) ( )f x g x< [ ],x a , ( ) ( )a a

f x d x g x d x

OE

E

EM

ATA 2

007

2007

2

2

2

xeaxxf += )()( 2 , x . 2 2y x= +

f ( )( )0, 0f : ) : 2a = .

7

) f .

6

) : i) ( )lim

x

f x

ii) ( )limx

f x+

6

) ( ) 2007f x = .

6

3 ,z w 0z w :

z w z w+ = .

:

) ( )Re 0z w = . 6

) z

w

.

5

) ,z w

, 0.

7

) f [ ],a 0 a < <

( ) ( ),z a i f a w f i = + = ( ) ( )'x f x f x =

( ),a .

7

4

( )2

0

1

1

x

g x d tt

=

+ t,x .

) g .

4

) : ( )2

1

xg x x

x

+ 0x

7

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

6

OE

E

EM

ATA 2

007

2007

3

3

) g , xx 0, 1x x= =

( )1

1 ln 22

g = ..

8

2008

2008

1

1

'

&

1

. . f, g .

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

c , x :

f (x) = g(x) + c

6

. f (x) = x, IN{0, 1}

IR :f(x) = x1

5

. 21

z,z .

() ():

. z1 z2

.

2

. : 2121

zzzz +=+

2

. : 212121

zzzzzz ++

2

. 21

zzzz = 21

zz

(z1) (z2).

2

2008

2008

2

2

4 10 x

4

y

A

B

. F(x) = dtf(t)x

0 , f

. () = 36 ..

:

. F(0) = . F(4) = . F(10) =

6

2

f

+

>+=

0x1,1)x(

0x,xf(x) , IR

. , f .

6

. , f x0 = 0.

8

. f 1-1.

3

. = 1 = 2, dxf(x)

2

.

8

3

f f (x) =x

e1e

, xIR .

. i. .

4

ii. f (x) =x

ex1x e1)(e + , f

.

5

. f.

6

. f.

4

.

f (x), xx, yy x = ln2

1.

6

2008

2008

3

3

4

f, g: IR IR x

:

=x

0

x

1

dtg(t)x2dtf(t) (1) g(x) 0 (2)

:

. f x0 = 0 f (0) = 2g(0)

6

. g(x) < 0 xIR

5

. 0

1

x

1

dtf(t)dtf(t) xIR

7

. H f (x) = 2g(x) + 2 (0, 1). 7

2009

2009

1

1

'

&

1

. f x0

. f x0

, : f(x0) = 0

9 . 1. = x +

f +;

3

2. f [, ];

3 . ()

():

1. 0

00

lim ( ) lim ( )x x h

f x l f x h l

= + =

2

2. 0

2009

2009

2

2

2

z + z

1 = 1, z C z1, z2 . :

A. z1z2 = 1 z1

3 =1.

4

B. (z1

2009 + z2

2009) R.

4

. z18 +

10

2z

1 + 1 = 0

4

. f(x) [0,1]

f(0)-2= 1 2

2 1

z z

z z

+ f(1)= 1 2

1 1 3

2 2 2z z+

x0 (0,1), f(x0)=3x0 2.

7

. w = 2z1 + 2z2 , z1 z2 , .

6

3

f(x) = x + 2 +2lnx .

. .

6

. f.

6

. 2

ln)(

+

=

x

xxxg x0 > 0 :

g(x) g(x0) x > 0.

7

. x > 2 : f(x 2) < 2f(x + 1) f(x + 4).

6

2009

2009

3

3

4

f (0, +)

:

f (x

1) =

x

e

x 1+ f(1) =

e

1

. f(x) = xe -1/x

.

8

. 1. f(x) x = 1.

2

2. 2

1

2( )f x dx

e> .

7

. g(x) = 3

)(

x

xf, (t)

Cg, xx x=1 x=t t> 1.

5

. E(t)limt +

.

3

2010

1

1

' .

&

1

. f [, ]. G

f [, ] , ( ) ( ) ( )f t dt G G

= . ( 10)

B.1. f x0

.

( 3)

B.2.

( )( )0 0,x f x f. ( 2)

. .

) 2 2

1 20z z+ =

1 2, Cz z

1 2z =z =0 .

) ( )g x x0 ( )0

limx x

g x

= ( )limy

f y l

=

( )( )0

limx x

f g x l

= .

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

) f

, ( ) 0f x > .

) f [ ]2, 5 ( ) 0f x [ ]2, 5 ,

( )2

5

0f x dx . ( 10)

2010

2

2

2

z, w 1 2

1

w

z

w

+=

w

( )1,0 = 1.

) z (0, 0) = 1.

( 6)

) 1z = (1) 1 2 3, , z z z

(1) :

i) 2 3 1 31 2

3 1 2

z z z zz z

z z z

+ ++= + + .

( 7)

ii) z1+z2+z3=0 :

31 2

2 3 1

3Re

2

zz z

z z z

+ + =

( 7)

) (): 3 4 12 0x y+ = .

w ().

( 5)

3

( ): 0,f R+ , x > 0

( )( )

1

1f x

xx f x

e

+ =

+ ( )1 0f = .

) ( ) xg x e x= + 1-1.

( 2)

) ( ) lnf x x= x > 0. ( 6)

) ( )( ) 1f x

h xx

=

.

( 6)

2010

3

3

)

x x

x x

e e

=

0,2

x

.

( 5)

) h 1 2,x x

2 10x x> >

( ) ( )2 15

2 1

1

2

h x h x

x x e

.

( 6)

4

:f R R ,

3 1

( ) 2 6

x u

f t dt du x x R . :

)

3

1

( ) 2f t dt= . ( 7)

) f (0, f (0))

4 3 0x y+ =

2 3

0

40

( )

lim

x

x

t f t dt x

x

.

( 5)

) 1x ( ) 0f x >

1

( ) ( )

x

h x f t dt= , x > 1

( )( )

1

h xh x

x >

.

( 7)

) ( )1,3 , ( ) 3 2f + = .

( 6)

M

2011

1

1

'

&

1

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7

. ;

4

. Rolle.

4

. () ().

1. z 2. 2

.

. 2

3. f : A R g : B R ,

, o .

2

4. , .

2

5. , , , . 2

M

2011

2

2

2

f : R R

4 12" " 1 , xR. R, = 1 . . i. " 1.

5

ii. . 5

.

lim

&

5

. i. 0, 1.

6

ii.

. 4

3

f : R R

x x 1, xR. . :

). *22 , 1 2 0 1 1

4 ii. 0, 1 , : x x 1

7

. , , - 2

, IR. i. .

/

0 . 6

ii. :

1 123 &

8

M

2011

3

3

4

. - 1, IR. 1;

3

. : 0, 0, . - 0 z, :

5 6

5 6

||2 5 66

8 1,

9 0. :

:. ).

; - 0, - 0. 5

ii. , - 0. 4

. . 5

. . 4

. (0, , , ,

2012

(....)

2012 _3.3()

: 1 3

:

: &

:

: . 11 2012

A1. , f x0, .

5

A2. f

x0;

4

A3. N f (x) = x, > 0 R.

f (x) = x ln

6

A4. :

i. 11, .

2

ii. i4 + 3

= i, .

2

iii. 0)x(flim0xx

>

, 0)x(f > x0.

2

iv. x, y y = f (x),

f x0, o y x x0 y = f (x0).

2

v. f ,

x0 , f (x0) 0, f.

2

2012

(....)

2012 _3.3()

: 2 3

f(x) = ex2

g(x) = lnx+2.

B1. fog gof .

6

B2. f f 1

.

6

B3. ex2

= lnx + 2 , ,

(e2

, 2).

6 B4. :

)x)(fog(

)x(glim

)x)(gof(

)x(flim

xx +

= = 0

7

f: IR IR x IR 1

x

2t f (t) dt2(1+3 ) f (x) = e ,

IR {0}.

1. :

i. f (x)f x 2(x) ' f 2= , x IR.

4

ii. 2 2

1f (x) =

x + 3 , xIR.

4

2.

0

t f (t) dt .

4 3. f.

8

4. ,

f x = , :

|| 3

1 E

|| 4

1

2012

(....)

2012 _3.3()

: 3 3

f IR f (0) = 2,

x+2

x 2

f (x) 2elim = 1

x + 2

f (x) < 0, xIR.

:

1. f (2) =1 f (x) x + 4, xIR.

6

2. f x0(2, 0).

6

3.

( )2(x 5)

0f f (t x)dt = f (0)

IR x = 5.

7

4. z

f(|z + i|) f(|z| + 1)

.

6

1

1 4

( ) 27 2013 - :

: (4)

A1. f [ ], . G

f [ ], , :

( ) ( ) ( )

f t dt G G =

7 A2.

( ...) 4

A3. f [ ], ;

4 A4. ,

, , , .

) 0z z , >0 = ( )0K z 2 , 0z, z .

) ( )0x x

lim f x 0

< , ( )f x 0< 0x

) : x x x

) : x 0

x 1lim 1x

=

) f f .

10

2

2 4

z :

( ) ( )z 2 z 2 z 2 2 + =

B1. z , ( )K 2,0 1= ( 5)

, z , z 3 ( 3)

8

B2. 1 2z , z 2w w 0+ + = , w , , ,

( ) ( )1 2z z 2 =Im Im :

4= 5= 9

B3. 0 1 2 , , 1. v :

3 22 1 0v v v 0+ + + =

:

v 4<

8

f,g : , f :

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

( )2

3 3xg x x 12

= +

3

3 4

1. :

( ) 2f x x 1 x= + , x 9

2.

( )( )f g x 1= 8

3. 0x 0, 4

, :

( ) 0 00

0

x 4

f t dt f x x4

=

8

( )f : 0, + :

f ( )0, + ( )f 1 1=

( ) ( )

h 0

f 1 5h f 1 hlim 0

h+

=

( ) ( )x

f t 1g x dt

t 1

= , ( )x 1, + 1>

:

1. ( )f 1 0 = ( 4), f 0x 1= ( 2).

6 2. g ( 3), ,

2 4

2 4

8x 6 2x 6

8x 5 2x 5g(u)du g(u)du

+ +

+ +

> ( 6) 9

4

4 4

3. g ,

( ) ( ) ( )( ) ( )x

f t 1 1 dt f 1 x , x 1

t 1

= >

. 10

( )

1. . - . - . .

2. . . .

3. . , , .

4. . 5. : (3)

. 6. : 10.00 . .

K

1

1 5

13 2013 :

: (5)

A1. f 0x ,

f . 7

A2. Fermat. 4

A3. f . f ;

4 A4. ,

, , , . ) z z z=

( 2)

) f 11 , f .

( 2)

) ( )0x x

lim f x

= , ( )( )0x x

lim f x

= +

( 2) ) f , g 0x

:

( ) ( ) ( ) ( ) ( ) ( )0 0 0 0 0f g x f x g x f x g x = ( 2)

) f

, f . ( 2)

10

2

2 5

z, w

22x w 4 3i x 2 z , x =

, x 1=

B1. z

1 1, =

w

(4,3) 2 4 =

8

B2. N , .

5

B3. z , w 1 :

z w 10 z w 10+

6

B4. z 1 , :

22z 3z 2zz 5 =

6

f : :

( ) ( )22x f x x f (x) 3 f (x) + = x

( ) 1f 12

=

3

3 5

1. :

( )3

2xf x ,

x 1=

+ x

f 6

2. f 1.

4

3. :

( ) ( )2 3 2 2f 5(x 1) 8 f 8(x 1)+ + 7

4. , , ( )0, 1 , :

( ) ( ) ( )3

2 3

0

f t dt 3 1 f

= 8

[ )f : 0, + , [ )0, + , :

( ) ( )( )( )

x 2

1

u

1

f t 1f x x dt du

f t

= + x 0>

( ) ( )f x f x 0 x 0> ( )f 0 0=

:

( )( )

f xg(x)

f x

= x 0> ( ) ( )( )3h x f x= x 0

4

4 5

1. N :

( ) ( ) ( )( )2f x f x 1 f x + = x 0> 4

2. . f f ( )0, + ( 4)

. ( )f 0 1 = ( 3) 7

3. g ( )0, ,+ :

. ( )g x 2 x x ( )0, + ( 2)

. ( ) ( )1

0

2 x f x dx 1

5

5 5

3. . , , .

4. . 5. : (3)

. 6. : 18:00

K

2013

(....)

2013 _3.3()

: 1 3

:

: &

:

: . 30 2013

: 3

1. f (, ),

x0, f .

, ( )f x >0 0 0

( , ) ( ,)a x x , 0

( )f x

f (,).

9

2. . (x0, f(x0)) f.

3

. f, g , , f g ;

3

3. , , , .

) ,f g [,] ,

[ ]( ) ( ) ( ) ( ) ( ) ( ) = aa a

f x g x dx f x g x dx f x g x

) z *, v

v

z z = .

) 11 , .

) 0 1a< < lim loga

x

x

+

= + .

) * ( ) vf x x= * 1( ) vf x vx = .

10

2013

(....)

2013 _3.3()

: 2 3

z w :

2

( 2) 1 3z z i z+ = 2 ,w z i=

1. z.

z ;

7

2. zz z

.

6

3. z 2= zz

0)Im( >z , 2013

2

zz.

6

B4. w z w

z (0,1).

6

, 0

( ) 1

ln , =0

x

xx

f x e

a x

=

.

1. (0, )a + f

1(0)

2f = .

7

= e.

2. . f . 6

. , .

6

3. 0

1 12

( ) 1 2013

x

x dtf t

=

+

(0, 1).

6

2013

(....)

2013 _3.3()

: 3 3

f, G F, [0, +) f G .

: f (0) = 1, G(0) = 0 x 0

( )f x > 0, '( ) 1G x > 0

( ) ( ) dx

F x f t t= .

1. ( ) 0F x ( )G x x x 0 .

5

2. [ ]0

lim ( ) lnx

F x x

(0, 1)

, ( )

( ) ln 0F

f

+ = .

7

3. , ,

[ ] [ ] 22

( ) ( ) ( ) ( ) ( ) ( ) 1f x F x f x G x G x x G x + = + , x 0.

:

. ( ) ( )F x G x x= , x 0.

7

. x0 >0, F GC , C

( )( ) ( )( )0 0 0 0, ,x F x x G x , yy ( 3)

, F G

C , C 0

x x=

( 3).

6