: 8 2015

:

http:// lisari.blogspot.com

2

00

0

20

14

30 2000

TEX :

1

. ) ; 2,5

) (x,y) , u (,) (x,y)

u , x ,y

u . 5

) ; N

. 5

1. I

.

1 :

x x

2 :

/2

1. 1 1

1 1

2. 0 1

1 0

3. 1 0

0 1

3

2. = 12 - 21 , 1 , 2

1 , 2 , 1 .

) . 4,5

) : 2x - y + 5 = 0 .

5

2

. 5 i

z 2 3i

) z + i , , R. 4

) z . 5

), )

.

) = Argz , iz :

.

- 4

.

2

.

- 2

. + 3

) 4 z : . 4 . 4 i . - 4i . -4 3

. , z ,

:z - 1

1 z - i

. 10

3

f :

2

2 2 5-x

x - 8x 16 , 0 x 5 f(x)

( ) ln(x - 5 e) 2( 1) e , x 5

. , x 5

lim f(x)

, x 5

lim f(x)

. 6

. , R , f x0 = 5. 10

. , x lim f(x)

. M 9

4

. f(t)

t , t 0 . f(t)

8 - 2 t 1

) f(t). 6

) t , ,

; 6

) t = 8

, t = 10

. ( ln11 2,4). 13

15 2000

TEX :

1

.1. z1 , z2 . : ______ __ __

1 2 1 2z . z z . z 5

.2. z1 = 1 (1 + i1) z2 = 2 (2 + i2)

,

, :

. 1

2

z

z 1. 12(1+2)+i(1+2)

. z1z2 2. 1 [(1)+i(1)]

. 1z 3. 1

2

(1+2)+i(1- 2)

4. 1

2

(1- 2)+i(1- 2)

5. 12(1+2)+i(1+ 2)

6.

1 [(1)-i(1)]

7,5

.1. .

:z1 = 2 (2 2

i 3 3

) z2 = 5 5

i 3 3

.

1

2

z z

: : 2 : 2i : -2 : -2i : 2 (1-i)

4,5

B.2. z = 1 + i . z1 6 . 8

2

. (2 - 2 - 1) ( + 2 - 3 )

( + 1) (3 - 2 + 2), .

. ,

5

. ,

10

. = 0 1

1 0

. : . 2 = -, 2

2 . 4

. 22004 + 2001 + 1999 = 2 , 2 2 . 6

3

f f(x) = 2x - 3x 2

x -

, .

. , f

x = 4. 5

. , f

(1,0) (-2,3). 10

. > 2, x0 (1,2) ,

f x0 xx. 10

4

,

1.000 .

. 4 25 .

200 .

4.000 .

,

10.000 .

. (x)

, : (x) = 10 (x + 16

x

+ 40) x

. 13

. ,

; 8

. .

. 4

9 2000

( )

1

. z= + i ,R .

z. K

A. Re(z)

. z

. z

. z z

. z + z

. z - z

Z. Im(z)

1. 2 2

2.

3. i

4. - i

5. 2 + 2

6.

7. 2

8. 2i

9. - + i

, ,

. 14

. z = 3-4i . :

) Re(z) Im(z) z

3

) z 4

) z z. 4

2

= 1 2

0 1

, = 1 0

0 1

.

) 2 = 2 - . 9

) (2 - ) = . 8

) X 2 X - = 2 . 8

3

f(x) = 2

x x 1

.

) x 1

limf (x)

. 12

) f . 13

4

600 .

, .

) (x)

(x) = -2x2 + 600x ( 0 < x < 300). 6

) x (x)

. 14

) . 5

x xE(x)

12 2000

:

1o

A1.A f ' x0 ,

f

(x0 , f(x0)). M 4

2. , f ' x0

, . 8,5

1.

.

. f x0 , f x0 .

. f x0 , f x0 .

. f x0 , f x0 . 4,5

2.

x0 .

. f(x)=3x3 , x0=1

1. y=-2x+

. f(x)=2x, x0=

2 2. y=

1 4

x+1

. f(x)=3 x , x0=0

3. y=9x-6

. f(x)= x , x0=4

4. y=-9x+5

5.

8

2

f(z)=2z i

, z C z -2i, z 2i

z o z.

. :

w1=f(9-5i) . . 6

. 1

1

02 M

3 0

w

w

1 w

w1 .

.

:

. 4

. x x

. yy

. y=x

. 2

. 3

5

. , :

=

. 2

8

3

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

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

. y=3 f '

x0(0,1). 7

. x1(0,1), f(x1)=(1 / 5) (2 / 5) (3 / 5) (4 / 5)

4

f f f f 12

. x2(0,1), f

(x2 ,f(x2)) y=2x+2000. 6

4

t=0 ' .

f(t)=2

1

t

t

, t0

t

. 15 6

.

. . 15

. ,

12 ,

. 10

16 2000

:

1o

A. f, .

. f(x)0 x , f

. M 8

. f(x)0 x , f ;

4,5

.1.

.

. f(x) =e1 -x

. 2,5

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

1

x + 3, x

2,)

. 2,5

. f(x) = g(x) + 3 x, h(x)=f(x)-g(x)

. 2,5

.2. f

-2,6.

f

. 5

2

. z1 , z2 z2+2z+2=0, 20

1z - 20

2z = 0.

12

. z1 . ,

, 1z

. 8

-2 1 3 6x

y

.

z1 z2 . 5

3

f , ,

: 2x

x 0

f(x)- e 1

2xlim

= 5.

. f(0). 7

. f x0=0. 9

. -xh x e f(x) ,

f h (0,f(0)) (0,h(0)) .

9

4

( ) , t

, P(t) = 4 + 2

t-6

25t

4

.

. .

2

. , .

10

. .

8

.

,

. 5

12 2000

1

. . , :

= -1X A 6,5

. ( 1...1

1...2) , .

1.

.

0

. = = 0 .

0

.

0

. = = 0 3

2.

A

D

. ,

, :.-1

- 1 A

- D

.

-1 -

A -

.

-1 -

A D -

.

-1 1

A D

.

-1

A

3

. 2 3

A -2 4

1 0

B . 1 1

. 3

1 2 2

A 1 1

6

. = 6,5

2

z 1 = 7 + 8 i z 2 = 4 - 5 i .

. z 1 z 2 . 8

. 1

2

z

z. 8

. 1 2 z z z z

z4 . 9

3

:

2x x, x 1

x 1

f (x)

x 2 3, x 1 .

. f x0 = 1

13

. x 2 x 2

lim f (x), lim f (x).

12

4

100 , x

, 3 21 1

f (x) x x x 10 , 9 3

1< x < 5.

. x ,

. 13

. x 1 = 2

x 2 = 4 ( 2.000 4.000

) . 12

2 2001

:

1o

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

7,5

.2. ,

.

z :

. 2

z z z . 2 2 z z . z - z . z z . i z z 5

.1. 1 2 z 3 4 i z 1 - 3 i,

,

.

1. 1 2 z z . 4

2. 2

1 z . 2

3. 2

2 z . 25

4. 1 z . 5

5. 2 i z . 2

. 5

. 10

7,5

.2. z z 1, 1

z z

. 5

2

f :

2

x-3

x , x 3

f(x) 1-e , x 3

x 3

. f , = 1/9. 9

. Cf f

(4, f(4)). 7

.

f , x x 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) = 1

2 2

0 1 - 2 x t f (xt)dt , x R.

g 21

g(x) - x f(x)

, x R.

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

. g . 4

. f :2

1 f(x)

1 x

. 4

. x

lim

(x f(x) 2x). 7

E

5 2001

:

1o

A.1. f . F

f ,

:G(x)=F(x)+C, CR f

G f : G(x)=F(x)+C, CR

6,5

.2.

.

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

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

. . . . .

,R f,g [,] 6

.1. f , f(x)=6x+4, xR

(0,3) 2. 6,5

.2.

. xe x dx

2

. 4 2

1

3xdx

x

2

. 2

0

2 x 3 x dx

2

2

. z

: z 16 4 z 1 9

. z

: z 1 z i 9

.

() (). 7

3

x , x 1

f (x) 1 e ln(x 1), x 1,2x 1

, R. .

. x 1

1 elim

x 1

x 1

7

. R f xo=1. 11

. =-1 (1,2) ,

f (,f()) xx.

7

4

f , (0,+) :

x2

1

tf t1f x dt x 0

x x

. f (0,+). 3

. f :1 ln x

f (x) , x 0x

7

. f. 6

. f. 4

.

f , x x x=1, x=e. 5

25 2001

:

1

. ) z1 = 1 ( 1 + i 1) z2 = 2 ( 2 + i 2)

, :

z1 z2 = 1 2 [ (1+2)+i (1+2)] 6,5

) z = + i , R , ,

,

.

A. Re(z)

. Im(z)

. -z

. z

. z

. z z

1. - - i

2. - i

3. +

4.

5. 2 2

6. 2 + 2

7.

6

. z1 = 1 + i z2 = i .

) z 1 z2 . 8

) z1 z2 . 4,5

2

f(x) = x2 - 4x + 3, x R .

) f xx

yy. 7

) f

(3, f(3)) . 9

) f. 9

3

f: RR , 2 - x4 f(x) 2 + x4 ,

x R . :

) f(0) = 2 6

) H f x0 = 0 . 9

) f x0 = 0 . 10

4

625 km x km .

90 km .

160 , 2x

5,5200

2000 .

) (x) : 1800000

K (x) 500 x ,x

0 x 90 . 12

) .

13

6 2001

:

1

. ) , z1 = + i z2=+ i , , , , R

, 1 2 1 2 z z z z 6,5

) z = + i, , R , ,

,

.

A. z z

. z z

. z z

. z

1.

2. 2

3. 2i

4.

5. 2 2

6. 22

6

. ) z1 = k + 15i z2 = 5+i, k , R .

k , z1=5 2 z . 6

) z , z z + ( z z ) = 5 +2i.

6,5

2

f(x) = x2 - kx + 1, x R .

) k, f

(1,0). 12

) f

(0, f(0)), k=17. 13

3

2

z 1 i

) z z=x+yi , x, y R. 8

) z. 8

) z

(2,0) = 2 . 9

4

2

1f (x)

1 x

, x R

) f(x). 5

) f. 12

) ( )

f. 8

30 2002

:

1o

A. f ' [, ]. G

f [, ],

f (t) dt G() G() . 12

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

f(x) = x . 8

.2. ,

.

. f [, ] (, ], f

[, ] . 1

. , 1 -1 , .

1

. f x0 x x

0

lim f(x) 0 ,

x x

0

lim f(x) 0 .

1

. f R , f (x)dx xf (x) xf (x)dx

1

. x x

0

lim f(x) 0 ,

f(x) > 0 x0 . 1

2

z f() = i z , IN*.

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

. z= Arg(z) = , f(13) = i 2 2

.

8

. z= 2 Arg(z) = 3

,

0, z

f(13). 10

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 [, ],

h(x)dx g(x)dx . 2

. R f , :

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

i) f f. 5

ii) f(x) x f (x) ,x2

x > 0. 12

iii) f,

x = 0, x = 1 x x , 1 1

E f (1)4 2

. 6

E

8 2002

:

1o

A. z1 = 1(1 + i1) z2 = 2(2 + i2) ,

: z1 z2 = 12 [(1+2) + i(1+2)]. 15

. ,

.

.

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

. f()

f . 2

. f IR. ,

[, ] , f

Rolle. 2

. f [, ]

x0[, ] f . f(x 0)=0.

2

. f [, ] x 0(, )

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

2

x

x

e 1f x , x

e 1

. f f 1 .

10

. f 1 (x) = 0 . 5

. 1

21

2

f x dx 10

3

f, R. ,

22

22

x z x zf x

x z

z z = + i, ,R , 0.

. x x

f x , lim f (x)lim

. 8

. f, z 1 z 1 . 9

. f. 8

4

f, R. ,

: f(x)f(x) + (f(x ))2 = f(x)f(x) , xR. f(0) = 2f(0) = 1.

. f. 12

. g

[0,1],

x

2

0

g t2x dt 1

1 f t

[0,1]. 13

5 2002

1

. f, g x o ,

f+g x o :(f+g) (xo) = f (xo)+g (xo) 9

.

, , , , ,

.

1. f ' x o ,

. 2

2. f ' x o ,

. 2

3. f ' f(x ) = 0

x , f . 2

4. f ' f(x) > 0

x , f . 2

5. f g x o , :

o o ox x x x x x

lim f x g x lim f x lim g x( ) ( ) ( ) ( )

2

6. f g x o , :

o o o

x x x x x x

lim f x g x lim f x lim g x( ) ( ) ( ) ( )

2

7. z i :2 2z 2

8. i : i 4 = 1 . 2

2

z 1 = -1+i , z2 = 3-4i

. z 1+5z2 6

. 2

1

z

z 6

. z 1 : rg(z1)=3

4

6

. 81

z . 7

3

f(x) = x3- 6x2+9x-2 .

. f .

10

. f

A , f ( )1 1 . 5

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

10

4o

:

3

2

x 4x , x 2x 2

f x

x k , x 2

( )

kR . :

. k, f x0 = 2 , 7

. x 1lim

f(x) , 5

. f x 0 = 4 5

. f x

g xx 3

( )( )

.

8

29 2003

:

1o

A. , f x0 ,

. 8

. ;

7

. ,

.

. z _z , z z z

2

. f

. f(x)>0 x , f

. 2

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

,cIR 2

. f ,

f

. 2

. f x0

. f x0 f(x0)=0, f

x0 . 2

2

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

i z +4, z

z.

. Re(w)=3+4 , m(w)=3. 6

. , w

y=x12, z y=x2.

9

. 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() 0. 9

. f. 8

8 2003

:

1o

A. f . F f

, :

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

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

10

. ,

.

. z1 , z2 ,

1 2 1 2 1 2 z z z z z z . 2

. f ' (, ),

x0 , f .

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

f . 2

. f : R 1 1 , x1 ,

x2 A : x1 = x2 , f(x1) = f(x2) . 2

. f, g , :

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

. x = x0

f ; 7

2

. () z

: z 2 m (z) 0 . 12

. , z (),

1 4

w z 2 z

x x . 13

3

2f(x) = x 1 - x .

. x lim f(x) 0

. 5

. f , x

. 6

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

. 1

2 0

1 dx ln 2 1

x 1

. 8

4

f IR ,

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

. f . 8

. f(x) = 0 . 8

. f(x)

g(x) f (x)

.

g xx ,

45 . 9

4 2003

:

1

. f(x) = x . f

R1 = IR {xx = 0} f(x) = 2

1

x. 10

.

, , (), , (),

.

1. z = x + yi , x, y ,

2 2 z x y .

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

x0 , y x

x0 f(x 0) .

3. f (, ),

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

(x0 , ), f(x0) f .

4. z = x + yi , x, y ,

_

z = x + yi .

5. f g x 0 ,

0

0

0

x x

x x

x x

lim f xf (x)

lim g(x) lim g x

( )

( )

, 0x x

lim g(x) 0

. 15

2

2x - 3x

f(x) x - 2

, x IR {2} .

. x 0

f(x) lim

x. 7

. y = x 1

f + . 8

. f (2, +). 10

3

2x , x 5

f(x) 10x - 25, x 5

x0 = 5 .

. f x0 = 5. 5

. f x0 = 5 f(5) . 8

. f

(5, f(5)). 4

. f . 8

4o

z = x + yi , x, y i (i z)

w i z

z i . : .2 2

2 2 2 2

2x 1 - x - y w i

x (y 1) x (y 1)

, 8

. w , z

(0 , 0) 1 = 1 8

. z , w

(0 , 0) 2 = 1 . 9

27 2004

:

1

. f ' x 0

. f x 0

, f (x 0 )=0 10

. f x 0

; 5

.

.

.

. 2

. 0x x

lim f (x) l

, 0x x

lim f (x)

0x x

lim f (x) l

2

. f , g x 0 , f g

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

. f , . f (x )>0

x , f .

2

. f [ ,] . G

f [ , ] ,

f(t)dt G() G() 2

2

f f (x )=x 2 lnx .

. f ,

. 10

. f .

8

. f . 7

3

g(x )=e x f (x ) , f IR

f (0)=f(3

2) = 0 .

. (0, 3

2) f ( )=f( ) .

8

. f (x )=2x 2 3x, () = 0

g(x)dx , IR

8

. lim ()

9

4

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

3 x

1

1g(x) z f (t)dt 3 z (x 1) 0

z z=+iC, , IR * , :

. g IR

g . 5

. N 1

z zz

8

. Re(z 2 ) = 1

2

6

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

f (x 0 )=0. 6

5 2004

:

1o

A. f .

f

f(x) = 0 x ,

f . 9

. ,

.

. f x 0 ,

. 2

. .

2

. f , g IR fog

gof , . 2

. C C f f 1

y = x xOy xOy. 2

. f x 0 , 0 0

kk

x x x xlim f(x) lim f(x)

, f(x) 0

x0 , k k 2. 2

. f

(, ) [, ]. 6

2

f: IR IR f(x) = 2 x + 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) |.

1

z f ()z

2 22

1z f ()

z , :

. |z |= 1 11

. f2() < f2() 5

. x 3f() + f() = 0 (1, 1).

9

4

f [0, +) IR , 12

2

0

xf(x) 2xf(2xt)dt

2 .

. f (0, +). 7

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

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

. xlim f(x)

xlim f(x)

. 6

4 2004

:

1

. + i, +i , , , , IR +i 0,

: 2 2 2 2

i i

i

9

. ,

( ).

I

. i1

B. i2

. i3

. i4

1. i

2. + 1

3. i

4. 1

5. 0

6. 4

,

. 4

, , ,

, , ( ), ,

( ), .

. f , g . f , g

f (x) = g(x) x ,

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

. f

, x 1 , x2 x 1 < x 2 : f(x 1 ) < f(x 2 ) .

3

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

2

f (x)x

3

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

C f f , x 0

= f (x 0 ) . 3

2

, 24x 3, x 1

f(x)6x k, x 1

, k IR .

. k, f x 0 = 1. 10

.

f ( 1, f(1)) . 8

. , : f(5) + f (5) + 34 = 0.

7

3

f(x) = 2x 3 3x2 + 6x + , x IR ,

. f x 0 = 2

f(2) = 98.

. = 6 = 54 . 6

. f . 9

. f . 4

. f(x) = 0

(1, 2) . 6

4

z = x + yi , x , y ,

IR :

2 2

z z z zi (1 )i

2 2i

:

. Im(z) = 0, = 1. 5

. = 0, z 2 + 1 = 0. 5

. : 0 1 . 7

. z

, . 8

7 2004

:

1

. 1z = + i 2z = + i , ,,, IR , ,

: 1 2 1 2z z z z

,,,

(), , (), .

. 1(, ) 2(, ) + i + i

, +i

+i . 3

. z = + i, , IR, z = + i . 3

. f(x) = x , x IR . H f

f (x) = x. 3

. f .

f

f(x) = 0 x ,

f . 3

. f, . f (x) < 0

x , f .

3

2

: 2f x 4x 1 2x , x IR .

) :

i)

x 0

f x 1lim 2

x

10

ii) f (0)= 2 f (0). 5

) : xlim f x

10

3

2x x 3

f xx 2

, xIR{2} , ,

. Cf f (1, 4)

f(3) + 3 f(1) = 0.

) = 1 = 0. 9

) f

(1,4) . 8

) y = x + 2

f + . 8

4

z = x + yi , x, y ,

k IR : x = 3k y = 2k+1 .

:

) 3 Re(z) + 4 Im(z) = 3, k = 2. 9

) z 1 5 , z 10 . 10

) z

, . 6

31 2005

:

1

.1 f, [, ].

f [, ]

f() f()

f() f() ,

x0 (, ) , f(x

0) = . 9

.2 y = x +

f +; 4

. ,

.

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

f() > 0. 2

. 0x x

lim f(x) g(x)

0x x

lim f(x)

0x x

lim g(x)

2

. f f - 1

f

y = x ,

f - 1

. 2

. 0x x

lim f(x)

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

1lim

f(x)

2

. f

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

. f ,

x x ,

. 2

2

1z , 2z , 3z

1 2 3z z z 3 .

. : 1

1

9z

z 7

. 1 2

2 1

z z

z z . 9

. : 1 2 3 1 2 2 3 3 1

1z z z z z z z z z

3 . 9

3

f f(x) = e , > 0.

. f . 3

. f,

, y = ex.

. 7

. () ,

f, yy,

() =e 2

2

. 8

. 2

()lim

2

. 7

4

f IR ,

2 f(x) = ex f ( x )

x IR f(0) = 0.

. : f(x) = x1 e

ln2

. 6

. :

x

0

x 0

f(x t) dtlim

x

. 6

. : h(x) = x

2005

x t f(t) dt

g(x) =

2007x

2007 .

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

. x

2005

x t f(t) dt

=

1

2008 (0 , 1).

6

6 2005

:

1

.1 f f(x) x . f

(0,+) : 1

f (x)2 x

9

.2 f : A IR 1 -1; 4

. ,

.

. , f

0 , f .

2

. f (, )

x o . f ( , x o ) (x o , )

, (x o f (x o ) )

f . 2

.

. 2

. f , g fog gof ,

fog gof . 2

. z , z

x x . 2

. f IR * , :

f(x)dx f(x)dx 2

2

. 1z , 2z 1z + 2z =4+4i 2 1z - 2z

= 5+5 i , 1z , 2z . 10

. z ,w

z 1 3 i 2 w 3 i 2 :

i . z , w , z = w

10

i i . z w . 5

3

f , IR f (x )0

x IR .

. f 1 -1. 7

. C f f (1 ,2005)

( -2,1) , 1 2f -2004 f(x 8) 2 . 9

. Cf,

Cf

( ) : 1

y x 2005668

. 9

4

f : IR IR , 2x 0

f(x) xlim 2005

x

.

. :

i . f (0)=0 4

i i . f (0)=1. 4

. IR , :

22

22x 0

x f(x)lim 3

2x f (x)

. 7

. f IR f (x )>f(x )

x IR , : i . xf(x )>0 x0. 6

i i .

1

0

f(x)dx f(1) . 4

8 2005

:

1

. 1. , f x ,

. 12

2. (x,y) z = x+yi .

z; 3

.

(), , (), .

1. f : R. 1-1, x1, x2

: x1 x2, f(x1) f (x2). 2

2. f xA () , f(x),

f(x) < f (x) xA. 2

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

0x xlim f(x)

> 0x x

lim g(x)

2

4. z1 z2 , 1 2 1 2z z z z 2

5. f [, ]

(, ) , , (, ) , :

f() = f()-f()

2

2

: z = 2- 2 + (3-2)i , R w = k+4i, k > 0.

z, w : Re(z) + Im(z) = 0 w = 5.

. z = -1+i. 8

. k = 3. 8

. R , z z 3i w 9

3

f(x) = x 3+kx2+3x-2, xR , kR ,

(1,1). :

. k = -1. 5

. f . 10

. f(x) = 0 (0, 1). 10

4

22 x kx 2

f(x)x 3

, k R x 3.

. y = x

f +, = 1 k = 3. 10

. (1, 2),

f xx.

8

. f

x = 1. 7

8 2005

:

1

. 1. f . f

f (x ) = 0 x , f

. 12

2. R.

; 3

.

() , , () ,

.

1. z = x+yi , x , y R. , : z z . 2

2. z = +i , : z z , , R . 2

3. x 0 , 2x 0

1lim

x . 2

4. f (x ) = x. f

R 1 = R. {x / x = 0} :

2

1f (x)

x . 2

5. f x0 R, :

o ox x x xlim k f(x) k lim f(x)

k R . 2

2

x 3i

z2 i

, x R .

. x , z . 10

. x = 6, z . 6

. x = 4 , z . 9

3

f :

3

4

, x 1x 1

f(x)

x 1 , x 1

. f . 6

. f . 10

. , f Rol le

[ 1,2] . 9

4

2kx x

f(x)4

, x R,

(0,0) = 1 .

. k = 4 . 7

. f , .

8

. (2 ,4) ,

f

, (2 , f (2) ) (4 , f (4) ) . 10

27 2006

:

1o

A.1 f , .

:

f (x )>0 x , f

.

f (x )

i i .

f f - 1 . 7

3

1 2 3z ,z ,z 1 2 3z z z 1 1 2 3z z z 0

. :

i . 1 2 3 1 2 3z z z z z z . 9

i i . 2

1 2z z 4 21Re(z z ) 1 8

. 1 2 3z ,z ,z ,

. 8

4

x 1

f(x) ln xx-1

.

. f . 8

. f (x )=0 2 .

5

. g(x )=lnx

(, ln) >0

h(x )=e x

( ,e ) IR ,

f (x )=0. 9

. g h

. 3

5 2006

:

1o

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

.2 f .

f ; 5

B. ,

.

. z 1 , z 2 , : 1 2 1 2z z z z 2

. f , g x g(x )0 ,

f

g x

:

o o o o

o 2

o

f f(x )g (x ) f (x )g(x ) x

g g(x )

.

2

. x0 1

ln x x

2

. f : IR 1 1 , y

f (x )=y x . 2

. f [ ,] . G

f [ , ] ,

f(t)dt G() G() 2

2

x

x 1

1 ef(x)

1 e

, x IR .

. f IR . 9

. 1

dxf(x)

. 9

. x

z , (4 z)1 0

= z 1 0

f f (x ) = x2

+x+, IR .

. z x=2.

7

. ( ) f

x=2 yy y =3,

i . ( ) . 9

i i .

f , ( ) , x x

3x

5 . 9

4

f(x) xln(x 1) (x 1)lnx x>0.

. i . : 1

ln(x 1) lnx , x 0x

.

i i . f (0 ,+). 12

. x

1lim xln(1 )

x . 5

. (0 ,+)

(+1)

= + 1 . 8

31 MA 2006

:

1

. 1 2z ,z , : 1 2z z = 1 2z z . 7

.

, , ,

.

1. f x 0.

f (x )0 x. 0x x

lim f(x)

0x x

1lim

f(x) . 3

2. , . (, )

(,) z i z i

. 3

3. f x 0 ,

x 0 . 3

4. f(x) x = [0 , +) , 1

f (x)x

x (0 , +) . 3

5. 0x x

lim f(x)

,0x x

lim f(x)

+ ,

0x x f . 3

6. f , g .

f , g

f (x ) = g(x ) x ,

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

3

2

x 2

4x + 13 = 0 (1)

. (1) . 9

. 1 2z ,z (1) ,

2 2006

1 1 2 2 z 2 z z 13 z i . 9

. z1

= 2+3i ,

z : 1z z 5 7

3

2

3x , x 1

4f(x)

x 8x 4, x 1

4x

R.

. R. f

x 0 =1. 10

. =0

. f R . 7

. f

+ . 8

4o

k R 3 2f(x) 2x kx 10 x R

. k R

f (1 , f (1)) x x .

5

. k = 3

. f . 8

. f ( , 0] . 5

. ( 14 ,15) f (x ) = 5

(0 ,1) . 7

24 2007

:

1

A.1 z1 , z2

, : 1 2 1 2z z = z z . 8

.2 f, g ; 4

.3 y f

+; 3

B. ,

, , ,

, .

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

f(x)dx 0 . 2

. f

x . f

f(x) > 0 x . 2

. f x 0 g x 0 ,

gof x 0 . 2

. f ,

g(x)

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

. 2

. > 1 xxlim 0

. 2

2

2 i

z 2i

.

. z (0,0)

=1. 9

. z1 , z2

2 i

z 2i

= 0 =2

.

i . z 1 z2

. 8

ii . 2v v

1 1z z . 8

3

f(x) = x3 3x 22 IR

+2

, Z

. f ,

. 7

. f(x) = 0 .

8

. x1 , x2 x 3

f, (x 1 , f(x1)), B(x2 , f(x2)) (x 3 , f(x3))

y = 2x 22. 3

.

f y = 2x 22. 7

4

f [0, 1]

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

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

: F(x) =x

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

x

0g(t)dt , x [0, 1].

. F(x) > 0 x (0, 1]. 8

. : f(x) G(x) > F(x) x (0, 1]. 6

. : F(x) F(1)

G(x) G(1) x (0, 1]. 4

. :

2x x2

0 0

xx 0 5

0

f(t)g(t)dt t dt

lim

g(t)dt x

7

3 2007

:

1

A.1 f x0,

. 10

.2 Rolle ; 5

. ,

.

. f() f .

2

. f, g, g [, ],

f(x)g'(x)dx f(x)dx g'(x)dx 2

. f ,

/x

f(t)dt f(x)

x. 2

. f

(, ), (,) =

x a

lim f x

= x lim f x

2

. f, g . f , g

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

x. 2

2

2

3x, x 0

x

f x

x x x ,x 0

. x 0

lim f x 3

8

.

f ' 2

f x

0=0,

= = 3. 9

. = = 3,

0

f(x)dx . 8

3

f(x) = ex

e lnx, x > 0.

. f(x) (1, + ).

10

. f(x) e x > 0. 7

.

2 2

2 2

x 2 x 2 4

2x 1 x 3

f(t)dt = f(t)dt f(t)dt

(0, +). 8

4

z1

= +i 121

2 zz

2 z

, IR 0.

z2

z1 IR .

. z2

z1

= 1. 9

. z1

. 6

. 21z >0, 1z

2020

1 1z 1 i z 1 i 0 10

30 MA 2007

:

1

. 1. f x x , {0,1}. f

R 1f x x 10

2. N f

. 5

.

(), , (), .

1. z z z z . 2

2. f 1-1, ( xx)

. 2

3. f x0R

0x xlim f x 0

, f(x)0.

.

i) x

f ' xlim

f x ii)

2

x 2

xf xlim

x 2 8

. N f (0,0)

. 9

. N f,

y=-2x+6. 8

4

f, R. A x0 xf(x)=x+2x, :

. f(0). 7

. f(x)

4 2007

:

1

. 1. : f,g x0

, f + g

x0

: (f+g) (x0) = f(x

0) + g(x

0). 12

2. f g ; 5

.

() , , () ,

.

1. +i +i

. 2

2. f , xx,

f. 2

3. f, g, h h (g f), (h g) f h (g

f) = (h g) f. 2

4. 2 . 2

2

z z 1 i iz .

. i) M z. 10

ii) (0,0) 5 . 10

. Re(z)=0, z=i. 5

3

:

2

2

1 1x , x 2

8 2f(x)

x 5x 6, x 2

2 x 1

. f x0=2. 12

. f

(0,f(0)). 6

. y =1

2x-2

f +. 7

4

f, R, 3 3 2x f x 8x 12x 8x 2f ,

xR.

. f 1-1. 8

. f(x)=0 (0, 1). 9

. g: RR 2f g x 3x f x 2 , xR ,

x0 g . 8

24 2008

:

1

A.1 f (x) ln x , x IR* IR*

: 1

ln xx

10

A.2 f [, ];

5

. ,

, , ,

, .

. f:A IR 11, f -1

:

1f (f (x)) x , x A 1f (f (y)) y , y f(A) 2

. f

f . 2

. z 2+z+=0 ,,IR 0 ,

C . 2

. f IR

, f( x ) > 0 x.

2

. f ,,

f(x)dx f(x)dx f(x)dx

2

2

z w

(i 2 2)z 6 w (1 i) w (3 3i) :

. z . 6

. w . 7

. w 6

. z w 6

3o

x ln x, x 0

f (x)0, x 0

. f 0. 3

. f

. 9

.

xx e

. 6

. f(x+1) > f(x+1)f(x) , x > 0 . 7

4

f 2

3

0f (x) (10x 3x) f (t)dt 45

. f(x)=20x 3+6x45 8

. g IR .

h 0

g (x) g (x h)g (x) lim

h

4

. f () g ()

2h 0

g(x h) 2g(x) g(x h)lim f (x) 45

h

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

i . g(x)=x 5+x3+x+1 10

ii . g 11 3

3 2008

:

1o

A. [, ]. G

f [, ],

f(t)dt G()- G() 10

. ;

5

. ,

, ,

, .

. 11, . 2

. f ,

f

, . 2

.

f(x)dx

xx

xx. 2

. , , : +i=0 =0 =0 2

. (, x 0) (x0 , )

. : 0 0x x x x

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

2

2

1

1 i 3z

2

z 2+z+=0,

.

. =1 =1. 9

. 31z 1 8

. w,

: 11w z z 8

3

f(x)=x2 2lnx, x > 0.

. : f(x)1 x>0. 6

. f. 6

.

ln x , x 0

f(x)g(x)

k , x 0

i . k g . 6

i i . 1

k2

, g , , (0,e).

7

4

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

x 0. :x

0F(x) f(t)dt ,x [0, +), x

0

F(x)h(x)

tf (t)dt

,x(0,+)

. 1

t 1

0 e [f (t) F(t)]dt F(1) 6

. h (0, +).

8

. h(1)=2, :

i . 2

0 f (t)dt

2

02 t f (t)dt 6

ii . 1

0

1 F(t)dt F(1)

2 5

2 8 2 00 8

:

1

.1. z1 = + i z 2

= + i ,

1 2 1 2z z z z . 7

2. f x . f

x ; 6

.

, , ,

.

1. z1 , z2 , : 1 2 1 2z z z z . 3

2. x IR : (x) = x. 3

3. f ,

x x,

. 3

4. f

[, ]

(, )

f() = f()

, , (, ) , : f () = 0. 3

2

3z 2

+ z + = 0, , .

. z1 = 1 + i , = 6, = 6

z 2 . 14

. : . 2 2

1 2z z 0 6

.2008 2008 1005

1 2z z 2 5

3

f 2

1 x , x 1f (x)

(x-1) , x 1

A. f :

. x = 1 8

. x = 1. 10

. f

(2, 1). 7

4o

f 2x 2x k

f (x)x

, k .

. f. 3

. f (1, f(1))

xx, k. 8

. k = 1,

. f. 8

. f [1, +). 6

20 2009

:

1o

. f . f

x , f . 10

. f x0 ; 5

. ,

, , ,

.

. z1, z

2 , 1 2 1 2z z z z 2

. f () x0A,

f(x)f(x0) xA 2

. x 0

x 1lim 1

x

2

. f

. 2

. f [, ] f(x)

xf (x) a ln(x 1), x>-1 >0 1

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

. = e,

. f . 5

. f 1,0

0, 6

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

(1, 2) 6

4

f [0, 2] 2

0t 2 f (t)dt 0

x

0

x

0

2

2t o

H(x) tf (t)dt, x 0,2

H(x)f (t)dt 3, x 0,2

xG(x)

1 1 t6lim , x=0

t

. G [0, 2]. 5

. G (0, 2)

2

H(x)G (x) , 0

9 2009

:

1o

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

1

f (x)2 x

9

B. f xo . f

xo ; 6

. ,

, , ,

.

. z z z 2

. f 1-1, f

. 2

. ox x

lim f (x) 0

f(x) < 0 xo

ox x

1lim

f (x) 2

. f(x) = x. H f 1R R x / x 0

2

1f (x)

x 2

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

. 2

2

z : (2 i)z (2 i)z 8 0

. N z = x+yi

. 10

. N 1z 2z

. 8

.

2 2

1 2 1 2z z z z 40 7

3

f(x)=ln[(+1)x2

+x+1] - ln(x+2), x > 1 -1.

. , xlim f (x)

.

5

. = -1

. f . 10

. f 6

. f(x) + 2

= 0 0

4

4

f:[0,2]R

2xf (x) 4f (x) 4f (x) kxe , 0 x 2 , f (0) 2f (0) , f(2) = 2 f(2)+12 e4

, f(1) = e2

k .

. 22x

f (x) 2f (x)g(x) 3x , 0 x 2

e

Rolle [0,2]. 4

. (0,2) , f () 4f () = 6 e2

+ 4 6

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

. 3 2xf (x) x e , 0 x 2

5

. 2

2 1

f(x)dx

x 4

26 MA 2009

:

1

. 1. x = x0

f ; 5

2. f , g x 0 ,

f + g x 0 :(f+g)(x0)=f(x0)+g(x0)

8

B.

, ,

, .

1.2 2z = z , z. 3

2. +i, ,

( ,). 3

3. 0

limx

x=0

x. 3

4. f [, ]

(, ), (, )

, : f() - f()

f () =-

. 3

2

1z =2+3i 20092

2z =(1- i) +3i +1

. z 2 = 1 + i . 8

. 1 2z z . 7

. 1

2

z

z +i , , . 10

3

2x + , x 1f(x) =

2x +3 , x >1 , .

. f x0 = 1, + =5. 5

. f x0 = 1 , = 1

= 4. 10

. = 1 = 4 ,

f(x)

g(x) =x

, x0 + . 10

4

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

I. f x 0 = 1 ,

. 4

. = 0

. f . 8

.

f y = 9x. 8

. 0f(x) - x

( 0 , 1 ) . 5

( )

19 2010

:

A1. f . F

f , :

G(x) F(x) c, c f

G f G(x) F(x) c, c

6

A2. x=x 0

f ; 4

A3. f

. f ;

5

4. ,

, ,

, .

) +i +i

.

) f

. f ,

.

) f

(, ), (,),

x A lim f (x)

x B lim f (x)

) (x)= x, x

) 0xx

lim f (x) 0

, f (x) 0 x 0 10

2

z 2z

z z 0

B1. z 1 z2 . 7

B2. z1 2 0 1 0 +z2 2 0 10 =0 6

B3. w 1 2w 4 3i z z

w . 7

B4. w 3 , 3 w 7

5

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

1. f. 5

2. :

2

2

4

3x 2 12 x 3x 2 ln

x 1

7

3. f

f .

6

4.

1

1

xf (x)dx

7

f : x :

f (x) x x

0

tf (x ) x 3 dt

f ( t ) t

1. f

f (x)f (x) , x

f (x) x

5

2. 2

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

7

3. 2f (x) x x 9, x 6

4.

x 1 x 2

x x 1

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

7

7 2010

:

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

(x)= x. 8

A2. f [, ]

; 4

A3. f x0A () , f(x0);

3

4. ,

, , ,

.

) f(x) = x, > 0, (x) =xx1

) fog gof, fog = gof

) o

x xlim f(x) =+

, o

x x

1lim =0

f(x)

) f [,] f(x) 0

x [,],

f(x)dx 0

) zC |z |2 =z z 10

z 1 , z2

z1 + z2 = 2 z1 z2

= 5

B1. z 1 , z2

5

B2. w |w z1 |2 +|w z2 |2 = | z1 z2 |2

w

(x+1) 2

+ y2

= 4 8

B3. w 2

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

B4. w1 , w2 w 2

|w1 w2 |=4, |w1 + w2 |=2. 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

f: IR IR IR f(0)=1

f(0)=0

1. f(x) 1 x 4

2.

13

03x

x f(xt)dt +xlim

x0 f(x) + 2x = 2x(f(x)+x2) ,

x, :

3. 2x 2f x =e x- , x 8

4. x+2

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

2

2

x +2x+3 4

6x +2x+1

f(t)dt f(t)dt 0 7

( )

25 MA 2010

:

1. , f x 0 ,

. 10

2. f

; 5

3. (5) , . .,

,

, , .

. f

C f .

. f

c, : cf (x) f (x) , x .

. z1 , z2 z 20, : 11

2 2

zz

z z

. f

[, ] [m, M], m .

. 0x x

lim f (x)

f(x)

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

10

3. : 2f (x 8) f (6x) 5

4. : x 0

f (x) 1lim

x

5

2x 3

f (x) 2x, x 0x

. :

1. f. 8

2. f. 8

3. f

(1, f(1)). 4

4. (, f()), >0, C f f,

Cf

(1, f(1)),

B(3, f(3)). 5

8 2010

:

1. f, [, ]. f

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

x0(, ) f(x

0)=. 10

2. f ;

5

3. ,

, , ,

.

) ,,, : +i=+i = =

) f f

C f , xx, ,

xx, C f , xx.

) f, g x o, f(x)g(x) x o ,

: 0 0

x x x xlim f(x) lim g(x)

) f, g x o g(xo)0,

f

g x o

:

0 0 0 0

0

0

x x x xx

x

2

f g f gf

g g

) P(x), Q(x) . P(x)

Q(x),

P(x) ,

. 10

f: [, ] , ,

B3. f 5

2(+) 5

z 26z+=0 ,

z1 , z2 Im(z1) > 0 |z1 | = 5.

1. =25. 8

2. =25, . 5

3. w |w z1 | = |w z2 | , w .

6

4. (z123i)8 + (z24+5i)8 . 6

f(x) = (x+3) 29 x-

1. . 4

2. f:

. (3, 3) ( 3)

. xo = 3 ( 3) 6

3. f. 9

4. f. 6

( ) 16 2011

:

A1. f x0 .

f x0 ,

: f (x0) = 0 10

A2. f R . y= x+

f + ; 5

A3. ,

, ,

, .

) z 0 z0=1

) f:A R 1-1, x1,x2A

: x1 x2, f(x1) f(x2)

) x R1= R {x | x=0} : (x)=2

1

x

) : x

x

xlim

=1

) C C f f1

y=x xOy x Oy . 10

z w z3i , :

1z 3i z 3i 2 w z 3i

z 3i

B1. z 7

B2. 1

z 3iz 3i

4

B3. w -2 w 2 8

B4. : z w z 6

f : R R , R , f(0)=f(0)=0,

: xe f x f x 1 f x xf x x R.

1. : xf x ln e x x R 8

2. f . 3

3. f .

7

4. xln e x x

0,

2

7

f, g : R R, x R

:

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

ii)

2tx

2x 0

1 f x edt

e g x t

iii)

2tx

2x 0

1 g x edt

e f x t

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

x R . 9

2. : f(x) = ex, x R 4

3. :

x 0

ln f x

1f

x

lim

5

4.

x

2

1

F x f t dt x x y y x=1.

7

6 2011

:

A1. f(x)=x x

( x) = x 10

A2. f, .

f . 5

A3. ,

, , ,

.

i. z = + i, , z z =2

ii. f x0A () f(x0),

f(x) f(x0) xA

iii. f , 1-1

.

iv.

0x x

=0lim f(x) f(x)>0 x0,

0

x x=

1lim

f(x)

v. f x0

. 10

z, w, : z-i =1+m(z) (1)

w w 3i i 3w i (2)

B1. z

21

y = x4

7

B2. w

(0, 3) =2 2 . 7

B3. ,

z, w z =w. 5

B4. N , ,

u ,

, , , . 6

y = x , x0.

(0, 1) xy

, .

t, t0

x(t)=16m/min.

1. , t, t0 :

x(t)=16t 5

2.

(4, 2) , , .

6

3.

. 6

4. t0

10,

4 d=()

. 7

xy.

f : , 3 , :

i)

x 0

=f(x)

lim 1 f(0)x

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

iii) f(x) 0 x

1. f

x0=0. 3

2. f . 5

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

3. g : 0x

xlim

xg(x) 6

4. 2

0f x dx >2 5

5. g,

xx x=0 x=1 ()=e5

2

1

0f x dx (1, 2) , 6

( ) 16 2011

:

1. f x0 . f

x0 , :

f(x0) = 0 10

2. f . y=x+

f +; 5

3. ,

, , ,

.

i. z 0 z0=1

ii. f:A 1-1, x1, x2A

: x1x2, f(x1) f(x2)

iii. x 1= {x/x=0} : 21

x = x

iv. :

x

xlim 1

x

v. C C f f-1

y=x xOy xOy. 10

z w z3i, :

z 3i z 3i

1w =z 3i

z 3i.

B1. z 7

B2.

1z 3i =

z 3i 4

B3. w 2 w 2 8

B4. : z w z 6

22

f(x) =xx

, x0

1. f . 6

2. f

A 2,f(2) . 6

3. f. 6

4. :

2x 1

1f 3

xlim

x 1 7

f:, f(0)=0, f(x) xf (x) =x

x.

1. g(x) =xf(x) x , x . 6

2. : 1 x

f(x) =x

, x x0 6

3. 1 x xx

, 3

2 2

6

4. (0,) : 22

2

7

6 2011

:

1. f(x)=x

x (x) = x

10

2. (x,y) z=x+yi

.

z

5

3. ,

, , ,

.

i. z=i, , zz=2

ii. f

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

iii. f

, 11 .

iv.

0x x

=0lim f(x) f(x)>0 x0,

0x x

=1

limf(x)

v. f x0

.

10

z, w,

:

z-i =1+m(z) (1) w w 3i i 3w i (2) 1.

z 21

y = x4

7

2.

w (0, 3)

=2 2. 7

3. ,

z, w z =w.

5

4. u=i ,

, , , .

6

y = x, x0.

(0, 1)

xy ,

.

t, t0 x(t)=16m/min.

1. ,

t, t0 : x(t)=16t

5

2.

(4, 2) ,

, .

6

3. y(t)

t, t>0

4m/min.

6

4. t0

10,

4

d=()

. 7

xy.

2

1f(x) =

x x , .

f

52,

12

5

18.

1. =1 =4.

5

2. f

.

6

3. f.

7

4. :

x3+(14)x2x+4=0 (1)

f(x)=, , , (1)

. 7

( )

28 2012

:

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

f

7

A2. f [, ];

4

A3. f . f

x0A ; 4

A4. , ,

, , , ,

.

i.

ii. f 1-1,

y f(x)=y

x.

iii.

0x x

lim f(x)= , f(x)

B3. w

2 2x y

19 4

|w|. 6

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

1 |z w| 4 6

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

1. f

1=(0,1]

2=[1,+). f. 6

2. x-1 2013x e , x>0 .

6

3. x1, x1 x1< x1

2, x 0( x1, x2) ,

f( x0) + f(x0) = 2012

6

4.

g(x) = f(x) + 1

x>0, xx x=e.

7

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

f(x) 0

2 2x -x+1

1

x xf(t)dt

e

x

1

nt tnx x= dt e f(t)

f(t)

1. f

.

10

f(x) = ex(nx x), x>0, :

2. :

2

x 0

1lim f(x) f(x)

f(x)

5

3. nxx1, x>0,

x

F(x) f(t)dt , x>0

>0, ( 2).

:

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

6

4. >0.

0(,2) : F() + F(3) = 2F()

4

( ) 28 2012

:

1. f

. f(x) > 0 x

, f

7

2. f

[, ];

4

3. f . f

x0A ; 4

4. ,

,

, ,

, , .

i.

ii. f 1-1,

y f(x)=y

x.

iii.

0x x

lim f(x)= , f(x)

9

3.

w

= 1. 8

22

f(x) x x

, x>0 ,

1.

14 2012

:

1. f (,

), x 0, f

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

4. u,

i

u ui = ww

, w0 x2y2=1

6

f:, : xf(x)+1= ex, x.

1.

x

e 1

, x 0f(x)= x

1 , x =0

6

2. o f 1

.

6

3.

f (0,f(0)). ,

f ,

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

8

4.

x 0

lim x( nx) n f(x)

5

f:A A=(0,+) :

f() = (,0]

f (0,+ ),

x

f(x) f(t)

1

1 12f(x) x e = e f (t) t dt 2

x t

x

1F(x) f(t)dt , x>0

1.

2

2xf(x) n

x 1, x>0

8

2. F

(x0,F(x0)), x0>0,

.

(x0,) >x0, F M(,F())

: F()x(1)y2012(1)0 6

3. >1,

35F() 1 )f() x 1) x 10

x 1 x 3

, x, (1,3)

5

4.

2 x x

x 1

tf dt tf t dt

x, x>0

6

14 2012

:

1. z1 , z2 , :

|z1z2|=|z1||z2| 7

2. f g ;

2

3. Rolle.

6

4. ,

,

, ,

, , .

i. f , xx, f.

ii. +i

+i .

iii. P(x)=x+-1x-1+1x+0

0 : 0xlim P x

iv. f x0,

x0

v. f (, ),

x0, f

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

f : ,

3

2

x , x 1f x

x , x 1

,.

x0=1

1. 2 2 = 1 6

2. 11, f(x)=0

[1,1] 8

3. f x0 = 1, a

6

4. =5

4 =

1

2

f (1, f (1))

5

(1,1) f

f(0) = 3 21

g x f x1 x

= , x(1,1) g(x)

x3, x(1,1), f f

(1,1 ).

1.

f g x0=0

.

6

2. g(0)=

f g

x0=0 y=x3 8

3. f (x)=, x(1,1), 0

4

4. f(x) x3, x(1,1) 7

( )

27 2013

A1. f [, ] . G

f [, ] , : f (t)dt

=G () G(). 7

A2. (...) 4

A3. f [, ] ;

4 A4. , , , , .

) oz z , > 0 K (z o)

2 , z, zo .

) ox x

lim f (x) 0

< , f (x) < 0 x o

) : x x xR

) : x 0

x 1lim 1

x

) f f .

10

z :

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

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

z 3 . ( 3)

8 B2. z 1 , z2 w2 + w + = 0, w , , R ,

1 2Im(z ) Im(z ) 2

: = 4 = 5 9

B3. o , 1 , 2 1 . v :

v3 + 2 v2 + 1 v + 0 = 0 : v 4

8

f,g :R R , f :

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

f (0) = 1

g (x) = x3 + 23x

2 1

1. : f (x) = 2x 1 x , x R 9

2. f (g(x)) = 1 8

3. x 0 0,4

, :

o

0

x4

f (t)dt = f (x0 4

) x0

8

f : (0, + ) R :

f (0, + )

f (1) = 1

h 0

f (1 5h) f (1 h)lim 0

h

g (x) = x f (t) 1

dtt 1

, x (1, + ) > 1

:

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

( 2).

6

2. g ( 3), ,

R 2 4

2 4

8x 6 2x 6

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

( 6)

9

3. g ,

( 1) x f (t) 1

dtt 1

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

.

10

13 2013 :

1. f x 0 , f

. 7

2. Fermat. 4

3. f . f;

4

4. , , , , , , .

) z z z

( 2)

) f 1 1 , f .

( 2)

) 0x x

lim f (x)

, 0x x

lim f (x)

( 2)

) f, g x 0 : (f g)(x0) = f(x0)g(x0) f(x0)g(x0)

( 2)

) f , f .

( 2) 10

z w

22x w 4 3i x 2 z , x

, x = 1 1. z

1= 1, w (4,3) 2= 4.

8

2. , .

5

3. z, w 1 :

z w 10 z w 10

6

4. z 1 , :

22z 3z 2zz 5

6

f: :

22xf (x) x f (x) 3 f (x) x

1

f (1)2

1. 3

2

xf (x) , x

x 1

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

0f (t)dt 3 1 f ( )

8

f: [0,+) , ,

[0,+), :

2

1 1

u(f (t))

dt

f (t)

duf (x) x

x > 0

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

:

f (x)g(x)

f (x)

x>0

3h(x) f (x) x0

1.

2

f (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

( 4)

6 4.

h, x x x = 0 x = 1 8

( )

27 2013

:

A1. f , [, ].

:

f [, ] f () f ()

, f () f()

x 0 (, ) , f(x0) =

7 A2. (...)

4 A3. f [, ]

;

4 A4. ,

,

, , .

a) |z z0 |=, > 0 K (z o)

2 , z, zo .

b) 0

0 (z0) , z0, z .

( 2)

ii. f (,x0)(x0,)

o 0 0

x x x x x xlim f x lim f x lim f x

( 2)

iii. 0 < < 1 , xxlim 0

.

( 2) iv. f

. f , f (x) > 0 .

( 2)

v.

g(x)

f(t)dt f g(x) g (x)

. ( 2)

10

z w :

2z i

w2z i

,

iz

2

w

1 . z,

=1

2,

1

M 0,2

10

2 . z, 1,

|w|= 1.

8

3. 1

z2

, w 4 + i w7 = 0

7

ln x

xe , x 0f (x)

0 , x 0

1. f x 0 = 0

4

2. f

7

3. i) , x > 0, f(x) = f(4) x4 = 4x

( 2)

ii ) N x 4 = 4x , x > 0, , x 1 =2

x2 = 4

( 6)

8

4. , , (2,4) ,

2

f ( ) f (t)dt f ( ) 2 f ( )

6

f: , = (0,+)

f (A) = , ,

f (x) 2e f (x) 2f (x) 3 x 1 . N f

f -1 f .

2 3

1 x 2f (x) e x 2x 3 , x 7

2 . f -1 . ,

f -1 , f -1

yy , x = 1

9

3. x 1A x,f (x) , 1B f (x), x f -1 f .

) , x,

f -1 f A

B , 1

( 3)

) x A, B

, .

( 6)

9

( )

2 IOYNIOY 2014 :

1. f . f

f x 0 x , f

.

8

2. f

. f

;

4

3. f A . f

0x A () , 0f x ;

3

4. , ,

, , ,

, . ) z C z z 2Im z

) ox x

lim f x ox x

1lim 0

f x

) f () ,

.

) 2

.

) f

. f ,

.

10

2

2 z z z i 2 2i 0, z C

1. .

9

2. 1z 1 i 2z 1 i ,

39

1

2

zw 3

z -3i

8

3. u

1 2u w 4z z i w, z1 , z2

2

8

2

f x x 3 x 1 , x R

1. f

f .

8

2. f

) y = 4x + 3

) f

.

8

3. g x x 1 f x , x R

.

9

h 2x x 2

h(x)x 1

x1 .

y= x - 2 h +,

1 . = 1.

7

2. ) y = x - 2

h - .

) h .

9

3.

4x 3

h x 0x

( 1, 0)

9

30-5-2000

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