1

8

2

3

--

--

22000011 ..

--

..

..

..

..

..

..

4

5

- . - , ... .

= 5 1

1,6180339887498948482045868342

+= .

- . - . - (. ). , -, . . 3, 5, 8, 13, 21, 34, 55, 89 -.

. .. 89

55=0,61. Fibonacci .

-. . Rhind . - - . ..

, (

=) -

-

(

=).

6

7

:

= { i, i= -1}

.. 2

1i , 3i , -2i , 2 i , 4,8i , -

4

3i , - 3 i

yy . yy - xx .

. i

: 1 = 4, i = 4 +1, -1 = 4 +2, - i = 4 +3,

.. i2003

= i3+500 4

= -i

: i2003

= ( 2i )1001

i= (-1)1001

i = -1i= - i

i-=1

i=

i

i=

ii

i1=

i

12

-

, 0i =1.

= { +i, , } . , .

= Re(z) z= +i

= Im(z) z= +i :

z1= +i z2= +i :

z= x+yi:

(x, y), (x, y). z (z).

z1= z2 = =

{ vi =

8

z1= +i z2= +i :

. , - .

, (+i)(+i) - .

i+

i+

(-i) . z= +i

. z=+i z z =-i

z1 + z2 = (+)+(+)i

z1 - z2 = (-)+(-)i

z1 z2 = (-)+(+)i

i+

+

+

+=

z

z -

2

1

i+

+

+

=

z

1 -

9

1. = +

2. = -

3. =

4.

5.

6.

7. z z =z z z = -z

z1. -.

( )

z1-z2= z1+(-z2)

( 8, , .96)

10

z=+i

xx

z+z+=0 , , 0 z 0

( Vietta,

- =z+z 21 1z 2z =

)

22 y+x |z|, z=x+yi

|z|:

( |z|= , |z| ())

1. z-=z=z

2. zz=z2

3. 2121 zz=zz

4. vv

z=z

5. 21 z-z 21 z+z 21 z+z

6. 2

1

2

1

z

z=

z

z

2

-i-=z 2,1

z20

11

z=x+yi :

Re(z)=0 (yy)

Im(z)=0 (xx)

Re(z)=Im(z) y=x 1 3

Re(z)+Im(z)=0 y= -x 2 4

Re(z)0 yy yy

|z|=, >0

|z-z0|=, >0 z0 = x0+ y0i .

z0 , (x0, y0) .

|z1-z2|

| z-z1|= |z-z2| z1 z2

1|z-z0|2, 1>0 2>0

|z+|+|z-|=2, >>0 (, 0) (-,0)

||z+|-|z-||=2, 0

12

.

1. . 95: 8, 12, 14 : 3 9 2. . 101: 4 9 : 3. . 123: 1, 2, 3, 4 .

1. 0=i

1+

i

1+

i

1+

i

1=i+i+i+i 3+v2+v1+vv

3+v2+v1+vv , *.

2. 1, 2, 3, 4 4 :

) 4321vvvvi=i=i=i ) 1=i 4321

v+v+v+v

3. -

=1+ v321 i+...+i+i+i . [.: {3, 4, 7, 8}]

4. : ) 200420032001i

1+

i

1+

i

1, ) i 2i

3i 2004i

[.: )1, )-1]

5. 2=xyi+1

xyi+

xyi1

i+xyv4v4

-

-, x, y *.

6. 3 1)i-(-

i2+=z -

. [.: =0 = -1]

7. x, y, :

)i7+y(+)i5+x(+)i+1(+i+1

yi+x2-yi)-(x=yi)+i)(x-(1-)yi+x(+4+2-

2004221002

[. x=1 y=3 x=3 y=1]

8. :

) 0=1+z+z2

) 0=1-z3

) 0=z-z3

z=x+yi

[.:) i2

3-

2

1-=z1 , i2

3+

2

1-=z2 ) 1z =1 , i2

3-

2

1-=z2 ,

i2

3+

2

1-=z3 ) 1z =0 , 2z =1 , 3z = -1 , 4z =i, 5z = -i]

13

9. z =5+12i. (: w=x+yi - z w = z . :

.)

[.: 2i+3=z1 2i--3=z2 ]

10.

) 0=i)-(1-iz 22-z)i+1(2 , z

) 0=6+i+z

i-z5-

i+z

i-z2

, z

[.:) i2

2+

2

2=z=z 21 ) -2i=z1 , -3i=z2 ]

11. 0=2i-i)z-1(2-z2 , z .

[.: i 2-=z1 , 2=z2 ]

12. . (: ) 6=i)y-2(+i)x+(2

8=2i)y-3(+i)x2+(3

[.: x=2+i y=2-i]

13. :

) z 3 z =z

) : 1.zz+1

z2+

zz+1

z2=w , 2. 2121 zz+zz=u

3. 22212111 zz+)zz+zz(2004+zz=v , . z, z1, z2

14. : ) z I z = -z

) |z|=1 z-1

z+1=w

15. :

) |z1|=|z2|=1 21

21

zz+1

z+z=w

) |z+i|=|z-i| z

) |z+IzI|+|z-IzI|=2|z| z

16. z1, z2 |z1+z2|=|z1-z2| 2

1

z

z I.

17. z1, z2 0=1+z+z2

:

) |z1|=|z2|=1

)

2

21

21

z-z

z+z=w

) = ( ) ( ) 20042001212002

21 ++ zzzz

14

18. z1, z2 |z1| +|z2|= |z1-z2| |z1+z2|=|z1-z2|.

19. )|z|+1)(|z|+1(+x|z-z|2+x=)x(f2

2

2

121

2, z1, z2 . f(x) 0

x . ; ( ...)

20. z1, z2, z3 |z1|=| z2|=|z3|=1 :

) 321

321 z

1+

z

1+

z

1=|z+z+z|

) z1+z2+z3 =2 2111

321

=++zzz

21. :

) i2+1=z-|z|

) 0=-3|z|8+z422

[.: ) i2-2

3=z , ) i

2

3=z1 , i2

3-=z2 , 2

1=z3 , 2

1-=z4 ]

22. z1, z2, z3 )z+z(z=z 321 , )z+z(z=w 312 )z-z(z=u 213 .

z, w , u

23. 4=1+z

16+z z -

4.

24. |2z+3|=1 1=|1+z|+|2+z|22

25. |z1|=5, z2=5+12i. = |z1+ z2|

[.: :18, :8]

26. z w 1=wz-1

w-z -

.

27. |z-1+i|=2 =|z-5+4i| (- : |z-1+i|= 2 3 |z-5+4i| 7)

[.: :7, .:3]

28. z+2(1+)z+2(1+)=0 .

[.: z1= -1--i , z2= -1-+i |z1|=|z2|=22

]

29. z z-1

zi+2=)z(f , z1

) f(2)

) 2004

]f[=w )2( .

15

) |z|=i+f

2-f

(z)

)z(

) |z|=1 f(z) .

[.: )2 2 )w = -23006 ) (2, 0) (0, -1)]

30. z :

) 0=z

1+zlm ) |z -1+i|=2 ) 1=

i+z

1-z ) |z -i -1|=|z -1+i| ) 2|z -2 -i|5

[.: )xx x2+y2=1 ) (1, -1) =2 ) (1, 0) (0, -1) ) (1, 1) (1, -1) )

(2, 1) 3]

31. :

) 1+z

i3+z=w , z z -1

) 3-z

i2+z=u z z3

[.: ) 2

3-,

2

1-

2

10= ,

(-1, 0).

) 1-,2

3 =

13

2

(3, 0).

32. (z) z ) |z+4|=2|z+1|. :

) )z+z(4=)z-z(-)z+z(22

) )i+(1+)-1(=z , (0,2]

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

33. - z=x+yi :

) |z -3+2i||z -5i| ) |z+3|+|z -3|=8 ) |Iz -3I-Iz+3I|=4

) 1>z

z-2 ) |4z -8 +12i|=16 ) |z -i|(1+i)

8

[.: ) (3, -2), (0,5) (3, -2) .

) (-3, 0) (3, 0) 2=8 2= 72

16

(2=

2-

2) ) (-3, 0), (3, 0) (2, 0), (-2,0)

) (2, 0), (0, 0) (0, 0), )

(2, -3) =4 )

(0, 1) =24 .]

34. ) z=4-3i,

1=9

y+

16

x22

, ) w=(-1) 2

+4(1-)i,

y2=16x.

[.: 1)x=4, y= -3 2+

2=1

2)x=(-1) 2

, y=3(1-) y2=]

35. z |z-3-4i|=2 : ) . ) =|z-6-8i| ) ) ;

[.: (3, 4) =2, )37, ) i25

96+

25

72=z1 , i25

104+

25

78=z2 ]

36. z |z-6-8i|=5 : ) z ) z )

[.: ) (6, 8) = 5 ) .:|z|=15, .:|z|=5 ) z1=3+4i, z2=9+12i]

37. t - (z) z=(1+t)+(3+2t)i. : ) z ) =|z-7| ) z ) ;

[.:) y=2x+1 )A= 53 3)z=1+3i )t=0]

38. z1, z2 |z1+3-7i|=2 |z2 -1-4i|=1. -: ) z1 z2 ) =|z1-z2|

[.:) (-3, 7) 1=2 (1, 4) 2=1, ) 28]

39. z1=x1+y1i z2 =x2+y2i:

) (z1) z1 (0, 0) -

=4 (z2) z2=z1+1z

8 -

4

y+

36

x22

=1

17

) (z1) z1 1

z2=z1+1z

8 16=y-x 22

40. 4+z

i3-2=w . :

) z w ) z w

[.:) 3x+2y+12=0, ) 2x-3y+8=0]

(: )

41. z :

=)4+|z(|2

i+i)z+z(

4

1)z+z(

222 -

f. i2+)zz(i+)z+z(2

3=w -

2,

w Cg g. : ) f ) g

) fC gC

[.:)f(x)=x3+2x )g(x)=3x

2 )E=

2

1..]

42. i2)+z-z(+)z+z(=w 2 , z=x+yi, x, y 3. :

) w z y=2x2

) xx yy, Cf x= -1 ) z=2+6i - .

[.:)=3

2.. ) 1 :y=4x-2 2 : y=12x-18]

43. z=+i, (0, 2) z1

z1w

+

= .

: 1) z

1z = 2) w

3)

2

dw .

[.:ln2]

44. z=x+yi 2223

z9-)z3(lm+)zRe(3+)1-z3(Re+)z(Re=)x(f :

) f ) Cf - .

[.: x= -1.., x=1 ., x=0 ..]

[ ]

18

45. z=1--i [0, 2]

) :=

0

diz

) 2

=

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

1-w

[.:1)=4 2) (-3, 0) = 22 ]

46. : = +

01

i

d

[.: =2]

47. z=x+yi |z|=1. :

) f(x)= 2+z-z3

) ;

[.:): 13 : ) ]

48. z=f(0)+i w=1+f(1)i |z+w|=|z-w| f [0, 1]. ) f -

[0, 1]. ) Izwu = 49. z=3+(x-3)i w=1+ilnx.

x (1, e), .

50. f [, ] )(if+=z2

2i+)(f=w 0

222z-w=z+w . :

) 0=wz+zw ) x0 (, ) f

xx

19

- (1984 2005)

1984

51. z=x+yi y0. 1-z

z=w

2

w

z .

1986 52. z=(2x-3)+(2y-1)i x, y . -

(x, y) |2z-1+3i|=3 . - .

[.: 4

1,-

4

7 =

4

3]

1989 53.

0=)1+z(+z+z2+z2+z2+z223456

[.: 1-=z=z 21 , i2

3+

2

1=z3 , i

2

3+

2

1-=z4 , i2

3-

2

1-=z5 , i2

3-

2

1=z6 ]

1991

54. +iz

i+z=w , 3* z i

) w z ) |w|=1 z

1993

55. f(z)= z+z

1)+z1)(-z( z Re(z) 0

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

z=x+yi , , x, y 3 x 0, Re[f(z)]=0

[.: : 1=y

+x

2

2

2

2

1

1]

1994

56. . z 1z , 2z 0=1+z+z2

2

2

1 z=z , 12

2 z=z , 1=z3

1 , 1=z3

2 0=1+z+z 21 .

: 0=z+z+)1+z(322

0=1+z2+z1416

20

. z, w 1w w=z-zi i+

1=w1

.

1w=w P z

1=y-x22

[.:. z=i z=-i]

1995

57. i) 1z , 2z 2

21

2

2

2

1 z-z=z+z Re )zz( 21 =0

ii) f: [, ] 3 [, ] z= 2 + if(a)

w=f()+i 2 0. 222

z-w=z+w f(x)=0 -

[, ].

1998 58. . z0, Im(z0)< 999 z,

zz0 z 0z , :

0000 z-zz-z

1998=

z-z

1+

z-z

1

. ; z= 0z .

1999 59. z=x+yi, x,y .

) , , (x, y),

6=2i-3-z+1-z22

, . -

. ) 1, 2 - . .

2000

60. i3+2

i+5=z

) z +i, , ) z, -

1=i-z

1-z.

21

2001

61. 1. 1z , 2z . : | 1z 2z |=| 1z || 2z |

2. , .

:

. zz=|z| 2 . 22 z=z . |z|-=|z|

. |z|=|z| . |z|=|zi|

1. 1z =3+4i 2z = i3-1

, .

2. z |z|=1, z

1=z

2002 62. f, :

22

22

z+x

z+x-z-x=)x(f , z z=+i, , , 0.

. : lim ( ),x

f x+

lim ( )x

f x

. f, 1+z > 1-z .

. f. 2003

63. z=+i, , w=3z-i z +4, z - z. . Re(w)=3-+4 m(w)=3-. . , w y=x-12, z y=x-2. . z, y=x-2, .

22

2004 64. :f 1)1( =f . x ,

0)1(1

3)()(3

1+= xzzdttfzxg

x

,

,Ciaz += , *, : . g g.

. z

zz1

+= .

. Re(z2)=2

1 .

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

2005

65. z1, z2, z3 =1z =2z .3z3 =

. : .z

9z

1

1 =

. 2

1

z

z+

1

2

z

z .

. : .zzzzzz3

1zzz 133221321 ++=++

23

1o 1

. 1z , 2z 0=1+z+z2

:

) 0=1+z+z 21 ,

) 22

1 z=z 12

2 z=z

) 1=z3

1 1=z3

2

) 1=zz 21

. 1z , 2z , 3z 0=-)z-1(+z)-1(+z23

,

v

3

v

2

v

1 z+z+z=w

2

. w -w=w

. 1-2z

2i-z=w , z=x+yi x -{ 2

1}, y *.

w z

16

17=1)-y(+

4

1-x

2

2

3

1z , 2z >1 v

1z =1+2i v

2z = 2-i :

) 2

1

z

z=w

) 1-w

1+w=z ,

2

1

z

z=w z-=z

) w-u+w+u=f )u( , u

4

t (z) z=(1+t)+(3+2t)i. : ) z ) =|z-7| ) z ) ;

24

2o

1 . .

1. Re(z1+z2) = Re (z1) + Re (z2) 2. Re(z2) = (Re(z))2

3. Im(z1z2) = Im(z1)Im(z2)

4. Im )Im(

1

2

1

z=

5. Re(z1z2) =Re(z1) Re(z2) - Im(z1)Im(z2) 6. z 2 = z2

7. 21 zz + = 1z + 2z

8. 21 zz = 1z 2z

9. z = z

1

10. z1 2z = 1z z2

. :

1. 2121 zzzz = ( ) ( )vv zz =

2. 1z2 = z z

2 . z = x + yi, x,y

w = ( ) ( ) izzizz 22

3 2+++ ,

y=3x2.

. z =1 =z

z

+

1

1 .

3

z = x+yi, x,y3 zzzzz 2=++ z .

4 z |z-6-8i|=5 : ) z ) z ) .

25

3o

1

. z I z z= .

( 5) . .

1. 1 2 1 2Im(z z ) Im(z ) Im(z ). =

2. i i.

3. 2 2z z=

4. z i= + - .

5. 1z 2z

.

6. 0z z = -

- -

0z .

7. z = ( 3)i + 7, ( -1, 5), = = 8.

8. 1, 2 z1 z2 yy

12, z1 = 2z .

9. 2x x 0 + + = 2 i+ 2 i.

10. z

x 3,= ( )Re z 1 2. =

( 10x2=20)

26

2

5 3 3

1 2 3

iz

i

+=

) z z i = + . ( 9)

) 3z . ( 9)

) , ,z z (2, 0) ,

. ( 7)

3

2 4

( ) ,iz i

f z z iz i

+=

.

) ( ) 1f z i= . ( 6)

) (z), Im[ ( )] 0f z = . ( 6)

) u z i= ( ) ,w f z i= 3 4u w i = + u w .

( 7)

) (z) C (0,1) ,

( )( )f z C . ;

( 6) 4

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

(1,-1) = 2 . ( 9)

) ( )w 4wz = 2y x= + .

( 7)

) A z w= .

( 9)

27

1 :

-

, - , . . 6 .. Leibniz Newton *. - , ... ... , .

.

- , - - , .. f(x0)=0 x0 - xx. **.

f(x)= x+

* - . - , Leibniz Newton, , . , Leibniz Newton . ** . . , .

x

y

x

y

x

y

>0

28

f(x)= ax2, 0.

f(x)= ax3, 0.

f(x)= x

, 0

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

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

>0 0 0

29

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

f(x)= ax, 0

30

x1 < x2 f(x1) < f(x2) x1, x2 , f x1 < x2 f(x1) > f(x2) x1, x2 , f

: x1 < x2 f(x1) f(x2) x1, x2 , f x1 < x2 f(x1) f(x2) x1, x2 , f x1 < x2 f(x1)= f(x2) x1, x2 , f 1. f(x)=x+: >0 , 0 (-,0] (0,+),

0 , 0 (-, 0) (0, +],

1 , 0

31

lim f(x)= xx0

XX0

. , - . 158-160. : .

1. )x(flim0xx

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

2. )x(flim0xx

32

10. h(x) f(x) g(x) x0 )x()x( glimhlim00 xxxx

= = )x(f0xx

lim

=

( )

11. xlim

0xx

=x0

12. xlim0xx

=x0

13. x

lim

0x

x

=1 ( |x||x| x )

14. x

lim

0x

1-x

=0

15. )g( )x(flim0xx

= )u(flim0uu

u=g(x) u0= )x(glim0xx

X0.

16. )x(flim0xx

=+ f(x)>0 x0, )x(flim0xx

= - f(x)0 x0 )x(f

1lim

0xx=+, f(x)

33

X0 :

x0 , f : + - + -

g : + - + - - +

f+g : + - + - ; ;

x0 ,

f : >0 0

34

- >1,

00 1 (0, +)

6.

.

7.

8. f g g(x)0 x A

g

f

9. c f -

10.

11.

12. -

0lim-

=

x

x , +=

+

x

xlim

=-xloglim

0x, +=+

xloglimx

+=xloglim

+=

x

-xlim , 0lim x

x=

+

+=xloglim

0x, =+

-xloglimx

35

f [, ] f()f() n f() f() x0(, ) f(x0)= n

x1 x2

x3

f()

f()

y

x O

[, ] BOLZANO

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

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

3) [, ] - ' x [, ] - x [, ]

4) f f .

1) 2) f .

3) f() f

f [, ] [, ] f()f() < 0 x0 (,) f(x0) = 0. Cf xx ' x0(,) f(x0) = 0 x0(,).

f ()

f ()

y=

B(, f())

(, f())

x0 x0 x0

f()

f()

36

1) 2) f [, ] [m, ] 1

. : ) f [, ]

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

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

( )x(flimx +

, )x(flim-

x )

) f (, )

( )x(flim-

x , )x(flim

x +)

: ) f [, )

( )x(flimx

, f()]

) f (-, ]

( )x(flim-x

, f()]

2

, x0 f(x)=0, - f(x)=0

f [, ] f [, ] - m.

m

f()

f()

37

. . 1. .145: -

.147:

2. .156: . 2, 3, 4 3. .165: . 4, 5 4. .174: . 3, 6, 7, 8, 9

.175: . 1, 2, 4 5. .181: . 1, 2

.182: . 1, 2, 3, 4 6. .186: . 3

.187: . 1, 2, 3, 4 7. .198: . 4, 6, 7, 8, 9, 10

.199: . 1, 2, 3, 4, 5, 6, 8 . 1. :

) f(x)= 3-4+x

7+x3 ) f(x)=

x+x

x2

) f(x)= 3+x-2 ) f(x)= 22

x-4+1+x+x

) f(x)= x-x3

) f(x)= 2

2

x-16

)6+x5-xlog(

) f(x)= 3+x-3+x2

x-1 ) f(x)=

2-x

1-2x-1

) f(x)= 2-2-4

xxx ) f(x)= x+3

x-3log

) f(x)= xx2

2

e-e

x ) f(x)= 1-x2

) f(x)= 1-x+x2

x2 ) f(x)= x)-ln( 2

3

) f(x)= 1)-x(x

1+x ) f(x)= lnx-xln

2

) f(x)= 3+32+9-xx

) f(x)= x-1-2

x-1+2

) f(x)= 1+x2 ) f(x)= 5x-5

10-2x

) f(x)= x-x

x) f(x)= x2ln(

2-3x +1)

) f(x)= )xln+1ln( ) f(x)= 1-2x-1-x

38

) f(x)= 2-x+x2

: ) =[-4, 5) U (5, +) ) A=3* ) A=[-5, -1] ) A=[-2, 2] ) A=[-1, 0] U [1, +) ) A=(-4, 2) U (3, 4) ) A=[-1, 0) U (0, 1] ) A=[-1, 1] ) A=3-{1} ) : A=(-3, 3) ) A=3*

) 2+6

, 2+

6

5, Z ) {x 3 x2+3

2

, x2-

2

, x2+

6

5, x2+

6

,

Z} ) = 6

5+2, 2+2 U 2, 2+

6

Z ) =[-1, 0] U [1, +)

) =(0, 1] U [e, +) ) =(-, 1] ) =(-, -3) U (-3, 1) ) =[2 -3

2, 2 +

3

2] Z

) =(5, +) ) =(-, 0) ) =[3

+2,

3

5+2] Z ) =(

e

1, +) ) =[0,

3

2]

) =(-, -1] U [1, +)

, : 1) 2) , .

2. ) 3 f

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

3.

) , f f(x)= +1)x-2(+x2

: ) 1 ) >0

3. fog gof

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

x-1

: (fog)(x)= 2

x-1 = [-1, 1]

(gof)(x)= 1 - x = [0, +)

) f(x)= 1-3x = [-1, 7], g(x)= 3+x2

= [-7, 8]

: (fog)(x)= 8+3x2

= [-2, 2]

(gof)(x)= 4+x6-9x2

= [-1, 3]

) f(x)= 2

x-1 , g(x)= 2+x3

: (fog)(x)= 3-12x-9x-2

= [-1, 3

1- ]

(gof)(x)= 2+x-132

= [-1, 1]

) f(x)= 1-x , g(x)= x-3

: (fog)(x)= 1-x-3 = (-, 2]

(gof)(x)= 1-x-3 = [1, 10]

39

) f(x)=1+x

3-x, g(x)=

x

5

: (fog)(x)=x+5

3x-5 = - {-5, 0}

(gof)(x)=3-x

1)+5(x = - {-1, 3}

) f(x)=3x+5, g(x)=

: (fog)(x)=

(gof)(x)=

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

: (fog)(x)=

(gof)(x)=

4. f(x)= . fof

: (fof)(x)=

5. (fog)(x)= (x-2)(x-1) g(x)=

2

3-2 x . f.

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

1 2

6. f(x)=3x+2 g(x)=x+. N g fog=gof

g(1)=2

: g(x)= 2

1+x

2

3

7. f g - gof : ) f, g ) f, g

8. :

i) f , -f -

{ x-1 x

40

ii) f, g , - f+g iii) f, g f(x)>0

9. , - .

10. :

) f(x)=e x

) f(x)= 3

x2

) f(x)=(x) ,

2

,0

x

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

3x , 2

2

) f(x)=ex+x

3

11. - :

) f(x)= 2-x

: 2+x=f21-

)x(

) g(x)=4+x

3+x2

: 2-x

4x-3=g )x(

1-

) h(x)= 2-x-1

: 3+x2-x=h41-

)x(

) (x)= 3

1-x43

: = )x(1-

) (x)= x

x

2+1

2

: x-1

xlog= 2

1-)x(

) (x)= )3-2x-1log(

: 2x1-

)10-1(2

1+

2

3= )x(

12. f(x)= 1+x2

g(x)= 9+6x-x2+1+2x-x22

. :

) ) )

3

4

1+x3 x

3

1-

3

4

1-x3- x0 x x 4

f(x) 2lim 1

x 4

=

:

) x 4lim f(x)

) 2

2x 4

3x f(x) x 6x 4f(x)lim

3x 11x 4

+

18. f: 2x+2xf(x)f2(x)2x+x(x+2f(x)) x .

x 0(x)lim f

= f(0).

19. 1 x x

1 x 1

+

+ f(x) x

x1 ex+ x

x 0(x)lim f

.

: 1

20. 2xx+f2(x)2xf(x)+

2x x

x 0(x)lim f

.

: 0

21. (x)3x + 3 (x 2)f 3 3 x 7 6 + + x (1, 3)

x 2

(x)lim f

.

: 1

2

45

22. 2(x)f x 2x x x i) x 0

(x)lim f

ii) x 0

(x)xf xlim

2x x

+

: i) 2 ii) 3

23. 2 2(x)2x x f 2x x + x :

i) x 0

(x)lim f

ii) x 0

(x)flim

x iii)

x 0

(x)

(x)

2f 4xlim

5f 3x

++

: i) 0 ii)2 iii)8

13

24. + + x x f(x) 8 x 4 16 x>-4

x 0

(x) (0)f flim

x

A. 2

46

1

( ) ( )

>

+

+++

=1x

1xxxx

52x2x

1x

)x(f

222

. =2, =-1, =-10

, , .

2

2xx)x(f x

=

=0x

x

)x(f

.0x33

)x(g

2

g xo=0.

. =4, =-1,

3

47

4

>

=

2x5x2

2x2x

xx)x(f

2.

, f x0=2. : =5 =-6

5

48

9 f,g: f2(x)+g2(x)+2f(x)+54g(x)+2x x .

f,g x0=2

.

10 f: (0,1)

yx)y(f)x(f x, y . 1) f . 2)

f(x)=x .

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

0 1x

)x(fim

0x=

i) f .

ii) =

2

x

)(f)x(fim

x .

12

f 2 2 4(x)1 1

f x x x x x

x 0(x) 0lim f

= .

13

f x0=0 |xf(x)-2x|x4 x

f x0=0 : f(0)=2

14

h 0

f(3 h)lim

h

+=5 f 3,

x 3

f(x) f(3)lim

x 3

.

: 5

15

f, g: f 2(x)+g2(x)+2f(x)+5 4g(x)+2x x x0=

2.

49

[, ] : BOLZANO - -

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

17 f g [, ] :

. f(x)0 ++1

50

23

f(x)=x2+x+ g(x)= -x

2+x+ 0. 1 f

2 g 1

51

3

1

f(0)

=

=0

12

00

)(x

xx

x

xf

2

f x0=0 x3 x3 + x2 + 2 f(x) x4 +2x2 + 2. 3

f

>+

=

02

02

)(

xxx

xxx

xf

.

N : 1. f x0=0. 2. f x0=0. 3. f x0=0. 4

f : x0=0 f(0) = 2005 -

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

5

f x0=0 0

limx

10021)(

2=

x

xf

:

1. )(lim0

xfx

2. f(0) 3. f(0). 4. f x0=0.

52

6

f : x,y3 f(x+y) = f(x)y+f(y)x

2005)(

lim0

= x

xf

x f x0=0.

7

f : x0=0 f(0)=0 f(0)=1 -

g 1+f(x) g(x) f(x)+1+x2 x .

f g x0=0 45.

8

f x0=0

2

,2

x

f(x)2-2xf(x)+xx=0. 1 3

f x0=0.

9

f g f2(x)-2xf(x)+x2+g2(x)-

-2xg(x)+2x2=0 x . f g

.

10

f f(1+h)=2+3h+3h2+h3 h3 :

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

53

RROOLLLLEE

1 , ,

2

2

x x x 0

x 4 x 2 x 0f (x )

+ + 0.

6) :

1) xex-1=0 (0,1).

2) x3-3x+=0 3 (-2,2). 7) f(x)=x2-2x+x nx- nx.

:1) f.

2) x2-2-(1-x)( nx-2)=0.

3) (1-x)( nx-2)x2-1 x>0.

59

8) 3 f(2006)=f(2006)=f(2006)=0 f(x)>0 x . f - f(x)=0 f(x)=0 .

9) f(x)= x -x2

nx, x>0

( 1991) 10) f,g R. fog 1-1. . g 1-1. . g(f(x)+x3-x)=g(f(x)+2x-1) . ( 2002) 11) f(x)=x5+x3+x. . f f . . f(ex) f(1+x) x . . f (0,0) f f -1. . f -1, x x=3. ( 2003)

12) z i4zi2z +=+ w= ++

i2z

i4z .

z1=(x-1)+( i)2z z2=2 nx+( i)1z .

x u= 1z z2 .

13) f f(x)=22

x

x

e

+

>1. :

1. f .

2. x0 ex1+ ,x

2

>1.

60

- FERMAT 1) f, -

3, : f 3(x)+f 2(x)+f(x)=x3-2x2+6x-1 x , , - 2

=

0xx

0xx

1)x(

2

.

3) x u=2

1m/sec

. - f f(x)=e-4x, Oy . :

1. E(x) (t) x

.

2. t0=2

3

sec. 3.

;

4. +x

lim (x) 0x

lim

(x).

61

4) :

1. x1-2

x 2 xR, 2. -2xlnxx2-4x+3 x>0.

5) f, f(0)=0

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

6) f 3 f3(x)+3f(x)=ex-x+2

x 2

x .

7) f(x)=3x4-8x3-6x2+24x-12 f(x)=0.

8) f(x)=ex-1- n(x+1), x>-1.

1) f.

2) xx.

3) 1+ n(x+1)ex x>-1

4) x1+ n(x+1) x>-1 =e

9) f f 3(x)+x3=3xf(x)-1

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

10) f(x)=x3-x2+1, 0 -.

11) f:(0,1) 3 f(x)2006 x(0,1). x1,x2(0,1) x1

62

14) f(x)=x7e2-x 0 , x>0. f .

15) f(x)=x3+x2+x+ , , 1

63

DEL HOSPITAL

1)

) x

1xeim

x

0x

) )1xe(im x

x

+

2)

) x1

xxim

0x

)

nx3x

nxx2im

x

++

+

3)

) xx

eeim

xx

0x

)

30x x

xxim

4)

)

1x

1

nx

1im

1x )

x

x2xx1im

20x

+

5)

) 1xn

1nxxnim

2

2

x +

+++

)

2x2xe2

xim

2x

3

0x

6) , , 4xx

xeeim

xx

0x=

++

7) f

+

=+=

)0,1(xx

)1x(n

)1,0(x)xe(

0x33)x(f

x

2

.

.

8) xxe-x-x-e0 x(0,) >0 = x0x

)x1(im +

+ .

9) n(x+1)x+2

x 2 x>-1 = x

0x

xim+

.

64

10)

x n(x 1) , x 0x

f (x)1

( x), x 02

+ >

=

.

f x0=0.

11) f f 5(x)+f 3(x)+f(x)=x-x

x 6

1

x

)x(fim

30x=

.

65

1. f f(x) = 4e2x, x , 2)x(flim0x

=

2x

)x(flim

x=

: i f. ii e2x-2x-10 x . iii e2x-2x=2x2+1 .

2. . y=2x+5 f + .

:

i x

)x(flim

x + ]x2)x(f[lim

x

+

ii , 1x3x2)x(xf

x4)x(flim

2x=

+

++

.

.

i ex-x+1>0 x . ii H 2ex+2x=x2+2 x=0.

4 1994

3. P(x) f )x(P

3xx)x(f

2 +=

1)x(flimx

=

, x=1 x=-2

x0=-1.

4. f 2x+32

23

x

1x3x2)x(f

++ x *.

y=2x+3 f x+ x- .

5. :

1) f(x)=2x

6x5x 2

++

2) g(x)= xx 2 +

6.

1) f(x)=1e

xx

2) g(x)=2x

ex x

66

7.

1) f(x)=x

)1x(n 2 + 2) g(x)=

x

x2

8. f g f(x)-g(x)=x-4 x y=3x-7 Cf x + :

1)x

)x(gim

x + 2)

1x3)x(xf

x2x3)x(gim

2x +

+++

2000

9. f(x)= ,2x

xx 2

+

x -{ }2 , . : y=2x-1

Cf x + . 2001

10. y=3x+5 f +

+xim

232

2

x2x3)x(fx

1x)x(xf

+

++.

11. f(x)=)2x(x

3xx)1(x)1( 232

+++

x + y=3x+2. 12. , y=x-2

f(x)=1x4x2

1x2x3x2

223

++

++ x .

13. f(x)= x1x3x4xx 22 +++ - x .

14. f(x)=2x

x6x2 2

++

g(x)=x2+1.

15. P(x)

f(x)=)x(p

2x)1(x 2 +:

1) =1 Cf 2) x=0 x=2 Cf 3) x0=1

New York University

67

- -

1) f

f(x)= ( ) 232234

5x7x2

52

3

x2

3

x++

++

+

f . 4 1990

2) f(x)=(+1) nx -(+3)x2-3x+ g(x)=1x

1x3x2 2

+++

. f x0=2

g + (1,0)

Cf .

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

f(0)=f

2

3=0.

.

2

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

. f(x)=2x2-3x, . ()= 0

,dx)x(g

.

im ()

4) f(x)=-x4+2(-1)x3-6(2++2)x2+2+1 - , f(x)=x4+6(+2)x3+6(22+8+7)x2+12x+2 2 . 5)

f(x)=xe x g(x)=( )

+

2

,2

x,x1

xn

2 .

6) f, g f(x)>0 x, g(x)>0 g(x)>0 xf() gof . 7) f f(0)=0

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

68

8) .1. f (0,+ ). g g(x)=Inf(x), x f(x)f(x) [ ]2)x(f x. 2. , g g(x)=In(x2+2) . .1. f f(x)=x-x, x 0

69

1

-

1

. 1. f [,] 1 x0(,) f(x0)=0 f()f()0 1 x0(,) f(x0)=0

3. f [,] m, [,] m ML x0(,) f(x0)=L

4. f(x)

70

3

,, f :

f(x)=

=

+

+++

1x

1x1xxxx

5)2(x22x)22(

.

(25 )

4

f =(,) (,] -

[,), (,). f()=-1, 2)x(fimx

=

3)x(fimx

=

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

(25 )

71

2

1

.

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

2. )x(f)x(fim 00

xx=

3. f [1, 5] f(1)=10 f(5)=50 f()=[10, 50].

4. ( ))x(g)x(fim0

xx+

= )x(fim

0xx

+ )x(gim0

xx

. .

1. =

+ 230x xx

xxim .0, .+ , .1, . .

2. =

3

3

x )1x(

)x31(im .+ .- , .9 .27 .0

3. 42x

)x(fim

2x=

f . f(2) :

.2 .0 .4 .1

4. xx

xxim

3

4

0xx +

:

.x0=-1 . x0=1 . x0=0 . x0=2

(4x4=16 )

2 . ) f(x)=(x-1)(x-2) (x-2001)+2004 1 1.

) f ( ) ( ) 3x2)x(f)x(ff 3 +=+ 1 1.

72

.

=

=3x

3x

3xx

3x4

)x(f

2

x0=3. (25 )

3

A. :

1.

+ x

1xim

x

2.

+ x

xim

x

3.

2xx

x1im

2

2

2x

4. 38x

23xim

1x +

+

(4x4=16 )

. ) ( ) 2004)x(fxim 4x

=+

)x(fimx + .

) ( ) 52xx)x(fim 21x

=+ )x(fim

1x .

(2x5=10 )

4 (0, 1) f, g, [, ]. f(x)

73

3

---- 1 (z) z (0,0) 1=1

(w) w=i23

i)1z(2)1z(3

++

(1,0) 2=1.

A= .wz

(25 ) 2 . : f -1(f(x))=x (4 ) . f: f 3(x)+3f(x)-x=0 x f(3)=3 :

1) f (8 ) 2) f -1 (8 ) 3) f -1(x)=4x2 (5 )

3 . f:[-2,2] f(-2)=f(2) f(0) (- 2,0) f()=f(+2). (12,5 ) . (-2,2)

5f()=f(-2)+f(2)+f(0)+f(-1)+f(1). (12,5 )

74

4

1. f [5,7] f(6)f(5)=f(7)-f(6) (5,7) - f xx.

(8 ) 2. x+1exxe+1 x(0,1) (8 )

3. f

,2

,2

f(x) 0

x

,2

f()=0.

,2

f()=f().

(9 )

4. f x0=1 4h

)h1(fim

0h=

+ :

1) f(1) 2) f x0=1 3) f x0=1

(8 )

75

4

1 . f x0 - f (x0, f(x0)).

( 4)

. , f x0 , .

( 8,5)

. x=x0 f;

( 4,5)

. - . . f x0, f x0. . f x0, f x0. . f . f(x)>0 x , f . . f x0 . f x0 f(x0)=0, f x0. (4*2=8 ) 2

f f(x)=

2

x-3

, x 3

1 - e , x 3

x - 3

x

>

. f , : = - .9

1

( 11)

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

( 14)

76

3 A. xx e-x ax-e 0 x (0, + ) >0 =e

( 9)

B. A f f(0)=f(0)=0 f(x)>0, x *, : i) f (- , 0], [0, + ). ( 8) ii) f . ( 8) 4 ) f(x)=lnx+x-1 i) . ii) f(x)=0 x=1 f(x). ( 10) )

(x)=2xlnx+x2--4x+3. ( 10) ) g(x)=xlnx

h(x)= - ,2

3x2x

2

1 2 + -

. ( 5)

77

5

2001

1 . f - x0 f (x0)=0 ( 10) B.1. f g f(x)=g(x) x f g . .2. f f (x0)=0 x . .3. f (A) f . .4. f f -. .5. f f(7)=5 f(5)=7. .6. f x0 x0 f . .7. f(1)

78

2 : . f f(x)>0 x * f(0)=f(0)=f(0)=0 1. f. ( 10) 2. f x0=0. ( 2) 3. g(x)=x3 f. ( 3)

. xx e-x ex x (0, + ) >0. = lim (1+x) ( 10) 3 . f : f(2)-f(1)=f(3)-f(2). x0 (1,3) f x0 xx. ( 12,5) . f [1,e] 0

79

6

1

. 1. f

f(x)=0 x. (7 )

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

f(x)=g(x)+c. (c: ). (6 )

.

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

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

3. f x0 x0 .

4. f f(x)>0 x .

(4x3=12 )

2

f g : i) f x0=2

ii) 32

3)(2

2=

x

xfimx g(x)=(x3+2x)f(x),

: . f(2)=3/2 (10 ) . g x0=2

(15 )

80

3

f f(x)=4e2x, 2)(0

=

xfimx 2

)(=

x

xfim

x , :

1. f f(x)=e2x-2x+1. (7 )

2. e2x-2x-10 x . (6 )

3. e2x-2x=2x2+1 1 . (6 )

4. y=-2x+1 f x-. (6 )

4

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

1. f(1)=f(2) (6 )

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

3. f (x)=0 1 (1, 2). (6 )

4. f (x)=0 2 (1, 2). (7 )

81

7

1 . . 1. z=3-4i 12. 2. z z =z2 z.

3. f x 4 f(x)= 4x

12x7x 2

+

f(4)=1.

4. f [-1,1] f(-1)=4, f(1)=3 x0(-1,1) f(x0)= 5. f (,) f(A)=(f(),f())

6. limxx

x2xx3

23

x0=1.

7. f =[0,3] f(0)=2 f(3)=-1 x0(0,3) f(x0)=0. 8. f g [,] f()=g() f()=g() x0(,) f(x0)=g(x0). 9. f [,] f()

82

3

. 1 (0,1) f()=g() -

f g [0,1] f(0)g(1). ( 12)

. f x0=0 x * 2xx)x(gx

( 12) 4

. ex 1xe + x (0,1) .

. f(x)=ex- 1x2

1 2

: 1). . 2). f(x)0 x 3). f(x)+f(x) f(x)

83

( )f x dx

84

85

.

1. - .

1. f(x) = 3

4

x . F(x) = cx +3 26

2. f(x) = xexe xx + . F(x) = exx + c

3. f(x) = 4(3x2+1)(x3+x+1)3 . F(x) = (x3+x+1)4+c

4. f(x) = 84

2

2 ++

+

xx

x . F(x) = 842 ++ xx +c

5. f(x) = (x+)x-1 . F(x) = ( )

( )1

1

++

+

x+c

6. f(x) = x

xxx2

. F(x) = x

x

+c

7. f(x) = x+3, . F(x) =

+

+

+

=+

414

4

42

ln

cx

cx

8. f(x) = 2

5x

, . F(x) =

=

33

5

3ln5

3

x

x

9. f(x) = x

x3

. F(x) = cx +2

2

1

10. f(x) = xx

1 . F(x) = n x - n x +c

11. 2

( )x xx e e

f xx

= A. F(x) =

x

e x+c

12. ( )x

f xx

= . F(x) = -2 x +c

86

13. 2( )f x x x = . F(x) = x33

1 +c

14. ( ) 53

f x x

= +

. F(x) = -

+3

55

1 x +c

15. 2

1( )

nxf x

x

=

. F(x) =

x

nx+c

16. 2

2

4( )

4

x xf x x

x

=

>2 . F(x) = 42 x +c

17. 3( )f x x x x = + + . F(x) = 4

4x-x+ x+c

18. ( ) 3f x x x= . F(x) = 6

5x2 x +c

19. 3 23 5 1

( )x x x

f xx

+ + += , x>0 . F(x) =

3

3x+

2

3 2x+5x+ nx+c

20. 3

( ) 2 2xf x e xx

= + , x>0 . F(x) = 2ex-3 nx-2

12x+c

21. 3 27

( )3

xf x

x

+=

+ . F(x) =

3

3x-

2

3 2x+9x+c

22. 2

1( )f x

x= -

2

1

x . F(x) = x+x+c

23. 5

( )1

xf x

x

+=

+ . F(x) = x+4 n(x+1)+c

24. ( ) x xf x e e= . F(x) = ex+c

25. ( )5

x

x

ef x

e=

+ . F(x) = n(ex+5)+c

26. 1

( )f xx nx

=

, x>1 . F(x) = 2 nx +c

27. ( )( 1) ( 1)

x

x x

ef x

e n e=

+ + . F(x) = n( n(ex+1))+c

87

28. 2

1

( )x

f xx

= . F(x) =+

x

1+c

29. 2

2( )

1

xf x

x

=+

. F(x) = - n(1+2x)+c

30. ( )f x x= n (x), x>0 . F(x) = -2

1 n2(x)+c

31. ( )f x x= ex . F(x) = ex+c

32. ( )f x x= , x>0 . F(x) = - n(x)+c

33. 2( )f x x= , . F(x) = 2

x-

4

2x+c

34. ( ) '( )f x f x= . F(x) = f(x)+c

2. :

1. 3/

0

3xdx . 24

5

2.

+2/

4/2

1

x

x

dx . 2

3.

2/

3/ x

dx

. 3n

4. 1

0dxee

xex . ee-e

5. +2

1

23 15

x

xxdx . n2-

6

31

6. ++3

13

235 47352

x

xxxxdx .

9

40+3 n3

7. )3

(9

1 xx dx . 3

16

8. 1

1

22 )13( xx dx . 0

88

9. 2/

0(x+xx)dx .

2

10.

0

32 )3( xxxx dx . 3

11. +

02x

xxx

dx .

12. +

0

1

23 dxe x . -3

1(e2-e5)

13. 4/

0

3 xdxx . 16

1

14. +++1

1

22 )43)(32( xxx dx . 168

15. 2/

0 x

x

dx . 2

16. ++

+1

0 2 4322

34

xx

xdx . 1

17. +1

02 1x

xdx .

2

1 n 2

18. 5

3 2x

dx . n 3

19.

3/

6/3x dx .

3

1

20.

+

2/

6/)2534( xx dx .

12

13

21. 2/

0

2xdx . 4

22. dxx

xx

+6

4

2

3

752 . 22+10 3n

23. ( ) +1

035 dx .

3

2

5

4

nn +

89

24. ++e

dxx

x

0 1

2 . e+ n(e+1)

25. e

dxx

nx

1

)( . 1-1

26. 6/

0xdx . -

2

3n

27. dxx

x

+3/

4/

21

. 3n

28.

4

3 2

2

4

4dx

x

xx . 2 3 - 5

29. dxx

x

3/

4/

3

. 8

1

2

3n

30. dxx

xn

x

+

0 3

33

. 2 ( )213

31. +1

0 1 xe

dx dx

ee

eexx

xx

+

21

0.

. 1+ n2- n(e+1) n(+1)- n2-2

1.

3.

.

1. dxxex x .

2. +1

0

2 )9( dxxnx . 59 1

n10 n92 2

3. 1

0dxex x .

e

e 2

4. dxx

x

4/

02

.1

n24 2

5. xdxx

4

3 . - 2

1

90

6.

2/

0

2xdxx . 16

42

7.

0xdxe x .

2

1+

e 1991

8. dxxx

2

. -1

9. xdxx 22/

4

. -8

2+

10.

2/

0

2 xdxx . 24

2

11. 1

0dxxe x . 1

12. 1

1

2 dxex x . e

e 52

13. 2e

enxdx . e2

14. 2

14xdxnx . 5n2

4

3

15. 2

2

1nxdxx .

8 7n2

3 9

16.

2/

0

2 xdxe x . 5

21 + e

17. ( ) +1

1

2 1 dxx . 3

18. ( ) +2

0

2 12 dxxx . 1

19. +7

42 65xx

dx .

5

8n

91

20.

=

1122

145)()(

2

0

xx

xxxfdxxf

. 3

4

21.

92

2. = e

xdxn1

: =e--1 3 , *.

(. 3=-2e+6)

3. = 2/

0xdx = 2

1

-

3 4 , 2, *.

4. =

,2/

0xdx * -2, >2 -

2.

5. = ,31

0dxx * -1

2.

6. =

,

2/

4/xdx * : = 2

1, 2

1 >

-

5. (. 5=- 24

1n+ ).

6. f [1, e] f [1, e]

f(1)=f(e). : =e

fefedxxxf1

).1()()(

7. f [0,2

] f

[0, 2

] f

2

=2. A =

=+2/

01])]()([ dxxxfxf : f(0)=1.

8. f :

1. +

+=

t

tadxxfdxtxf

)()(

2. x1 t

f dt f (x)dxaxx x

=

9. :

1. =1

)()1(0

1

0dxxfdxxf

93

2. =2004

20012005)( dxxf

x

xdt

x

tf

2004

2001)( =X2005

3. =2

12005)( dxxf 2005

2

1)( =

++

t

tdxtxf

4. =1

0

1

0)1()1( dxxxdxxx a

10. .

1. g(x)= xxtdt

2. h(t)= 2 x1

tt e dx

3. G(x)= 2

1F(xt)dt+

x

tx1

)( F(t)dt

11. F : [-1, 3] F(x)=2x 1 2t 1

x 1e dt

+

+ .

. F(x)=- 12)12(12)1( 2 +++ + xx xee .

12. :

) dtt

exfx

x+

=1 2

)(

) +++=x

x

t

dt

t

dtxg

1

0 20 2 11)( .

13. F(x)= x

dttf1

)( f(t)= +2 4

1

1tdu

u

u F(2).

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

dtnttn1

2 2 .

.

15. : dttimx

x +2

00 =0

3

2

0

0 x

dtt

im

x

x

+

.

3

2

16. f(x)= ,0

)( x

tft dte

x .

94

17. f(x)=x+1+1

1

+x.

C f, x,

x=2, x=5. . 2

27+ n2..

18.

C f f(x)=-x2+5x-6 xx,

x=4. . =3

17..

19. xx

C f f(x)=x3+x2-6x

. =12

253..

20. f f(x)=x2-4x g g(x)=-x.

Cf, Cg :

1. x=-2 x=-1

2. x=1 x=2

3. x=-1 x=4

4. Cf, Cg.

. 1)=6

41..2)=

6

13..3)=

6

49..4)=

2

9..

21.

Cf, Cg f f(x)=x3-x2-6x g g(x)=3x-x2

. =2

81..

22. / f(x)=(2x+2

) [-

4

,

4

].

f x0=8

f .

95

23. : C1:f(x)=x2 c2: g(x)=

x

1 :

1) C1,C2 x=2

x.

2)N :=

1|f(x)-g(x)|dx, >0

3) lim+

().

. 1)3

1+ n2..2) 1 - n+

3

1(3-1) 01. ) +a

lim E().

. ) -a2

1+

2

1 )

2

1

96

28. / f f(x)=x3-x2, ,

) (1,-2) f.

) /

. . )

f

.

. ) )

29. f(x)= .21 x

xx, x=0, x=1 f-1.

. 38

15 ..

30.

f:f(x)=-x2-2x+3, f (2,-5) yy.

. 3

8 ..

31. x=o,

x=2

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

. 2 2 -2..

32. ) f [-,] =a

adxxf 0)(

) f [-,] =a

a

a

dxxfdxxf0

)(2)(

33. f(x)= ex,

x=0, x=1 xx, x= .

34. f f(x)= xe

x 12 f(0)=-1.

f g g(x)=12

)(

+xxf

yy

x=1.

97

35. f f(x)= 21

x,

x=1, x=3 xx. y= - .

36.

f(x)= x , x0=1 xx.

x= , . 37. :

1. ++ 232 xxdx

2. ++ dxxxx

)2)(1(

3. ++ dxeee

xx

x

)2)(1(

38.

+x t

x tu

x ududt

dudte

im

0 0

0 0

0 2

. +

39. dxex

x x

++2

0 1

1

. 2

e

98

.

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

1.

f (x0,y0).

2. f,

.

3. x u=1m/sec,

f

, =

, to =1m.

2 . f [,]

. :

=

)(

)()()()(

f

ffffdxxf

-1(x)dx

3. 1 2

1:(2)x+()=4

2:(2)x+()=42, [0,).

1. [0,).

2. ,

.

3.

:2x+=0.

4. :

=0 2

)(

dxxxf dxxf )(0

dxx

xx

+0 23

. 4

n3

5. 1.

.

2. f:

. f (2,3) xx

45. = ++2

2

2 )1)(( dxxxxf .

99

6. f,g [,]. f(x)< g(x) x(,)

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

f

g .

7. f f(x)=9

2

2 +x

x

1. f -1.

2. 1= 1

1( ) .f x dx

3.

f -1, x=-1 x=1 xx.

8. f =(-1,+ )

f(x)=2005ex+2)1(

2001

+x

(0,50) -1.

9. ) f x

f(x)-f(x)

100

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

g(x)=3

1

xz f(t)dt-3|z+ z

1|(x-1)0, z=+C, , *, :

. g

g.

. |z|=|z+z

1|

. Re(z2)=-2

1

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

f(x0)=0.

2004

12. f f *, g

* G(x)= +2

1)( dtxtf

+ ++

1

1

12

,)()(xx

x needttgxt x *.

G(x)G(1) x *, 2f(2)-f(1)=f(2)-f(1)-3g(1)-1.

13. f g

h(x)=

+x x

xxdttgdttxf1

12

0

2 1)()( h(x)h(1)

x . :

1. f(1) = 2g(0)+3

2. xf(x)-1 = 2xg(x) (0, 1).

3. 2 x=1/2 f, g -

x1,x2(0, 1)

f(x1)+4g(x2)=2.

14. f,

, :

i) f(x) 0, x

ii) f(x) = 1-2x2

1

2

0

tf (xt)dt, x .

101

g g(x)= 2

)(

1x

xf ,

x .

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

. g .

. f : f(x)=21

1

x+.

. ( )xxxfimx

2)( +

.

2001

15. f f(x)2 x .

g(x)=x2-5x+1-

xx

xdttf52

0,)( . g(x)=0

(-3, 0).

16. . f .

. =+3

0

7

1)(

2

1)12( dxxfdxxf .

. +=+3

0

7

12004)()12(4 dxxfdxxf

(1,7) , f()=334.

. f: :

+x

tft0

2 )()1( dt=x2+ +1

0

2 ,)(6 dtttx x .

) f(x)=1

522 +

+

x

x

) f

(0,f(0)).

17. g (x) = x

1 2

1 dt

z t+ z= + i, , *

1. g g.

2. g (x) 1- 21

x1x

2

+

++ x 1=z

102

3. g, xx, yy x=1.

. 1) g g(x)>0 x>1 g(x)

103

1

1

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

( 15)

B. , .

) f x0 x0 - .

( 2) ) f [,]

( 2)

) f ( )

( ( ) )' ( ( ))g x

af t dt f g x= .

( 2)

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

) z 2zz z= . ( 2)

2

f, ,

52

1)(lim

2

0=

+ x

exf x

x

) f(0). ( 7)

) f xo=0. ( 9)

) )()( xfexhx= , -

f h ))0(,0( fA ))0(,0( hB , .

( 9)

104

3 z1, z2

*N >1 1 2z i = + 2 1 2z i

= + .

:

1) 1

2

zw

z= .

( 7)

2) +

=1

1

w

wu

( 7)

3) wzwzzf ++=)( .

( 8)

4) edxwzzvve

5210

= ( 3)

4

. : (0, )f + f(x)=2004(x-1)+x

dtt

tf

1

)( x>0.

) f(x)=2004 xx ln . ( 10) ) Cf, xx x e=

501( )12 +e .. ( 10)

. f: =1

0))(()(

x

u

dudttfxf .

f (1)=0. ( 5)

105

2

1 . f f(x)>0

x, f . ( 10)

. x0 A, A f.

( 5) . :

1. f .

2. .

3. .

4. .

5.

=+++

0dx f(x) dx f(x) dx)x(fdx )x(f < < <

f .

( 5x2=10)

2

z=x+yi 8i512z = :

1. z .

( 9)

2. = i77z +

( 8)

3. z.

( 8)

106

3 f f (0) = f(0) = f(0) = 0

f(x)>0, x * : 1. f (-, 0] [0, +).

( 8)

2. f . ( 9)

3. f x0 = 0. ( 8)

4 ) f x

f(x)-f(x)

107

3 1

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

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

( 13)

. .

1). f [,]

[,].

2). ziizzzz ====

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

>

xfimxx

4). f -1(f(x))=x

5). f x0 .

6). =+

0)()( dxxfdxxf

( 6)

. .

( 6)

2

f [1, 3] f(1)=3 f(3)=1.

:

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

( 9)

2). x1,x2 (1,3) f (x1) f (x2)=1.

( 8)

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

(1,3).

( 8)

108

3

f [,] z=2+if() w=f()+i2 0

222

z-w=z+w . :

) w +z zw=0 ( 15)

) x0[,] f xx .

( 10)

4

g(x)+

x

1 2tz

dt z=+i, x .

1. g g(x)=0 .

( 10)

2. g.

( 5)

3. g(x) 121

x1x 2

+

++ x3 .1z =

( 5)

4. g, xx , yy x=1.

( 5)

109

4 1 . : f x0 -

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

(10 ) . (5 ) . 1. .

2. 0xx

im g(x)=0

0xxim

)x(g

1=

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

4. 3 5. f, xx,

x= x= =

( )f x dx

(5x2=10 )

2

:

1) z z =z zI z =-z (10 )

2) 1z = 2z =1 w=1

1

z1

z1

+I u=

2

21

21

zz

zz

+

(10 )

3) 1z = 2z = 3z =1 321 zzz ++ =321 z

1

z

1

z

1++

(5 )

110

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

1. f(1)=f(2) (6 )

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

3. f(x)=0 1 (1, 2). (6 )

4. f(x)=0 2 (1, 2). (7 )

4 f f(x)= ex, x

21

( ) ( )x

F x f t dt= xx yy x = 1.

( 12)

. f [0,1] 1

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

(0,1) f()=p(). ( 13)

111

5 1

. , - f (x)=0 x.

(10 ) . f x0.

(5 )

. : 1. f 2. f: g: gof . 3. f [,] f

f() f()

4. f (x0,f(x0)) f f

5. (x,y)

x 0yf(x) =

dx)x(f f .

(2x5=10 )

2 . z=f(0)+i w=1+f(1)i |z+w|=|z-w| f

[0, 1]. f - [0, 1].

(5 ) . z=3+(x-3)i w=1+ i nx .

x (1,e), wz . (5 )

. z=2+i2, (0,) w=z1

z1

+

6

5

2

w d

(10 )

112

3

f:(0, +) f(x)-f(y)=fxy

( ) x, y (0, +) -

f(x)=0 , : ) f 1-1 (10 )

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

2-2)+f(x+1) (10 )

) x

113

6 1 . f [, ]. G f

[, ], :

= )(G)(Gdt)t(f

B. x+.

.

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

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

3. F f

= )(F)(Fdx)x(f .

4. 0x x

im f (x)

0x xim g(x)

( ))x(g)x(fim0xx

+ .

5. ii

1i == .

2

f f(x) - f (x)=3x2+1 x .

i) P(x), 2 P(x) - P(x)=3x2+1.

ii) f f(0)=8

iii) f(x)= ex+3x2+6x+7 .

114

3

z2-z+1=0, , z z1,z2

f(0)=21 z

1

z

1+ f(2)= 22

2

1 zz +

:

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

3) =3 1 (0,2) f ()=2 4) =-2 1 f(x)=0 (0,2).

4

f(x)=x- nx+ex, x1

i) f ii) f(A).

iii) x+ex= nx+2005 [1,+ )

iv) I= +

+

+e

1

ee1e

e1

1 dx)x(fdx)x(f

115

7

1o

. f f (x)=0 x.

.1) z1,z2 z2+z+=0 ,, 0 z

z1+z2=

z1z2=

.

2) x0 )x(fim0xx

)x(fim0xx

=f(x0)

. 3) ( ) - y=x .

4) )x(gim)x(fim))x(g)x(f(im000 xxxxxx

+=+ .

5) f g (f(x)g(x))=f (x)g(x) x . .

2o

f (3,2) (5,9) -: )

) f(3+f(x2+2x))=9 f(3x-1)-2

116

3

f:(0,+ ) z1=2+i f()

z2= i)(f

112 +

0

117

118

119

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