Download - Askhseis Mathimatika g Lykeiou
-
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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