-
-
- 1 -
-
-
- 2 -
.
-
-
- 3 -
-
-
- 4 -
, ,
.
1 !
2 ( )
.86: ( C )
.87: ( ,
)
.88-90: C. 2 .89
.90: ( )
.90: ( i )
.91: ( )
.91: : ( 1 2 1 2z +z = z +z )
.92: ( z2 + z + = 0)
.93:
.97: ( )
.97:
.98:
.98: : 1 2 1 2
z z = z z
.99: : 0 1 2
z - z = , > 0 z - z = z - z
.124-5:
1 ( )
.133: ()
.141: ( )
.142: ( )
.143: ( )
.149: ( , )
.150: ( )
.151: ( 1 -1)
.152: +
.153-154: ( )
.155:
.160:
.161:
.162:
-
-
- 5 -
.165: 1
.166:
.167: : 0
0
0x x
x x
0
0
limP(x) = P(x )
limP(x )P(x)
Q(x) Q(x )
.169: ( )
.170: ( )
171: ( )
.173:
0 0x x u u
limf g(x) = limf(u)
.178:
.179:
.183: (
)
.184:
.185:
.186:
.188: ( x0 )
.189: f .
.
.190:
.191:
.192: Bolzano (
)
.192: ( )
.194: ( )
.194: +
.195: ( ) +
.196: ( )
.201-03:
2 ( )
.212:
.213: ( f x0 )
.213: + ,
.214:
.217: ( )
.218:
.222: . f .
-
-
- 6 -
.223: ( c ) =0 (x )=1
.224: : (x ) = x - 1 , 12 x
.226:
.229: ( )
.230: ( ) +
.231: ( )
.231: (x - )=-x - - 1
.232: (x )=1
2 x
+
.234: + (x )= x - 1 ( x ) = x ln
.235: 1 ln x
+
.241: ( )
.241-242: . . . x0
.246: Rolle +
.246: .. +
.251:
.251:
.252:
.253:
.254:
.258: ( )
.259: ( ) -
.260:
260-1: (Fermat)
.261: ( - )
262:
.264:
.273: (-)
.274:
.275: ( )
.275:
.276:
.279: ( )
.280:
.281:
.282:
.283:
.287:
.295-9:
-
-
- 7 -
3 ( )
.303: ( )
.304:
.329-330:
.330:
.332:
.334:
.334-5:
.336:
.337:
.342-345:
.346:
.348:
.354-9:
-
-
- 8 -
.
-
-
- 9 -
-
-
- 10 -
, , , ,
2 .93-94 2 . 99-100
8/95, 3/96, 4/96, 7/96
- Vieta 14A/96
z: z :
11/96, 6/96, 8/96
9/101, 1/101, 7/102, 10/102
12/96, 9/97, 4/101, 5/101, 6/101, 8/101, 2/101, 3/101, 4/102,
5/102, 6/102, 9/102,
1/123, 6/103
7/101, 8/102, 3/123
, , , ,
2 .226-227 , 3 . 247-248, .252, 2 .254-256 , 3 . 265-267,
.335-336 1 2 .346-347
- 6/148, 2/156
+ 2/176, 4/176, 3/182, 4/182, 1/187, 3/187,
4/187, 3/102, 1/285, 2/286
2/199, 3/199, 6B/286
Bolzano + . +
4B/199, 5B/200,8B/200
6/200, 4/257
7/200
9/200
x0 3A/220, 2B/220, 4B/220, 6B/221, 7B/221, 8B/221,
1B/228, 5B/286, 7/240
2/228, 3/228, 4/228, 5/238, 7/239,
10/239, 11/239, 1/240, 2/240, 3/240, 4/240, 6/240, 8/24011/241,
12/241
12/239, 14/239
1/244, 2/144, 4/244, 5/244
.Rolle .. 3/249, 1/249, 3/250, 4/250, 5/250, 6/250,
7/250
+ +
1/256, 1/257, 11/293, 4/308, 1/308, 3/309, 4/309, 11/351
2/257, 6/257, 2/291
7/258, 8/258, 3/269
6/256, 7/ 256, 5/257, 2/267, 1/269, 2/269
, 8/268, 4/269, 6/270, 8/270
-
-
- 11 -
. Fermat 5/268, 5B/270, 7/292
- 3/278, 4/278, 5/279, 8/292
+ +
1/338, 7/339, 8/3399/339, 3/338, 4/338, 6/339, 11/340,
12/340
6/352
5/338, 1/339, 2/339, 3/339, 4/339, 5/339, 6/339, 5/352
10/353
3/349, 5/349, 1/349, 3/350, 5/350, 8/351,
9/351, 10/351, 12/3518/351,9/351
-
-
- 12 -
. -
-
-
- 13 -
-
-
- 14 -
-
1
+ i + i .
2
+ i + i .
3
z , : z = (z)
4
z , : 2 z z z
5
z , : 22zz
6
z = 3i 7 z =3i+7 .
7
z1 , z2C 2 21 2z +z =0 , : z1 = z2 = 0.
8
z , : z = z
9
z , z z 2Im(z)
10
z , z z 2Re(z)
11
iz z , z
12
.
-
-
- 15 -
14
z1 , z2 , :
1 2 1 2 1 2z - z z - z z + z
15
: 1 2z - z = z - z
12 , 1 , 2 z1 , z2 .
16
: 0
z - z = , > 0
(x0 ,y0 ) z0 .
17
z = 0 z = 0 , z C
18
1 2 1 2 1 2z = z z = z z , z C
19
zC, 2014 2014iz z
20
z z , zC .
21
f , g g f fg,
22
f , g , h h (g f ) , (h g) f h (g f )= (h g) f
23
f : R 1 1 , x1 , x2 A : f (x1 ) = f (x2 ) , x1 = x2
24
f 1-1 y f (x) = y x.
25
f 1-1 .
-
-
- 16 -
26
, 1 -1.
27
f .
28
f : R. 1 ( ) , f f x x x A
29
f : R. 1( ) = , ( ) f f y y y f A
30
f f -1 y = x.
31
f f - 1 y = x.
32
xf(x) = 10
g (x) = logx.
33
1 -1 .
34
f 1-1 , f (x ) = 0 .
35
f : . 1( ) = , f f y y y A
36
0 0
( , ) ( , )x x l .
: 0 0
x x x x
limf(x) l lim(f(x) l) 0
.
37
0
lim ( ) 0
x x
f x , f (x ) > 0 x0 .
-
-
- 17 -
38
0
lim ( ) 0
x x
f x , f (x ) < 0 x0 .
39
f (x ) < 0 x0 0x x
lim f(x) 0
.
40
f x0 0
lim ( ) 0
x x
f x ,
0
lim ( ) 0
x x
f x .
41
f g x0 :
f (x )g(x ) x0 , 0 0
lim ( ) lim ( )
x x x x
f x g x .
42
0
lim( ( ) ( ))
x x
f x g x ,
0
lim ( )x x
f x 0x x
lim g(x)
.
43
:. 0
1lim 1
x
x
x
44
: lim 1
x
x
x
45
0
lim ( )
x x
f x , f (x )>0 0 .
46
0
lim ( )
x x
f x , f (x )
-
-
- 18 -
50
0
lim ( ) 0
x x
f x f (x )>0 x0 , 0
1lim
( )
x x f x
51
0
lim ( ) 0
x x
f x f (x ) 1 : lim
x
56
11 1 0 (x)= ... , 0 : lim ( ) lim
x x
x x x a x x
57
0lim ln
xx
58
0
1lim ln
x x
59
x 0
7xlim
x
= 7.
60
f () f .
61
f () f .
-
-
- 19 -
62
f
( ) 0f x x f ()>0 .
63
f f .
64
f [ , ] [ m , M ] m .
65
f [ , ] f () f () > 0 f ( , ) .
66
f [ , ]
x0 ( , ) f (x0 ) = 0, f () f () < 0.
67
f x0 , x0 .
68
f x0 g f (x0 ) , g f
x0 .
69
f x0
g x0 , g f
x0 .
70
f f .
71
f x0 , x0 .
72
f x0 , x0 .
-
-
- 20 -
73
f Bolzano , f .
74
f x0 , f x0 .
75
f , g x0 ,
f g x0
0 0 0
( ) ( ) ( ) ( ) f g x f x g x .
76
f , g x0
0
( ) 0g x , f
g x0
:
0 0 0 0
0
0
( ) ( ) ( ) ( )( )
( )
f x g x f x g xfx
g g x .
77
0x 1
ln xx
.
78
: 1
(7 ) 7
x xx , xR.
79
f R , [ , ] , f Rol le .
80
f [0,1] ,
fC ,
0, (0) , 1, (1)f f .
81
f , f
.
-
-
- 21 -
82
2 .
83
f ( , ) x0 , f . f (x ) ( , x0 ) (x0 , ) , f (x0 ) f ( , ) .
84
f , . f , f (x ) < 0 x .
85
f x 0 . f x 0 f (x 0 )=0, f x 0 .
86
f x
. f ( ) 0 f x
x .
87
f . f (x) 0 x ,
f .
88
f , g . f , g f (x) g (x)
x , f (x ) = g(x )
x.
89
, f 0, f .
90
f [ , ] x0 [ , ] f . f (x0 ) = 0.
-
-
- 22 -
91
f ( , ) , x0 , f . f (x0 )>0 ( , x0 ) f (x0 )0 x , f .
93
f f (x ) > 0 x .
94
C f .
95
f , C f C f .
96
f ( , ) , 0 . f ( , x0 ) (x0 , ) , (x0 , f (x0 ) ) c f .
97
3
23 2f(x)dx f( ) f( )
98
5
5
2 2
17
7dx ln x
x
99
f , ,
( ) ( )( )a
f ff x dx
.
100
f [ , ] R ,
( ) ( ) f x dx f x dx
.
-
-
- 23 -
101
f [ , ] ,
( ) ( )( ) ( ) f ff x dx xf x dx
.
102
f , g R, :
( ) ( )( ) ( ) ( ) ( ) f x g xf x g x dx f x g x dx
.
103
f , ,
:
f(x)dx f(x)dx f(x)dx .
104
f [ , ] [ , ]
f (x ) 0 ( ) 0 f x dx
.
105
( ) 0 f x dx
, f (x ) 0
x [ , ] .
106
f [ , ] . G f [ , ] ,
( ) ( ) ( ) f t dt G G
.
107
f(x)g (x)dx f(x)g(x) f (x)g(x)dx ,
f , g [ , ] .
108
f , g , g [ , ]
, ( ) ( ) ( ) ( ) f x g x dx f x dx g x dx
.
109
f , :
( ) ( ) ( ) x
f t dt f x f
x.
-
-
- 24 -
110
f , :
( ) ( ) x
f t dt f x
x.
111
f
, ( ) ( ) ( ( )) ( ) g x
f t dt f g x g x
.
112
( ) f x dx
xx xx.
-
-
- 25 -
-
-
- 26 -
.
-
-
- 27 -
-
-
- 28 -
-
-
- 29 -
-
-
- 30 -
..1. A1.A f ' x0
,
f (x0 , f (x0 ) ) .
2. , f '
x0 ,
.
3 .
.
. f x0 , f
x0 .
. f x0 , f
x0 .
. f x0 , f
x0 .
4 .
x0 .
. f (x )=3x 3 , x0=1
1. y=-2x+
. f (x )=2x, x 0=
2 2. y=
1 4
x+1
. f (x )=3 x , x 0=0
3. y=9x-6
. f (x )= x , x 0=4
4. y=-9x+5
5.
-
-
- 31 -
..2. 1. Fermat.
2. f ,
' 0f x . f .
3 .
.
1. :f A
, f .
2. 0
lim 0x x
f x
, f x 0x
0
lim 0x x
f x
.
3. f 0f a f 0f x
,x a , f , . 4. f , g
' 'f x g x x , .
5. f ,
: .
6. f ,g
,x , a
f x dx g x
.
..3. 1. ,f g .
,f g
' 'f x g x , c , x
: f x g x c
2. , 0, 1vf x x v
: .
3 . 1 2,z z .
() () .
. 1z 2z
.
. : 1 2 1 2z z z z
. :
f x x
f x g xx
f x dx f x dx ,a
f x g x
1' vf x v x
1 2 1 2 1 2z z z z z z
-
-
- 32 -
. 1 2z z z z 1 2z z
1z 2B z .
4 . 0
x
F x f t dt , f
.
.
:
.
.
. 10F
..4. 1. f 0x
. f
0x , :
0' 0f x . 2 . x
f ;
3 . f
, ;
4 .
() () ;
1. 0
limx x
f x l
00
lim
h
f x h l .
2. 0 1a lim 0xx
a
.
3. f , f
f a f . 4. f g
,a : ' 'a
f x g x dx f x g x dx f x g x
5. f x ,
f x x f 1-1 .
36 . .
0F
4F
-
-
- 33 -
f [, ] . G
f [,] , ).()()( aGGdttf
..5. 1. f
0x .
2.
)(, 00 xfxM f .
3 .
.
) 02
22
1 zz 1, 2z z C .021 zz
) axg )( 0x axgxx
)(lim0
( )y aim f y l
0
( ( )) .x xim f g x l
) f [, ] f () , .0)(' f
) f , .0)('' xf
) f [2,5] 0)( xf
[2,5] , .0)(
2
5
dxxf
..6. 1. f ,
, . :
f ,
f f
, f f ,
0 ,x a , 0f x .
2 . 0x x
f ;
3 .
Rolle .
4 .
() () .
-
-
- 34 -
1. z 2z z Re z .
2.
xxim e .
3. : : f A R : g B R ,
f
g, .
4. f 0x
, 0x .
5. ' 'f x g x dx f x g x f x g x dx
', 'f g
, .
..7. 1. , f
' x0 ,
.
A2. f
x0A;
A3. f (x ) = x , >0
R f (x ) = x ln.
A4.
:
i . 1 -1
.
ii . 4v 3i i , .
iii . 0x xlim f (x) 0
, f (x )>0 x0 .
iv. x , y y =
f (x ) , f x0 , o
y x x0
y = f (x0 ) .
v. f ,
x0 , f (x0 )0 ,
f .
..8. 1. f,
.
. f (x)0 x ,
f .
-
-
- 35 -
. f (x)0 x ,
f ;
2 .
.
. f (x) =e1 - x
.
. f f (x) = -2x+2
1
x + 3, x
2,)
.
. f (x) = g (x ) + 3 x, h(x)=f(x)-
g(x ) .
3 .
f -2,6 .
f
.
..9. A1. z1 , z2 .
: z1 z2 = z1 z2 .
2. ,
.
z :
-2 1 3 6x
y
-
-
- 36 -
. 2
z z z . 2 2 z z . z - z . z z .
i z z
3 . 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
4 . z z 1,
1 z
z .
..10. A1. f(x )= , >0
R xR f (x ) = ln .
A2. f, .
f .
-
-
- 37 -
A3. ,
, , ,
.
. z = + i , , z z =2
. f
x 0 A () f (x 0 ) , f (x ) f (x 0 ) xA
. f ,
1 -1 .
. 0x xlim f(x) 0
f (x )>0 x 0 ,
0x x
1lim
f(x)
.
. f x 0
.
..11. A1. f x 0
. f
x 0 ,
: f (x 0 ) = 0.
A2. f . y=x+
f +;
A3. ,
, , ,
.
) z 0 z 0=1
) f:A 1 -1,
x 1 , x2A : x 1x2 ,
f (x1 ) f (x 2 )
) x 1= {x/x=0} : 21
x x
.
) :x
xlim 1
x .
) C C f f - 1
y=x
-
-
- 38 -
xOy xOy.
..12. A1. f . f (x ) > 0 x , f . A2. f [, ] ; A3. f . f x 0A
; A4. , , , , , , . . . f 1 -1, y f(x)=y x .
.
0x x
lim f(x)= , f (x )
-
-
- 39 -
. .
..14. A1. f
(, ) , x 0 , f . f (x)>0 (, x 0 ) f (x)
-
-
- 40 -
..16. 1. f x x , R Z .
f (0 , +) :
-1f ' x x .
2. f , g
: f (x)=g (x) x .
c :
f x g x c x .
3. f .
: f .
4.
.
) f :A 1f ,
f .
) f 0x ,
.
) f
, 0x , 0f ' x 0 . ) f
. f '' x 0 x .
) f , f x 0 ,
.
) f
, , f
.
..17. A. vf x x ,v IN 0, 1 .
f v 1f ' x v x .
B. f ,
,
0x
0 0f x
0f x x 0x
0f x dx
0 ,x a 0 0f x
0f x dx
-
-
- 41 -
.
, 1 2 3I , I , I
.
.
.
1. x 0
xlim
x
2. x 0
1lim x
x
3. x 0lim ln x
4. xx
1lim
e
.
. 0
. 1
.
..18. A. f , g x0 ,
f + g x0 :
( f + g ) (x0 ) = f (x0 ) + g (x0 ) .
. f .
f ;
3
10
I f x dx
3
20
'I f x dx
3
30
''I f x dx
-
-
- 42 -
.
.
1. f : R 1 - 1
1 2x ,x 1 2x x 1 2f x f x .
2. 0 0x x x x
lim f x lim g x
f x g x 0x .
3. f ,
, .
4. f ,
, 0f ' x 0 .
5.
f x dx 0 f
, .
..19. 1. f
0x . f
0x ,
: 0f ' x 0 .
2.
f ;
3.
.
. f :A 1 2x , x
: 1 2f x f x 1 2x x .
.
.
. 0x x
lim f x
f x 0 x
0x .
. f
,
f()0 x .
. 0f x dx
0f x
,x a .
0 ,x a 0 0f x 0f a f
0 ,x a
, 0f x ,x a
0x x
0
limx x
f x g x
0
limx x
f x
0
limx x
g x
-
-
- 43 -
..20. A.1 f ,
. :
f (x )>0 x , f
.
f (x )
-
-
- 44 -
,
.
A. Re(z)
. Im(z)
. -z
. z
. z
. z z
1. - - i
2. - i
3. +
4.
5. 2 2
6. 2 + 2
7.
..22. A1. f ' [, ] .
G f [, ] ,
f (t) dt G() G() .
2 . f .
f ;
3 . ,
.
. f [, ] (,
] , f [, ] .
. , 1 -1 ,
.
. f x0 0x x
lim f (x)
=0,
x x0
lim f(x) 0 .
-
-
- 45 -
. f R ,
f (x)dx xf (x) xf (x)dx , f [,] .
. x x
0
lim f(x) 0 ,
f (x ) > 0 x0 .
..23. 1 . , f
x0 , .
2 .
;
3 . ,
.
. z z ,
z z z .
. f
. f (x)>0
x , f .
. f ,
[ , ] ,
f (x)dx f (x) .
. f ,
f
.
. f x0
. f x0
f (x0 )=0, f
x0 .
..24. 1 . f .
F f , :
-
-
- 46 -
. G(x) = F(x) c ,c R
f
. G f
G(x) = F(x) c ,c R .
2 . ,
.
. z1 , z2 ,
1 2 1 2 1 2 z z z z z z .
. f ' (, ) ,
x0 , f
.
f (x ) > 0 (, x0 ) f (x) < 0 (x0 , ), f (x0 )
f .
. f : R 1-1 ,
x1 , x2 A : x1 = x2 ,
f (x1 ) = f (x2 )
. f , g ,
:
f(x) g (x) dx f(x) g(x) f (x) g(x) dx .
3 . y =
f + ;
..25. 1. f ' x 0
. f
x 0 ,
f (x0 )=0.
2 . f
x 0 ;
-
-
- 47 -
3 .
.
.
.
. 0x x
lim f (x) l
, 0x x
lim f (x)
0x x
lim f (x) l
. f , g x 0 ,
f g x 0 :
( f g) (x 0 ) = f (x 0 ) g(x 0 ) .
. f,
. f (x)>0 x , f
.
. f [,] . G
f [, ] ,
f(t)dt G() G() .
..26. A1. z1 , z2 , :
1 2 1 2z z z z .
2 . ,
.
. f x 0
, .
.
.
. f , g IR
f og gof ,
.
. C C f f 1
y = x
xOy xOy.
. f x 0 , 0 0
kk
x x x xlim f(x) lim f(x)
,
f (x ) 0 x 0 , k k 2.
-
-
- 48 -
3 . f
(, )
[, ] .
..27. 1. + i , +i ,
, , , IR +i 0, :
2 2 2 2
i i
i
.
2 . ,
(
) .
I
. i1
B. i2
. i3
. i4
1. i
2. + 1
3. i
4. 1
5. 0
6. 4
, .
3. 1, 2, 3 4,
, ,
( ) , , ( ) ,
.
1. f , g .
f , g f (x) = g (x)
x , c , x
: f (x ) = g(x) + c.
2. f
, x 1 , x2
x1 < x2 : f (x 1 ) < f (x 2 ) .
-
-
- 49 -
3. f(x ) = x . H f
(0,+) 2
f (x)x
.
4. , ,
(x0 , f (x 0 ) ) , C f f,
x 0
= f (x 0 ) .
..28. 1. : 1 , 0,2
x xx
.
A2. f ;
A3. ,,,
() , , () ,
.
1. 1 (, ) 2 (, ) + i + i
,
+i +i
.
2. z = - + i, , IR,
z = - i .
3. f(x ) = x , x IR . H f
f (x ) = x.
4. f R * .
f R *
f (x ) = 0 x R * ,
f R * .
5. f,
. f (x ) < 0 x , f
.
-
-
- 50 -
..29. 1 . : 1 * * , x x x R .
2 . y = x +
f +;
A3. ,
.
. f [, ] f () < 0 (,
) f () = 0, f() > 0.
. 0x x
lim f(x) g(x)
0x x
lim f(x)
0x x
lim g(x)
. f f - 1
f y = x,
f - 1
.
. 0x x
lim f(x)
= 0 f (x ) > 0 x 0 , 0x x
1lim
f(x)
. f
, x f(t) dt f(x) f()
x.
. f
, x
x ,
.
..30. 1 . (x ) :
0
0lim ( ) ( )x x
P x P x
.
2 . f: A IR 1 -1;
-
-
- 51 -
A3. ,
.
. , f
0,
f .
. f (, )
x o . f (, x o )
(x o , ) , (x o f (x o ) )
f .
.
.
. f , g fog gof,
fog gof.
. z , z
xx.
. f [ , ] IR,
:
f(x)dx f(x)dx .
..31. 1. i 1 , i , -1 , - i .
2. (x,y) z = x+yi
. z;
3 .
( ) ,
, () , .
1. f : R. 1 -1,
x 1 , x2 :
x1 x2 , f (x 1 ) f (x 2 ) .
-
-
- 52 -
2. f
x A () , f (x ) , f (x ) < f (x ) xA.
3. f , g x f (x ) g (x )
x , 0x x
lim f(x)
> 0x x
lim g(x)
.
4. z1 z2 , 1 2 1 2z z z z .
5. f [, ]
(, ) ,
, (, ) , : f () = f() - f()
.
..32. 1. : ()= 1.
2. R.
;
3 .
() ,
, () , .
1. z = x+yi , x, y R. , : z z .
2. z = +i, : z z , , R .
3. x 0, 2x 0
1lim
x .
4. f(x ) = x. f
R 1
= R. {x / x = 0} :
2
1f (x)
x .
5. f x0
R, :
o ox x x x
lim k f(x) k lim f(x)
k R .
..33. A1. f .
F f , :
-
-
- 53 -
:G(x)=F(x)+C, C R
f G f
: G(x)=F(x)+C, C R .
2.
.
.
f (x)dx = . . . . .
f (x) g(x) dx = . . . . .
f (x) g(x) dx = . . . .
, R f ,g [,] .
3 . :
. 1
x
0e x dx
. 2
4
1
3x dx
x
.
2
02x 3x dx
..34. A1. f
( , ) , x0 , f
. :
f (x ) > 0 ( , x0 ) f () < 0 ( 0 , ) , f (0 )
f .
2 . f .
f ;
3 . ,
.
. z1 , z2 , : 1 2 1 2z z z z .
-
-
- 54 -
. f , g x g(x )0,
f
g x
:
o o o oo 2
o
f f(x )g (x ) f (x )g(x ) x
g g(x )
.
. x0 1
ln x x
.
. f:R 11,
y f(x)=y
x.
. f [,]. G
f [, ] ,
f(t)dt G() G() .
..35. A1. f ( , ) , x0 , f . :
f () ( , x0 )(0 , ) , f (x0 ) f ( , ) . 2. .
3 .
,
, , .
1. f x 0 .
f (x )0 x. 0x x
lim f(x)
0x x
1lim
f(x) .
2. , .
(, ) (,) z i
z i .
3. f
x 0 , x 0 .
-
-
- 55 -
4. f(x) x = [0, +),
1f (x)
x x (0, +).
5. 0x x
lim f(x)
,0x x
lim f(x)
+ ,
0x x
f .
6. f , g .
f , g f (x ) = g (x)
x , c , x
: f (x ) = g(x) + c.
..36. A1. z1 , z2
, :
1 2 1 2z z = z z .
2 . f , g ;
3 . y
f +;
A4. ,
, , , ,
.
. f [, ] x
[, ] f (x ) 0
f(x)dx 0 .
. f
x .
f f(x) > 0
x .
-
-
- 56 -
. f x 0 g
x 0 , gof x 0 .
. f
, g(x)
f(t)dt=f g(x) g (x)
.
. > 1 xxlim 0
.
..37. A.1 f
x0, .
.2 f ;
. ,
.
. f ()
f .
. f, g, g [, ] ,
f(x)g'(x)dx f(x)dx g'(x)dx .
. f
,
/x
f(t)dt f(x)
x.
. f
(, ) ,
(,) = x lim f x
=
x lim f x
.
. f , g .
f , g f (x) = g(x)
x , f (x ) = g(x) x .
-
-
- 57 -
..38. 1. f x x , {0,1} .
f R
1f x x .
A2. N f
.
A3.
( ) ,
, () , .
1. z z z z .
2. f 1 -1,
( xx)
.
3. f x0
R 0x x
lim f x 0
,
f (x )
-
-
- 58 -
1. +i +i
.
2. f ,
xx, f .
3. f, g, h h (g f ) ,
(h g) f h (g f ) = (h g) f .
4.
2 .
..40. A1. f (x) ln x , x IR*
IR* : 1
ln xx
.
A2. f
[, ] ;
3 . ,
, , , ,
.
. f:A IR 11,
f - 1
: 1f (f (x)) x , x A
1f (f (y)) y , y f (A) .
. f
f
.
. z 2+z+=0 ,, IR
0 , C
.
. f IR
,
f ( x ) > 0 x.
-
-
- 59 -
. f ,,
f(x)dx f(x)dx f(x)dx .
..41. A1. [, ] .
G f [, ] ,
f(t)dt G() - G() .
2 .
;
3 . ,
, , ,
.
. 11,
.
. f ,
f
,
.
.
f(x)dx
xx
xx.
. , , : +i=0 =0 =0
.
(, x 0 ) (x0 , ) .
: 0 0x x x x
lim f (x) lim (f (x) ) 0
.
..42. 1. z1 = + i z 2
= + i ,
1 2 1 2z z z z .
-
-
- 60 -
2. f x
. f x ;
3 .
,
, , .
1. z1 , z2 , : 1 2 1 2z z z z .
2. x IR : (x) = x.
3. f
, x
x, .
4. f
[, ]
(, )
f () = f ()
, , (, ) , : f ( ) = 0.
..43. 1. f .
f x
f (x ) = 0 , f
.
2 . f x0
;
3 . ,
, , ,
.
. z1, z
2 , 1 2 1 2z z z z
. f
() x0
A, f (x)f(x0) xA.
. x 0
x 1lim 1
x
.
-
-
- 61 -
. f
.
. f [, ]
f (x )
-
-
- 62 -
2. f , g x 0 ,
f + g x0
: ( f+g ) (x0 )=f(x0 )+g(x0 ) .
3.
, , ,
.
1.2 2z = z , z.
2. +i, ,
( ,) .
3. 0
limx
x=0
x.
4. f
[, ] (, ),
(, ) , : f() - f()
f () =-
.
..46. A1. f .
F f , :
G(x) F(x) c, c
f
G f
G(x) F(x) c, c .
A2. x=x0
f ;
A3. f
. f
;
-
-
- 63 -
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
..47. A1. : 1 , 0 x x R Z x .
A2. f ;
A3. f
x0A () , f (x0 ) ;
4. ,
, , ,
.
) f(x ) = x , > 0, ( x ) =x x 1 .
) fog gof,
fog = gof
) 0x x
lim f (x)
, 0x x
1lim 0
f (x) .
-
-
- 64 -
) f [,]
f (x ) 0 x [,] ,
f(x)dx 0 .
) zC |z|2 =z z .
..48. 1. , f
x 0 , .
2. f (x0 , f (x0 ) ) C f . C f ;
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)
-
-
- 65 -
2. f
;
3. ,
, , ,
.
) ,,, : +i=+i = =
) f f
C f , xx,
, xx,
C f , xx.
) f , g x o , f (x )g(x)
x o , : 0 0x x x x
lim f (x) lim g(x)
.
) f , g x o g(x o )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)
,
.
..50. A1. f
x0 . f
x0 ,
: f (x0 ) = 0.
A2. f R . y= x+
f + ;
-
-
- 66 -
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
xlim 1
x .
) C C f f 1
y=x
xOy xOy .
..51. A1. f .
f
f (x ) = 0 x ,
f .
A2. f
. f
;
A3. f A. f
x o () , f (x o ) ;
A4. ,
, , ,
.
) z z z = 2Im(z)
-
-
- 67 -
) ox x
lim f (x)
, ox x
1lim 0
f (x)
) f () ,
.
) f , ,
,
f (x)dx f (x)dx f (x)dx
) f
. f
,
.
..52. 1. f
(, ) , x 0 , , f
. f (x) (, x 0 ) (x0 , ) ,
f (x 0 ) f
(,)
2. Bolzano.
3. f .
f ;
4. ,
,
, , , ,
.
) |z z0|= , >0
(z0 ) , z0 , z .
) f
(,x 0 ) (x0 ,)
0x xlim f(x) = (
0x xlim f(x) =
0x xlim f(x) = )
) 0 < < 1 ,
x
xlim =0 .
) f
. f
, f (x ) > 0 .
-
-
- 68 -
)
g(x)
f(t)dt f g(x) g (x)
.
-
-
- 69 -
-
-
- 70 -
-
-
- 71 -
-
-
- 72 -
..1. z 2 1 3 2 i , R. 1 . z
.
2 . 3 z 2 i 7 .
3 . z
.
..2. z
: .z 6 3i 8
f (z )= 6 2 z i , z :
1. f (z ) .
2. f (z ) .
3. z.
..3. :
49 41
1 33
1 i 2 1 iz
4 1 i
.
1. , R : 3 31 z i .
2. z2 = + i ,
1 .
z , : 1 2
z z z ,
.
..4. f [0,] ,
0f(x)dx 2 F f .
1. F(0) - F () .
2. (0,) f ( ) = .
..5. f (x ) = x-x , 0
-
-
- 73 -
[0 , 1] .
ii ) (0,1) 1
f(x)dx f() .
..8. f [0 , 1]
(0 , 1) f (0) = 2011 f (1) = 0.
) x0 (0 , 1) , f (x0 ) = 2011x0 .
) x1 , x2 (0 , 1)
f (x1 ) f (x2 ) = 22011 .
..9. f [1,3]
f (x) x -2
x (2,3) .
..10. f : RR
: 2 xf x 2e f x , xR.
..11. f
[0,4] , f (0) = 5 f (4) = 1.
1. f .
2. f (x ) = ,
[1 , 5] .
3. (0 , 4) : (1) 2 (2) 3 (3)
( )6
f f f
f .
..12. f ,g (0,+ )R
x > 0 :
f (x ) = ( )
1
g t
xe dt g(x) =
( )1
f t
xe dt .
1. f , g .
2. h (x) = e - f ( x ) x , x>0
f .
3. : ( )
limx
f x x
x
,
0
( )lim
x
f x x
x
.
..13. f R , : 2014
40
1996f (x) [f(x)] dx = 0 .
f (x ) = 0
-
-
- 74 -
(1996,2014).
..14. f R x :
e x + x
0f(t) dt - 2 x - 1 0.
R c f
.
..15. f [0,2],
f (0) = f (2).
[0 , ] f ( ) = f ( + ) .
..16. :
i ) 1 0
lim tx
x tdt
e ii )
0lim tx
x tdt
e
..17. :
i ) F(x)=2
1
2
x x t
dtt
. ii ) G(x) =
2
2 32
1 4
x
xdtt .
iii ) 12
3x H(x) dt1
lnt.
..18.
f , g: [0 , 1] R , f (x)0 x[0 , 1] .
(0,1) : 0 1
( ) ( ) f t dt g t dt
.
..19. f g
R : f (x ) g (x) = x xR.
, g < 0 < :
) f (x ) = 0 ( , ) .
) f (x ) = 1 ( , ) .
..20. : 31f(x) x x2
.
i ) f
1f .
ii ) : 1 .( ) 64 f x iii ) f - 1 ,
: 1f (1).
..21. f :RR
f (x)
f (x)e dx 0, , R < .
-
-
- 75 -
:
i ) f () = f () .
ii ) f (x) = 0
(,) .
..22. f R
f (1) = 0 f (1) 0. z :
( ) 1 ( ) f xe z f x x R .
z .
..23. : 4
f(x) 2x , x 0.x
i ) ( ) f , xx
x = , x = +1, >0, ()=2+1+4 ln(1+1
) .
ii ) ()
.
..24. : [ , ] ,f [,] ,
(,) f () = 2 , f () = 2 .
i ) f(x) 2x
( , ) . ii ) 1 , 2 ( , ) :
f (1 ) f (2 )=4.
..25. f R '(0) 1f
: x x
0f(t)dt x e , xR.
f (0, f (0) ) .
..26. : [0,1] (0, )f
f (0)=1, f (1)=2 : =2
1
0
f (x)dx
f (x) f(x)
.
..27. : e
f(x) lnx xx
.
) f .
) : 1f (x) x .
-
-
- 76 -
..28. f R
f (0) = 2 xR , : x 2 .e 1 f(x) ln x 1 4
f x0 = g (0) , : g (x) = x
0xf (t)dt .
..29. f f (1) = 1,
f (x ) > 0 1
0f(xt)dt 2004f(x) x[0 , +) .
..30. f :[,] R f (x ) > 0
f (x)2004f(x)
f(x) .
f () = 2004 f () I =
f (x )dx .
..31. : 21x
2f(x) e x .
( , f ( ) ) ,
(1 , 2) , C f , C f ( , f ( ) ) ,
: x + 2y = 1.
..32. z2 + z + = 0 , , R ,
z1 = 3 + 2i z2 , :
) , , z2 .
) f (z )= z-z1 + z-z2 ,
zC.
..33. f [ , ]
2
1z f() i 2
2z f () i .
2 1 2 1z z z z [ , ]
f ( ) = 0.
..34. f , g [ , ]
( , ) g(x) g (x) 0 ( , ) .
z1 = f () + ig () z2 = g () + i f ()
: 1 2 1 2
z z z z (,)
f () f()0
g () g() .
..35. (z 2) z 0 , 1z .
Re(z1 ) = -1.
-
-
- 77 -
..36. x0 > 0
, 1 2
z ex i , z ln x i
.
..37. zC z 1
z 1 2i .
..38. f f(x y) f(x) f(y) 2xy
x, y R
x 0
f (x)4
xlim
.
1. f .
2. f . 3. f (1) f ( 1) 0 .
..39. f R 0
xtf (x) e f (x t)dt
, xR . f .
..40. z 2
z 1 2i2
.
) w 2z 1 i .
) 1 2
w , w )
.
..41. z ( 1) ( 2)i , R .
z
.
..42. : z2 + (-4)z + (+5) = 0. (1) , R .
z1 (1) 1 1z z 2 1z 2 ,
:
1. .
2. z1 z2 .
3. : 100 1001 251z z 2 .
..43. f : R R 3f (x) f (x) x 5 0 x R .
) f
.
) : 1lim ( )xx
e f x
.
-
-
- 78 -
..44. ) z : z 1 2i z 7 2i
. ) , R
2x 5x 10f (x)
x 1
.
..45. : z 3 (2 1)i , R .
) z .
) .
..46. f [ , ]
> 0 ( ) 2 f x dx
.
f (x ) > 1 , x ( , ) :
+ + f(t)dt = xx
( , ) .
..47.
f :R R (2 , 5) , ( -1 , 3).
1. f .
2. f .
3. : f(2x 1) f(5) f ( f (x ) ) = f (5) . .
4. : f - 1 (5) , f - 1 (3) .
5. : f (3+f - 1 (x+1)) = 5.
6. C f (9 , 9)
f - 1 .
..48. : f (x ) = x + xe - x . 1.
fC
(0,f (0))
2x y + 7 = 0.
2. = 1, f
y = x fC .
3. () fC
, y = x x = 0 , x = > 0 , = 1.
4. :
E()lim .
..49. . f
(0,+ ) f (x ) > 0 x >0. f ,
x>0 :e+1
x f(x)f(t)dt =
x.
-
-
- 79 -
. : 2004 2004 2005 2005
1 1 1 1 i i i i .
..50. :
f (x ) =
t
t0
lnx t e dt , x 0.1 + e
) f .
) :x 1
f(x)lim
x 1 .
..51. f :R R
0
x
f (t)dt xf (2004) ,x R
(0,2004) f () 0.
..52. f [1 , 3] .
) f (1) = f (3) , x1 , x2 1 0 xR.
-
-
- 80 -
z w z w xR , : = e.
. R f g
: ( ) ( ) (2 )
( ) ( ) (4 )
g x f x f x
x f x f x
.
g(1)=5 g (2)=7 , (1) .
..58. f 1,0 f 0 f 1 . :
. 1
2
f x f x .
. 1
f x f x3
.
..59. . f [ ,
] x0 ( , ) f (x ) = 0,
: f() f()
x x
0 00 .
. f ,0 f
,0 , x>1 :
f x f xf(x)
1 1
2.
..60. f :R R f (1) = 0,
f (1) 0 z C ( )2 2 ( ) f xe z f x x R .
1. z
.
2. w =4
zz
.
..61. z = 1
x
xi
ey
e x , y R y > 0
z2 .
) y x
) y : R R
y(x) y - 1 (x) .
..62. f R ,
f (x )2 xR.
-
-
- 81 -
2
25
0( ) 5 1 ( ) , .
x x
g x x x f t dt x R
:
. g(-3) g(0) < 0.
. g(x ) = 0 ( -3 , 0) .
( 4 1997)
..63. h: [1 , + ) R
:x h(t)
h(x) (x ) dtt
11999 1 1.
:
. h(x) = 1999x lnx , x 1.
. h [1 , + ) .
( 1 1999)
..64. f R .
2 2 2 4
1
0( ) ( ) 2 ( )
I x f t xt f t x t dt , R
x0 = 2
1
05 ( ) t f t dt.
( 1 2000)
..65. f [ ,]
( ,) z = e f () + 3i w = - f () i .
Re(z - w ) = 2f() ,
x o ( ,) f (x o ) + f (x o ) = 0.
..66. z w :
z 2 z 2 4 2 2
iw 3 2i w 2 3i 2 .
.1 z.
B.2 N w.
.3 | w|.
.4
z , w.
..67. x
x
e 1f x , x R
e 1
.
. f
1f .
-
-
- 82 -
. 1f (x ) = 0
.
. 1
21
2
f x dx .
2 2002
..68. z=+i, , IR
w=3z iz +4, z z.
. Re(w )=3+4 , m(w )=3.
. , w
y=x12, z
y=x2.
. z ,
y=x2,
.
2 2003
..69. . ()
z : z 2
m (z) 0 .
. , z
() ,
1 4 w z
2 z
xx .
2 2003
..70. f f (x )=x 2 lnx .
. f,
.
. f
.
-
-
- 83 -
. f .
2 2004
..71. f: IR IR f (x ) = 2 x + m x 4 x 5 x ,
m IR , m > 0.
. m f (x ) 0 x IR .
. m = 10,
f, xx
x = 0 x = 1.
2 2004
..72. z = + yi , x, y
, IR
:
2 2
z z z zi (1 )i
2 2i
. :
. Im(z) = 0, = 1.
. = 0, z 2 + 1 = 0.
. : 0 1 .
. z
,
.
..73. 1z
, 2z
, 3z
1 2 3z z z 3 .
. : 11
9z
z .
. 1 2
2 1
z z
z z .
. : 1 2 3 1 2 2 3 3 1
1z z z z z z z z z
3 .
2 2005
..74. . 1z , 2z
1z + 2z =4+4i 2 1z - 2z = 5+5i , 1z , 2z .
. z,w
z 1 3i 2 w 3 i 2 :
-
-
- 84 -
i . z, w ,
z = w .
ii . z w .
2 2005
..75. f (x ) =2+(x -2)2
x 2.
. f 1 -1.
. f - 1
f
.
. i .
f f - 1
y = x .
i i .
f f - 1 .
2 2006
..76.
x
x 1
1 ef(x)
1 e
, x IR .
. f IR .
. 1
dxf(x)
.
. x
-
-
- 85 -
.
2 2007
..78. 2
3x, x 0
x
f x
x x x ,x 0
.
. x 0
lim f x 3
.
.
f ' 2
f x
0=0,
= = 3.
. = = 3,
0
f(x)dx .
2 2007
..79. z=( -2)+2i, R.
. z.
. z z 2 1
Rez
.
. z 2 Im(z)0, .
..80. 4
f xx
, x>0.
. : i ) x
f ' xlim
f x ii )
2
x 2
xf xlim
x 2
. N
f (0,0) .
. N
f,
y= -2x+6.
-
-
- 86 -
..81. f, R . A
x0 xf(x )=x+2x, :
. f (0) .
. f (x )
-
-
- 87 -
xR , x0 g .
..85. z w
(i 2 2)z 6 w (1 i) w (3 3i) :
. z .
. w .
. w .
. z w .
2 2008
..86. 11 i 3
z2
z2+z+=0, .
. =1 =1.
. 3
1z 1 .
.
w, : 11w z z .
2 2008
..87. z=(2+1)+(21) i , R.
. .
z,
R.
.
z0=1-i .
. w
2
ow w 12 z , oz
.
2 2009
..88. z
: (2 i)z (2 i)z 8 0 .
-
-
- 88 -
. N
z = x+yi .
. N 1z
2z .
.
2 2
1 2 1 2z z z z 40 .
2 2009
..89. 2
z 2z
z z 0 .
B1. z1
z2
.
B2. z1 2 0 1 0 +z2 2 0 1 0 =0.
B3. w 1 2w 4 3i z z
w
.
B4. w 3 ,
3 w 7 .
2 2010
..90. z 1 , z2
z1 + z2 = 2 z1 z2
= 5 .
B1. z 1 , z2
B2. w
|w z1|2 +|w z2|2 = | z1 z 2|2
w
(x+1) 2
+ y2
= 4.
B3. w 2
2 Re(w) + Im(w) = 0.
B4. w1 , w2 w
2 |w1 w2|=4,
|w1 + w2|=2.
-
-
- 89 -
2 2010
..91. z x yi x,y .
B1. 2z i z 3 , z.
B2. z 2 i ,
w : 2w z z .
B3. z 2 i z iz
uz 1
, : 2010u 1 .
..92. f: [, ] , ,
-
-
- 90 -
1.
z 21
y = x4
.
2.
w (0, 3)
=2 2 .
3. ,
z, w z =w.
4. N
, , u
,
, , , .
2 2011
..95. z w
:
|z - 1|2 + |z + 1|2 = 4 (1)
|w 5 w |= 12 (2)
1. z = 1. 2. z1 , z2
z
|z1 - z2|= 2 , |z 1+ z2|.
3. w :
2 2x y
19 4
|w|. 4. z,w (1) (2) : 1 | z - w| 4. 2 2012
..96. z, z-1
z 1w=
z 1 .
:
1. |z|=1
2. O
41
zz
.
-
-
- 91 -
3.
1 2
1 2
1 1z z 4
z z z1 , z2
z.
4. u,
i
u ui = ww
, w0 x 2-y2=1.
2 2012
..97. f 0,4
24
0
2( )
32 2 f x dx
.
) 0,4
,
f ( ) = .
) 2
( )lim
( )
x
x f x
x .
..98. z
: (z 2) ( z 2 ) + z 2 = 2.
B1.
z , K (2,0) = 1.
, z
, z 3 .
B2. z 1 , z 2
w2 + w + = 0 , w
, , R , 1 2Im(z ) Im(z ) 2
: = 4 = 5
B3. o , 1 , 2
1 .
v :
v3 + 2 v2 + 1 v + 0 = 0 : v 4 .
-
-
- 92 -
..99. z w
2
2x - w - 4 - 3i x = -2 z , x R ,
x = 1
1. z
1= 1,
w (4,3)
2= 4.
2. ,
.
3. z, w
1 : z - w 10 z + w 10 .
4. z
1 , :
|22 3 2| = 5. 2 2013
..100.
22
z + (z + z ) i - 4 - 2i = 0, z .
1. .
2. z1=1+i z 2=1- i ,
39
1
2
zw 3
z
-3i
3.
u
1 2u w 4z z i
w, z1 , z2 2.
2 2014
..101. z w
:
2z iw
2z i,
iz
2
-
-
- 93 -
w
1.
z,
=1
2, M(0, -
1
2) ,
2. z,
1, |w|= 1.
3. z=1
2 ,
w4 + i w7 = 0.
2 2014
..102. ln x
f x , 0x
.
A. f
(1, f 1 ) x y 0 , .
B. = 1:
. f .
. .
. : 1
1
8 .
2 2003
..103. f , g
x .
, 0
0.
. i ) L.
ii )
f g . . g .
. : x .
2 2004
' ' 1, ' 1f x g x f x
2lim
2x
g xL
f x x
4f x g x x
-
-
- 94 -
..104. f x 2 x ln x 2 , x 0 .
) : ln x
f ' x , x 0x
.
) x 0lim f ' x
.
) f
.
)
ln x
g xx
,
.
2 2005
..105. z 1
z iw
i z z i .
) : .
) w ,
'x x . ) w z .
) f , 1f a
z = f () i w=f() i . 0f x
.
2 2006
..106. 2 ,xf x x a e x . f :
) : 2 .
) f .
) :
i . xlim f x
ii . xlim f x .
) 2007f x . 2 2007
..107. f
, 0
1 1, 0
x xf x
x x
, .
. , f .
. , f
'x x
1x
e
2x e
w iz
w i
1z
,
2 2y x
0, 0f
-
-
- 95 -
0 0x .
. f 1-1.
. 1 2 , 2 f x dx
.
2 2008
..108. 1
1,z z Cz
1 2,z z .
:
. 1 2 1z z 3
1 1z .
. 2009 20091 2z z .
. 8
1 10
2
11 0z
z
. f x 0, 1 1 22 1
0 2z z
fz z
1 2
1 1 31
2 2 2f
z z 0 0,1x
0 03 2f x x .
. 1 22 2w z z ,
1z 2z ,
.
2 2009
..109. z, w w
wz
1
21
w ( -1,0)
=1.
) z
(0,0) =1.
) z 1 (1) 321 ,, zzz
(1) :
i ) 2 3 1 31 2
3 1 2
z z z zz z
z z z
.
ii ) 0321 zzz :
31 2
2 3 1
zz z 3Re
z z z 2
.
) () : .01243 yx
w
() .
2 2010
-
-
- 96 -
..110. :f : 3 24 12 1f x x x x ,
x R R 0 1x .
. i . =1.
ii . f .
. :
3xf x
imf x
.
. i . f
0, 1 . ii .
f 'x x .
2 2011
..111. f (x ) = e x - 2 g(x) = lnx+2.
B1. f g g f .
B2. f f - 1 .
B3. x 2e lnx 2 ,
2e ,2
.
B4. : x x
f(x) g(x)lim lim 0
g f (x) f g (x)
.
2 2012
..112. z w
: 2
z z 2 1 i z 3 w=2z i ,
1. z.
z ;
2. z z z
.
3. z
z z 2 Im(z) > 0 , : 2013
z z
2
.
4. w
z w
z (0 , 1) .
2 2013
-
-
- 97 -
..113. z , z1 w z = + i , ,R
, 1 =1 + (2)
+2
|(1 ) ( 1
2)| = 2 | +
1
2|.
1. :
) z
x y + 4 = 0.
) w
x2 + 2y = 0.
2. :
| | 72
4.
B3. ) C
.
)
x y + 4 = 0 C
.
2 2014
-
-
- 98 -
-
-
- 99 -
-
-
- 100 -
..1. ,, z,w,u , .
1. : z w w u z u .
2. . 3. :
. 2 2 2 22 2 . z w z w z w
. z w z 3 .
..2. : ( 1) 6( )
xf x
x
x( -1 , +) ,
R , y = 2 x = -1.
1. f : 2x 6
x 1f(x) , x 1
.
2. G(x ) G (x) = f (x ) , x >-1, (0,2) . 3.
G(x)h(x)
x 1
, x >-1.
..3. : f (x )= . , 1
x
x
t
te t
dt x Re
1. f .
2. f (x)= x + x xR. 3. N f 1 : [0,] [0,] . 4. C f , C f - 1 x =
0, x = = 4..
..4. f [ 0, ] . :
1.
0 0
1f(x)dx f(x) f(-x) dx
2 .
2. x
( - x )x 0
2011 dx =
22011 2011 .
..5. F f :RR
xR : 2 2 22 ( ) ( ), F x F x 0.
: 1. F(0) = F(1) = . 2. f (x ) = 0 R.
..6. z f
R . :
x 0 x 1
zf(x) 3 3 zf(x) 1 1lim , lim
x x 1
R
-
-
- 101 -
[0 , 1] f () = 0.
..7. f : (0, + ) R
lnt f ( t ) t 1.
N :
1. f (1) = 1
2.
2 1
2
1
1
2 ( ) 1 21
( 1)lim
x
x
xf t dt x e
x
3. 2 + 2
1( ) 2 ln
xf t dt x x
(1 ,e) .
..8. f [ , ] ,
z x = x + i f (x ) , x[ , ] .
Imz = Im z , f (x) = 0
( , ) , f ( , ) .
..9. f [ , ] ( , ) , ( , ) . :
z1 = + f() i , z 2 = + ()f i , z3 = + ()f i .
: 1 2 3z z z ,
( , ) : f () = -2.
..10. : F(x)=1
4x2 (2lnx-3)-x(lnx-2)
x > 0. 1. F F(x ) = 0 (0 , + ) .
2. :2x
f(x) x lnx2
23xg(x) 2x
4
: 2004
( )
( )
1
1f x
g xdt
t >0 x >0.
..11. 'f R ,
R : 5
( ) ( ) 5 ( ).
x
xg x f t dt f x
..12. f :R *R 1 -1 f f (x) f(x)
xR * , 0, : 1. f f (x) x 0.
2. f .
-
-
- 102 -
3. f .
4. f .
5. 1f .
..13. :f (0, ) R x
f(x) f(y) fy
x, y > 0. f (x ) = 0 , :
1. f 1-1.
2. :2 2f(x 3) f(x) f(x 1) f(x 1) .
3. f (x) 0 x > 1 f
.
..14. f : R R f (x+f(y)) = f (x+y )+2 x, y R .
:
) f (x ) = x + f (0) .
) f (x ) = x + 2 xR.
..15. f : RR , :
f ( lnx) = x x>0 f (0) = 0. 1. f . 2. : f (x ) > x 2
xR.
3. : 32
1
2f(x)dx .
..16. :f R R 2 2(f f )(x) x (2 1)x x R , .
f 'f () 1
C f ( , f ()) .
..17. f : R R : 1
0
x f(xt)dt f(x) 1 x R .
..18. : f (x) 2x x,x [0,].
1. 1f
.
2. 1f (x) x.
3. =2 1
0f (x)dx .
-
-
- 103 -
..19. 1. : 2 2
1
11
1 1 0
1 1
x
xdy dy
y y
x (0, ) .
2. 2
1( )
1
f x
x.
x = ,
C f , xx x = 1
2 , x = 2
.
..20. i
z xx i
.
1. 0
x
z .
2. :
xlim Im(z) x .
3. Re(z) Im(z) 9z 2 i .
..21. f (0, )
4
3
f (x)dx 1 4
5
f (x)dx 3.
:x 2
x 1g(x) f(t)dt
, >0.
1. g .
2. (2 , 3) f (+2) f (+1) = -4.
..22. 1. , :
1 1
0 0
x (1 x) dx x (1 x) dx .
2. 2004f (x) x(1 x) , 2003g(x) x(1 x)
f , g.
..23. . :
2
2
x
1
x 1
( t)dt
2
(1 x)lim
.
. g(x) = x lnx.
i ) : x
g(x)lim 0
.
ii ) :e lnx
I dxg(x)
2
1.
..24. 1. : xx , x 0e x , x 0f (x) . )
0x 0 .
-
-
- 104 -
)
f x=0 , x=
2.
2. ) :
)
f( ) , f( ) ,f()2 2
.
) f .
) : x
f (x)
xlim
.
..25. : 3 5
= 2i z z i , z +i , , R2 2
.
1. Re() , Im(). 2.
1
y x3
.
3. 3 i .
4. -3
102
.
5. 10
2
z i
x 3y 1 , x 3y 1 . (..)
..26. : x
2
f (x) x xtdt 14 , , ,x 0.
1. , f (x) f (2) fC
M(2 , 6) .
2. , 1.
)
.
)
.
) .
..27. f 3f x ln x 1 x x e , x>-1.
1 . f
f - 1 .
2 . f - 1 (x ) = 0.
3 . 1f ,
-
-
- 105 -
1f e .
..28. f ,g R : 1
1( ) ( )
x
xf t dt g t dt = x2 -2x +1 xR.
C f ,C g f (x ) = 0
1 , 2 1 < 1 < 2 .
1.
i ) H g(x) = 0 ( 1 , 2 ) .
ii ) (1 , 2 ) g() = -2.
2. g R
i ) f .
ii ) H f R ,
x o = ) i i ) .
iii ) N
C f , C g y.
..29. f R
() f
: ( , f () ) , ( , f () ) ( , f () ) .
x0R f (x0 ) = 0.
..30. f
f : (0 , + ) (0 , + ) f (1) = 1 :
1 1xf
x f (x)
x > 0.
1. :f (x) 1 f (x)
f (x) x f(x) ,x > 0.
2. f .
..31. 2( ) 4 , f g x x x x R
g (x) = 2x 1 , xR .
1. f .
2. 2x
1
h(x) t f g (x) dt
Ch (1 , h(1)) .
3.
f g f .
-
-
- 106 -
..32. : f (x ) = x e- x
+ x . 1. R , c f (0 , f (0))
( ) : 2y - 4x - 5 = 0 .
2. f
.
3. c f .
4. f .
..33. f 2
lnxf(x)
x .
1. C f . 2. .
3. , lnx
g(x)x
f .
4.
E()lim
()
C f x = 1 , x = >1 y=0.
..34. f R , f ( 3 ) (x ) > 0
xR. 1. C f . 2. (x1 , f (x1 ) ) (x2 , f (x2 ) ) C f .
..35. f R :
f2 (x ) 4 e x f (x ) = 1 xR f (0) = 2 - 5 . 1. f . 2. f .
3. : limx
f x .
..36. f [2 , 5] .
f (x ) < 0 x [2 , 5] 5
2f (x)dx 6 :
1. 2
xg(x ) f (t ) dt [2 , 5] .
2. 2
50 g(x)dx 18 .
..37. f : R R F
f R. F (x) 0 F(x) = F(2 x ) xR , f (x ) = 0 .
..38. f [0 , 4]
2 2
f 0 7 f 4 7 0 .
-
-
- 107 -
1 . f . 2 . [0 , 4] , :
1 1 12 3 4
2 3 4
9
f f f
f .
3 . f (x ) 0 x [0 , 4]
: 7 2xlim f(3) 1 x 2x 1
.
..39. . f :RR f (x ) 0 xR. f (5) + f (6) + f (7) = 0 , f . . f
R
f (1) = f (7) f (x7 ) f (7x ) xR. f (x ) = 0 (1 , 7) .
..40. f : R R , :
f (1 x ) + 2 = x f (x ) , x R. 1. f R .
2. : 21
( ) ln 1 ef x dx e e .
..41. f R *
: 1
( ) - ( ) xxf x f x e , xR * f (1) = e .
1. f .
2. : 2
1/
31/.
( )
e
edx
f x
x
..42. f : (0 , + )R f (x) 0 x > 0 . :
z1 = f () + i , z2 = 1 1
if ( )
, , > 0 2 2 2
1 2 1 2z z z z .
:
1 . 1 2Re z z 0 . 2 . x0 ( , ) , C f . 3 . f (1) = 3 , :
2006
2005x
3x 2x 1
f 1988 x 3lim
.
..43. 2x x 1
f (x)x 1
.
1 . ()
C f , C f + x = 2 x =
>2.
-
-
- 108 -
2 . E( )lim
.
3 . 3 ,
t = 4.
..44. f f (x ) = x2 ( + )x + , < .
C f
xx.
1. .
2. 1 2
C f xx ,
: 1
2
3
2
E
E.
..45. f : RR , R
: x 0
xf(x) 1988xlim 18
x
2006x 7 x
xlim f(x) lim
x .
1. f (0) . 2. f ( -7) . 3. f y = -x ( -7 , 0) .
..46. . f [0 , 1] .
(0 , 1) : 20
f ( ) tf (t)dt
.
. f [ , ] 2f (x)dx
.
( , ) f ( ) = .
. f R 2003 2005
2002 2004f(x)dx f(x)dx .
(2002,2004) f ( + 1) = f ( ) .
..47. f [1 , 3] .
1 . 1
3
2lim
x
f x
f x f (1) .
2 .
1
ln3
e xf dx
x , :
9x
f x
(1 , 3) .
3 . : 3 4 f x 1 , 2x :
1 2 2 f .
-
-
- 109 -
..48. f :RR ,
:
f (x ) > 0 xR R.
f .
..49. f 0,fD R.
x f x
f xx
0x ( ) : y = 2x e
Cf 0 0 x , y , : 1. .
2. f .
3 . ln f x x x , f . 4. f .
5. 1
ln 02
xx
0,x .
..50. . f [1 , 7]
: f (2) < f (1) < f (7) < f (5) .
(1 , 7) , f () = 0.
. f (x ) = xe - x , xR
.
. f
.
. : 2/ x
1/2 22 e xe dx e
.
( 4 1993)
..51. z = e x + (x 1) i , xR.
1 . : Re(z) > Im(z ) xR.
2 . x0 (0 ,1) ,
w = z2 + z + 2i .
3 . z
.
..52. f : R R ,
: f (x)1
f (x)e 1
xR f (0) = 0.
1 . : x
xf (x) f (x)2
x >0.
2 . f , xx x = 0 x = 2 : > f (2) .
-
-
- 110 -
..53. f (0 , + )
e f (1) = 1
2f x
f xx
x > 0.
1 . (x) = f (x ) e 1 / x
x > 0.
2 . f .
3 . N
3
f xh x
x , xx
x = 1 x = 2.
..54. f , g R
:
) f (x1 + x2 ) = f (x1 ) f (x2 ) x1 , x2 R.
) f (x ) = 1 + x g(x ) xR.
) x 0
g x 1lim
.
. f R.
. f (x) = e x . f )
) .
(New York University)
..55. 1 . f x
2 2
ef x
x
, >1
.
2 . x0 :
2x xe 1
, > 1.
(. )
..56. f 2 .1f x ln x 2x 1 , 0 , 22
f .
f = .
(. )
..57. . N f , xR
: f (x )=2x 3 + 0
( ) u
xe f x u du .
. f R.
) A
-
-
- 111 -
. : 2 21 1 1
0 02 1
x xe e dx e dx e .
..58. f [1 , + )
f (1) = 1 1 < f () < 2008
2007 x > 1.
: < f (x ) < 2008 1
2007
x x > 1.
..59. . : x 1f(x) 2 ln2 g(x) ln 2x .
, (1,ln2) (2,ln4).
. f [ -,] : f (x )1
x ( -,) >0.
f()= f ( -)=- f (0)=0.
..60. f :R R f (1) = 1. A z C
x R 2
1
1 12 5 ( ) 5 12( 1)
txx
z i f t dt z i e dt x .
1. C z
.
2. h
C.
3.
(x) = 1
( )xh t dt , xx
, yy x = 1.
..61. R f , :
f (x )0 R
f(x) dx 0 , , R.
1 . : = .
2 . zC z -1 tz
t e dt 22010
10
zw
z
1
1 .
..62. N
R.
)
3
1
1 (5 5)lim
1
x
z i x z x x
x
)
3 2
1
1 2 8lim
2
x
z i x x
x
-
-
- 112 -
..63. z1 , z2
(z1 + z2 )2 0 0 1 = (z1 - z2 )2 0 0 1 . f (x ) = 1 2xz z , x
R.
N :
1. 1 2 1 2z z z z = 0.
2. f R
3. x o ( -1,1) f (x o ) = 3x o 2 1.
4. f 2z .
..64. f : RR :
f 2 (x ) +2f (x )x = x2 + 2x xR f (0) = 1.
1. :
g (x) = f (x ) + x , xR .
2. f .
3. 0
( ) 1lim
x
f x
x lim ( )
xf x .
..65. f [2,3] , (2,3)
f (x) 0 x (2,3) . :
1. f (2) f (3)
2. (2,3) :
5f ( ) = 2 f (2) + 3f (3) .
3. 1 ,2 (2,3) : f ( 1 ) f ( 2 ) > 0.
..66. f :R R
f (x) = -4x3e f ( x ) xR f (0) = -1.
1. N f .
2. N f .
3. : 20
1
1
my
dxx
ym
f .
..67. f : RR :
(x2 + x +1)f (x) = e x (2x+1)f (x) xR f (0) = 1.
1. N f .
2. N f .
3. : 0 1
11( ) ( )
y ef e x y dx f x dx
e yM
f .
-
-
- 113 -
..68. f : R R
g (x) = 0 1 ( ) 1 x u x
f t dt du e , x R. A g(x ) 0 x R,
:
1. 1
0( ) 1 f t dt
2. x o (0 ,1) 0
0
13 ( ) 1
1
x
f t dtt
.
3. (0,1) f ( ) = 4 - 1.
..69. f f (x ) = 21
1 ( ), 0
x tf tdt x
x x f
(0,+).
1. f .
2. f (x ) = 1 ln
, 0
x
xx
.
3. f .
4. f .
5. C f
x =1, x = e xx.
..70. f f (x) = 4e 2 x xR.
1. N C 1 , C2 f (x) = e 2 x + C1x +
C2 .
2. 0
( ) 2lim
x
f x ( )
2lim
x
f x
x, f .
3. e 2 x 2x 1 0 xR
4. e 2 x 2x = 2x 2 + 1
5. N Cf - , ()
6. Cf , () ,
x =0 , x = -1
..71. f [0,1]
f (x )>0 x(0,1) . A f (0)=2 f (1)=4,
:
. y=3 f '
x0 (0,1) .
. x1 (0,1) , f (x1 )=
1 2 3 4
5 5 5 5
4
f f f f
. x2 (0,1) ,
-
-
- 114 -
f (x2 , f (x2 ) )
y=2x+2000.
3 2000
..72. f ,
R , :
f3 (x ) + f2 (x ) + f (x ) = x3 2x2 + 6x 1 x R , ,
2 < 3.
. f .
. f .
. f (x ) = 0
(0,1) .
3 2001
..73. x , x 1
f (x) x 1 1 e ln(x 1), x 1,2
R.
. x 1
1x1 elim
x 1
.
. R f x o=1.
. =-1 (1,2) ,
f ( , f ( ) )
xx.
3 2001
..74. f , g R .
fog 1-1.
. g 1-1.
. :
g ( f (x ) + x3
