Μεθοδική Επανάληψη_2
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Μεθοδική Επανάληψη Συλλογή Λυμένων Ασκήσεων για Μεθοδική Επανάληψη Ερωτήσεις Θεωρίας – Ενδεικτικές Μεθοδολογίες – Λυμένα Συνδυαστικά Θέματα Κων/νος Παπασταματίου Φροντιστήριο «Αιχμή» Κ. Καρτάλη 28 τηλ. 6948065522 2421302598
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Transcript of Μεθοδική Επανάληψη_2
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/ . 28 . 6948065522 2421302598
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| . 28 .2421302598
2
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...........................................................................................................................................28
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.................................................................................................................................................31
.............................................................................................................................................31
( 0x ) ...................................................................33
() ............................................................................34
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| . 28 .2421302598
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, , . , :
. . . , .
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| . 28 .2421302598
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1.
, :
, , (0) (1) .
i 2 1i
z z a i ,
,a 2. .
i , ,a
3. 1z a i 2z i , , , ,
1 2 z z a i i a
4. i ;
i 0a 0 5.
i i .
1 ,M 2 ,M i
i ,
i i a i ,M a .
1 2 .
6. i i .
-
| . 28 .2421302598
5
i i a i
,a . 1 2 7.
i i i
: 2i i a i i i i i i i i
i
8. i
a i i .
x x .
9. i
i
i
:
2
2 2 22
2 2 2 2
i ii i i i i i
i i i i
i
2 2 2 2
ii
i
10. i:
:
0 1i , 1i i , 2 1i , 3 2i i i i
4 , 4, :
4 4 41, 0
, 11
1, 2
, 3
vi
i i i i i i i i
i
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| . 28 .2421302598
6
11. 1 2 1 2z z z z
1z a i 2z i , , ,
1 2 1 2z z a i i a i a i a i i z z
12. 2 0z z , , , , 0a 0
, , :
2
22 4z
a
2 4 .
2
2
2 2 2
1
4 4 4 2
ii
,
:
22
2 2
iz
a
.
1,22
iz
.
13. , ,z x yi x y
z ,
2 2z OM x y
14. z , 2
z z z
z x yi
:
2
2 2 2 2 2z x y x y
2 2z z x yi x yi x y
2
z z z
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| . 28 .2421302598
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15. 1 2 1 2z z z z
:
22
1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 1 2 2z z z z z z z z z z z z z z z z z z z z z z z z
16. . f
f x A
() x A
y . y f x f x
17. :f A ;
| f y y f x x A f x A .
18. :f A ;
fC ,M x y y f x ,
, ,M x f x x A . 19. ;
f g x A
f x g x
20. f, g f g , f g , f g f
g .
:f A :g B
f g , f g , f g f
g ,f g
-
| . 28 .2421302598
8
f g x f x g x , f g x f x g x , f g x f x g x
f xfx
g g x
f g , f g f g A B
f g , f
g A B ,
x g x , | , 0x x A x B g x .
21. :f A , :g B g f f
g.
g f x g f x f
f x g.
1 |A x A f x B 22. f . f , , , f :
, 1 2,x x 1 2x x 1 2f x f x
, 1 2,x x 1 2x x 1 2f x f x
, 1 2,x x 1 2x x 1 2f x f x
, 1 2,x x 1 2x x 1 2f x f x
23. f, , ; f :
, x : x f x f x
, x : x f x f x
24. f 0x
.
-
| . 28 .2421302598
9
f :
0x () 0f x , 0f x f x x
0x () 0f x , 0f x f x x 25. f; () () f ( ) () f. 26. 1-1;
:f 1-1, 1 2,x x
1 2x x , 1 2f x f x
27. f x ax , 0a 1-1.
1 2f x f x , :
1 2 1 2 1 2ax ax ax ax x x
28. 1-1 ;
1-1 :f A . y f , f
x f x y
:g f A y f
x f x y . g f 1f .
29. 11 1 0...v v
v vP x a x a x a x a
0x
0
0limx x
P x P x
:
-
| . 28 .2421302598
10
0 0
0 0 0 0
0 0 0 0
1
1 1 0
1
1 1 0
1
1 1 0
1
0 1 0 1 0 0 0
lim lim ...
lim lim ... lim lim
lim lim ... lim lim
...
v v
v vx x x x
v v
v vx x x x x x x x
v v
v vx x x x x x x x
v v
v v
P x a x a x a x a
a x a x a x a
a x a x a x a
a x a x a x a P x
30.
P xf x
Q x 0x 0 0Q x
0
0
0
limx x
P x P x
Q x Q x
P xf x
Q x , P x , Q x x 0x
0 0Q x .
0
0
0
0
0
limlim
lim
x x
x x
x x
P xP x P x
Q x Q x Q x
31. .
h x f x g x 0x
0 0
lim limx x x x
f x g x
, 0
limx x
f x
32. f 0x ;
0
0limx x
f x f x
33. f 0x ;
:
) 0x
) 0x , , 0f x
34. f ,a ;
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| . 28 .2421302598
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,a
35. f ,a ;
f ,a
,a
limx a
f x f a
limx
f x f
36. Bolzano.
f, ,a .
f ,a
0f a f
0 ,x a , 0 0f x
: , 0f x ,a 37. Bolzano. f Bolzano
f x x 0 ,x . 38. ;
: f, ,a . :
f ,a
f a f
, f a f
0 ,x , 0f x
: f a f .
f a f .
g x f x , ,x a :
g ,a
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| . 28 .2421302598
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0g g 0g a f a 0g f
Bolzano, 0 ,x , 0 0g x f x ,
0f x 39. f
,a f y
0 ,x . 40. .
f ,a , f ,a
m. , 1 2, ,x x , , 1m f x 2M f x ,
m f x M , ,x . 41. , , f
,a ;
,m M , m . 42. ( )
,a ;
,A B ( ,B A ) limx a
A f x
limx
B f x
43. fC 0 0,A x f x ;
f 0 0,A x f x fC .
0
0
0
limx x
f x f x
x x
, fC ,
0 0: y f x x x .
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| . 28 .2421302598
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44. f 0x
;
0
0
0
limx x
f x f x
x x
.
f 0x 0f x . :
0
0
0
0
limx x
f x f xf x
x x
45. fC 0 0,A x f x , f 0x ;
f 0x , 0f x , f 0x fC
0 0,A x f x .
46. f 0x ,
.
0x x :
00 00
f x f xf x f x x x
x x
0 0
0 0
0
0 0
0
0
0 0
0
lim lim
lim lim 0 0
x x x x
x x x x
f x f xf x f x x x
x x
f x f xx x f x
x x
f 0x .
0
0limx x
f x f x
, f 0x .
47. f x x 0 0x ,
.
f x x . f 0 0x , ,
0 0
0lim lim 1
0x x
f x f x
x x
,
0 0
0lim lim 1
0x x
f x f x
x x
.
-
| . 28 .2421302598
14
48. f :
1) f ; 2) f (, ) ; 3) f [, ] ;
1) f 0x
2) f (, )
0 ,x a
3) f [, ]
0 ,x a
limx a
f x f a
x a
limx
f x f
x
49. f ;
1 . 1x
f x ,
1:f A , x f x
f f.
50. f x c , c
0f x .
0x , 0x x 0
0 0
0f x f x c c
x x x x
.
0
0
0
lim 0x x
f x f x
x x
, 0c
51. f x x , x
1f x
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| . 28 .2421302598
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0x , 0x x 0 0
0 0
1f x f x x x
x x x x
.
0 0
0
0
lim lim1 1x x x x
f x f x
x x
, 1x
52. vf x x , v
1vf x v x .
0x , 0x x
1 2 3 2 10 0 0 00 1 2 3 2 100 0 0
0 0 0
......
v v v vv vv v v v
x x x x x x x xf x f x x xx x x x x x
x x x x x x
.
0 0
0 1 2 3 2 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0
lim lim ... ...v v v v v v v v vx x x x
f x f xx x x x x x x x x x vx
x x
,
1v vx vx
53. f x x 0,
1
2f x
x . 0 .
0x 0, , 0x x :
0 000
0 0 0 0
2 2
0 0
00 0 0 0
1
x x x xx xf x f x
x x x x x x x x
x x x x
x xx x x x x x x x
0 0
0
0 0 0
1 1lim lim
2x x x x
f x f x
x x x x x
, 1
2x
x
f 0 0x
0 0
0 1lim lim
0x x
f x f
x x
, f 0 0x .
54. f, g 0x ,
f g 0x : 0 0 0f g x f x g x
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| . 28 .2421302598
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0x x , :
0 00 0 00 0 0
f x g x f x g xf g x f g x f x g x f x g x
x x x x x x
0 0 0 00 0 0
f x f x g x g x f x f x g x g x
x x x x x x
f, g 0x , :
0 0 0
0 0 0
0 0
0 0 0
lim lim limx x x x x x
f g x f g x f x f x g x g xf x g x
x x x x x x
0 0 0f g x f x g x
55. vf x x ,*v
* 1vf x v x .
*v
11
2 2
1 11v v v
v v
v vv
x x vxx vx
x xx
56. af x x , a 0,
1af x a x
ln lnaa x a xf x x e f x e
ln ln ln 11
ln lna x a x a x a af x e e a x a e x ax axx
57. xf x a , 0,a
lnxf x a a
ln lnxx a x af x a e f x e
ln ln lnln ln lnx a x a a x xf x e e x a a e x a a
-
| . 28 .2421302598
17
58. lnf x x , *x *
1
ln xx
ln , 0ln
ln , 0
x xf x x f x
x x
0x 1
ln lnf x x f x x f xx
0x 1
ln lnf x x f x x f x xx
1 1
1f x f xx x
1
f xx
*x
59. y f x x ;
y x 0x 0f x .
60. Rolle . f :
[, ]
(, )
f a f
, ,a , 0f
, ,a , fC
,M f x x . 61. . f :
[, ]
(, )
, ,a , f f a
fa
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| . 28 .2421302598
18
, ,a , fC
,M f ,A a f a ,B f 62. f
f
0f x x
f
1 2,x x 1 2f x f x .
1 2x x , 1 2f x f x
1 2x x , 1 2,x x f .
1 2,x x , :
2 1
2 1
f x f xf
x x
(1)
, , 0f , , (1), 1 2f x f x .
1 2x x , 1 2f x f x .
, , 1 2f x f x . 63. f, g
f, g
f x g x x
c , x : f x g x c
f g x
0f g x f x g x . , , f g . , c , x
f x g x c , f x g x c . 64. f , .
0f x x f
0f x x f
0f x .
1 2,x x 1 2x x . 1 2f x f x .
-
| . 28 .2421302598
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1 2,x x f .
1 2,x x , :
2 1 2 1 2 12 1
f x f xf f x f x f x x
x x
0f 2 1 0x x , 2 1 0f x f x , 1 2f x f x
0f x . 65. f.
f, , 0x A (
), > 0, , 0f x f x ( 0f x f x )
0 0,x A x x . 0x (
), 0f x ( ), f. 66. f (, ),
0 ,x a , .
f (, ), 0 ,x a f .
0f x 0,a x 0f x 0 ,x , 0f x f.
0f x 0,a x 0f x 0 ,x , 0f x f.
f 0 0, ,a x x , 0f x f.
67. Fermat.
: f 0x . f
0x , 0 0f x
: f 0x . 0x
f , > 0 ,
0 0,x x 0f x f x 0 0,x x x (1)
, , f 0x ,
0 0
0 0
0
0 0
lim limx x x x
f x f x f x f xf x
x x x x
,
-
| . 28 .2421302598
20
0 0,x x x , (1), 0
0
0f x f x
x x
,
0
0
0
0
lim 0x x
f x f xf x
x x
(2)
0 0,x x x , (1), 0
0
0f x f x
x x
,
0
0
0
0
lim 0x x
f x f xf x
x x
(3)
(2) (3) 0 0f x . . 68. f ; f :
f x f
, .
f x f
, . 69. f ; f :
0f x x f , .
0f x x f ,
70. f (, ),
0x . 0 0,A x f x f;
0 0,A x f x f : f 0,a x 0 ,x , ,
fC 0 0,A x f x
-
| . 28 .2421302598
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71. 0x x f
0x ;
0x x f 0x
0
limx x
f x
, 0
limx x
f x
0
limx x
f x
72. y f
( );
limx
f x
( limx
f x
) y
f ( )
73. y x f ( );
lim 0x
f x x
, ( lim 0x f x x )
74. y x f ( ) , ;
limx
f x
x
lim
xf x x
limx
f x
x
lim
xf x x
75. De l Hospital.
0
0
: 0
lim 0x x
f x
, 0
lim 0x x
g x
0 ,x 0
limx x
f x
g x
( ), :
0 0
lim limx x x x
f x f x
g x g x
: 0
lim 0x x
f x
, 0
lim 0x x
g x
0 ,x 0
limx x
f x
g x
( ), :
0 0
lim limx x x x
f x f x
g x g x
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| . 28 .2421302598
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76. f . f ; f F
F x f x , x . 77. f . F f , :
G x F x c , c f
G f G x F x c , c .
G x F x c , c f
G x F x c F x f x x G f . x F x f x
G x f x , G x F x , x . c ,
G x F x c , x .
78. f [, ] G f
[, ]. : af x dx G G
x
aF x f t dt f [, ]. G
f [, ], c ,
G x F x c (1) (1), x a ,
a
aG a F a c f t dt c c
c G a . , G x F x G a , , x ,
a
G F G G f x dx G
, a f x dx G G
79. f g, [, ] 0f x g x
,x f, g
x a x . E
a
E f x g x dx
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| . 28 .2421302598
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:
1 2a a a
E E E f x dx g x dx f x g x dx
a
E f x g x dx
80. f g, [, ] f x g x
,x f, g x a
x . E a
E f x g x dx
, f, g [, ], c
0f x c g x c , ,x . (. )
(. ). , : a
E E f x c g x c dx
a
f x g x dx
. aE f x g x dx
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| . 28 .2421302598
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2Rez z z
2Imz z z i
2 2Im 0z z z z z z
2 2Re 0z I z z z z z
z x yi 2 2z x y
2
z z z
2 0az z , , < 0
1,2
2
iz
.
1 2z z .=, 1 2z za
, 1 2z z
a
( Vietta)
z z z x yi
, ,v v
f z w g z w
. (. .) z , z x yi , :
0Ax By 0A 0B ()
2 2 0x y Ax By
2 2 4 0A B 2 2 4
2
A B
,
2 2
A BK
2 2y px 2 2x py ()
2 2
2 21
x y
a ()
2 2
2 21
x y
a ()
w . .
w a i .
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| . 28 .2421302598
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0z z a 0z
1 2z z z z 1z
2z
1 2 2z z z z a 1 2,z z 2
1 2 2z z z z a 1 2,z z
2.
2
2 2 2
2
zz
z z z z
zz
1 2 1 2 1 2z z z z z z
1 2, ,M M M 1 2, ,z z z .
: () :
() : 0Ax By
2 2
min ,z d OA B
z : () () (0,0).
11
: y x
1 1
11
1
1: y x
1
: 0
1:
Ax By
y x
z .
1M 1z
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| . 28 .2421302598
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1 11 12 2
min ,Ax By
z z d MB
z : () ()
1 1M z 1 1 1z x y i
11 1 1
: y y x x
1 1
11
1 1 1
1: y y x x
1 1 1
: 0
1:
Ax By
y y x x
z .
C:(, ) 0 0,K x y 0z : :
min z OK
max z OK :
C x, y 2 2 2
0 0:C x x y y
. .
(0,0) : y x
: 00 0
0
yy x
x
00
:y
y xx
2 2 2
0 0
0
0
:
:
C x x y y
yy x
x
.
.
1M 1z
1 1 1
1 1 1
min
max
z z M A M K
z z M B M K
:
C x, y 2 2 2
0 0:C x x y y
.
() 1M
-
| . 28 .2421302598
27
1 01 11 0
:y y
y y x xx x
2 2 2
0 0
1 01 1
1 0
:
:
C x x y y
y yy y x x
x x
.
.
2 2
2 2: 1x y
Ca
2 2 2 :
min z OB OB
max z OA OA a z i
z a
2 2
2 2: 1x y
Ca
2 2 2
min z OA OA a z a
1 2,M M
1 2, ,
1 2 1 2min ,z z d 1 1: y x 2 2: y x
1 21 22
,1
d
C(, ), :
1 2max 2z z
() C(, ), :
1 2min ,z z d
1z 2z :
2 2 2
0 0:C x x y y
: y x
() () , 0 0,x y .
-
| . 28 .2421302598
28
1
1
0 01
: y y x x
0 0
2 2 2
0 0
1:
:
y y x x
C x x y y
0 0
1:
:
y y x x
y x
.
1 1 1,C K 2 2 2,C K :
1 2 1 2K K , :
1 2 1 2 1 2min z z K K
1 2 1 2 1 2max z z A K K
1 2 1 2K K
1 2 1 2max 2 2z z
1 2 1 2K K
1 2 1 2max 2 2z z
1 2 1 2K K 1 2
1 2 1 1 2min z z K K
1 2 1 1 2max z z K K
1 2 1 2 1 2 : :
1 2 1 2 1 2max z z K K
g f , :
f
g
: |g f A x A f x B g f : g f x g f x
-
| . 28 .2421302598
29
1f :
f.
, f 1 1 ( 1 2,x x A 1 2 1 2f x f x x x )
1f ( y f x x ).
f x y f x f y f x y f x f y
0x y 0f y x
f x y f x f y f x y f x f y
1x y 1f y x
0)(lim0
xfxx
, 0)( xf 0x
0)(lim0
xfxx
, 0)( xf 0x
gf , 0x )()( xgxf 0x ,
)(lim)(lim00
xgxfxxxx
)(lim0
xfxx
, 0)( xf 0x ,
)(lim0
xfxx
, 0)( xf 0x .
-
)(lim0
xfxx
,
))((lim0
xfxx
,
)(lim0
xfxx
,
))((lim0
xfxx
.
)(lim0
xfxx
, 0)(
1lim
0
xfxx
.
0)(lim0
xfxx
0)( xf 0x , )(
1lim
0 xfxx, 0)(lim
0
xfxx
0)( xf 0x , )(
1lim
0 xfxx.
)(lim0
xfxx
,
|)(|lim0
xfxx
.
)(lim
0
xfxx ,
k
xxxf )(lim
0 .
-
| . 28 .2421302598
30
01
1)( xxxP
, 0
)(lim)(lim xx
xxP
)(lim)(lim
xx
xxP
01
1)( xxxP
11 1 0...Q x x x x
lim limv
v
x x
P x a x
Q x x
lim limv
v
x x
P x a x
Q x x
0
,0
de l Hospital.
0 :
0 0 0
lim lim lim1 1x x x x x x
f x g xf x g x
g x f x
.
: 1) 2) 3)
x x , 0
0 ,
, :
h x f x g x 0x
0 0
lim limx x x x
f x g x
, 0
limx x
f x
!!! 0
lim 1x
x
x
, 0
1lim 0x
x
x
,
0
1lim 0x
xx
x x 1x , x x , 1x
, , ,
0 0
1limx x x x
0 0
1limx x x x
.
. : 0
0
1limx x x x
, 0
2
0
1lim
vx x x x
:
limx
f x g x
limx
f x g x
: x .
, ,0 ,0
, .
, .
-
| . 28 .2421302598
31
0
0
,
,
g x x xf x
a x x
0x x f .
0x x :
0
limx x
f x
, f 0x
0
limx x
f x
0f x , f 0x , 0f x , f
0x .
1 0
2 0
,
,
g x x xf x
g x x x
f
0 0
0lim limx x x x
f x f x f x
. .
, f 0x ,
: f 0 0
0 0lim limx x x x
x f x f x f x
, ,
0x 0
0limx x
f x f x
.
f x y f x f y
f x y f x f y : 0
00
0lim lim
x x h
x x hf x f x h
f x y f x f y f x y f x f y
: 0
00
1lim lim
x x h
x x hf x f x h
x0
1 0
2 0
,
,
g x x xf x
g x x x
:
0x x 1f x g x
-
| . 28 .2421302598
32
0x x 2f x g x
0x x
0 0
0 0
0 0
lim limx x x x
f x f x f x f x
x x x x
.
f 0x
f
0 0
00
limh
f x h f xf x
h
,
0 0
0 01
0
lim , 01h
f x h f xf x x
x h
, f 0x ,
:
0 0
0 0
0
0
0 00
0 0
lim lim
lim lim
x x x x
x x x x
f x f x f xf x
f x f x f x f xf xx x x x
0f x fC
0 0,M x f x , 0f x fC x x
0
0
0
limx x
f x f x
x x
0x x fC 0x .
fC 0 0,M x f x
0 0 0: y f x f x x x
fC 0 0,M x f x x x , 0 0f x
fC 0 0,M x f x y x ,
0f x
fC 0 0,M x f x y x ,
0 1f x
fC 0 0,M x f x y x ,
0
0 0
f x
f x x
fC gC 0 0,M x y ,
0 0
0 0
f x g x
f x g x
fC gC ,
0 1
0 0 0 1 1 1
f x g x
f x x f x g x x g x
-
| . 28 .2421302598
33
f
0f x fx D . , f x c c fx D , c
, :
o o o
o 0f x o f.
f 0x
, Fermat 0 0f x
: o . o
o f x
o 0f x
o .
f 0x
, 0 0f x
( 0x )
0x
:
( )
Bolzano
0f a f ,
!!! Bolzano , .
0f a f
o 0f a f Bolzano
o 0f a f x a x
Rolle :
o f , f ,
f a f
-
| . 28 .2421302598
34
o 0f x , Bolzano, f, F
F a F
...
.. , 0 ,...f a f
!!! . , , .
ff D . 0f x .
Fermat -
( 0f x 0f x ). 0f x . 0x
0 0f x . 0f x f x 0f x f x ,
f 0x . Fermat 0 0f x !!! Fermat
, 0x .
( 0x
)
. .
Rolle .
( 0x
) .
()
( ) :
o f x a f x f a
o f x a f x f a f,
0f a .
o f
0
0
x a f x f a f x
x a f x f a f x
-
| . 28 .2421302598
35
o f
0
0
x a f x f a f x
x a f x f a f x
o f 0x 0f x f x fx D .
0 0f x 0 0 0f x f x f x fx D
o f 0x 0f x f x fx D .
0 0f x 0 0 0f x f x f x fx D
f ,ff D m M .
m f x M fx D ( ,ff D m M , m f x M )
f f
. .. . ... .
Jensen f
f ,
2 2
f a faf
f ,
2 2
f a faf
( .. ,2
aa
,2
a
)
: f f
o f 0 0 0: y f x f x x x fC
0 0,M x f x 0 0 0f x f x x x f x o f 0 0 0: y f x f x x x fC 0 0,M x f x
0 0 0f x f x x x f x
!!! 0x x
.
f 0f x ,x Bolzano f
o 0 0: 0x f x 0f x x
o 0 0: 0x f x 0f x x Bolzano f .
, :
-
| . 28 .2421302598
36
o f [, ] 0f x , 0af x dx
o f [, ] 0f x ,x f
[, ], : 0af x dx
f :
0
limx x
f x
, 0
limx x
f x
0
limx x
f x
0x x fC
limx
f x
( limx
f x
) y
f ( )
fC
o
limx
f x
x
lim
xf x x
o
limx
f x
x
lim
xf x x
y x fC
.
.
y x fC lim 0
xf x x
c cx c f x c f x
2
a a
x x
f g x g x f g x
f x g x f x g x f x g x
1
1
vv axax
v
1
1
v
vf x
f x f xv
2
f x g x f x g x f x
g x g x
2
1f x
f x f x
1
ln xx
lnf x
f xf x
-
| . 28 .2421302598
37
2
xf x f x f x
x x
2
2
f xf x f x
2
f x f x f x f x f x
2
2
f x f x f x f x
f xf x
2
2
f x f x f x f x
f xf x
axax ee
a
g x g xf x g x f x f x e e
g x g xf x g x f x f x e e
ln
xx aa
a
0a f x xf x xf x
x xf x f x e f x e
x xf x f x e f x e
f
x
aF x f t dt
F f ( )
aa
f x dx F x
a af x g x dx f x g x f x g x dx
.
a
x g x dx
f
f x x . :
1) h xe
x x f x f x f x
x x f x f x f x
x xe e f x f xe f x e
21
xx
2f x
f xf x
21
xx
2f x
f xf x
, 0
2
f xf x f x
f x
ln1 x
x
axax
x
ax
-
| . 28 .2421302598
38
2) ,h x h x
3) 11 1 0...v vv va x a x a x a 4) ln h x
a a af x dx x f x dx x f x x f x dx
o P(x) Q(x)
.
a
x
2x
ax x
-
| . 28 .2421302598
39
fC , x x :
2
1
E f x dx
1 2
0f x
fC , gC x a x :
aE f x g x dx
.
f x g x .
fC , gC , x x :
2
1
E f x g x dx
1 2
0f x g x
f x g x f x g x c
, , . , . :
o 0a f x dx
o xa f t dt f x
o g xa f t dt f g x g x
o
21
2 2 1 1
g x
g xf t dt f g x g x f g x g x
o
x x x x
a a a a
x
a
f x g t dt f x g t dt f x g t dt f x g t dt
f x g x f x g t dt
o xa a xf x t dt f u du
( u x t )
o xa a xf x t dt f u du
( u x t )
!!!
f 2f x g x , 0g x x A ( ) :
o f
o 0f x
-
| . 28 .2421302598
40
Bolzano f x g x x A
f x g x x A . f .
o 0 0: 0x A f x f x g x
o 0 0: 0x A f x f x g x
-
| . 28 .2421302598
41
1. z w
: 2 2
1 1 4z z (1) 5 12w w (2)
i. = 1.
ii. 1 2,z z z 1 2 2z z ,
1 2z z .
iii. w
2 2
19 4
x y
w
iv. ,z w (1) (2)
: 1 4z w
2. z, w, : 1 Imz i z (1)
3 3w w i i w i (2)
i. z
21
4y x
ii. w
(0, 3) 2 2
iii. , z, w z w
iv. , u , , , , .
3. 1 2,z z
1 2 2z z 1 2 5z z
i. 1 2,z z
ii. w 2 2 2
1 2 1 2w z w z z z
2 21 4x y
iii. w ii
2Re Im 0w w
-
| . 28 .2421302598
42
iv. 1 2,w w w ii
1 2 4w w , 1 2 2w w
4. z : 2 2 8 0i z i z
i. z x yi
ii. 1z 2z
.
iii. 1 2,z z
2 2
1 2 1 2 40z z z z
5. 1
1 3
2
iz
2 0z z ,
.
i. 1 1
ii. 31 1z
iii. w,
: 1 1w z z
6. 1z x yi 1
2
1
2,
2
zz
z
,x y 0y .
2 1z z .
i. 2 1 1z z
ii. 1z .
iii. 21z 0xy , 1z
20 20
1 11 1 0z i z i
7. z 2 2 2 2z z z (1)
i. z (2, 0) = 1. , z
, 3z
ii. 1 2,z z
2 0w w , w , , , 1 2Im Im 2z z
: 4 5
iii. 0 1 2, ,
(i). v : 3 22 1 0 0v a v a v a
: 4v
-
| . 28 .2421302598
43
8. z, 10 104 z z
f 2f x x x a , a . i. z 2x
ii. () f
2x y y 0 3y ,
a. () b.
f, (), x x
3
5x
9. i. 1 2,z z 1 2 4 4z z i
1 22 5 5z z i , 1 2,z z .
ii. z w 1 3 2z i 3 2w i :
a. z, w z w
b. z w
10. : ,f [, ] 0f x
,x z Re 0z , Im 0z Re Imz z .
1
z f az
2 221
z fz
, :
i. 1z
ii. 2 2f a f
iii. 3 0x f a f 1,1
11. z w 3z i , :
3 3 2z i z i 1
33
w z iz i
i. z
ii. 1
33
z iz i
iii. w 2 2w
iv. z w z
12. 2
2zz
z 0z .
i. 1z 2z
ii. 2010 20101 2 0z z
iii. w 1 24 3w i z z
w .
iv. w iii, 3 7w
-
| . 28 .2421302598
44
13. 2 1 2 1z i , i. .
z,
. 0 1z i
.
ii. w 2
012w w z
0z , .
14. z w 2 2 6i z 1 3 3w i w i :
i. z ii. w
iii. w
iv. z w
15. 2
2
aiz
a i
, a
i. z (0, 0) = 1
ii. 1 2,z z 2
2
aiz
a i
a 0a
2a .
a. 1z 2z .
b. : 2
1 2
v vz z .
16. 1 2 3, ,z z z 1 2 3 1z z z 1 2 3 0z z z .
i. :
a. 1 2 3 1 2 3z z z z z z
b.2
1 2 4z z 1 2Re 1z z
ii. 1 2 3, ,z z z ,
.
17. 1 2 3, ,z z z 1 2 3 3z z z
i. : 1
1
9z
z .
ii. 1 2
2 1
z z
z z .
iii. : 1 2 3 1 2 2 3 3 11
3z z z z z z z z z
-
| . 28 .2421302598
45
18. :f 1 1f . x ,
3
1
13 1 0
x
g x z f t dt z xz
, z i *,a , :
i. g g .
ii. 1
z zz
iii. ii 21
Re2
z
iv. 2 0f a , 3f > , 0 2,3x
0 0f x
19. z a i , ,a 3 4w z iz , z z
i. Re 3 4w a Im 3w a ii. , w
12y x , z 2y x .
iii. z,
2y x , .
20. z :4
Re 2Re( )z zz
i. z. ii. Re( ) 0z ,:
a. z
zw4
,44 w
b. 3 4c z i
c. b c
21. 1 2 3, ,z z z : 1 2 3z z z , 2 3 1z z z ,
1 3 2z z z
i. 1 2 3 0z z z
ii. 1 2 3z z z , , 1 2 3, ,z z z
:
a. 2
1
1Re
2
z
z
b. 3 2
1 1
z z
z z
c. .
-
| . 28 .2421302598
46
22. ,z w : 3 4
1
z iw
i
i. 1 1
2 2w i z.
ii. 2
2w
a. z . b.
.
23. z w
3 4 3 4 20 0i z i z 1w w . : i. z
ii. w
iii. z
iv. z w
24. f z i z , z
2f z f z (1). i. .
ii. 5 4
3 3z i z (1)
iii. 5g z z , z (1).
g z
25. z, w 22 4 3 2x w i x z (1) , x 1x
i. z
1 1 ,
w (4, 3) 2 4
ii. , .
iii. z, w (i) 10z w
10z w
iv. z (i)
: 22 3 2 5z z zz
26. : , ,0f a a , z a if a
w if . f [, ], f (, ) z iw z iw :
i. zw ,
-
| . 28 .2421302598
47
ii. af f a
iii. 0 ,x a 0
0
0
f xf x
x
27. 1 2,z z 1 2 0z z 1 2 1 2z z z z
i. 1
2
zw
z
ii. 1 2
1 2 1 2
1iz z
z z z z
iii. 1z 2 1z i
28. 1xz e x i , x
i. Re Imz z x
ii. x 2 1w z z .
iii. z .
29. z, 1z , 1
1
zw
z
. :
i. 1z
ii.
41
zz
iii. 1 21 2
1 14z z
z z
, 1 2,z z z
u , i
u ui ww
, 0w ,
2 2 1x y
30. 1 ln 1f x x x , 0x
i. f 1 0,1
2 1, . f
ii. 1 2013xx e , 0x .
iii. 1 2,x x 1 2x x ii,
0 1 2,x x x , 0 0 2012f x f x iv.
1g x f x 0x , x x x e
-
| . 28 .2421302598
48
31. : 0,f , 0x
: 0f x , 2 21
1
x x x xf t dt
e
1
lnln
x t tx x dt e f x
f t
i. f .
lnxf x e x x , 0x , :
ii. :
2
0
1limx
f x f xf x
iii. ln 1x x , 0x ,
x
aF x f t dt , 0x , 0a , .
3 2 2F x F x F x 0x .
iv. 0 .
, 2 3 2F F F
32. :f , ,
:
0
lim 1 0x
f xf
x , 0 1 0f f f 0f x x
i. f
0 0x
ii. f
g x f x x , x , :
iii. g 0
limx
x
xg x
iv. 2
02f x dx
v. g, x x 0x 1x
5
2E e
1
0f x dx
1,2 , 0
2f t dt
33. y x , 0x .
0,1 Oxy
, .
-
| . 28 .2421302598
49
t, 0t
16 / minx t m i. , t, 0t
16x t t
ii.
4,2A , ,
. iii.
.
iv. 01
0,4
t
,
d M .
Oxy
34. 2 ln 3f x x x x , 0x
i. f
ii. f 0,1
1,
iii. 0f x .
iv. 1 2,x x iii 1 2x x ,
1 2,x x , 0f f
f ,M f .
35. :f
0 1f 0 0f
i. 1f x x
ii.
13
0
30limx
x f xt dt x
x
22 2f x x x f x x , x , :
iii. 2 2xf x e x , x
iv. 2x
xh x f t dt
, 0x
2
2
2 3 4
62 1
0
x x
x x
f t dt f t dt
36. 2ln 1 1 ln 2f x x x x , 1x 1
i. , limx
f x
.
-
| . 28 .2421302598
50
1 ii. f .
iii. f
iv. 2 0f x a 0a
37. : 0,2f
24 4 xf x f x f x kxe , 0 2x
0 2 0f f , 42 2 2 12f f e , 21f e k .
i. 2
2
23
x
f x f xg x x
e
, 0 2x
Rolle 0,2
ii. 0,2 ,
24 6 4f f e f
iii. 6k 0g x 0,2x
iv. 3 2xf x x e , 0 2x
v. 2
21
f xdx
x
38. 2 2lnf x x x , 0x
i. : 1f x 0x . ii. f.
iii. ln
, 0
, 0
xx
f xg x
k x
a. k g
b. 1
2k , g , ,
0,e .
39. f 0, 0f x
0,x . :
0
x
F x f t dt , 0,x
0
x
F xh x
tf t dt
, 0,x
i. 1
1
01te f t F t dt F
ii. h 0, .
iii. 1 2h , :
a. 2 2
0 02f t dt tf t dt
-
| . 28 .2421302598
51
b. 1
0
11
2F t dt F
40. 2
3 , 0
, 0
xx
f x x
x ax x x
i. 0
lim 3x
f x
ii. 2
f
f 0 0x ,
3a .
iii. 3a , 0f x dx
41. lnxf x e e x , 0x
i. f 1,
ii. f x e 0x
iii. 2 2
2 2
2 2 4
1 3 2
x x
x xf t dt f t dt f t dt
0, .
42. 11
1
x
x
ef x
e
, x
i. f
ii.
1
0
1dx
f x
iii. 0x : 5 7 6 8x x x xf f f f
43. ln 1 1 lnf x x x x x , 0,x i. :
a. 1
ln 1 lnx xx
, 0x
b. f 0,
ii. 1
lim ln 1x
xx
iii. 0,a , 11a aa a
44. f, 0f x
x i. f 1 - 1
ii. fC f 1,2005A 2,1B
1 22004 8 2f f x
-
| . 28 .2421302598
52
iii. fC ,
fC 1
: 2005668
y x
45. :f ,
20lim 2005x
f x x
x
i. :
a. 0 0f
b. 0 1f
ii. , :
22
220lim 3
2x
x f x
x f x
iii. f f x f x x , :
a. 0xf x *x
b. 1
01f x dx f
46. :f 2 4 5x x x xf x m , m , 0m
i. m 0f x x ii. 10m ,
f, x x 0x 1x .
47. f 0, , 12
2
02 2
2
xf x xf xt dt
i. f 0,
ii. 1xf x e x
iii. f x 0,
iv. limx
f x
limx
f x
48. 2 1f x x x
i. lim 0x
f x
ii. f, x
iii. 2 1 0f x x f x
iv. 1
20
1ln 2 1
1dx
x
49. f , : 2f x f x 0f x x .
i. f .
ii. 0f x .
-
| . 28 .2421302598
53
iii.
f xg x
f x
.
g x x ,
45o
50. :f , 0 0 0f f
, : 1xe f x f x f x xf x x
i. : ln xf x e x , x ii. f
iii. f
iv. ln xe x x
0,2
51. , :f g , x
0f x 0g x
2
2 0
1 txx
f x edt
e g x t
2
2 0
1 txx
g x edt
e f x t
i. f g
f x g x x
ii. : xf x e , x
iii. :
0
lnlim
1x
f x
fx
iv.
21
x
F x f t dt x x , y y 1x .
52. 22 ln 1f x x x , x i. f.
ii. :
2
2
4
3 2 12 3 2 ln
1
xx x
x
iii. f
f , y y
iv. 1
1I xf x dx
-
| . 28 .2421302598
54
53. :f x
: f x x 0
3x t
f x x dtf t t
(1)
i. f
f xf x
f x x
,
x
ii. 2
2g x f x xf x x ,
iii. 2 9f x x x , x .
iv. 1 2
1
x x
x xf t dt f t dt
x
54. ln 1xf x a x , 1x 0a 1a
i. 1f x 1x , a e ii. a e
a. f .
b. f 1,0
0,
c. , 1,0 0,
1 10
1 2
f f
x x
1,2
55. f 0,2
2
02 0t f t dt . 0
x
H x tf t dt , 0,2x
0
2
20
3 , 0,2
1 16lim , 0
x
t
H xf t dt x
xG x
tx
t
i. G 0,2
ii. G 0,2
2
H xG x
x , 0 2x
iii. 0,2a 0H a
iv. 0,a
20 0
a
a tf t dt f t dt
56. ln , 0
0 , 0
x x xf x
x
i. f 0. ii. f .
-
| . 28 .2421302598
55
iii. a
xx e .
iv. 1 1f x f x f x , 0x .
57. f
2
3
010 3 45f x x x f t dt
i. 320 6 45f x x x ii. g .
0limh
g x g x hg x
h
iii. f (i) g (ii)
202
lim 45h
g x h g x g x hf x
h
0 0 1g g , :
a. 5 3 1g x x x x b. g 1 1
58. 3 23 2f x x x , 2
,
i. f , .
ii. 0f x .
iii. 1 2,x x 3x f,
1 1,A x f x , 2 2,B x f x 3 3,x f x
22 2y x iv.
f 22 2y x
59. f [0, 1]
0 0f . g [0, 1]
0g x 0,1x . : 0
x
F x f t g t dt , 0,1x
0
x
G x g t dt , 0,1x
i. 0F x x (0, 1].
ii. : f x G x F x x (0, 1].
iii. :
1
1
F x F
G x G x (0, 1].
iv.
2
2
0 0
0 5
0
lim
x x
xx
f t g t dt t dt
g t dt x
-
| . 28 .2421302598
56
60. 2
2 2f x x 2x .
i. f 1 1.
ii. 1f f .
iii. f 1f
y x .
iv.
f 1f .
61. 1
ln1
xf x x
x
i. f.
ii. 0f x .
iii. lng x x
, lnA a a 0a
xh x e ,B e , 0f x .
iv. g h .
62. f xf x e , 0 . i. f .
ii. f, , y ex .
. iii. () ,
f, y y ,
2
2
eE
iv. 2
lim2
E
63. f ,
2 x f xf x e x 0 0f
i. : 1
ln2
xef x
ii. :
0
0lim
x
x
f x t dt
x
iii. : 2005x
xh x t f t dt
2007
2007
xg x .
h x g x x .
-
| . 28 .2421302598
57
iv. 20051
2008
x
xt f t dt
0,1
64. f 2 lnf x x x i. f,
. ii. f .
iii. f.
65. xg x e f x , f
3
0 02
f f
i. 3
0,2
f f
ii. 22 3f x x x 0
aI a g x dx , a
iii. lima
I a
66. , :f g , f :
1f x x f x x , x 0 1f
2
3 3 12
xg x x
i. : 2 1f x x x
ii. 1f g x
iii. 0 0,4
x
, :
0
0
0 0
4 4xf t dt f x x
67. : 0,f :
f 0,
1 1f
0
1 5 1lim 0h
f h f h
h
1
1
x
a
f tg x dt
t
, 1,x 1a . :
i. 1 0f , f 0 1x ii. g ,
2 4
2 4
8 6 2 6
8 5 2 5
x x
x xg u du g u du
-
| . 28 .2421302598
58
iii. g ,
1
1 11
x
a
f ta dt f a x a
t
.
68. :f , : 1 xxf x e , x .
i. 1
, 0
1 , 0
xex
f x x
x
ii. 1f .
iii. f 0, 0A f . , f , 2 2f x x , x .
iv. 0
lim ln lnx
x x f x
69. :f A 0,A , :
,0f A
f 0, ,
1
1 12 2
xf x f tf x x e e f t t dt
x t
, 0x (1)
1
x
F x f t dt , 0x
i. 22
ln1
xf x
x
, 0x
ii. F 0 0,x F x ,
0 0x . 0 ,x
0x , F ,M F : : 1 2012 1 0F x y
iii. 1 ,
351 1 10
1 3
F f x x
x x
, x (1, 3).
iv. 2
1
x x
x
tf dt tf t dt
x
0x
70. :f :
22 3xf x x f x f x x
1
12
f
i. : 3
2 1
xf x
x
, x
ii. f (i).
-
| . 28 .2421302598
59
iii. :
3 22 25 1 8 8 1f x f x iv. , 0,1 , :
3
2 3
03 1f t dt f
71. : 0,f ,
0, , :
2
1 1
1x u f tf x x dt du
f t
0x
0f x f x 0x 0 0f :
f xg x
f x
0x
3
h x f x 0x
i. : 2
1f x f x f x 0x
ii. f f 0, 0 1f
iii. g 0, , :
a. 2g x x 0,x
b. 1
02 1x f x dx
iv.
h, x x 0x 1x .
-
| . 28 .2421302598
60
72. f 0, 0f x 0,x . f, x x , y y
x u uE u e f u 0u :
i. xf x f x e 0,x
ii.
0
1lim 0x
f x
x
iii. f.
iv. 1
2f x x
73. f ,e
0
1x f t
f x e dt
,x e (1) .
i. lnf x x c ii. f .
iii. fC Ox Oy
iv. z Re 0z
,
2 2f x xe e x z e ,x e
74. : 0,f , 0f x 0x ,
1 1, 2 3f f 2 2
1 1f xt dt f x dx
i. 2 2
1
1 x
xf t dt f x dx
x
ii. f x x 1x 2x
75. f : , :
x x u
0 0 02x tf t dt f t dt du xx 2 x (1)
f 0 0 (2)
f 0 (3)
. f .
. f x 1 x , x
. :
i. () fC f
2
3
-
| . 28 .2421302598
61
ii. fC f ,
() fC
2x
3
4x
3
76. z , w , :
2
z 1i z i
(1)
2w z i 2zi 2 0 (2) , 0 1
. z
. w , .
. 2z w
v2z w
w 2z ,
. 1 2z ,z ,,
(1) w ,
(2), :
13
3
77. f , g : 0, , :
f x f x
g x g x x (1)
2g x g x 0 x (2)
f 1 2
g 1 1
. f g
. f gC ,C x 1 x 2
.
1
g x
xlim f x
.
x 12e 2 l
g x
n x 30
f x 1 1 1,e .
78. f 0f x
1
1x
te f t dt x x . :
i. 1
1fe
ii. 0f x x
-
| . 28 .2421302598
62
iii. f x x (0, 1)
iv. z 5
2 12
z e f i
.
79. :f :
f 1, 1A f
f 0x xf xf x xex
(1)
i. : 2
12
x x xf x xe e e
ii. 2004f e 2004f iii. g ,
1
00 1
2
eg g f x dx g
80. : 0,f
2
11
x
x
f t xf x dt
x
. :
i. f .
ii. f : ln 1f x x iii. f .
iv. , , 0 a : f f f f
81. : 0,f , :
0, 0x y 2 2f xy x f y y f x (1)
f 1 1 1f
i. 0x :
1
2limh
f xh f x f xx
xh x x
ii. f 0x : 2 32x f x xf x x
iii. 0x : 2 lnf x x x
iv.
0limx
f x
x
v. f, x x 1x x e
-
| . 28 .2421302598
63
82. : ,f , ,
0 , f 0f x ,x .
i. f 1f
ii. 1f
1 0
f
f a af x dx f x dx
a. f
b. 0 ,x fC
0 0,A x f x 1 : 2006 0x y iii. :
a. , , f
b. 1 2, , 1 2 1f f
83. z , ,a , a 1v a i
izai
(1)
i. z . ii. z
. iii. z .
iv. 4 3 4 7z i
v. 1 2,z z (1) 1 2 2z z
84. 2 0z az , z , ,a 1 2,z z ,
1 2z i
i. , .
ii. 2008 20081 2z z
iii. 1 2 3, ,A z B z z 1 2 3, ,z z z
132
117
5
zz i
z , :
a. .
b. 1 1w z w z , w
c. w,
2 2 10w z w z , .
85. :f : 2
5 1
1 2
x f x
f x
x 0 3f
i. 2 9f x x x , x
ii. lng x f x : a. g
-
| . 28 .2421302598
64
b. g 4
20
1
9I dx
x
c. 9J I K , : 2
4
20 9
xJ dx
x
42
09K x dx
d. 20J K e. J K
f. g 1g .
86. z x i x , 21
w zz
f
.
i. 0 1,0x , 2 1z
z
.
ii. a , 0f x , 1f 0f a , ,
0f
iii. 1
0Re w dx
87. :f
1
2 1 7lim 10
1x
f x
x
.
i. 3 7f 3 5f
ii. () f 3, 3M f . a. () 5 8y x
b. 3, (). 2 m/sec, .
88. 2 332
ln 1
1
x tf x dt
t
i. f. ii. f .
iii. :
5lim 0
5x
f x
x
iv. fC x x , .
v. 3 2z f i . z .
89. : 0,f :
11 1
x f tf x dt
t
0x
-
| . 28 .2421302598
65
i. lnf x x x x , 0x 0 0f ii. f .
iii. fC .
iv.
11 21
e
f x dxe
e
90. :f 2
1
x
f tx
tf x dt
e
x . i. :
a. f 4016
02008I f x dx
b. f
c. 3
3
xf x , x
ii. 1
z iw
z
, 1z . w
: a. z 1y x
b. ,M a f a fC 1 0a z.
91. f 0,f ,
1f x f x x . i. :
a. f b. f
c. g x f x .
ii.
2 210
0 1
2
f x f fdx
f x
iii. 0 1f f.
92. 201
1
x
f x dtt
i. f .
ii. 2 1x
x f xx
0x
iii. f x x ,2 2
x
1f
iv. fC
0y , 0x 1x .
-
| . 28 .2421302598
66
93. , : 0,f g , 0,x :
0f x 0g x
01
1x
f t dtg x
01
1x
g t dtf x
:
i. 0f x 0g x 0,x
ii. f x g x 0,x
iii. 1
2 1f x
x
0,x
94. :f 1
x
f x f x t dt x . F f , :
i. 2F x F x F x x
ii. 0 , 2 , 0kF F
iii. 0f x ,0
95. 1 1z i i , . i. z.
ii. , z ;
iii. 0 . 2 2z 3
zw
i
, :
a. 3
b. , 2vw
96. , ,z w u . 1z w u (1),
0z w u (2) 2 2 2 0z w u (3). :
i. 2 2 2 2 2 2z w z u u w
ii. 2 2 2
1 1 10
z w u
iii. , , ,z w u zwu zw wu uz
z w u
.
iv. 2z w u
97. :f 1
2z
, :
2 2 2f x x x f x x (1)
0limx
f x
x
2
2 1
z
z
-
| . 28 .2421302598
67
i. :
a. 2 2 1z z
b. z 2 2: 1C x y
ii. 20
limx
f x
x x
iii. g x f x x
,0 0, . iv. f.
v. 33 4 5 10z i x x ,
1,2
98. :f 0f x x .
1
u if a
1
w aif
.
i.
1Re uw a
f a f
Im
auw
f f a
ii. Im 0uw 0 , , , ,
f
f
iii. Re 0uw , , .
iv. 21
lim 2004 2010x
f f xx
99. i. , f 1-1 ,
.
ii. f 3 0f x x .
2 0f , 3 2 0f x 4 3f x f x , : a. f
b. f , 3f .
c. 2 3f x
d. gC
f xg x
f x
x x
, gC x x
45 .
e. 2x 2 1 2f x f x f x .
-
| . 28 .2421302598
68
100. : ,f 0 a ,
1z a if a 2z if 12
zw
z .
i. 1 2 1 2z iz z iz
ii. f x
g xx
Rolle [, ]. iii. f,
.
iv.
lim 1
x
ax a
f x a tdt
x a x a t
, 1f x
(, ).
101. : 0,f
1
1
1
x
f t
tf x dt
t e
0,x .
i. lnf xe f x x x , 0,x
ii. lnf x x , 0,x
iii. 0 a f f a f f
a
iv. g, 2xe
g x fx
0,x 1 1g .
102. :f , 0 0f ,
2xf x xe x .
i. 0 1f
ii. 2
0lim 1x
f x
x x
iii. 1
01xf x e dx , f [0, 1].
103. f,
0 20xf x f t xdx dt x
, x (1)
i. f ii. f.
iii. 21 1
2 2
xf x e x , x :
a. f, b. f.
-
| . 28 .2421302598
69
c. x , 0f x
104. z a i ,a :
21 1
1z z x
e e x e x
i. z
1,0K 1
ii. 2 3z i
iii. f (0, 2),
2 2 2f x x x (1) 0f x 0,2x a.
fC
b. fC z.
c. ln
1
x
G x f t dt
105. :f f ,
01 1
f xte dt x x .
i. f xe f x x x .
ii. f 1f
iii. f . iv. f,
x x 1x 1x e
v. 1
12
xx f x f x
, 1,x
106. : 0,f 0 1f
4 3 0f x f x 0f x 0,x . i. f .
ii. 3
1
1f x
x
, 0x
iii. () f
0, 0A f . iv. , f,
x x , () 7x .
v. 22 1 2f a f f a 0,2
a
107. :f , x
2 23 3x xf x f x e e (1)
i. 2xf x e , x
-
| . 28 .2421302598
70
ii.
1
x e f xf x
, x
.
iii.
1 0
0 1
11dx e f x dx e
f x
iv.
2
1
1xh x dt
f t
.
v. 1
2
0
1
4
eh x dx
108. : ,f
xz f x xi , ,x 0a . i. Re Imaz z Re Im az z ,
f x x , 0 ,x a
ii. 0
1x
af x dx
0
3x
f x dx
, : a. f,
x x x a x
b. 1, ,a ,
2 1
3 1
f f
109. : ,2 2
f
0 0f
0 1f , 2f x f x f x , ,2 2
x
i. 2 1f x f x , ,2 2
x
ii. 201
1
x
g x dtt
1 1 .
iii. h x g f x g x ,2 2
.
iv. f x x , ,2 2
x
v. g,
x x 0x 1x , ln 2
12
g E
110. 310 10
i zf z
z i z i
. :
-
| . 28 .2421302598
71
i. f z
ii. 10 10 2z i z i z
iii. 1f z
iv. z,
1
6f z z , .
111. z,w, :
2
1zi z i
a a (1)
22 2 0w z i zi a (2) a 0 1a i. z.
ii. w
iii. 2
2
z wv
z w
2w z , .
iv. 1 2,z z ,
(1) w ,
(2),
13
3
112. , :f g ,
2 2 2 2f x g x f x g x , x (1) xz f x g x i
i. 1 4f 1 3g , :
a. xz ,
b. 2 2f x g x ,
xz ,
ii. a 1az i , :
a. 2012 2012a aA z z
b. f, g
113. 4
f z zz
, z 0z
i. 1z 2z , 2f z
1 2,z z 4 4
1 2 1 23
32
z z z zz
0,0O 2
-
| . 28 .2421302598
72
ii. f z z , :
a. lim ln 1hh
z h
b.
2 20122 4 40
2
t f z t f z
t t
t (0, 2).
iii. f z z , *z , :
a. z
b. 4 5z z 4 5,z z z
(iii a.) 4 5Im Im 0z z
114. :f , :
xf x e , x (1)
1f x f x x (2)
i. 0f xf x e , x
ii. f x
g xx
, 0,x
a. g b. g
c. , , ln ln ln 1a , 33a aae e e e
115. : 0,f 0 0f , 0a :
f xy f x f y , 0x y (1)
0f x 0x (2) i. :
a. 1 1f
1 1f
x f x
0x
b. 0f x 0x
c. f 0,
xf x af a
f x f a
0x
ii. : 2 0x ay a f
,M a f a , f.
116. :f , x
f a f x f (1) ,a 1 0a (2). : i. f .
-
| . 28 .2421302598
73
ii. g x f x f a x 2
ff a
iii. 0f x ,a
iv. 1 2 3, ,x x x ,
3 2 1
2 4 1f f a f
f x f x f x
117. f 1 ln
, 0x
f x xx
i. f .
ii. ,M x f x 1 /sec, E t , ,0 , 0,0 , 0,A x O B f x , 0t 0 4x t
iii. 0t (1,1), :
a. 1x t t
b. E t 1,e
118. , : 0,f g ,
1ln , 0,1
-1 , 1,x
x x xf x
e x
1
2
x
g x f t dt
i. f 0 1x 1 1f ii. g
iii. 0
limx
g x
limx
g x
iv. g
v. 0g x 0,
vi. g 1
0,e
1
,e
119. :f , :
2
2
12 0
1f x f x
x
, x (1)
1 1 1f f (2)
0 1f (3)
i. 2 1f x x x , x
ii. 0f x x iii. f
iv. lnh x f x , x , :
-
| . 28 .2421302598
74
a. 2
1
1h x
x
, x
b. f, x x 0x 1x
c. 1f x x
h xx
0,x
120. : 0,f , :
1 1 1ln 1 2ln 2
4 1
x x u f txf t dt x x dt du
t
, 0x
i. f 1 1
1 1
f x
x x x
, 0x
f
ii. 1
1 lnx
f x xx
, 0x :
a. f , fC
f.
b. 2012
11 xx xe , 0x
c. 1
2ln 2 22
xf x f
, 0x
d. 2012 2012
x xtf t dt x f t dt , 0,2012x
121. , :f g ,
xf x xe a
21
22
x xg x e e , ,a
,f gC C f, g ,
(), , :
i. : 0a , 3
2 () y x
ii. 1E , fC
f, () x t , 0t
iii. 2E ,
gC g , () x t , 0t
iv. 1 2tE t E t te 0t
122. :f , : 2
1
1ln ln 1
f x f xte t dt e x xt
x
i. 0f x , x
-
| . 28 .2421302598
75
ii. 2 1xf x e x , x
iii. 0 0,1x , 0 3f x iv. :h :
3 3h f x x f h x h x , x , h y x
.
v. 0,1 , 2 1f e
123. :f , 2
1
x t t
t
x
e ef x dt
e
i. f 2
1 xf x e , x
ii. f, .
iii. 3
1
x
h x f t dt , x , :
a. 1 1
3
0 00h x dx x f x dx
b. 0 0,1x , 0
03
0
h xf x
x
124. : 0,f , :
2 2x y
f xy f x f yxy
, 0,x y (1)
limx
f x
i. 0f x , 0,x
ii. 1f
iii. 1
f x xx
, 0,x
iv.
2
1lim
x
xf x
x
v. 1x x xx
, 0,x
vi. : 21 1
1 11
1
x x
e e
dt dtt f t
ft
, ,6 3x
vii. g 0a ,
1
lna
a
xI g f x dx
x
125. :f , :
-
| . 28 .2421302598
76
f x f x x , x (1)
0 1f (2)
f xg x
f x
, x
i. 0 0f ii. f , .
iii. g .
iv. 2 1f x x , x
v. 1
x
h x f t dt
vi. hC
0, 1x x x x .
-
| . 28 .2421302598
77
126. z , 1z 1
1
zw
z
. :
i. 1z
ii. 1
zz
4
1z
z
iii. 1 21 2
1 14z z
z z
1 2,z z z
127. f 2 2
2 2
x z zf x
x z z
, ,x z
i. fC
ii.
20
Relimx
f x z
x z
iii. z fC
2,1A
128. 1 2,z z 1 2z z ,
2003 2003
1 2 1 2z z z z . :f 1 2f x x z z , x . :
i. 1 2Re 0z z
ii. f
iii. 0 1,1x : 2
0 03 1f x x
iv. f 2z
129. . f,g , f x g x
x , , :
f x dx g x dx .
. f : 0, xf x x ln 1 e
i. f f x 0, .
ii. : y x fC .
iii. x 0, x
x x
x
eln 1 e e
e 1
.
-
| . 28 .2421302598
78
iv. fC , : y x x 0 ,
x 1 , : 11
2ln E 1 e
1 e
130. x
2
2 0
1f x ln x x 1 dt
1 t
.
. f .
. 3
4
2 0
10 dt ln 2
1 t
.
. x
2 0
1dt x
1 t
,
x 0,
2.
. fC , x x x 0
x 1 .
131. f : , , , f ' f ' 0
f f .
. f x f
, x g x x
0 , x
, .
. g x , g 0 .
.i. g x g 0 , g x
, .
ii. , : f f
f
.
.
f
0
f
, ,
f f
. f x ,
g x , .
132. . i. o u , u u
ii. z w 1 , z w
1 z w
.
. z 2i 1 , :
i. .
ii. z 2 2i .
-
| . 28 .2421302598
79
133. f : 3f x 5f x x 0 ,
x . f
. f
. 3g x x 5x , f
1f
. f
. f 0x
0
0x xlim f x f x
. f x 19 x 1
.
1
x 0
flim
x
x
134. f f ,
x 0 1
2
0 x 0t 1 f t dt 2 tf t dt 4 xtf x dt , x , f 0 0 f 0 2
. 2
2xf x
x 1, x .
. E fC , xx x 0
x , 0 . cm10sec
, E , 3cm .
. g g x x 2 f x , x .
i. y x 2 gC .
ii. gC ,
x 0 x 2 , E ln5
135. z
3z 1 x z 2 xf x , x 1,
x 1,
x 1
lim f x .
:
. i. z 1 z 2 .
ii. z, 3
2.
. i. f 1, .
-
| . 28 .2421302598
80
ii. f 2012 z 1 .
.
f(x) , x 1
g(x) 4 , x 1
4(x 1), x 1
x 1
1, :
i. f 1 4, .
ii. 3 15
z i2 2
3 15
z i2 2
136. . A f , g 1-1, f g
1-1.
. f , , 1-1,
3
h x f x 2f x 3 1-1.
. 3g x g xh x e 2e 3 , g 1-1 .
g g 2 ln 2 , :
i. h 1h 0 0 1h 9 2 .
ii. 1 2h 2 h x 8x 0 .
137. f : f 1 0
f 1 f 0 4 3z i
2 f 1 , zz 1 i 3
2.
:
. f 1 f 0 f 1 8
. 2Re z z
. f 1 2 f 1
. 11 f z 0
138. z z 4 4i 2 Im(z) 4 .
. ..
.
2f(x) 4 x 8x 14 .
. .
-
| . 28 .2421302598
81
. .
139. f, 1, e , f 1 2 , f e e 1 1,4 . : . 1 2x , x 1,e , 1 2x x , 1 2f x f x 0
. 1, e , f 0
. 0x 1,e , 4
0 0 0 0f x f x 4f x x
. y x e 2 fC
1, e
. 1 2 , 1, e , 1 2 , 1 2f f 1
140. f : f 1 0
f 1 f 0 4 3z i
2 f 1 , zz 1 i 3
2.
:
. f 1 f 0 f 1 8
. 2Re z z
. f 1 2 f 1
. 11 f z 0
141. 1 2z , z f :
0
1 2 2
f x x z xt z dt
f x x x :
. 2x
1 2 0
f x z t z dt
. 21
z2
. f x 2013 0, .
. x : x
21
0
z xz t dt
2 4
142.
i. : 2x
x ln 1 x x2
x 0 .
;
-
| . 28 .2421302598
82
ii. 2ln 1f xx .
E fC , x x x 0 x 1 ,
7 1
E30 3
143. f,g 0, :
f 1 g 1 f 1 g 1 0
g xf x e 1 (1)
f xg x e 1 (2)
x 0 .
. f x g x x 0 .
. f xh x e x , x 0 .
. 0 x 1 : xh x 1 3 (*) , 3 e .
144.
. f : , , , f f . :
i. , f
f f
2
ii. 1 2 , ,
1 2
2 1 1
f f f f
. f : 0, e
fx
x xln
, ,
1
2.
i. 1
2
.
ii. 1
2
f x 0 0, ,
2
1,1
e
21,e .
145. f : , 0 ,
1z if 2z if 12
zw
z.
-
| . 28 .2421302598
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. 1 2 1 2z iz z iz
. Rolle
f x
g xx
,
. f
.
x
x
f x tlim
x dt
x t1 , f x 1
,
146. f :
x
2 0
4f x dt
1 f t, x .
. f .
. fC .
. f fC , 1fC
. fC , 1fC
147. f 1,2
f (x) 0 x 1,2 f 1 =0 , f 2 =2 , f 2 1 . . f .
. y x f.
. 0x 1,2 0f x 1 .
. :
i. f x f 1 f 2 f x
x 1 2 x
x 1,2 .
ii. f x 2 x 1 x 1,2 .
iii. 2
1f x dx 1 .
. ex y 2
f.
. 1 2 , 1,2 1 2 1 2 1f f f 2 .
148. f , 0x 0 .
g : , 1
0g x f xt dt , t, x z : z 2 i z 2 i (1),
:
-
| . 28 .2421302598
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. , () y 2x .
. , fC
2
x 0
1 1xf 5x
x xlim 10
1 2f 2
x x
.
. :
x
0
1f u du x 0
g x x
f 0 x 0
.
. g .
. 2 2
1 0z f t dt i f t dt (1) 1,2
f g .
149. . x 1f x x e 1 , x z x i x 1w e i .
. f
. x z w .
. 3 2
2
x x 2f x
x 1
.
fC
: y 2 x 4 , , * .
150. f , f x 0 x z
1f x dx 0 f 1 1 .
. f x 0
. z .
.
3
2x
z z 3 x xlim
z z 3 x x
. f x'x x 0 x 1 z 2z ,
x
2
0f t dt 3x 6x 6 0,1
151. .
1 2z ,z 1 2 1 2z z z z .
1 2w z z .
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| . 28 .2421302598
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. f x
1z 1 i 2z 1 f x i () f f 0 0 f 0 0 . 2 3
. g 1 2g x Im(z z ) Rolle ,
f
f
1 f e
1 f e
, f 1
152. f : * f x 0 x ,
: 22f x f x x 0 x 1
f 02
. f 2
f x3 x
, x .
. 1
f x 12 x .
. g : g xf x eg x , x , :
i. g xh x e ,2 .
ii. : g x 1 g x 11 e e 2
, x .
iii. g xe 2x 2012 0 ,
153. 2 1 tx
0F x e dt . :
. .
. F 0 0x 0 .
. F 0 .
. F.
. 2x1 F x e , x 0 .
.
x 0
F xlim
x,
x
F xlim
x.
154. f x :
2x-1
x
x2xf 2x t t 1 f 2x t dt e ex , z
f zz 4 f 4 z .
. f.
. z.
-
| . 28 .2421302598
86
. z.
i. x M t y,
y t 0 x 0t
-
| . 28 .2421302598
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. 6
0
111I P t dt
2 .
158. 2t 3
f tt 2
, t 1,4
2
t4
x
1
x 2g x f(t) e dt
x 1
, x 0 .
. 4
1I f t dt .
. : 2 2 2
1 t 4
x x xe e e , t 1,4 x 0 .
. xlim g x
.
159. : 1,f ,
1 1, 1 0f f
1
1, 1
, 11
x
x
g x f t dtx
x
i. g 1, ii. g
iii. g 1,
iv. ,a 1 a
1
1
1
1
a
f t dt a
f t dt
160. :f :
3 f x f x x , x i. f .
ii. 1f .
iii. 1
lim3x
f x
x
iv. fC , x x
1x 3
2x
161. 2
1
1
1
x
f x dtt x
1,2x
i. f 1
ln2
xf x
x
, 1,2x
ii. f .
iii. fC
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| . 28 .2421302598
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1. i. (1)
2 2
2
1 1 4 1 1 1 1 4 1 1 4
2 2 1 1 1
z z z z z z zz z z zz z z
zz zz z z
z
ii. 1 2,z z z,
. 1 1z 2 1z
:
22
1 2 1 2 1 2 1 2 1 1 1 2 1 2 2 22 2 2 2z z z z z z z z z z z z z z z z
2 2
1 1 2 1 2 2 1 2 1 2 1 2 1 22 1 1 2 0z z z z z z z z z z z z z z
1 2z z
.
2 2 2
1 2 1 2 1 2 1 1 1 2 1 2 2 2 1 20 1 1 2z z z z z z z z z z z z z z z z
2
1 2 1 22 2z z z z
iii.
w w x yi , ,x y
:
2 2
2 2 2 22 2
5 12 5 12
5 5 12
4 6 12 4 6 12
16 3616 36 144 1 1
144 144 9 4
w w x yi x yi
x yi x yi
x yi x y
x y x yx y
w 2 9 3a a 2 4 2
w max max 3w a w
w min min 2w w
iv. w : 2 3w z 1z
:
1 1 1 2 1 1 1 3
1 4
z w z w z w w z w w w z w w
z w
2. i. z z x yi , ,x y
(-,0)
(0,-)
(,0) (0,0)
(0,)
-
| . 28 .2421302598
89
(1) :
22
22 2 2 22 2
2 2 2 2 2
1 Im 1 1 1
1 1 1 1
12 1 2 1 4 3
4
z i z x yi i y x y y
x y y x y y
x y y y y y x y x
ii. (2)
223 3 3 3 3 3 1 0 3 1 0w w i i w i ww iw iw i ww iw iw w i w w
w x yi ,x y . 2 2 2w x y 2w w yi
2 2 2 23 2 1 0 6 1 0x y i yi x y y (4)
2 2
22 2 2
6 1 8 8
6 9 8 3 8
x y y
x y y x y
w (0,3) 8 2 2
!!! z .
, (4) = 2 2 0x y Ax By ,
: 2 21 4
2B ,
2 2
A BK
.
iii. z w (3) (4)
2 2 2 2
2 2 2 22 2
2 2 2
2
14 4 4
46 1 0 4 6 1 0 2 1 0
6 1 0
4 24 4 4
11 0 1 11 0
y x x y x y x y
x y y y y y y yx y y
x y xx y x x
yy y yy
2,1A 2,1B
iv. .
2 2 2 2
2 2 1 1 2B A B AAB x x y y
2 2 2 2
0 2 3 1 8 2 2K A K AAK x x y y
2 2 2 2
2 0 3 1 8 2 2B K B KKB x x y y
() = () .
2 22 2 2 24 2 2 2 2 16 16AB AK KB
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| . 28 .2421302598
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. u ,, . () .
3 1
10 2
y y
x x
: 1 1 2 1 2
1
B By y x x y x y x
y x
() .
3 1
10 2
y y
x x
: 1 1 2 1 2 1A Ay y x x y x y x y x () () u.
1 1 1 0
1 2 2 1 1
y x y x y x x
y x y y y
0, 1 0 1u i u i
3.
i. Vietta : 2 0x Sx P 1 2S z z (
) 1 2P z z ( )
2 22 5 0 4 5 0x x x x
1 2,z z 2 24 2 4 1 5 16
1
1,2 1,2 1,2
2
2 16 1 22 4
1 22 2 1 2
i i z iiz z z
z i
ii. :
2 2 22 2 2
1 2 1 2
2 2 2 2 2
1 2 1 2 1 2 1 2
1 2 1 2 4 1 2 1 2 16
w z w z z z w i w i i i
w i w i i w i w i
w x yi ,x y .
2 2 2 2
2 2
2 22 2 2 2 2 2 2 2
2 2 22 2 2 2
1 2 1 2 16 1 2 1 2 16
1 2 1 2 16
1 2 1 2 16 1 2 1 2 16
2 1 4 4 4 4 16 2 1 2 8 1 4 1
w i w i x yi i x yi i
x y i x y i
x y x y x y x y
x y y y y x y x y
w 1,0K 2
()
()
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| . 28 .2421302598
91
iii. w x yi ,x y
(ii) : 2Re Im 0 2 0 2w w x y y
