ekfoniseis liseis 1-200
TRANSCRIPT
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200
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200 , . forum mathematica.gr , , (, , ) . 200 2014. , , . (568 ). . , , [email protected].
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: 2015
- 200 - :
1
EMA 1
( . / )
z, w w 2= 4 w
z .w 1
=
i. z .
ii. :
a. z w 4
b. 2w 4 4 w 1
iii. : ( )4 2w 8 2 w 32Re(w) 80+ +
iv. z, w .
1
, w 2 w 1= , z .
i)
( ) ( )4 w z 4z 4 w z w 1 4 w zw z zw w z 4 w z 1 z 4 ww 1 z 1
+= = = + = + + = + =
+
: z 1 ,
= = = =
4 wz 1 1 4 w 1 w 4 1
w 1, .
,
( ) ( ) ( ) ( )2 2z 4w 2 2 z 4 2z 2 z 4 2z 2 z 4 z 4 2z 2 2z 2z 1
+= = + = + + = + + + = + +
+
2zz 4z 4z 16 4zz 4z 4z 4 z 4 z 2 + + + = + + + = = .
ii)
a. z w z w z w 2 2 4 + = + = + = .
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: 2015
- 200 - :
2
b. 2
2 w 4w 4 4 w 1 4w 1
.
2 2 2w 4 w w w 4 w w w 4 w 4 w 4w w
w 1 w 1 w 1 w 1 w 1 w 1
+ = = + = + + =
2 z 2 2 4+ = + = .
iii)
( ) ( )+ + + + + +
4 2 4 2 2 2
2 2
w 8 2 w 32Rew 80 w 8w 16 32Rew 80 w 4 32 Rew 80
w 4 32 Rew 80
, ii : 2 2 2 2w 4 4 w 1 w 4 16 w 1 . , ,
:
( )( )( )
+ + +
+ + +
2 2
2
16 w 1 32Rew 80 w 1 2Rew 5 w 1 w 1 2Rew 5
w w w 1 2Rew 5
( )2 22 w w 2Rew 4 2 2Rew 2Rew 4 4 4 + + , .
iv) z , w 2 2x y 4+ = , z w 2= =
(, 2 2y 4 x= ). :
( )( )2 2 22 22 2
z 4 z 1 z z z 4z 44 z z z 4 z z 4z w z
z 1 z 1 z 1 z 1 z 1
+ + + + = = = = = =
+ + + + +
( ) 222 2
4 z z z 44z 4z z 4
z 1 z 1
+ + = = =
+ +
( )( )
2 2
2 2
4 2yi x yi 4 2x 8 2x 8iy
2x 5 2x 5x 1 y
+ + += +
+ ++ +.
( 2 2y 4 x= )
22 4 xz w 4 , x 2
2x 5
= +
.
( )24 x
f x , x 2 , 22x 5
= +
, ( ) ( ) ( )( ) ( )22 x 1 x 4
f x , x 2 , 22x 5
+ + =
+.
( ) ( )f x 0 x 2 , 1 >
( ) ( )f x 0 x 1 , 2 <
, f 1 . , z w
x 1= . x 1= ( )2 2y 4 1 3 y 3= = = . , z 1 3 i= + z 1 3 i=
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: 2015
- 200 - :
3
4 z
w 1 3 iz 1
+= =
+ w 1 3i= + . ,
( ) ( )z , w 1 i 3 , 1 i 3= + ( )1 i 3 , 1 i 3 + .
: ( ) ( )z w 1 i 3 1 i 3 2 2i 3 4 12 4 = + = + = + = = 2 2x y 4+ = z , w
2 2x y 4+ = .
(, ( ) ( )z w 1 i 3 1 i 3 4 = + = .)
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: 2015
- 200 - :
4
EMA 2
)f : 1, +
f(1) 1, f (1) 0.= = g,
x
1 f(t)dt , x 1g(x) .x 11 , x 1
>= =
:
A. g )1, + .
B. g.
C. g )1, + .
D. a
1
b
1
f(t)dt a 1, 1 .
1f(t)dt
< < =
. f ( )
( )x
1
f t dt g ( )1 , + ,
g ( )1 , + . g 0x 1= :
( )( )( ) ( )
( )( ) ( ) ( )
x
1
xx 0f 0 f
11
1 1x 1 x 1 x 1 x 1
f t dtf t dt f xlim g x lim lim lim f 1 1 g 1
x 1 1x 1
= ========= == = ========== = =
, g )1 , + .
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: 2015
- 200 - :
5
) x 1 ,> ( )( ) ( ) ( ) ( )
( )
x x
1 1
2
f t dt x 1 f x f t dtg x
x 1 x 1
= =
.
,( ) ( )
( )x
1
x 1 x 1
f t dt1
g x g 1 x 1lim lim
x 1 x 1
= =
( ) ( )( )
( )( )
x0 0
f 0 01
2 1x 1 x 1
f t dt x 1 f x 1lim lim
x 1 2 x 1
= == ========= ==
( )( )( )( )
( ) ( )f 1x 1 x 1
f x 1 f x f 1lim lim 0
2 22 x 1
= = ========= =
.
, ( )( ) ( ) ( )
( )
x
1
2
x 1 f x f t dt, x 1g x x 1
0 , x 1
> = =
C) g 1, ( )g x 0 > ,
x 1> , ( ) ( ) ( )x
1
x 1 f x f t dt 0 > , x 1> .
1
( ) ( ) ( ) ( )x
1
G x x 1 f x f t dt= , )x 1 , + . G
( )1 , + , ( ) ( ) ( ) ( ) ( ) ( ) ( )G x f x x 1 f x f x x 1 f x = + = .
, f f )1 , + ,
( )x 1 f x> > ( )f 1 0 = )( )f 1 , + . , ( ) ( ) ( )G x x 1 f x 0 = > ,
x 1 G> )1 , + . ,
( ) ( ) ( ) ( ) ( )x
1
x 1 G x G 1 x 1 f x f t dt 0> > > .
2
( ) ( )x
1
F x f t dt , x 1= . ( ) ( ) ( )x 1 f x F x > ,
x 1> .
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: 2015
- 200 - :
6
... F 1 , x ( ) 1 , x :
( ) ( ) ( ) ( ) ( ) ( )F x F 1 F x F x
F f x 1 x 1 x 1
= = =
. ,
( ) ( ) ( ) ( ) ( ) ( ) ( )f F x
1 x f f x f x F x x 1 f xx 1
< < < < <
.
D) 1 a < < .
( )
( )
( ) ( )( ) ( )
a
1 1 1
1
f t dt f t dt f t dta 1g a g
1 a 1 1f t dt
< < .
( )x xx x xx
x x 1 1lim lim lim 0
e 1 ee 1
+ +
+ + +
==== = = =
+ ,
xx
xlim 0
e 1=
.
0= . , y 0=
f
C + . :
( ) xxx
f x 0 0 xx 0 x 0 x , e 1
= = = = =
. ,
( ) , 0 .
iii. ) g , . :
( ) ( )x
x x
xe x, x 0
g x e f x e 1
0 , x 0
= = =
( ) ( ) ( ) ( ) ( ) ( )x x xx , x 0g x f x e f x f x e 1 f x0 , x 0
= = =
=.
( ) ( )g x f x xx , x = . f, g 0 , ,
( ) ( ) ( ) ( ) xx 0
0 0 , 0 0 0 0 0
E g x f x dx xx dx xxdx x x dx x x x dx
= = ======= = = =
0 x = + = .
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: 2015
- 200 - :
9
) : ( ) ( ) ( )xx
x x
xe xxe xg x g x xx , x 0
e 1 e 1
+ = + = =
. ,
x 0 , ( ) ( )g x g x xx+ = . x 0= , ( )g 0 0= .
, ( ) ( )g x g x xx+ = , x . , ( )
I g x dx
= =
( )( ) ( )
xx g x dx xxdx g x dx
= = . ( )
g x dx
u x du dx= = . , ( )
g x dx
( ) ( ) ( )
g u du g u du g x dx I
= = = = . ,
( )
( ) ( )
= = =
= + = + =
1 1I xxdx I I xxdx x x dx
2 2
1 1x x xdx 2 0
2 2
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: 2015
- 200 - :
10
EMA 4
( .)
A. ( )h(x) lnx x, x 0, .= + +
i. h
1 e1
1h (x)dx.
+
ii. lnx x 0+ = ,
( ) 0, . +
iii. x 0
xlim
h(x)+
2x
xlim .
h (x)+
B. 2x
f(x) xln x x , x 0.2
= + >
i. f ( ) 0, + (.ii) .
ii. f ( )1,f(1)
g, 3
2x 11g(x) x .3 6
=
iii. x
f (x)lim ,
x
( ) 0, + (.ii) .
4
)
i. h ( )0 , + , ( ) 1h x 1 0 , x 0 hx
= + > >
( )+ 0 , h 1 1 . ( )1 e
1
1
I h x dx+
=
( ) ( ) ( )1u h x x h u dx h u du = = = .
( ) ( ) ( )= = = =x 1 h u 1 h u h 1 u 1 .
( ) ( )= + = + = =x 1 e h u 1 e h e u e .
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: 2015
- 200 - :
11
, ( ) ( ) ( ) ( )e
2 2e e e
1 1 11
1 u 1 e 41I uh u du u 1 du 1 u du
u 2 2
+ + = = + = + = =
.
ii. h , lnx x 0+ =
.
h:
( )( ) ( ) ( ) ( )h
xx 0h 0 , lim h x , lim h x ,
+ +
+ ========= = + =
0 ,
( )h x 0= 0> .
iii. ( )x 0 x 0 x 0
x x 1 1lim lim lim x 0 0 0 0
x lnx x lnxh x+ + +
= = = = =
+ + + ,
( ) ( )2 2x xx x
lim lim 0h x x lnx + +
= =+
,
( ) ( ) ( )x
2 2 2
x 1 10
x lnx x lnx
+
=+ + +
.
)
i. f ( )0 , + ,
( ) ( ) ( )2x
f x xlnx x lnx x h x , x 02
= + = + = >
.
( ) ( ) ( )h
x h x h 0 h x 0
> > = > . ( ) ( ) ( )h
0 x h x h 0 h x 0
< < < = < .
0 +
( )f x +
f
min
f h . f
.
ii. ( ) 1f 12
= ( )f 1 1 = . , 1 3 : y x 1 : y x
2 2+ = = .
:
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: 2015
- 200 - :
12
( )
( ) ( )
=
= + + + = + + + = =
+ + + =
3 32 2 3 2
3
y g xx x11 3 1
x x x x 0 x 3x 3x 1 03 3 6 2 3 3y x2
x 1 3x x 1 0
( )( ) ( ) ( )( ) ( ) ( )( )( )
+ + + + = + + + = + + + =
+ + = =
2 2 2
2
x 1 x x 1 3x x 1 0 x 1 x x 1 3x 0 x 1 x 2x 1 0
x 1 x 1 0 x 1
: g
C ( )( )1 , g 1 3 : y x2
= [
, ( ) ( )( )y g 1 g 1 x 1 = + ]. ,
Bii .
iii. ( ) ( ) ( )
( )
0
0
x x x x x
11f x h x x lnxx lnx 1xlim lim lim lim lim 1
x x x 1 x
+ ++= = == = = +
.
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: 2015
- 200 - :
13
5
z,w z,w 0, z w = 2w z zi.= +
:
i. w .
ii. z, w .
iii. z, w .
5
( )z w 0 0 , z w 0= = > .
i. w , w i , 0= . :
2 2w z zi i i zi= + = +
( ) ( )( )
= +
= + = + = + + = +
= + =
z yi2 2 2 2
2
i zi i x yi i i xi y x i y
x 0 y 0
( )2x , y = = . , 2z i= . ,
= + = + = = =2 4 2 4 2 4z w i 0 0 , .
ii.
( ) ( )= + = + = + = + = + = + + =2 2w z zi w z zi w zz zi w z z i w z z i z i z i 1.
, z x yi= + , x , y , z
( ) 0 , 1 1. , ( ) ( )2 2x y 1 1 1+ + = .
w a i , a , = + .
( )= + + = + + + = + + + 2 2 2 2 2w z zi i x y x yi i a i x y xi y
( )2 2a x y y
2 x
= + +
= . ( ) + = 2 21 x y 2y . ,
( ) ( ) = =
a y2 3
x
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: 2015
- 200 - :
14
, z x yi= + w y xi= + .
z : ( )2 21C : x y 1 1+ + = .
( )3 w
( )2 22C : 1 x y 1 + = . ( ( ) 1 , 0 1). 1 2C , C
, ( ) 2 1 1 2= < + = .
iii. z , w , ( ).
( ) 0 , 0 ( ) 1 , 1 . [ ( ) , 1 , 1 ( ) ( )1 , 2 ].
, z w 0= = ( z w 0 )
z w 1 i= = ( = + 2w z zi
( )1 i 2 1 i i i 0 , = + + = ).
:
( ) =
= + + =
2w 1 1
w z zi *z i 1
. ,
. ( )* ,
( ) ( )1 2C , C .
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: 2015
- 200 - :
15
6
h, :
xh (x) h(x) x e 1, x (1) = + + h(0) 0.=
A.
i. ( )x x1 xe e dx + .
ii. h.
iii. ( )xh(x) x e 1 , x ,= h(x) 0, x .
B. f 0,1 , :1
0
f(x)dx 1= (2)
( )1
f(x)
0
1 f(x) e dx 0 (3). f(x) 1, x 0,1 =
6
A)
i.
( ) ( ) ( ) ( ) + = = = = +
x x x x x x x1 xe e dx 1 e xe dx x xe dx x xe dx x xe c
ii. ( ) ( ) ( ) ( )( ) ( )
= + + = + +
= +
xe
x x x xx
x x x x
h x h x x e 1 e h x e h x xe 1 e
e h x e h x xe 1 e
( )( ) ( )( ) ( ) = + = i)
x x x x xe h x xe 1 e e h x x xe ,
( ) = + x xx e h x x xe c , x .
= =x 0 0 c . , ( ) ( ) ( ) ( ) = = = x x x xe h x x xe h x xe x h x x e 1 , x .
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: 2015
- 200 - :
16
iii. xxe x , x .
x 0= 0 0 .
( )>
> > > >
x x 0ex 0 x xx 0 e e e 1 xe x .
( )
x x 0ex 0 x xx 0 e e 1 e 1 xe x .
) ( ) ( )g x 1 f x , x 0 , 1 = . ,
( ) ( )( ) ( )1 1 1 1
0 0 0 0
g x dx 1 f x dx dx f x dx 0= = = . ,
( ) ( ) ( ) ( )= = g x 1 f x f x 1 g x . , ( )( ) ( ) ( ) ( )1 1
f x g x 1
0 0
1 f x e dx 0 g x e dx 0
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( )
1 1 1 1g x g x g x
0 0 0 0
1g x
0
1g x e dx 0 g x e dx 0 g x e dx g x dx
e
g x e g x dx 0
( ) ( )( ) ( )( ) ( )1 1
g x
0 0
g x e 1 dx 0 h g x dx 0 1 .
hog 0 , 1
( )( )hog x 0 , x 0 , 1 . hog
( )( ) > 1
0
0 , 1 h g x dx 0 , ( )1 . , ( )( )h g x 0 , x 0 , 1 = ,
( )g x 0= , x 0 , 1 ( ( )h x 0= x 0= ),
( )1 f x 0 = , ( )f x 1 , x 0 , 1 = .
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: 2015
- 200 - :
17
7
( )
2x 1
20
1F(x) dt
4 t 1
= +
a. F.
b. : ( ) F x x , x , .6 2 2
=
c. ,
f
( )21
f(t) ,4 t 1
= +
x= 3 1.
d. 3
21
1dx.
4 x
7
a) ( )( )21
t4 t 1
= +
( ) ( )2 24 t 1 0 t 1 4 + > + <
t 1 2 2 t 1 2 3 t 1+ < < + < < < . , ( )3 , 1 .
F 0 , 2x 1 ( )3 , 1 . ,
( )3 2x 1 1 2 2x 2 x 1 , 1 < < < < . , F ( )1 , 1 .
( )3 , 1 , , F
( )1 , 1 ,
( ) ( ) ( ) ( )( )
2x 1
20
1F x t dt 2x 1 2x 1 2
4 2x 1 1
= = = = + 2 2
2 1
4 4x 1 x=
.
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: 2015
- 200 - :
18
b) g : ,2 2
, ( ) ( ) g x F x x6
= + . g
,2 2
, :
( ) ( ) = = = = 2 21 1
g x F x x 1 x 1 x 11 x x
x 0
,
2 2
x x1 1 0
xx
>
= =========== = ,
x2
< . , g
,
2 2
. ( )0
0
g F t dt 0
6 6 6 6
= + = =
. ,
( ) ( ) = = , ( )t 3 , 1 .
3
2x 1 3 1 2x 3 x2
= = = .
:
( ) ( )3
2 13 1 )2
0 0
3 E f t dt f t dt F F
2 3 3 6 6
= = = = = =
..
d) 3
21
1I dx
4 x=
x t 1 dx dt= + = .
x 1 t 0= = .
x 3 t 3 1= = .
, ( )
3 1 )
20
1 I dt
64 t 1
= = +
.
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: 2015
- 200 - :
19
8
f R
( ) ,f(a) ( ) ,() , 0 < < C f
.
A. :
i. f(a) f()
a =
ii. f ( ), ( )f (x) 0, x , ,
fC
fC .
B. f ( , 0 :
i. x
f (x)lim L
x
= ( )
x
f x 1 f(x)lim L
x
=
ii. f(x)
g(x) , x 0x
= < f(0) 0= .
C. ( )( )B ,f fC ( )( )A ,f a
( ) , :
( )( ) ( ) ( ).f f f a =
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: 2015
- 200 - :
20
8
A)
i. ( ) ( )f f a
a
=
. ( )
: y x= .
( ) ( ) ( )
( ) ( ) ( )( ) ( )
=
= =
= =
f a A f a a
af
B f
f a
a
f .
ii. ( ) ( )f x
: a , , xx
= . a , ,
.
( )a , , .
( ) ( ) a = ( i)
, Rolle ( ) ( )0 0x a , : x 0 = . , ( )( ) ( )
2
x f x f x x
x
= .
, ( ) ( ) ( )0 0 0 0 x 0 x f x f x = = . , ( ) ( )( )0 0 0 : y f x f x x x =
fC ( )( )0 0M x , f x ( ) 0 , 0 .
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: 2015
- 200 - :
21
( ) fC ( )( )1 1N x , f x .
( ) ( )0 1f x f x = . Rolle f
( ) ( ) = 0 1 0 1x , x x , x : f 0 , .
)
i. f ( ( ) , 0 f , 0 .
ii.
x 0<
... f
( ) ( ) ( ) ( )( ) ( ) ( )x xf x f x 1
x 1 , x x 1 , x : f f x f x 1x x 1
= =
.
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )
< < > > > >
< <
f
x xx 1 x f x 1 f f x f x 1 f x f x 1 f x
f x 1 f x 1 f x f x
( ) ( ) ( ) ( ): x 0 f x 1 f x 1 f x f xx x x
< > > . ,
( )
=
x
f xlim L
x
x x 1
) (
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: 2015
- 200 - :
22
( )( )
x x
f x 1
f x 1 x 1 Llim lim L
x x 1
x 1
= = =
( ) ( )x
x 1
x
f x 1 f lim lim L
x 1
=
======= =
. ,
( ) ( )
=
x
f x 1 f xlim L
x.
iii. g ( ), 0 , ,
( ) ( ) ( )2xf x f x
g x ,x
= x 0< .
... f ( ) ( ) ( ) ( ) ( )x xf 0 f x f x
x , 0 x , 0 : f 0 x x
= =
( ( )f 0 0= ).
, ( ) ( ) ( ) ( ) ( ) =
A. :
i. ( ) ( ) (x
0f x f t dt,x 0,3 .>
ii. ( ) ( )x
x
0h x e f t dt=
0,3 .
iii. ( ) (x
0f t dt 0,x 0,3 .>
B. :
i. ( )2
x
0 x f(t)dt
= 0,3 .
ii. ( ) ( )1
12
0 02f t dt f t dt.
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: 2015
- 200 - :
24
( ) ( ) ( ) ( )g x f x f x 0 , x 0 , 3 = > g 0 , 3 g 0 , 3 . ,
( ) ( ) ( ) ( ) ( > > = > x
0
x 0 g x g 0 0 f x f t dt 0 , x 0 , 3 .
ii) h 0, 3 , , ( )0 , 3 , :
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )x x x x
x x x x x
0 0 0 0
h x e f t dt e f t dt e f t dt e f x e f x f t dt 0 = + = + = >
, ( )0 , 3 ( )) xe 0 > ). , h 0 , 3 .
iii) ( ) ( ) ( ) ( )0h
0
0
x 0 h x h 0 e f t dt 0 h x 0
> > = = > , (x 0 , 3 .
, ( ) ( ) ( ) ( >
> > >
xx xe 0 A)a)
x
0 0
e f t dt 0 f t dt 0 f x 0 , x 0 , 3 * .
) i) ( )0 , 3 ,
( ) ( )2
x
0
x f t dt
= =
( ) ( ) ( ) ( )
= =
x x x
0 0 0
2 f t dt f t dt 2 f t dt f x ( ) ( ) ( ) ( )x
2
0
x 2 f x f t dt f x 0
+
= + >
,
(*) ( ) ( )f x f x > ( )f x 0 , > ( )x 0 , 3 . ,
( ) ( )x
0
f t dt 0 , x 0 , 3> . , ( ) x 0 > , ( )x 0 , 3 0 , 3 .
ii) ( ) ( ) ( ) ( ) ( )
< <
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: 2015
- 200 - :
25
... ( )( )
( )
= =
2 2
1 1
21 1 1, 1 x , 1 : x 2 1 2
2 2 1 21
2
.
,
( ) ( ) ( ) ( ) ( )
< < < < + > > + 3 3 3
32
22 2 2
x 2 f 2 x 1 4 f 2 dx 0 x dx f 2 x 1 4f 2 0 x dx f 2 1
( ) ( ) > + = 3
2
E f 2 1 , x dx .
-
: 2015
- 200 - :
26
10
f : ,
( ) ( ) ( ) ( ) ( )x
2
a
3f 3,f 2,f x f f t dt,x ,
8 = = = + + .
i. .
ii. ( )f x 0 ,> x a, .
iii. ( )
af x dx 1.=
iv. ( ) a, ( ) 1f . a
=
10
i. x a= ( ) ( )= + =33 f 0 f 88
. f
+ 2 f a , ,
( ) +x
2
a
f t dt a , . , f a , ,
( ) ( )2f x f x = + , x a , .
( ) ( )= = + = + = + =2x a f a f a 2 3 4 3 1 .
ii. ( ) ( )2f x 1 f x 0 = + > , ( )x a , f a , f a , .
, ( ) ( ) ( ) ( ) ( ) >a x f a f x f 3 f x 8 f x 0 , x a , .
iii.
( ) ( ) ( )= = = + = + =+
2 2
2
1u f x du f x dx du 1 f x dx du 1 u dx dx du
1 u.
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: 2015
- 200 - :
27
( )( )
= = =
= = =
x a u f a 3
x u f 8
: ( ) ( ) 8 8 8
2 2
2 3a 3 3
1f x dx u du 1 u du 1 u
1 u
= = + = + = +
2 21 8 1 3 9 4 1= + + = =
iv. F : a , , ( ) ( )x
a
F x f t dt= . F a ,
( )a , . , ( ) ( ) ( ) ( )F F a
a , : F a
=
,
( )( )
iii)a
f t dt1
f a a
= ==
. , f a , , 1 1.
-
: 2015
- 200 - :
28
EMA 11
) )0, 0, , + +
( ) ( ) )1f x f x , x 0, . = +
i. f .
ii. ( )x 0lim f x
( )xlim f x .+
iii. ( )( ) ( ) )x
0
f f x tf t dt, x 0, .= +
iv. )x 0, + ( )f x x> ( )f x x.<
v. ( ) ( )f x x,x 0,= + .
vi. ( )f x x, x > > ( ) ( )f x x, x 0, .<
vii. ( ) 21f x x , x 2
> > ( ) ( )21f x x , x 0, .2
<
viii. ( ) ( ) )1xf x f x , x 0, , = + ,
f.
11
i.
) ( ) ) ( )( ) ( ) )1f
1
f
f x 0 , x 0D 0 , f A 0 , f x 0 , x 0 f 0 ,
f x 0 , x 0
= + = + > > +
.
ii. ( ) ( )x 0lim f x f 0 0
= =
( )xlim f x +
= + , )1fD 0 , = +
-
: 2015
- 200 - :
29
iii.
( ) ( )( )( ) ( )( )
( ) ( )
x
0
x f f x
x h xh x f f x dt
=
==
( ) ( ) x h x C= +
C 0=
iv. )x 0 , + . ( ) ( )f x x hf x x> < , ( )f 0 0= .
v. ( ) ( )g x f x x , x 0= >
( ) ( ) ( )1g x f x 1 f x 1 = =
( ) ( )( ) ( )( )1x 0 x 0 x 0lim g x lim f x 1 lim f x 1 0 1 1 0
+ + +
= = = = < .
, ( )g x 0 < 0 ( )a 0 : g a 0 > <
( ) ( )( ) ( )( )1x x xlim g x lim f x 1 lim f x 1 0 + + +
= = = + >
, ( )g x 0 > + , , ( ) a 0 : g 0 > > >
( ) ( ) . .
a , a , : g 0 =
( ) ( )1
1
g x f x 1 , x 0g
f
= >
( ) ( )g
0 g g 0 0
> < = . , ( )g 0<
( ) ( )( ) ( )
( ) ( ) ( )
( )x x x
x
1
DLHx x x
f xlim g x lim f x x lim x 1
xlim g x
f xlim lim f x lim f x
x
+ + +
+ + +
+ + +
= =
= +
=== = = +
.
, ( )g x 0> + . , ( ) 0 : g 0 > > > .
( ) ( ) . .
, , : g 0 =
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
g
g
g
x g x g 0
x g g x g 0
0 x g 0 g x g
> > =
< < < < =
< < > >
( ) x 0= .
x
1
2 +
g - +
g
..
( )g
-
: 2015
- 200 - :
30
vi. ( ) ( ) ( )x g x g 0 f x x> > = > .
( )( ) ( ) ( )
0 x g x 00 x g x 0 f x x
x g x 0
< < <
( ) x 0 <
viii. ( ) ( )1xf x af x , x 0 =
( ) ( ) ( )a 1x
axf x af x 0 f x c x
= = , x 0>
( ) aa
1 a
x f c
c
c
= =
=
=
, ( )
( ) ( )1 a a
1 a af x x , x 0
f x xf 0 0
= >
= =
x 0 +
( ) x + -
( ) x
-
: 2015
- 200 - :
31
EMA 12
f , :
( )f a 1 a 1,f(a) a > < ( )f a 1 a 1,+ > + .
i. , f 1 -3
.
ii. , f (x) 1 f(x) x =
( ) 1, 1 . +
iii. f
( ) 1, 1 + : ( )f 0. >
12
i) ( ) ( )g x f x x= . Bolzano g a 1 , a
a , a 1 + .
g a 1 , a a , a 1 + , f , g
.
( ) ( ) ( )( ) ( ) ( ) ( )
g a 1 f a 1 a 1 0g a 1 g a 0
g a f a a 0
= >
-
: 2015
- 200 - :
32
1 2
x , x .
( )1 2x , x ( ) ( ) ( ) x xx e h x e h x =
( ) ( ) ( )( ) 1 1x x1 1 1 1x e h x e f x x 0 = = =
( ) ( ) ( )( ) 2 2x x2 2 2 2x e h x e f x x 0 = = =
( )0 1 2x x , x ( ) ( ) ( ) 0 0x x0 0 0x 0 e h x e h x 0 = =
( ) ( )( ) ( )
0 0
0 0 0
h x h x 0
f x x f x 1
=
=
iii) ... f a 1 , a .
( )1 a 1 , a : ( )( ) ( )
( ) ( ) ( )1f a f a 1
f f a f a 1a a 1
= =
.
... f a , a 1 + .
( )2 a, a 1 + : ( )( ) ( ) ( ) ( )2
f a 1 f af f a 1 f a
a 1 a
+ = = +
+ .
... f 1 2
, .
( )1 2 , : ( )( ) ( )2 1
2 1
f f f
=
.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 2 1
f a 1 f a f a f a 1 f a 1 f a 1 2f af 1
+ + + + + + = =
.
( )( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( )
2 1
, f a 1 a 1f a 1 f a 1 2a
f a 1 a 1
f a a 2f a 2a
f a 1 f a 1 2f a 0
0.
+ > + + + >
> < >
+ + + >
>
, ( ) ( )1 : f 0 > .
-
: 2015
- 200 - :
33
EMA 13
f : f(x) 0
x 1
2
0
t f(t)dt 1 .
( )1
2 2 4
0
g(x) f(t) 2xt f(t) 5x t dt,= + x g(0) 0.>
a. f(x) 0> x .
b. g(x) 0= .
c. 1
z g(x) 0= 1
z 3= :
i. f,
xx x 0= x 1=
ii. ( )12 Im z 3 < .
13
) ( ) ( )f
f 1f x 0 , x
: ( ) ( ) ( )
= +
1 1 1
2 2 4
0 0 0
g x f t dt 2 t f t dt x 5x t dt
( ) ( ) ( )
= +
151 1
2 2
0 0 0
tg x f t dt 2 t f t dt x 5x
5
( ) ( ) ( )
= +
1 1
2 2
0 0
g x f t dt 2 t f t dt x x .
( ) ( )( )
( )
= > > 11
g 0 f t dt 0 f x 0 , x . (
( ) ( )1
0
f x 0 , x f t dt 0<
-
: 2015
- 200 - :
34
b) ( ) =g x 0
( ) ( )1 1
2 2
0 0
x 2 t f t dt x f t dt 0
+ = .
( ) ( ) ( ) ( )
< <
-
: 2015
- 200 - :
35
EMA 14
2
x xg(x) e 1 x , x 02
= 2xf(x) e , x 0=
i. g .
ii. x 0 g(x) 0. ;
iii. f .
iv. , f 0,1
( )f 0,1 .
v. 2
1 ex
0 1
e dx ln xdx e+ =
vi. 2
1x
0
ee dx .
2>
14
i) g ) + 0 , ( )+ 0 ,
:
( )
= = >
2x xxg x e 1 x e 1 x , x 0
2.
, : lnx x 1 , >x 0 . x xe : x xlne e 1
xx e 1
xe 1 x 0 ( =
=x 0 )
, ( ) >g x 0 , >x 0 . , g ) + 0 , ( g 0).
-
: 2015
- 200 - :
36
ii)
( )
( ) ( ) ( )( )
= =
> > >
g
x 0 g 0 0g x 0 , x 0
x 0 g x g 0 g x 0.
=x 0 .
iii) f ) + 0 , , . f ( )+ 0 , , :
( ) ( ) = =2xf x e
( )= = >22 xx 2e x 2xe 0 , >x 0 , >2xe 0 , x . , f
) + 0 , f . 2xy e= 2ln y x=
ln y 0 ,
= =x 0
x ln y x ln y . ( ln y 0 y 1 ). ,
) ( ) + = 1 1f : 1 , , f x lnx .
iv) 1f fC , C =y x .
+ xe x 1 , x , x 2x : +2x 2e x 1 . , + >2x 1 x (
). , ( ) = >2xf x e x . , fC
=y x .
( ) ( ) ( )
======= =
f
f 0 , 1 f 0 , f 1 1 , e
-
: 2015
- 200 - :
37
v) = e
1
I lnx dx .
( )=
= = = =
2u
2
x e
u
1 1 1 1u lnx du lnx dx du dx du dx
2u x 2ue2 lnx.
= =x 1 u 0 . = =x e u 1 .
,
( ) = = = = + = 2 2 2 2 2 2
11 1 1 1 1u u u u x x
0 0 0 0 00
I u 2ue du u e du ue e du e e dx I e dx e
()
2
1x
0
e dx 3
f
C .
e
1
lnx dx 2 1fC
, , 2 1= .
, ( ) ( ) ( ) + = + = + = = = = 2
1 ex
3 2 3 10 1
e dx lnx dx 1 e e .
vi) ii) : ( ) g x 0 , x 0 . x 2x : ( ) 2g x 0 .
h ( ) ( )= 2h x g x 0 0 , 1 . ,
( )
> >
2
41 1x 2
0 0
xh x dx 0 e 1 x 0
2
> + + = + + = >
2
13 51
x
0 0
x x 1 1 43 ee dx x 1
3 10 3 10 30 2.
-
: 2015
- 200 - :
38
EMA 15
)f : 1, , +
)x
3 2
1
f (x) x f(x) 2 tf(t)dt x 1, x 1,+ = + .
i. f(1) 0=
ii. f .
iii. )x 10 f(x) , x 1,2
+ .
iv. f .
v. f.
vi. 3
1
2 3 f(x)dx > .
15
i. x 1= :
( ) ( ) ( ) ( ) ( ) ( )( )+ = + = + = 1
3 3 2
1
f (1) f 1 2 tf t dt 0 f 1 f 1 0 f 1 f 1 1 0 ( )f 1 0 = ,
( )2f 1 1 0+ > .
ii. f )1 , f + ) ( )1 , t f t +
) ( )x
1
1 , tf t dt + )1 , + , ( ) ( )x
1
tf t dt xf x =
.
, f 3f ( )2x f x . ,
, :
( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )x x
3 2 3 2
1 1
f x x f x 2 tf t dt x 1 f x x f x 2 tf t dt 1 + = + =
-
: 2015
- 200 - :
39
( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )
( ) ( )2 2 2 2
2 2
13f x f x 2xf x x f x 2xf x 1 f x 3f x x 1 f x 0
3f x x+
+ + = + = = >+
, x 1 . , f 1 ( ). , f )1 , + .
iii.
( )
( ) ( )( )
f
x 1 , f 1 0f x 0 , x 1
x 1 f x f 1 0
= =
> > =
. ( = x 1= ).
, ( ) ( ) 22 21 1
f xx3f x x
= < > + < >
.
, )g : 1 , + , ( ) ( )1
g x f xx
= + )1 , + (
1).
, ( ) ( ) ( ) ( ) ( ) x 11 1x 1 g x g 1 f x 1 f x 1 f xx x x
> < + < < < .
: x 1 x 1
, x 1x 2
< > .
[, ( ) ( )
22
2
x 1x 1 x 1 x 1 x 1 x 1 x 1 1x 1 0
x 2 x 2 2 2 2x
< < < <
( )2
2
2x 2 xx 1 0
2x
< , ,
2 2x 1 0 , 2x 0 , x 2x 2 0 > > + < ( )]. , ( ) x 1f x , x 12
< >
( ) 1 1f 1 02
= = . , ( ) x 10 f x , x 1
2
.
iv. ( ) ( )2 21
f x , x 13f x x
= +
, f
( )1 , + , .
( ) ( ) ( )( )( )( ) ( ) ( )
( )( )
+ = = + = < > + + +
2 2
2 22 22 2 2 2
6f x f x 2x1 1f x 3f x x 0 , x 1
3f x x 3f x x 3f x x,
-
: 2015
- 200 - :
40
( ) ( )2x 0 , f x 0 , f x 0> > > . , f )1 , + , f 1. f
)1 , + .
v. f : )( ) ( ) ( ) (f
xf 1 , lim f x , f 1 0 , 1
+
+ ====== = ,
( ) 1f 1 13 0 1
= = +
( ) ( ) 22 21 1
f xx3f x x
= .
,
( ) ( ) ( ) ( )( )( )
( ) ( ) ( ) ( )
+
= == = = + =
0
0
x 1 x 1 x 1 x 1
xF xG x G 1 xF x F x xf xlim lim lim lim F 1 f 1 e
x 1 x 1 1x 1,
-
: 2015
- 200 - :
43
( ) ( ) ( ) ( ) ( )( ) ( ) ( )0
2 0
x 1 x 1 x 1
G x G 1 x f xlim lim lim 2xf x 2x f x 2f 1 2f 1 0 e e
x 1 x 1+ + +
= = + = + = + =
.
, ( )G 1 e = .
, ( ) ( )x
x
1x
f t dt e , 0 x 1G x
xe , x 1
+ < x 1> , ( ) ( )x x xG x xe e xe 0 = = + > .
, G 1. (
( ) ( ) ( ) ( )x
0xx0
1
x 1 x 1 x 1
f t dt e eG x G 1 f x e
lim lim lim 2ex 1 x 1 1
+ += == =
( ) ( )0
x x x0
x 1 x 1 x 1
G x G 1 xe e e xelim lim lim 2e
x 1 x 1 1+ + +
+= == =
. , ( )G 1 2e 0 = > .
, ( )G x 0 > ( )0 , G+ ( )0 , + .
t : t
t e 1e t 1 1 , t 0t t
+ + > .
,
+ = + = +
t t1 1 x
1
xx x 1
e e1dt 1 dt t lnt 1 x lnx dt x lnx 1
t t t,
( )x 0 , 1 .
, ( )+ +
+ = = tx
x 0 x 01
elim x lnx 1 lim dt
t.
:
G (0 , 1 .
( ) ( )+ +
= + =
x
x
x 0 x 01
lim G x lim f t dt e ( ) ( )
-
: 2015
- 200 - :
44
, Bolzano G 1
x , 1 ( ) ( )0 1 0x x , 1 : G x 0 = (
0x , G ).
( ) ( )
< < = G
0 0x x G x G x 0 G ( )00 , x .
( ) ( )
> > = G
0 0x x G x G x 0 G ( )0x , 1 .
, G 0
x .
iii) ) ( )( )G 0 , + .
iv) G ,
G. G ( )( )A 1 , f 1 :
( ) ( )( ) : y G 1 G 1 x 1 =
( ) : y 0 e x 1 =
( ) : y e x 1=
, ( ) ( )G x e x 1 , x 0 ( = x 1= ).
-
: 2015
- 200 - :
45
EMA 17
f x x x
2 2 2
0 0 02ttf(t)dt t f (t)dt tdt +
x 0, ). +
a. xf(x) x= x 0, ). +
b. f .
i. f .
ii. 0, .
iii. 2 3 2009f(x) f(x ) f(x ) f(x )+ = + 0, .
c. f
C , x1= x
1.+
d. N xf(x), 1= = x.
17
a) ( ) ( ) ( ) ( )x x x x
22 2 2
0 0 0 0
2f t dt t f t dt tdt tf t t dt 0 1 +
( )2
tf t t ( )2
tf t 0 ,
( ) ( )x
2
0
tf t t dt 0 , x 0 2 .
( )1 ( )2 ( ) )x
2
0
tf t zt dt 0 , x 0 , = +
( ) ( ) ( )x
2 2
0
tf t t dt 0 xf x x xf x x,x 0
= = .
-
: 2015
- 200 - :
46
b)
i. ( ) ( )( )
x, x 0
xf x f x xf 0 , x 0
>
= =
.
)0 , + 0,
( ) ( )x 0 x 0
xf 0 lim f x lim 1
x+ + = = = .
, ( )x
, x 0f x x
1 , x 0
>
= =
x 0> f ( ) 2xx xx
f xx x
= =
.
0
x 0= , :
( ) ( ) ( )
( )
0
0
2 DLH x 0x 0 x 0x 0 x 0 2
x1f x f 0 x xx x x 11 1xlim lim lim lim lim 0 0
x 0 x 2 x 2x x+ + + +
= = == = = =
.
, 0, ( )f 0 0 = .
( ) 2xx x
, x 0f x x
0 , x 0
> =
=
.
ii. ( )g x xx x , x 0= 0, ( )g 0 0 0 0 0= =
( ) ( ) ( )g x xx x xx 0 , x 0 , = = < , g
0 , .
( ) ( ) ( ) ( )x 0 g x g 0 g x 0 f x 0> < < < , f
0 , .
,
0.
( )f 0 1= ( ) f 0
= = .
iii. 0 1 ( ) ( ) ( ) ( )2 3 2009f x f x f x f x+ = + .
-
: 2015
- 200 - :
47
( ) ( )( ) ( ) ( ) ( ) ( )
( )+ > + < + > + ,
( )0 , 1 .
0 1.
: 0 1
)0 , + .
c) f y x 1= , x > ,
x 1 1 x x x 1
x x 2 2 6 < < < < < .
( ) ( )h x f x x 1 , x 0=
0 , .
( )h 1 0= ( ) ( )h x f x 1 0 = < , h , 1
.
f x x ( ) *f x 0 x , = = .
1= , x = .
y x 1= x x y 0 x 0= = .
-
: 2015
- 200 - :
48
f
x x , C y x 1= ,
( )E : ( ) ( )
1
1E 1 1 f x dx
2= + .
x 1
, x 1 , x x
1 x
0 , x 1 , x x
(
x2
= )
( ) ( )
11 1 1
1 xdx 0 f x dx lnx f x dx ln 1
x x
> < < <
.
, ( ) 1 1 1 12
< + < +
.
d)
( )s x x= y x 1= .
( )w x x x 1 , x 0=
1, ( )w 0 0 0 1 0= = , ( )w 1 1 11 0= = .
( ) ( ) ( ) ( )x 0 , w x 0 f x 1 f x f 1 x 1 = = = = .
0 1 .
0 x 1< < ( ) ( ) ( )xf x f 1 1 x x 1 0 w x 0x
> > > > .
,
( ) ( )11 1 2
0 0 0
xE w x dx w x dx x 1
2
= = = =
11 1
2
=
.
-
: 2015
- 200 - :
49
EMA 18
A. 0> ln 3 0.+ =
B. ( )f : 0, ,+ ( )1f(x) 1 lnx 2 .x
=
i. f(x) .
ii. , :
a. x 0,> : ( )2 1
f(x) 0.
+
b. o
x > o o
f(x ) f (x ) 0.+ =
18
) ( )g : 0 , + ( )g x lnx x 3= + . g
1 , e ( ) ( ) ( )( )g 1 g e 2 e 2 0 = < . , Bolzano
( ) ( ) 1 , e : g 0 = , ln 3 0+ = . g + ,
( ) 1g x x 0 , x 0x
= + > > . , g ( )0 , + . , .
)
i. f ( )0 , + ,
( ) ( ) ( ) ( )21 1 1 1 1
f x 1 lnx 2 1 lnx 2 lnx 2 1x x x xx
= + = + =
( )2 2
g xlnx 2 lnx 2 x 1 lnx x 31 1 11 , x 0
x x x x x x x
+ + = + = = = >
. ,
( ) ( )f x 0 g x 0 x = = = ( ) ( ) ( ) ( )g
f x 0 g x 0 g x g x
= > > > .
-
: 2015
- 200 - :
50
ii.
- 0 +
( )f x - +
f
f (0 , ) , + . f .
iii. f :
( ) ( ) ( ) ( ) ( ) ( )
> + + =
2 2 1 1
f x f , x 0 f x f
( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( )
2 2 2
)
ln 3
2 2 2
1 1 1 1 1 111 ln 2 3 2
1 1 10. , f x 0 , x 0.
=
= + ========= + = + =
= + = + >
) ) : , + , ( ) ( ) ( ) x f x f x= + . ,
( ) ( ) ( ) f f = + = ( ) ( )( ) 1 ln 21
1 ln 2 0 0
= + = < < > > > < = 0.
, ( ) ( ) ( ) + + +
= + = + + = + x x xlim x lim f x lim f x 0 (
( ) ( ) ( )xlim f x 1 0 2 +
= + = +
( ) ( )
( )2x x x 2
lnx x 3lnx x 3lim f x lim lim
x x
+
+
+ +
+ + = === =
x
11 0 1xlim 0
2x
+ += = =
+ ).
, ( ) +
= + xlim x ( )> > : 0
Bolzano ,
( )0 0x : x 0> = , ( ) ( )0 0f x f x 0+ = .
O
-
: 2015
- 200 - :
51
EMA 19
2f(x) x z 4 3i x 2010, x ,= + + z
z 4 3i. + f(x) 1 1
,f ,2 2
xx, :
i. z.
ii. z z .
iii. 9 z 4 3i 11. + +
19
i. f , , ( )f x 2x z 4 3i , x = + .
1
f 02
=
. 1
f 1 z 4 3i2
= +
. , 1 z 4 3i 0 + =
( )z 4 3i 1 z 4 3i 1 = + = .
z ( ) 4 , 3
1= .
ii. ( )1c : z 4 3i 1 + = , ( ) ( )2 2
1c : x 4 y 3 1 + =
( z).
z ( )2c
x x , ( ) ( )2 22c : x 4 y 3 1 + + =
( ( ) 4 , 3 , 1= ).
: ( )maxz z 2 6 2 8 = + = + =
( )minz z 2 6 2 4 = = =
-
: 2015
- 200 - :
52
iii. : ( ) ( )z 4 3i z 4 3i 8 6i z 4 3i 8 6i 1 10 11+ + = + + + + = + = .
, ( ) ( )z 4 3i z 4 3i 8 6i z 4 3i 8 6i 1 10 9+ + = + + + = = .
, 9 z 4 3i 11 + + .
-
: 2015
- 200 - :
53
EMA 20
2009 2007f(x) x x x, x= + +
g(x).
a. f f
.
b. ( )( )fog 2009 ( )( )fog 2008 .
c. f (0,0)
f 1f .
d. : 1f(x) f (x)< x 0< 1f(x) f (x)> x 0.>
20
) f . f , , :
( ) 2008 2006f x 2009 x 2007 x 1 = + + ,
( ) = 2007 2005f x 2009 2008 x 2006 2007 x ,
( ) ( )2005 2f x x 2009 2008 x 2007 2006 = + .
, ( )f x 0 > , x f f 1 1 .
: ( ) 2005f x 0 x 0 x 0 = = =
( ) 2005f x 0 x 0 x 0 > > >
- 0 +
f(x) + +
f(x) - +
, f ( )0 , + ( ), 0 . f
( )( ) ( )A 0 , f 0 A 0 , 0= .
O
-
: 2015
- 200 - :
54
b)
f
f
fog .
[ 1 2
x , x
( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )
< > > >g f
1 2 1 2 1 2 1 2x x g x g x f g x f g x fog x fog x
, fog .]
, ( ) ( ) ( )( )fog
2008 2009 fog 2008 fog 2009
< > .
c) f
C
( ) ( ) ( ) ( ) ( ) ( ) ( )A 0 , 0 : : y f 0 f 0 x 0 , : y 0 1 x 0 = = ,
( ) : y x := 1f fC , C .
d)
x 0> f y x=
( ) >fC f x x , ( )x 0 1> . ( ) ( ) ( )1f x f x x 0 2> > .
: a 0> ( ) ( )1f a f a
( )( ) ( )( ) ( )( )( )
( )( ) ( )
( )( ) ( ) ( )
1f1
f
f f a f f a f f a a f f a a f a
f f a f a f a a
<
< >3f f
1f f a f f a f f a a f f a a f a f f a f a f a a ,
( )3 . , ( )3 .
-
: 2015
- 200 - :
55
21
f : [0, ) R+ f(0)=0 f
(0, )+ . f(x)
g(x) ,x 0x
= > x
2
F(x) g(t)dt,x 0.= >
() g.
() F .
() 4
3
F(3) g(x)dx.<
() 1 2 1 2
x ,x 0 x x 4,> + = 1 2
F(x ) F(x )+
.
21
) x 0> .
( ) . . .
f 0 ,
f , x
x
0
( ) ( ) ( ) ( ) ( ) ( ) ( )f x f 0 f x
0 , x : f f x x f x 0 x
= = =
, x 0> .
( ) ( ) ( ) ( ) ( ) ( )f
0 x f f x x f x f x f x x f x
< < < < < , ( )x 0 *> .
g ( )0 + , ,
( ) ( ) ( ) ( )( )*
2
f x x f x f xg x 0
x x
= = >
, x 0> . , g ( )0 , + .
) g , F ( )0 , + , ( ) ( )F x g x =
( ) ( )F x g x 0 = > , x 0> . , F ( )0 , + .
) ( )G : 0 , + , ( ) ( )x 1
x
G x g t dt+
= .
0 x
-
: 2015
- 200 - :
56
G ( g )
( ) ( ) ( ) ( )x 1 x 11
x x 1
G x g t dt g t dt g t dt+ +
= = + =
( ) ( ) ( ) ( ) ( ) ( ) ( )x 1x
1 1
g t dt g t dt g x g x 1 x 1 g x 1 g x+ = + = + + + = +
.
, ( ) ( )g
x 0 x x 1 g x g x 1 G
> < + < + ( )0 , + .
( ) ( ) ( ) ( ) ( ) ( )3 4 4G
2 3 3
2 3 G 2 G 3 g t dt g t dt F 3 g t dt
< < < 1 2 2 1
x x 4 x 4 x+ = = .
( )H : 0 , + , ( ) ( ) ( )H x F x F 4 x= + .
H , ( )F x ,
( )F 4 x , ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )H x F x F 4 x F x F 4 x 4 x g x g 4 x = + = + = .
( ) ( ) ( )g
1 1H x 0 g x g 4 x x 4 x x 2
= = = =
( ) ( ) ( )g
H x 0 g x g 4 x x 4 x x 2
> > > > .
x 0 2
+ ( )H x +
( )H x
min
H () x 2= , ( ) ( ) ( )H 2 F 2 F 2 0 0 0= + = + = .
( ) ( )1 2F x F x+ 1 2x x 2= =
.
O
-
: 2015
- 200 - :
57
22
x
1
ln tf(x) dt.
t 1=
+
i. ( )
( )2x 1f(x) x 1
lim .x 1
ii. : 1 1
2 3
1 1
ln t ln tdt dt.
t 1 t 1+
.
( )f x 0 lnx 0 x 1 = = =
( )+ >
> > >x 1 0
f x 0 lnx 0 x 1
, f ( )0 , 1 . , 1 1 1 1f f3 2 3 2
< <
1 1
3 2
1 1
ln t lntdt dt
1 t 1 t= >
+ + .
iii. g ( )0 , + , :
( ) ( ) ( ) 2 21
ln 1 lnx 11 1 1 lnx lnxxg x f x f f x fx x x x 1 1 x 1 x 1x x1
x x
= + = + = + = + = + + + +
( ) +
= + = + = + = = = + + + + ++
2
2
lnx x 1 ln xlnx xlnx 1 lnx lnx 1 lnx lnx1
x 1 x 1 x 1 1 x x 1 x x 2x x x 1.
, ( )2ln x
g x2
=
x 0> . , ( )
2ln xg x c , x 0
2= + > .
( )= = + =x 1 g 1 0 c c 0 . , ( )2ln x
g x , x 02
= > .
iv. ( )g x 0 , x 0 > . : ( )e e 2
1 1
ln xE g x dx dx
2= = =
( ) ( )e e e e e
e2 2 2
11 1 1 1 1
1 1 1 1 1 1 eln xdx x ln xdx xln x x 2lnx dx e lnxdx x lnxdx
2 2 2 2 x 2 2 = = = = = =
ee
11
e 2e exlnx dx e e 1
2 2 2
= + = + = .
0 1 +
f(x) - +
f
min
-
: 2015
- 200 - :
59
v. : ) ( ) ( )g : 1 , , g x f x x 1 + = + . g
)1 , + ( )1 , + ,
( ) ( ) lnx x 1lnxg x f x 1 1 0x 1 x 1
= = = ,
lnx x 1 . , ( )lnx x 1 1 1 2 0 = < . , ( )g x 0 < .
, g )1 , + , ( ) ( ) ( ) ( )> < = + < < x 1 g x g 1 0 f x x 1 0 f x x 1 .
x 1= ( )f x x 1 0 0 .
: x 1= 0 0 .
x 1> . ... f
( ) 1 , x 1 , x : ( )( ) ( ) ( )
= = +
f x f 1 f xlnf
x 1 1 x 1.
ln1
1 < > . , ( ) ( )g x 0 , x 2 , 2 g
( ) 2 , 2 g ( )2 , 2 .
iv) ( )1 : ( ) ( )( ) ( )( ) ( ) ( )2f x f x 1 1 f x , x 2 , 2 2 = + . g
( )2 , 2 , ( ) ( )g 0 f 0 1 2 0= = > . ,
( ) ( )g x f x 1 0 , x 2= > < . , ( )2 : ( )( )( )
( )
2
1 f xf x 0 , x 2
f x 1
+ = <
= < = +
f x 1 02 2
2 , 2f x 1 4 x , x 2 f x 1 4 x , x 2 , 2 . , )
: ( ) ( )2 2f x 2f x x 3 0 , x 2 , 2 + = .
( ) ( ) ( )( ) ( )22x 2 f 2 2f 2 1 0 f 2 1 0 f 2 1= + = = = ., x 2= ,
( )f 2 1 = . , ( ) 2f x 1 4 x , x 2 , 2 = + .
vi) ( )z i 2 z 0 i 2 = + = . z ( )K 0 , 1
2= .
f ( ) 2y f x 1 4 x= = + ,
( ) = + = 22 2y 1 y 1 4 x y 1 x 4 , y 1 . , fC
z i 2 = .
z z max=
z x yi , 1 y 3= + . , ( ) ( )2 22 2x y 1 4 x 4 y 1+ = = .
, ( )22 2 2z x y 4 y 1 y 3 2y= + = + = + .
y ( )g : 1 , 3 , g y 3 2y = + .
+ + 1 y 3 2 2y 6 1 3 2y 9 1 3 2y 3 . , g
0
y : 0 0 o 0
3 2y 3 3 2y 9 2y 6 y 3+ = + = = = .
( )220 0x y 1 4+ = 20 0x 0 x 3= = . , ( ) ( )0 0x , y 0 , 3= .
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: 2015
- 200 - :
63
f 0 ( )f 0 . x 0= ( ) ( ) ( )f x f x f x x =
: ( ) ( ) ( ) ( ) ( ) ( )f 0 f 0 f 0 3 f 0 f 0 f 0 0 = = = .
, f
C ( )0 , 3 x x .
vii)
( )( )2 2
2
0 0
I f x 1 dx 4 x dx= = .
= =
x 2 dx 2d , ,
2 2.
= = =x 0 0 0 .
= = =
x 2 1 2
.
,
2 2 2 22 2
0 0 0 0
1 2 2 I 4 4 2d 4 d 4 d 2 d 2 0
2 2 2
+
= = = = + = =
.
2
2
0
I 4 x dx= = 2 2x y 4+ = . ,
2 2I
4
= = .
-
: 2015
- 200 - :
64
EMA 24
f : 1
z2
10z 1
2 2f (x) x 2xf(x)+ = x 0
f(x) z 2lim .
x 2z 1
=
i. z 1=
ii. ( )10
10
z iw
z 1
+=
.
iii. ( )2x 0
f xlim
x x
iv. ( ) 35 z 3 4i x x 10,+ + = +
1,2 .
24
i) ( ) ( )2 2f x x 2xf x+ = , x . x o 0 :
( ) ( )+ =
2 2
2 2 2
f x 2xf x x
x x x
( ) ( )2 2
f x f xx2
x x x
+ =
, x 0. (1)
( )
x 0
f xlim
x
( )2
x 0
f xlim
x
. (1), :
( ) ( ) ( ) ( )
+ = + =
2 22
2
x 0 x 0 x 0 x 0
f x f x f x f xxlim lim 2 lim 1 2 lim
x x x x x
+ = = = = =
222 2z 2 z 2 z 2 z 2
1 2 1 0 1 z 2 2z 1 z 2 2z 12z 1 2z 1 2z 1 2z 1
( ) ( ) ( ) ( ) = + = + = = =2 2 2 2z 2 z 2 2z 1 2z 1 z 2z 2z 4 4 z 2z 2z 1 3 z 3 z 1 z 1.
-
: 2015
- 200 - :
65
ii) ( ) ( )= = = = =2 1z 1 z 1 z z 1 z * z 0 , z 1z
. , ( ) + = =
10
10
z iw
z 1
( ) ( ) ( )( )
( ) ( ) ( ) ( )
+ + + + = = = = = = = = = =
10 1010
10 1010 10 10 1010* 10
10 10 10 10 10 10 10 1010
10
1 zi 11 i ziz i z i 1 zi i i z i z z iiz zw
1 z 1 z 1 z 1 z 1 z z 1z 1 1z 1 1zz
. , w .
iii) ( ) ( ) ( ) ( )
= = = =
2 2x 0 x 0 x 0
f x f x f xx x 1lim lim lim 1 1 1 1
x x x x 1x x x x,
( ) ( )
>
=
= ========= =x 0
u x a)
x 0 x 0 u 0u 0
f x f uxlim 1 , lim lim 1
x x u.
iv) : z 3 4i z 3 4i 1 5 6+ + = + = .
, ( )6 z 3 4i 0 1 + . , + = =z 3 4i z 3 4i 1 5 4 .
, + 2 z 3 4i 8 ( )8 2 z 3 4i 0 2 + . g : 1 , 2 ,
( ) ( )3g x x 10 5 z 3 4i x= + + + . g 1 , 2 , .
, ( )( )1
g 1 6 z 3 4i 0= + ( ) = + (2)
g 2 8 2 z 3 4i 0 .
( )g 1 0= , 1 1 , 2 ( )g x 0= .
( )g 2 0= , 2 1 , 2 ( )g x 0= .
( )g 1 0> ( ) < g 2 0 Bolzano, ( ) =0 0x (1,2) : g x 0 .
, ( )0 0x 1 , 2 : g x 0 =
-
: 2015
- 200 - :
66
EMA 25
( )0,+ f, :
xf(x)ln f(x) 1, x 0. = >
i. : f(x) 1> x 0.>
ii. f ( )0, .+
iii. : ( )f(x)f f(x) f(1)> x 0.>
iv. 1 2
x ,x 0> : 1
f(x ) e= 22
f(x ) e .=
v. 2
1
x
x
f(x)dx.
25
i) ( ) ( )( ) ( )x f x ln f x 1 , x 0 1 = > ( ( )1
( )f x 0 , x 0> > ).
a 0> : ( )f a 1= .
x a= ( )1 ( ) ( )( )a f a ln f a 1 a 1 ln1 1 a 0 1 0 1 = = = = , .
( ) ( )ln
a 0 : f a 1 ln f a 0
> < < . , ( ) ( )
f a ln f a 0 < ,
( )1 .
, ( )f x 1> , x 0> .
ii) f ( )( )ln f x , .
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )1 : x f x ln f x 1 f x ln f x x f x ln f x x f x 0 = + + =
-
: 2015
- 200 - :
67
( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( )( )
f x ln f xf x x ln f x x f x ln f x f x 0 , x 0 f
xln f x x
+ = = < > +
( )0 , + .
iii) x ( )f x ( )1 : ( ) ( )( ) ( )( )( )f x f f x ln f f x 1 = . ,
( ) ( )( )( )( )( )
1f x f f x
ln f f x = , x 0 > (
( ) ( )( ) ( )( )( )f x 1 , x 0 f f x 1 ln f f x 0> > > > ).
, ( )( )( ) ( )
1f 1
ln f f x> ( ) ( )( )( )f 1 ln f f x 1 < .
, ( ) ( )1 f 1 ln f 1= ( ( )1 x 1= ).
, ( ) ( )( )( ) ( ) ( )( )f 1 ln f f x f 1 ln f 1 <
( )( )( ) ( )( )ln f f x ln f 1< ( ( )f 1 0> ) ( )( ) ( )f f x f 1< ( ln ) ( )f x 1> ( f ). , , . ,
( ) ( )( ) ( )f x f f x f 1 , x 0 > > .
iv) 1
x x= ( )1 : ( ) ( )( )1 1 1 1 1 1x f x ln f x 1 x eln e 1 x e = = = .
2
x x= ( )1
: ( ) ( )( ) 2 2 22 2 2 2 2 2 21
x f x ln f x 1 x e lne 1 x e 2 1 x2e
= = = =
v) f , , ( )2
1
x
x
f x dx .
( ) ( )( )22
1
1
x 2e 1
x 1
e
f x dx f x dx= = ( )( )
2
1
2e
1
e
1dx
x ln f x .
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: 2015
- 200 - :
68
( ) ( )( ) ( )( )
( )( )( ) ( ) +
= = = = = ++
f x ln f x u lnu 1 lnudxu f x du f x dx du dx du dx du
x ulnux 1 lnux 1 ln f x
.
= = =
)1 1x u f e
e e.
= = =
)2
2 2
1 1x u f e
2e 2e.
, : ( )
22
2
1e e2e
21 e ee
1 lnu 1 lnu1 dx 1I du du
x lnu ulnu uln ulnf x
+ += = = .
= =1
t lnu dt duu
.
= =u e t 1 .
= =2u e t 2 .
, +
= = + = + + =
11
22 2
1 t 1 1 1I dt lnt 1 0 ln2 ln2
t 2 2t.
-
: 2015
- 200 - :
69
EMA 26
: 2f(x) 2ln x x 1.= +
i. f .
ii. f .
iii. f.
iv. f.
v. : 2 5 10f(x) f(x ) f(x ) f(x ).+ = +
vi. ( )( )
2
, 0, e ,
+ > = .=
26
i) ( ) 2f x 2lnx x 1= + , ( )0 , + . f
( )0 , + , ( ) 2f x 2x 0 , x 0.x
= + > > , f ( )0 , + .
ii) f , ( )2
2 2
x 12 2f x 2x 2 2 , x 0
x x x
= + = + = >
.
( )x 0
2f x 0 x 1 0 x 1>
= = =
( )x 0
2f x 0 x 1 0 x 1>
> > >
0 1 +
f(x) - +
f
..
O
-
: 2015
- 200 - :
70
f ( )1, + ( )0 , 1 .
( )( ) ( )A 1 , f 1 A 1 , 0= .
iii) ( ) ( )
< < < =f
0 x 1 f x f 1 0 .
( ) ( )
> > =f
x 1 f x f 1 0 .
iv) , ( ) ( )2x x
lim f x lim 2ln x x 1 + +
= + = +
( ) ( ) ( )2x 0 x 0
lim f x lim 2ln x x 1 2 0 1+ +
= + = + = .
, ( )( ) ( ) ( ) ( )
+ +
+ = = + =
f
xx 0
f 0 , lim f x , lim f x , .
v) ( ) ( )
( ) ( )( )
( ) ( ) ( ) ( )5f5
5 10 2
10 2 10 2
f x f xx x0 x 1 f x f x f x f x
x x f x f x
+ + > + > >
x 1= ( ) ( ) ( ) ( )2 5 10f x f x f x f x+ = + ( ( )f 1 0= ). , x 1= .
vi) ( )( )
2
2 2 2 2
2
2 2 2 ln 1 1 a a a 2 a 2
2 2 a
a aa ee e a e e
e
+ = = = =
( ) ( )2 2 2 22 a 2 2 a 2 2 2ln a e ln e ln a ln e ln lne 2ln a a 2ln = + = + + = +
( ) ( )f
2 2
f 1 12lna a 1 2ln 1 f a f a
+ = + = = .
0 1 +
f(x) - + O
-
: 2015
- 200 - :
71
EMA 27
f : ( )3 f(x) 3f(x) f(x) e 2 x ,+ + + = x .
i. : f .
ii. : g ( ) ( )g(x) f 2 3x f 3 2x ,= + x .
iii. : ( )( )( ) ( )( )( )2f g x x 1 f g x 4 x . >
27
i) a , a < . ( ) ( )f a f < . ( ) ( )f a f . ( ) ( )3 3f a f ( 3x ) ( ) ( )f a f e e (
xe ).
( ) ( )( ) ( )
( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( )
+
+ + + + + +
3 3f a f 3 3 3 3
f a f
: f a f
f a f f a f a e 2 f e f 2 a a
e e
2 2
.
, ( ) ( )f a f < .
ii) a , a >
+ > + + > +
f f 2 3a f 2 32 3a 2 3
3 2 3 2a f 3 2 f 3 2a
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
+ + > + +
+ > + >
f 2 3a f 3 2 f 2 3 f 3 2a
f 2 3a f 3 2a f 2 3 f 3 2 g a g
, g .
iii) ( )( )( ) ( )( )( ) ( )( ) ( )( ) ( ) ( )f g
2 2 2f g x x 1 f g x 4 x g x x 1 g x 4 x x x 1 x 4 x
> > <
( ) ( ) ( ) ( ) ( )( ) ( )
2 2
2 2
x x 1 x 4 x 0 x x 1 x x 4 0 x x x 1 x 4 0
x x x x 4 0 x x 4 0 x 2 0 x 2.
< + < + <
+ < < < < <
- -2 0 2 +
x - - + +
2x 4 + - - +
( )2x x 4 - + - +
O
O O
-
: 2015
- 200 - :
72
EMA 28
A. f g , f(x) g(x),
x , , :
f(x)dx g(x)dx.
B. f , f (x) 0, >
x . , , ,< :
i. f(a) f()
f .2 2
+ +
-
: 2015
- 200 - :
73
... f a
, :2
+
( )( ) ( )
2 2
a a f f f f
2 2a x , : f x
2 a a
2 2
+ +
+ = = +
.
,
( ) ( )( ) ( )
( ) ( )
+ +
+ < < < +
f
1 2 1 2
a a f f a f f
2 2 a x x f x f x 2f f a f
a a 2
2 2
( ) ( )+ +
-
: 2015
- 200 - :
74
EMA 29
1,1 f (-
1,1) : 3 2z f(1)z z f(0) 0 (1)+ + = . (1)
1 i,+ :
i. ( )1 0,1 : 1f( ) 0.=
ii. ( )2 0,1 : 22f ( ) 7 0. + =
iii. ( )3 4 , 0,1 : 3 4 21 1
3f ( )f ( ) .
2 2 =
iv. f , ( ) 0f x =
(-1,1).
29
i) 1 i+ , :
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( )
( ) ( ) ( ) ( )
3 2 2 31 i f 1 1 i 1 i f 0 0 1 3i 3i i f 1 2i 1 i f 0 0
1 3i 3 i 2if 1 1 i f 0 0 f 0 3 i 2f 1 1 0
1f 0 3 0 2f 1 1 f 0 3 f 1
2
+ + + + = + + + + + =
+ + + = + + =
= = + = =
Bolzano f 0 , 1
( ) ( )1 1 0 , 1 : f 0 = .
ii) f 0 , 1 ( )
. . .
0 , 1
( ) ( ) ( ) ( )2 2f 1 f 0 1 7
0 , 1 : f 31 0 2 2
= = =
. , ( )22f 7 0 + = .
iii) ... f 10 ,
( ) ( ) ( ) ( ) ( )13 1 31 1 1
f f 0 f 0 3 0 , : f
0
= = =
.
-
: 2015
- 200 - :
75
... f 1 , 1 ( ) ( )( ) ( )1
4 1 41 1 1
1f 1 f 12 , 1 : f
1 1 2 2
= = =
.
, ( ) ( )3 4 21 1 1 1
3 1 3f f
2 2 2 2
= =
.
iv) : ( ) ( )3 4 21 1
3f f 0
2 2 = >
,
10 1< < .
, ( ) ( ) ( ) ( ) ( ) ( )
> < +
x 0.>
B. ( )1
x1 x , x 0f(x)e , x 0
+ >= =
x 2g(x) e x 1,= x .
i. f 0
x 0.=
ii. f
.
iii. g .
iv. h , f(x) , x 0
h(x) .g(x) , x 0
=
+ + + +. ,
) 0 , + .
( ) ( ) ( )> > = + > >+
xx 0 x 0 0 ln x 1 , x 0
x 1.
( ) ( )h t ln t 1= + ...
( ) 0 , x 0 , x : ( )( ) ( ) ( )h x h 0 ln x 11
h x 0 1 x
+ = =
+.
,
< < < + < + > > + +
1 10 x 1 1 x 1 1
1 x 1
( ) ( )>+
> > + >+ +
x 0ln x 1 1 x1 ln x 1
x x 1 x 1.
-
: 2015
- 200 - :
77
) i) ( ) ( )x 0lim f x f 0
+= , ( )
1
x
x 0lim 1 x e
++ = .
( )+
+ + x 01
x1 x 1 : . ( ) ( )( )1
x
ln 1 x1ln 1 x xx1 x e e
++
+ = =
( ) ( )( )0
0
x 0 x 0 x 0
ln 1 xln 1 x 1lim lim lim 1
x x x 1+ + +
++=== = =
+.
, ( )+
+ = =1
1x
x 0lim 1 x e e . , f
0x 0= .
ii) f ( )0 , + , ( ) ( )( ) ( ) ( )ln 1 x ln 1 x1
x xxln 1 x
f x 1 x e ex
+ + + = + = = =
( )( )
2
xln 1 x
x 1f x 0 , x 0
x
++
= < > , ( ) ( )2 xf x 0 , x 0 , ln 1 x 0x 1
> > + f 0x 0= . , f )0 , + . ,
( ) ( ) =x 0 f x f 0 e , x 0 . , f 0, e.
iii) ( ) xg x e 2x , x =
( ) xg x e 2 , x =
( ) xg x 0 e 2 x ln2 = = = ( ) xg x 0 e 0 x ln2 > > > .
ln2 +
( )g x - +
g
min
, g
( ) ( )ln2ln2 , g ln2 e 2ln2 2 2ln2 2 1 ln2 = = = .
, ( ) ( )g x g ln2 , x
O
-
: 2015
- 200 - :
78
( ) ( ) ( )g x 2 1 ln2 0 1 ln2 > > .
, ( ) > g x 0 , x g .
iv) ( ) ( ) 1 2 1 2 1 2x , x x x h x h x .
1 2
x x 1 2
x , x 0> , ( ) ( ) ( ) ( )1 2 1 2h x h x f x f x , f 1 1 ,
.
1 2
x x 1 2
x , x 0< , ( ) ( ) ( ) ( )1 2 1 2h x h x g x g x , g 1 1 ,
.
1 2
0 x x 0 < . ( ) ( )1 2h x h x
( ) ( )1 2f x g x .
.
)( ) ( ) ( ) (f
1xf 0 , lim f x , f 0 1 , e A
+
+ ======== = =
( ) ( )( )ln 1 x1
xx
x x x lim f x lim 1 x lim e
+
+ +
= + =
( ) ( )( ) ( ) 0x x x x
ln x 1ln x 1 1lim lim lim 0. , lim f x e 1
x x x 1
+
+
+ + + +
++ === = = = = +
( )( )g
g , 0
======== ( ) ( ) ( ) 2x x 0lim g x , lim g x , 0 A
= =
.
( ) ( )x 2x x
lim g x lim e x 1
= = ( )0 1= + =
( ) ( )x 2x 0 x 0lim g x lim e x 1 0.
= =
, 1 2
A A = ( ) ( )1 2f x g x . , h 1 1 .
-
: 2015
- 200 - :
79
31
f 0,1 (0,1)
:
0 f (x) 4,< x (0,1).
f(0) f(1) 0+ =
:
i. 0
x (0,1) , 0
f(x ) 0.=
ii. (0,1) , f () 2f(1). =
iii. 2x 2 f(x) 2x, < < x (0,1).
iv. 1
0f(x)dx 1. f 0 , 1 . , f 0 , 1 . ,
( ) ( )< f 1 f 0
f 0 f 0 2f 0 0 f 0 0 f 1 0 .
Bolzano f 0 , 1 ( ) ( )0 0x 0 , 1 : f x 0 = . 0x
f .
ii) g : 0 , 1 , ( ) ( ) ( )g x f x 2f 1 x= . g
0 , 1 , ( )0 , 1 .
( ) ( )
( ) ( ) ( ) ( )( ) ( )
( )f 0 f 1 0
g 0 f 0
g 1 f 1 2f 1 f 1 f 0+ =
=
= = ==========
, . Rolle, ( ) ( ) 0 , 1 : g 0 = . ,
( ) ( ) ( ) ( ) = =f 2f 1 0 f 2f 1 .
-
: 2015
- 200 - :
80
iii) 0 x 1< < .
... f 0 , x ( )1x 0 , x : ( )( ) ( )
1
f x f 0f x :
x
.
:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
< < < < +1
f x f 00 f x 4 0 4 0 f x f 0 4x f 0 f x 4x f 0 1
x
... f x , 1 : ( )2x x , 1 : ( )( ) ( )
2
f 1 f xf x :
1 x
.
:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
< < < < 2
f 1 f x0 f x 4 0 4 0 f 1 f x 4 1 x f 1 f x 4 1 x f 1
1 x
( ) ( ) ( ) ( ) ( )f 1 4 1 x f x f 1 2 < .
:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( )+ =
+ + < < + + < < f 0 f 1 0
1 2 f 0 f 1 4 1 x 2 f x f 1 f 0 4x 2x 2 f x 2x , x 0 , 1
iv) )
( ) ( )( )
( ) ( )( ) ( )
( )< =
= < < < 0 f x 4 f 1 f 0
f 2f 1 0 2f 1 4 0 f 1 2 0 f 0 2 ( ) > = = 11 1 1
2
0 0 0 0
x dx 0 f x dx 2x 2 dx x 2x 1 . ,
( ) ( ) < <
-
: 2015
- 200 - :
81
32
( )f(x) ln 1 x x,= 1.<
i. f .
ii. y x=
( )h(x) ln 1 x= , (0,0).
iii. f ( )1f 2008 f (x) 0.+ =
iv. xg(x) 1 e , x ,= ( )fog ,
( )gof ( ) ( )xfog (x) e gof (x).=
v. f,
xx x 1 e.=
32
i) f ( ), 1 ,
( ) ( )( ) ( ) 1 x 11 1f x ln 1 x x 1 x 1 11 x x 1 x 1
+ = = = = =
2 x
0x 1
= =
( )( ) ( )( ) ( ) ( )( ) ( )x x x x x xfog x f g x f 1 e ln 1 1 e 1 e lne 1 e x 1 e= = = = + = + .
( ){ } ( ){ } ( )gof f gA x A : f x A x 1 , f x , 1 = < = .
( )( ) ( )( ) ( )( ) ( )( )ln 1 x x
ln 1 x x
x x x
e 1 x e 1 xgof x g f x g ln 1 x x 1 e 1 1
e e e
+
= = = = = = .
( )( ) ( )( )xfog x e gof x , x 1= < .
v) ( ) ( )
< > =f
x 0 f x f 0 0 .
, :
( ) ( )( ) ( )0 0 0 0
1 e 1 e 1 e 1 e
E f x dx ln 1 x x dx ln 1 x dx xdx
= = = =
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( )
0 220 00
1 e1 e 1 e1 e
2 20 0
1 e 1 e
2 20 00
1 e1 e 1 e
2 2
1 ex xx ln 1 x dx xln 1 x dx
2 1 x 2
e 1 1 1 x e 1x1 e lne dx e 1 dx
1 x 2 1 x 2
e 1 e 11e 1 dx dx e 1 ln 1 x e 1
1 x 2 2
e 1 e 2e 31 .
2 2
= = + =
= + + = + + =
= + + = + + =
+= + =
.
-
: 2015
- 200 - :
83
33
f ( )x 1,1
f(x) 0 x
0 f(t)dt x.
=
A. :
i. x
0
1 f(t) dt .
f(x)
=
ii. f(x) 0> ( )x 1,1 .
iii. ( )2
1f(x) , x 1,1 .
1 x=
B. 4 3 2g(x) 3x 4x 12x f(0), x .= + +
i. g.
ii. g.
33
)
i) , f ( ) ( ) x
0
1 , 1 f t dt ( ) ( )
x
0
1 , 1 f t dt
( )1 ,1 . , :
( ) ( ) ( ) ( ) ( ) ( ) = = < = <
x x x x
0 0 0 0
f t dt x , x 1 , 1 f t dt x , x 1 f t dt f t dt 1 , x 1
( ) ( )( )
( ) ( ) ( )
= < =
-
: 2015
- 200 - :
84
ii) x 0= ( )1 : ( ) ( ) ( ) ( ) ( )
= = = = 0
0
1 1 1 f t dt 0 1 f 0 1
f 0 f 0 f 0.
, ( )
( )
>
-
: 2015
- 200 - :
85
34
F txx
1
eF(x) dt.
t=
i. F F (x).
ii. x 1
F(x)lim .
x 1
iii. xe ln x F(x) ln x < 0 x 1.< <
iv. F.
34
i) u t x= : du x dt= .
= =t 1 u x .
= = 2t x u x .
, ( )2 2x xu u
x x
e eduF x du
u x u
x
= = .
g ( )ue
g uu
= ( ) ( )A , 0 0 ,= + .
F 2x , x . :
( ) < >
+ < >
2 2
x 0 x 0 x 0 ,
x 0 x 0. , F ( )0 , + .
: ( ) ( )= + = > 2 2x xu u u1 xu
x 1 1 1
e e eeF x du du F x du du , x 0
u u u u.
g ( )+ ux
1
e0 , du
u ( )0 , + . ,
ux2
1
edu , x
u ( )+ 2x u
1
e0 , du
u ( )0 , + (
). , F ( )0 , +
.
-
: 2015
- 200 - :
86
:
( ) ( )2 2 2 2x x x xu u u x xx xu
2
21 1 1 1
e e e e e 2e eeF x du du du du x , x 0
u u u u x xx
= = = = >
ii) ( ) ( ) ( ) ( )
x 1 x 1
F x F x F 1 2e elim lim F 1 e
x 1 x 1 1
= = = =
.
iii) ( )txx
1
eF x dt , x 0
t= > . ( )x 0 , 1 . 0 x t 1< . ,
< < 0 x t te e e e 1 e e
( )
( ) ( )
>
x 1tx x tx x1 1 1x tx x tx x
x x
tx xx x xx
1 1 1
e e e e1 11 e e 1 e e dt dt dt
t t t t t t
e e1dt dt dt lnx F x e lnx , x 0 , 1
t t t
iv) ( )x 0lim F x
+= + . :
( ) ( )x
x 0
x 0
lim e lnx 1
lim lnx
+
+
= =
=
)
( )+
= x 0lim F x .
-
: 2015
- 200 - :
87
35
( )f : 0,+
2
1 1f (x) f(x) ,
x x + = x 0>
1
e, .e
i. f .
ii. 3 3
3 3x e
2 2
e dx x dx
iii. x
1
g(x) f(t)dt, x 0.= >
( )h : 0,+ 21h(x) g(x) g ln xx
= + ( )0, .+
1x x
1 1
ln t 2 ln tdt 2 2xf(x) dt, x 0.
t x t+ = >
35
i) x 0> :
( ) ( ) 1xf x f xx
+ =
( )( ) ( )xf x lnx , x 0 = >
, ( )xf x lnx c , x 0= + > (c : ) (1)
f
C ( ) =
1 1A e , f e
e e.
x e= (1) : ( ) = + = + =1e f e lne c e 1 c c 0e
. ,
( )x f x lnx , x 0 = > ( ) lnxf x , x 0x
= >
-
: 2015
- 200 - :
88
f ( )0 , + , ,
( ) ( ) 2lnx x lnxlnx
f xx x
= = =
2
1 lnx, x 0
x
= > .
( )f x 0 1 lnx 0 lnx 1 x e = = = =
( )ln
f x 0 1 lnx 0 lnx 1 lne 0 x e
> > < < <
0 e +
( )f x + -
f
max
f (0 , e )e , + . f e,
( ) ( )lne 1f e 2e e
= =
ii) ( )2 ( ) ( )f x f e , x 0 > , x 0
e 0
lnx lne, x 0 e lnx x lne , x 0
x e
>
> > >
lne x e xlnx lne , x 0 x e , x 0
> > .
32 , 3 .
( ) ( )
< < < < < < f ln
2 3 2 3
3 , 2
ln 3 ln23 2 f 3 f 2 2ln 3 3ln2 ln 3 ln2 3 2
23
33 2 < .
e xx e , x 0 > .
,
3 3
3 3 3 3
2 2 3 3 3 3e x 3 e x e x e x
3 3 2 2 2 2
x e , x 3 , 2 x dx e dx x dx e dx x dx e dx
.
O
-
: 2015
- 200 - :
89
: > 3 3
3 3e x
2 2
x dx e dx , m ( ) e xm x x e=
33 , 2
.
iii) ( ) ( ) ( )x
2 2x x x
1 1 1 1
ln t ln xlntg x f t dt dt lnt lntdt , x 0
t 2 2
= = = = = >
. ,
( )2ln x
g x , x 02
= > . ,
( ) ( ) ( ) = + = + = + =
+ = =
222 2
2 2 2
2 22 2 2
1ln
lnxxln x ln x1h x g x g ln x ln x ln x
x 2 2 2 2
ln x ln xln x ln x ln x 0
2 2
,
( ) ( ) ( ) = > + =
21h x 0 , x 0 g x g ln x 3x
.
iv)
( ) ( ) ( )( )
13x x
1 1
ln t 2 lnt lnx 2 1 1 2dt 2 2xf x dt g x 2 2x g g x g 2 2lnx
t x t x x x x x
+ = + = + + =
22
ln x 2 2lnx 0 , x 0x
+ = > ( )
( ) : 0 , + , ( ) 2 2 x ln x 2 2lnxx
= + .
( ) 1 0= . , 1. ( )0 , + ,
( ) ( )2 22lnx 2 2 2
x xlnx x 1 , x 0x x x x
= + = + > .
( ) : 0 , + , ( ) x xlnx x 1= + ( ) x lnx , x 0 = >
( ) x 0lnx 0
x 1
>
> >
, ( ) ( ) x 1 0 , x 0 = > .
, ( ) ( ) ( ) ( )22
x x 0 , x 0 , 1 1 ,x
= > + 1. ,
( )0 , + .
0 1 +
( ) x - +
-
: 2015
- 200 - :
90
36
f,g 2002, a 2002>
( ) ( )f 2002 g 2002 1= = ( ) ( ) ( ) ( )a
2002
f x g x f t g t dt+ =
2002, , :
i. ( ) ( )g x 2 f x .=
ii. ( )( )a
2
2002
f t 1 dt a 2004. =
iii. a 2004.
iv. a) ( ) ( )( )x
2
2002
h x f t 1 dt,= ( )0x 2002,2004
( )0h x 0. =
) f , x'x .
) f .
36
i. ( ) ( )a
2002
f t g t dt = . ( ) ( )f x g x , x 2002 , + = .
x 2002= , : ( ) ( )+ = + = =f 2002 g 2002 1 1 2 . ,
( ) ( ) ( ) ( ) + = = f x g x 2 g x 2 f x , x 2002 , a .
ii. ( ) ( ) ( ) ( )( ) ( ) ( )( ) = = = a a ai)
2
2002 2002 2002
f t g t dt 2 f t 2 f t dt 2 2f t f t dt 2
-
: 2015
- 200 - :
91
( ) ( )( ) ( ) ( )( )
( )( )
= + =
+ =
a a2 2
2002 2002
a2
2002
f t 2f t dt 2 f t 2f t 1 1 dt 2
f t 1 dt a 2002 2
( )( ) = a
2
2002
f t 1 dt a 2004 .
iii. ( )( )2f t 1 0 , x 2002 , a . ,
( )( ) a ii)
2
2002
f t 1 dt 0 a 2004 0 a 2004
iv. ) f 2002 , 2004 , , ( )( )2f t 1
2002 , 2004 h ( )2002 , 2004 ( ) ( )( ) ( )2h x f x 1 1 = .
, h 2002 , 2004 , ( )2002 , 2004 :
( ) ( )( )2002
2
2002
h 2002 f t 1 dt 0= = ,
( ) ( )( )2004 ii)
2
2002
h 2004 f t 1 dt 2004 2004 0= =
, Rolle, ( ) ( )0 0x 2002 , 2004 : h x 0 =
) ( )( )
( )( ) ( ) = = =1
20 0 0
h x 0 f x 1 0 f x 1 . f
Rolle 0
2002 , x . , ( ) ( )0 2002 , x : f 0 = , fC
x x ( )( ) ( )A , f , 0 = .
) f
C :
( ) ( ) ( )( ) ( )( )
=
=
=
: y f f x
: y f 0 x
: y f
-
: 2015
- 200 - :
92
37
1 2 3
z ,z ,z ,, .
( )2009 20093 1 2z i z 1 i z ,+ = + :
i. .
ii. ( ) ( )22 23 1 3 12 z z z z .+ +
iii. 2
z 2 2,= ( )3 1m z z 4.
iv. 2 2 2
3 1 2z z 2 z ,+ = ( ) ( )1 3z a f a i,z f i= + = + a. 0>
( ) ( )f x
g x ,x 0,x
= , :
) 3 1
z z .
) f
.
37
i. ( ) ( )502 12009 4 4 502 502i i i i 1 i i+= = = = .
, ( ) ( )2009 20093 1 2 3 1 2 3 1 2 2z i z 1 i z z i z 1 i z z i z z i z+ = + + = + + = + .
( ) ( ) ( ) ( ) = = = = 3 2 2 1 3 2 2 i 3 2 2 1z z i z z z z i z z z z z z
.
, ( ) ( ) ( )+ = + = + 3 1 2 3 2 2 1z i z 1 i z z z i z z 1
( )( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
= = = + = + + =
+ =
= = = + = +
12 2 2 2 2
1 3 1 2 2 1 1 2 2 1 1 2
2 21 2
2 2 2 2 2 21 2
z z z z i z z z z i z i z 1 i z 1 i z
1 i z z
2 z z 2 AB AB AB AB B
, ( ) ( ) ( ) 2 2 2= + , ,
, B 90= .
-
: 2015
- 200 - :
93
ii.
( ) ( )2 2 2 2 2 2 2 2 23 1 3 1 3 1 3 1 1 3 3 1 1 32 z z z z 2 z 2z z z 2 z z z z 2 z z 0+ + + + + + ( )21 3z z 0 , . , .
iii. ( )+ = + + = = = + =3 1 2 3 1 2 3 1z i z 1 i z z i z 2 z 2 2 2 4 z i z 4 .
,
( )= + + = + + + ii)
23 1 3 1 3 1 3 1 3 1
4 z i z z iz z z z z 4 z z 16
( ) ( ) + + 2 2 3 23 1 3 12 z z 16 z z 8 2
,
( )( )+ = + = + = + + = 2 2 23 1 3 1 3 1 3 1 3 1 3 1 3 1z iz 4 z iz 16 z iz z iz 16 z iz z iz z z 16
( ) ( )( )
( ) ( ) = + 2
2 2 21 3 1 3 3 1 1 3 1 3 1 3 1 3
i z z z z 16 z z i z z z z 16 8 i 2Im z z 8 Im z z 4
iv. ) ( )+ = + = + + =2 2 2 2 23 1 2 3 1 2 1 3 1 3 2z iz 2 z z iz 2z z z 2Im z z 2 z .
2 2 23 1 2
z z 2 z+ = ( )1 3Im z z 0= ,
1 3z z .
)
( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) = + + = + = + + a)
1 3z z a f a i f i a f a i f i a f a f i af f a
( ) ( ) ( ) ( ) ( ) ( ) = = =f a f
af a f 0 g a g a
.
a ( a = ,
( )= = = =1 3 1 3 1 3z z z z 0 z z 0 A 0 , ). > a 0 a , , .. 0 a < < .
g a , , g ( )a , . , ( ) ( )g a g = . , Rolle , ( ) ( )0 0x a , : g x 0 = ,
( ) ( ) ( ) ( ) ( )
= =0 0 00 0 02
0
f x x f x0 x f x f x 3
x
g
C ( )( )0 0A x f x ( ) ( ) ( )0 0 0 : y f x f x x x = . ( )O 0 , 0 ( ) ( ) ( ) ( ) ( )0 0 0 0 0 0 f x x f x f x x f x = = , ( )3 . .
-
: 2015
- 200 - :
94
38
( . .)
f : : ( ) ( )x
2
0f x 1 f t dt,= + x .
:
i. f .
ii. ( )x xe e
f x ,x .2
=
iii. f .
iv. f ( ) ( )x
1
0f x h t dt, = h
, 1
20
1 dx.
x 1=
+
38
i. f 2f + 21 f ,
. , f ( ) ( )x
2
0
f x 1 f t dt= + ,
( ) ( )2f x 1 f x , x = + . , f + 21 f
, . ,
( ) ( )( )( )
( )( ) ( ) ( )( )
( ) ( )( ) ( )
2 2
2 2
f x f x f x f x1f x 1 f x 1 f x f x , x
f x2 1 f x 1 f x
= + = + = = = + +
.
ii. ( ) ( ) ( ) ( ) ( ) ( ) = + = + f x f x f x f x f x f x , x .
( ) ( ) ( )g x f x f x , x= + , ( ) ( )g x g x , x = ,
, ( ) ( )xg x c e , x 1= . ( ) ( ) ( )g 0 f 0 f 0 0 1 1= + = + = (
( ) ( )0
2
0
f 0 1 f t dt 0= + = ( ) ( )2f 0 1 f 0 1 = + = )
, x 0= , ( )1 : ( ) = =0g 0 c e 1 c .
-
: 2015
- 200 - :
95
, ( ) ( ) ( )( )
( ) ( ) ( )( ) = + = + = =
xe 2xx x x x 2x x eg x e f x f x e e f x e f x e e f x , x
2
( ) = +2x
x ee f x 2
, .
( )= = + = 1 1x 0 f 0 2 2
.
, ( ) ( ) ( )
= = = = x2x 2x x2x
x x
x
e 1 e 1 e ee 1e f x e f x f x , x
2 2 2 22e.
iii. f f , f
( ( ) ( )0
0 0x xlim f x f x , x
= )
+
( ) ( )( )
( )
2x
2x2x 2x 2xx
x x xx x x x x xx
2xx
xx x
e 1e 1f x e 1 2e 4e2elim lim lim lim lim lim
x x 2xe 2e 2x 2e 22xe
8elim lim 4 e
2e
+ + + + + +
+ + + + + +
+ +
= = ==== = ===== ====
+ +
= = = +
, f
C + .
( ) ( )( )
( )
+
= = ==== = =
+ + = +
xx
xxx xx x
x x x x x
x
e ee ef x e e e e2lim lim lim lim lim
x x 2x 22x
0lim
2
,
fC .
iv. ( ) ( )2f x 1 f x 0 = + > , x , f f 1 1 .
1f
( )y f x= ( )2x
x
e 1f x
2e
=
( ) ( )2x
22x x x x
x
e 1y e 1 2ye e 2y e 1 0
2e
= = = : xe .
: 24y 4 0= + > .
-
: 2015
- 200 - :
96
,2
x 22y 4y 4
e y y 12
+= = + . , x x 2e 0 , e y y 1> = + + .
[ 2y y 1 0 , y + < . , y 0= , 1 0 < .
y 0< , ( ).
2 2 2 2y 0 , : y y 1 0 y y 1 y y 1 0 1 , > + < < + < + < ]
( )x 2 2e y y 1 x ln y y 1= + + = + + .
, ( ) ( )1 2f x ln x x 1 , x = + + .
, ( ) ( ) ( ) ( )( )
= + + = + +
x x
2 2
0 0
h t dt ln x x 1 , x h t dt ln x x 1
( )( )
+ + +
+ + + + = = = =
+ + + + + + +
2
22 2
2 2 2 2
x x 4x1
x x 1 x 1 x 1 1h x
x x 1 x x 1 x x 1 x 1. ,
( )2
1h x , x
x 1=
+ .
( )22
1ln x x 1
x 1
+ + = +.
, : ( ) ( )1 1
2
2 00
1I dx ln x x 1 ln 1 2
x 1
= = + + = + + .
-
: 2015
- 200 - :
97
39
f : , ( )f x 0, x .
1f f
( )
( ) ( )f x x
2 1
3 1
2x 3x f t dt 5 f 1 t dt,+ + = + x , :
i. ( ) ( )f 1 x 4x 3 xf x , = + + x .
ii. ( ) 0,1 ( ) ( )f f 1 4. + =
iii. H f .
iv. f
y 3,x 0= = x 1= 1= ..
39
i. 1f ( ) x
1
3
f t dt f
( )( )
f x
1
3
f t dt . , ( )f 1 t ,
( ) x
1
f 1 t dt . ( )x
1
1 t dt
= = u 1 t du dt .
= =t 1 u 0 .
= = t x u 1 x .
, ( ) ( )( ) ( )( ) ( )1 xx
1 0
f 1 t dt f u du f 1 x 1 x f 1 x = = =
.
( )( )
( ) ( )( ) ( ) ( )
+ + = + + + =
f x x12 1
3 1
2x 3x f t dt 5 f 1 t dt 4x 3 f f x f x f 1 x
( ) ( ) + + = 4x 3 x f x f 1 x , x (1)
-
: 2015
- 200 - :
98
ii. x 1= (1) ( ) ( )f 0 7 f 1= + .
x 0= (1) ( )f 1 3= .
... f 0 , 1
( ) ( ) ( ) ( ) ( )( ) ( )f 1 f 0
0 , 1 : f 3 7 f 1 4 f 11 0
= = + =
. , ( ) ( )f f 1 4 + = .
iii. ( )
f
f x 0 , x f .
ii) f . ,
( )f 0 < ( )f 1 0 < . , ( )f x 0 , x f < .
( )
( )f x
1
3
f t dt ( ) ( ) ( ) = = =1u f t t f u dt f u du .
( )( ) ( ) ( )
= = =
= = =
t 3 f u 3 u 1
t f x f u f x u x ,
( )( )
( ) ( ) ( )f x x x
x1
13 1 1
f t dt u f u du u f u f u du = = = = ( ) ( ) ( )1
x
xf x f 1 f t dt= + .
,
( ) ( ) ( )1 x1
2
x 0
2x 3x xf x 3 f t dt 5 f t dt
+ + + = , x .
x 0= ( ) ( ) ( ) ( ) + = = 1 1 1
0 0 0
3 f t dt 5 f t dt f t dt 4 2
( ) ( )
< > = >f
x 1 f x f 1 3 0 . , ( )f x 3 0 , x 1 > < .
: ( )( ) ( )( )21 1
0 0
E f x 3 dx f x dx 3 4 3 1= = = = ..
-
: 2015
- 200 - :
99
40
f,g, ( ) ( ) ( )f x 0,g x 0, x 0 , ,
( )( )
( )
x
0x
0
t t dt
h x
t dt
=
, 0x , ( )
( )
( )h 1
h 2
f x dx 0.
i. h ( ),0
( )0, .+
ii. ( )h 0 0,= h 1-1 .
iii. f g ( )a, ,
( ) ( )0 a ,f g < < > ( ) ( )
a a
f x dx g x dx,
-
: 2015
- 200 - :
100
( )( )
( )
( ) ( ) ( ) ( )
( )
= = =
x x x xx
0 0 0 00
x 2x
00
t t dt t dt t t dt t dtt t dt
h x
t dt t dt
( ) ( ) ( ) ( )
( )( )
( ) ( )
( )( )
( ) ( )
( )
= = = >
x x x x x
0 0 0 0 0
2 2 2x x x
0 0 0
x x t dt x t t dt x t dt t t dt x t t dt
x x 0
t dt t dt t dt
, x 0> [ ( ) ( )2
x
0
x 0 , t dt 0
> > 0 t x ,
( ) ( )x t 0 x t t 0 ( ) ( ) ( )m t x t t=
0 , x . , ( ) ( )x
0
x t t 0 > , x 0> ]. , h ( )0 , + .
x 0< h ( ), 0 , ( ) ( )t t , t
( ), 0 .
( ) ( )( ) ( )
( )
x
0
2x
0
x t t dt
h x x 0
t dt
= >
[ ( ) ( )2
x
0
x 0 , t dt 0
> >
x t 0 x t 0 ( ) ( ) ( ) ( )0
x
x t t 0 x t t dt 0 .] . , h ( ) ( ), 0 , 0 , + .
ii. ( )( )
( )
x
0
x
0
t t dt
, x 0h x
t dt
0 ,x 0
= =
h 0. ( ) ( )x x
0 0
t t dt , t dt
0 ( , ( ) ( )t t , t ).
-
: 2015
- 200 - :
101
, ( )( )
( )
x
0
xx 0 x 0
0
t t dt
lim h x lim
t dt
=
:
0
0.
DL : ( ) ( )( ) ( )x 0 x 0 x 0x x
lim h x lim lim x 0 h 0 x
= = = = . , h
0.
, ( ) ( )
) > +
+ +
h x 0 , x 0 ,h [0, )
h 0 ,.
, ( ) ( ) (
>
h x 0 , x ,0h ,0
h ( ,0].
, h h 1 1 .
iii. ) : ( ) ( ) ( ) a , : f g < .
[
( ) ( ) ( )( ) ( ) ( ) ( )( ) x
a a
: a , , x x f t g t dt a f t g t dt = .
a , ( ( )x
a
f g a , ).
, ( ) ( ) ( )( ) ( ) ( ) ( )( )
a a
a a f t g t dt a 0 a f t g t dt= = .
( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( )
( ) ( )( ) ( ) ( )( ) ( ) ( )( )
= =
+ =
a a a
a a a
f t g t dt a f t g t dt f t g t dt
f t g t dt a f t g t dt a f t g t dt
.
, ( ) ( ) a = ( )a , . , Rolle,
( ) ( ) a , : 0 =
( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( )
( ) ( )
= = < .
, ( ) ( )1 2g x g x < .
,
( ) ( )( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 1 1 2 2 1 2
1 2
f x f xf x g x f x g x y x y x y a ,