ΕΚΦ-ΛΥΣΕΙΣ-ΓΕΛ-3 (1).pdf
Transcript of ΕΚΦ-ΛΥΣΕΙΣ-ΓΕΛ-3 (1).pdf
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10-30-40-50-60
3 2015
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(1) 260 (2) 247 (3) 280 (4) -
Z, W
Z, W ( 1 ) Z W WZ ( 2 )
Z W 1 ( 3 ) Z
WZ 1
( 4 )
(1) : Z W
WZ 1
( 5 )
(2) 1 1 ( 6 )
(3) 1 Z WIm( ),Im( ) Z, W
, :
(3) Z WIm( )Im( ) 0 . ( 7 )
(3) : 1Z W(ln( 1))Im( ) (1 e )Im( ) 0xx ,
x (0,1) ( 4 )
(4) Z, W , 1 2 3
,u ,u u u
Z W u , 1 2 3 1
u u u u
: 1 2 3
,u ,u u
. ( 3 )
(1) Z
W ZW W Z ZW Z WZ 1
( 1)
( 4 )
ZW 1 0 , ZW
1 , ( 3 ) WW 1 , WZ
( 2 ). ZW 1 0 . Z W
WZ 1
-
2 6
(2) 1 , ( 4 ) Z
WZ 1
1
, 1
1 , ( 4 ) Z
WZ 1
1
, 1
(3) Z W 2
Z W WZ Z W WZWZ
1 11
1 1
Z W WZ 2Z W WZ Z W Z W1 1( )( ) ( )( ) 4 Im Im 0 Im Im 0i
(3) 1Z W( ) (ln( 1))Im( ) (1 e )Im( )xx x , 0x
. 1 W(0) (1 e )Im( ) , 1 11
1 e 1 e 0e
Z(1) ln2Im( ) , 2 1 ln2 0 , (0) (1) 0
Bolzano
[0,1] . 0 (0,1)x 0( ) 0x
0x
1Z W(ln( 1))Im( ) (1 e )Im( ) 0xx
(4) Z W Z Wu u ( 4 ) u 1 .
1 2 3
,u ,u u
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3 6
(1) e 1 ln( 1)x x , [0, ) ( 7 )
(2) e 1 ln( 1)x x , [0, )x ( 3 )
f : [0, ) . x 0
: e 1 f ( ) 2e 1 ln( 1) 1xx x x .
(3) f ( ) 2e 2xx x , [0, )x ( 5 )
(4) ( ):y 2x , f
C
f ( 5 )
(5) ( )
f
C f , ( )
(4) ( 5 )
(1) ( ) e 1 ln( 1)xx x , 0x . (0)
[0, )
1
( ) e1
xxx
. e 1x 1
11x
0x , ( ) 0x , 0x ,
[0, )
e 1 ln( 1)x x , [0, )
(2) (1) [0, )
( ) (0) 0x , e 1 ln( 1)x x , [0, )x
(3) e 1 f ( ) 2e 1 ln( 1) 1xx x x , [0, )x
e 1 ln( 1) 0( ) ( ) .
, (2), ( ) (0) 0x 0( ) ,
0( ) ( ) , 0 .
1 1 f ( ) 2e 1 1xx x x , f ( ) 2e 2xx x
(4) f ( ) e 2
lim lim [ 2 1 ] 1x x
xx
x x x
lim [f ( ) ] lim [ 2e 2] 2x x
xx x
( ) :y 2x , fC
f
-
4 6
(5) f ( ) 2e 1xx , ( )
f
C f , 2e 1x .
( ) ( ) / / 2e 1 1x , e 0x ,.
( ) fC
f , ( )
f , g : (0, ) ,
e
2 21 1
f (t) e f (t)( ) dt dt
e 1t t
x
g x
f f ( )f ( ) 0x x
x
f ( ) 0x 1x
2 2 f ( )
(f ( )) f ( )f ( ) f ( )(f ( )) f ( )ex
x x x x x x
, 1x
(1) f
f ( 6 )
(2) f (1) f
( 6 )
f ( ) lnx x ,
(3) g [1, e] ( 5 )
(4) ( )
Cg
g , x x
1x ex ( 8 )
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(1) f (0, ) ,
f ( ) 0 , .
f ( ) 0x , f .
f (0, ) ,
1 1
2 2 f ( )lim[(f ( )) f ( )f ( ) f ( )(f ( )) lim[f ( )ex x
xx x x x x x
,
2
(f (1)) f (1) f (1) 1 , f ( ) 0x , f
(0, )
-
5 6
(2)
f ( ) 2 2f ( )e f ( )(f ( )) (f ( ))
f ( )f ( )
xx x x x
xx
f ( )x
f ( ) 2 2
1lim[f ( )e f ( )(f ( )) (f ( )) ] 0
x
xx x x x
1
f ( )lim 0x
x
De l Hospital
f ( ) 2 f ( ) 3
0
f ( )e (f ( )) e (f ( )) 2f ( )f ( )f ( ) 2f ( )f ( )lim f ( ) lim
f ( )1
x x
x
x x x x x x x xx
xx
f (1) f (1) 1 1 2f (1) f (1) 1 f ( ) 0x f ( ) 0x
x , f
f ( ) lnx x 1 1
2 2
ef (t) e f (t)( ) dt dt
e 1t t
x
g x
(3) 2 2
f ( ) ln( ) 0
x xg x
xx
, [1, e]x g
[1, e]
(4) 1 1
2 2
e ee(e) dt dt
e 1t t
f (t) f (t)g
,
12
ee(e) (1 ) dt 0
e 1 t
f (t)g
( ) 0g x , [1, e]x
1 1 1
21 1
e e ef (t) e f (t)( ) ( )d dt d dt 1d
2 e 1t t[ ][ ]x x
g x x x x
2
1 1 12
e ef (t) f (t)e( ) dt d (e 1) dt
e 1t t
[ ][ ]x
x x
e1
2 2 2
e e
1 1 1
f (t) f ( ) f (t)( ) dt d e dt
t t
[ ]x x
x x x
x
2
1 1
e eln 1 1( ) d (ln ) d
2 2
xx x x
x
-
f ( ) 2 f ( ) 2 f ( )
e (f ( )) f ( )f ( )e f ( )(f ( )) e f ( )x x x
x x x x x x , 1x
f ( )
f ( )f ( )e f ( )( )xx x x , f ( )f ( )f ( )e f ( ) Cxx x x , f , f 1x f (1) 0 .
1 1
f ( )f ( )f ( )e f ( ) Clim lim( ( )
x x
xx x x
, C 0 , f ( )f ( )f ( )e f ( )xx x x
f ( ) f ( )
f ( )e 1 e( )x xx x , f ( )1
e Cx
x 1 1
f ( )
1f ( )e Clim lim( )
x x
xx x
-
6 6
1
C 0 f ( )
ex
x , f ( ) lnx x .
( 1,2,3 )
1 1 1 1
1
12 2
1 1 1tdt dt t ln tdt t ln t t t dt
t t
f (t) ln( ) ]
x x x xx
,
12
dtt
f (t)x ln
1 11
x xx ,
12
ee( ) ln dt
e 1 t
1 1 f (t)1g x
x xx
..