H άσκηση της ημέρας -...
date post
19-Jan-2017Category
Education
view
11.276download
4
Embed Size (px)
Transcript of H άσκηση της ημέρας -...
:
lisari.blogspot.gr
http://lisari.blogspot.grhttp://lisari.blogspot.gr
___________________________________________________________________________ 2015 http://lisari.blogspot.gr
1
http://lisari.blogspot.gr/
___________________________________________________________________________ 2015 http://lisari.blogspot.gr
2
8
(1-11-2015)
8/11/2015
f 0,1 ,
:
2x x 1 f(x) 2x 1, x 0,1
f 0,1
: x 3x , f
C
. .
. f(x) 0 ,
1 2
, .2 3
.
1 1
10
2
f(x)dx f(x)dx
.
2
x 0
x ln f(x) 1lim
f(1 x) 1
. , 0,1 , , :
2f() f() 2 .
http://lisari.blogspot.gr/
___________________________________________________________________________ 2015 http://lisari.blogspot.gr
3
1 ( )
. x 0 : 1 f(0) 1 f(0) 1. :
x 0
2
x 0 x 0 x 0
f(x) 1 f(x) 1x x f(x) 1 2x x 1 2 lim x 1 lim lim 2
x x
1 f (0) 2.
x 1 :1 f(1) 1 f(1) 1. :
x 1
2
x 1 x 1 x 1
f(x) 1 f(x) 1x x 2 f(x) 1 2x 2 2 x 2 lim 2 lim lim x 2
x 1 x 1
2 f (1) 3.
: x fC ,
0,1 , f () 1 .
f 0,1 ,
f 0,1 , :
1 f (0) f () 1 f (0) 1.
: 3x fC ,
0,1 , f () 3 .
f 0,1 ,
f 0,1 ,
:3 f () f (1) 3 f (1) 3.
fC 0, 1 1,1
, f(0) 0 1 f(1) 3 2.
. f 0,1 ,
f (x) f (0) 1 f . 0,1 , f
f 0,1 f(0),f(1) 1,1 .
0 1,1 , 0,1 , f() 0 .
.
f .. 0,
,1 , 1 0, 2 ,1 , :
1
f() f(0) 1f ( )
2
f(1) f() 1f ( )
1 1
.
f .
1 2 1 2 f ( ) f ( )
1 1 1
1 . 1 2
f .
2 2 1 f ( ) f (1)
1 1 2
3 1 .1 3 3
http://lisari.blogspot.gr/
___________________________________________________________________________ 2015 http://lisari.blogspot.gr
4
. 1
x 0,2
:
1 1
12 2f .
0 0 1
10 x f(x) f() 0 f(x)dx 0 f(x)dx f(x)dx 0
2
1
1 12
10 1
2
f(x)dx f(x)dx f(x)dx.
. 2 x 0
x 0 x 0
x 0 x 0
lim f(x) 1 ln f(x) 1x ln f(x) 1 f(x) 1 ln f(x) 1lim lim
f(x) 1 f(1 x) 1 f(x) 1 f(1 x) 1f(1 x) 1lim lim
x x x x
00,
1 ( 3)
:
f(x) 1
DLHx 0 0 0 0
2
1
ln lim f(x) 1 ln f(x) 1 lim ln lim lim 0
1 1
.
x 0
f(x) 1lim f (0) 1.
x
1 x u
x 0 u 1
f(1 x) 1 f(u) 1lim lim f (1) 3.
x 1 u
. g h :
2g(x) f(x) x x,x 0,1
h(x) f(x) 2x,x 0,1
: g(0) f(0) 1 g(1) f(1) 2 3 . g
0,1 , Bolzano 0,1 ,
g() 0 2f() 0. (1)
: h(0) f(0) 1 h(1) f(1) 2 3 . h
0,1 , Bolzano 0,1 , h() 0 f() 2 0. (2)
(1) (2) : 2f() f() 2 .
. : 2 2f() f() 2 0 0
1 , , 0,1 . .
http://lisari.blogspot.gr/
___________________________________________________________________________ 2015 http://lisari.blogspot.gr
5
2 ( )
A. = + (1, (1)) (1) = 1
(1) = 1 + . ,
[0,1] . 1 0 ( 1)
(0) ()
= 3 + (2, (2)) (2) = 3
(2) = 32 + . ,
[0,1] . 2 1 ( 2)
(1) ()
2 + 1 () 2 1 , (1) [0,1]
= 0 (1) 1 (0) 1 (0) = 1
= 1 (1) 1 (1) 1 (1) = 1
0 (0) = 0+
()+1
1 (1) = 1
()1
1
2 + 1 () 2 1 2 + () + 1 2 + 1 ()+1
2
,
0+
( + 1) 0+
() + 1
0+
2 1 (0) 2 ()
:
{ 1 (0)
1 (0) (0) = 1(0) = ( 1)
11 1 = 0
: (1) = 1 + 1=0 (0) = 0 + =
2 + 1 () 2 1 2 + 2 () 1 2 2
2 + 2
1() + 1
12 2
12 2
1() + 1
12 + 2
1
,
http://lisari.blogspot.gr/
___________________________________________________________________________ 2015 http://lisari.blogspot.gr
6
1
(2 2
1)
1
() 1
1 1
2 + 2
1 2 (1) 3 ()
: { 3 (1)
3 (1) (1) = 3(1) = ( 2)
11 2 = 3
: (2) = 32 + 2=3 (1) = 3 + =
. , (
) = . () = () 2 + 1 [0,1]
() 2 1 () 2 + 1 0 () (1
2) .
1
2 Fermat (
1
2) = 0 (
1
2) 2 = 0 (
1
2) =
2
( 1
2 , (
1
2)) : = 2 1
: () 2 1
: { () 2 1
() 2 1
() = 2 1
(
) .
=1
2 (1) (
1
2) 0 (
1
2) < 0
=2
3 (1)
1
9 (
2
3) (
2
3) > 0
{
[1
2,2
3]
(1
2) (
2
3) < 0
Bolzano () = 0
(1
2,2
3)
:
0 1 (0) () (1) 1 () 3 () > 0
() > 0 [0,1] 1-1
() = 0 [0,1]
http://lisari.blogspot.gr/
___________________________________________________________________________ 2015 http://lisari.blogspot.gr
7
, () = 0 (1
2,2
3) .
. () 2 1 ()1
20
(2 1) =1
41
2< 0
1
20
()1
20
< 0
()
12
0
< 0 ()
1
0
+ ()
12
1
< 0 ()
1
0
< ()
1
12
.
0+
(1)1
=
1
()1
1= (1) = 3
2 + 1 () 2 1 2 + () + 1 2
(2 + ) (() + 1) (2) (2 + ) (() + 1)
(2)
{
0+
((2 + )) = = 0
0+
((2)) = = 0
.. 0+
((() + 1)) = 0
0+
2(()+1)
(1)1= 0+
(()+1)(1)1
=0
3= 0
. () = () + 2 + () 2
(1
2,2
3) () = 0 , =
() = () + 2 + () 2 () = () + 2 + 2
(1) = (1) + 2 2 = 3 2 > 0
() = () + 2 + 2 = 2 = ( 1) < 0
{ [, 1]
()(1) < 0 Bolzano (, 1) () = 0
() + 2 + () 2 = 0 () + 2 + = () + 2
http://lisari.blogspot.gr/
___________________________________________________________________________ 2015 http://lisari.blogspot.gr
8
3 ( )
A.
+ () ()
= 0 (1) (0) = 1
= 1 (1) (1) = 1
> 0 (1) 2 + () + 1 + 1 ()(0)
, f
lim0( + 1) lim
0
()(0)
()
< 1 (1) + 2 ()(1)
1 , f
lim1( + 2) lim
1
()(1)
1 ()
[0,1] [0,1] 1-1
- (1, (1)) = + ,
(1) = 1
1 0 (1)
(0) 1 (0) () () =
=
(0,-1) 1 = 0 + =
- (2, (2)) = 3 + ,
(2) = 3
2 1 (2)
(1) 3 (1) () () =
=
(1,1) 1 = 3 + =
B.
[ 1
2 ,2
3 ]
(1) = 1
2 (
1
2) 0 ,
(1
2) = 0 [ 0,
1
2 ] [
1
2 ,1 ] f
, 1 (0,1
2) , 2
http://lisari.blogspot.gr/
___________________________________________________________________________ 2015 http://lisari.blogspot.gr
9
(1
2, 1) , 1 < 2 :
(1) =(1
2)(0)
1
20
= 2, (2) =(1
2)(1)
1
21
= 2 ,
, (
) <
(1) = 2
3
