Πεδίο ορισμού της παραγώγου
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Transcript of Πεδίο ορισμού της παραγώγου
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2 2017
1) f ;
2) f x ln x 1
f xx
*xR ;
3)
2
3 23x ,x 0
f x x , g x x , h x xx, k xx x 0
;
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( )
f , x A f
. , x B
h 0
f (x h) f (x)f (x) lim
h
.
() (derivative) f f .
:
B A
f : A R f : B A R
1) f x ln x 0, . 1
f xx
,
* ,0 0, R . ;
, f.
f 0, .
, :
x 0, : 1
f x ln xx
2) f x x 0,
0, 1
f x2 x
.
-
f
f. ; f 0,
1
f x2 x
f 0
x 0 .
:
x 0 x 0 x 0
f x f 0 x 1lim lim lim
x 0 x x
f ( ) 0
x 0 .
3) f x x x 0,
0, . f 0
x 0 x 0
f x f 0 x xlim lim 0 1 0
x 0 x
f :
x
x x, x 0f x 2 x
0 , x 0
4) f : R R f R .
x , x 0
f (x)x 1 , x 0
R ,
x ,0 0, :
x , x 0f (x)
x 1 , x 0
x , x 0
f (x)x , x 0
f 0
x 0 :
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x 0 x 0
f x f 0 xlim lim 1
x 0 x
x 0 x 0
f x f 0 x 1lim lim 0
x 0 x
f 0
x 0 .
, f *,0 0, R .
5)
23f x x .
:
f R , xR :
2 2 1
123 3 3 3
2 2f x x x x x
3 3
23 x 2
3x
x. x.
, 2 23 3f x x x
:
x 0, 2 2 1
123 3 3 3
2 2f x x x x x
3 3
x ,0
2 2 1
2 13 3 3 3
2 2f x x x x x x
3 3
0
x 0 :
2 2 23 33 3
333x 0 x 0 x 0 x 0 x 0
f x f 0 x x x 1lim lim lim lim lim
x 0 x x xx
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f 0 .
f :
1
3
3
1
3
3
2 12, x 0x , x 0
3 x3f x
2 2 1x , x 0 , x 0
3 3 x
6)
.
, , f, 2
3f x x 1 .
,
x 0, . f
A 1, x 1 0 x 1 .
, x 1, :
2 2 1
13 3 3
3
2 2 2 1f x x 1 x 1 x 1 x 1
3 3 3 x 1
:
:
2 2
ln x233 3f x x , g x x , h x e ; ,
R f x g x h x x A .
7) , xf x x
0, .
, x 0, : x x ln x x ln x xf x x e e x ln x x ln x 1
-
.
.
1) f : , R f : , R f : , R
f .
2) f : , R f : , R f : , R ,
f .
f
.
3)
.
,
1
2
g x , xf x
g x , x
1
2
g x , xf x
g x , x
( 1 2
g ,g ) f
0
x .
g x , x
f x, x
f x g x x , ,
( g ) f
0x .
f x x , x 0,2 .
