lisari tea

23

!

: lisari team

:

: 2015-2016

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lisari tea 2016 http://lisari.blogspot.gr

1: ,

.

1

f f x 4x 3, x 1R f R R

, f 1 1 . fR R ,

0y R , 0x R , :

(: 0y R

0

0

y 3x f

4

0 0 00 0y 3 y 3 y 3

f x f f f f 4 3 y f4 4 4

R R )

2: 2f x dx 0

f x 0

, . : f , , f x dx 0

f x 0

f ,

1

f : 0,1 R

1

x 2

0

14 e f x f x dx 1

e

1 1 1

21x 2 x 2 x x x

00 0 0

14 e f x f x dx 1 4 e f x f x dx e ... e 2e f x 1 dx 0

e

2

x xx e 2e f x 1 0,1

1

2x x

0

e 2e f x 1 dx 0, .

f 1 1 0

0 0 0 0 0 0 0

3 f yf x y f f x f y 4x 3 f y x

4

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lisari tea 2016 http://lisari.blogspot.gr

x 0 0,1 x1

f x , x 0,12e

.

2

, 1 2 2

2 x 1dx2

,

2 2

2 x 1dx 2 x 1dx xdx x 2 x 1 dx 0 12

x 2 x 1 0, x 1 ( x 2 )

, . . x 2 x 1 dx 0,

1

3:

1

1

0

f x f x xf x dx 1 f 0 1

: 1

0

A xf x dx 1 :

xe

xf x f x A ... f x A A 1 e

1 1

x x

0 0

A xf x dx 1 A x A A 1 e dx 1 ... A 0 f x e

4: ,

.

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lisari tea 2016 http://lisari.blogspot.gr

(H )

5:

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1

1

0

f x 1 1 3xt f t dt

: 1 1

0 0

A f t dt tf t dt

1 1 1

0 0 0

f x 1 1 3xt f t dt 1 f t dt 3x tf t dt 1 A 3 x

1 1 1

0 0 0

2 5 8A f t dt 1 A 3 t dt ... B tf t dt ... A f x 2x

3 3 3

6: ( ) f x g x , f

0x k g 0x k .

1

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lisari tea 2016 http://lisari.blogspot.gr

7:

1

f : [0,1] R 0,1

x 1 x 1 x 1x e 1 f x e 0,1 x f , 1e x x

f 0,1

x 0 x 1 f 0 0 f 1 0 . :

x 1 x 1 x 1e 1 xe f x 1 0x e f x

0, x 0,1f x :

x 1 x 1 x 1

2

e 1 xe f x x0

f x

e 1 f x

x 1x e 1

f x0

Rolle

x 1x e 1g x

f x

0,1 , 0,1 :

g 0 . f 0,1

8: Fermat

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lisari tea 2016 http://lisari.blogspot.gr

1

f : , R , , ,

f , 1,2 f 0,f 1.

1 2x ,x , ,

1 2f x f x 0

f , , .

f , 1,2 , f 0, f 1

1 2 1 2x ,x , : f x 1 f x 2

Fermat, 1 2f x f x 0

9:

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1

f : R R

f 0 2 , f 0 0 2 4 2f x 2f x x 2x , xR

2 24 2 4 2

22 2 2

g x

f x 2f x x 2x f x 2f x 1 x 2x 1

f x 1 x 1 f x 1 x 1, x

R

g R (;) g 0 1 0 .

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lisari tea 2016 http://lisari.blogspot.gr

3 3f 0 0

2 2 x xf x 1 x 1 f x x 2 f x 2x c f x 2x3 3

,

10: , .

1

ln x

f x , x ex

. :

) 1

f e, 0,e

)

x

3x

2

x x xf e e 0

x 2016 ln xlim

) f e, ,

x

1f e, f x , f e ... 0,

elim

) x e

2

x0

3 3x 2016x x

2 2

x x 1 x x x 1 x0 f e e 0 f e e

ln x e x 2016 ln x e x 2016

11: ,

.

1

f

2f x x 1 x

) f

)

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lisari tea 2016 http://lisari.blogspot.gr

x 0

2016 1x f x 0lim

)

fD 0,1 2

1 xf x 0

2 x 1 x

0,1

) f

,

f 0,1 f 0 ,f 1 1,1

, 1f 1,1 ,

x 0

2016 1x f xlim

x 1,1

1 2016 1 20160 f x 1 0 x f x x

.

12: ()

1

2f x x 1 x ,x R . R

2

3

2

3x 1f x 1 1

2 1

f R , 1-1. 2

3 3xg x x 12

1 f g x f g x

1 1

g , 1 , , g 1,0 1, , g 0, 1,2 2

1 1

2

11

2

1 1

2

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lisari tea 2016 http://lisari.blogspot.gr

13: o o

x x x x

2f x 0 f x 0lim lim

.

1

f : r r

2xlim f x 4f x 1 3

xlim f x

22

x xlim f x 4f x 4 0 lim f x 2 0

2

x xlim f x 2 lim f x 2 0 0

f x 2 f x 2 f x 2

x xlim f x 2 0 lim f x 2

14: f / /

/ f fC (

).

4 . 277 !

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lisari tea 2016 http://lisari.blogspot.gr

15:

.

1

xR

1 1 1

0

2 2

0

2

0x dt ttdt dt2x t t 1 1 0

:

1

0

2 dt2

t t 13

1 1 1 1

0

22 2 2

0

2

0 0x dt t dttdt 2x x dt t 1 t1 t t 1

xR

x xR 0 . :

1 11 1 1 1

0 0 0 00 0

1

2 32 22 2

1

0 0

2

22 2

t t 1 tdt 1 0 t tt t

4 dt 4 t 1 0

2

dt dt t2 3

2dt dt t 1 t0 t t 1

33

16:

.

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lisari tea 2016 http://lisari.blogspot.gr

1

xxe x 0

f x0 , x 0

f 0 1!

17: (

lisari team).

1

f : 1, R

x

2 22 2016 2014

2015

2f 2 ln 1 x x 10,

xlim

1

22x x

f x x 1 ln x 1 , x 12

1

2 222f 2 ln 1 0

x x

2 22 2016 2014

2 22

2015

2 22

2 22

2f 2 ln 1 x x 12f 2 ln 1 x

x

, 2f 2 ln 1 0,

, 2f 2 ln 1 0

lim lim

. 22x x

f x x 1 ln x 1 , x 12

.

18: .

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lisari tea 2016 http://lisari.blogspot.gr

1 ( 11 . 340 . )

f f 0

f x f x xdx 2f 1

f .

0

f x xdx

1 f 0 f Rolle!

19: 1f f 0f x (

lisari team).

1

f 1 1 ,

1 3f x x 2x 3 , x R .

3 2

x 0

f x x 3x 1lim 12

f x 1L

1

f 05

1 3f x x 2x 3 , x R , x f x , :

3f x 2f x 3 x , xR

:

x 0

u 1

23 3 3

u f x

x 0u 1

23 3 3

f x f x 2f x 3 3 f x 2f x 3 1

f x 1

u u 2u 3 3 u 2u 3 1... 12

u 1

lim

lim

1f 1 0 f 0 1 .

1f f x x , x R

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lisari tea 2016 http://lisari.blogspot.gr

1x 0 x 0f x f 0 f x 1

lim lf

imx 0 xf

u f x , x 0 x 0limu limf x f 0 1

, f 0. :

1 1 1 11x 0 u 1 u 1 u 1

1

f x 1 u 1 1 1 1 1lim lim lim lim

5f fu u u 1f x f fff 1

u 1 u 1

20: (

lisari team).

1

f [0,1] .

3 2015 2 4 2016f x f x ... f x f x f x .. f x , x [0,1] (1)

(1) 1 2x 0,x 1 .

f [0,1] ( )

0,1 , :

2 2

3 4 3 4

5 6 5 6 3 2015 2 4 2016

2015 2016 2015 2016

f f

f f

f f f f ... f f f .. f , !

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