lisari team €²»­ˆµ¹‚ ³¹± „±...

download lisari team €²»­ˆµ¹‚ ³¹± „± œ±¸·¼±„¹¬  ƒ±½±„»¹ƒ¼

of 16

  • date post

    19-Jan-2017
  • Category

    Education

  • view

    49.763
  • download

    4

Embed Size (px)

Transcript of lisari team €²»­ˆµ¹‚ ³¹± „±...

  • lisari tea

    23

    !

    : lisari team

    :

    : 2015-2016

  • __________________________________________________________________________________

    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

  • __________________________________________________________________________________

    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: ,

    .

  • __________________________________________________________________________________

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

    (H )

    5:

    .

    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

  • __________________________________________________________________________________

    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

  • __________________________________________________________________________________

    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:

    .

    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 .

  • __________________________________________________________________________________

    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

    )

  • __________________________________________________________________________________

    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

  • __________________________________________________________________________________

    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 !

  • __________________________________________________________________________________

    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:

    .

  • __________________________________________________________________________________

    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: .

  • __________________________________________________________________________________

    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

  • __________________________________________________________________________________

    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 , !

    .........................................