Η άσκηση της ημέρας -...

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    14-Apr-2017
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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    M

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    ) 2xe 0 x R :

    2x 3 2x 32x

    3 3

    e 1 e f x lim f x lim f x e

    2016 x R

    ) f R g

    f , g

    g R

    x x

    g lim g x , lim g x ,R R

    0 g R 0x R : 0g x 0

    g 1 1 0

    x

    )

    g

    0 0 0 0 0

    1 1 1x x g x g x g x 0

    2016 2016 2016

    1

    1

    lnx x 1 x 0 ( x 1 )

    , :

    )

    )

    ) :

    7 : (30/10/2016)

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    2xe 0 x R , x 2xe :

    2x 2x 2xlne e 1 e 2x 1 x R

    2xe 1 x 0

    x g x : 2g xe 2g x 1 0 x R

    0g x 0 x x

    0

    x x :

    2g x 2g x

    1 1g x g x

    2016 2016 1 1g x 2

    2016f g x 2g x 1 e 2g x 1 e 2g x 1

    1g x

    2016 2g xe 2g x 1 0x ,

    00x x

    1 1limg x g x 0

    2016 2016

    0

    0

    2g x 2g x

    0x xlim e 2g x 1 e 2g x 1 0

    2g xe 2g x 1 0 0x ,:

    02g xx x

    1lim

    e 2g x 1

    :

    02g xx x

    1 1lim g x

    2016 e 2g x 1

    2 :

    0 02g xx x x x

    1g x

    2016 1 1lim lim g x

    2016f g x 2g x 1 e 2g x 1

    :

    2

    0 02g xx x x x

    1g x

    2016 1 1lim lim g x

    2016f g x 2g x 1 e 2g x 1

    .

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    ,

    :

    2f x x 1

    x 1

    f x f 1f 1 lim ... 2

    x 1

    , :

    x 1

    f x f 1lim ... 2

    x 1

    x 1

    f x f 1lim

    x 1R f 1 2

    .

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    ) x R 2xe 0

    2x 32x

    3

    e f(x) lim e

    .

    ) f < R ( < ) g < R

    ,

    R . , 0

    x

    )

    00x x

    1 1lim x x

    2016 2016

    1u x

    2016

    0

    01u x

    2016

    1lim g u g(x )

    2016

    g

    00x x

    1 1limg x g x

    2016 2016

    g

    0 0 0 0

    1 1x x g x g x 0

    2016 2016

    1

    t t 1s , 7

    1 cm / s12

    .

    ) 1

    t 0 ,

    .

    :

    i 0t i1

    t s2

    , , i

    t 0 .

    .

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    x 0 :

    3 3 32 2 2 2 2 2

    4 4 4 2 2 2

    3x 4 8 4x 3x 4 2 8 4x 2 1 3x 4 2 8 4x 2

    x x x x x x

    2u x u 0 x 0 :

    x 0 u 0 u 0

    x 0 u 0 u 0

    2

    2

    3 2 3

    2

    f u f 03x 4 2 3u 4 2 3f 0

    u u 0 4x

    g u g 08 4x 2 8 4u 2 2g 0

    u u 0 3x

    lim lim lim

    lim lim lim

    :

    f u 3u 4

    3f u

    2 3u 4

    3f 0

    4

    3 2g u 8 4x

    2

    3

    4g u

    3 8 4u

    1g 0

    3

    :

    x 0 x 0 x 0

    3 32 2 2 2

    4 2 2 2

    3x 4 8 4x 1 3x 4 2 8 4x 2 3 1

    4 3x x x xlim lim lim

    1 0 (20/11/2016)

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    x 0 :

    k 2 k m 4 m 4 m 4k k2 k 2 km m m

    6 6 6 4 2 2 4

    x x x x x 1 x 1

    x x x x x x x

    2u x u 0 x 0 :

    x 0 u 0 u 0

    x 0 u 0 u 0

    k k2 k k

    2 k 1

    m 4 mm m

    4 m 1

    f u f 0x u f 0

    u u 0x k

    g u g 0 x u g 0

    u u 0x m

    lim lim lim

    lim lim lim

    :

    k kf u u

    11

    k k

    f u u k

    k 1

    f 0k

    mmg u u

    11

    m m

    g u um

    m 1

    g 0m

    :

    x 0 x 0 x 0 x 0 x 0

    k 2 k m 4 m 4k 2 km m

    6 4 2 2 4

    ,,,m,k 0

    k 1 m 1

    x x x 1 x 1

    x x x x x

    k m

    lim lim lim lim lim

    :

    :

    k,m,, 2,3,... ,, 0 :

    x 0

    k k mm

    , ,

    x x

    , x

    ,

    lim

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    0

    0 2 x 2 , x 0

    3 32 2 2 2

    4 4 4

    2 2

    34 2 4 2 2 23

    32 2 2 2 2 23

    3x 4 8 4x 3x 4 2 8 4x 2f(x)

    x x x

    3x 4x

    x 3x 4 2 x (8 4x ) 2 8 4x 8

    3 4

    x 3x 4 2 x (8 4x ) 2 8 4x 8

    2x 0

    1lim

    x

    2x 0

    3 3lim

    43x 4 2,

    32 2 2x 0 3

    4 1lim

    4(8 4x ) 2 8 4x 8

    x 0limf(x) .

    0

    0

    m m

    4 4

    x

    , x 0

    k 2 k m 4 m 4k 2 km m

    6 6 6

    x x x x f(x)

    x x x

    42

    k 1 m 16 2 k k 1 6 m 4 m 1k m

    4 2k 1 m 12 k k 1 m 4 m 1k m

    xx

    x x ... x x ...

    1 1

    x xx ... x ...

    :

    k 1 m 1x 01 1

    limf(x)k m

    ,

    .

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    :

    32 2

    4

    3x 4 8 4x

    x=

    2

    4

    3x 4 2

    x -

    3 2

    4

    8 4x 2

    x=

    =2

    4

    3x

    x

    2

    1

    3x 4 2+

    2

    4

    4x

    x

    2

    2 23 3

    1

    8 4x 2 8 4x 4

    =

    =2

    3

    x

    2

    1

    3x 4 2+

    2

    4

    x

    2

    2 23 3

    1

    8 4x 2 8 4x 4

    2 2x 0

    3 1lim

    x 3x 4 2= ,

    2 3 2x 0

    4 1lim

    x 8 4x 2=

    x 0lim

    32 2

    4

    3x 4 8 4x

    x=

    x 0 ,

    k 2 k m 4m

    6

    x x

    x=

    k 2 k

    6

    ax

    x -

    m 4m

    6

    x

    x=

    2

    6

    x

    x

    k 1 k 2

    2 k 2 k 2 2 k k 3 k 2 2 k k 1kk k

    1

    x x (x ) ..... x

    +

    4

    6

    x

    x

    m 1 m 2 m 3

    m 4 m 4 2 m 4 m 2 m 4 m 1mm m m

    1

    x x x ..... x

    =

    4

    x

    k 1 k 2

    2 k 2 k 2 2 k k 3 k 2 2 k k 1kk k

    1

    x x (x ) ..... x

    +

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    2

    x

    m 1 m 2 m 3

    m 4 m 4 2 m 4 m 2 m 4 m 1mm m m

    1

    x x x ..... x

    x 0lim (

    4

    x

    k 1 k 2

    2 k 2 k 2 2 k k 3 k 2 2 k k 1kk k

    1

    x x (x ) ..... x

    )= ,

    x 0lim (

    2

    x

    m 1 m 2 m 3

    m 4 m 4 2 m 4 m 2 m 4 m 1mm m m

    1

    x x x ..... x

    )= ,

    :

    x 0lim

    k 2 k m 4m

    6

    x x

    x=

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    :

    232 2

    32 2

    4x 0 x 0 34 2 2

    3x 4 8 4x3x 4 8 4xlim lim

    x x 3x 4 8 4x

    3 22 2

    2x 0 2 223 3 34 2 2 2 2 2 2

    3x 4 8 4xlim

    x 3x 4 8 4x 3x 4 3x 4 8 4x 8 4x

    6 4 2

    x 0

    27x 108x 144x 64lim

    4 216x 64x 64

    22 22

    3 3 34 2 2 2 2 2 2x 3x 4 8 4x 3x 4 3x 4 8 4x 8 4x

    2 4 2

    2x 0 2 223 3 34 2 2 2 2 2 2

    x 27x 92x 208lim

    x 3x 4 8 4x 3x 4 3x 4 8 4x 8 4x

    4 2

    2x 0 2 223 3 32 2 2 2 2 2 2

    27x 92x 208lim

    x 3x 4 8 4x 3x 4 3x 4 8 4x 8 4x

    4 2

    x 0

    2x 0 2 223 3 32 2 2 2 2 2 2

    lim 27x 92x 208

    1lim

    x 3x 4 8 4x 3x 4 3x 4 8 4x 8 4x

    208 ,

    22 22

    3 3 32 2 2 2 2 2 2