Mathematica gr μαθ θετ κατ...

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  • ( )

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    18 MA 2016

    1 (18/05/2016, 23:00)

  • 2

    mathematica.gr mathematica

    http://www.mathematica.gr/forum/viewtopic.php?f=133&t=54288

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    mathematica.gr

    http://www.mathematica.gr/forum/viewtopic.php?f=133&t=54288

  • 3

    ( )

    18 MA 2016

    :

    ()

    1. f (, ), 0x ,

    f .

    f (x) 0 0(, x ) f (x) 0 0(x , ) , 0f(x )

    f . 7

    2. f, g .

    4 3. -

    . 4

    4. , , , , , , .

    ) f :[, + IR , G f [,+ ,

    f(x)dx G() G() .

    ) f,g 0x f(x) g(x) 0x ,

    o ox x x xlim f(x) lim g(x)

    .

    ) f , f (x) 0 0 0x (,x ) (x ,)

    0 0(,x ) (x ,) .

    ) f 1 1 , y ,

    y f(x) x .

    ) f *, +, f [,+ M m.

    10

    1. . 262, ( i)

    2. . 141

    3. . 246

  • 4

    4. ) , . 334

    ) , . 166 ) , . 252

    ) , . 152 ) , . 195

    B

    2

    2

    xf(x)

    x 1

    , x IR .

    1. f , - f .

    6 2. f , f

    . 9

    3. f . 7

    4. 1, 2, 3 f . ( )

    3

    :

    1. f IR 2 2

    2xf (x)

    (x 1)

    .

    2 2

    2xf (x) 0 0 x 0

    (x 1)

    .

    x 0 f (x) 0 f x0 = 0, f

    [0, ) .

    x 0 f (x) 0 f x0 = 0, f

    ( ,0] .

    f x 0 f(0) 0 .

    2. f R 2

    2 3

    6x 2f (x)

    (x 1)

    .

    2 3

    2 3

    2 3

    2 (x 1) 02 2

    2 3

    2 (x 1) 02 2

    2 3

    2 (x 1) 02 2

    2 3

    6x 2 1 3 3 3f (x) 0 0 6x 2 0 x x x x

    (x 1) 3 3 3 3

    6x 2 1 3 3 3f (x) 0 0 6x 2 0 x x x

    (x 1) 3 3 3 3

    6x 2 1 3 3 3f (x) 0 0 6x 2 0 x x x x

    (x 1) 3 3 3 3

  • 5

    f 3

    ,3

    ,

    3 3,

    3 3

    3

    ,3

    .

    3 3

    A ,f3 3

    3 3

    B ,f3 3

    3 1A ,

    3 4

    3 1B ,

    3 4

    .

    3. f , R.

    2 2

    2 2x x x

    x xlim f(x) lim lim 1

    x 1 x

    xlim f(x) 1

    .

    f y 1

    . 4. , , f :

    x

    f(x)

    f(x)

    0- 33

    0

    0 0

    + +

    f(x)

    33

    14

    14

    + +

    11

    1, 2, 3, f.

    : f .

  • 6

    1. 2x 2e x 1 0 , x IR .

    4 2. f :IR IR

    2 22 x 2f (x) e x 1

    x IR . 8

    3. 2x 2f(x) e x 1 , x IR , f .

    4 4. f 3 ,

    f x 3 f x f x 3 f x x [0, ) . 9

    :

    1. 2x 2f x e x 1 , IR

    2xf x 2x e 1 .

    2 22 x xx 0 e 1 e 1 0 x IR ,

    f(x) < 0 x < 0, f(x) > 0 x > 0 f(x) = 0 x = 0.

    ,

    f(x) x=0 f(0) = 0,

    f x f 0 f x 0 ,

    x = 0.

    2x 2e x 1 0 , x 0 .

    2. 2x 2e x 1 0 , x IR :

    2x 2f(x) e x 1 , x IR .

    , 2 22 x 2 2 x 2f(x) 0 f (x) 0 (e x 1) 0 e x 1 0 x 0 .

    , f x 0 .

    f (0, ) .

    f (0, ) .

  • 7

    :

    f(x) 0 (0, ) , 2x 2f(x) e x 1,

    f(x) 0 (0, ) , 2x 2f(x) e x 1.

    , f(0) 0 , : 2x 2f(x) e x 1 , x [0, )

    2x 2f(x) e x 1 , x [0, ) .

    , f ( ,0) .

    f ( ,0) .

    :

    f(x) 0 ( ,0) , 2x 2f(x) e x 1,

    f(x) 0 ( ,0) , 2x 2f(x) e x 1 .

    , f(0) 0 , 2x 2f(x) e x 1 , x ( ,0]

    2x 2f(x) e x 1 , x ( ,0] .

    ,

    f :

    ) 2x 2f(x) e x 1, x IR )

    2x 2f(x) (e x 1), x IR

    )

    2

    2

    x 2

    x 2

    e x 1, x 0

    f(x)

    (e x 1), x 0

    )

    2

    2

    x 2

    x 2

    e x 1, x 0

    f(x)

    (e x 1), x 0

    3. H f IR,

    2 2x xf (x) 2xe 2x 2x e 1

    2 2 2 22 x x 2 x xf (x) 4x e 2e 2 4x e 2 e 1 .

    2x 0 x R 2xe 1 .

    22 x4x e 0 x IR , f (x) 0 x IR f (x) 0

    x 0 .

    , f x 0 , f .

    4. h x f x 3 f x 0, .

    h 0, , :

    h x f x 3 f x .

    f IR, f 0, IR .

    x 0, x 3 x f -

    :

  • 8

    f x 3 f x f x 3 f x 0 h x 0 .

    h 0, , 1 1 .

    f x 3 f x f x 3 f x , x 0, :

    1 1

    h x h x x x

    .

    x x , x IR , x 0 .

    x 0 x x , x 0 .

    x 0 .

    f R , ,

    :

    0

    f(x) f (x) xdx

    f(IR) IR x 0

    f(x)lim 1

    x

    f(x) xe x f(f(x)) e x IR .

    1. f() ( 4) f (0) 1 ( 3)

    7 2. ) f IR ( 4) ) f IR . ( 2)

    6

    3. x

    x xlim

    f(x)

    6

    4.

    e2

    1

    f(lnx)0 dx

    x

    6 :

    1.

    0

    0 0

    0 00 0

    f(x) f (x) xdx

    f x x dx f (x) xdx

    f x x f x xdx f x x f x xdx

    f f 0 0 f f 0 0

    f f 0 .

  • 9

    , x 0 ,

    x 0 x 0

    f(x)limf(x) lim x 10 0.

    x

    f x0 = 0, ( ), f(0) 0 , f() = .

    x 0 x 0

    f(x) f(0) f(x) xlim lim 11 1.

    x 0 x x

    f (0) 1 .

    2.

    ) f 0x IR .

    f 0x , 0x 0x -

    . Fermat 0f (x ) 0 .

    ( f -

    R, f(x)e IR ( ), f(f(x))

    IR ( ) xe IR) -

    .

    :

    f(x) xe f (x) 1 f (f(x))f (x) e

    0x x :

    0 0f(x ) x

    0 0 0e f (x ) 1 f (f(x ))f (x ) e

    0x

    01 e x 0 ,

    , 0f (x ) f (0) 1 0 . f .

    ) f x 0 x IR f , f -

    IR

    f 0 1 0 f x 0 x IR

    f IR.

    3. f(x) IR, x x

    f(x),f(IR) lim lim f(x)

    R (), , x

    f(lim x)

    x

    2lim 0

    f(x) .

    x(, ), > 0, :

    x x |x| |x| 2 2 x x 2

    f(x) f(x) f(x) f(x) f(x) f(x)

    x x

    2 2lim lim 0

    f(x) f(x)

    , :

  • 10

    x

    x xlim 0

    f(x)

    .

    4.

    e

    1

    f lnxdx

    x

    1u lnx du dx

    x . :

    x 1 u 0 x e u .

    :

    e

    1 0

    f lnxdx f u du: 1

    x .

    f IR *0, + f(0) = 0, f() = ,

    0 u f 0 f u f 0 f u

    , , (

    ),

    e2 2

    0 0 1

    f lnx0 f u du du 0 dx

    x

    .

    :

    1. ye y 1 y = 0, 2x 2e x 1

    x = 0. 2x 2e x 1 0 , x 0 .

    1. lnx x 1 x 0 , x 1 .

    , yx e , y IR , y ylne e 1 , ye y 1 , y IR , -

    y 0 .

    , 2x 2e x 1 , x IR , 2x 0 ,

    x 0 .

    , 2x 2e x 1 0 , x 0 .

    1. 2g(x ) 0 xg(x) e x 1 xg (x) e 1 .

    x 0 g (x) 0 , x 0 g (x) 0 x 0 g (0) 0 .

    g R ( ,0]

    [0, ) 0 g(0) 0 .

    g(x) 0 x 0 . 2 2g(x ) 0 x 0 x 0 .

    : g(x) 0 0

    xe x 1 0 , .

  • 11

    3. 2 2x 2 xf (x) (e x 1) 2x(e 1), x R .

    2 2 2 21 2 1 2x x x x2 2

    1 2 1 20 x x 0 x x 1 e e 0 e 1 e 1 (1)

    1 20 2x 2x (2) (1),(2)

    2 21 2x x

    1 2 1 22x (e 1) 2x (e 1) 0 f (x ) f (x )

    f [0, ) .

    2( x)f ( x) 2x e 1 f (x), x R

    1 2 1 2x x 0 x x 0

    1 2 1 2 1 2f ( x ) f ( x ) f (x ) f (x ) f (x ) f (x )

    f ( , 0] f x 0 f

    IR f R.

    4. x 0 .

    , , x 0x 0 .

    x 0 0|x | x ( |x| |x| x 0 )

    0 0|x | |x | 3 0 0x x 3 .

    :

    0 0|x | 3 x 0 0 0 0|x | |x | 3 x x 3 x

    0 0 0 0|x |,|x | 3 , x ,x 3 x

    1 0 0 2 0 0 |x |,|x | 3 , x ,x 3 :

    1 23f ( ) 3f ( )

    1 2f ( ) f ( ) f ( ) 1-1

    1 2 , 1 2 , .

    0 0x |x | 3 0 0 0 0|x | x |x | 3 x 3 .

    :

    0 0 0 0f(x ) f |x | f(x 3) f |x | 3

    x

    0 0 0 0|x |,x , |x | 3,x 3 x 1 0 0 2 0 0 (|x |,x ), |x | 3,x 3