σπιράλ...

Download Σπιράλ 7_Ολοκληρώματα (2012-13).pdf

Post on 14-Apr-2018

218 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

  • 7/30/2019 7_ (2012-13).pdf

    1/41

    : 3 2012-2013

    -1-

    &

    ()

    3:

    555 1

    3.1

    1.

    . -

    .

    2. H

    .

    , 1f(x)x

    (x) F ln x ,

    A ( ,0) (0, ) F(x) = f (x) x A , :

    ( ,0) , f 1 1

    F(x) c , c .

    (0, ) , f 2 2F(x) c , c .

    A ( ,0) (0, ) , f F(x) c , c

    3. 1 2

    F ,F f, ,

    c. :

    1 2F (x) F (x) c, c R .

    4. , .

    1, 1 x 0

    f x1, 0 x 1

    , = (1,1),

    F = (1,1) F(x) = f (x) .

    5. : f , :

    ) .

    ) .

    6. f , . ()

    7. f ,

    1 1

    2x , x 0f(x) x x

    0, x 0

    ,

    :2

    1x , x 0F(x) x

    0, x 0

    .

    31

  • 7/30/2019 7_ (2012-13).pdf

    2/41

    : 3 2012-2013

    -2-

    8. f ,

    , 0

    x .

    9. -

    .

    x

    .

    1

    /

    1 f(x) 0 G(x) c, c R

    2 f(x) 1 G(x) x c, c R

    3 1f(x)x

    G(x) ln x c, c R

    4 f(x) x 1x

    G(x) c, c R 1

    51

    f(x)2 x

    G(x) x c, c

    6 f(x) x G(x) x c, c R

    7 f(x) x G(x) x c, c R

    8 21

    f(x) x

    G(x) x c, c R

    9 21

    f(x) x

    G(x) x c, c R

    10 xf(x) e xG(x) e c, c R

    11 xf(x) , 0 1 xG(x) c, c R

    ln

    10.

    :

    F G f g * ,:

    i) F G f g

    ii) F f .

  • 7/30/2019 7_ (2012-13).pdf

    3/41

    : 3 2012-2013

    -3-

    2

    /

    1 k(x) f (x) g (x) K(x) f(x) g(x) c, c

    2 *k(x) f (x), *K(x) f(x) c, ,c

    3 k(x) f (x) g(x) f(x) g (x) K(x) f(x) g(x) c, c

    4 2f (x) g(x) f(x) g (x)

    k(x)g (x)

    f(x)k(x) c, c

    g(x)

    5 k(x) g f(x) f (x) K(x) g f(x) c, c

    11. -

    -

    f .

    :

    x .

    3

    /

    1 g(x) f (x) G(x) f(x) c, c

    2f (x)

    g(x)f(x)

    G(x) ln f(x) c, c

    3 g(x) f (x)f (x), 1 1f (x)

    G(x) c, c , 1 1

    4f (x)

    g(x)f(x)

    G(x) 2 f(x) c, c

    5 g(x) f (x) f(x) G(x) f(x) c, c

    6 g(x) f (x) f(x) G(x) f(x) c, c

    7 2f (x)

    g(x) f(x)

    G(x) f(x) c, c

    8 2f (x)

    g(x) f(x)

    G(x) f(x) c, c

    9

    f(x)g(x) f (x) e f(x)

    G(x) e c, c

    10 f(x)g(x) f (x) f(x)

    G(x) c, cln

  • 7/30/2019 7_ (2012-13).pdf

    4/41

    : 3 2012-2013

    -4-

    [2-7, 1-4 308-9]

    1. f(x) 2x ,

    yy 2e .

    2. , -

    A(0,1) .

    ) xf

    f(x) e 2x x, D

    ) 3 f1

    f(x) x x , D ( 1, )2 x 1

    ) x ff(x) 3 4 ln 4 2, D

    ) x x ff(x) e x e x, D

    3. ( ) -

    6

    5 10N (t) t , (0 t 6 )9 3

    .

    6

    3 7,5 .

    4. t sec

    21(t) 1 m / sect 1

    0 t 5 . t 1 sec ,

    .

    5. :) 3 2

    1f(x) 8x 3x 1 , x 0

    x

    )2

    x 2 g(x) x ,x ,

    7 2 2 x

    ) x(x) (x 1)e

    )2

    xx xf(x)

    x

    ) xx x

    g(x) e

    ) (x) (3x 5)

    ) 5x 3f(x) 4e

    ) 42xg(x) ( )1

    )1 1

    (x) , x ,2x 1 2

    )3

    1f(x)

    (2x 1)

    ) g(x) 2x 1

    )1

    (x)2x 1

    ) x 3f(x)x 2

    6. F 25 xf(x) 4x 3xe , F( 1) F(2) .

    7. f , f(1) 3 , f -

    0

    x 1 , 4f (x) 30x 2, x .

    8. f : xf x f x( ) ( ) 1 e , x ,

    0

    x 0 2.

  • 7/30/2019 7_ (2012-13).pdf

    5/41

    : 3 2012-2013

    -5-

    9. ( ) -

    15 4x , x -

    2x 5 . :

    ) ,

    ) .

    10. -:

    ) 2 3 2 33x 2xf(x) x x 1( )( )

    )3ln x

    g(x)x

    )2

    x 2(x)

    x 4x 7

    )2

    2xf(x)

    1+ x

    ) x,g(x) x 0,

    )x

    1(x)

    1 e

    ) 3

    f(x) x x, x ,2 2

    )x

    1 xg(x)

    e x

    11. f : f(0) 0 f(1) 1 .

    0

    x (0,1) 0x

    0f (x ) e 2 e .

    12. 1

    2x 2 02 x

    (0,1) .

    13. f(x) x 1 2 .

    [ f F], F f .

    14. f : F .

    f(1) 2 x 2x F(x)f(x) 2xe , f.

    15. *f : ,

    F 2F(x)F(1 x) F(x ) x .

    16. f : f(0) 1 F -

    : ( )fx 1(F x) , x .

    ) F(0) .

    ) (x)g(x) F F x( ) .

    ) f .

    17. f : F .

    A f(1) 1 x f x F 2( ) ( x 1) , :

    ) f 2 x F 1( ) (x) x .

    ) F 2 x F 1( ) (x)

    x

    .) f .

  • 7/30/2019 7_ (2012-13).pdf

    6/41

    : 3 2012-2013

    -1-

    &

    ()

    3:

    555 1

    3.4

    3.5 -

    1. . -

    f(x)dx 0

    .

    2. .

    .11 2

    0 02xdx x 1 0 1

    11 2

    0 02xdx x 1 2 1 1 .

    3. f [,] .

    4.

    .

    f (x)dx f() f()

    f (x)dx f () f ()

    5. f,g [,] f(x) g(x) ,

    f(x)dx g(x)dx .

    : ( )

    f(x) g(x) 0 f(x) g(x) .

    f(x) g(x) dx 0 f(x)dx g(x)dx 0 f(x)dx g(x)dx .

    6.

    f(x)dx

    f

    f.

    .

    f(x)dx f(t)dt f(y)dy .

    7.

    1

    f (x)dx

    f(x)dx .

    f 1f

    : x f u (1), dx f (u)du (2).

    1

    u 2

    u (3), 1

    u 2

    u -

    1

    f(u ) 2f(u ) f

    1-1.

    32

  • 7/30/2019 7_ (2012-13).pdf

    7/41

    : 3 2012-2013

    -2-

    1f f

    : 1(v)x f (1), 1dx f (v)dv (2).

    1v

    2v (3),

    1v

    2v -

    1 1f (v ) 1 2f (v )

    -

    1f 1-1.

    : 2

    1

    0f (x)dx 5 3f(x) x x .

    f(u) x (1) dx f du(u) (2)

    f(u) 0 f(u) 2 .

    5 3 3 2 3f(u) 0 u u 0 u (u 1) 0 u 0 u 0 5 3f(u) 2 u u 2

    u =1 f 1-1.

    1

    u 0 2

    u 1 (3)

    (1),(2),(3) :

    2 1 1 1

    1 1 4 2

    0 0 0 0

    19f (x)dx f f(u) f (u)du u f(u)du u (5u 3u )du

    12 .

    . [ ]

    f(x)dx F (x)dx F(x) F() F()

    . :

    f (x)g(x)dx f(x)g(x) f(x)g (x)dx ,

    f , g .

    :

    i)

    x

    P(x)e dx , P(x) x ,

    * .

    ii)

    P(x)(x )dx , P(x) x ,

    * .

    iii)

    P(x)(x )dx , P(x) x ,

    * .

    iv)

    P(x)ln(x )dx , P(x) x ,

    * , x 0 .

    v)

    x

    e (x )dx ,

    *, , .

    vi)

    x

    e (x )dx

    , *

    , , .

  • 7/30/2019 7_ (2012-13).pdf

    8/41

    : 3 2012-2013

    -3-

    -

    , :

    [1] x e

    [2] (x ) (x )

    [3] P(x) ()

    1x

    x

    [4] ln(f(x)) , -

    .

    , :

    x

    P(x)e dx P(x) -

    0

    -

    , ,

    -

    .

    P(x)(x )dx

    P(x)(x )dx

    P(x) -

    0

    -

    , ,

    -

    P(x)ln(x )dx P(x) -

    0

    -

    ,

    x

    I e (x )dx

    x

    I e (x )dx

    0 -

    , 2 ,

    -

    -

    ,

    I.

    , ,

    I.

    2

    P(x)dx

    (x )

    2

    P(x)dx

    (x )

    P(x) -

    1 0

    -

    , 1 ,

    -

    2

    1

    x .

    21

    x .

    , :) ,

    ) .

  • 7/30/2019 7_ (2012-13).pdf

    9/41

    : 3 2012-2013

    -4-

    .

    f g(x) g (x)dx -

    :

    2

    1

    u

    uf g(x) g (x)dx f(u)du ,

    f,g , u g(x), du g (x)dx 1 2u g(), u g() .

    , [

    3, 31] u f(x) .

    *******

    .

    P(x)dxQ(x) , -

    :

    [1] P(x) Q (x) , ,

    u Q(x)

    Q(x)dx ln Q(x)

    Q(x)

    .

    [2] , , P(x) Q (x) ,

    ) < :

    Q(x) x *(x ) , N 2x x 2 4 0 .

    P(x)

    Q(x)

    :

    x A

    x .

    (x ) :

    1 2

    2

    AA A

    ...x (x ) (x )

    .

    2x x

    2

    Ax B

    x x

    .

    , -

    .

    :

    331 2

    3 2 2 3 2

    x x 2 x

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

    , [1].

  • 7/30/2019 7_ (2012-13).pdf

    10/41

    : 3 2012-2013

    -5-

    ) :

    P(x):Q(x) : P(x) Q(x) (x) (x) ,

    (x),(x) , , (x)

View more