Γ.ΚΩΤΣΑΚΗ ΕΙΣΗΓΗΣΗ Ημερίδα 18-04-10

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Transcript of Γ.ΚΩΤΣΑΚΗ ΕΙΣΗΓΗΣΗ Ημερίδα 18-04-10

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    2

    O

    y

    fC

    ( ,0)x

    x

    f , , = , . f f 0 , : = ft

    dt .

    , .

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    F = ft dt 3

    O

    y

    1 4xx

    f( ) 1 x x +

    (1,0) B( ,0)x

    ( , 1)x x

    (1,2)

    ()

    = + , . , , .

    , 0 1 4. , ,

    = 1 , : = ft

    1

    dt = F (1)To

    : = = +

    2 =

    2 + + 1 12

    =1

    22 + 2 3 > 0. (2)

    1 2 : F = 12

    2 + 2 3.

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    4

    O

    y

    x

    fC f( ) 2x

    1 4

    2

    f( ) 2x x

    F = 12

    2 + 2 = + 1 = f ,

    F,

    f

    1,4

    .

    = , < 1, , , . f :F = 2, < 1

    2

    1,

    1

    F = f , 0,4. : f

    , o ft

    dt , F f .

    :

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    F = ft dt 5

    f

    f , :

    F = ft

    dt ,

    F = ft

    dt

    = f .

    O

    y

    x hx

    x

    fC

    , , : F: F = lim

    h0F + h F

    h(1)

    : = Z BEZ

    =

    f

    t

    +h

    dt

    f

    t

    dt = F

    + h

    F

    . (2)

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    6

    h , , : = . = h. f . (3)

    2 3 F + h F = h. f .

    (1) :F = lim

    h0h. f

    h= lim

    h0 f = f.

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    F = ft dt 7

    : =

    : f.

    F

    : . = 1 2 ,

    1, 2 .

    :F = ft

    dt = f.

    =

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    8

    =

    ..

    , .

    .

    .. f = ln , = 0, +.

    1 0, +, F = 0, +.

    F : F = ln , > 0 .

    F1 = lnt

    t

    1

    1d t = 0 F1 =

    ln1

    1 = 0 ,

    : = 0.. F : F = ln =

    1 ln2 , > 0.

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    F = ft dt 9

    : F (0,e e, +). = e , F Fe = lnt

    t

    e

    1

    dt = lnt lnte1

    dt =1

    2 ln2t1e = 12

    e, 1/2.. : lim0F = lim0 ft

    dt , 0 DF .

    F , , : lim0F = F0 = ft

    0

    dt.

    . = F

    d , , : = F

    d = F F

    d =

    F

    f

    d

    ,

    f

    d

    .

    + + 0 12

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    10

    : =

    g F = ft dt ,

    : = Fg. : F g , : = g .

    = D

    H,

    H

    , : , . = 1 2 , : , , . :

    = ftg

    dt = fgg.

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    F = ft dt 11

    =

    +

    .

    : f = , , 0 0, +.

    F:

    1,

    2 + 2

    , 0 0, +. 1, 2 + 2 , 0 , 1 , 0, :2 + 2 , 0 2 + 2 > 0 0,2. : DF = 0,2. F :

    F = tt2

    +21

    dt = 2 + 22 + 2 2 + 2

    =2 + 2

    2 + 2 2 + 2 =2 22 2

    2 2 ,

    0,2

    .

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    12

    : =

    : f. :

    g1

    g2

    ,

    .

    f 1 2, :g1, g2 1 g1, g2 2.

    H : .

    : = ftg1 dt + ftg2

    dt =

    ftg2

    dt ftg1

    dt .

    :

    = f

    g2

    g2

    f

    g1

    g1

    .

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    F = ft dt 13

    f , F : F = f (1) = Fg1g2 = Fg2 Fg1 (2)

    2 , :

    = Fg2 g2 F g1 g1 =

    fg2 g2 fg1 g1 .: , F, , .

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    14

    : G = et1 2

    lnt 12dt

    f

    = e ln

    1

    2,

    < 1

    > 1.

    : = , 1 1, + :

    2 < 1 1 < 1 2 > 1 1 > 1 0,1 , 1 , G = , 1 0,1. G: :

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    F = ft dt 15

    1 = , 1 , G :

    G

    =

    et

    2ln

    t

    1

    2dt +

    et

    1

    ln

    t

    1

    2dt =

    et1

    lnt 12dt et2

    lnt 12dt (1)

    G :

    G = et1

    lnt 12dt et2

    lnt 12dt

    = e1 ln

    1

    1

    2

    1

    e 2 ln

    2

    1

    2

    2

    =

    e1 ln2 2 e 2 ln2 12. (2) 2 = 0,1 , (2).

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    16

    :

    =

    ,

    .

    ,

    ,

    :

    = , .

    .. .

    = ,

    ..

    f ,g,h . : g, t = u u t

    , :

    =

    ( )

    1 2

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    F = ft dt 17

    1 : = + , g =

    , . :

    .

    .. =

    = .

    .

    g

    = 2

    f

    2

    t

    2

    1

    dt

    6

    2

    2

    ,

    .

    2 f2t21

    dt , 2t = u . 2 dt = du.

    2, 4, 2 f2t2

    1

    dt :

    f2t21

    2 dt = fu42

    du.

    1 2 2 4

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    18

    g , :g

    =

    f

    u

    4

    2du

    6

    2

    2

    =

    fu2

    du + fu4

    du 62 2

    g = fu4

    du fu2

    du 62 2, . (1)

    :g = 4 f4 2 f2 12 2 =22 f4 f2 12 2 = 26 + 1 12 2 = 0 .

    c , g = c g0 = 0 , : g = 0.. g = 0 , :

    fu42

    du 62 2 = 0 fu42

    d u = 62 + 2 (1)

    = 1 = 2, 1 : fu4

    2

    du = 8 , fu84

    du = 28 , :

    fu42

    du + fu84

    du = 8 + 28 f82

    d = 36.

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    F = ft dt 19

    2 : , +) : = > , +).. : , .

    .

    + + , , , + .

    .

    + = + ,

    ,

    .

    . : fe 1 , 0.

    g =f

    e 0, +) 0, +g = e f e f

    e2 =f f

    e > 0.

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    20

    0, +), 0 :g g0 f

    e f0e0

    fe 1 f e

    . :h = ft

    0

    d t + 2 + e 0.

    h0 = ft00

    d t + 20 + e0 = 3 , : h h0. h0:

    0,

    :

    h = ft0

    d t + 2 + e = f + 2 ln + e .: h0 = f0 + 2ln + 1 = 1 + 2 l n + 1 = 2ln + 1.

    h

    Fermat , :h0 = 0 2ln + 1 = 0 ln + 1 = 0 ln = 1 = 1/e .

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    F = ft dt 21

    . : ft0

    d t + e 2 = 0

    : = ft

    0

    d t + e 2 , 0,1. 0,1.

    0 = ft0

    0

    d t + e0 2 = 1 2 = 1 < 0 .

    1 = ft10

    d t + e 1 2 = ft10

    d t + e 3.

    .

    : f

    e

    f

    e

    0 ,

    f e 10

    d 0 f10

    d e10

    d

    f10

    d e 01 0 f1

    0

    d e 1 0

    f10

    d e 1 +e3 f10

    d + e 3 2e 4 > 0 1 > 0. : 0. 1 < 0 , Bolzano 0,1: = 0.

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    22

    3 f , : = = . f . 0, e , : f = 0 .

    F

    =

    f

    u

    0

    du ,

    0, e

    .

    F 0, e F = f , 0, e. F 0, e

    0, e :

    F = Fe F0e 0 =

    fue0

    du fu00

    du

    e=

    e

    e= 1,

    f = 1. (1)

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    F = ft dt 23

    fe = 1. (2) 1 2 : f = fe. (3) f , e Rolle , , e f = 0 .

    f

    . 4 : :

    =

    ,

    , , . f.

    = 0 , : ft0

    dt 1

    2f0 f f

    0

    =1

    ft0

    dt 12

    1 f ft0

    dt 12

    f 1

    f

    t

    0

    dt

    1

    2

    f

    1

    (1)

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    24

    = 0 , :

    f

    t

    0

    dt

    1

    2

    f

    f

    0

    f

    t

    0

    dt

    1

    2

    f

    1

    ft0

    dt 12

    f 1 (2)

    A 1 2 : ft0

    dt =1

    2f 1 (3)

    3 :f = 1

    2f f = 2f f 2f = 0 e2

    f e2 2e2 f = 0 e2 f = 0 . .. e2 f = c, c . (4) = 0, 4 : f0 = c c = 1 . 4 :e2 f = 1 f = e2 , .

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    F = ft dt 25

    5 = , .. f

    .. , :

    +

    +

    .

    . : + +

    .

    . f 0, + :f = 1

    2e

    2 + 1

    2 e

    2 =1 2 e

    2 , > 0.

    - f:

    f = 0 1 2 e2 = 0e2>0 = 1

    f > 0 1 2 e

    2 > 0

    , e2>0 < 1

    f

    < 0

    > 1.

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    26

    = 1, f1 = 1e .. 1, +) f t + 1 , : f + 1 ft f

    f + 1+1

    dt ft

    +1

    dt f

    +1

    dt

    f + 1 ft+1

    dt f. (1). : lim

    +

    f , lim

    +

    f + 1:

    lim+f = lim+e

    2

    ++ , De L =

    lim+

    1

    21

    2e

    2

    = lim+1

    e2

    = 0.

    : lim+f + 1 = 0 , 1 :

    lim+ ft+1

    dt = 0.

    + + 0 e1/

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    F = ft dt 27

    6. : : = + ,

    = .

    .

    = .. :

    ( ) .

    . : + = .. t = u dt = du : f = 1 fu0

    1du = 1 + fu1

    0

    du (1)

    f = fu10

    du + f 1, . (2) 2 :f1 = f0 = 1, = 45.

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    28

    . : lim1 f 1( 1)3 = lim1

    fu10

    du

    ( 1)3 0

    0De L =

    lim1 fu10 du + f 1

    3( 1)2 = +.. g = e fe 1:: fe = 1 + e fue

    10

    du

    fe 1 = e fue 1

    0

    du , g :

    g = e fe 1 = fue 1

    0

    du (3)

    lim+g = lim+ fue 1

    0

    du (4)

    , e = , 0 :lim+g = lim0 fu1

    0

    du

    = fu10

    du = fu01

    du = 2 = 2.