S4E ±„ 1.µ‰¯± [Edit by Lisari Team 2015-16]

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Transcript of S4E ±„ 1.µ‰¯± [Edit by Lisari Team 2015-16]

  • l i s a r i . b l o g s p o t . g r

    2015-16

    study4exams.gr

    lisari team

  • lisari team / lisari.blogspot.gr study4exams.gr

    K

    1

    1: - -

    1: - .

    I - -

    [ - .1.2 ].

    . "" "" .

    (A R R )A O / .

    , ( )

    f x A ( )x A () y ( )yR R .

    :

    f : A R x y R ( )y f x=

    x f yR f x y.

    i. x ( ) . y , , f x .

    ii. ( )f x ( )y f x= .

    iii. xR f f x , ( ){ }fA D x : y f x= = = R R

    1 / 230

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    K

    2

    iv. ( )f A f R ( )y f A x A ( )y f x= , {f (A) y := R x A }y f (x)=

    v. (x,y) x =

    y = x f f

    fC

    . x fC y

    .

    :

    f: :

    ) .

    ) f (x) x A .

    :

    f (x) f , R f (x) .

    2 / 230

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    3

    1.

    fC f, f fC

    xx', fC .

    [ ]fD ,=

    2. f ( )y f x= , :

    ( ){ }fA D x : f x= = R R

    :

    ( )f x R :

    i. ( ) 11 1 0f x x x ... x = + + + + , i R, i 0,1,..., = 0, N ( ( )f x

    fA D= = R

    ), :

    ii. P(x)f (x)Q(x)

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

    { }fA D x : Q(x) 0= = R .

    iii. f (x) g(x)= { }fA D x : g(x) 0= = R 2, N .

    iv. g(x)f (x)h(x)

    = ( )( )f

    g xA D x : h(x) 0 0

    h x = =

    R 2, N .

    v. [ ]f (x) ln P(x)= { }fA D x : P(x) 0= = >R .

    vi. [ ]f (x) P(x)= fA D= = R .

    vii. [ ]f (x) P(x)= fA D= = R .

    3 / 230

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    4

    viii. [ ]f (x) P(x)= ( )fA D x : P x , Z2 = = +

    R .

    ix. [ ]f (x) P(x)= ( ){ }fA D x : P x , Z= = R .

    :

    " " .

    ( , , ..) .

    fC f, f fC

    yy' ( )f A fC .

    ( ) [ ]f A ,=

    4 / 230

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    K

    5

    1. fC ( )y f x= , :( ) ( )0 0 f 0 0M x , y C y f x = .

    2. x A yR

    fC .

    fC , ..

    3. f 0x 0x x= fC .

    4. ( )f x 0=

    '.

    5. ( ) fC yy'

    ( )f 0 y= .

    5 / 230

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    K

    6

    6. ( )f x 0> fC xx', ( )f x 0< fC xx'.

    7. ( ) ( )f x g x= fC

    gC , f, g .

    8. ( ) ( )f x g x> ( ) ( )g x f x> fC gC gC fC .

    o g f> [ ) ( ), ,

    o f g> ( ) ( ], ,

    o [ ]f gD D , =

    6 / 230

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    K

    7

    .

    9. ( )f x fC xx'.

    10. ( )f x fC ( ) fC xx' ( ) xx'.

    : f : A B R ,

    o ( ) ( )1 2f x f x 1 2x x 1 2x , x A . o 1 2x x= ( ) ( )1 2f x f x= 1 2x , x A

    :

    yy'.

    (0,0) .

    7 / 230

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    8

    :

    f : A R g : B R .

    f g ( )f g= :

    1. ( )A B= . 2. x A ( ) ( )f x g x= .

    : f gD D A

    f gf (x) g(x) x A

    = = = =

    1. f ,g

    .

    2. f g f gD D x

    ( ) ( )f x g x= , f, g , . ..

    ( ) ff x x , D= = R ( ) gg x x, D= = R

    [ )0, = + . ( ) ( )f x x x g x= = = x . . ( )

    0x R ( ) ( )0 0f x g x , ( ) ( )f 8 g 8 . 3.

    ( )f g , . ' ( ) ( )f x g x= x A .

    4. f g= , .

    8 / 230

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    9

    5. f g "" R . ' ""

    R , , f, g. f gE D D ( ) ( )f x g x= x E . f g= .

    9 / 230

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    K

    10

    :

    f, g , A B O / .

    f g+ ,

    f g ,

    fg

    fg

    f, g :

    f g+ ( )( ) ( ) ( )f g x f x g x+ = + A B

    f g ( )( ) ( ) ( )f g x f x g x = A B

    fg ( )( ) ( ) ( )fg x f x g x= A B

    fg

    ( ) ( )( )f xf x

    g g x

    =

    ( ){ }A B x A B : g x 0 =

    1.

    A B . A B O = / , . A B O / .

    2. f g+ , f g , fg , fg

    , .

    .

    3. ' f . :

    ( )( ) ( )f x f x = fD A = R ( )( ) ( )f x f x = fD A = *N

    10 / 230

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    11

    f : A , g : B R R

    . :

    ( ) ( )( )gf

    x f x g f x (1)

    f (A) B O / .

    :

    1

    f : A , g : B R R , ( )f A B .

    gof : A R , ( )( )x g f x

    f g.

    :

    2

    f : A R g : B R

    ( )f A B , ( )f A B O / .

    ' ,

    11 / 230

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    12

    ( )f x B , ( ){ }x A : f x B = .

    :

    gof : R ( )( )x g f x , {x A : f (x) B} =

    f g.

    :

    : fog 1. g R , x A ( )f x R .

    f : A R g A .

    2. , .

    3. f : R R ( )g x x= . fog gof= , : () gof , fog R () xR :

    ( )( ) ( )( ) ( )fog x f g x f x= =

    ( )( ) ( )( ) ( )gof x g f x f x= =

    12 / 230

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    13

    ' g I .

    4. fog gof . . . :

    ( ) ( ) ( )( )fog oh x fo goh x=

    5. . .

    6. .

    : 14/2/2012

    13 / 230

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    K

    14 / 230

  • lisari team / lisari.blogspot.gr study4exams.gr

    K

    1

    1: - -

    2:

    [. 1.3: - ].

    :

    f :

    ,

    1 2x , x 1 2x x< :

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

    ,

    1 2x , x 1 2x x< :

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

    15 / 230

  • lisari team / lisari.blogspot.gr study4exams.gr

    K

    2

    f , f .

    f :

    , 1 2x , x

    1 2x x< :

    ( ) ( )1 2f x f x

    ,

    1 2x , x 1 2x x< :

    ( ) ( )1 2f x f x

    16 / 230

  • lisari team / lisari.blogspot.gr study4exams.gr

    K

    3

    :

    f A :

    0x A () , ( )0f x , ( ) ( )0f x f x xA .

    0x A () , ( )0f x , ( ) ( )0f x f x xA .

    f () () f .

    17 / 230

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    4

    -

    ( )f x x= + , 0 > = =fA D R

    ( )( )f =A R .

    3.

    ( )f x x= + , 0 < = =fA D R

    ( )( )f =A R .

    4.

    18 / 230

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    5

    ( ) 3f x x= , 0 > = =fA D R

    ( )( )f =A R .

    5.

    ( ) 3f x x= , 0 < = =fA D R

    ( )( )f =A R .

    6.

    19 / 230

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    6

    ( ) 2f x x x= + + , 0 >

    ,2

    ,2 +

    .

    x2

    =

    f2 4 =

    .

    7.

    ( ) 2f x x x= + + , 0 <

    ,2

    ,2 +

    .

    x2

    =

    f2 4 =

    .

    8.

    20 / 230

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    7

    ( ) xf x = , 1 > = =fA D R ( ) ( )( )f 0,= +A .

    ,

    ( ) xf x e= , 9.

    ( ) xf x = , 0 1< < = =fA D R ( ) ( )( )f 0,= +A .

    10.

    21 / 230

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    8

    ( )f x log x= , 1 > ( )0,= = +fA D

    ( )( )f =A R .

    11.

    ( )f x log x= , 0 1< <

    ( )0,= = +fA D ( )( )f =A R .

    12.

    22 / 230

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    9

    ( )f x x= , =A R ,

    0,2

    ,

    3,2 2

    3 , 22

    T 2= f

    2 , 22 +

    ,

    32 , 2

    2 2 + +

    ( )32 , 2