Θεματα μαθηματικα κατευθυνσησ γ λυκειου

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  • 0

    : . . 0

    2012

    : ,

  • 1

    : . . 1

    1. , :f g \ \ ( ) 5( ) 1 ( ) (1)f g x x x g x= + + + x\ . ) g 1-1.

    ) 0x \ , ( )0 0f x x= . ) f g ,

    g xx.

    ) ( ) ( )1 2 1 2g x g x x x= = ( ) ( ) ( ) ( ) (1)1 2 1 25 5

    1 1 1 2 2 25 5

    1 1 2 2

    ( ) ( )

    1 ( ) 1 ( )

    1 1 (2)

    g x g x f g x f g x

    x x g x x x g xx x x x

    = = + + + = + + +

    + + = + +

    : 5( ) 1x x x = + + 4( ) 5 1 0 ( ) ( ) "1 1"x x x x = + > /

    "1 1"

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

    ) ( )x

    [ ] ( ).( 1) (1) 0( 1) 1

    1,1 : ( ) 01,1(1) 3

  • 2

    : . . 2

    2. , :f g \ \ : (1) (2) ... (2006) (1) (2) ... (2006) (1)f f f g g g+ + + = + + + . [ ]0 1, 2006x , : 0 0( ) ( )f x g x= .

    ( ) ( ) ( ) ( )1 (1) (1) (2) (2) ... (2006) (2006) 0 (2)f g f g f g + + + = : ( ) ( ) ( )h x f x g x= [ ]1,2006 , - [ ]( ) , 1, 2006m h x M x

    [ ][ ]

    [ ]

    ( ) (2)

    1 1, 2006 (1)

    2 1, 2006 (2)2006 (1) (2) ... (2006) 2006

    ...2006 1, 2006 (2006)

    m h M

    m h Mm h h h M

    m h M

    +

    + + +

    2006 0 2006 0m M m M 0 ( )h x , ( )h x ,

    [ ]0 0 0 01, 2006 : ( ) 0 ( ) ( )x h x f x g x = = 3. f [ ],a

    ( ) ( ) 0f a f > . ( )0 ,x a ( )0 0f x = . [ ],x a ( ) ( ) 0f x f a .

    [ ],x a ( ) ( ) 0f x f a , [ ], : ( ) ( ) 0a f f a <

    [ ] ( ). .

    1 1

    ( ) ( ) 0, : ( ) 0

    ,f f a

    a ff a

    < =

    , [ ] [ ). .

    2 2

    ( ) ( ) 0( ) ( ) 0, ( ) 0

    ,( ) ( ) 0f ff a f

    fff f a

    = <

    f ( ),a , .

  • 3

    : . . 3

    4. [ ]: ,f a \ : 2( )f > . : 3 33 ( )f x dx

    < , 2( )f x x= .

    3 3

    3 :

    32

    3x x dx

    = : 2( ) ( )g x f x x= [ ]0 0, : ( ) 0x a g x =

    2 2( ) ( ) 0 ( ) 0f f g > > >

    3 33 3

    2 2

    3 ( ) ( )3

    ( ) ( ( ) ) 0

    ( ) 0

    af x dx a f x dx

    f x dx x dx f x x dx

    g x dx

    < < < < .

    f f f f f ( [ ],a )

    ( ) 0( )

    ( )f x

    f x f x

  • 4

    : . . 4

    ( ) 0 ( ) ( )(1) ( ) ( ) ( ) ( ) (2)( ) ( )

    f x f a ff a f f f af a f

    = =

    : [ ]( )( ) ,( )

    f xg x af x

    = ,g :

    ( )( )

    ( )2

    2 2

    ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) (3)( ) ( )

    f x f x f xf x f x f x f xg x g xf x f x

    = =

    (2)

    ( )( )( )

    ( ) ( )( )( )( )

    f ag af a

    g a gfgf

    = ==

    [ ]( ) ( ). . 0 0

    ,

    , , : ( ) 0( ) ( )

    Rolleg

    g x g xg a g

    =

    =

    ( ) ( ) ( )( ) ( )

    20

    2(3)0 0 0

    0 20

    ( 02

    0 0 0 0 0

    0

    ( ( ) ( ), : 0

    (

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

    f x f x f x x a

    f x

    f x f x f x f x f xx

    >

    =

    = >

    6. f \ ( ) ( ) 0 (1)f x f x+ + = , x\ , * \ . ( ) 0f x = .

    (1)

    (1) ( ) ( ) 0 ( ) ( )

    (1) ( ) ( ) 0 ( 2 ) ( )

    ( 2 ) ( ) (2)

    x x

    x x

    f x f x f x f x

    f x f x f x f x

    f x f x

    + + + = = + + + + = + = +

    + =

    ( ) ( )2(1) 2 2 0 ( ) ( 2 ) 0( ) ( 2 ) 0 ( 2 ) ( ) (3)

    (2), (3) ( 2 ) ( ) ( 2 ) (4)

    x xf x f x f x f x

    f x f x f x f xf x f x f x

    + + = + = + = =

    = = +

    ( )f x 2T = . 0 (0) (2 )x f f = = ,

  • 5

    : . . 5

    [ ]( ) ( ). .

    0, 2

    0, 2 0, 2 : ( ) 0(0) (2 )

    Rollef

    f ff f

    =

    =

    (4) ( 2 )( 2 ) ( ) ( 2 )( 2 )( 2 ) ( ) ( 2 )

    f x x f x f x xf x f x f x

    = = + + = = +

    f 2T = . , f ( )0, 2 , , 2 ( ).

    ( ) 0f x = , .

    7. f [ ]1, 4 . (1) 1, (2) 2, (3) 3f f f= > < (4) 4f = , ( )0 1, 4x

    0( ) 0f x = .

    (1) 1(4) 4

    ff

    = = : ( ) ( )g x f x x= , ( ) ( ) 1g x f x =

    (1) (1) 1 0(4) (4) 4 0(2) (2) 2 0(3) (3) 3 0

    g fg fg fg f

    = == == >= , \ : ( ) ( )f f = .

    ( )1 i A + , 1 1

    11

    1 1 1

    11

    : ( ) (1 ) ( )

    ( ) (1)( )

    x x

    xx

    x e if x i e if x i

    ee f x

    f x

    + = + + = += ==

    \

    ( )1 i A + ,

  • 7

    : . . 7

    22 2

    2 2 22

    : ( ) (1 ) ( ) (2)( )

    xx xex e if x i e f x

    f x

    = + = + ==\

    1

    1 2

    2

    1

    1 2

    2

    ( ) 1( ) ( )(1), (2)

    ( ) 1

    x

    x x

    x

    f xf x f xe

    f x e ee

    = =

    =

    : ( )( ) xf xg xe

    =

    2

    ( ) ( ) ( ) ( )( ) ( ) (3)( )

    x x

    x xf x e f x e f x f xg x g x

    e e = =

    1

    2

    11

    1 22

    2

    ( )( )( ) ( )

    ( )( )

    x

    x

    f xg xe g x g x

    f xg xe

    = =

    =

    [ ]( )

    1 2(3).

    1 2 1 2

    1 2

    ,( , ) , : ( ) 0 ( ) ( )

    ( ) ( )

    Rolleg x xg x x g f fg x g x

    = =

    =\

    10. :f \ \ ( ) ( ) 1 (1)f x f x = x\ . (0) 1f = , :

    ) ( ) ( ) 1f x f x = x\ ) ( ) xf x e= x\ ) fC : 1y x = +

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

    f x f x =

    ( )( )

    (1), (2) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 0

    ( ) ( ) ( ) ( ) 0

    ( ) ( ) 0( ) ( )

    f x f x f x f xf x f x f x f x

    f x f x f x f x

    f x f x xf x f x c

    = =

    + = =

    =\

    0 (0) ( 0) 1x f f c c= = = , ( ) ( ) 1 (3)f x f x = , ( ) 0, ( ) 0f x f x

  • 8

    : . . 8

    ) 1(3) ( )( )

    f xf x

    =

    1

    01 1

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

    0 : (0) 1

    xf x f x f x x f x C ef x

    x f C e C

    = = == = =

    \

    ( ) xf x e=

    ) ( )0 0, ( )A x f x . : 00( ) xf x e= , 0

    0( )xf x e = .

    ( )1 0 0 0 0 0 0 0: ( ) ( ) ( ) ( ) ( )y f x f x x x y f x x f x x f x = = + , 2 : 1y x = + . 1 2

    0

    0 0

    0 00 0

    0 0 0 0

    ( ) 1 1 0( ) ( ) 1 1 0 1 0 1 1

    x

    x x

    f x e xf x x f x e x e e e = = = + = + = + = + =

    1y x= + ( ) ( )00, 0,1e = 11. f [ ],a ( ), ( ) ( )f a f = . ( )1 2, ,..., ,a ,

    1 2( ) ( ) ... ( ) 0 (1)f f f + + + =

    i (1) [ ],a

    .

    : x + =

  • 9

    : . . 9

    1 1

    1 1 1

    x

    x

    x

    x x x

    = = + =

    = = =

    f .

    f ...

    ( )

    1 1

    2 2

    3 3

    ( ) ( ), : ( )

    ( 2 ) ( ), 2 : ( )

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

    ...

    ( ) (1 , : ( )

    f a f aa a f

    f a f aa a f

    f a f aa a f

    f f aa f

    + + = + + + + = + + + + =

    + + = ( )1 )

    +

    ( )1 2

    1 2

    ( ) ( ) ... ( )

    ( ) 2 ... ( ) 1

    ( ) ( ) 0( ) ( ) ... ( ) 0

    f f f

    f a f f f a f f a

    f f af f f

    + + + = + + + + + + + =

    + + + = = =

  • 10

    : . . 10

    12. f [ ],a , ( ) 0f = . ( ) 0f > , [ ],a f

    .

    f f f f f ( ) 0( ) ( ) ( )( ) 0 lim 0 lim 0

    f

    x x

    f x f f xfx x

    =

    > > >

    ( ) 0f xx > .

    ( )0 : , x > ( ) 0f xx > .

    ( ),x ( ) 0f x < 0x < , . ( )( ) 0 ,f x x < f , [ ]( ) ,f x x 0 .

    13. 1x

    x xI dxe

    = + . : i)

    1

    x

    x

    xe xI dxe

    = + ii) I =

    i) : ( )x y x a y = = + x y dx dy dy dx= = = x y yx y y

    = = == = =

    ( ) ( )( ) ( ) 11 1 1x y yy yx x y yI dx dy dy

    e ee

    = = = =+ + +

    1 1 1

    y x

    y y x

    y

    y y ye y xe xdy dy dxe e e

    e

    = = =+ + +

    ii) 1 21

    1

    xx

    xx

    x

    x xI dxx x xe xe I dx

    exe xI dxe

    +

    = ++ = += +

  • 11

    : . . 11

    ( ) ( ) ( )( ) ( )( ) ( ) [ ]

    (1 )2 21

    2 2

    2 ( ) ( )

    2 ( 1) ( 1)

    2 ( ) 2 2

    x

    x

    x x eI dx I x xdxe

    I x x dx I x x x dx

    I xdx

    I x

    I I I

    + = =+ = =

    = + = + + = + + = =

    14. : /2

    0

    ( )( ) ( ) 4

    x

    x xxI dx

    x x

    = =+ .

    : 2

    x y dx dy dy dx= = =

    0 02 2

    02 2 2

    x y y

    x y y

    = = = = = =

    2

    2

    x y y

    x y x

    = = = =

    , ( )( ) ( ) ( )0

    /2

    y

    y y

    yI dy

    y y

    = +( )

    ( ) ( )/2

    0

    y

    y y

    yI dy

    y y

    = + ( )

    ( ) ( )( )

    ( ) ( )

    ( )( ) ( )

    /2

    0 ( ) /2

    0/2

    0

    ( )2

    x

    x x x x

    x xx

    x x

    xI

    x x x xI dx

    x xx I dx

    x x

    += + + = += +

    /2

    02 2 0

    2 4I dx I I

    = = =

  • 12

    : . . 12

    15. f \ , 2( )f x x > x\