Ορόσημο...

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    15-Apr-2017
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  • R 9

    R

    K

    { }0,1,2,3,...= .

    To { }..., 3, 2, 1, 0, 1, 2, 3,...= .

    *

    0 . { }* 1,2,3,...= .

    Q

    ,

    *

    -

    .

    .. 4 1 1

    2, 0,25, 0,333... 0, 3 ...2 4 3= = =

    , .

    . .. 2 1,41, 3 1,73, 3,14 ...

    R

    .

    , .

    R .1

    R

    , = : 0= =

    :

    i. =

    ii. = =

    iii. = = =

    , :

    0 ( 0 )

    : i.

    ii. =

    iii.

    : 0< < , .

    .

    , :

    ( ) = + 1

    : , 0

    = =

  • A - 110

    H -

    -

    ,

    .

    -

    -

    ,-

    ,

    .

    , -

    , -

    .

    ( ) ( ) + + = + + = + +

    ( ) ( ) = =

    -

    .

    , -

    :

    ... , *

    = + + +

    ... , *

    =

    1= : 1 =

    1 = .

    1. + +

    2. , 0 >

    3. , 0

    iii v :

    .

    ! .

    ! .

    5. i. 0 0 =

    ii. 0 0 0= = =

    iii. 0 0 0

    6. i.

    = ,

    = ,, , 0

    ii.

    = ,

    :

    = = ,, , 0

    7. i. 0> 0 0> + >

    ii. 0< 0 0< +

    0>

    ii. , 0 <

    0

    ii. , , : 1 1

    < * :

    i. = =

    ii. < > +

    ) x 1 y xy 1 x y< < + < +

    A6. ,,

    0> > 0> .

    :

    + > x x>

    . 0 x 1< < > x x , = + , .

    :

    ( ) x x x x x 0 x x 1 0+ > > >

    x 0 x 0> >

    x 1 x 1 x 1 0> > >

    . ().

  • 13

    .2

    2 . -

    :

    . . .

    = , 2

    1 1 : = =

    0 , 0 1=

    1

    =

    : = = .

    , .

    .. ( )2 2

    3 3 = 3 3 .

    o :

    i)

    +

    = ii) : =

    iii) ( )

    = iv) ( ) =

    v)

    =

    vi)

    =

    ,

    :

    .

    :

    0 0

    0

    0 0. .. 19600 0=

    0 .

    .. ( )4

    2 , ( )4

    3

    , 4

    7

    , .

  • 14 A - 2

    A7. : x, y

    x 3, y 2> > : x 1 y 12 5 20 = .

    A8. () :

    ( ) ( ) ( ) ( ) 1 2 3

    1 1 1 1 0+ + +

    + + + = .

    A9.

    :

    ( ) ( )

    ( )

    1 22 6

    1 1

    2,5 5 2 0,4

    8 0,1

    =

    A10. ,y ,

    ( ) ( )x y3 5

    2 1 12 2

    = + .

    A11.

    :

    i) ( )100

    x 2

    2 1

    =

    iii) ( )4x 6

    1 1

    =

    ii) ( )x 1

    5

    1 1

    +

    =

    iv) ( )2x 10

    1 1+

    =

    A12. A * x, y, -

    :

    x y y xx y

    y x

    1

    + + +

    =

    A13. 0, x, y x x=

    x y= :

    y

    1y

    =

    A14. 2 1 0, 0+ + = -

    :

    i) 3

    1=

    ii) 2007 2007 2+ =

    2

    -

    ( -

    ).

    i) ( )

    22 1 2

    3 2 4

    ii)

    3 4

    2 2

    2:

    5

    i) ( )

    22 1 2 4 2 4

    7

    3 2 4 3 2 4

    = =

    ii) 3 4 3 2 3 2

    2 2 2 4 2 4

    2 5 5 5:

    2 5 2 2

    = = =

    1A ( )

    43 2

    =

    ( )3

    4 5

    2 3 2

    =

    3 2 :

    ( ) ( )3

    4 123 3 2 3 2 36 24 12

    = = =

    2 212 3 15 10 20

    2

    2 3 2 13 26

    = = =

    36 24 12 26

    3 2 16 24 38

    20

    :

    = =

  • 15

    .3

    . (

    ).

    .

    ( )2 2 2 2 + = + +

    ( )2 2 2 2 = +

    ( )( ) 2 2 + =

    ( )( ) ( )2x x x x + + = + + +

    ( )3 3 2 2 3 3 3 + = + + +

    ( )3 3 2 2 3 3 3 = +

    ( )( )3 3 2 2 + = + +

    ( )( )3 3 2 2 = + + :

    ( )( ) 1 2 2 1 ... = + + + +

    :

    Euller

    ( ) ( ) ( ) ( )2 2 23 3 3 1 3

    2 + + = + + + +

    : 0+ + = 3 3 3 3= = + + =

    Lagrange

    ( )( ) ( ) ( )2 22 2 2 2 + + + =

    .

  • 16 A - 3

    A15. N :

    i) ( ) ( )2 2

    4x 3 3x 4+

    ii) ( ) ( )3 3

    3 2 3 2+

    iii) ( )( ) ( )2

    2 4x 2 4x 4x 1+

    iv) ( )( ) 2x 1 x 1 x 5x + +

    3

    2

    1 0 + + =

    3

    + + ++ + + = + +

    3

    3

    1 1 1 3

    + + ++ + + =

    = + + + =

    = + + + = + +

    i) N :

    ( ) ( ) ( )

    ( ) ( ) ( )

    3 3 3

    3

    + + =

    =

    ii) :

    ( ) ( ) ( )3 3 3

    x 2 3x 4 6 4x 0 + + =

    i) E ( ) ( ) ( ) 0 + + =

    Euller :

    ( ) ( ) ( )

    ( )( )( )

    3 3 3

    3

    + + =

    =

    ii) O

    ( ) ( ) ( )x 2 3x 4 6 4x 0 + + = :

    ( ) ( ) ( )

    ( )( )( )

    3 3 3x 2 3x 4 6 4x

    3 x 2 3x 4 6 4x

    + + =

    =

    :

    ( )( )( )3 x 2 3x 4 6 4x 0 =

    x 2 0 = 3x 4 0 = 6 4x 0 =

    x 2= 4

    x3

    = 3

    x2

    =

    ( )2 2 2 2

    + + = + +

    0 1 1 1

    0 + + =

    ( ) ( )2 2 2 2 2 + + = + + + + + ,

    , :

    0+ + =

    0 :

    1 1 10

    + + =

    A16. N :

    i) ( ) ( )2 2

    x y x y 4xy+ =

    ii) ( ) ( ) ( )2 2 22 2 5 2 3 4+ + + = +

    iii) ( )( ) ( )( )3 4 4 2 + + = +

    iv) ( ) ( ) ( )2 2

    2 2 2 2 2 2 4 2+ + = +

  • 17

    .4

    -

    . ,

    .

    ( )( )2

    x 1 x 1 x 1 +

    .. ( )2x 4 2 x 2 = . -

    2.

    1.

    . 3 2xy 3xy 2x 6xy + . 2 22x y 6xy 8xy +

    :

    . x.

    :

    ( )3 2 3 2xy 3xy 2x 6xy x y 3y 2 6y + = + . 2xy.

    :

    ( )2 22x y 6xy 8xy 2xy x 3 4y + = +

    2.

    . xy 3x 2y 6 + . 3 23x x y 6x 2y +

    . .

    ( ) ( ) ( )( )xy 3x 2y 6 x y 3 2 y 3 y 3 x 2 + = + = +

    . ( ) ( )3 2 3 23x x y 6x 2y 3x x y 6x 2y + = + =

    ( ) ( ) ( )( )2 2x 3x y 2 3x y x 2 3x y= + = +

    3. ( )

    ( )

    2 2 2

    2 2 2

    2

    2

    + = + +

    = +

    3.

    i) 2 225x 20xy 4y + ii) 2 29x 24xy 16y +

    iii) 2 216x 40xy 25y+ +

    -

    1 :

    (

    )

    2 :

    (

    )

    3 :

  • 18 A - 4

    ( ) ( ) ( )2 2 22 2

    25x 20xy 4y 5x 2 5x 2y 2y 5x 2y + = + =

    ii) ( ) ( ) ( )2 2 22 2

    9x 24xy 16y 3x 2 3x 4y 4y 3x 4y + = + =

    iii) ( ) ( ) ( )2 2 22 2

    16x 40xy 25y 4x 2 4x 5y 5y 4x 5y+ + = + + = +

    3. ( ) ( )2 2 = +

    3.

    i) 2 225x 4y ii) 4 816x y

    i ) ( ) ( ) ( )( )2 22 2

    25x 4y 5x 2y 5x 2y 5x 2y = = +

    ii) ( ) ( ) ( )( )

    ( ) ( ) ( ) ( )( )( )

    2 24 8 2 4 2 4 2 4

    222 4 2 2 4 2 2

    16x y 4x y 4x y 4x y

    4x y 2x y 4x y 2x y 2x y

    = = + =

    = + = + +

    3. - ( ) ( )

    ( ) ( )

    3 3 2 2

    3 3 2 2

    + = + +

    = + +

    3.

    i) 3x 64 ii) 38x 27+

    i ) ( )( ) ( )( )3 3 3 2 2 2x 64 x 4 x 4 x 4x 4 x 4 x 4x 16 = = + + = + +

    ii) ( ) ( ) ( ) ( )( )33 3 2 2 2

    8x 27 2x 3 2x 3 2x 2x 3 3 2x 3 4x 6x 9 + = + = + + = + +

    3. 2 ( ) ( ) ( )2x x x x + + + = + +

    , ,

    2 , 2x x , 0+ + .

    : 2x x + + ,

    : ( ) ( )( )2x x x x + + + = + + , , -

    .

  • 19

    3.

    i) 2x 5x 6 + ii) 2x 4x 3+ + iii) 2x x 2

    i ) 2, 3 ( ) ( )2 3 5 + = ( ) ( )2 3 6 = .

    ( )( )2x 5x 6 x 2 x 3 + =

    ii) O 1,3 1 3 4+ = 1 3 3 = .

    : ( )( )2x 4x 3 x 1 x 3+ + = + +

    iii) O 2,1 ( )2 1 1 + = ( )2 1 2 = .

    : ( )( )2x x 2 x 2 x 1 = +

    4.

    i) 2

    2x 18 ii) 2 2x 2xy y 9 + +

    iii) 2 2x y x 2xy y + + iv) 9 7 5 3x x x x +

    i) ( ) ( ) ( )( )2 2 2 22x 18 2 x 9 2 x 3 2 x 3 x 3 = = = +

    ii) ( ) ( ) ( )

    ( ) ( ) ( ) ( )

    2 22 2 2 2 2x 2yx y 9 9 x 2xy y 9 x y 3 x y

    3 x y 3 x y 3 x y 3 x y

    + + = + = = =

    = + = + +

    iii) ( ) ( ) ( ) ( ) ( )( )22 2

    x y x 2xy y x y x y x y 1 x y x y 1 x y + + = + = + = +

    iv) ( ) ( ) ( )

    ( )( ) ( )( )( )

    ( ) ( ) ( )

    9 7 5 3 3 6 4 2 3 4 2 2

    3 2 4 3 2 2 2

    2 23 2

    x x x x x x x x 1 x x x 1 x 1

    x x 1 x 1 x x 1 x 1 x 1

    x x 1 x 1 x 1

    + = + = =

    = = + =

    = + +

    4 ( )

    i) 2 2x 2xy 3y+ ii) 4 2

    x 5x 9+ +

    i ) 23y 2 2y 4y

    ( ) ( ) ( )( )

    ( )( )

    2 22 2 2 2 2x 2xy 3y x 2xy y 4y x y 2y x y 2y x y 2y

    x 3y x y

    + = + + = + = + + + =

    = +

    ii) 2x

    4 :

  • 20 A - 4

    ( )

    ( )( ) ( )( )

    24 2 4 2 2 2 4 2 2 2 2

    2 2 2 2

    x 5x 9 x 5x 9 x x x 6x 9 x x 3 x

    x 3 x x 3 x x x 3 x x 3

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

    = + + + = + + +

    A17. :

    i) ( )4x x 2y x 2y +

    ii) 2 2 2 25 x y x 5 y+

    iii) 2 236x 49y

    iv) 4 416x y

    v) 225x 20x 4 +

    vi) 3