Alykeioy2014teliko 140826083236-phpapp01

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  • o888oo

    222000111444

    111333 --- 222555 222000111444 OOOLLLYYYMMMPPPIIIAAANNN BBBAAAYYY ---

    MasterText Box

    MasterText Box

  • 2014. . . , . , , . .

    2014

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  • . , . ............................1 - 44 . -, . - . ............ 45-90

    . , . ..........................................................91-109 IV. , . ...........................110-120

    V. , . ............121-164 VI. , . ....................................165-204

  • ...

    ...

    , . , . - . 1. . - . . . , - . () . 1. )32(32 22 = 2. 520105 221 =+ ++ )421( 22 + 3. ]1)1(21[)1(3)1(3)1(6)1(3 222 +=+ yxyyxyxyxyx

    3 ( 1)( 1 2 2 1)x y y xy x= + + 3 ( 1)( 2 2 ).x y y xy x= +

    () , - . , - .

    MasterText Box

  • 2014

    2

    1. )()( bybxayaxbybxayax +++=+++ )()( yxbyxa +++= )()( bayx ++= . 2. 2 2 2 2 2 2 2 2 2 2 22 2 ( ) 2 ( )a x b x xa xb x a b x a b+ = + + )2()( 222 xxba += = ).)(2( 22 baxx += () , , :

    2 2 ( ) ( )A B A B A B = +

    1. 6 4 3 2 2 2 3 2 3 2( ) ( ) ( )( ).x y x y x y x y = = + 2. 4 4 2 2 2 216 81 (4 ) (9 )x y x y = 2 2 2 2(4 9 ) (4 9 )x y x y= + )94(])3()2[( 2222 yxyx += ).94()32()32( 22 yxyxyx ++= () : 22 2 BABA + , , :

    2222 2 BABABABABA +++=++ )()( BABBAA +++= 2( ) ,A B= +

    2 2 22 ( )A AB B A B+ + = +

    222 )(2 BABABA =+

    1. 2222 )5(532)3(25309 yyxxyxyx ++=++ .)53( 2yx += 2. =+ 22 2 yyxx 22 )(2)( yyxx + = 2)( yx 3. 2 2 2 2 2 2 2 2(4 5 ) (3 6 ) (4 5 3 6 ) [4 5 (3 6 )]x y y x x y y x x y y x = + )6354()22( 222 xyyxyx += )810()(2 22 yxyx += ).45()(4 22 yxyx += ()

    ++= xxxf 2)( 0 ,, .

  • 3

    42 = )(xf - :

    > 0, )(xf )(xf ),)(( 21 xxxx = 1x , 2x

    ,21

    +=x

    22

    =x .

    , 2 4 > 0,

    2( )f x x x = + + =

    ++

    x

    ax 2

    +

    +=

    ax

    2

    22

    42

    +=

    22

    22 x

    ++

    +=

    2222

    2

    xx =

    ))(( 21 xxxx = ,

    21+

    =x ,

    22

    +=x .

    = 0, : 2

    2)(

    +=

    xxf .

    , = 0 :

    =+++ xxxf 2)(

    ++

    x

    ax 2

    += 2

    2

    42 x

    2

    2

    +=

    xa .

    A < 0, )(xf . , < 0,

    +

    +=++= 2

    22

    42)(

    xaxxxf

    +

    +=

    22

    22 x ,

    )(xf .

    1. 2( ) 4 3f x x x= + . : ,1= ,4= 3 = 04314)4(4 22 >=== ,

    ,32

    4421

    =+

    =+

    =

    x 24 4 1.

    2 2x

    = = =

  • 2014

    4

    )1)(3(34)( 2 =+= xxxxxf . 2. 12025)( 2 += xxxf ,

    03002514)20(,1,20,25 2 >=====

    532

    5031020

    25230020

    22,1

    =

    =

    =

    =

    x

    12025)( 2 += xxxf

    +=

    532

    53225 xx .

    3. 1)( 2 ++= xxxf , ,1= ,1= ,1=

    03412

  • 5

    45)( 3 += xxxf 443 += xxx )1(4)1( 2 = xxx

    )1(4)1)(1( += xxxx ]4)1()[1( += xxx )4)(1( 2 += xxx

    +=

    2171

    2171)1( xxx .

    3.

    43)( 23 += xxxf .

    43)( 23 += xxxf 44 223 ++= xxx )1(4)1( 22 += xxx

    2( 1) [ 4( 1)]x x x= + )44()1( 2 ++= xxx 2)2)(1( += xx . () : , N*.

    , N 0 12322221 ... ++++++ .

    )...)(( 1221 ++++= .

    2= , N*, 2222 )()( == )()( += . , 1. ))(( 2233 ++= a 2. ))(( 43223455 +++= a 3. ))(( 222244 += a ))()(( 22 ++= . 12 += , , + + - 0 :

    )...)(( 122321 ++++=+ .

    , 1. ))(( 2233 ++=+ a 2. ))(( 43223455 +++=+ a 3. 7 7 2 2 6 5 4 2 3 3 2 4 5 6( )( )a + = + + + +

  • 2014

    6

    12 += , , + + - 2 , . + . , - = 2 , , 2 2 + = + = 2 2( ) ( ) + = 2 2 2 1 2 2 2 2 1( )[( ) ( ) ( ) ] + + +iii . , : 1. 6 6 2 3 2 3 2 2 2 2 2 2 2 2( ) ( ) ( )[( ) ( ) ] + = + = + + 2 2 4 2 2 4( )( ) = + + . 2. 52521010 )()( +=+ )()( 8624426822 +++= 3. 34341212 )()( +=+ )()( 844844 ++= 2= , , 1> :

    ( ) ( )222222 11 +=+=+ ( ) ( ) 111111 22222222 22 ++= ( ) ( )2211 22222 2 +=

    ( )22112211 22222222 22 +

    ++=

    , : 1. 22224444 22 ++=+ 22222 )2()( += )2)(2( 2222 +++= 2. 44448888 22 ++=+ 222244 )2()( += 4 4 2 2 4 4 2 2( 2 )( 2 ) = + + + () :

    + + =2 2 2 2 2 2 ( +) - = ( + +)( + -)

    2 2 2 2 2 - 2 + - = ( -) - = ( - +)( - -)

  • 7

    1. 22222 )2()34(492416 +=++ )234)(234( +++= 2. 22222 )()5(225 =+ [ ] [ ])(5()(5( += )5)(5( ++= . 3. 2222222244 44)2()(4 ++=+ 2222 )2()2( += )22)(22( 2222 +++= . 4. 4222244224 2 ++=++ 224224 2 ++= 222 )()( += ))(( 2222 +++= . () . () 2 > 11 1 0( ) ...f x x x x

    = + + + + , 0 .

    , - 0)( =f , x )(xf .

    )(x , Horner. )(xf = )( x )(x .

    , )(xf - 0 . )(xf 0 .

    , = )(xf -

    0 .

    . 6116)( 23 += xxxxf . : 1, 2, 3, 6 .

    0)1( =f , Horner 1 -6 11 -6 | 1 1 -5 6 1 -5 6 0

    )65)(1(6116)( 223 +=+= xxxxxxxf )3)(2)(1( = xxx ,

  • 2014

    8

    652 + xx 1= , 5= , 6= , 01 >=

    22

    151 =

    =x , 3

    215

    2 =+

    =x .

    2. . - . . () ()

    222 2)( ++=+ 222 2)( +=

    ()

    22))(( =+ () ()

    32233 33)( +++=+

    3 3 2 2 3( ) 3 3 = +

    () ()

    ))(( 2233 ++=+ ))(( 2233 ++=

    () , 3

    222)( 2222 +++++=++

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

  • 9

    ,

    )...()...()...( 21212

    21 ++++++=+++ )...(...)...()...( 21212211 ++++++++++++=

    )...(2... 1312122

    221 +++++++= 1232 2...)...(2 +++++

    ).........(2... 12322122

    221 ++++++++++= .

    ji ji

    ii

    =

  • 2014

    10

    : 1. )1)(1)((31)1( 222266322 ++++++=++ . 2. 3 3 3 3(2 3) (2 ) ( ) ( 3) 3(2 )( 3)( 3 2 ) = + + + + . )32)(3)(2(3278 33 ++= . () (Euler) 3 3 3 2 2 2( )( ) + + = + + + + + 3

    [ ]222 )()()()(21 ++++= + 3

    3)(3)(3 33333 +++=++ )(3)( 33 ++++= [ ][ ] )(3)()()( 22 +++++++= [ ] 3)()( 22 ++++= )()( 222 ++++= .

    )222222(21 222222 ++=++

    [ ]222 )()()(21 ++= a .

    : 1. )4)(2(3)4()2(24648 333333 ++= )482164)(42( 222 ++++= .

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

    [ ] 2 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) = + + + + + + + + + + 2 2 2 2 2 2 2 2 2( ) ( 2 2 ) = + + + + + + + + + + +

    )34( 22 += . () Lagrange

    (i) 212212

    221122

    21

    22

    21 )()())(( =+++

    22, (i) :

  • 11

    21 22

    1 2 2 11 2

    ( )

    = .

    (i) . (ii) 232211

    23

    22

    21

    23

    22

    21 )())(( 3 ++++++