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  • : , ,

    3 & .. 10683, .: 210 3834533

    :/ . . .- : @ .

    http www archimidis edu gre mail spiliotis yahoo gr

    . 4

    :

    2013

    2013

    15

    20

    13

  • 1

    A

    4 2 1 2 - 1 10 (15) 3

    :1. - .

    2. .3. .4. 15 .

    5. 2 .6. .

    l l

    , .

  • 2

    B

    . 3 . 2 . 1 . 2 . . 1 2 14 (15) 3

    2 1 1 . . . 2 . . 3 12 (15) 3

    :x . , - .

    x -.

    x .x .x 15 .

  • 3

    () 2

    4 2 2 2 1 2 1 13 (15) 3

    x tests .x .x - . .

    x 15 . .

    x . - .` ` `

    x 7 12 .

    . . 2 . 3 . . 1 . 2 . 2 . . 1 . 1 2 14 (15) 3

    . . 2 . 3 . . 1 . 2 2 1 2 13 (15) 3

  • 4

    2013

    2013

    .1 f [, ]. G f [, ], : f (t)dt G() G() .

    7.2 (...) .

    4A3. f [, ] ;

    4A4. , - , , , .. 0| z z | , 0! 0K(z ) - 2 , 0z, z .

    . 0x x

    lim f (x) 0o

    , f (x) 0 0x .. : | x | | x |d , x\ .. :

    x 0

    x 1lim 1xo .

    . f f .

    10

    z : (z 2)(z 2) | z 2 | 2 .1. z, - (2, 0) = 1 . ( 5) , z , - | z | 3d . ( 3)

    82. 1 2z , z 2w w 0 , w , , \

    1 2| Im(z ) Im(z ) | 2 : 4 5 9

    3. 0 1 2 , , 1. . v :

    3 22 1 0v v v 0

    : | v | 4 . 8

  • 5

    f , g : o\ \ , f :z f (x) x f (x) 1 xc , x\zf (0) 1 z

    23 3xg(x) x 1

    2

    1. : 2f (x) x 1 x , x\ 92.

    f g(x) 1 83. 0 x 0, 4

    , :

    0

    0 0 0x4

    f (t)dt f x x4

    8 f : (0, )f o \ :z f c (0, )f z f (1) 1 z

    h 0

    f (1 5h) f (1 h)lim 0ho

    xf (t) 1g(x) dt

    t 1 , x (1, ) f 1!

    :1. f (1) 0c ( 4) , f 0x 1 ( 2)

    6

    2. g , \2 4

    2 4

    8x 6 2x 6

    8x 5 2x 5g(u)du g(u)du

    !

    9 3. g ,

    xf (t) 1( 1) dt f () 1 (x )

    t 1 , x 1! ( 6)

    . 10

  • 6

    1, 2, 34. )o , )o , )o , )o , )o ( )

    1. : 2z 2 z 2 z 2 2 z 2 z 2 2 z 2 2 2 0 1 2 z 2 2 , z 2 1 . M 2 2,0 1 z z 2 2 z 2 2 1 2 3 d 1maxz z z 3d .

    2. 1 2Im z Imz 2 1z , 2z xxc , 1z 2 i , 2z 2 i , - :

    2 2z 2 i z 2 i 0 x 2 i 0 2z 4z 5 0 2w w 0 , : 4 5

    - z x iy :

    1 2Im z Imz 2y 2 y 2 y 1 r z x i r (1) :

    z 2 1 x i 2 1 x 2 i 1 r r 2x 2 1 1 x 2 0 x 2 (1): z 2 i r . Vieta.

    1 2

    1 2

    z z 4 42 i 2 i 6z z

    3. :

    3 22 1 0 3 22 1 0

    2 2 22 1 0 2 1 0 3 3 3 d d

    x

    y

    0A(3)

  • 7

    :3 2 3 3 3d

    4 , 24 :2 2 3 4 3 3 d 2 2 4 16 16 3 3 0 d 2 4 4 4 3 4 1 0

    2 4 4 3 1 0 2 4 1 1

    4 0 4 .

    : f x x f x 1 xc f x x f x x xc 2 2f x x xc c

    2 2 f x x x c f 0 1

    2f 0 c 1 c 2 2f x x x 1 2f x x x 1 r (1) x 0 (1) f 0 0 1 r , f 0 1 r . f 0 1 f x x - :

    2f x x x 1 2f x x x 1 2. f g x f 0 ( f 1-1 ).

    f x :

    2f x 1c x2

    2

    2 2

    x 1 x

    x 1 x 1

    22

    x x 10

    x 1

    !

    f p , 1-1 (2) 1-1 :

    f g x f 0 g x 0 23 3xg x x 1 0

    2 (3).

    3 23 2x 3x 2 0 ( Horner ). :

  • 108

    3 2h x 2x 3x 2 0 2h x 6x 6x 6x x 1 0 x 0, 1c @ > h x 0 x ,0 1,c ! f f

    3xlim h x lim 2xof

    f 3xlim h x lim 2xof

    f ., , h x : h h ( g) 0,f .

    3. Bolzano 0 x4

    g x f t dt f x x4

    0,4

    . g x 0,4

    .

    0 0 4 4

    g 0 f t dt f 0 f t dt 04

    ! f ,0

    4

    f x 0!

    00

    g f t dt f 0 14 4

    Bolzano 0,

    4

    .

    1. :

    h 0

    f 1 5h f 1 f 1 f 1 hlim 0

    ho

    f 1 5h f 1 f 1 h f 1lim 5 0

    5h h

    (1)

    :

    h 0 y 0

    f 1 5h f 1 f 1 y f 1

    5lim y 5h 5lim 5f 15h h

    lim y lim5h 0o o

    c

    (2)

    h 0 y 0

    h 0

    f 1 h f 1 f 1 y f y

    l lim y h lim f 1h y

    lim y lim h 0o o

    o

    c

    (3).

    (1), (2), (3) : 5f 1c , f 1 0c f 1 0c . f c :

    @0 x 1 f x f 1 0 f 0,1c c p > x 1 f x f 1 f x 0 f 1,c c c! ! ! n f

    x+ +0 0

    1

    1 2

    0h

    h

  • 119

    :

    f x 1 x 1 f 1 1 .

    2. g x ( ) : f x 1g x

    x 1c f. :

    f x 1g xx 1

    c x >

    1

    x < 1

    g x 0c ! , x 1, f g x n .

    2. 1 . : x 1

    xh x g u du

    , x 1! . : c x 1x ch x g u dt g u du cc g x g x 1 g x 1 g x

    g > @x, x 1 : g x 1 x g 0 h c c ! n . ( x 1! ):

    2 4 2 4h 8x 5 h 2x 5 8x 5 2x 5 ! ! 2 4 2 24x x x x 4 0! x 0

    2x 4 0z

    x 2 x 2 0 ^ `x 2, 2 0 .3. 2

    f x x 1 f x 1f x 1g x

    x 1 x 1

    c c cc

    2f x x 1 f x f 1

    x 1

    c

    2

    f x 1 f 5 x 1x 1

    c

    f x f x 1

    c c ( f c n ) 0 ! g

    1 x. x .. : 1 g x f 1 x .(, g). g fC , x z .

    g x g g x g g x c! ! f 1g x x 1

    ! f 1 1 g x x 1

    ! :

    f 1 x g x x 1 ...

    x .

    x+ 0

    10f

    f

  • 1210

    .1 f 0x , f - .

    7.2 Fermat.

    4A3. f . - f ;

    4A4. , - , , , .. z | z | | z | . ( 2). f 1 1 , - f . ( 2)

    . 0x x

    lim f (x)o

    f , 0x x

    lim f (x)o

    f . . f , g 0x :

    0 0 0 0 0(f g) (x ) f (x )g(x ) f (x )g (x )c c c . ( 2). f , f . ( 2)

    10 z, w

    22x | w 4 3i | x 2 | z | , x\ , , x 1 .1. z - 1 1 , - w K(4, 3)

    2 4 . 8

    2. , .

    53. z, w 1. :

    | z w | 10 d | z w | 10 d . 6

    4. z 1. , : 2| 2z 3z 2zz | 5 . 6

  • 111

    f : o\ \ : z 22xf (x) x f (x) 3 f (x)c c , x\ z 1f (1)

    2

    1. :3

    2

    xf (x)x 1

    , x\ f - \ .

    62. f 1.

    43. :

    2 3 2 2f 5(x 1) 8 f 8(x 1) d 7 4. , , (0, 1) , :

    3 3 30

    f (t)dt (3 1) f ( ) 8 f : [0, )f o \ , [0, )f , :

    z 2

    x u

    1 1

    f (t) 1f (x) x dt du

    f (t)

    c , x 0!

    zf (x)f (x) 0c z , x 0! f (0) 0 :

    f (x)g(x)f (x)c x 0! 3h(x) f (x)c x 0t

    1. : 2f (x)f (x) 1 f (x)cc c x 0! . 42.. f f c (0, )f . ( 4). f (0) 1c . ( 3)

    73. g (0, )f , :.g(x) 2 xt , x (0, ) f . ( 2). 1

    0(2 x)f (x)dx 1 ( 4)

    6 4. - h , xx x 0 x 1 .

    8

  • 112

    & 2013

    1. f (x) x f (x) xc ,

    x\ . 7

    A2. f . f ox A ;

    4 3. () .

    4 4. , ,

    , , , -, .) 1f (x) , x 0

    x z 21f (x) xc . ( 2)

    ) f, g :(f (x)g(x)) f (x)g(x) f (x)g (x)c c c

    ) . ( 2)

    ) , . ( 2)

    ) A B , P(A) P(B)! ( 2)

    10

    ^ `1 2 3 4 , , ,

    ^ `1 4 , ^ `1 3 , ^ `1 ^ `3 :

    2

    1 3 2x 1

    1 x x 1 1P( ) lim2 x xo

    3P( ) f (x) x, x=1,

    xf (x) ln, x, x 03

    !

    B1. 1 1P( ) 4 31P( )3

    10

    2. 1 3P(A )3 4

    cd d , Ac . 7

    3. 3P(A )4

    c , > @2 4P( ),( ), ( ) ( ) ( )c c , c .

    8

  • 113

    , - 4 . :

    50 4x 85

    75 x 74

    1. c 10 . 42. .

    ix if[ , ) [ , ) [ , ) [ , )

    3. 1 2 3 4f 0,1, f 0,3, f 0,2 f 0, 4 . -, 80, 200

    3.

    4. 1 () f (1, f(1)), , - ,

  • 114

    3. 1 ee 7 e , R

    1f (), f (), f (), f (e), fe

    c f (x) x ln x 2 . 7

    4. :n 1 2 10 11 30

    1 t , n 1, 2,3......,30 : 0 t t ..... t t ....t 1e

    , A {t : f (t, f(t)), x xc },B {t : f (t) f (t) 1}c ! , f (t) t ln t 2 . :) ( 3) ) ( 4)

    7

  • 115

    1, 2, 3 , 4. _ _ _ _

    1. 2 2

    1 x 1 3 2 2

    x x 1 1 x x 1 11P lim2 x x x x 1 1o

    2

    2 2

    x x 1 11 lim2 x x x x x 1 1

    21 x xlim2

    2x x x 214x x 1 1

    1 1 1f x ln x f 12 3 3

    c c 3 1P f 1 3c 2. ^ `2 3A , c . ^ ` ^ ` ^ `3 2 3 4 1 1 , , cc :

    3 1P P 1 P cd d 1 3P 3 4cd d3. ^ ` ^ ` ^ `2 3 2 33 3P P , P P 4 4c ^ ` ^ `2 3

    3 3 1 5P P 4 4 3 12

    ^ ` ^ ` ^ ` ^ `4 1 2 3 1 5 1P 1 P P P 1 04 12 3 . ^ ` ^ ` ^ `4 3 3 4 , ^ ` ^ `3 4 1P A B B A P P 3 ^ ` ^ ` ^ `2 3 2 4 3 1A B , , P 3c c .

    1. c :

    > 50, 50 c , > 50 c, 50 2c , > 50 2c, 50 3c , > 50 3c, 50 4c Ax 85 : 50 3c 50 4c85 170 100 7c c 102

    .2. : 1 2 3 4f f f f 1 (1)

    4 3f 2f (2) 1 2 3 4x 74 55f 65f 75f 85f 74 (3)

    31 2

    f 1 75 f f2 2

    (4) (1), (2), (3), (4) ; : (1), (2) 1 2 3f f 3f 1 (5)

    (4) 1 2 21 1f f f2 2

    : 3 33 1 1f f 0,22 2 5 . (2): 4f 0, 4 .

  • 116

    (1), (3) 1f 0,1 2f 0,3 . :

    ix if> 50, 60 55 0,1 > 60, 70 65 0,3 > 70, 80 75 0,2 > 80, 90 85 0,4

    if if c c .

    1

    1 11

    ff 0,6

    c cc

    22 ff 0,6c 3

    3f

    f0,6

    c .

    1 1 2 2 3 3x f x f x f xc c c c 1 2000,1 55 0,3 65 0, 2 750,6 3 .4. .

    x3s x+3sx2s x+2sxs x+sx

    16% 2,35%34%

    : x 2s 74 x s 68

    : x 70 , s 2 s 1 1CVx 35 10

    . f x ln x 1c f 1 1c , f 1

    : y f 1 f 1 x 1c y x 1

    x 0 (1) : y 1 . y 0 (1) x 1 .

    x

    y

    0

    E

    1

    1

  • 117

    : 21 2 1 1 2 1 4

    2 1 2 1 3

    2 ] y x 1 x y 1 .2. . x y 1 31 1 30 .2. . :

    1 2 20 21 22 25 26 27 50x x ... x 20 3 x x ... x x x ... x 15 3150

    1 50x ... x 60 15 3150

    60 15x 3150

    60 1530 3150 2

    3

    3. R f. : 1f x 0 ln x 1 0 x

    ec 11 1 1f ln e 2 2

    e e e

    .

    , ,

    1e e f f f f e 1f 0

    e c

    1R max min f e f e 2 0e

    c :

    ln ln ln e ln e 2 4x5

    eln ln ln ln e 8

    5

    eln e 85

    7 eln e 8 7 8 e 15 e e35 5 5 5

    4. . 0 2

    d 0 f t 1c d 1 te

    : 20P A30

    .

    f t f t 1 ... t 0,1c! t 30t . ^ `11 29A t ,..., t 19P A 30 .

    : 3 3

    ..

    x+ 0

    1/e

    f(1/e)

    0f

    f

  • 2018

    1. :P(A ) 1 P(A)c

    7

    2. . 4

    3. f ox ; 4

    4. , , , , , -, .)

    o0x x

    lim (x) xo

    ( 2)) (cf (x)) cf (x)c c ( 2)) .

    ( 2) ) , 10%.

    ( 2) ) ,

    A B z ( 2 10

    xf (x) 2e (2x 3), x \ .

    1P(A) x 1f (x )P(B) 6 e f 1xB1. f .

    6 2. 1P(A)

    2 2P(B)

    3

    6 3. .

    5 4. 1 2P(A B )

    6 3c cd d

    8

  • 2119

    v X - 5 c, :

    ix if % iF iF %[ , ) [ , ) 3 + 10 [ , ) [ , ) 2 2 + 10

    2 3 + 30

    3F 5F :25x 8x 3 0 , x\ \

    1. 1 10 . 82. 1 2 3 4f % 10, f % 30, f % 20, f % 30 5f % 10 53. 25% 16 25%

    24, =10 c=4 ( 4)

    4. 22 800, .

    4

    2xf (x) 1, xx 1 \ ^ `1 2 3 4 , , , , 1 2 1, 0 3 41

    , , i i 1( ) f ( ) 3 , i=1, 2 3 x 11 f (x)P( ) lim6 x 1o

    c .1. ,

    ^ `A / f () 0c d , ^ ` / f () 1 ! 2 1 / x x x

    4 t

    \) 1 2 3 4( ), ( ), ( ) ( ) ( 8)) (), (), () (-) ( 8)

    16 2. () f, -

    x xc o45 . 43. M ( , y ) , =1, 2, 3, 4 (): y = x+1 y2

    yR 5 , 3 4 , : ,y : yR :

    5

  • 2220

    2013

    A 1. 1 2z , z :

    1 2 1 2| z z | | z || z | 10

    2. Bolzano. 5

    3. , , -, .. 1 2z , z 1 2z zz ,

    1 2| z z | | z z | ( 1z ) ( 2z ). C C f 1f y x .

    . 0 1 xxlim of

    f. f 0x. > @f : , Ro f x 0t

    > @x , , : f (x)dx 0t

    10

    B z :

    1 1Rez 1 2

    1. z -

    K(2,0) 1 , ( 7). - ( 2).

    9 2. 1 2z , z 1, :

    1 2| z z 4 | 2 d 8

    3. z 1, :

    | z | 5 8

  • 221

    2xf (x) n x x, x 02

    !A1. f (0, )f ,

    f . 8

    2. , (, +1) 4f (x 2x) f (4)

    . 9

    3. (0, )f 2x n x 2 2x A

    8

    f : 0, Rf o :x 3 2

    13 2tf (t)dt x 3x f (x) 3x 8, x 0 !

    1. f (0, )f 2

    2

    x 1f (x)xc

    6 2.

    2x 1f (x) , x 0x !

    ( 3) y x f f x 1 2x e

    8 4.

    f (x) 2f (x)x 1

    c ! x 1! 5

  • 222

    . . . - -. .

    . ..

    :) , UFO - .

    ) .) .

    -!

  • 2

  • 2

  • 2

    , , . , . - , , . . , , .

    , , . . , . - .

  • 2

  • 2

  • 0

  • ISSN 1107-1877-4

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    .

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