Mathhmatika Kai Stoixeia Statistikhs-Biblio Mathiti

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    , .

    . /

    :

    /

    /

    :

    /

    :

    :

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    1:

    1.1 9

    1.2 19

    1.3 271.4 39

    2:

    2.1 58

    2.2 62

    2.3 83

    2.4 104

    2.5

    117

    3:

    3.1 - 138

    3.2 146

    3.3 157

    3.4 - 165

    - 179

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    -

    ,

    1999-2000.

    .

    ,

    . ,

    ,

    .

    .

    . . , -

    -

    . -

    -

    . -, -

    .

    , -

    , ,-

    .

    .

    . -

    , .

    , -

    ,

    . -

    , 396, 15310

    .

    1999

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    1

    17

    ,

    .

    . ,

    t )(tfx ,

    t

    )(tfx t , ,

    )(tfx . , ,

    ,

    . Newton (1642-1727) Leibniz (1646-

    1716),

    ,

    . 17

    , 18

    Jacob Bernoulli (1654-1705) Johann Bernoulli (1667-1748), Euler

    (1707-1783), , Lagrange (1736-1813)

    . ,

    19 Bolzano (1781-1848),

    Cauchy (1789-1857) Weierstrass (1815-1897).

    1.1

    (function)

    .

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    10

    fBA

    ,

    , R

    , R.

    . f, g, h,, .

    .

    f .

    Ax By ,

    )(xfy

    y f x. )(xf f x.

    x, ,

    , y, x

    x, .

    ,

    2

    1)( xxf . : f

    21)( xxf 21)( xxf

    21 xy , , 21 x .

    )(xf ,

    R )(xf

    . , 2

    1)( xxf

    01 2 x ,

    ]1,1[ , 2

    3)(

    x

    xg

    }2{ RA , R 2, 13)( xxh

    R.

    f

    xy

    ,

    1

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    11

    . , , 2

    2

    1)( gxxf 2

    2

    1)( gtts .

    f, g,

    :

    gfS , )()()( xgxfxS , Ax gfD , )()()( xgxfxD , Ax gfP , )()()( xgxfxP , Ax

    g

    fR ,

    )(

    )()(

    xg

    xfxR , Ax 0)( xg .

    , 1)(2 xxf 1)( xxg ,

    )1(11)(2 xxxxxS

    )1)(2(211)(22 xxxxxxxD

    )1()1()1)(1()(22 xxxxxP

    11

    1

    )(

    )()(

    2

    x

    x

    x

    xg

    xfxR , 1x .

    f .

    f

    Oxy ))((,( xfxM

    Ax

    . , ),( yxM f, )(xfy . )(xfy

    ),( yx

    f

    f.

    .

    .

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    12

    O

    y

    x

    y=x

    -2 -1 O 1 x

    y

    2

    1

    2

    3

    y=x2

    ()

    xxf )(

    13.

    () 2)( xxf .

    -2 -1

    O 1

    y x1

    y

    x2

    -2

    -1

    1

    2

    -2 O-1 x1

    1

    2

    3

    y

    y=ex

    ()

    xxf

    1)( .

    () x

    exf )(

    xx , 0xe Rx .

    O

    -1

    y

    1

    1 x

    y=lnx

    2

    y=x

    O x

    y

    2

    y=x

    O x

    y

    ()

    xxf ln)(

    yy ,

    0x .

    () xxf )(

    xxg )(

    2.

    2

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    13

    x

    xf 1)(

    0x . f .

    -

    2

    y=x

    3/2

    /2 O x

    y

    xxf )( , ]2,0[ x 21 ,xx

    2,0

    21 xx 21 xx .

    xxf )(

    2,0

    .

    2,

    23 .

    21,xx

    2

    3,

    2

    21 xx , 21 xx .

    xxf )(

    2

    3,

    2

    . :

    f

    , 21

    ,xx 21

    xx

    )()( 21 xfxf , ,

    21 ,xx 21 xx )()( 21 xfxf .

    .

    ,

    3

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    14

    ]2,0[ x 2

    1 x 2

    31 x . , ,

    xxf )( (maximum) 2

    x

    (minimum) 2

    3x .

    g 4

    1xx g

    gx

    1x , ,

    1x .

    g

    1x . 3xx . )( 1xg )( 3xg

    . , 4xx )( 4xg

    gx

    4x . g 4x .

    2xx . )( 2xg )( 4xg

    .

    . ,

    )( 1xg )( 4xg .

    :

    f :

    Ax 1 , )()( 1xfxf x

    1x , Ax 2 , )()( 2xfxf x

    2x .

    , ,

    .

    1

    1)(

    2

    x

    xxf , 1x .

    f x 1.

    x

    y

    O x4x3x2x1

    y=g(x)

    4

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    15

    )(xf x 1.

    x 1 )(xf 1x )(xf

    0,5

    0,9

    0,99

    0,999

    0,9999

    1,500000

    1,900000

    1,990000

    1,999000

    1,999900

    1,5

    1,1

    1,01

    1,001

    1,0001

    2,500000

    2,100000

    2,010000

    2,001000

    2,000100

    x

    1 ( 1), )(xf 2.

    , 1x

    11

    )1)(1(

    1

    1)(

    2

    x

    x

    xx

    x

    xxf ,

    x 1

    )1( x , 1)( xxf

    2 )21( x . f

    1 (limit) 2

    2)(lim1

    xfx

    .

    f

    0x , ,

    0x .

    0x

    . , 23

    1)( xxf ,

    R. 0x , 2)( xf ,

    2)(lim0

    xfx

    . , 0lim2

    0

    x

    x 0lim

    0

    x

    x.

    23

    1 xy

    2

    y

    x

    O

    x

    y

    y=x2

    y x

    y

    x

    () () ()

    yx

    x

    2 1

    1

    x

    2

    1O

    y

    5

    6

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    16

    fg 0x , 1)(lim

    0

    xfxx

    2)(lim0

    xgxx

    1 2 ,

    :

    21))()((lim0

    xgxfxx

    1))((lim0

    kxkfxx

    21))()((lim0

    xgxfxx

    2

    1

    )(

    )(lim

    0

    xg

    xf

    xx

    xx

    xf 1))((lim0

    xx

    xf 1)(lim0

    .

    , , 93)(2 xxxf

    19649limlim3lim)93(lim)(lim22

    2

    2

    2

    22

    xxxxxxxxxxf .

    93)(2 xxxf )2()(lim

    2fxf

    x

    .

    f 20 x .

    f,

    Ax 0 )()(lim 00

    xfxfxx

    .

    ,

    .

    , ,

    , , ,

    .

    0lim0

    xxxx

    , 0lim0

    xxxx

    0lim0

    xxxx

    ( 0 0 x ).

    :

    i)x

    x

    x 4

    53lim

    5

    ii) 14lim8

    xxx

    iii)3

    9lim

    2

    3

    x

    x

    x.

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    17

    i) 12020

    54553

    453lim

    5 x

    xx

    ii) 53294184814lim8

    xxx

    iii) 6)3(lim3

    )3)(3(lim

    3

    9lim

    33

    2

    3

    x

    x

    xx

    x

    x

    xxx.

    1. xxxf 3)(3 , )1(f , )2(f , )1(f .

    2. 65)(2 ttt , )0( )1( .

    t 0)( t ;

    3. h )( , )0(h

    2

    h .

    ]2,0[ 0)( h ;

    4. 2ln2

    1)( xxf , )1(f )(ef .

    5. )2)(1(

    2)(

    xx

    xxf ;

    6. x )7)(3()( xxxf ; )7)(3()( xxx ;

    7. 123)( 2 xxxf 12)( xxg , )()( xgxf , )()( xgxf ,

    )(

    )(

    xg

    xf.

    8. :

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    18

    i) )43(lim 20

    xxx

    ii) )]4)(12[(lim2

    xxx

    iii)

    xx

    x

    1lim4

    iv) )32(lim0

    xxx

    v) )3(lim

    4

    xx

    x

    .

    9. :

    i))2(3

    4lim

    2

    2

    x

    x

    xii)

    1

    5lim

    2

    2

    1 x

    x

    xiii) ])1[(lim

    0xx

    x

    iv)4

    16lim

    2

    4

    x

    x

    xv)

    5

    25lim

    2

    5

    x

    x

    xvi)

    2

    232lim

    2

    2

    x

    xx

    x.

    1. xe

    xf1

    1

    1)(

    , 1)()( xfxf .

    2. 100 m,

    .

    . x,

    x.

    3. , , 20 cm.

    h cm, h.

    r,

    r.

    4. 10 . , , .

    5. i)

    5

    5lim

    5

    x

    x

    x 52

    1 ii)

    h

    h

    h

    11lim

    0

    2

    1 .

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    19

    1.2

    ),( RO

    . .

    ,

    o90 .

    , ,

    .

    . ),( RO

    , ,

    .

    -

    .

    f

    ))(,( 00 xfxA

    C.

    ))(,( 00 hxfhxM C 0h .

    C ,

    0h , (tangent) C.

    h

    xfhxf

    A

    M

    )()(

    00 ,

    C

    h

    xfhxf

    h

    )()(lim

    00

    0

    .

    B

    xx0+hx0

    f(x0)

    y

    f(x0+h)

    C

    7

    8

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    20

    , S t sec (s)

    2

    2

    1)( gttS ,

    2

    m/s81,9g .

    , 5t s;

    , s5t s1,5t . :

    1,0

    )5()1,5(

    SS

    m/s5405,491,0

    5905,4)1,5(905,422

    .

    .

    65 t 1,55 t

    05,55 t

    01,55 t

    001,55 t

    0001,55 t

    00001,55 t

    53,955

    49,5405

    49,29525

    49,09905

    49,054905

    050491,49

    050049,49

    ,

    49,05 m/s.

    5t s5t . 5 s m/s05,49 .

    , 0t

    O

    B t0+h

    A t0

    9

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    21

    , h . 0t

    2001

    2

    1)( gttSOAS

    ht 0

    )2(2

    1

    2

    1)(

    2

    1)(

    20

    20

    2002 hhtggthtghtSOBS .

    , h

    )2(

    2

    1 2012 hhtgABSSS

    0t ht 0

    ghgth

    hhtg

    tht

    S

    2

    1)2(

    2

    1

    )(0

    20

    00

    .

    h ,

    , 0gt .

    0t 0t .

    , 0t

    00

    00

    0lim

    )()(lim gt

    h

    S

    h

    tShtS

    hh

    .

    50 t , m/s05,49581,9 , -

    .

    ,

    , ,

    t )(tfx .

    O t0

    A

    t0+h

    B

    0t ,

    0t ht 0 0h . h

    )()( 0012 tfhtfxxx . ,

    10

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    22

    h

    h

    tfhtf

    h

    x

    )()( 00 .

    ,

    0t

    h

    tfhtf

    h

    x

    hh

    )()(limlim 00

    00

    .

    ,

    .

    f x x0

    , -

    ,

    ,

    h

    xfhxf

    h

    )()(lim 00

    0

    .

    , ,

    , , , ,

    .

    ,

    f 0x .

    f x0, )( 0xf

    f 0x . :

    h

    xfhxfxf

    h

    )()(lim)( 00

    00

    , 2

    3)( xxf 4, :

    )4()4( fhf :22

    43)4(3)4()4( hfhf

    )484(3222 hh

    )8(3 hh

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    23

    0h h fhf )4()4( :

    hh

    hh

    h

    fhf324

    )8(3)4()4(

    .

    h

    fhf

    h

    )4()4(lim

    0

    :

    24)324(lim)4()4(

    lim00

    h

    h

    fhf

    hh.

    , 24)4( f .

    f 0x (rate of change)

    )(xfy x, 0xx . ,

    :

    f ))(,( 00 xfx

    )( 0xf , )(xf x 0xx .

    )(tfx

    0t )()( 00 tft ,

    )(tf t 0tt .

    . , ,

    ||)( xxf 00 x .

    0h ,

    h

    fhf

    h

    )0()0(lim

    0

    1lim

    0

    h

    h

    h,

    0h , 1lim)0()0(

    lim00

    h

    h

    h

    fhf

    hh,

    h

    fhf

    h

    )0()0(lim

    0

    .

    O

    y

    x

    y=|x|

    11

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    24

    1. tttx 2)( , t .

    ) :

    (i) ]2,0[ (ii) ]1,0[ (iii) ]5,0,0[ (iv) ]1,0,0[ .

    ) 0t .

    ) )(txx .

    ) )0,0(O

    ().

    ,

    )(txx 0t .

    )

    i) m/s32

    6

    2

    )0()2(

    xx ii) m/s2

    1

    2

    1

    )0()1(

    xx

    iii) m/s5,15,0

    )0()5,0(

    xx iv) m/s1,1

    1,0

    )0()1,0(

    xx

    ) 0t ,

    m/s1)1(limlim)0()0(

    lim0

    2

    00

    h

    h

    hh

    h

    xhx

    hhh.

    )

    t )(tx ,

    4

    1

    2

    1)(

    2

    2

    ttttx ,

    ,

    4

    1,

    2

    1

    2

    1t . ,

    .

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    25

    -1 1 2

    1

    2

    3

    4

    5

    (2,6)

    t

    x

    6

    x=t()

    x=1,1t

    x=1,5t

    x=2t

    x=t2+t

    x=3t

    ) )0,0(O

    3, 2, 1,5 1,1, tx 3 ,tx 2 , tx 5,1 tx 1,1 .

    .

    0t

    0t , 1.

    , tx , .

    2.x

    xf3

    )( .(i) )3(f .(ii) f

    ))3(,3( f .

    (i) h

    hh

    hh

    fhf

    33

    3333

    33)3()3( 0h

    hhh

    h

    h

    h

    h

    h

    fhf

    3

    1

    )3(

    3)3()3( .

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    26

    )3(f31

    31lim)3()3(lim

    00

    hhfhf

    hh.

    (ii) f

    3x )3(f . ,

    xy 3

    1.

    )1,3())3(,3( f ,

    33

    11

    11

    2 .

    , 23

    1 xy .

    1.

    i) 13)( xxf 3x

    ii) 5)(2 xxg 2x

    iii) xxx 2)( 2 4x

    2. 1

    1)(

    ttf 1t .

    3. i) L r rL 2 . L r, 3r .

    ii) r 2rE . r, 2r .

    y x 1

    32

    yx

    3

    x

    y

    O

    3

    1

    654321

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    27

    4. i) x x 5x .ii) x

    x, 10x .

    5. :

    i)2

    )( xxf ))3(,3( fA

    ii) xxf 2)( , ))4(,4( fA .

    1.3

    f , Ax f . ,

    Bx h

    xfhxfxf

    h

    )()(lim)(

    0

    .

    () (derivative) f f ., 23)( xxf , :

    )2(3)2(33)(3)()(22222 hxhxhxhxxhxxfhxf ,

    0h

    hxh

    hxh

    h

    xfhxf36

    )2(3)()(

    .

    , xhxxfh

    6)36(lim)(0

    .

    , f 0x

    . ,

    23)( xxf 40 x xxf 6)(

    40 x , 2446)4( f .

    f f

    f .

    )(tx t,

    )()( txt .

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    28

    , t,

    )()( tt )()( txt .

    f

    h

    xfhxfxf

    h

    )()(lim)(

    0

    .

    .

    cxf )(

    0)()( ccxfhxf

    0h ,

    0)()(

    h

    xfhxf,

    0

    )()(

    lim0

    h

    xfhxf

    h .

    0)( c . xxf )( hxhxxfhxf )()()( , 0h ,

    1)()(

    h

    h

    h

    xfhxf.

    11lim)()(

    lim00

    hh h

    xfhxf.

    1)( x . xxf )( 2)( xxf .

    hhxxhxhxxhxxfhxf )2(2)()()(22222 ,

    O

    y

    c

    x

    y=c

    O

    y

    x

    y=0

    () ()

    O

    y

    x

    y=x

    O

    y

    x

    y= 1

    ()

    ()

    12

    13

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    29

    0h , hxh

    hhxh

    xfhxf 2)2()()( .

    , xhxh

    xfhxf

    hh2)2(lim

    )()(lim

    00

    .

    xx 2)(2

    O x

    y

    y=x2

    O

    y

    x

    y=2x

    () ()

    1)(

    xx , ..

    2

    2111 11)(

    1

    xxxx

    x

    ,

    3

    3122

    2

    222)(

    1

    xxxx

    x

    x

    xxxx2

    1

    2

    1

    2

    12

    11

    2

    1

    2

    1

    .

    1

    )( xx , .

    x x .

    xxf )( ( 15).

    )(xf 0xx

    f ))(,( 00 xfx ,

    f .

    f

    x .

    14

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    30

    y

    y=x

    y=(x)

    x

    x

    y

    2

    2

    /2

    45o135

    o135

    o

    45o

    , xxf )(

    xx )( . xxg )(

    xx )( . xe xln , e,

    xx ee )( xx 1)(ln .

    )(xcf )()( xcfxF .

    ))()(()()()()( xfhxfcxcfhxcfxFhxF , 0h

    hxfhxfc

    hxfhxfc

    hxFhxF )()()()(()()( .

    )()()(

    lim)()(

    lim00

    xfch

    xfhxfc

    h

    xFhxF

    hh

    .

    )())(( xfcxfc .

    15

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    31

    4455

    1553)(3)3( xxxx ,

    22

    414

    14

    4

    xxxx

    xx

    xx61

    6)(ln6)ln6( .

    )()( xgxf )()()( xgxfxF .

    ))()(())()(()()( xgxfhxghxfxFhxF

    ))()(())()(( xghxgxfhxf ,

    0h ,h

    xghxg

    h

    xfhxf

    h

    xFhxF )()()()()()(

    .

    ).()()()(

    lim)()(

    lim)()(

    lim000

    xgxfh

    xghxg

    h

    xfhxf

    h

    xFhxF

    hhh

    )()())()(( xgxfxgxf 34)3()()3(

    344 xxxxx

    2

    22 112

    1)()(

    1

    xx

    xxx

    xxx

    .

    )()( xgxf )(

    )(

    xg

    xf

    :

    )()()()())()(( xgxfxgxfxgxf

    2))((

    )()()()(

    )(

    )(

    xg

    xgxfxgxf

    xg

    xf

    .

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    32

    xxxxxxxxx )()()(

    34

    2

    4

    22

    22

    22

    2

    )2(222

    )(

    )()1()1(1

    x

    x

    x

    xx

    x

    xxx

    x

    xxxx

    x

    x

    x , x, x,x

    e xln . , ,

    23

    )1( x 33 )1( x

    )1(6)1(3)1(3))1)(1(())1((3232323323 xxxxxxxxx

    23222333232333)1(93)1()1)(1(6])1()1[())1(( xxxxxxxxxx

    1)(

    2

    xxF ; )(xF xxf )(

    x 1)(2 xxg . , , ))((1)( 2 xgfxxF .

    F g f.

    :

    )())(())(( xgxgfxgf . ))(( xgf ,

    f g(x)

    g.

    ,

    23222332333)1(93)1(3))1()1(3))1(( xxxxxxx .

    , , x

    x2

    1

    , :

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    33

    12

    12

    1)1(12

    1122

    2

    2

    2

    x

    xxx

    xx

    x .

    , )12(2)12()12())12(( xxxx .

    .

    0)( c

    1)( x 1

    )(

    xx

    xx

    2

    1)(

    xx )(

    xx )( xx

    ee )(

    xnx

    1)(

    )())(( xfcxcf

    )()())()(( xgxfxgxf

    )()()()())()(( xgxfxgxfxgxf

    2))((

    )()()()(

    )(

    )(

    xg

    xgxfxgxf

    xg

    xf

    )())(())(( xgxgfxgf

    1. i) xxf )( ii) xxf 3)( 2 .

    i)

    )()( xxf

    xx

    xx

    x

    xxxx

    x

    x22

    22

    2

    1

    )()(

    .

    x

    x2

    1)( .

    ii)

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    34

    )3(332)3(32])3[()(2

    xxxxxxxf xxxx 63)32(33323 ,

    22 .

    2. , ttttxx 96)(

    23 , t x .

    i) t.ii) 2 s 4 s;iii) () ;iv)

    ;

    v) 5 s.

    i)

    9123)96()()(223 ttttttxt .

    ii)

    st 2 m/s3921223)2( 2

    st 4 m/s9941243)4( 2 .

    iii) , 0)( t ,

    091232 tt

    0342 tt

    1t 3t .

    , 1 s 3 s.

    iv) , 0)( t ,

    091232 tt

    0342 tt

    0)3)(1( tt

    1t 3t .

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    35

    , 1t 3t ( )31 t .

    :

    x=x(t)

    t=0

    t=3

    t=1

    40

    v) :

    m4|04||)0()1(|1 xxS .

    1t 3t m4|40||)1()3(|2 xxS

    3t 5t m20|020||)3()5(|3 xxS

    , S 5s

    m282044321 SSSS .

    ( 1-18)

    1. i) 5)( xf ii) 4)( xxf iii) 9)( xxf .

    2. i)2/3

    )( xxf ii) 3)( xxf iii) 5)( xxf .

    3. i) 3)( xxf ii) 5 2)( xxf , 0x .

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    36

    4. i)x

    xf 1)( ii) 31)( xxf iii)

    5 21)(x

    xf , 0x .

    5. i)3

    4)( xxf ii) 56)( xxf iii) 205

    2)( xxf .

    6. i)4

    6)(

    xxf

    ii) xxxf 6)( .

    7. i)24

    3)( xxxf ii)x

    xxf 35)(2 iii)

    xxxxf 12)(

    2

    .

    8. i) 58)( 3 xxxf ii) )(86)( 2 xxxxf .

    9. i) )1)(1()(43 xxxf ii) )1()( xxxf .

    10. i) )1)(1(3)( xxxxxf ii) xxxxxf 34)( 22 .

    11. i)1

    )(2

    x

    xxf ii)

    x

    xxf

    )( iii)

    x

    xxxf

    1

    )(

    .

    12. i)x

    xf1

    1)(

    ii)

    2)1(

    3)(

    xxf .

    13. i)5

    )1()( xxf ii) 5)12()( xxf iii) 52 )32()( xxxf .

    14. i) xxf3)( ii) 3)( xxf iii) xxxf 4)( iv) xxf 3)( .

    15. i) xxxf 22)( ii) xxf 1)( .

    16. i)xexf 3)( ii)

    2

    )(xexf iii) axexf )( iv)

    xx

    x

    ee

    exf

    )( .

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    37

    17. i) xxf 2ln)( ii) 31ln)( xxf .

    iii) )ln()( xxf iv) 1ln)( xxf .

    18. i)x

    xxf

    ln)( ii) xexf x ln)( .

    19. i)

    1

    )(2

    3

    t

    ttf ))3(,3( fA .

    ii)

    f

    )(

    3,

    3

    f

    .

    20. t

    2

    )2(4

    11)( ttB ,

    8t . : i) t ii) 1, 2 8 .

    21. )3,0(A )0,(xB x

    10x .

    22. )1()( xxxf ))0(,0( fO

    xx 060 .

    1. 1

    3)(

    x

    xxf

    53 xy ;

    2.

    O

    x

    y

    B(x,0)

    A(0,3)

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    38

    496)(23

    xxxxf xx .

    3.

    1)(

    x

    xxf yx0 .

    4. t

    ttttx 23 2)( .

    t .

    ;

    5. xBxAxf )( , 0)()( 2 xfxf .

    6. pxpx eexf )( , )()( 2 xfpxf .

    7. xexf )( , 0)(4)(3)( xfxfxf .

    8.

    xxxf 2)( 3x .

    9. P

    I

    IIP

    )( , 0I , , . i)

    )(IP , , ,

    )0(P . ii) 2)(11

    )( IP

    IP .

    10. tAty )( , t , .

    i)

    t.

    ii)y.

    iii), 0,

    .

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    39

    1.4

    . ,

    , .

    .

    .

    , ,

    m/s200 . 2

    m/s10g , , , h

    t

    2520)( ttth tt 205 2 . (1)

    h(t) :

    t 0t 4t .

    22

    t

    20)2( h .

    2t 2t .

    2010)( tth ]4,0[ .

    0)( th , 2

    , ,

    2 , .

    O

    h

    t

    20

    2 4

    16

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    40

    t 0 2 4

    )(th 0

    .

    :

    f

    0)( xf , f

    .

    f

    0)( xf , f

    .

    2t h, h , 2t h .

    :

    f 0)( 0 xf ),(0 x , 0)( xf ),( 0x 0)( xf ),( 0 x , f

    ),( 0xx .

    f 0)( 0 xf ),(0 x , 0)( xf ),( 0x 0)( xf ),( 0 x , f

    ),( 0xx .

    O

    y

    x

    f(x0)=0

    f(x)>0 f(x)

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    41

    1. 0,)( 2 xxxf .

    :

    xxxxf 2)()( 2 .

    xxxf

    2020)( .

    xxxf

    2020)( .

    , 0 , 0)( xf

    x

    2 0)( xf

    x

    2 .

    x

    2

    )(xf 0 0 , 0)( xf

    x

    2 0)( xf

    x

    2 .

    x

    2

    )(xf 0 , 0,)( 2 xxxf

    x

    2

    0 0 .

    f

    4

    4

    222

    22

    .

    2. 1000

    2100

    10001000)(

    t

    ttp ,

    t.

    ;

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    42

    22

    2

    22

    2

    )100(

    )100(1000

    )100(

    21000)100(1000)(

    t

    t

    t

    ttttp

    .

    10100)100(10000)( 2 ttttp .

    t, 10t .

    010001000)100(

    )100(10000)(

    22

    22

    2

    tt

    t

    ttp

    10100)10)(10( ttt .

    0)( tp 100 t .

    0)10( p , 0)( tp (0, 10) 0)( tp , (10, ).

    10

    1050200

    1010001000

    10100

    1010001000)10(

    2

    p .

    f, 0xx .

    , 0x ,

    0xx 0xx .

    f ,

    . f ,

    , f .

    , 0)( 0

    xf .

    f, 0xx .

    , x 0x ,

    0xx 0xx .

    f ,

    O x0 x

    y

    y=f(x)

    O

    x0x

    y

    y=f(x)

    18

    19

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    43

    . , f , f ., 0)( 0 xf .

    :

    f 0)( 0 xf 0)( 0 xf , f () 0xx .

    f 0)( 0 xf 0)( 0 xf , f () 0xx .

    f 0)( 0 xf 0)( 0 xf ,

    2

    f.

    1. . 30 .

    .

    x. x2 . y ,:

    3022

    12 xyxy

    xxy 2302 (1)

    ,

    .

    ,

    222

    2230

    2

    1)230(

    2

    12 x

    xxxxxxxyE

    ,

    BA

    yy

    2x

    xx

  • 8/3/2019 Mathhmatika Kai Stoixeia Statistikhs-Biblio Mathiti

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    44

    22

    230)( xxxE

    .

    xxE )4(30)(

    xxE

    4

    300)( .

    0)4()( xE ,

    x

    4

    30

    . x (1)

    y

    4

    60

    4

    30)2(302

    4

    30

    y .

    m2,44

    30

    yx , .

    2. P t:

    3,250

    20020)(2

    ttt

    tP .

    .

    t

    ttP 2

    25020)(

    2.

    512502250

    0)( 32

    tttt

    tP .

    5t .

    02500

    202500

    202250

    20)(332

    ttt

    ttP , 3t .

    5t

    250012520)2550200(20)5( P .

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    45

    1.

    i) xxxf 2)(2 ii) 63)( 2 xxf iii) 42)( 2 xxxf .

    2.

    i) 56)(23 xxxf ii) 13)( 3 xxxf .

    3.

    i)3

    2)( xxf ii) 16)( 3 xxf iii) 1033)( 23 xxxxf

    iv) 1153)(23 xxxxf .

    4. 40.

    .

    5. 2m100

    ;

    6.

    32 3dm .

    ,

    .

    7.

    12 2dm ,

    ;

    8. 32 xy

    .

    9. c

    c ,

    c .

    ;

    10. :

    10

    .

  • 8/3/2019 Mathhmatika Kai Stoixeia Statistikhs-Biblio Mathiti

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    46

    1. qrrp 100)ln1(100 , pq,

    q

    pr .

    2.

    xx

    1ln

    2, ,

    e

    x1

    .

    3. 60 cm ,

    ,

    .

    ,

    .

    4. 160002

    m

    .

    900 ./m

    600 ./m.

    ,

    .

    5.

    .

    6.

    .

    ,

    .

    7. ,

    ty

    tktk eekk

    Aty 21

    12

    )(

    , , 1k 2k

    12 kk . t

    .

    60

    60

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    47

    8. km/h,

    30001,06 .

    i)

    1000km .

    ii)

    ,

    1000km.

    ii)

    .

    9. 450.

    ;

    10. 20

    . 40 km/h,

    20 km/h. 10 km,

    ;

    1. ,

    m

    (kgr/m).

    x

    )(xfm , x

    h

    xfhxf

    h

    )()(lim

    0

    , .

    x+hx

    xxfm )( , x

    ,

    i) ]21,1,1[

    ii) 1x .

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    48

    2. C x v :

    32353)( vvxxxC .

    v16 .

    , .

    3.

    123)( 23 xxxxf

    ;

    4. ),( A 1.

    0x0y p q .

    qp 2)( .

    5. 20 cm ;

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    .

    7. R .

    .

    R

    R

    O

    8. )(xC x , C ,

    x

    xCxc

    )()(

    h

    xChxC

    h

    )()(lim

    0

    .

  • 8/3/2019 Mathhmatika Kai Stoixeia Statistikhs-Biblio Mathiti

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    49

    ) x, :

    =.

    ) () x

    260021000

    1)( 2 xxxC .

    i) ,

    1000 , 2000 3000 .

    ii)

    ;

    9. x , )(xp .

    x ,

    )()( xpxxR . R

    R . x

    )()()( xCxRxP . P

    P .)x,

    .

    )

    , 2

    001,053800)( xxxC

    xxp 01,050)( ;

    10. 1 2 .

    Fermat,

    .

    i)

    2

    21

    2

    1

    21

    2)(

    )(

    dxd

    dxxt

    ii) )(xt .

    iii) 2

    1

    .

    d2

    d1

    a

    d

    x d-x

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    50

    1.

    )3)(2()( xxxxV .

    :

    . ),0[ . )2,0( . ]0,( . ]3,2[ .

    2.

    . x .2

    x . 12 xx . 21 xx .

    3.

    63.

    . 8 . 2 . -6 . 10.

    4.

    22

    6)(

    xxf

    .

    x

    22

    62

    x

    :

    . 2x . 11 x . 22 x . 2x .

    5.

    xxf )( 2

    1)( xg .

    :

    .2

    .

    3

    .

    3

    2 .

    6

    x

    3x

    2x

    O

    B

    A

    x x

    y

    y=x2

    y=x1

    x

    y

    B(k,4)A(1,4)

    (k,-3)(1,-3)

    O x

    y

    3

    2

    1

    321-1-2-3

    1 2/ B

    x

    y

    A

    O

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    51

    6. f .

    )()( 2 xfxf :

    . 2 . 3 . 4 . 5 . 6

    7. f g .

    )2()2( gf :

    . 5 . 4 . 3 . 2

    8. y

    f 4 .

    :

    . 1 . 1

    . 2 . 12 .

    9.

    ]3,0[/2)(2 Axxxf

    f .

    10.

    .

    x

    y

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    O x

    y=g(x)

    y=f(x)

    y

    4321

    1

    2

    3

    4

    y=f(x)

    (1,-1) x

    y

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

    O x

    y

    3

    2

    1

    -1

    321-1

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    52

    i) ii)

    x

    y

    y

    x

    x

    2

    x

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    x

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    0lim

    O

    yx

    x

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    y

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    x

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    0lim

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    x

    y

    y

    x x

    x

    | |

    2

    x

    xx

    x

    ||lim

    0

    yx

    1

    1 2( )

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    21 )1(

    1lim

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    1

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    x

    y

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    3

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    1

    1

    12lim2

    xx

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    53

    11. i) 2

    3)( xxf 12)( f , ;

    ii) x

    xf1

    )( 9

    1)( f , ;

    iii) xxf )( 2

    3)( f , ;

    12. f g 4)3( f , 2)3( g , 6)3( f

    5)3( g 3x

    ) gf ) gf ) gf )g

    f

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    )3(h .

    14.

    .

    .

    O x

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

    O

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    y

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    y

    O

    x

    y

    () () () ()

  • 8/3/2019 Mathhmatika Kai Stoixeia Statistikhs-Biblio Mathiti

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    2

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  • 8/3/2019 Mathhmatika Kai Stoixeia Statistikhs-Biblio Mathiti

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    56

    , 11,

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    57

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    58

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    59

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    60

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    61

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    117

    ) .

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    5,3 yx , 46,1xs , 66,1ys

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    123

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    123

    :

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    y 10 8 3 4 1 6 5 12

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    .

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    125

    1. ( ) () ()

    1982-87 :

    1982 1983 1984 1985 1986 1987

    , x 15,6 15,2 14,7 14,3 13,4 12,8

    , y 10,8 11,1 11,5 12,1 12,8 13,2

    ) .

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    126

    , :

    1 2 3 4 5 6 7 8

    55 62 71 66 63 56 72 51

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    127

    3. 200 . ,

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    128

    :

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    5:12.1

    1956

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    4:27.3

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    : The World Almanac and Book of Facts, 1994.

    8. 2

    km 1960 1974 :

    , 63 64 64 64 66 65 65 66 66 67 67 67 67 68 68

    , 1960 61 62 63 64 65 66 67 68 69 1970 71 72 73 74

    ),

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    km . ;

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    129

    (: 1960 1x ).

    9. (...)

    1931-1964 .

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    130

    (

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    1

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    : =21 2x =3.311 x =2312

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    1995, 2000.

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    131

    1-10 () ().

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    132

    .

    15-24 .

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    133

    xy :

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