ΜΑΘΗΜΑΤΙΚΑ Γ΄ ΛΥΚΕΙΟΥ ΕΠΑΝΑΛΗΨΗ 2016

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Μιχάλης Μάγκος Επαναληπτικά Θέματα 2016 Πληροφορική Οικονομικό Θετικό Μαθηματικά Γ΄ Λυκείου

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Transcript of ΜΑΘΗΜΑΤΙΚΑ Γ΄ ΛΥΚΕΙΟΥ ΕΠΑΝΑΛΗΨΗ 2016

  • 2016

  • -

    - 1 -

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    - 2 -

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    - 3 -

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    - 4 -

    .

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    - 5 -

  • -

    - 6 -

    , ,

    . 1 !

    1 ( )

    .133: ()

    .141: ( )

    .142: ( )

    .143: ( + )

    .144:

    .149: ( , )

    .150: ( )

    .151: ( 1 -1)

    .152: +

    .153:

    .153-154: ( )

    .154: f - 1 ( f (x ) )=x, xA f ( f - 1 (y ) )=y, y f (A)

    .155: .

    .159:

    .160:

    .161:

    .162:

    .163:

    .165: 1

    .166:

    .167: : 0

    0

    0x x

    x x

    0

    0

    limP(x) = P(x )

    limP(x )P(x)

    Q(x) Q(x )

    .169: ( )

    .170: ( )

    171: ( )

    .173:

    0 0x x u u

    limf g(x) = limf(u)

    .178: ( )

    .179:

    .183: (

    )

    .184:

    .185:

    .186: ()

    .188: ( x0 )

    .189: f .

    .

  • -

    - 7 -

    .190:

    .191: ( )

    .192: Bolzano (

    )

    .192: ( )

    .194: ( )

    .194: +

    .195: ( ) +

    .196: ( )

    .201-03:

    2 ( )

    .212: ( C f )

    .213: ( f x0 )

    .213: + ,

    .214:

    .217: ( )

    .218:

    .222: . f .

    .223: ( c ) =0 (x )=1

    .224: : (x ) = x - 1 , 1

    2 x

    .226:

    .229: ( )

    .230: ( )

    .230:

    .231: ( )

    .231: (x - )=-x - - 1

    .232: (x )=1

    2 x

    +

    .234: ()

    .234: (x )= x - 1 ( x ) = x ln

    .235: 1 ln x

    .235:

    .241: ( )

    .241-242: , -- .246: Rolle + .246: .. + .251: .251: .252:

  • -

    - 8 -

    .253: .254:

    .258: ( )

    .259: ( ) -

    .260:

    260-1: (Fermat)

    .261: ( - )

    262:

    .264:

    .264: ( )

    .273: (-)

    .274:

    .275: ( )

    .275:

    .276:

    .279: ( )

    .280:

    .281:

    .282:

    .283:

    .287:

    .295-9:

    3 ( )

    .303: ( )

    .304:

    .328:

    .329-330:

    .330:

    .332:

    .334:

    .334-5:

    .336:

    .337:

    .342-345:

    .346:

    .348:

    .354-9:

  • -

    - 9 -

    .

  • -

    - 10 -

  • -

    - 11 -

    , , , ,

    (

    ,

    )

    2 .226-227 , 3 . 247-248, .252, 2 .254-256 , 3 . 265-267 1 2 .346-347

    -

    - 6/148, 2/156

    + 2/176, 4/176, 3/182, 4/182, 1/187, 3/187, 4/187, 3/102, 1/285, 2/286

    2/199, 3/199, 6B/286

    Bolzano + . +

    4B/199, 5B/200,8B/200

    6/200, 4/257

    7/200

    9/200

    x0 3A/220, 2B/220, 4B/220, 6B/221, 7B/221,

    8B/221, 1B/228, 5B/286, 7/240

    2/228, 3/228, 4/228, 5/238, 7/239, 10/239, 11/239, 1/240,

    2/240, 3/240, 4/240, 6/240, 8/24011/241, 12/241

    12/239, 14/239

    1/244, 2/144, 4/244, 5/244

    .Rolle .. 3/249, 1/249, 3/250, 4/250, 5/250,

    6/250, 7/250

    + +

    1/256, 1/257, 11/293, 4/308, 1/308, 3/309, 4/309, 11/351

    2/257, 6/257, 2/291

    7/258, 8/258, 3/269

    6/256, 7/ 256, 5/257, 2/267,

    1/269, 2/269

    , 8/268, 4/269, 6/270, 8/270

    . Fermat 5/268, 5B/270, 7/292

    - 3/278, 4/278, 5/279, 8/292

    + +

    1/338, 7/339, 8/339, 9/339, 3/338, 4/338, 6/339, 11/340,

    12/340, 1/352, 2/352, 4/352, 7/353

    6/352

  • -

    - 12 -

    10/353

    1A/349, 2A/349, 3/349, 4A/349, 5/349, 1/349, 2B/349, 3/350, 4B/350, 5/350, 8/351, 9/351, 10/351, 12/351, 8/353,9/353

  • -

    - 13 -

    . -

  • -

    - 14 -

  • -

    - 15 -

    -

    1

    f (x )=2lnx g(x) = lnx2 .

    2

    .

    3

    x = x x = 0

    4

    0 0

    00

    ( ) ( )( ) lim

    h

    f x h f xf x

    h

    5

    f x0 , .

    6

    f

    7

    2

    0

    2 = 0

    8

    f R :

    () + () = (1)1

    0

    1

    0

    9

    f R, : (2 + 1) =1

    2 ()

    7

    1

    3

    0

    10

    f f(x) 0 [ , ] :

    () > 0 () < 0

    11

    f , g g f fg,

    12

    f , g , h h (g f ) , (h g) f h (g f )= (h g) f

  • -

    - 16 -

    14

    f : R 1 1 , x1 , x2 A : f (x1 ) = f (x2 ) , x1 = x2

    15

    f 1-1 y f (x) = y x.

    16

    f 1-1

    .

    17

    , 1 -1.

    18

    f .

    19

    f : R. 1 ( ) , f f x x x A

    20

    f : R. 1( ) = , ( ) f f y y y f A

    21

    f f -1 y = x.

    22

    f f - 1 y = x.

    23

    xf(x) = 10

    g (x) = logx.

    24

    1 -1 .

    25

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

  • -

    - 17 -

    26

    f : . 1( ) = , f f y y y A

    27

    0 0

    ( , ) ( , )x x l .

    : 0 0

    x x x x

    limf(x) l lim(f(x) l) 0

    .

    28

    0

    lim ( ) 0

    x x

    f x , f (x ) > 0 x0 .

    29

    0

    lim ( ) 0

    x x

    f x , f (x ) < 0 x0 .

    30

    f (x ) < 0 x0 0x x

    lim f(x) 0

    .

    31

    f x0 0

    lim ( ) 0

    x x

    f x ,

    0

    lim ( ) 0

    x x

    f x .

    32

    f g x0 :

    f (x )g(x ) x0 , 0 0

    lim ( ) lim ( )

    x x x x

    f x g x .

    33

    0

    lim( ( ) ( ))

    x x

    f x g x ,

    0

    lim ( )x x

    f x 0x x

    lim g(x)

    .

    34

    :. 0

    1lim 1

    x

    x

    x

    35

    : lim 1

    x

    x

    x

    36

    0

    lim ( )

    x x

    f x , f (x )>0 0 .

    37

    0

    lim ( )

    x x

    f x , f (x )

  • -

    - 18 -

    38

    0x x

    lim f(x)

    , 0

    lim ( )

    x x

    f x

    39

    0

    lim ( )

    x x

    f x , 0

    lim ( )

    x x

    f x

    40

    0

    lim ( )

    x x

    f x , 0

    1lim 0

    ( )

    x x f x

    41

    0

    lim ( ) 0

    x x

    f x f (x )>0 x0 , 0

    1lim

    ( )

    x x f x

    42

    0

    lim ( ) 0

    x x

    f x f (x ) 1 : lim

    x

    47

    11 1 0 (x)= ... , 0 : lim ( ) lim

    x x

    x x x a x x

    48

    0lim ln

    xx

    49

    0

    1lim ln

    x x

    50

    x 0

    7xlim

    x

    = 7.

  • -

    - 19 -

    51

    f () f .

    52

    f () f .

    53

    f

    ( ) 0f x x f ()>0 .

    54

    f f

    .

    55

    f [ , ] [ m , M ] m .

    56

    f [ , ] f () f () > 0 f ( , ) .

    57

    f [ , ]

    x0 ( , ) f (x0 ) = 0, f () f () < 0.

    58

    f x0 , x0 .

    59

    f x0 g f (x0 ) , g f

    x0 .

    60

    f x0

    g x0 , g f

    x0 .

    61

    f f .

    62

    f x0 , x0 .

  • -

    - 20 -

    63

    f x0 , x0 .

    64

    f Bolzano , f .

    65

    f x0 , f x0 .

    66

    f , g x0 ,

    f g x0

    0 0 0

    ( ) ( ) ( ) ( ) f g x f x g x .

    67

    f , g x0

    0

    ( ) 0g x , f

    g x0

    :

    0 0 0 0

    0

    0

    ( ) ( ) ( ) ( )( )

    ( )

    f x g x f x g xfx

    g g x .

    68

    0x 1

    ln xx

    .

    69

    : 1

    (7 ) 7

    x xx , xR.

    70

    f R , [ , ] , f Rol le .

    71

    f [0,1] ,

    fC ,

    0, (0) , 1, (1)f f .

    72

    f , f

    .

  • -

    - 21 -

    73

    2 .

    74

    f ( , ) x0 , f . f (x ) ( , x0 ) (x0 , ) , f (x0 ) f ( , ) .

    75

    f , .

    f , f (x ) < 0 x .

    76

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

    77

    f x

    . f ( ) 0 f x

    x .

    78

    f

    . f (x) 0 x ,

    f .

    79

    f , g .

    f , g f (x) g (x)

    x , f (x ) = g(x )

    x.

    80

    , f 0,

    f .

    81

    f [ , ] x0 [ , ] f . f (x0 ) = 0.

  • -

    - 22 -

    82

    f ( , ) , x0 , f . f (x0 )>0 ( , x0 ) f (x0 )0 x , f .

    84

    f f (x ) > 0 x .

    85

    C f .

    86

    f , C f C f .

    87

    f ( , ) , 0 . f ( , x0 ) (x0 , ) , (x0 , f (x0 ) ) c f .

    88

    3

    23 2f(x)dx f( ) f( )

    89

    5

    5

    2 2

    17

    7dx ln x

    x

    90

    f , ,

    ( ) ( )( )a

    f ff x dx

    .

    91

    f [ , ] R ,

    ( ) ( ) f x dx f x dx

    .

    92

    f [ , ] ,

  • -

    - 23 -

    ( ) ( )( ) ( ) f ff x dx xf x dx

    .

    93

    f , g R, :

    ( ) ( )( ) ( ) ( ) ( ) f x g xf x g x dx f x g x dx

    .

    94

    f , ,

    :

    f(x)dx f(x)dx f(x)dx .

    95

    f [ , ] [ , ]

    f (x ) 0 ( ) 0 f x dx

    .

    96

    ( ) 0 f x dx

    , f (x ) 0

    x [ , ] .

    97

    f [ , ] . G f [ , ] ,

    ( ) ( ) ( ) f t dt G G

    .

    98

    f(x)g (x)dx f(x)g(x) f (x)g(x)dx ,

    f , g [ , ] .

    99

    f , g , g [ , ]

    , ( ) ( ) ( ) ( ) f x g x dx f x dx g x dx

    .

    100

    ( ) f x dx

    xx xx.

  • -

    - 24 -

  • -

    - 25 -

    .

  • -

    - 26 -

  • -

    - 27 -

  • -

    - 28 -

  • -

    - 29 -

    ..1. A1.A f ' x0

    ,

    f (x0 , f (x0 ) ) .

    2. , f '

    x0 ,

    .

    3 .

    .

    . f x0 , f

    x0 .

    . f x0 , f

    x0 .

    . f x0 , f

    x0 .

    4 .

    x0 .

    . f (x )=3x 3 , x0=1

    1. y=-2x+

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

    2 2. y=

    1 4

    x+1

    . f (x )=3 x , x 0=0

    3. y=9x-6

    . f (x )= x , x 0=4

    4. y=-9x+5

    5.

  • -

    - 30 -

    ..2. 1. Fermat.

    2. f ,

    ' 0f x .

    f .

    3 .

    .

    1. :f A

    , f .

    2. 0

    lim 0x x

    f x

    , f x 0x

    0

    lim 0x x

    f x

    .

    3. f 0f a f 0f x

    ,x a , f , . 4. f , g

    ' 'f x g x x ,

    .

    5. f [ , ] ,

    : ( ) ( )f x dx f x dx

    .

    6. f ,g

    ,x , a

    f x dx g x

    .

    ..3. 1. ,f g .

    ,f g

    ' 'f x g x ,

    c , x

    : f x g x c

    2. , 0, 1vf x x v

    : .

    3 . 0

    x

    F x f t dt , f

    .

    .

    :

    f x

    x

    f x g x

    x

    ,a

    f x g x

    1' vf x v x

    36 . .

  • -

    - 31 -

    .

    .

    . 10F

    ..4. 1. f 0x

    . f

    0x , :

    0' 0f x .

    2 . x

    f ;

    3 . f

    , ;

    4 .

    () () ;

    1. 0

    limx x

    f x l

    00

    lim

    h

    f x h l .

    2. 0 1a lim 0xx

    a

    .

    3. f , f

    f a f .

    4. f g

    ,a : ' 'a

    f x g x dx f x g x dx f x g x

    5. f x ,

    f x x f 1-1 .

    ..5. 1. f

    0x .

    2.

    )(, 00 xfxM f .

    3 .

    .

    ) f (x ) = ln(x2+1) [0 , +)

    0F

    4F

  • -

    - 32 -

    ) axg )( 0x axgxx

    )(lim0

    ( )y aim f y l

    0

    ( ( )) .x xim f g x l

    ) f [, ] f () , .0)(' f

    ) f , .0)('' xf

    ) f [2,5] 0)( xf

    [2,5] , .0)(

    2

    5

    dxxf

    ..6. 1. f ,

    , . :

    f ,

    f f

    , f f ,

    0 ,x a , 0f x .

    2 . 0x x

    f ;

    3 .

    Rolle .

    4 .

    () () .

    1. : 0

    1lim lnx x

    2. :

    xxim e .

    3. : : f A R : g B R ,

    f

    g, .

    4. f 0x

    , 0x .

    5. ' 'f x g x dx f x g x f x g x dx

    ', 'f g

    , .

    ..7. 1. , f

    ' x0 ,

  • -

    - 33 -

    .

    A2. f

    x0A;

    A3. f (x ) = x , >0

    R f (x ) = x ln.

    A4.

    :

    i . 1 -1

    .

    ii . f (x ) = e x+1 .

    iii . 0

    limx x

    f (x)>0, f (x)>0 x0 .

    iv. x , y

    y = f (x ) , f x0 , o

    y x x0

    y = f (x0 ) .

    v. f ,

    x0 , f (x0 )0 ,

    f .

    ..8. 1. f ,

    .

    . f (x)0 x ,

    f .

    . f (x)0 x ,

    f ;

    2 .

    .

    . f (x) =e1 - x

    .

    . f f (x) = -2x+2

    1

    x + 3, x

    2,)

    .

    . f (x) = g (x ) + 3 x,

    h(x)=f(x)-g(x ) .

  • -

    - 34 -

    ..9. A1. f [, ] .

    G f [,] , :

    ).()()( aGGdttf

    3. f .

    f x0A ;

    2.

    f -2,6 .

    f

    ..10. A1. f(x )= , >0

    R xR f (x ) = ln .

    A2. f, .

    f .

    A3. ,

    , , ,

    .

    . f (x )=logx, x>0 g(x)=10 x .

    . f

    x 0 A () f (x 0 ) , f (x ) f (x 0 ) xA

    -2 1 3 6x

    y

  • -

    - 35 -

    . f ,

    1 -1 .

    . 0

    lim ( ) 0x x

    f x

    f (x )>0 x0 , : 0

    1lim

    ( )x x f x .

    . f x 0

    .

    ..11. A1. f x 0

    . f

    x 0 ,

    : f (x 0 ) = 0.

    A2. f . y=x+

    f +;

    A3. ,

    , , ,

    .

    ) 1

    0( ) (1) (0)f x dx f f , f [0 , 1]

    ) f:A 1 -1,

    x 1 , x2A : x 1x2 ,

    f (x1 ) f (x 2 )

    ) x 1= {x/x=0} : 21

    x x

    .

    ) :x

    xlim 1

    x .

    ) C C f f - 1

    y=x

    xOy xOy.

    ..12. A1. f . f (x ) > 0 x , f .

  • -

    - 36 -

    A2. f [, ] ; A3. f . f

    x 0A ; A4. , , , , , , . . , . . f 1 -1, y f(x)=y x .

    .

    0x x

    lim f(x)= , f (x )

  • -

    - 37 -

    3. Rolle .

    4. , , , , , , . . - f , xx, f . . x>0 lnx x + 1> 0.

    . 0

  • -

    - 38 -

    2. f , g

    : f (x)=g (x) x .

    c :

    f x g x c x .

    3.

    .

    ) f :A 1f ,

    f .

    ) f 0x ,

    .

    ) f

    , 0x , 0f ' x 0 . ) f

    . f '' x 0

    x .

    ) f , f x 0 ,

    .

    ) f

    , , f

    .

    ..17. A1. vf x x , v IN 0, 1 .

    f v 1f ' x v x .

    2 . f ,

    ,

    .

    0 0f x

    0f x x 0x

    0f x dx

    0 ,x a 0 0f x

    0f x dx

  • -

    - 39 -

    , 1 2 3I , I , I

    .

    3 .

    .

    1. x 0

    xlim

    x

    2. x 0

    1lim x

    x

    3. x 0lim ln x

    4. xx

    1lim

    e

    .

    . 0

    . 1

    .

    ..18. A1. f , g x0 ,

    f + g x0 :

    ( f + g ) (x0 ) = f (x0 ) + g (x0 ) .

    2 . f .

    f ;

    3 .

    .

    3

    10

    I f x dx

    3

    20

    'I f x dx

    3

    30

    ''I f x dx

  • -

    - 40 -

    1. f : R 1 - 1

    1 2x ,x 1 2x x 1 2f x f x .

    2. 0 0x x x x

    lim f x lim g x

    f x g x 0x .

    3. f ,

    , .

    4. f ,

    , 0f ' x 0 .

    5.

    f x dx 0 f

    , .

    ..19. 1. f

    0x . f

    0x ,

    : 0f ' x 0 .

    2.

    f ;

    3.

    .

    . f [2 , 5]

    .

    .

    .

    . 0x x

    lim f x

    f x 0 x

    0x .

    . f

    ,

    f()0 x .

    . 0f x dx

    0f x

    ,x a .

    ..20. A.1 : ( ln|x| ) = 1

    x.

    .2 f

    . f

    0 ,x a 0 0f x 0f a f

    0 ,x a

    , 0f x ,x a

    0x x

    0

    limx x

    f x g x

    0

    limx x

    f x

    0

    limx x

    g x

  • -

    - 41 -

    ;

    3 . ,

    .

    . f f () 0 x R

    . 1

    ln( ) , ( ,0) x xx

  • -

    - 42 -

    ..22. A1. f ' [, ] .

    G f [, ] ,

    f (t) dt G() G() .

    2 . f .

    f ;

    3 . ,

    .

    . f [, ]

    (, ] , f [, ] .

    . , 1-1 ,

    .

    . f x0 0x x

    lim f (x)

    =0,

    x x0

    lim f(x) 0 .

    . f R ,

    f (x)dx xf (x) xf (x)dx , f [,] .

    .

    ,

    .

    ..23. 1 . , f

    x0 , .

    2 . y =

    f + ;

  • -

    - 43 -

    3 . ,

    .

    . f ' (, ) ,

    x0 , f

    .

    f (x ) > 0 (, x0 ) f (x) < 0 (x0 , ), f (x0 )

    f .

    . f

    . f (x )>0

    x , f .

    . f ,

    [ , ] ,

    f (x)dx f (x) .

    . f ,

    f

    .

    . f x0

    . f x0

    f (x0 )=0, f

    x0 .

    ..24. 1 . f .

    F f , :

    . G(x) = F(x) c ,c R

    f

    . G f

    G(x) = F(x) c ,c R .

    2. f

    x 0 ;

  • -

    - 44 -

    3. ,

    .

    . 0x x

    lim f (x) l

    , 0x x

    lim f (x)

    0x x

    lim f (x) l

    . f , g x 0 ,

    f g x 0 :

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

    . f ,

    . f (x)>0 x , f

    .

    . f , g ,

    :

    f(x) g (x) dx f(x) g(x) f (x) g(x) dx .

    ..25. A1. : 1 , 0,2

    x xx

    .

    2 . ,

    .

    . f x 0

    , .

    . , ,

    (x0 , f (x 0 ) ) , C f f,

    x 0

    = f (x 0 ) .

    . f , g IR

    f og gof ,

    .

    . C C f f 1

    y = x

    xOy xOy.

    . f x 0 , 0 0

    kk

    x x x xlim f(x) lim f(x)

    ,

    f (x ) 0 x 0 , k k 2.

  • -

    - 45 -

    3 . f

    (, )

    [, ] .

    ..26. 1 . : 1 * * , x x x R .

    2 . y = x +

    f -;

    A3. ,

    .

    . f [, ] f () < 0

    (, ) f ( ) = 0, f() > 0.

    . 0x x

    lim f(x) g(x)

    0x x

    lim f(x)

    0x x

    lim g(x)

    . f f - 1

    f y = x,

    f - 1

    .

    . f R * .

    f R *

    f (x ) = 0 x R * ,

    f R * .

    . f

    , x

    x ,

    .

    ..27. 1 . (x ) :

    0

    0lim ( ) ( )x x

    P x P x

    .

  • -

    - 46 -

    2 . f: A IR 1 -1;

    A3. ,

    .

    . , f

    0,

    f .

    . f (, )

    x o . f (, x o )

    (x o , ) , (x o f (x o ) )

    f .

    . f [ , ] IR,

    :

    f(x)dx f(x)dx .

    . f , g fog gof,

    fog gof.

    . f (x ) = x, xR f ()=-.

    ..28. A1. f .

    F f , :

    :G(x)=F(x)+C, C R

    f G f

    : G(x)=F(x)+c, c R .

    2.

    .

    .

    f (x)dx = . . . . .

    f (x) g(x) dx = . . . . .

    f (x) g(x) dx = . . . .

    , R f ,g [,] .

    3 . :

    . 1

    x

    0e x dx

    .

    24

    1

    3x dx

    x

  • -

    - 47 -

    .

    2

    02x 3x dx

    ..29. A1. : ()= 1.

    2 . .

    3 . ,

    .

    . 0x x

    lim f(x)

    ,0x x

    lim f(x)

    + ,

    0x x

    f .

    . f , g x g(x )0,

    f

    g x

    :

    o o o oo 2

    o

    f f(x )g (x ) f (x )g(x ) x

    g g(x )

    .

    . x0 1

    ln x x

    .

    . f:R 11,

    y f(x)=y

    x.

    . f [,]. G

    f [, ] ,

    f(t)dt G() G() .

    ..30. A1. f ( , ) , x0 , f . :

    f () ( , x0 )(0 , ) , f (x0 ) f ( , ) . 2. R.

  • -

    - 48 -

    ;

    3 .

    ,

    , , .

    1. f x 0 .

    f (x )0 x . 0x x

    lim f(x)

    0x x

    1lim

    f(x) .

    2. x 0, 2x 0

    1lim

    x .

    3. f

    x 0 , x 0 .

    4. f(x) x = [0, +),

    1f (x)

    x x (0, +).

    5. f , g .

    f , g f (x ) = g (x)

    x , c , x

    : f (x ) = g(x ) + c.

    ..31. A.1 f

    x0, .

    .2 f ;

    . ,

    .

    . f ()

    f .

    . f , g, g [, ] ,

    f(x)g'(x)dx f(x)dx g'(x)dx .

  • -

    - 49 -

    . f

    ,

    /

    f(t)dt f() - f()

    x.

    . f

    (, ) ,

    (,) = x lim f x

    =

    x lim f x

    .

    . f x0

    R, :

    o ox x x x

    lim k f(x) k lim f(x)

    k R .

    ..32. 1. f x x , {0,1} .

    f R

    1f x x .

    A2. N f

    .

    A3.

    ( ) ,

    , () , .

    1. f x 0 g

    x 0 , gof x 0 .

    2. f 1 -1,

    ( xx)

    .

    3. f x0

    R 0x x

    lim f x 0

    ,

    f (x )

  • -

    - 50 -

    ..33. 1. : f , g

    x0

    , f + g

    x0

    : ( f + g) (x0) = f (x

    0) + g (x

    0) .

    2. f g ;

    3 .

    ( ) ,

    , () , .

    1. f [, ]

    x [, ] f (x ) 0

    f(x)dx 0 .

    2. f ,

    xx, f .

    3. f, g, h h (g f ) ,

    (h g) f h (g f ) = (h g) f .

    4.

    2 .

    5. > 1 xxlim 0

    .

    ..34. A1.

    .

    A2. f

    [, ] ;

    3 . ,

    , , , ,

    .

  • -

    - 51 -

    . f:A IR 11,

    f - 1

    : 1f (f (x)) x , x A

    1f (f (y)) y , y f (A) .

    . f

    f

    .

    . f

    x .

    f f(x) > 0

    x .

    . f IR

    ,

    f ( x ) > 0 x.

    . f ,,

    f(x)dx f(x)dx f(x)dx .

    ..35. A1. [, ] .

    G f [, ] ,

    f(t)dt G() - G() .

    2 . f .

    3 . ,

    , , ,

    .

    . 11,

    .

    . f ,

    f

    ,

    .

    .

    f(x)dx

    xx

  • -

    - 52 -

    xx.

    . f

    , x

    x, .

    .

    (, x 0 ) (x0 , ) .

    : 0 0x x x x

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

    .

    ..36. 1. f .

    f x

    f (x ) = 0 , f

    .

    2 . f x0

    ;

    3 . ,

    , , ,

    .

    . f 1 -1,

    f .

    . f

    () x0

    A, f (x)f(x0) xA.

    . x 0

    x 1lim 1

    x

    .

    . f

    .

    . f [, ]

    f (x )

  • -

    - 53 -

    ..37. A1. f .

    F f , :

    G(x) F(x) c, c

    f

    G f

    G(x) F(x) c, c .

    A2. .

    A3. f

    .

    f

    ;

    4. ,

    , , ,

    .

    ) 0

    limx

    x=0

    x

    ) f

    . f ,

    .

    ) f

    (, ) ,

    (,), x

    A lim f (x)

    x

    B lim f (x)

    ) (x)= x, x

    ) 0xx

    lim f (x) 0

    , f (x) 0 x 0

    ..38. 1. , f

    x 0 , .

    2. f (x0 , f (x0 ) ) C f . C f ;

  • -

    - 54 -

    3. (5) , . .,

    , , ,

    .

    . f

    C f

    .

    . f

    c, : cf (x) f (x) , x.

    . f(x ) = x , > 0, ( x ) =x x 1 .

    . f

    [, ] [m, M],

    m .

    . 0x x

    lim f (x)

    , 0x x

    1lim 0

    f (x) .

    ..39. 1. f,

    [, ] . f [, ]

    f ()f() , f () f ()

    x0

    (, ) f (x0)= .

    2.

    .

    3. ,

    , , ,

    .

    ) : x

    xlim 1

    x .

    ) f f

    C f , xx,

    , xx,

    C f , xx.

  • -

    - 55 -

    ) f , g x o , f (x )g(x)

    x o , : 0 0x x x x

    lim f (x) lim g(x)

    .

    ) f , g x o g(x o )0,

    f

    g x o

    :

    0 0 0 0

    0

    0

    x x x xx

    x

    2

    f g f gf

    g g.

    ) P(x) , Q(x) .

    P(x)

    Q(x), P(x)

    ,

    .

    ..40. A1.

    f f - 1 y = x.

    A2. Bolzano

    .

    A3. f A. f

    x o () , f (x o ) ;

    A4. ,

    , , ,

    .

    ) f

    . f

    , f (x ) > 0 .

    ) ox x

    lim f (x)

    , ox x

    1lim 0

    f (x)

    ) f () ,

    .

  • -

    - 56 -

    ) f

    , , , :

    f (x)dx f (x)dx f (x)dx

    ) f

    . f

    ,

    .

  • -

    - 57 -

  • -

    - 58 -

  • -

    - 59 -

    ..1. f [0,] ,

    0f(x)dx 2 F f .

    1. F(0) - F () .

    2. (0,) f ( ) = .

    3. 0

    ( ) 1x

    f t dt x

    (0 , ) .

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

  • -

    - 60 -

    2. f

    C f (0 , f (0) ) .

    B3. f (x )+ 2x 2 xR.

    B4. : ln ( ) ( ) 1f x f x .

    ..7. f

    [0,4] , f (0) = 5 f (4) = 1.

    1. f .

    2. f (x ) = ,

    [1 , 5] .

    3. (0 , 4) :

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

    6

    f f f

    f .

    ..8. f ,g (0,+ )

    x > 0 : g xf x e f xg x e .

    f (1) = g(1) = 0:

    1. f , g .

    2. h (x) = e - f ( x ) x , x>0

    f .

    3. : ( )

    limx

    f x x

    x

    ,

    0

    ( )lim

    x

    f x x

    x

    .

    ..9. f R , : 2016

    40

    0f (x) [f(x)] dx = 0 .

    1. f (x ) = 0 (0 , 2016).

    2. [0 , 1008 ]

    f ( ) = f (1008 + ) .

    ..10.

    f :R R (2 , 5) , ( -1 , 3).

    1. f .

    2. f .

    3. : f(2x 1) f(5) f ( f (x ) ) = f (5) . .

    4. : f - 1 (5) , f - 1 (3) .

    5. : f (3+f - 1 (x+1) ) = 5.

    6. C f (9 , 9)

    f - 1 .

  • -

    - 61 -

    ..11. : 31f(x) x x2

    .

    1. f

    1f .

    2. : 1 .( ) 64 f x 3. f - 1 ,

    : 1f (1).

    B4. :

    1

    6

    ( ) 64lim

    6x

    f x

    x

    .

    ..12. f f(x y) f(x) f(y) 2xy

    x , y R

    x 0

    f (x)4

    xlim

    .

    1. f 0.

    2. f .

    3. f .

    ..13. : 4

    f(x) 2x , x 0.x

    1. ( ) f ,

    xx x = , x = +1, >0,

    ( ) 2 1 4ln( 1) 4ln .

    2. ()

    .

    3. : lim ( )

    .

    ..14. . :

    i ) 1 0

    lim tx

    x tdt

    e

    ii ) 0

    lim tx

    x tdt

    e

    . :

    i ) F(x)=2

    1

    2

    x x t

    dtt

    .

    ii ) G(x) =2 3

    2

    3 4

    xdtt

    .

    iii ) 12

    3x H(x) dt1

    lnt.

  • -

    - 62 -

    ..15. f g

    R : f (x ) g (x) = x xR.

    , g < 0 < :

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

    2. f (x ) = 1 ( , ) .

    ..16. . f :RR

    f (x)

    f (x)e dx 0, , R < .

    :

    f (x) = 0 (,) .

    . f R

    : 3( ) ( ) 1 xf x f x e xR .

    : f ( lnx) = f (1 x ) x > 0.

    . f R

    () f

    : ( , f () ) , ( , f () ) ( , f () ) .

    x0R f (x0 ) = 0.

    ..17. . : [ , ] ,f

    [,] , (,) f () = 2 , f () = 2 .

    i ) f (x )=2x

    ( , ) . ii ) 1 , 2 ( , ) :

    f (1 ) f (2 )=4.

    . : [0,1] (0, )f

    f (0)=1, f (1)=2 : =2

    1

    0

    f (x)dx

    f (x) f(x)

    .

    ..18. . : e

    f(x) lnx xx

    .

    i ) f .

    ii ) : 1f (x) x .

    B. f :[,] R f (x ) > 0

    f (x)

    2016f(x)f(x)

    .

  • -

    - 63 -

    f () = 2016f () I =

    f (x )dx .

    ..19. :21( )

    2

    xf x e x .

    B1. ( , f ( ) ) ,

    (1 , 2) , C f , C f ( , f ( ) ) ,

    : x + 2y = 1.

    B2. f R.

    B3. R g ,

    : ( ) ( ) ( )f x g x f x xR.

    g(0)=0 g.

    ..20. f : R R R 3f (x) f (x) x 5 0

    x R .

    B1. f

    .

    B2. : 1lim ( )xx

    e f x

    .

    B3. f R

    f (x ) = 0.

    4. C f

    (3 , f (3)) .

    B5. (3 , 5) , : 1

    ( )2

    f .

    ..21. : f (x ) = x + xe - x . 1.

    fC

    (0, f (0))

    2x y + 7 = 0.

    2. = 1, f

    y = x fC .

    3. () fC

    , y = x x = 0 , x = > 0 , = 1.

    4. :

    E()lim .

    ..22. : f (x ) = x e- x

    + x . 1. R , c f (0 , f (0) )

    ( ) : 2y - 4x - 5 = 0 .

    2. f

    .

    3.

  • -

    - 64 -

    c f .

    4. f .

    ..23. f [1 , 3] .

    1. f (1) = f (3) , x1 , x2

    1

  • -

    - 65 -

    f x f xf(x)

    1 1

    2.

    ..28. f R ,

    f (x )2 xR.

    2

    25

    0( ) 5 1 ( ) , .

    x xg x x x f t dt x R

    :

    . g(-3) g(0) < 0.

    . g(x ) = 0 ( -3 , 0) .

    4 1997

    ..29. h: [1 , + ) R

    : h(x) (x ) F(x)1999 1 1, F

    ( )

    ( )h x

    f xx

    h(1)=0.

    B1. h.

    B2. h [1 , + ) .

    ..30. f R .

    2 2 2 4

    1

    0( ) ( ) 2 ( )

    I x f t xt f t x t dt , R

    x0 = 2

    1

    05 ( ) t f t dt.

    1 2000

    ..31. f f (x ) = x 2 lnx , x>0.

    1 f .

    2. 1

    ( )

    x

    f x x x , x > 0.

    3.

    f , xx 1

    xe

    x = e .

    ..32. x

    x

    e 1f x , x R

    e 1

    .

    . f

    1f .

  • -

    - 66 -

    . 1f (x ) = 0

    .

    . 1

    21

    2

    f x dx .

    2 2002

    ..33. f f (x )=x 2 lnx .

    . f,

    .

    . f

    .

    . f .

    2 2004

    ..34. f: IR IR f (x ) = 2 x + m x 4 x 5 x ,

    m IR , m > 0.

    . m f (x ) 0 x IR .

    . m = 10,

    f, xx

    x = 0 x = 1.

    2 2004

    ..35. f (x ) =2+(x -2)2

    x 2.

    . f 1 -1.

    . f - 1

    f

    .

    . i .

    f f - 1

    y = x .

    i i .

    f f - 1 .

    2 2006

  • -

    - 67 -

    ..36.

    x

    x 1

    1 ef(x)

    1 e

    , x IR .

    . f IR .

    . 1

    dxf(x)

    .

    . x

  • -

    - 68 -

    ..39. f, R . A

    x0 xf(x )=x+2x, :

    B1. f (0) .

    B2. f (x)

  • -

    - 69 -

    : x5y+2010=0

    B3. f 5

    2(+).

    ..43. f :RR f (1) = 1

    xR : e x f (x ) + e x f (x) + f (x) = 0

    1. f

    2. N 1

    0( ) f x dx

    ..44. f :RR R

    ( )( ) 0

    f x

    xf x

    e f (0)=1.

    B1. N f f (x ) > 0

    xR.

    B2. A (0, 2 )

    2

    2

    2 2 2ln(1 )

    2

    ee

    e

    B3. : 20

    2 2

    2

    xdx

    x e.

    ..45. f 0,4

    , F f

    2 2(0)

    4 32 2F F

    .

    B1. 0,4

    ,

    f ( ) = .

    B2. 2

    ( )lim

    ( )

    x

    x f x

    x .

  • -

    - 70 -

    ..46. : x

    2

    f (x) x xtdt 14 , , ,x 0.

    1. , f (x) f (2) fC

    M(2 , 6) .

    2. , 1.

    )

    .

    )

    .

    ) .

    ..47. f : [0 , 5]R f (0) = 0 ,

    f (3) = f (5) = 6.

    B1. x0 (0 , 5) , : 5 f (x0 ) 6 = 0.

    2. x1 (0 , 5) , : 5f (x1 ) - 2 = 0.

    ..48. ln x

    f x , 0x

    .

    1 . f

    (1, f 1 ) x y 0 , .

    B2. = 1:

    . f .

    . .

    . : 1

    1

    8 .

    2 2003

    ..49. f , g

    x .

    , 0

    0.

    1 . i ) L.

    ii )

    f g . 2 . g .

    3 . : x .

    2 2004

    ' ' 1, ' 1f x g x f x

    2lim

    2x

    g xL

    f x x

    4f x g x x

  • -

    - 71 -

    ..50. f x 2 x ln x 2 , x 0 .

    1. : ln x

    f ' x , x 0x

    .

    2. x 0lim f ' x

    .

    3. f

    .

    4.

    ln x

    g xx

    ,

    . 2 2005

    ..51. 2 ,xf x x a e x . f

    :

    1. : 2 .

    2. f .

    3. :

    i . xlim f x ii .

    xlim f x .

    4. 2007f x

    . 2 2007

    ..52. f

    , 0

    1 1, 0

    x xf x

    x x

    , .

    1 . , f .

    2 . , f

    0 0x .

    3 . f 1-1.

    4 . 1 2 , 2 f x dx

    .

    2 2008

    ..53. :f : 3 24 12 1f x x x x ,

    x R R 0 1x .

    1 . i . =1.

    ii . f .

    2 . :

    3xf x

    imf x

    .

    'x x

    1x

    e

    2x e

    2 2y x

    0, 0f

  • -

    - 72 -

    3 . i . f

    0, 1 . ii .

    f 'x x . 2 2011

    ..54. f (x ) = e x - 2 g(x) = lnx+2.

    B1. f g g f

    .

    B2. f f - 1 .

    B3. : 2 ln 2xe x ,

    (e - 2 , 2) .

    B4. :

    ( ) ( )

    lim lim 0.( ) ( )x x

    f x g x

    g f x f g x

    2 2012

    ..55. : f (x ) = - e 3 x x3 + 1.

    B1. f

    .

    2. :

    0

    ( ) 1lim

    ( )x

    f x

    f x

    .

    B3. ) f

    .

    ) 3 3 2015 1

    xe xe , .

    4. g (0 , +)R :

    33 ( ) 3 3 6( ) ln 2g xe g x x e x , x>0,

    g g(x)=lnx+2 .

    2016

    ..56. R f

    :

    f ()=e , f ()=e , f ()=e , ,,R

  • -

    - 73 -

    1. f (x) f (x) = e 2 x 2 ( , ) .

    2. x0 ( , ) , :

    ( f (x0 ) )2 + f (x0 ) f(x0 ) = 2e 2 x 0 .

    University of Bristol,

    ..57. f [0 , 1] ,

    f (0)=0, f (1)=1 0 f (x )1 x [0 , 1] .

    :

    1. x0 (0 , 1) , f (x0 ) = 1 x0 .

    B2. , (0 , 1) , f () f ()=1.

    ,

    ..58. f R , :

    f (0)=0 (x2 1) ( f (x) x3 ) 0 xR.

    :

    1. C f xx (0,0) .

    2. f (1)=1 f ( -1)=-1

    B3. ( -1 , 1) : f ( )=1.

    University of Oxford,

    ..59. P(x) :

    2 3( )

    ( )

    x xf x

    P x

    , R.

    P(x) R ,

    f :

    lim ( ) lim ( ) 1x x

    f x f x

    C f x=1 x=-2

    f x0=-1.

    University of New York

    ..60. >0 : 1

    lim 1( )

    x

    xdt

    t t

    .

    University of Oxford,

  • -

    - 74 -

    ..61. 1

    0

    xx e dx * .

    1. 1 .

    2. : 11

    ( 1)e

    , * .

    3. :

    ) 1

    3

    0

    xx e dx ) 1

    01 1 xx x x e dx

    BAC. ,

    ..62. :

    4

    ( ) ln1

    x

    x

    e ef x

    e

    .

    B1. : f (4-x)=4-f(x ) , xR.

    2. : 4

    0( )f x dx .

    BAC. ,

    ..63. f [0 , 1]

    f (x)1, x [0 , 1] 1/2

    0( ) 0f x dx .

    :

    1. 1

    00

    2

    xf dx

    2. ( )2 2

    x xf x f

    B3. 1

    0

    1( )

    4f x dx

    ,

    ..64. 2 2

    ( )( )

    txf t

    t x

    , tR x>0.

    :

    1. 2

    1( )f t

    x tR x>0.

  • -

    - 75 -

    2. 0lim ( ) 0x x f t dt

    .

    BAC. ,

    ..65. R f

    :

    1

    0( ) ( ) 0f x f x dx

    12

    0( ) ( ) 18f x f x dx .

    :

    14

    0( ) ( ) .f x f x dx

    . . . 2004, Harvard University

  • -

    - 76 -

  • -

    - 77 -

  • -

    - 78 -

  • -

    - 79 -

    ..1. : ( 1) 6( )

    xf x

    x

    x( -1 , +)

    , R , y = 2 x = -1.

    1. f : 2x 6

    x 1f(x) , x 1

    .

    2. G(x ) G (x) = f (x ) , x >-1, (0,2) . 3.

    G(x)h(x)

    x 1

    , x >-1.

    ..2. g:RR ( )1

    x

    x

    e xg x

    e

    .

    R f

    : ( ) ( ) ( ) , f x g x g x x R f (0)=0.

    1. f . 2. N f 1 : [0,] [0,] . 3. C f , C f - 1 x = 0, x = = 4..

    ..3. f [ 0, ] . :

    1.

    0 0

    1f(x)dx f(x) f(-x) dx

    2 .

    2. x

    ( - x )x 0

    2011 dx =

    22011 2011 .

    ..4. F f :RR

    xR : 2 2 22 ( ) ( ), F x F x 0.

    : 1. F(0) = F(1) = . 2. f (x ) = 0 R.

    ..5. : F(x)=1

    4x2 (2lnx-3)-x(lnx-2)

    x > 0. 1. F F(x ) = 0 (0 , + ) .

    2. :

    2xf(x) x ln x

    2

    23xg(x) 2x

    4

    : 2004

    ( )

    ( )

    1

    1f x

    g xdt

    t >0 x >0.

    ..6. A. 'f R F

    f R.

  • -

    - 80 -

    N R :

    5 ( ).( ) ( 5) ( ) f xg x F x F x

    B. f : R R f (x+f(y)) = f (x+y )+2 x, y R .

    :

    ) f (x ) = x + f (0) .

    ) f (x ) = x + 2 xR.

    ..7. f :R *R 1 -1 f f (x) f(x)

    xR * , 0, : 1. f f (x) x 0.

    2. f .

    3. f .

    4. f .

    5. 1f .

    ..8. :f (0, ) R x

    f(x) f(y) fy

    x, y > 0. f (x ) = 0 , :

    1. f 1-1.

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

    3. f (x) 0 x > 1 f

    .

    ..9. f : RR , :

    f ( lnx) = x x>0 f (0) = 0. 1. f . 2. : f (x ) > x 2 xR.

    3. : 32

    1

    2f(x)dx .

    ..10. : f (x) 2x x,x [0,].

    1. 1f

    .

    2. 1f (x) x.

    3. =2 1

    0f (x)dx .

    ..11. 1. , :

    1 1

    0 0

    x (1 x) dx x (1 x) dx .

  • -

    - 81 -

    2. 2004f (x) x(1 x) , 2003g(x) x(1 x)

    f , g.

    ..12. 1. : xx , x 0e x , x 0f (x) . )

    0x 0 .

    )

    f x=0 , x=

    2.

    2. )

    :

    )

    f( ) , f( ) ,f()2 2

    .

    ) f .

    ) : x

    f (x)x

    lim

    .

    ..13. f 3f x ln x 1 x x e , x>-1.

    1. f

    f - 1 .

    2. f - 1 (x ) = 0.

    3. 1f ,

    1f e .

    ..14. f

    f : (0 , + ) (0 , + ) f (1) = 1 :

    1 1xf

    x f (x)

    x > 0.

    1. :f (x) 1 f (x)

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

    2. f .

    ..15. f R *

    :

    1

    ( ) - ( ) xxf x f x e , xR * f (1) = e , f ( -1)=1/e.

    1. f .

    2. : 2

    1/

    31/.

    ( )

    e

    edx

    f x

    x

  • -

    - 82 -

    ..16. :f R R 2 2(f f )(x) x (2 1)x x R , .

    f 'f () 1

    1. f ().

    2. C f

    ( , f ()) .

    ..17. f 2

    lnxf(x)

    x .

    1. C f .

    2. .

    3. , lnx

    g(x)x

    f .

    4.

    E()lim

    ()

    C f x = 1 , x = >1 y=0.

    ..18. f R , f ( 3 ) (x ) > 0

    xR. 1. C f . 2. (x1 , f (x1 ) ) (x2 , f (x2 ) ) C f .

    ..19. f R :

    f2 (x ) 4 e x f (x ) = 1 xR f (0) = 2 - 5 . 1. f . 2. f .

    3. : limx

    f x .

    ..20. :f

    1

    2 1 7lim 10

    1x

    f x

    x

    .

    1. :

    ) 3 7f

    ) 3 5f

    2. f

    3, 3f . ) 5 8y x . ) , 3,

    . 2 m/sec ,

  • -

    - 83 -

    . ... 2008

    ..21. . f :RR f (x ) 0 xR. f (5) + f (6) + f (7) = 0 , f . . f R

    f (1) = f (7) f (x7 ) f (7x ) xR. f (x ) = 0 (0 , 7) .

    ..22. . f : R R F f R.

    F (x) 0 F(x) = F(2 x ) xR , f (x ) = 0 . . f : R R , : f (1 x ) + 2 = x f (x ) , x R. 1. f R .

    2. : 21

    ( ) ln 1 e

    f x dx e e .

    ..23.

    2x x 1f (x)

    x 1

    .

    1 . ()

    C f , C f + x = 2 x = >2.

    2 . : E( )lim

    .

    3 . 3 , t = 4.

    ..24. f 0,fD R.

    x f x

    f xx

    0x ( ) : y = 2x e

    Cf 0 0 x , y , : 1. .

    2. f .

    3. ln f x x x , f . 4. f .

    5. 1

    ln 02

    xx

    0,x .

  • -

    - 84 -

    ..25. f f (x ) = x2 ( + )x + , < . C f xx. 1. . 2. 1 2 C f xx ,

    : 1

    2

    3

    2

    E

    E.

    ..26. f [0 , 4]

    2 2

    f 0 7 f 4 7 0 .

    1 . f . 2 . [0 , 4] , :

    1 1 12 3 4

    2 3 4

    9

    f f f

    f .

    3 . f (x ) 0 x [0 , 4]

    : 7 2xlim f(3) 1 x 2x 1

    .

    ..27. f : RR , R

    : x 0

    xf(x) 1988xlim 18

    x

    2006x 7 x

    xlim f(x) lim

    x .

    1. f (0) . 2. f ( -7) . 3. f y = -x ( -7 , 0) .

    ..28. f [1 , 3] .

    1 . 1

    3

    2lim

    x

    f x

    f x f (1) .

    2 .

    1

    ln3

    e xf dx

    x , :

    9

    xf x

    (1 , 3) .

    3 . : 3 4 f x 1 , 2x :

    1 2 2 f .

    ..29. . f [1 , 7]

    : f (2) < f (1) < f (7) < f (5) .

    (1 , 7) , f () = 0.

    . f (x ) = xe - x , xR .

  • -

    - 85 -

    . f .

    . : 2/ x

    1/2 22 e xe dx e

    .

    4 1993

    ..30. f : R R ,

    : f (x)

    1f (x)

    e 1

    xR f (0) = 0.

    1 . : x

    xf (x) f (x)2

    x >0.

    2 . f , xx x = 0

    x = 2 : > f (2) .

    ..31. f (0 , + )

    e f (1) = 1

    2f x

    f xx

    x > 0.

    1 . (x) = f (x ) e 1 / x

    x > 0.

    2 . f .

    3 . N

    3

    f xh x

    x , xx

    x = 1 x = 2.

    ..32. f , g R

    :

    ) f (x1 + x2 ) = f (x1 ) f (x2 ) x1 , x2 R.

    ) f (x ) = 1 + x g(x ) xR.

    ) x 0

    g x 1lim

    .

    1 . f R. 2 . f (x) = e x . f ) ) . New York University

    ..33. g(x) = x lnx.

    1. : x

    g(x)lim 0

    .

    2. :e lnx

    I dxg(x)

    2

    1.

    3. : g (x ) x x > 0.

    4. : 1

    ln 0xxe

    x > 0.

  • -

    - 86 -

    ..34. f :RR .

    f f (3) = f (2) :

    1. : f (x+1) = f (x ) .

    2. : f (3x+1) > f (3x) .

    3. : f (x6 + 2) = f (x6 + 1) .

    4. f .

    ..35. 1 . f x

    2 2

    ef x

    x

    , >1

    .

    2 . x0 :

    2x xe 1

    , > 1.

    .

    ..36. f 2 .1f x ln x 2x 1 , 0 , 22

    1. f .

    2. f = .

    .

    ..37. . : x 1f(x) 2 ln2 g(x) ln 2x .

    , (1,ln2) (2,ln4).

    . f [ -,] : f (x )1

    x ( -,) >0.

    f()= f ( -)=- f (0)=0.

    ..38. f : RR :

    f 2 (x ) +2f (x )x = x2 + 2x xR f (0) = 1.

    1. :

    g (x) = f (x ) + x , xR .

    2. f .

    3. 0

    ( ) 1lim

    x

    f x

    x lim ( )

    xf x .

    ..39. f [2,3] , (2,3)

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

    1. f (2) f (3)

  • -

    - 87 -

    2. (2,3) :

    5f ( ) = 2 f (2) + 3f (3) .

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

    ..40. R f , :

    f (x )0 R

    f(x) dx 0 , , R.

    1. : = .

    2. R

    e

    t

    t e dt

    22010

    10 .

    ..41. f :R R

    f (x) = -4x3e f ( x ) xR f (0) = -1.

    1. N f .

    2. N f .

    3. : 20

    1

    1

    my

    dxx

    ym

    f .

    ..42. . f

    [1 , +) f (1) = 1 1 < f () < 2008

    2007

    x > 1. : < f (x ) < 2008 1

    2007

    x x > 1.

    . f :RR ,

    :

    f (x ) > 0 xR R.

    f .

    ..43. f : RR :

    (x2 + x +1)f (x) = e x (2x+1)f (x) xR f (0) = 1.

    1. N f .

    2. N f .

    3. : 0 1

    11( ) ( )

    y ef e x y dx f x dx

    e yM

    f .

    ..44. f f (x) = 4e 2 x xR.

    1. N C 1 , C2 f (x) = e 2 x + C1x +

    C2 .

  • -

    - 88 -

    2. 0

    ( ) 2lim

    x

    f x ( )

    2lim

    x

    f x

    x, f .

    3. e 2 x 2x 1 0 xR

    4. e 2 x 2x = 2x2 + 1

    5. N Cf - , ()

    6. Cf , () ,

    x =0 , x = -1

    ..45. :f

    :

    ' 1 1f 0

    (8 ) (3 )lim 5

    5x

    f x f x

    x

    .

    fC 0, 0f 2,8 :

    1. 0 2f . 2. 'f .

    3. 1 1 3f .

    4. 0,2 :

    2 3 1 3f f

    ..46. :f R R

    :

    (1) ( 1)( )

    2

    f ff x x

    , x R .

    : 1. f (1) f ( -1) = 2

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

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

    4. 1 2, 1,1x x 1 2x x 1 2

    1 12

    ( ) ( )f x f x

    ..47. : 0,f

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

    x 0, .

    1. lng x f x x lnh x x x 0x .

    2. f .

    3. f 0, .

  • -

    - 89 -

    4. fC ,

    .

    5. 1

    2015f x ,

    (1,2) .

    ..48. :f

    : 24 4xf x x f x e

    44

    f xf x x

    .

    1. f .

    2. 24 xf x x e ,

    limx

    x

    f x

    .

    3. fC

    y y .

    4. f

    1fC 11, 1f .

    5. 0 1,2x ,

    fC 0 0,M x f x

    1,5K .

    ..49. :f :

    0

    2lim 0x

    f x

    x

    .

    1. 0f .

    2. 2

    20limx

    x f x

    x

    .

    3. f :

    2 2 1,x xf x e f x e x ,

    . : ,x xf x e e x .

    . : limx

    f x

    limx

    f x

    .

    . f ,0

    0, , f x k 2 2k .

  • -

    - 90 -

    ..50. f [0,1]

    f (x )>0 x(0,1) . A f (0)=2 f (1)=4,

    :

    1 . y=3 f '

    x0 (0,1) .

    2 . x1 (0,1) , f (x1 )=

    1 2 3 4

    5 5 5 5

    4

    f f f f

    3 . x2 (0,1) ,

    f (x2 , f (x2 ) )

    y=2x+2000.

    3 2000

    ..51. f ,

    R , :

    f3 (x ) + f2 (x ) + f (x ) = x3 2x2 + 6x 1 x R , ,

    2 < 3.

    1 . f .

    2 . f .

    3 . f (x ) = 0

    (0,1) .

    3 2001

    ..52.

    x , x 1

    f (x) x 1 1 e ln(x 1), x 1,2

    R.

    1 . x 1

    1x1 elim

    x 1

    .

    2 . R f

    x o=1.

    3 . =-1 (1,2) ,

    f ( , f ( ) )

    xx.

  • -

    - 91 -

    3 2001

    ..53. f , g R .

    fog 1-1.

    1 . g 1-1.

    2 . :

    g ( f (x ) + x3 - x ) = g(f (x ) + 2x -1)

    .

    3 2002

    ..54. f (x ) = x5+x3+x .

    1 . f

    f .

    2 . f (e x ) f (1+x ) x IR.

    3 .

    f (0,0)

    f 1f .

    4 .

    f 1 , x

    x=3.

    3 2003

    ..55. 2f(x) = x 1 - x .

    1 . x lim f(x) 0

    .

    2 .

    f , x - .

    3 . 2 f (x) x 1 f(x) 0 .

    4 . : 1

    2 0

    1 dx ln 2 1

    x 1

    .

    3 2003

  • -

    - 92 -

    ..56. g(x)=e x f (x ) , f

    R f (0)=f(3

    2) = 0.

    1 . (0, 3

    2)

    f ()=f() .

    2 . f (x )=2x 23x, :

    () = 0

    g(x)dx ,R.

    3 . lim ()

    .

    3 2004

    ..57. f f (x) = e , > 0.

    1 . f .

    2 .

    f , , y = ex.

    .

    3 . () ,

    f ,

    yy, () =e 2

    2

    .

    4 .

    2

    ()lim

    2

    .

    3 2005

    ..58. f, IR

    f (x)0 x IR .

    1 . f 1 -1.

    2 . C f f

    (1,2005) ( -2,1) , 1 2f -2004 f(x 8) 2 .

    3 . Cf,

    Cf () :

    1y x 2005

    668 .

  • -

    - 93 -

    3 2005

    ..59. f(x ) = ex

    e lnx, x > 0.

    1 . f(x )

    (1, +).

    2 . f (x ) e x > 0.

    3 .

    2 2

    2 2

    x 2 x 2 4

    2x 1 x 3

    f(t)dt = f(t)dt f(t)dt

    (0, +).

    3 2007

    ..60. f(x ) = x 3 3x 22 IR

    + 2

    , Z.

    1 . f ,

    .

    2 . f(x ) = 0

    .

    3 . x1 , x2 x 3

    f , (x 1 , f (x 1 ) ) ,

    B(x2 , f (x 2 ) ) (x 3 , f (x 3 ) ) y = 2x 22.

    4 .

    f

    y = 2x 2 2.

    3 2007

    ..61. xln x, x 0

    f (x)0, x 0

    .

    1 . f 0.

    2 . f

    .

    3 .

    xx e .

  • -

    - 94 -

    4 . f (x+1) > f (x+1)f(x) , x > 0 .

    3 2008

    ..62. f (x )=x2 2lnx, x > 0.

    1 . : f (x )1 x>0.

    2 .

    f .

    3 .

    ln x , x 0

    f(x)g(x)

    k , x 0

    i . k g .

    ii . 1

    k2

    , g , ,

    (0,e) .

    3 2008

    ..63. xf (x) ln(x 1), x > -1 >0

    1 .

    1 . f (x) 1 x>-1 = e .

    2 . = e ,

    . f .

    . f

    1,0 0, .

    . , 1 0 0 , , ,

    f () 1 f () 10

    x 1 x 2

    (1, 2) .

    3 2009

    ..64. f (x )=ln [ (+1)x2

    +x+1] - ln(x+2) , x > 1

    -1.

    1 . ,

  • -

    - 95 -

    xlim f (x)

    .

    2 . = -1

    . f

    .

    .

    f

    . f(x ) + 2

    = 0

    0 .

    3 2009

    ..65. f(x )=2x+ln(x 2+1), x .

    1. f .

    2. :

    2

    2

    4

    3x 2 12 x 3x 2 ln

    x 1

    .

    3. f

    f

    .

    4.

    1

    1

    xf (x)dx

    .

    3 2010

    ..66. f(x ) = (x 2)lnx + x 3, x > 0

    1.

    f .

    2. f

    (0,1] [1, +).

    3. f(x ) = 0

    .

    4. x1 , x2 3 x 1

    < x2 ,

    (x1 , x2 ) ,

    f ( ) f ( ) = 0

    f (, f ( ) )

    .

    3 2010

  • -

    - 96 -

    ..67. f : R R ,

    R , f (0)= f (0)=0, :

    xe f x f x 1 f x xf x x R.

    1. : xf x ln e x x R.

    2. f

    .

    3. f

    .

    4. xln e x x

    0,2

    .

    3 2011

    ..68. y = x , x0.

    (0, 1)

    xy ,

    .

    t, t0 x (t)=16m/min.

    1. ,

    t, t0 : x(t )=16t.

    2.

    (4, 2) ,

  • -

    - 97 -

    , .

    3.

    .

    4. t 0 1

    0,4

    d=()

    .

    xy.

    3 2011

    ..69. f(x ) = (x - 1) nx - 1, x>0. 1. f 1=(0,1] 2=[1,+ ) . f .

    2. 1 2013xx e , x>0

    . 3. x1 , x2

    x 1< x2

    2, x 0 ( x1 , x2 ) , f ( x0 ) + f (x0 ) = 2012. 4. g(x) = f (x) + 1 x>0, xx x=e. 3 2012

    ..70. f:RR, :

    xf (x )+1= e x , xR .

    1. :

    1 , 0

    ( )

    1 , 0

    xex

    f x x

    x

    .

    2. o f 1

    .

    3.

    f (0,f (0) ) . ,

    f , 2f(x)=x+2,

    x .

    4. 0

    lim ln ln ( )x

    x x f x

    .

    3 2012

  • -

    - 98 -

    ..71. f ,g :R R , f

    :

    ( f (x ) + x) ( f (x) + 1) = x , x R

    f (0) = 1

    g (x) = x 3 +

    23

    2

    x 1

    1. : f (x ) = 2 1x x , x R.

    2.

    f (g (x)) = 1.

    3. x 0(0, 4

    ) ,

    :o

    0

    x

    4

    f (t)dt = f (x0 4

    ) x0 .

    3 2013

    ..72. f :R R :

    2xf (x ) + x2 ( f (x) - 3) = - f (x ) xR

    f (1)= 1

    2

    1.

    3

    2( )

    1

    xf x

    x

    , xR

    f R.

    2.

    f 1.

    3. :

    3 2

    2 25 1 8 8 1f x f x .

    4. , , (0, 1) ,

    : 3

    2 3

    0( ) 3 1f t dt f

    . 3 2013

    ..73. h(x) = x - n(e x + 1) , x

    1. h .

    2. : ( 2 ( ) )

    1

    h h x e

    ee

    , x

  • -

    - 99 -

    3.

    h + , - .

    4. (x) = e x (h(x) + n2), x

    (x) , xx x = 1.

    3 2014

    ..74. :

    ln

    , 0( )

    0 , 0

    x

    xe xf x

    x

    .

    1. f x 0 = 0.

    2. f .

    3. i ) , x > 0,

    f (x ) = f (4) x4 = 4 x

    ii ) N x 4 = 4 x , x > 0,

    , x 1=2 x 2=4

    4. , , (2, 4) ,

    : 2

    ( ) ( ) ( ) 2 ( )f f t dt f f

    .

    3 2014

    ..75. f :RR xR

    12 ( )

    21 3 ( ) xtf t dt

    f x e , R-{0} .

    1. :

    i . f f () = -2 f2 (x ) xR.

    2 2

    1 ( ) .

    3.

    f x x R

    xii

    2. :

    0( )tf t dt

    .

    3. f .

    4.

    , f x = ,

    :

    1 1

    4 3E

    a a

    .

    3 2005

  • -

    - 100 -

    ..76. .

    1.

    f 0, 0f . 2. f

    .

    3.

    f , 0, 0f 1x a .

    i . : .

    ii . .

    3 2006

    ..77. f 1xef x e , x R .

    1 . i ) .

    ii ) 1'' 1 xxx ef x e e , f

    .

    2 .

    f .

    3 . f .

    4 .

    , .

    3 2008

    ..78. : 2 2lnf x x x . 1 .

    .

    2 . f .

    3. ln

    2

    x xg x

    x

    0 0x : 0g x g x

    0x .

    4 . 2x : 2 2 1 4f x f x f x . 3 2009

    ..79. : (0, )f ,

    0x ( )

    1 ( )

    1f xx

    x f xe

    0)1( f .

    ) ( ) x

    g x e x 1 -1.

    ) xxf ln)( x>0.

    1xf x e a x 1

    2

    12

    a aE a e a

    alim E a

    'f x ' , 'x x y y1

    ln2

    x

  • -

    - 101 -

    ) ( ) 1

    ( )f x

    h xx

    .

    ) 0 , .2

    x xx x

    xe e

    ) h 21,xx

    012 xx : 2 1

    2 1

    5

    h(x ) h(x ) 1

    x x 2e

    .

    3 2010

    ..80. : f R R :

    1f x f x , x R . 1 . :

    i . 2 1

    2 2f

    0 1 1f f

    ii . 0 0, 1x , : 0 0 1f x x

    2 . , , f 1

    22

    f x x ,

    x R .

    i . 2

    '2

    f

    fC 2

    2.

    ii . :

    0

    1

    x

    f f xim

    x

    .

    3 2011

    ..81. : xx

    , x 0f(x) e 1

    ln , x 0

    .

    1. (0 , + ) f

    1

    f (0)2

    .

    2. . f .

    .

    , .

    3. x

    0

    1 12x dt

    f(t) 1 2013

    (0 , 1) .

    3 2013

  • -

    - 102 -

    ..82. f , g R f (1) = 1,

    g(1)=0 :

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

    1. :

    () = g(x)+1.

    2. ) g (1) .

    ) :

    lim+

    [( + 1) ( + 2

    + 1)] = 0.

    3. () = ( 1)2 xR,

    ) f .

    ) R, (1 , )

    h : () = (1 ) + 1.

    3 2014

    ..83. f 0,2

    ,

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

    2 6 2 6f x xf x x x

    .

    1. : f (x )=x-x , x 0,2

    .

    2.

    ( ) 1 , x 0,2

    ( )( )

    1, x < 0

    f x

    g xx

    x

    R.

    , g

    .

    3. =2, g(x) = 0

    ,02

    .

    4. =2, g 1 -1.

    2016

    ..84. f :RR ,

    xx f (0)>0.

    f (x)

  • -

    - 103 -

    4. : 1 1

    0 02 ( ) ( )xf x dx f x dx .

    5. :

    1

    0(0) 2 ( ) ( )

    01

    f xf x dx f x

    x x

    ,

    (0 , 1) .

    ..85. f : (0 , +)R

    : f (1)=0 ( )f xe x e x>0.

    1. f .

    2. ( )ln 0f xx e x>0.

    3. : ( )

    ( )f x

    g xx

    , x>0.

    M t = 0

    ( , g()) , (0,1) y = g(x) , x x=x(t) ,

    y=y(t ) t0.

    ( t )

    ( t )=2 (t) ,

    xx,

    g ,

    e.

    4. 2 2

    ( ) 1 ( ) 2h x x g x , x>0.

    h

    .

    5. : 2

    1

    1lim

    ( )

    x

    xxdt

    g t .

    ..86. f ,g : [0 , 1]R .

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

    0 0( ) ( )f x dx g x dx .

    2. m f [0 , 1]

    : 1

    0( )m f x dx M

    3. [0 , 1] , : 1

    2

    0

    1( ) ( )

    3x f x dx f .

  • -

    - 104 -

  • -

    - 105 -

  • -

    - 106 -

  • -

    - 107 -

    ..1. f :

    f (x) > f (x ) x 0.

    f (0) = f (0) = 0 h(x) = f (x) e - x , :

    1 . h .

    2 . 2( ) ( )x f x .

    3 . x f (x ) > 0 x > 0.

    4 . 7

    0f(x)dx f(7).

    ..2. R f

    : 2( ) = x 1 , xf x e xR f (0) = 3 .

    1. : f ( 1998 ) < f ( 2016 ) .

    2. (0 , 2016) , :

    f 1821 2f 1940 3f 2016

    f 6

    .

    3. 20

    1xe x dx

    f .

    4. y = 4

    f x0(0 , 1) .

    5. : f ( lnx) = f ( -2x + 2) x > 0.

    6. : 12xxf e x 2 02

    , x > 0.

    ..3. f : [ , ] R

    [ , ]. :

    1. 1

    f()

    f()xf (x)dx f (x)dx .

    2. 1( )

    ()( ) ( ) ( ) ()

    f

    ff x dx f x dx f f

    3. 1 12

    11

    1 2ln

    x

    e

    e

    e ee dx dx

    x e e.

    4. f () = f () = :

    1()

    () ( ) = (2 f(x))dx

    f

    ff x dx x .

    5. : xeI lnxdx e dx 21

    01.

    ..4. f : R R

    :x 2

    2f (x) 7x3

    x 2lim

    f (5) =

    x

    x + 7x

    xlim

    .

  • -

    - 108 -

    1 . f .

    2 . C f (2 ,

    f (2) ) .

    3 . f :

    . f (x ) 5x + 3 0 xR.

    . (2 , 5) f

    .

    ..5. 1 . f

    f (x ) = lnx

    g(x) = x x > 0. 2 . ()

    f g x = 1

    x = , > 1.

    3 . : E( )lim

    .

    4 . : x

    2f (x) x 2x 1lim .

    ..6. :

    x xx

    g x e x , x f x e , x2

    2

    1 0 02

    1 . g .

    2 . x 0 g x 0 ,

    ;

    3 . f

    .

    4 . 21 ex

    0 1e dx lnx dx e .

    5 . :21 x

    0

    ee dx

    2 .

    ..7. : t

    f(t)t

    2 3

    2, t [1,4] .

    . : I f(t) dt. 4

    1

    . t

    xx

    g(x) f(t) e dtx

    24

    1

    2

    1 , >0.

    i ) :

    t

    x x xe e e 2 2 2

    1 4

    t [1,4] >0.

    ii ) : xlim g(x).

    1 1999

  • -

    - 109 -

    ..8. 1 . g(x) = x3 + x.

    g ,

    :g - 1 (x+1) = x+1.

    2 . R f

    : f3 (x ) + f (x ) x xR.

    : 2

    0

    5( )

    4 f x dx .

    ..9. f , g: (0 , + )R :

    f (1) = g (1) = 0 f (x) + e g ( x ) = g (x) + e f ( x ) = 0 x>0.

    1 . f g .

    2 . h(x)=e - f ( x ) x, x>0

    : f (x ) = - lnx.

    3 . : e x > 1 f (1+x ) x>0.

    ..10. f : (0,+ )R xf (x ) f(x) = x

    x >0 >1 .

    1. f(1) = 0 , f .

    2. f (x) - 1

    e v x >0.

    3. A Cf 2

    3

    e

    .

    4. Cf , xx

    x= e , x = 3

    1

    e 3

    .

    ..11. f : R R

    ( )lim

    x

    f x , ( )lim

    x

    f x f (x ) = ( )

    2

    1f x

    e, xR ,

    f (0) = 1.

    1. f R

    , R

    x =0.

    2. :

    i ) f (x ) + e f ( x ) = 2x +1

    ii ) H f f 1

    iii ) O f f 1

    .

    3. Cf -

    Cf + .

  • -

    - 110 -

    ..12. f (x ) = e x ()

    C f , x = ,

    > 0.

    1. () .

    2. x o (0,) x = x o

    .

    3. 00

    lima

    x

    a.

    4. : 1

    0

    1ln

    2

    x

    x ee dx .

    ..13. A. x > 0 , : 2 2ln

    0 0

    /

    t txx e

    e dt e dt .

    B. g R ,

    g(x )>0 2

    ( ) ( ) ( ) 0g x g x g x xR.

    :

    . g

    g

    .

    . : 1 2 1 2( ) ( )2

    x xg g x g x

    x1 , x2R.

    1 1997

    ..14.

    f : [0,+ )R f (0) = f (0) = 0 ,

    f (x) > f (x) x [0,+ ) .

    N :

    1. h : [0,+ )R h(x) = f (x)e - x

    2. f 2 [0,+ ) .

    3. 1 (1)

    2 2

    ff .

    4. lim ( )

    x

    f x .

    ..15. f :

    (x2 +1) f (x) + 4xf (x) + 2f (x ) = 0 xR.

    1. g(x) = 2xf(x) + (x 2 +1) f (x )

    R.

    2. Cf (0,0)

    y 2x +3 =0 f

    3. f , ,

    .

    4.

    5. Cf , xx

  • -

    - 111 -

    yy .

    ..16. f : (0 ,+) R f (1) = 1

    e

    x > 0 , 2

    ( ) ( ) x f x f x .

    1. g(x) = f (x )

    1xe

    (0,+)

    2. f .

    3. f

    C f .

    4. 2

    31

    ( )

    f xdx

    x.

    ..17. f (x ) = x3 + x + 1.

    1 . f .

    2 . f - 1 : f ( ) 1 1 . 3 . f (x ) = 0 R.

    4 . : f (x)dx1

    3

    1.

    5 . : 1

    x 3

    f (x) 1lim

    x 3

    .

    ..18. :

    f (x ) =

    3 1 , x 0

    0 , x =0

    x xx

    || <

    1

    .

    1. f .

    2. lim ( )x

    f x .

    3. f .

    4. x o 1 1

    ,

    f (x o ) = 0.

    ..19. ),1(:g,f

    0 0 1f g , 1,x

    2 22 2 0f x f x g x g x g x f x (1) , 0f x 0g x .

    1. 0f x g x 1,x .

  • -

    - 112 -

    2. 1

    1f x

    x

    .

    3. f

    .

    4. f , xx

    x , 1x 0 ,

    lim

    .

    ... 2008

    ..20. 1. :

    f (x ) = 2 ( x-1) + lnx .

    2. : ln x

    g(x) x , > 02 x

    .

    3. :

    ln

    , 1, 4 .2

    xx x x

    x

    4.

    f , Ox

    x = 1 , x = 4.

    ..21. : f (x ) = ln

    , 0xe x

    xx

    .

    1. f .

    2.

    x

    x exa

    e

    x > 0

    = e .

    3. ) N 0<

  • -

    - 113 -

    ..23. f : R R

    : f (0) = f (0) = 1

    (x2 +1)f (x )+4xf (x) + 2f(x) = e x , xR

    1. f

    2. f

    3. : e x > x2 +1

    4. : 3

    3

    1

    ( )(1 ) x

    dxf x e

    .

    ..24. f :RR ,

    : x2 f (x) +4x f (x) +2f (x) > 0 xR.

    N :

    1. g : R R g(x) = x2 f (x )

    R.

    2. g .

    3. f (x) > 0 xR .

    ..25. f(x) = , 0

    2

    0 , 0

    x xx

    x

    .

    1. N f

    .Rol le 1

    0 , 2001

    .

    2. :

    x + xx = 0 2001 , 2

    .

    3. : 2

    2

    1

    1

    2dx

    x x

    .

    ..26. f : R R , 2 2

    ( ) ( )f x f y x y x , y R .

    1. f R.

    2. f 0.

    3. : 1

    0

    1 1(0) ( ) (0)

    3 3f f x dx f .

    4. f R , :

    2

    ( )lim .x

    f x

    x

  • -

    - 114 -

    ..27. :f )(f

    :

    f .

    f

    f xf x e x x

    :

    1. To limx

    f x

    2. H f .

    3. f