¸­¼±„± µ†µ 2001 2014

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Transcript of ¸­¼±„± µ†µ 2001 2014

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    2001 2014

    2014 15

  • 2 |

    2001

    . & :

    1

    . f, .

    . f (x)=0 f

    .

    9

    . f (x) > 0 , f;

    3,5

    . . f : f (x) f(x) = 0 x f(0) = f (0).

    :

    (i) H g(x) = [f(x)]2 [f (x)]2 + 2001

    3

    (ii) g(x) = 2001, x

    2

    . .

    (i) f

    , f (x) > 0 x.

    2

    (ii) f (x) > 0 x (1, 2) (2, 4)

    f.

    2,5

    (iii) f

    f (x) 0 x, f

    .

    3

    2

    z 2 iz

    f (z) , z 1.1 z

    A. f(2).

    4

  • 3 |

    . w = [f(2)]2004

    4

    . : f (z) 2

    zf (z) i

    8

    . z 1 f(z)

    .

    9

    3

    f(x) = ( 1)x 6

    x

    x(1, +), ,

    y = 2 x = 1.

    . f f(x) = 2x 6

    , x 1x 1

    9

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

    (0, 2)

    8

    . G(x)

    h(x) , x 1.x 1

    8

    4

    f, g :

    x 12

    1 xf (t)dt g(t)dt x 2x 1, x Cf, Cg .

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

    . :

    . g(x) = 0 (1, 2)

    4

    . (1, 2) g() = 2

    5

    . g :

    . f

  • 4 |

    4

    . f , xo =

    ..

    5

    . Cf, Cg y

    7

  • 5 |

    2002

    . & :

    1

    . f, [, ] f() f().

    f() f()

    xo(, ), f(xo) = .

    5

    . , , < f.

    .

    . f Rolle [, ],

    .

    . x1, x2[, ] f(x1) f(x) f(x2), x[, ].

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

    .

    f (x)dx ' f (x)

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

    10

    . z1, z2 0 ,

    . :

    . 1 2z z z z ,

    .

    2

    . z2 = iz1 (

    )

    3

    . f xo

    , .

    5

    2

    f: :

    x xim f (x) , im f (x)

    f(x)

    2f (x)

    1 e

    x f (0) = 1

    . :

  • 6 |

    . .

    3

    . .

    2

    . x = 0

    2

    . f(x) + ef(x) = 2x +1

    4

    . f

    2

    . f f 1

    4

    . . Cf

    f, .

    4

    . Cf +

    4

    3

    z 2i 2z

    f (z)z 2i

    .

    z + 2i.

    . zo ,

    : f(zo) = 3 + i.

    3

    . N f(z) 2.

    5

    . f (z) 2 2, z

    C, .

    7

    . Arg(f(z)2)=4

    , z

    .

    8

  • 7 |

    . C .

    2

    4

    f: f(x) = 2

    x

    1 x F, G

    x x

    21/e 1/e

    f (t)F(x) f (t)dt,G(x) dt, x 0.

    t

    :

    . . 1

    f (x) fx

    x*.

    2

    . 1

    f '(x) 18

    x.

    6

    . , 0 < < : 1 1

    f f

    4

    . g g(x) = F(x) + G(x), x > 0 g(x) = nx 1,x 0

    4

    . h x1 = 0, x2 =2

    h(x) = F((x)) + G((x)) 0 < x < 2

    , =

    0,2

    . h.

    5

    E. To

    f (x), h(x) x = 1 1

    2

    4

  • 8 |

    2003

    1

    A. ) f x x . f

    : f ' x x .

    ) f, g : ' 'f x g x

    x .

    c : f x g x c x .

    ) f . : f

    0x .

    B. .

    ) f :A 1f , f

    .

    ) f 0x 0 0f x , 0f x x

    0x .

    ) f ,

    0x , 0f ' x 0 .

    ) f

    . f '' x 0 x .

    ) f , f x 0 0f x dx

    , 0 ,x a

    0 0f x .

    ) f ,

    0f x dx

    , f .

    2

    ln a x

    f x , a 0x

    .

    A. f (1, f 1 )

    x y 0 , .

    B. a 1 :

    a. f.

    b. .

    c. : 1

    1

    8 .

  • 9 |

    3

    f , 0 z a i

    w f a i f f 0 .

    A. :

    a.

    1

    1 i zz

    1 f i w

    f a a .

    b. z iw z, w 0 ,

    .

    B. 2 2 2

    z iw z iw . :

    a. f f 0 .

    b. z, w 0 .

    c. 0x a, ,

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

    4

    f f '' x :

    x 0 1

    2

    0 1 0

    t 1 f '' t dt 2 t f t dt 4 x t f x dt x , f 0 0 f ' 0 2 .

    ) 2

    2xf x , x

    x 1

    .

    ) f,

    x 'x x 0 x a 0 .

    10

    / sec3cm , ,

    3cm .

    ) g : g x x 2 f x x .

    i. y x 2 g

    x .

    ii. g,

    x 0 2x , : ln5 .

  • 10 |

    2004

    1

    A. :

    f, , .

    f ,

    f a f

    , f a f , 0x a,

    : 0f x .

    B. f, ,

    .

    , 1 2 3I , I , I

    .

    3

    10

    I f x dx

    3

    20

    'I f x dx

    3

    30

    ''I f x dx

    . .

    1. x 0

    xlim

    x

    2. x 0

    1lim x

    x

    3. x 0lim ln x

    4. xx

    1lim

    e

    .

    . 0

    . 1

    .

    . vf x x ,v IN 0, 1 . f

    v 1f ' x vx .

    2

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

    x .

  • 11 |

    2lim

    2x

    g xL

    f x x

    ,

    0

    0.

    . i) L.

    ii) f g .

    . g .

    . : 4f x g x x x .

    3

    x x

    t0

    2g x dt, a 0

    a e

    z g x x i z i z 1 .

    A. ,

    i) g

    ii) z 1g .

    B. , :

    a. Re Imz z , x .

    b. 1 .

    c. 2 1

    2 t t0 0

    1 1 1 1dt dt

    1 e1 e a e a e

    .

    4

    f, g g 0 1

    2' 0f x g x , 2 2 1f x g x x .

    . :

    i) g ' x g x f x , x .

    ii) g , 0 , 0,

    1.

    . i) f .

    ii) f

    0, 0 .

    . , f

    y x. x 1 , 1

    E ln g 12

    .

  • 12 |

    2005

    1

    A. f . :

    f ' x 0 , f .

    B. af x x x 0 . : a 1f ' x a x .

    C. .

    1. Af : 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 , 0 ,x a 0 0f x ,

    0f a f .

    4. f , ,

    0 ,x a 0f ' x 0 .

    5.

    af x dx 0 f , ,

    0f x ,x a .

    2

    f x 2 x ln x 2 , x 0 .

    ) : ln x

    f ' x , x 0x

    ) x 0lim f ' x

    .

    ) f .

    )

    ln x

    g xx

    , 'x x 1

    xe

    2x e .

    3

    xz e x 1 i, x .

    ) : Re z Im z x .

    ) 0x 0, 1 2w z z 2i .

    ) z .

  • 13 |

    4

    f 1

    f 02

    xe f x f ' x x f ' x x .

    ) f x

    xf x , x

    1 e

    f x f x x x .

    ) xlim f x

    .

    ) /2

    /2

    I f x dx

    .

    ) : /2

    0

    04

    f x dx

    .

  • 14 |

    2006

    1

    A. ) f 0x .

    f 0x ,

    : 0f ' x 0 .

    ) 0x x f;

    B. .

    a. f :A 1 2x ,x :

    1 2f x f x 1 2x x .

    b. 0

    limx x

    f x g x

    0

    limx x

    f x

    0

    limx x

    g x

    .

    c. 0x x

    lim f x

    f x 0 x 0x .

    d. f

    , x .

    e. 0f x dx

    0f x ,x a .

    2

    z 1

    1

    zw

    i z

    z i .

    ) : w i

    zw i

    .

    ) 1z w ,

    'x x .

    ) w z .

    ) f , 1f a z f a i z f i

    . 0f x , .

    3

    1xf x e a x 1 .

    ) f

    0, 0f . ) f .

    ) f,

    0, 0f 1x a .

  • 15 |

    i. : 2

    12

    a aE a e a .

    ii. alim E a

    .

    4

    f 0f x 1

    0

    , ,g x t f xt dt t x .

    :

    ) 20

    1x

    g x t f t dtx

    0x .

    ) g 0 0x .