ΘΕΜΑΤΑ ΠΑΝΕΛΛΗΝΙΩΝ

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Όλα τα θέματα στις Πανελλαδικές Εξετάσεις

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  • 1

    1 4

    ( ) 27 2013 - :

    : (4)

    A1. f [ ], . G

    f [ ], , : ( ) ( ) ( )

    f t dt G G =

    7 A2.

    ( ...) 4

    A3. f [ ], ;

    4 A4. ,

    , , , .

    ) 0z z , >0 = ( )0K z 2 , 0z, z .

    ) ( )0x x

    lim f x 0 < , ( )f x 0< 0x ) : x x x

    ) : x 0

    x 1lim 1x

    =

    ) f f .

    10

  • 2

    2 4

    z :

    ( ) ( )z 2 z 2 z 2 2 + = B1. z ,

    ( )K 2,0 1= ( 5) , z

    , z 3 ( 3) 8

    B2. 1 2z , z 2w w 0+ + = , w , , ,

    ( ) ( )1 2z z 2 =Im Im :

    4= 5= 9

    B3. 0 1 2 , , 1. v :

    3 22 1 0v v v 0+ + + =

    :

    v 4< 8

    f,g : , f :

    ( )( ) ( )( )f x x f x 1 x+ + = , x ( )f 0 1= ( ) 23 3xg x x 1

    2= +

  • 3

    3 4

    1. :

    ( ) 2f x x 1 x= + , x 9

    2.

    ( )( )f g x 1= 8

    3. 0x 0, 4

    , :

    ( ) 0 00

    0

    x 4

    f t dt f x x4

    = 8

    ( )f : 0, + :

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

    h 0

    f 1 5h f 1 hlim 0

    h+ =

    ( ) ( )x

    f t 1g x dt

    t 1= , ( )x 1, + 1>

    :

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

    6 2. g ( 3), ,

    2 4

    2 4

    8x 6 2x 6

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

    + +

    + +> ( 6)

    9

  • 4

    4 4

    3. g ,

    ( ) ( ) ( )( ) ( )x

    f t 1 1 dt f 1 x , x 1

    t 1 = >

    . 10

    ( )

    1. . - . - . .

    2. . . .

    3. . , , .

    4. . 5. : (3)

    . 6. : 10.00 . .

    K

  • 1

    1 4

    ( ) 28 2012

    :

    : (4)

    A1. f . x , f

    0(x)f >

    7

    A2. f [, ];

    4 A3. f .

    f x0A ; 4

    A4. , , , , .

    )

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

    ) = + , f(x)

  • 2

    2 4

    ) ,x

    1)(x 2= x{x |x=0}

    ) ,(x)g(x)dxf[f(x)g(x)](x)dxgf(x)

    += g,f

    [,] 10

    z w :

    4=+z +1z 22 1_ (1)

    12= w 5_w (2)

    B1. z = 1

    6

    B2. z1, z2 z 2=zz 21

    _ , .zz 21 + 7

    B3. w

    14y

    9x

    22=+

    w

    6

    B4. z,w (1) (2) :

    1 wz 4 6

  • 3

    3 4

    f(x)=(x1) nl x1, x>0 1. f

    1=(0,1] 2=[1,+). f

    6

    2. x>0 .

    ,ex 20131-x = 6

    3. x1, x2 x10, x=e xx

    7

    f : (0,+) , x>0 :

    f(x) 0

    exx

    f(t)dt 21xx

    1

    2

    + xx = nl f(x)edt

    f(t)tnt

    x

    1

    + l

    1. f .

    10

  • 4

    4 4

    f(x) = ex( nl xx), x>0, : 2. : ( )( ) ( ) ( )

    +xf

    xf1

    xflim 2

    0 x

    5

    3. nl xx1, x>0,

    ( ) dt, f(t)xF x = x>0,

    >0, ( 2). :

    F(x) + F(3x) > 2F(2x), x>0 ( 4).

    6

    4. >0. (,2) :

    F() + F(3) = 2F()

    4 ( )

    1. ( , ) . .

    2. . . .

    3. . 4. .

    , .

    5. . 6. . 7. : (3)

    . 8. : 10 .30 . .

  • 1

    1 4

    ( ) 19 2010

    :

    : (4)

    A1. f . F f , :

    G(x)=F(x)+c, c

    f

    G f

    G(x)=F(x)+c, c 6

    A2. x=x0 f ;

    4 A3. f

    . f ;

    5 4. ,

    , , , .

    ) +i +i .

  • 2

    2 4

    ) f . f , .

    ) f (,), (,),

    )x(flimB)x(flimAxx +

    ==

    ) (x)=x, x

    ) 0)x(flim0xx

  • 3

    3 4

    f(x)=2x+ln(x2+1), x 1. f.

    5 2. :

    ( )

    +

    +=+1x

    1)2x3(ln2x3x2 42

    2

    7 3. f

    f .

    6 4.

    =1

    1

    dx)x(xfI

    7

    f: x :

    f(x)x

    f(x)x =3+ x

    0

    dtt)t(f

    t

    1. f

    f(x)=x)x(f

    )x(f , x

    5 2. g(x)= ( )2)x(f 2xf(x),

    x, . 7

  • 4

    4 4

    3.

    f(x)=x+ 9x2 + , x 6

    4.

    +

    +

    ++= 10 >

    A. 1)x(f ,1x > =e 8

    . =e,

    . f .

    5

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

    6

    . , ),0()0,1( + ,

    02x

    1)(f1x

    1)(f =+

    (1, 2) 6

    3 5

  • 4

    4

    f [0, 2]

    ( ) 0dt)t(f2t20

    =

    = x0 ],2,0[x,dt)t(ft)x(H

    =

    +=

    0 x,

    ttlim

    ],(x,dt)t(fx

    )x(H

    )x(G

    t

    x

    2

    2

    0

    0

    116

    203

    . G [0, 2].

    5

    . G (0, 2)

    2x0,x

    )x(H)x(G 2

  • 5

    1. ( , , ). .

    2. , . .

    .

    3. . 4.

    . 5.

    . 6. : (3)

    . 7. : 10.00 . .

    K

    5 5

  • 1

    24 2008 :

    : (5)

    1o

    A.1 f(x) = xln , x* * :

    ( )x1xln =

    10

    .2 f [,];

    5

    B. ,

    , , , , . . f:A 11,

    f1 : )A(fy y,))y(f(f A xx,))x(f(f 11 ==

    2

    . f f .

    2

    1 5

  • 2

    . z2+z+=0 ,, 0 , .

    2

    . f ,

    f( x ) > 0 x.

    2

    . A f ,,

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

    2

    z w

    3i)(3wi)(1w 6z)22i( ==+ :

    . z .

    6

    . w .

    7

    . w

    6

    . wz 6

    2 5

  • 3

    3

    =>=

    0x , 0 0x,lnx x

    f(x)

    . f 0. 3

    . f .

    9

    .

    x

    ex = . 6

    .

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

    7 4

    f

    += 203 45f(t)dt 3x)10x(f(x) .

    f(x)=20x3+6x45 8

    3 5

  • 4

    . g

    .

    hh)(xg(x)glim(x)g

    0h

    = 4

    . f () g ()

    45f(x)h

    h)g(x2g(x)h)g(xlim 20h+=++

    g(0)=g(0)=1,

    i. g(x)=x5+x3+x+1 10

    ii. g 11 3

    ( )

    1. ( , , ). .

    2. , . .

    .

    3. .

    4 5

  • 5

    4. . , .

    5. .

    6. : (3) .

    7. : 10.30 .

    K

    5 5

  • 1

    24 2007 :

    : (5)

    1o

    A.1 z1 , z2 , :

    2121 z zz z = . 8

    .2 f, g ;

    4

    .3 y = f +;

    A

    3

    B. , , , , , . . f [,]

    x[ , ] f(x) 0 . > 0 dx f(x) 2

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

    2

    1

  • 2

    . f x0 g x0 , gof x0.

    2

    . f , ( ) ( ) (x) g g(x)f dt f(t) g(x) = .

    2

    . > 1 . 0 lim x x

    = 2

    2

    2ii 2 z +

    += IR .

    . z (0,0) =1.

    9

    . z1, z2

    2ii 2 z +

    +=

    = 0 = 2 .

    i. z1 z2.

    8

    2

  • 3

    ii. :

    )z( )(z 22

    1 = .

    8

    3

    :

    f(x) = x3 3x 22

    IR + 2 , Z .

    . f , .

    7

    . f(x) = 0 .

    8

    . x1, x2 x3 f, (x1, f(x1)), B(x2, f(x2)) (x3, f(x3)) y = 2x 22 .

    3

    . f y = 2x 22 .

    7

    3

  • 4

    4

    f

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

    g [0, 1]

    g(x) > 0 x [0, 1]. :

    F(x) = , x [0, 1], x0 dt g(t) f(t) G(x) = ,