# 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) = ,