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Πανελλαδικ ∆ευτέρ ΜΑΘΗΜ Επιμέλεια: Νατσαρίδης Π Μαθημ κές Εξετάσεις ρα 25 Μαΐου 2015 ΜΑΤΙΚΑ Παναγιώτης ματικός
Λυσεις Μαθηματικά Γ΄ λυκείου
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Λύσεις Μαθηματικά Γ΄ Λυκέιου
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:
25 2015
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.. & . 2 1 2461025125 1
: 25 2015
1. . 194
2. . 188
3. . 259
4.
-
.. & . 2 1 2461025125 2
1.
( ) ( ) ( )( )
2 22
2
4 2 1 4 2 1
4 4 4 1 1
4 4 16 4 4 4 4 3 12 4
4
z z z z
z z z z
zz z z zz z z
zz
zz
z
= =
=
+ = +
=
=
=
2z =
z
2=
2.
21 1 1 1 1
1
42 4 4z z z z zz
= = = =
2
2 2 2 2 22
42 4 4z z z z zz
= = = =
) 1 2 1 2 2 11 22 1
2 1
4 42 22 2 2 2
4 4z z z z z z
w wz zz z
z z
= + = + = + = w
)
1 21 2 1 2
2 1 2 1 2 1
2 22 2 2 2 4 4 4 4z zz z z z
w wz z z z z z
+ + = + =
-
.. & . 2 1 2461025125 3
3.
( )( )
2 21 21 2 1 2
2 1
2 21 1 2 2
21 2
1 2
1 2
2 24 4 2
2 2 0
2 0 0
z zz z z z
z z
z z z z
z z
z z
z z
= +
+ + =
+ =
+ =
=
( ) 1 3 1 1 1 12 1 2 5 (1)z z z iz z i z = = = = ( ) 2 3 2 1 1 1
1 12 2 1 2 5 (2)z z z iz z iz z i z = = = = + =
(1) (2) ( ) ( ) =
-
.. & . 2 1 2461025125 4
1. ( ) 2 , 1xef x x
x=
+
f :
( )( ) ( )
( )( )( )
( )( )
22 2
2 2 22 2 2
1 2 2 1 10
1 1 1
x xx xe x e x e x x e xf x
x x x
+ + = = = >
+ + +
{ } - 1 1x f
f
( ) 2 21
0 0 01 1
xx
x x x
eim f x im im ex x
= = = = + +
( ) 2 1 2 2x x x
x x x x
e e eim f x im im imx x+ +
= = = = + +
: ( ) ( )0,f A = +
-
.. & . 2 1 2461025125 5
2.
( )( ) ( )( ) ( )( )
( )
23 2 3 2
3 2
3
2
3
1 1 25
1 2
1 2
2
x x
x
x
ef e x f e x f
e x
e e
x
ef x
+ = + =
+ =
=+
=
( ) ( )3
0,2e f A = + ( ) ,ox + ( )
3
2oef x =
f .
3.
0x > [ ]2 ,4x x
[ ]2 , 4t x x ( ) ( )4 4t x f t f x
( ) ( ) ( ) ( ) ( ) ( )4 4 4 4 4
2 2 2 2 2
4 4 1 4 2x x x x x
x x x x x
f t dt f x dt f t dt f x dt f t dt f x x< < <
-
.. & . 2 1 2461025125 6
4.
( )( )
4
2
1 , x>0
2 , 0
x
x
f t dtg x x
x
= =
0x >
( ) ( ) ( ) ( )4 4 2
2
0, :x x x
x
f t dt f t dt f t dt
+ =
( )( ) ( ) ( )
( ) ( )( ) ( )
( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )
4 2 4
22
4
22
4
22
4
22
4 4 2 2
2 4 2 4 2 2
2 4 2 2 4
x x x
x
x
x
x
x
x
x
f t dt f t dt x f t dt xg x
x
f x f x x f t dt
x
xf x xf x xf x f t dt
x
x f x f x xf x f t dt
x
=
=
+
=
+
=
( ) ( ) ( ) ( ) ( ) ( )( )4 2 4 2 4 2 0 2 4 2 0x x f x f x f x f
x x f x f x> > > >
-
.. & . 2 1 2461025125 7
3 ( ) ( ) ( ) ( )4 4
2 2
4 2 2 4 0x x
x x
f t dt f x x xf x f t dt< >
:
( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
( )
4
24
22
2 4 2 2 4 0
2 4 2 2 40
0
x
x
x
x
x f x f x xf x f t dt
x f x f x xf x f t dt
x
g x
+ >
+
>
>
g ( )0,+
g 0ox =
( )( ) ( )
( ) ( )
( ) ( )( )
4 2
0 0 g
4 4 2 2
1 4 0 2 0 2 0 2 1 2
x x
x x
x
f t dt f t dtim x im
x
f x f xim
f ff
+ +
=
=
=
=
=
=
g 0ox = . g [ )0,+
-
.. & . 2 1 2461025125 8
1. ( ) ( ) ( )( ) 2f x f xf x e e + = , ( )0 0f =
( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )
( )( )
( ) ( )
1 1
2 2 2
2 2
2 01
2
2 1
f x f x f x f x
f x f x
f xf x
f x f x
f x e e f x e f x ee e x C C
e xe
e xe x x
+ = = +
= + + =
= +
+ = +
( )( )( )
2 2
2
1
1 (1)
f x
f x
e x x
e x x
= +
= +
( )
( )
2
0
0 1 0 0 1 0
f x
fe x x
e
+
= >
( ) 0f xe x > x
(1) ( ) ( ) ( )2 21+x ln 1+xf xe x f x x= + = +
-
.. & . 2 1 2461025125 9
2.
( )( )22
2 22 2
211 12 1 0 x
1 11 1
x
x xxf xx x xx x x
++ ++ = = = >
+ + ++ + +
f .
( )( )( )
2
2 2 2
11 1 1
x
xxf xx x x
+ = =
+ + +
( ],0
[ )0,+
( )( ) ( )0, 0 0,0f
) fC ( )0,0
( ) ( )( )0 0 0y f f x y x = =
f [ ]0,1
( ) ( ) ( )( )
( )( )( ) ( )
( )( )
1 1
0 01
2
01 1
2
0 01 12 1
2
20 001
2
0
ln 1
ln 1
ln 12 1
1 ln 2 1
21
ln 2 2 12
1 2 ln 2
2
f x x dx x f x dx
x x x dx
xdx x x x dx
x xx x x dx
x
x
= =
= + +
= + +
= + + + +
= + +
= +
=
-
.. & . 2 1 2461025125 10
3.
0x >
( ) ( ) ( )0 0f x f f x> >
( )( )
( ) ( ) ( )
2
0
2 2
0
1x
x
e f t dth x
x
h x e f t dt f x
=
=
( )( )
2
0
0
10 0
x
x
e f t dtim h
x+
= =
( )( ) ( )( )( )( )
( )( )
2
0 0 0 02
lnln 0 1 01 1x x x x
f xf x f x xim x f x im im im f xf xx x
+ + + +
= = = = =
( ) ( )
2
0 0
2 0 01x x
x xim imf x f x+ +
= = =
( )( )
2
0
0
1ln 0
x
x
e f t dtim x f x
x+
=
-
.. & . 2 1 2461025125 11
4. ( ) ( ) ( ) ( ) ( )2
2 2
0 0
2 1 3 3 8 3x x
h x x f t dt x f t dt
= +
h [ ]2,3
( ) ( ) ( ) ( )2 2
2 2
0 0
2 1 8 3 8 3h f t dt f t dt = = +
( ) ( )1
2
0
3 1 3h f t dt=
[ ]2,3 f .
( ) ( ) ( )22 2 3
2 2 2 2
0 0 0
83 3xf x x f x x f t dt x dx < < < = =
( ) ( )2 2
2 2
0 0
8 3 8 03
f t dt f t dt<
( ) ( )2 3 0h h <
Bolzano ( )2,3 ( ) 0h x =
