Μαθηματικά Κατεύθυνσης Απαντήσεις Θέματων...
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Μαθηματικά Κατεύθυνσης Απαντήσεις Θέματων Πανελληνίων 2010
Transcript of Μαθηματικά Κατεύθυνσης Απαντήσεις Θέματων...
-
2010
| www.lazaridi.info | tel. 6977.385.358 |
www.lazaridi.info Page 1 of 4
| | 2010 | |
|
1. 2. 3. 4. / / / /
|
0,22 =+ zz
z
1.
0222222 22 =+=+=+ zzzzz
z iz +=11 , iz =12
2.
( ) iiz 21 221 =+= ( ) iiz 2122
2 ==
pi ( ) ( ) ( ) ( ) ( ) ( ) 02222 10051005100510051005221005212010220101 ==+=+=+ iiiizzzz
3.
2|)34(||20||)34(||11||34||||34| 21 =+=++=+=+ iwiiwiiiwzziw
)3,4( K 2=R
4.
5)(2534)( 222 ==+= OKOK
325)(|| min === ROKw
725)(|| max =+=+= ROKw
7||3 w
4
3 K
0
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2010
| www.lazaridi.info | tel. 6977.385.358 |
www.lazaridi.info Page 2 of 4
|
Rxxxxf ++= ),1ln(2)( 2 1.
01
)1(2)( 22
>+
++=
x
xxxf Rx f R
:
To 12 ++ xx < 0 Rxxx >++ ,012
2.
+
+=+
11)23(ln)23(2 4
22
x
xxx ( ) ( )++= 1ln1)23(ln)23(22 422 xxxx
( ) ( )++=++ 1)23(ln)23(21ln2 242 xxxx 112 )23()( = fxfxf = 232 xx
2,10232 ===+ xxxx
3.
( )22
2
1
)1(2)(+
=
x
xxf
1010)( 2 === xxxf
pi )2ln2,1( + : 2ln1)1(1)2ln2()1()1()1( +=+=++= xyxyxffy (1)
pi )2ln2,1( + : 2ln13)1(3)2ln2()1()1()1( +==+= xyxyxffy (2)
() (1) (2) pipi + )2ln1,0(
4.
340
322)1ln(2))1ln(2())1ln(2()(
1
1
21
1
21
1
221
1
21
1
=+=++=++=++==
dxxxdxxdxxxxdxxxxdxxxfI
+=1
1
21 )1ln( dxxxI ( ) pi, :
)( xf
)( xf
( )( )2ln2,1
)1(,1..
+
f
1x
+
1
( )( )2ln2,1
)1(,1..
+
f
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2010
| www.lazaridi.info | tel. 6977.385.358 |
www.lazaridi.info Page 3 of 4
A- )1ln()( 2 += xxxg pi pi ( )()( xgxg = ) [-1,1] pi pi
- =+
=+=
1
1
221
1
21 )1ln(2)1ln( dxx
xdxxxI
=
+=
+=
+
+=
1
12
1
12
31
12
21
1
22
110
12
2)1ln(
2dx
x
xxdx
x
xdxx
xxx
x
02
)1ln(01
1
1
21
12
1
1
=
++=
++=
xdxx
xxdx
- 12 += xu , xdxdu 2= 0ln21)1ln(
2
2
1
1
21 ==+=
ududxxxI
|
1.
++=x
dtttf
txxf
0 )(3)( 3)0( =f Rxxxf ,)(
Rxxxf
xfxxf
xxxfxxf
xxf
=
+
=
+= ,)()(
)()(
)(1)(
2.
( ) ==+== )(2)(2)()(2))()((2)()(2)(2)()( 2 xfxfxxfxfxfxxfxfxfxxfxfxg
0)(2)(2)(2))(()()(2)(2))(()(2
1==
==
xfxfxfxxfxxf
xfxfxxfxf cxg =)(
cg == 9)0( 9)( =xg , Rx
3. 09)(2)(9)(2)(9)( 22 === xxfxfxxfxfxg pi )(xf
9)( 2 += xxxf pipi 9)( 2 += xxxf pipi pi 3)0( =f pi
Rxxxxf ++= ,9)( 2
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2010
| www.lazaridi.info | tel. 6977.385.358 |
www.lazaridi.info Page 4 of 4
4.
+
=1
)()(x
x
dttfxh , Rx pi )1()( +< xhxh
++
+=+=1
00
1
0
0
)()()()()(xxx
x
dttfdttfdttfdttfxh
0)()1()1()()( >+=++= xfxfxfxfxh h R
pi
+
+
++=
=
x
xx
xxfxf
xf f R
0)()1()1()(1 >+++++
