Θεωρία Και Ασκήσεις Στις Μονότονες Συναρτήσεις -...
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Θεωρία και Ασκήσεις Μονότονες Συναρτήσεις - Αντίστροφη Συνάρτηση για το Μέρος Β Ανάλυση Κεφάλαιο 1 - Όριο - Συνέχεια Συνάρτησης, Μαθηματικά Κατεύθυνσης Γ' Λυκείου
Transcript of Θεωρία Και Ασκήσεις Στις Μονότονες Συναρτήσεις -...
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2)( xxf R .
]0,(, 21 xx 21 xx )()( 21 xfxf ,
),0[, 21 xx 21 xx )()( 21 xfxf . f
]0,( ),0[ .
f :
) ,
21, xx 21 xx )()( 21 xfxf
) ,
21, xx 21 xx )()( 21 xfxf
) , 21, xx
21 xx )()( 21 xfxf
) , 21, xx
21 xx )()( 21 xfxf
f
f , f .
f f
.
f f
.
x1 x1 x2 x2
f(x1)
f(x1) f(x2)
f(x2)
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) f g gf .
) f g gf .
:
) f g . 21, xx 21 xx
)()( 21 xfxf )()( 21 xgxg , :
))(())(()()()()( 212211 xgfxgfxgxfxgxf
gf .
) .
: f (
) ,
( ) .
x
xf1
)( , *Rx
)0,( ),0( ,
*R .
)0,(, 21 xx 21 xx )()(11
21
21
xfxfxx
)0,(f .
),0(, 21 xx 21 xx )()(11
21
21
xfxfxx
),0( f .
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*21, Rxx 21 0 xx )()(11
21
21
xfxfxx
.
f *R .
f
21, xx :
i. f )()( 2121 xfxfxx , f
)()( 2121 xfxfxx .
ii. fC ox xx
0)( xf .
1)
f .
i. f Axo () ,
)()( oxfxf Ax . )( oxf
f .
ii. f Axo () ,
)()( oxfxf Ax . )( oxf
f .
f f .
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1) H 2)( 2 xxf 0ox
2)0( f , Rx )0()( fxf .
2) H xxg )( 0ox
0)0( f , Rx )0()( fxf .
x x'
y'
y
0
y
O
y'
x
-2
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2)
f ],( A fC
( 1).
)()( oxfxf Ax ox .
,
Ax ),( oo xxx )()( oxfxf . f
Axo .
1
f A .
f Axo ,
0 )()( oxfxf
),( oo xxAx .
Axo ( ) )( oxf
f .
1 f Axo A .
2 f
],( A .
)()( oxfxf Ax ox . ,
Ax
),( oo xxx )()( oxfxf . f
Axo .
y
x O
y'
x
Cf
f(x)
x x- x+
f()
- +
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2
f A .
f Axo ,
0 )()( oxfxf
),( oo xxAx .
Axo ( ) )( oxf
f .
2 f Axo A .
3
.
y
x O
y'
x
Cf
f(x)
x x- x+ - +
y
x
O
y'
x x3 x1 x2 x4
x5 x6
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3 f 31, xx 5x
42 , xx 6x . )()( 61 xfxf
)()( 63 xfxf .
f ox
() . f .
f ox
() .
f .
. f
( ), (
) ( )
f . ( 3)
1) 1,
1,2)(
2
x
x
x
xxf . 4
f .
4 f 11 x
02 x .
f .
4
y
x O
y'
x 1 -1 -2
1
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2) 1,
1,
3
2)(
3
x
x
x
xxg . 5
g .
5 g 11 x
32 x .
g .
5
y
x O
y'
x
1
1 2
2
3
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(1-1)
f
(1-1), Axx 21, 21 xx )()( 21 xfxf .
1: f 1-1,
Axx 21, )()( 21 xfxf 21 xx .
.
2: f
, 1-1.
.
2 .
1-1, .
: x
xf1
)( , *Rx 1-1,
*R .
i) 1-1, xx
.
ii) f 1-1,
)(Afy Ax yxf )( .
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f A
1-1 AxxfyRyAf ),(/)( f . f 1-1, )(Afy Ax
yxf )( . g )(Af
A )(Afy Ax ,
yxf )( .
:
xygyxf )()( (1)
:
)(
:
xfyx
RAf
)(
)(:
ygxy
RAfg
g f 1f .
1
(1) :
xyfyxf )()( 1
)(xfy xyf )(1 , ( Ax )
xxff ))((1 , Ax .
)(1 yfx yxf )( , ( )(Afy )
yyff ))((1 , )(Afy .
),( yxM Ax )(xfy
f . ),( xyN , )(Afy )(1 yfx
1f . ),( yxM ),( xyN
x=f-1(y) y=f(x)
A f(A)
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xy , (
).
fC 1fC
xy .
2
0a 1a .
xaxf )( R
),0( .
f R ( 1a f
R , 10 a f R ).
f 1-1 1f
),0( R .
:
xyfyxf )()( 1
xyfya x )(1
:
xyya ax log
yyf alog)(1
xaxf )(
xxf alog)(1 .
y
x O
y'
x
Cf
Cf-1
y=x
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3 xaxf )(
xxf alog)(1 1a .
3
y
x O
y'
x
Cf
Cf-1
y=x
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f A .
) f , Ax Ax
)()( xfxf .
) f , Ax Ax
)()( xfxf .
:
) f A ,
Ax ))(,( xfxM ))(,( xfxN
. f
yy .
) f , Ax ))(,( xfxM
))(,( xfxN .
f .
.. xxf )( Rx ( xx )( ),
xxg )( Rx ( xx )( )
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1) 85)( 32 xexf .
) f .
) , , .
:
) f R . Rxx 21, 21 xx :
)()(8585
55323233
21
3232
32323232
2121
21
2121
xfxfee
eeeexxxx
xx
xxxx
Rf .
) Rf f 1-1. :
8,5
8ln
3
1
3
2
8,5
8ln23
8,5
8ln32
8,5
8lnln
8,5
8
8,8585)(
32
32
3232
yy
x
yy
x
yy
x
yy
e
yy
e
yyeyeyxf
x
x
xx
)8,(,5
8ln
3
1
3
2)(1
y
yyf
)8,(,5
8ln
3
1
3
2)(1
x
xxf .
2) 27
ln5)( x
exxf .
) f .
) xexx 27ln5 .
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:
f ),0( .
) ),0(, 21 xx 21 xx :
21 lnln xx 21
11
xx
21 ln5ln5 xx 21
77
x
e
x
e
:
)()(27
ln527
ln57
ln57
ln5 212
2
1
1
2
2
1
1 xfxfx
ex
x
ex
x
ex
x
ex
f ),0( .
) :
0)(027
ln527
ln527ln5)0(
xfx
ex
x
exxexx
x
(1)
027
ln5)( e
eeef (1) ex
),0( f ex .
3) ),5(: Rf R .
5)(
2)()(
xf
xfxh R .
:
Rx :
5)(
31)(
5)(
35)()(
5)(
2)()(
xfxh
xf
xfxh
xf
xfxh .
Rxx 21, 21 xx :
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)()(5)(
31
5)(
31
5)(
3
5)(
3
5)(
1
5)(
15)(5)()()(
21
2121
21
5)(
2121
xhxhxfxfxfxf
xfxfxfxfxfxf
xf
h R .
4) x
x
e
exf
3
3
2
1ln)(
.
) f .
) f .
:
003101 0333 xxeeee xxx . f
),0( .
)
xexf
32
1
2
1ln)( .
),0(, 21 xx 21 xx :
)()(2
1
2
1ln
2
1
2
1ln
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
12233
213333
3333
3333
21
2121
2121
2121
xfxfeeee
eeeeeeeexx
xxxx
xxxx
xxxx
),0( f .
) f ),0( , 1-1.
.
:
)1(12
1
1)12(122
1
2
1ln
3
333
3
3
3
3
y
x
xyxyx
x
xy
x
x
ee
eeeeee
ee
e
ey
:
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2
1ln
2
1120120
12
10
12
1
yeee
ee
yyy
yy
2
1ln,y .
(1) :
yyy
x
ex
ex
ee
21
1ln
3
1
21
1ln3
21
13
2
1ln,,
21
1ln
3
1)(1 y
eyf
y
2
1ln,,
21
1ln
3
1)(1 x
exf
x
5) ),0(: Rf
xxexf xf
3ln5))(ln( )( ),0( x . f
),0( .
:
5ln)( xexxg , ),0( x :
xxxfgexfxfg xf
3ln))((5))(ln())(( )(
g ),0( . ),0(, 21 xx
21 xx :
)ln()ln( 21 xx 21 xx ee :
)()(5)ln(5)ln()ln()ln( 2121212121 xgxgexexexex xxxx
),0(, 21 xx 21 xx :
)ln()ln( 21 xx 21
11
xx
)ln()ln( 21 xx 21
33
xx :
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))(())((3
)ln(3
)ln( 212
2
1
1 xfgxfgx
xx
x
)()( 21 xfxf ),0( f .
6) 13
)( xe
xf .
) f R .
) 1f .
) h : 3
)())((
1 xfxfh
.
:
) Rxx 21, 21 xx :
)()(13
1333
212121
21 xfxfeeee
eexxxx
xx
Rf .
) f R , 1-1. 1f . :
1
3ln
1
3
1
1
31
31
3)(
1
yx
ye
y
ey
eeyxfy x
xy
xx
),1(,1
3ln)(1
y
yyf
),1(,1
3ln)(1
x
xxf
) :
),1(,1
3ln
3
1))(()(
3
1))(( 1
x
xxfhxfxfh (1)
yxf )( xyf )(1 .
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(1) :
1)(
3ln
3
1)(
1
yfyh (2)
:
13313
1
1
3
ln1
3ln1
1
3ln1)(01)(
1
111
eyeyey
ey
eyy
yfyf
1y )13,1( ey .
(2) :
)13,1(,
11
3ln
3ln
3
1)(
ey
y
yh
)13,1(,
11
3ln
3ln
3
1)(
ex
x
xh
7) RRf : : 93))(( xxfof
Rx . f .
:
Rxx 21, )()( 21 xfxf :
2121
212121
33
9393))(())(())(())((
xxxx
xxxfofxfofxffxff
f 1-1 . :
3)(3
19)(3
93)())(()())(()()(
yfxyfx
xyfxfofyfxffyfxfy
: Ryyfyf ,3)(3
1)(1
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Rxxfxf ,3)(3
1)(1
