Mathimata Prosanatolismou G Lykeioy New 2
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Transcript of Mathimata Prosanatolismou G Lykeioy New 2
, 1 -
1
,
&
(. , . .
-
:
: 2015-2016
, 1 -
2
1 : -5
2 : .10
3 : .16
4 : - ....24
5 : ...............................................................................30
6 : - ( 1)....................................36
( 1-6)...............................................................40
7 : .............................................................................43
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8 : ..............................................................................49
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9 : 11......................................................................................52
10 : ........................................................................58
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11 : 2- ...............75
( 7-11)....................................86
.........................................88
2-
1-
, 1 -
3
12 : 0x .................................................................91
13 : .......................................................................................101
14 : 0x ........................................................123
15 : .................................................................135
16 : 3...................................................................152
........................................................................................154
3........................................156
3-
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4
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2
3
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7 A > 0, : 1 1
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:
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(, ] = {x R | < x } : - .
(. 3)
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7
R,
:
(, +) = {x R | x > }
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(, ) = {x R | x < }
(, ] = {x R | x }
(. 4)
R (,+) .
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8
1 2 2
2 2
3
4 ( 0)
5
6 ( 0)x x
7 x x x ( 0)
8 0 0 0x x x x x
9 0 0 0 x x x x x x
.
1. : 13, 4, , 23
2. 2 1 : 2 3 5 3
3. :
1 3 ...
2 1 ...2
3 3 5 ...
4 ...
5 ...
6 ...
7 ...
8 ...( 0)
9 2 ...
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9
4. :
1x
2x 113
x
2 2x 3 8x
5. :
( , 1)
[2, )
( 3,7]
2[0, ]5
3 ,2
.
1.
1 2/ 13 2
1/1 2 12
x xx
B x x
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R.
() f , x A
y. y f x f(x).
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x f (x)
x,
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f fD .
f x A,
f f A . :
/ ( ) f A y y f x x A -
fD f
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A xf xB x
, ( ) 0B x x
, f :
/ ( ) 0fD x B x
2
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2 : ( ) ( )f x A x , ( ) 0A x
f :
/ ( ) 0fD x A x
3 :
( ) ln ( )f x A x ( ) log ( )f x A x , ( ) 0A x
f :
/ ( ) 0fD x A x
.
- 1. :
) 21( )4
xf xx
) 22 3( )
3 4xg x
x x
(1 )
) :
2 24 0 4 2 2x x x x
f :
2, 2fD ) :
2 3 4 0 2 1x x x x
f :
1, 2gD
2. :
) 3( ) 3 1f x x ) 2( ) 4 3g x x x
, 1 -
12
(2 )
) :
13 1 03
x x
f :
1 , 3f
D
) :
2 4 3 0 1 3x x x x
f :
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3. :
) 2( ) ln 1f x x )
3 2( ) log 2 2g x x x x
(3 )
) :
2 1 0 1 1x x x
f :
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) :
3 2 2 22 2 0 1 2 1 0 1 2 0x x x x x x x x
2 1 2
1x - - + +
2 2x + - - +
3 2 2 2x x x - + - +
f :
2,1 2,gD
4. :
, 1 -
13
) 221( ) ln 5 4
9f x x x
x
) 3 23 2
1( )ln( 2 2)1
xg xx x xx
( )
) :
2 2
2
9 0 0 3 3
5 4 0 1 4
x x x x
x x x x
f :
, 3 3,1 4,fD
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2
3 2 2 2
1 0 1 1
2 2 0 2 2 0 2 1 0
2 0 2
x x x
x x x x x x x x
x x
f :
2,fD
.
1. :
) 21( )
9f x
x
)
2( ) ln 3 4g x x x
2. :
) ( ) 5 15f x x ) 5 2( ) 25g x x
3. :
) 2( )3 1xf xx
) 23( )
4 3xg x
x x
1/. ;
, 1 -
14
.
1. :
) 3 21( )
6 11 6xf x
x x x
) 3 3 2
2( )6 11 6xg x
x x x
2. :
)
3 2
3
2 2( )ln 8
x x xf xx
)
3 241( )
ln 2 2g x
x x x
3. :
) 3 221( ) ln 6 52 1
f x x xx x
) 4 25 21( )
ln( 2 1)4 4xg x
x xx x
4. :
) , 1
( ) 10, 1
x xf x x
x
)1, 0
( ) , 1 02, 1
x xg x x x
x x
5. :
1( ) 1 1 ln ( ) ln1
xx
x
ef x e x g xe
6. :
21( ) + ( ) 2
2 1 1xxf x g x x
x x
( ) 7.
2
1( )1 2 1 3
f xx x
,
8. ,
, 1 -
15
) 2( ) ln 1f x x x ) 2( ) 2 3g x x x
9. :
11( ) 3 log , ( ) ln (2)10
f x x a g x f f x f
, a
fC 2, 3A : i) a .
i) N f .
ii) N g .
10. :
2
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f xx x
2 5f 5 24f
i) N f
ii)
iii) 1 , 3f f f
iv) ( ) 3f x
( ) 11. : 0,f :
ln 1xf x f xe
0x
i) f
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1 1 1, , 2 2 2
f f f f f f
12. :f : 22 ( ) (1 ) 2 1f x f x x x , x
i) f
ii) ( ) ( 2)g x f x
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16
:
f Oxy
. M(x, y) y f x ,
, ( )M x f x , x A , f
fC . , , y = f(x) fC .
, y f x f .
x A y ,
f .
f (. 7).
, ,
. (. 7).
f , :
) f fC .
) f f A fC .
) f 0x A 0x x
fC (. 8).
3
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17
fC , f , ,
f f
:
) - f
, xx,
f ,
M(x, f (x))
M(x, f (x)),
xx. (. 9).
) f
fC
xx ,
xx, fC
. (.
10).
f(x) = x +
f(x) = x 2 , 0 .
, 1 -
18
f(x) = x 3 , 0 .
af xx
, 0 .
( )f x x ( )g x x .
, 1 -
19
, 0
( ), 0
x xg x
x x
,
. y x
yy.
: f(x) = x, f(x) = x, f(x) = x
, f (x) = x f (x) = x
T = 2, f (x) = x = .
f(x) = x, 0 < 1 .
:
, 1 -
20
1 x y x ya a a
x x ya a 2 x x
xa
3 yx xya a
4 x x ya a
5 0 1a
