Άλγεβρα Α' Λυκείου - Θέματα ΟΕΦΕ (2006-2013) - Ερωτήσεις και απαντήσεις
5. Παράγωγοι Α' (2013).pdf
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Transcript of 5. Παράγωγοι Α' (2013).pdf
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()
2:
555 1
2.1
1. f 0
x ,
:
0
x .
0 0, x x , 0, x 0x , -
.
0x .
2. H 0
x :
x = S(t)
0t
0 0(t ) S (t ) , -
.
: , 0t
00
S t S t0
t t
,
0
(t ) 0 .
, 0t
00
S t S t0
t t
,
0
(t ) 0 .
, (t) 0t
0 0
(t ) (t ) , .
( )
1. 0
x , 0
x .
[ . 217 ]
2. 0
x 0
x .
[ ]
1. , ... -
, , -
.
15
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2.
,, , , -
.
3. f 0 f
x D
f 0
x
0x .
4. ,
, 0
x
, :
0f x x 0x .
0
0
f x f x
x x
.
(.. )
0
0
0x x
0
f x f xlim f x
x x
.
5. f,
(.. )
, (. 0
x ), :
f
x
f x f lim f
x
0
0
f x f x
x x
0
x x
.
0x x h . :
f -
0x x h ,
g -
0
xh, h 0
x .
6. ,
0
0
x x
0
f x f xlim
x x
, f
.
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[ .219-221]
1.
) 2f(x) x x 0
x 1 0
x 2 .
) ( ) 1 x1 2g x , x , , 0x 0 .
2. 2
x , x 0f(x)
x x 4 , x 0
, , f 0.
3.
2
2
x x , x>1
f(x) x 1
x 2x , x 1
, , , f 0
x 1 .
4. g(0) 1,g (0) 2 . , 2g (x), x 0
f(x)x , x 0
,
0x 0
.5. f 1 x :
3f x 3 x 2 x f x . , , f x = 1 .
6. f 0
x 1 x 1
f(x) 2lim 3
x 1
,
f 0
x 1 .
7. f 0
x , x
f(x) f()lim
x
.
8.
A g 0
x , f x x) x( ) g(
0
x g() 0 .
9. f, g 0
x 0 f(0) g(0) 0
( )x)f g(x x , f (0) g (0) .
10. x f x 2) 1( )g(x 0
g(x ) 1 0
g 0(x ) , f
0
x .
11. f, g 0
x 1 f(1) g(1)
2( ) ( )f x x g x x , x , : f (1) g (1) 1 .
12. f : 2 22x x f (x) 2x f(x) x ,
x . f (0) 1 .
13. A x 2 2 6 3f (x) g (x) x 2x 1 f, g -
0
x 1 , :
) f( 1) g( 1) 0
)
2 2
f ( 1) g ( 1) 9
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14. f : f (1) 0 .
f(x 1) , x 2g(x)
f(3x 5) , x 2
2.
15. f 0
x :
) 0 00h 0
f(x 2h) f(x )lim =2f (x )
h
) 0 00h 0
f(x h) f(x 3h)lim = 4f (x )
h
)0
0 00 0 0
x x0
x f(x) x f(x )lim f(x ) x f (x )
x x
) 0
0
0
xxx0
0 0 0x x0
e f(x) e f(x )lim e f(x ) x f (x )
x x
) 0 00 0
h 1
f(x h) f(x )1f (x ) lim , x 0
x h 1
16. f 1, 2 3 3x f (x) 2f(x) x 1 ,
x . N f 1.
17. f : 3f (x) f(x) x x , x .
:
(i) f x x x , x R .
(ii) f 0.
18. f, g : 2 2 2 2f (x) g (x) x x ,
x . :
) f x x x g x x x , x .
) f g 0.
***********
19. f : ( ) ( )f x y f x f( )y 2xy x,y f(0) 0 .
N
) f 0, f .
) f f() 0 , f .
20. f : , 0
x 0
f( ) f() f() , , :
) f(0) 0
) f (x) f (0) x , x .
21.
f : 0, 0,
:f(x y) f(x) f(y) f(x) f(y) , x, y . f
x 0 , f .
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22. f x, y :
2 2f(x) y f(x y) f(x) y . f 0
x .
23. f , 0
x 0 ,
2 4xf(x) x x x . f
0
x 0 .
24.
f 1 f (1) 2
, x,y * - f(x y) f(x) f(y) (1) . f -
0 0
0
2x *, f (x )
x .
25. f : 0
x 0 .
x 0
f(x) f(x)lim ( ) f (0)
x
, , .
26. f x,y
2f(x) f(y) x y (1) . :
) 2
f(x) f(y) x y x,y .
) f 0
x ,0
f (x ) 0 .
27. f : (0, ) :f(y)f(x)
f(x y)y x
, -
x, y 0 . f (1) 1 , f
0x (0, ) .
28.
f : , f x y f x f y *x, y .
) f 1 0 .
) x
f f x f yy
.
) 1
f f yy
.
) f 1 ,
.
29. f 0 f (0) =1, :
f(x) 2 x (1)
f(x) 2f(y) 6f(x y)
f(y) 2
x,y (2).
f 0
x R , 0 0
f (x ) f(x ) 2 .
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&
()
2:
555 1
2.2
1. f (x) f(x) . -
f.
, 0
f (x ) 0f(x ) .
0
f (x ) f 0
x , 0f(x ) 0,
0f(x ) .
2. .
3. f , f .
4.
f , f
.
5. f(x) x 0, 0, .
6. f 0 f
x D ( ()0
f (x ) ), -
( 1)f 0
x 0 0
(x , x ) 0
(, x ]
0
[x ,), 0 .
7. ! ()f (x) ( ) f (x) (
), :() ( 1)f (x) f (x)
1f (x) f (x) f(x) .
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[ 1,2,4,5 1,2 .227-228]
1. :x
x 0
e 1lim 1
x
.
2.
: x 1ln x
lim 1x 1
.
3. f : , f(x y) f(x) f(y) x, y .
) f(0) 0 .
) f(x y) f(x) f(y) .
) f .
) f 0, f .
4.
f: 0, f(x y) f(x) f(y) , x, y (0, ) 1 x
1 x f(x)x
, f .
5. f : 0
x 1 f(1) 1 f (1) 1 . -
:
)
2
x 1
f(x) 1lim
x 1
)
x 1xf(x) 1lim
x 1
)x 1
xf(x) 1lim
x 1
6. f , g
f 0
x
g 0
x .
(f g)(x) 0
x , -
0
g(x ) 0 .
7. f : 2x , x 0
f(x)x, x 0
) f.
) f 0
x 0 .
) x 0
1lim
f (x) .
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()
2:
555 1
2.3 -
1. x
.
: .
2. -
f, g0
x , 0 0 0
(f g) ( x ) f (x ) g (x )
0 0
f(x ) g(x )
0 0f(x ) g(x ) 0
0 0f(x ) g(x ) .
.
3. f 0
x ,
f g, f g fg
0x .
0
x
.
4. f, g 0
x
f g f g f
g
0x .
:
x x 0
f x0 x 0
,x x x 0
g xx x 0
0
x 0, f g (f g)(x) x
0x 0 .
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5. f
, :
i) f , -
f .
ii) f ,
0
x ,
, f ,
,
.
: .
6. f,g f(x) g(x) f (x) g (x) . f (x) g (x)
f(x) g(x) .
7.
f , 1
f
f() f (x) 0, x f() , :
1
1
1f x , x f
f f x
:
x f() 1f f (x) x :
1 1 1f f x x f f x f x 1
1
1
1
(f ) (x) , x f()
f f (x)
(1)
.
(1) , 0 0
f(x ) y 0
f (x ) 0 , 10
0
1(f ) (y )
f x
.
( )
1. f , f -
.
2. f , f
.
******
: -
,
.
, ,
.
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( )
f fA f
f
1)
f(x) c
(c) 0
2) f(x) x (x) 1
3)f(x) x , {0, 1} 1(x ) x
4) *f(x) x ,
* 1
(x ) x *
5)f(x) x ,
[0, ) , 0,
(0, ) , 0
1(x ) x
[0, ) , 1,
(0, ) , 1
6)
f(x) lnx
(0, ) 1(lnx)x
(0, )
7) f(x) logx (0, )1
(log x)x ln10
(0, )
8) f(x) ln x *
1(lnx)
x
*
9) f(x) x [0, ) 1
x2 x
(0, )
10)x
f(x) e x x
(e e)
11)x
f(x) , 0x x
( ) ln
12) f(x) x (x) x
13) f(x) x (x) x
14) f(x) x
fA {x / x 0}
{x / x , }
2
2
2
1(x)
x(1 x)
f fA A
15) f(x) x f
A {x / x 0}
{x / x , }
2
2
1(x)
x
(1 x)
f fA A
16)1
f(x)x
* 2
1 1
x x
*
17)
f(x) x
1, x 0
x 1, x 0
*
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: *f(x) x , ,
.
, :
f(x) x f
A [0, ) .
, :
( x) , x 0f(x) x
x , x 0
f
A
.
,
.
!
f(x) x , g(x) x ,
fA [0, )
gA .
( )
( )
1) (f g) (x) f (x) g (x)
2)
( f) (x) f (x)
3)
1 1 2 2 1 1 2 2
( f f ... f ) (x) f (x) f (x) ... f (x)
4)
(f g) (x) f (x) g(x) f(x) g (x)
(f g h) (x) f (x) g(x) h(x) f(x) g (x) h(x) f(x) g(x) h (x)
( 3 -
)
5) 2f (x) g(x) f(x) g (x)f
(x)g g (x)
( )
g f g(),
f g :
(f g) (x) f (g(x)) g (x) f(g(x)) f (g(x)) g (x)
u g(x), : f(u) f (u) u
y f(u) u g(x), :dy dy du
dx du dx ( )
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1 2 3
y f(u (u (u (....u (x)....))))), : 1 2
1 2 3
dudu dudy dy...
dx du du du dx
( V)
f(x) ,
:
u f(x), f
, :
1) 1f (x) f (x) f (x), {0, 1} 1) 1u u u , {0, 1}
2)f (x)
f(x) , f(x) 02 f(x)
2)u
u , u 02 u
3) f(x) f(x) f (x) 3) u u u
4) f(x) f(x) f (x) 4) u u u
5)2 2
1 f (x)f(x) f (x)
f(x) f(x)
5)2 2
1 uu u
u u
6)2 2
1 f (x)f(x) f (x)
f(x) f(x)
6)2 2
1 uu u
u u
7)
f(x) f(x)e e f (x)
7)
u u
e e u
8)1 f (x)
ln f(x) f (x)f(x) f(x)
8)1 u
ln u uu u
9)
f (x)log f(x)
f(x) ln
9)
ulog u
u ln
10) f(x) f(x) ln f (x) 10) u u ln u
11)
1f (x) f (x) f (x), f(x) 0, {0,1}
11)
1u u u , u 0, {0,1}
! g(x)(x) [f(x)] f(x) 0, g(x) lnf(x)(x) e
g(x) g(x) lnf(x) g(x) lnf(x) g(x) (x) [f(x)] e e g(x) ln f(x) [f(x)] g(x) ln f(x) ...
:
-
.
,
.
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[ 1,2,3,4,6,12,13,14,15 7,9 .238-240]
1. :
)x x xe lnx
f x , g x x x , h x ,x 1 1 x1 x
)x
1 x 2 xlnx 2x 1f x , g x , h x , x
1 x x 2 1 x e
)2 2
2 x x x 1 x x 1 ef(x) x 2 2 e x
)xxxx x 3f(x) x , x 0, g(x) (x) , x 0, , h(x) (x 1) , x 1, (x) 3 , x
2
) 2 2x 1
f(x) x , x 0, g(x) log (x), x (1,2) (2,), h(x) ( x)( x)
2. -
Leibniz ( ):
)
(x) ln(x), x (0,)
) 4 2k(x) (3x 1)
3. ,
:
) xf(x) (e 1) ln(x 1)
) 2g(x) ln(1 x )
)
xe e
h(x) (2x 1) lnx
4. N :
) 3f(x) x
) 5 2g(x) x
)3 x
1h(x)
2 5
5. N f 0
x : f(x) x x 0
x 0 .
6.
f : 3 1f (x) xf (x)x
x 0 . N f (1) .
7. A f , :
) f (0)
) g (0) , f (0)=1 g(x) f(x)x f(x)
8. f(x)=2
2
x
1+ x
,
f 3f
4 4
.
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9. N :
) 2f x x ln x, 22f(x) xf (x) x 0
) x
xy
e
, dy
x x 1 y 0dx
) xy e x x , xy 2y 2e x 0
10. f x :
f(2x 3) f(x) . f 0
x 3
xx.
11. A64 27
( )
f xx x
, 0
x 0,2
0
f 0(x ) .
12. A 2f(x) x g(x) x,
) (g f) (1) g f(1) f (1) (g f) (1) .
)
(g f) (0) g f(0) f (0) . (g f) (0) ;
*****
13. P(x) x :2
P x 4P x
P(0) 4 .
14. ) (x) 2, 2
x -
-
. ,
2
x x x 0 .
) 2
x 1
1 1 2 2f x x x 1 x 2 , 2.
) ,
4 2 x x 4 x 1 x 3 4
2
x 2 .
15. 3 2f x x x x 1 2 3
, ,
. :
i)1 2 3
f x 1 1 1
x x x f x
1 2 3
x , ,
ii) 31 2
1 2 3
0
f f f
iii)1 2 3
f 0 1 1 1
f 0
iv)
1 2 3
22
2 2 2 2
f 0 f 0 2
1 1 1
f 0 f 0
*****
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16. :
) x 2x 3x x1
S 1 e e e e
) x 2x 3x x2
S e 2e 3e e , .
17. f : , y xf(x y) e f(x) e f(y)
(1), x,y, , :
) f(0) 0
)
xf (x) f(x) f (0)e
, x .
18. *f :
f(xy) f(x) f(y) *x,y R f (x) 0 . :
)yf (x)
f (y) x
) f (1) f ( 1) 0
19. f : , :
f (0) 1 f(x y) f(x y) 2f(x)f(y) x, y .
: f (x) f(x) x .
[ f() ]
20. :
) 1
f(x)x
()
1
( 1) !f )x(
x
,
) f(x) x ()
( ) +x2
f x
,
) xf(x) xe , () xf (x) e (x ) .
21.
1
1 1 0f x x x x ,
0 1 1 , , , ,
0 .
()
f (x) !
()f (x) 0 .
[ f-1]
22. . f : (,) R , . f -
0
x (,) 0
f (x ) 0 1f 0
f(x ) :
1f 0
f(x ) 10
0
1(f ) f(x )
f (x )
.
. xf(x) e x , 1(f ) (1) .
23. x 3f(x) e x x , x .
(i) f 1f-
(ii) 1f- 1
fD
-
,
1 1(f ) (1)
2
.
24.
A f(x) x,
x ,2 2
1
2
1
(f ) (x) , x ( 1,1)1 x
.
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()
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555 1
2.1 - 2.3
1. f
C f 0 0
(x ,y )
0 0 0 0
(x ,y ) (x ,f(x )) -
0
f (x ) f 0
x .
, : 0 0 0y f(x ) f (x )(x x )
:
f , :
) 0 0
A(x , f(x )) -
: .
) : , 0 0
M(x , f(x )),
0
x , 0
x -
.
1) (x0,f(x0)) Cf. [5 220, 7,B1,11239]
) 0
f (x ) 0
f(x ) .
) 0 0 0
y f(x ) f (x )(x x ) .
2) ( ) , Cf. [10 239]
)
0 0M(x , f(x ))o -
():0 0 0
y y f (x )(x x ) .
(, , 0
x )
)
.
) 0
x
.
3)
[3 228, 8,9,B2,6239])
0 0M(x , f(x ))o .
) 0
f (x ) , 0
x .
) , , .
18
x0
M
Cff(x0)
A
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4) [A11, 2 239]
: y x f
C
0 0
M(x ,f(x )) f
C -
:
) (),
0 0( )f x x .
) ()
f
C , 0
f (x ) .
5) [3 239]
0 0
M(x ,f(x )) . -
f g
0
x :
)0 0
f(x ) g(x ) ,
fC gC , y f(x)
y g(x) ,
)0 0
f (x ) g (x ) , f
C g
C
.
0
x -
.
6) ()
f g.
() f
C g
C ,
,f( ()) ,g( ())
:
) f () g ()
) f
C ,f( ()),
,g( ()) . , -
.
7) () [4,10 239]
() f
C
,f( ()) g
C . ,g( ()) g
C
:
) f () g () g
A g
C -
().
) g
C
,f( ()) .
x0
M
Cf
f(x0)=
y=x+:
x0+
x0
M
Cf
f(x0)=
=x+:
x0+
x0
M
Cf
f(x0 )
Cg
g(x0)=
Cf
Cg
A
B
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1. 2f(x) x x 3 0,1 .
f
C .
2. 2f(x) ln x x 3 , , R : 2x y 4 0 -
fC
A 1,f(1) .3. f
3 2 3 4f(x) x f(x) 2x 4x -
x . f
C A 1,f(1) .
4. f : 2xlnx f(x) x x x . -
0
x 1
f
C M 1,f(1) .
5. 2f(x) x x 1 . -
f
C :
) (1,1)
) (2,1) .
6. 2f(x) x 4x 33 , 2g(x) x 1 .
7. f 3 2f(x) x x 2x 5 () : 2x y 1 .
) ()
f .
) () f
C .
8. x1f(x) x e 3 2g(x) x 3x 5x . -
f
C A 1,f(1) , gC .
9. 2f(x) 2x x 2g(x) x 4x 1 .
10. f y 2x 1 f
C
1 . 2
x 1
f (x) 1lim
x 1
.
11. 2f(x) x 2x 6 7 , .
) , f
.
) , f
C xx.
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()
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555 1
2.4
1. y f(x) 0
x .
y x
0x
0f (x )
y x f (x).
2. x,y y f(x) f
x, :) y x , f (x) 0 .
) y x , f (x) 0 .
[ ]
3. S S(t) -
t. H S
t .
4. S t 0t
0S (t )
, S t 0t , ()
0t
0(t ) .
0 0=(t ) (tS ) .
, ,
(t) S (t) .
5. t 0t -
0
(t ) , t 0t , ()
0t
0(t ) .
, , 00 0
t =( S) = ( ) (tt ) .
, , -
. (t) (t) S (t) .
6. , :
) S(t) 0, .
) S(t) 0, .
) S(t) 0, .
)
S(t) 1
, ( ).) S(t) 2 , ().
) (t) S (t) 0 , .
19
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) (t) S (t) 0
, .
) (t) S (t) 0
, .
) (t) S (t) 1 , .
) (t) S (t) 2 , .
) (t) S (t) 0 , .
) (t) S (t) 0
, .
) (t) S (t) 0
, .
) (t) S (t) 1 , .
) (t) S (t) 2 , .
[ ]
7. , , ()
x .
8. : ( ) ( ) ( )P t t K t (1),
:
0
(x ) -
x, 0
x x 0
x .
0
E (x ) -
, x 0
x x 0
x .
0
P (x ) P
x, 0x x
0x
.
9. (1) : ( ) ( ) ( ) P t t K t .
10., :
x ,
K(x)K (x)
x .
( ) x ,
E(x)E (x)
x .
x ,
P(x)P (x)
x .
[ ]
11. y x[y y(x) ] x t( x x(t) ), y
t[ y(t) y(x(t)) ].
12. (g f) (x) g f(x) f (x)
dy dy du
dx du dx , y g(u) u f(x) , .
13.dy
dx
, x .
14. dy
dx x , y
.
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1.
( -)
2.
(x,y) (x, y) 0 -
t. :
x(t),y( (t)) 0 .
t.
3.
x ,y
y y
y, 2y 2px ,2 2x y 1 . , ,
) t: x x(t) , y y(t)
) t.
[ ]
3 m ' .
0,1 m/sec.
2,5 m :
) .
) .
:
, x, y, -
t. : x(t),y(t) (t), x (t) 0,1 m/ sec .
) (t) .
.
x 1 1 1
(t) x(t) (t) x(t) (t) (t) x(t)x (t) (t) ...3 3 3 3
) y (t) .
2 2
x y 9
[. ].
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, : 2 2 0 00
0
x(t ) x(t )x (t) y (t) 9 x (t) x(t) y (t) y(t) 0 y (t )
y(t )
,
0t -
0
y(t ) 2,5m . 20 0x(t ) 9 y (t ) ... 2,75 .
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[ .243-245]
1. t -
S 3 2S(t) 2t 21t 60t 3 S t sec. :
) .
) .
)
24 m/sec.
) 18 m/sec2.
) .
)
.
2. (x) (x), x
3 21
x x 20x 600x( ) 10003
x(x) 420
.
) N
.
)
.
) (x) (xP ) )K(x .
3. 22 cm / sec . -
1,8 m, .
4. p(t)
p(t) p (t) 0 .
5. 2 2x y 1 .
1 3
,2 2
, y 3 sec.
N x -
.
6. x y
() () 12cm . 0t -
8cm/sec () 3cm
:
) T
) T ()
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&
()
2:
555 1
2.1 - 2.4
() ()
1. f 0
x , -
0
0
x x0
f(x) f(x )lim
x x
.
2. f 0
x
, 00
0 x x0
f(x) f(x )
f (x ) lim x x
.
3. f 0
x ,
0
0
x x0
f(x) f(x )lim
x x
,
0
0
x x0
f(x) f(x )lim
x x
.
4. f 0
x ,
.
5. f 0
x , -
.6. f
0x ,
0
0 00 h x
f(x h) f(x )f (x ) lim
h
.
7. f 0
x , f
.
8. f 0
x , f
C
0 0x ,f(x ) .
9. f 0
f (x ) 0 , -
f
C 0 0x ,f(x ) y 0 .
10. H f 0 0A x ,f(x ) ,
f
C .
11. f 3 2xf (x) (x 1) e . f
C A 1,f(1) -
.
12. fC x x
, : 0f(x ) 0
0f (x ) 0
.13. () f,
.
19
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-2-
14. f ,0
x 0
x f (x) 1 0
0
x f
C y x 1 3
.
15. -
, .
16. , -
.
17. 2012f(x) x .
18. 2013f(x) x
f
C .
19. f(x) g(x) x (,) f (,), g
(,) f (x) g (x) x (,) .
20. f x , f (x)
x .
21. f : , f .
22. 0
x f(x), 0
x
f (x) .
23. f ,g : , 0
x f g
0x , f g
0x .
24. f ,g : , 0x f 0x g
0x , f g
0x .
25. .
26. xf(x) x , x 0 , f xf (x) x (1 ln x), x 0 .
27. 3f (x) 4x , 4f(x) x 5, x .
28. f 0
x g 0
g(x ), g f -
0
x .
29. f ,g :
, 0x
f, g 0
x , f g 0
x .
30. f ,g : , 0
x f, g
0
x , f g 0
x .
31. f ,g : , 0
x f 0
x ,0
f(x ) 0
f g 0
x , g 0
x .
32. f ,g : , 0
x f g 0
x ,
0g(x ) 0 , fg 0x .
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: 3 2012-2013
-3-
33. g 0
x f
0g(x ), f g
0x .
34. f f(x) 1 x .
35. f
x1
f(x)2
x1
f (x) ln 22
.
36. f : , h 2
h(x) f (x) f(x) .37. f 1 , f -
.
38. f (x) 22x , f(x) 2x 1 .
39. f 1
xf(x) x , x 0 1 f (1) 1 .
40. f ,g : f f(x) g (x), x , f f (1) (f g )(1) .
41. f :
f x x, x ,
2f(x) x, x .
42. f(x) x , f (x) f(x) x, x .
43. 2f(x) ln(x 1) , f (0) 1 .
44. f , f .
45. 0
f (x ) 0 , 0
0
f(x) f(x )0
x x
x
0x .
46. h 0
f(3 h) f(3)lim 10
h
, f (3) 10 .
47. f(x) 3
. f (x) 3
.
48. f f (3) 5 , f (3) (5) 0
49. x 5
d(x 5)5
dx
.
50. f , f
.
51.
2012 ,x 3
f x 2013 ,x 3
, f (x) 0
.
52. f g 0
x , 0 0 0
(f g) (x ) f (x ) g (x )
53. 0
x , -
0
x .
54. , -
0
x , -
0x .
55. , ,
.
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56. , ,
.
57. x
, 0x x
limK(x)
0
x .
58.
.
59. (t)
t. (t) .
) t 0,4 .
) t 2 .
) t 4 .
) t 6 .
) t 8 .
) t 2,6 .
1. , , , ,
f 0.
: :
: :
:
y
x
O
K
2. ()
( )y f x ( , ) 1 1 . ( ) f 1
: 1 : 1
: 2 : 2
:
y
xO
y=f(x)2
1
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