Σπιράλ 5_Παράγωγοι B' (2012-13).pdf
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Transcript of Σπιράλ 5_Παράγωγοι B' (2012-13).pdf
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555 1
2.5) Rolle
1. Rolle -
.
2. f Rolle
[,], , ,
:
0x (,) 0f (x ) 0 . f (x) 0 (,) . f (,) .
[ Rolle]
f xx 0
A(x , 0) 0
x (,) .
0
x (,) -
f 0 0
M(x ,f(x )) xx.
[ Rolle]
( f(x) c ) 0
f (x ) 0
0x (,) [ ].
3. , ,
1
t 2
t , 0
t
1
t ,2
t ( ).
[ Rolle]
4. Rolle ! , -
f
,
Rolle [ ].
20
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5. f (x) 0 (,) . -
.
6. Rolle .
7. [,], [,]
. Rolle f() f() .
[ ]
1. f , f .
2. f , f, -
f.
3. f , f -
f .
4. , f - ( , 1)
1 , ()f ( ) .
5. f (x) 0 x , f .
6. f (x) 0 x , f .
7. 1-1
f (x) 0 .
8. f (x) 0 x , f 1-1 .
1. Rolle
: f(x) 0
(,) (,) (,)
(,) (,) (,)
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!
, Rolle f ,
() F, F -
: F (x) f(x) f
x A .
2. .Rolle f, -
f . ,
f () 0
f () 0
, - Rolle f f .
f F f F
0 c f (x) f(x)
1 x f(x) f (x)21 f (x)
2
x+1x
+1f (x) f (x) 1
1f (x)
+1
1
x2 x
f (x)
f(x)
2 f(x)
x x f(x) f (x) f(x)
x x f(x) f (x) f(x)
xe xef (x)e f (x) f(x)e
1
xln x
f (x)
f(x)
ln f(x)
xx
lnf (x) f (x)
f(x)
ln
f (x)g(x) f(x)g (x) f(x)g(x) f(x) x f (x) x f(x)
2
f (x)g(x) f(x)g (x)
g (x)
f(x)g(x) 2
f(x) xf (x)
x
f(x)x
2[f (x)] f(x) f (x) f(x) f (x) g(x)[f (x) f(x)g (x)]e g(x)f(x)e
f (x) g (x) f(x) g(x)
2
1
x
1
x 2
f (x)
f (x)
1f(x)
2
1
xx
2
f (x)
f(x)
f(x)
2
1
xx
2
f (x)
f(x) f(x)
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3. (,) - . ,
x, , -
, . -
f Rolle.
I . Rolle
1) f f (x) 0 [f(x) x] 0 (x) f(x) x
2) f f
f (x) f(x) 0 f (x) f(x) 0
f (x) f(x) 0 f(x) 0
-x -x
-x -x -x
e e
e e ex(x) f(x)e
3) f ()( ) f() f (x)( x) f(x) 0 f (x)( x) f(x)( x) 0
f(x)( x) 0
(x) f(x)( x)
4) f ()( ) f() 2
f (x)(x ) f(x) 0 f (x)(x ) f(x)(x ) 0
f (x)(x ) f(x)(x ) f(x)0 0
x (x )
f(x)(x)
x
5) 1f () 1 f (x) x 0 f (x) (x ) 0 [f(x) x ] 0 (x) f(x) x
6) f () f()
2
xf (x) f(x) 0 xf (x) f(x) 0
x f (x) (x ) f(x) 0
x f (x) (x ) f(x) f(x)0 0
x x
-1 -1x x
f(x)
(x) x
7) f ()f () 0 2
f (x)f (x) 0 2 f (x)f (x) 0
f (x) 0
2(x) f (x)
8) f () f() 0 2 2 22
f (x) f(x) 0 2f (x) f (x) 2f (x) f(x) 0
f (x) f(x) 0 f (x) f (x) 0
22(x) f (x) f (x)
9) () ( 1)
f () g ()f ()
() ( 1)
g(x) g(x)() ( 1)
g(x) g(x)( 1) ( 1)
g(x)( 1)
f (x) g (x)f (x) 0
f (x)e e g (x)f (x) 0
f (x) e f (x) e 0
f (x)e 0
g(x)( 1)(x) f (x)e
, :
10)2
f () f () 0
f(x)(x) f (x)e
11)f()
f () 0
lnx f(x)(x) f(x)ex
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4. -
[,], . f() f( ) . -
,
, x
. Rolle. f( )f()
-
. Rolle f(x)(x)x
.
5. , , f(x) 0 , -
:
Bolzano
( 0 f() )
( f(x) 0 x )
Rolle f ( f g , g(x) =
f (x))
6. f(x) 0 ,
Rolle.
7. f(x) 0 -
:
Rolle
8. f(x) 0 ,
Rolle .
9. f(x) 0 (,),
.
10. (,)
f () 0 (3 )f () 0
Rolle -
.
11. Rolle F , -
Bolzano f , f
, f ( ).
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[ Rolle ] [1, 249]
1. 2x x ,x 0
f(x)3 ( )x ,x 0
,, -
Rolle [ 1,1] . f
C .
2. f f (x) 0 , x . f xx .
3. f [,],
(,) f() f() 0 . :
) f(x)
g(x)x c
, c [,] 0
x (,) 0
g(x ) 0 .
) 0
x (,) f
C 0 0 x ,f(x )
(c,0) .
4. f [,], (,) f (x) 0
x (,) . f() f() .
[ 1 ] [1,2,3, 249-250]
5. ,,
05 3
, 4 2x x 0 -
(0,1) .
6. f
0,2
f 1 f(0)2
.
0
x 0,
2
:
0 0f (x ) x .
7. f [,], [,] .
, , 21z f ()[f() 1] 2i 2
2z f () i . 3
1 2Re(z z ) f () , f (x) 0 -
(,) .
8. f [ 2,2] f(1) 0 2g(x) f(x)(x 4) ,
0x ( 2,2)
0g (x ) 0 .
9. f : [,] (0, ) lnf() lnf() .
(,) , f () f() .
10. f :[,] , [,] (,)
f() f() c( ) . (,) ,
: f () c , c .11. f : [,] , : 2 2 2 2f () f () . -
(,) f() f () .
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12. f (0, )
1f(e) f(1)
e (1).
0x (1,e)
2
0 0 0x f (x ) 1 ln x 0 (2).
13. f, g [,] f() f() 0 . -
(,) , f () f() g () .
14. f [1, e] f(1) 1 ,
2f(2) 4 ln2, f(e) e 1 . (1,e) , 21
f 2
.
15. f [0,] f(x) 0 x [0,] . -
[0,] f ()
0f()
.
16. f f(x) 0 x
f 2012e
f 2011 .
f (x) f(x) (2011,2012) .
17. f [,] (,) f() 0 . -
(,) f
f
.
18. f [1,e],
(1,e) f(e) f(1) 1 . x f (x) 1
(1,e) .
19. f : [,] , , ,
f() f() . , , :
c f f , c .
20. f : [,] , [,] (,)
f() f() 0 . (,) , -
: f () f() .
21. f : [,]
, [,]
(,)
f() f() 0 . (,) , -
: f () f() .
22. f : [,] , [,] (,)
f() f() . (,) ,
: f () f() .
23. f : [,] , [,] (,)
f() f() . (,) ,
: f () f() .
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24. f,g:[,] , [,]
(,) h()h()e f() e f() . (,) -
, : f () h () f() .
25. ) f .
f (x) 0 , f(x) 0
.
) f . -
f (x) 0 , f(x) 0 -
.
[ 1 ]26. 2x 2x 2ln(x 1) 0 . []
27. f , -
fC : 2x y 1 0 x
.
fC y 2x .
[ 1 ]
28. f : xf (x) e , x 0 f(x) 1, x [0,1] .
0x0 0x (0,1) : f(x ) e 1 .
[ ] 1
.
[ ]
29. x 2e x x .
[ ] [7, 249]
30. 2xx x x [ ,] .
31. 4 3 2x x 5x x 0 *, R 0
.
[ F F()=F() ( )]
32. f : [,] (0 ) f(x) 0
x [,] .
f() f() 0
x (,) 0 0 0 0
f(x ) ln f(x ) x f (x ) .
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555 1
2.5) ( )
1. (...)
.
2. f ...
[,], , , -
:
0x (,) 0 f() f()f (x ) .
f() f()f (x)
(,) .
[ ...]
( )f (x) [f() f()] 0 - (,) .
0
x (,) -
f 0 0M(x ,f(x )) , A(,f()) B(,f()) . [ ...]
3. -
1 2
[t , t ] 0 1 2
t (t ,t )
.
[ ...]
4. ... !
5. ... .
6. f [,], ...
f [,] f
.
7. f f() f() , - Rolle.
[ Rolle ]
8. f f() f() , -
(,) f () 0 . - f M(,f()) , -
xx .
21
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9. f f() f() , - (,) f () 0 . - f M(,f()) ,
xx .
1. ..., , :
) f() f()
f() f() .
) f () .
) -
,
.
) 1 2 1 2 f (x ),f (x ),..., f (x ) x ,x ,..., x (,) .)
1 2 1 2f (x ),f (x ) x ,x (,)
[,] . , , -
[,] [,], [,],
2
.
) .
2. ... f -
) . - , x x x x7 6 9 8 .) , -
f() f()
-
[,] .
) , f() f()
-
f() f()
f ..
3. f (,) [,] 0
x -
(,) ..
0[,x ]
0[x ,] .
4. [,] -
1 2
, , , 1 1 2 2
f() f() f ( ) f ( ) f ( )
1 1 2 2 f ( ) f ( ) f ( ) f () .
f. [,] -
.
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) , , [,]
1 2
.
, , , 1
,
1 1 2 , ..
1 2 1 ... ,
) ... .
[]1. f [,] -
:
f
[,],
f() f()
f () f ()
[ 3, 249]
f [,], f() f()
f () f ()
2. f [, ], (, )
f (,) + f() f()
f2 2
f (,) + f()+f()
f >2 2
3. f [,] .
f f f
2 2
, , f 0 .
[ ... ] [2 .249]
1. 2
3
x + , x 1f(x)
x -x+ , x 1
. ...
[ 1,2] :
) , .
) ,f() [ 1,2]
: 2x y 3 0
2. f [4,10] f(4) 6 f(10) 0 .
(4,10) , fC A(,f()) 0 135 xx.
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3. f f 1-1,
f
C
f
C .
[ ...]
4. x x x x7 6 9 8
[ ...] [3, 4,5 .249-250]
5. N : , .
6. f [1,5] f(1) 2 f (x) 2
x (1,5) , : 10 f(5) 6 .
7. f x 1 x f(1)=0, f x x 1 , x .
8. f R R . : f 2009 f 2012 f 2010 f 2011 .
9. f 2 f .
f(2x 3) f(2x 7) f(2x 1) f(2x 5) x .
10. N : x xx 1 e xe 1 , x .
11. : xe x 1 , x .12. : ln x x 1 , x (0, ) .
13. ) N :
x 1
ln x x 1x
, x 0, .
) :x 1
lnxlim 1
x 1
.
14. ) N :x
ln(x 1) xx 1
, x 1 x 0 .
) : i)x 0
ln(x 1)lim 1
x
ii)
3x
ln(x 1)lim 0
x
15. :e
2 ln e
.
16. : x 1x e 1 (x 1)e x (1, 2) .
[ [, ] ...]
17. f [1,11] f(1),f(6),f(11)
,
(1,11) , f () 0 .
18. f :
f(2x) 2f(x), x
.
) :f(2) f(0)
f(1)2
.
) (0,2) , : f () 0 .
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19. f [,] (,)
f() f() . 1 2
x ,x (,)
1 2f (x ) f (x ) 2 .
20. f ... [0,3], -
1 2 3
, , (0,3) 1 2 3
f ( ) f ( ) f ( ) f(3) f(0) .
21.
f f ln f ln
.
ln ln ln , ,, 0 2
e
,
1 2 ,
1 2f ( ) f ( ) 0 .
22. f , f( 1) 1 - , f(1) 1 .
)1 2
1 1 1 2
f ( ) f ( ) 2 .
) 1 21 1 1 2
1 1
2f ( ) f ( ) .
23. f -
A(4,11) B(19,5) .
1 2
, , 1 22f 3f 2 .
24. f [,] (,) f(x) 0
x [,] . 1 2 0
, , (,)
1 2 0
1 2 0
f f f 2
f f f
.
25. f [,] -
f
C , :
) f
C -
.
) (,) f () 0 .
) f(x)-f()
(
x)
x-
. Rolle
[,] .
) (,) f()-f()
f ()=-
.
26. f 2 [,] f() f() 0 . (,)
f() 0 , (,) f () 0 .
27. f 2 f (x) 0, x ,
f
C .
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[ [, ] Bolzano]
... [,] , -
Bolzano
28. f [,], (,) f() , f() , ,
:
) (,) f() .)
1 2 , (,)
1 2
1 2f ( )f ( ) 1 .
29. f [,], (,), -
: f() f() . :
) 0x ,
0
f f f x
, , .
) 1 2 , , , : 1 2
f f f
,
1 2 .
[ ... Bolzano Rolle]
30. . f :[0,2] . (0, 2) :
2f(0) 3f(2)f()
5
, f(0) f(2) .
. 1 , 1 2
x ,x (0,2) 1 2
x x 1 2
2f (x ) 3f (x ) .
. f (0,2) 2g(x) f(x) x x
. Rolle [0,1] [1,2], , .. f (0,2), p (0,2)
5f (p) f(0) f(2) .
31. f [,] f() f() , f () 0
f () 0 . f (x) 0 (,) .
[... - F()=F()]
32. f [,], (,), f(x) 0 x (,) .
(,) :( )f ()
f( )f() ef()
.
[... ]
33. f(x) f (x) , x ,
0 , xlim f(x)
.
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()
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2.6 ) ( )
1. .
2.
, f gC ,C
x -
,
, yy,
c ,
c 0 c 0 .
3. To
.
: , , . [ .252]
4. 1 2
f (x) g (x), x , :
1 1f(x) g(x) c , x
2 2
f(x) g(x) c , x
5. f (x) 0 fx A -
;
6.
c, , x,
, .
[]
1. f f (x) f(x) , x , ,
c xf(x) c e , x .
= 1 [ 252]
f ( ) ( ) f x f x , x xf(x) ce , c .
2.
f (x) 0 , f , f .
22
y
O x
y=g(x)+c
y=g(x)
22
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1.
f -
.
f f (x) 0
x . f , f(x) c x .
c f.
2.
.
.
.
3.
-
f, f , f -
x.
.
, , -
f(x) g(x)
x .
[: ,
]
, f(x) g(x) c , x .
1 2
U , , : 1 1
2 2
g(x) c ,x f(x)
g(x) c ,x
.
c 1 2c , c f.
4. f(x) 0 , ln(f(x)) .
5.
.
.
, ,
.
f .
f 0.
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[ ] [1, B1 .256-7]
1. f ,g : 2f (x) g (x)
2g (x) f (x), x .
) 3 3h(x) f (x) g (x) .
) h, f(0) 1 g(0) 2 .
2. f : f (x) 2f(x) , x f(0) 1
f(x) 0 , x . N :
) G(x) ln f(x) 2x .
) 2xf(x) e , x .
3. f, f (x) 0 x (0,1) (1,2)
[1,2], f [1,2] .
4. f : f(x)f (x) 2x
, *x .
5. f x, y 2f(x) f(y) (x y) .
:
) 2f(x) f(y) (x y) x, y .
) f .
6. f : f x f x 0( ) ( ) , x .
) f(0) f (0) 0 , :
i) H 2 2h (f ) f .
ii) f(x) 0 , x .
) g(x) g (x) 0 g(0) 0 ,g (0) 1 g(x) x
[ ]
7. N f : f (3x -1) 2x 1 f(2) 5 .
8. f
, f(0) 2 5
f (x x) 6x , x
,) 5 6 2f(x x) 5x 3x 2 .
) f
C 2,f(2) .
9. f : :1
f (x y) f(x)f(y)2
x, y . : y 2x 2 fC
0,f(0) , :
) f(0),) f.
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10. f:(0, ) 1 f (1) 1 .
f(x y) y f(x) x f(y) , x,y 0 , :
) f x
f x 1x
x 0 .
) f x x ln x , x 0 .
11. f : f(x y) f(x) f(y)
x, y . f (0) 1 .
12. f : f(x y) f(x 2y) 2f(x) 3y
x, y . f
C g
C , 1
g(x) 5x
, A(1,),
:
) f (x) 1 x ,
) f.
[ ]
13. f :
) xf (x) 3x 42x e f(0) 5
)2 2
2xf (x)
(x 1)
f(0) 1
) 2f (x) 2xx x x f(0) 1
) xf (x) f(x) 2 ,x 0 f(1) 0
) f
[0, )2
f (x) xf(x) x 2012
f6 2 12
) f (x) 2f(x) , x , f(0) 1 .
14. f (0, ) f(x) 0 . , ,
, :
)
xf(x)
,
) A(1,1)
) 1.
15. f R f(0) f (0) 1 :
2f (x) f (x) 6x 3x x . f(x) .
16. f : xf(x) 2f (x) f (x) e ,
x . g(x) f (x) f(x) g(0) 1 :
) h(x)=x
g(x)x
e g
) f f(0) 1
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17. N f : : f x 2x( ) (f x 2x) f(0) 1 .
18. f : 2(x 2)f (x) 2x 5x 2 , x
f(0) 1 . f.
19. f { 2} , 2(x 2)f (x) 2x 5x 2 ,
x { 2} , f(0) 1 f( 3) 7 .
[ ( ( ) ( )f x f x , xf(x) ce ) ]
20. *f ,g : f (x)g(x) f(x)g (x) f(x)g(x) ,
x f(0) g(0) 0 . N :
) xf(x) e g(x) , x
) g(x) 0 , x .
21. f : :
f (x) f(x), x f(0) 1, f (0) 1 . :
) xf(x) f (x) 2e , x .
) xf(x) e , x .
[ ]
22. f : f(0) 2f (0) 1 2
f (x)f(x) f (x) f(x)f (x), x
. :
) x1
f(x)f (x) e2
x
) 2 xf (x) e , x .
)x
2f(x) e , x .
23. f : : ( ) ( 1)f x f x
x . f(0) 1 , :
) f x f( ) ( )x 1 x .
) f x f 1( ) (x) x .
) xf(x) e x .
24. f : f (x) f(x) x f (0) f (0) 1 .
) x
f (x) f(x)g(x)
e
.
) xf (x) f(x) 2e , x .
) .
