Σπιράλ 4_Παράγωγοι A' (2012-13).pdf
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Transcript of Σπιράλ 4_Παράγωγοι A' (2012-13).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 ) , -
.
: , 0
t 0
0
S t S t0
t t
,
0
(t ) 0 .
, 0
t 0
0
S t S t0
t t
,
0
(t ) 0 .
, (t) 0
t
0 0
(t ) (t ) , .
[]
1. 0
x , 0
x .
[ . 217 ]
2. 0
x 0
x .
[ ]
1. , ... -
, , -
.
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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 x0
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
0x 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
,
0
x 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 2
2x 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
14. f : f (1) 0 .
f(x 1) , x 2g(x)
f(3x 5) , x 2
2.
15. f 1, 2 3 3x f (x) 2f(x) x 1 ,
x . N f 1.
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16. 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 0x x
0
x f(x) x f(x )lim f(x ) x f (x )
x x
) 0 00 h 1
f(x h) f(x )1f (x ) lim
x h 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, 0, :
f(x y) f(x) f(y) f(x) f(y) , x,y . f
x 0 , f .
21. f x,y :2 2f(x) y f(x y) f(x) y . f
0x .
22. f , 0
x 0 , 2 4xf(x) x x x . f
0
x 0 .
23. f 1 f (1) 2
, x,y *
- f(x y) f(x) f(y) (1) . f -
0 0
0
2x *, f (x )
x .
24. f : 0
x 0 .
x 0
f(x) f(x)lim ( ) f (0)
x
, , .
25. f x,y
2
f(x) f(y) x y (1) . :
) 2
f(x) f(y) x y x, y .
) f 0
x ,0
f (x ) 0 .
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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 ( () 0f (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 : 0
x 0 :
f( ) f() f() , . :
) f(0) 0
) f (x) f (0)x x
4. f : , f(x y) f(x) f(y) x, y .
) f(0) 0 .
) f(x y) f(x) f(y) .
) f .
) f 0, f .
5. f: 0, f(x y) f(x) f(y) , x, y (0, )
1 x1 x f(x)
x
, f
0x (0, ) .
6. f : 0
x 1 f(1) 1 f (1) 1 . -
:
)
2
x 1
f(x) 1lim
x 1
)x 1
xf(x) 1lim
x 1
)x 1
xf(x) 1lim
x 1
7. f , g :
f 0
x
g 0
x .
(f g)(x) 0
x , -
0
g(x ) 0 .
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&
()
2:
555 1
2.3 -
1. x
.
: .
2. -
f, g 0
x , 0 0 0
(f g) (x ) f (x ) g (x ) 0 0f(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 0g x
x 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() , :
11
1f x , x f
f f x
:
x f() 1f f (x) x :
1 1 1
f 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 ,
1
0
0
1(f ) (y )
f x
.
[]
1. f , f -
.
2. f , f
.
******
:
,
. , ,
.
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( )
f f 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
(logx)x ln 10
(0, )
8) f(x) ln x *1
(lnx)x
*
9) f(x) x [0, ) 1
x2 x
(0, )
10) xf(x) e x x(e e)
11) xf(x) , 0 x x( ) ln
12) f(x) x (x) x
13) f(x) x (x) x
14) f(x) xf
A {x /x 0}
{x / x , }
2
2
2
1(x)
x(1 x)
f f
A A
15) f(x) xf
A {x /x 0}
{x / x , }
2
2
1(x)
x
(1 x)
f f
A A
16)1
f(x)x
* 21 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
fA .
, -
.
( )
( )
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 ( )
1 2 3
y f(u (u (u (....u (x)....))))) , : 1 2
1 2 3
dudu dudy dy...
dx du du du dx
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( 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) uu , u 0
2 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 21 f (x)
f(x) f (x) f(x) f(x)
5) 2 2
1 u
u u u u
6) 2 21 f (x)
f(x) f (x) f(x) f(x)
6) 2 21 u
u u u u
7) f(x) f(x)e e f (x) 7) u ue 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) ln f(x) g(x) ln f(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 ln x
f x , g x x x , h x ,x 1 1 x1 x
) x1 x 2 xln x 2x 1
f x , g x , h x , x
1 x x 2 1 x e
)2 22 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 1f(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 )
)
x
e eh(x) (2x 1) ln x
4. N f 0
x : f(x) x x 0
x 0 .
5. f : 31
f (x) xf (x)x
x 0 . N f (1) .
6. A f , :
) f (0)
) g (0) , f (0) =1 g(x) f(x)x f(x)
7. f(x)=2
2
x
1+ x,
f 3f
4 4
.
8. xf(x) e (x) , 2 2f (x) 2f (x) f(x) 0 .
9. N :
) 2f x x ln x , 22f(x) xf (x) x 0
) x
xye
, dyx x 1 y 0dx
) xy e x x , xy 2y 2e x 0
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10. f x 2f(x )=xf(x), f (1) 0 .
11. f x :
f(2x 3) f(x) . f 0
x 3
xx.
12. A64 27
( )
f xx x
, 0
x (0, )2
0
f 0(x ) .
13. 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) ;
14. P 2
x x x .
15. P(x) x : 2
P x 4P x
P(0) 4 .
16. N P(x), P(0) 1 2(P (x)) P (x) 8P(x) .
17. ) (x) 2 , 2
x -
-
. , 2
x x x 0 .
)
2x 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 .
18. 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 21 1 1
f 0 f 0
19. :
) x 2x 3xx
1S 1 e e e e
) x 2x 3x x2S e 2e 3e e , .
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20. f : , y xf(x y) e f(x) e f(y) ,
x,y, , :
) f(0) 0
) xf (x) f(x) f (0)e , x .
21. *f : f(xy) f(x) f(y) *x,y R f 0 . :
)yf (x)
f (y) x
) f (1) f ( 1) 0
22. f : , :
f (0) 1 f(x y) f(x y) 2f(x)f(y) x, y .
: f (x) f(x) x .
[ f() ]
23. :
) 1
f(x) x
()
1
( 1) !f )x( x
,
) 1
f(x)x 1
, () 1
( 1) !f (x)
(x 1)
* x { 1} .
) f(x) x ()
( ) +x2
f x
,
) f(x) x , ()
f (x) x2
) xf(x) xe , () xf (x) e (x ) .
24. 1 1 1 0f x x x x ,
0 1 1 , , , ,
0 .
() f (x) ! ()f (x) 0 .
[ f-1]
25. . f : (,) R , . f -
0
x (,) 0
f (x ) 0 1f 0
f(x ) :
1f 0
f(x ) 1 00
1(f ) f(x )
f (x )
.
. xf(x) e x , 1(f ) (1) .
26. x 3f(x) e x x , x .
(i) f 1f-
(ii) 1f- 1fD - ,
1 1(f ) (1)2
.
27. 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,B1,11 239]
) 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,B2,6 239])
0 0M(x , f(x )) o .
) 0
f (x ) , 0
x .
) , , .
17
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 -
().)
gC
,f( ()) .
x0
M
Cf
f(x0)=
=x+:
x0+
x0
M
Cf
f(x0 )
C
g(x0)=
Cf
C
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. x 1f(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) 1limx 1
.
11. 2f(x) x 2x 6 7 , .
) , f
.
) , f
C xx.
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555 1
2.4
1. y f(x) 0
x .
y x 0
x 0
f (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 0
t 0
S (t )
, S t 0t , ()
0t
0(t ) .
0 0=(t ) (tS ) .
, ,
(t) S (t) .
5. t 0
t -
0
(t ) , t 0
t , ()
0
t 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 , .
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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, 0
x x 0
x .
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)) ].
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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.
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2.
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) t: x x(t) , y y(t)) t.
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0,1 m/sec.
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) (t) .
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x 1 1 1
(t) x(t) (t) x(t) (t) (t) x(t)x (t) (t) ...3 3 3 3
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1. t -
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3. 22 cm / sec . -
1,8 m, .
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1 3,
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.
6. x y
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