diaforikos logismos g lykeiou.pdf
TRANSCRIPT
-
- www.praxisgroup.gr 1.
1. x0 - ; x0 ;
f x0 , -
0
0
x x0
f (x) f (x )im
x xl .
f x0.
2. x0 ;
f(x0 ),
00
0 0 0 0
x x h 0 x 0x x0
f (x) f (x ) f (x h) f (x ) df (x ) f (x )df (x)im , im , , , imx x h dx dx x =
+ l l l
0
3. ; ( )
, - .
x x
y y
cf cf
x
y
cf
xo
f(x )o (x ,f(x ))o o
M x,f(x))(M x,f(x))(
( x0 , f(x0)) (x, f (x)) Cf .
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x>x0 ( ) xx (x ) =
+0
0
0
f (x) f (x )x x .xx
)x(f)x(fim
0
0
xx 0
+l
x
-
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7. T ;
x, y y = f (x) f x0 y x x0 f(x0) .
: , - .
8. x0.
f x0 - .
x : f (x) - f (x0x 0) = 0 00
f (x) f (x ) (x x )x x
[ ]
0 0
00 0x x x x
0
f (x) f (x )im f (x) f (x im (x x ) 1x x
= l l
f x0
00
'0(x )x x
0
f (x) f (x )im f Rx x = l 1
0 00
'o 0 0x x x xx x
im f (x) imf (x ) f (x ) im(x x )
= l l l
0
'0 0x x
im f (x) f (x ) f (x ) 0
= l
0xx)x(imf
l = f ( x0 ) f x0
!!
- ( ). f , .
f f ( ) f f ( ) f f ( ) / f f ( ) /
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9. ; f . f . 0x f 1 f . f(x) f: 1 IR xf(x) - f. f A2 A1 f(x) f : 2 IR x f(x) f. 3 .f )( )x(
10. ;
. :
c = 0 (x ) = ( x ) 2
1( x) , x 0x
= =/
x = 1 ( x ) = - x 2
1( x) , x 0x
= =/
(x) = x -1, - {0,1} (ex) = e ( x x) n = l
(x) = x-1, R - Z 0x,x1)nx( >=l 1( ogx) , x 0
x n10 = =/l l
1( x ) , x 02 x
= > 0x,x1)xn( =/=l
A f, g :
( f + g ) (x) = f (x) + g (x) ( cf (x)) = cf (x), c
( f ) (x)=f (x) (x)+f(x) (x) g g g ' ' '2
f f (x) g(x) f (x) g (x)(x) ,g(x) 0g g (x)
= =/
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(f h)(x)=f(x) (x) (x)+f(x) (x) g g h g h (x)+f(x) g (x) h (x) (f(g(x))) = f (g(x)) (x) g(x) = u g
( f(u)) = f (u) udxdu
dudy
dxdy = ( )
, , .
([ (xnl 2+4)])= [ (xnl 2+4)] (x[ nl 2+4)]= 2 2
21n(x 4) (x 4)
x 4 = + + = +l
'
22
1n(x 4) 2xx 4
= + +l
f(x) = x
g(x) = nxlh(x) = x2+4 (f(g(h(x)))) = f (g(h(x)) g (h(x)) = f (g(h(x)) g (h(x)) )x(h
dydx
.
( )
x
y
1 2 3
f( =f) ()
11. .Rolle ;
f :
- f [.]
- f / (, )
- f () = f ()
-
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(,) f () = 0
: C . f
: f [,] i) , ii) .
12. ( . Lagrange ) - ;
x
y
1 2
f()
f()
3
: f :
- [,]
- (,)
(,) ' f ( ) f ( )f ( ) =
:
.. - // (,f()) (,f())
13. .. .Rolle ;
.Rolle .. .
.Rolle :
i. f
ii. f .Bolzano .Rolle f F ( F F(x) = f(x)
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iii. A ( ) .
iv. f(x) = 0 , ( .Rolle )
.. :
v. < f ( ) f ( ) < -
vi. f(x) = A-B
vii. f () = 0 .. . Rolle.
viii. f (x) .
: 1) 2) 3) f x0 f (x0) = 0 x .Rolle (
.. ).
14. ;
:
f
- f
- f (x0) = 0 x .
f .
x1, x2 f(x1)=f(x2) x1 = x2 1 2f (x ) f (x )= 1x x=/ 2 x1 < x2 .. [x1,x2] ( f [x1,x2] , f / (x1,x2)
21 22 1
f (x ) f (x )(x , x ) . .f ( )
x x1 =
f () =0 2 12 1
f (x ) f (x )0
x x = f(x2)-f(x1)=0f(x2)=f(x1).
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15. 2 ; f , g , C f(x) = g(x) +C ( ).
f, g f - g (f-g)(x) = f(x)-g(x)=0 0 - f(x) - g (x) = C f(x) = g (x) +c.
16. ;
f(x) =0 f
>0 x f ( ) f(x)< 0 f ( ) .
x1,x2 x1 < x2 f(x) > 0 [x1,x2]
1 2
1 2
f [x , x ]
f (x , x )
: 2 1 2 1 2 12 1
f (x ) f (x )f '( ) f (x ) f (x ) f '( )(x x )x x
= = (1) x1 < x2 x2 x1 > 0 f( ) > 0 ( ) (1)f(x2)-f(x1)>0 f(x2) > f(x) x1 < x2 f(x1) < f(x2) f f(x) < 0
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18. f ( - ) f>0 ( f 0 xA (x0 , x0 + ) f(x) f(x0) f(x0) > 0 xA (x0 , x0 + ) f(x) f(x0) : .
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22. Fermat ;
f x0 f x0 f (x0) = 0.
x
y
f(xo)
xo x +o xo-
cf
f f(x
ox
0) x (x0 , x0 + ) f(x) f(x0) 1 f / 0x
0
0
xx0
0
xx0 xx
)x(f)x(fimxx
)x(f)x(fim)x('f
o0=
=+
ll
x (x0 , x0) x < x0 x- x0 < 0 (1) f(x) f(x0) 0 0
0
f (x) f (x )0
x x
o
'00
x x 0
f (x) f (x )0 f (x ) 0
x xim l (2)
x (x0, x0 + ) x > x0 x x0 > 0 (1) f(x) f(x0) 0 0
xx)x(f)x(f
0
0
0)x(f0xx
)x(f)x(fim 0
'
0
0
xx0
+l (3)
(2), (3) f(x0) = 0 f(x0) .
23. ;
( ) :
f f . .
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24. ;
f ( , ) x0 ( ).
f(x) > 0 ( , x0 ) f(x) < 0 ( x0, ) f(x0) . f(x) < 0 (, x0 ) f (x ) > 0 ( x0, ) f(x0 ) . f(x ) ( , x0 ) ( x0, ) f(x0) -
f ( , ).
) .
) - .
.
x
y
..
....
..
..
..
..
( )
) // . - .
x'x
25. ;
f - ( ) f ( ) f .
26. ;
f cf cf f cf - cf .
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x x
y y
xo xo
cf
cf
27. ;
: f 2 f (x) > 0 x f f(x) < 0 f .
28. f f(x) > 0 ; ( f f(x) < 0 ;
f f (x0) =0 x0 ( . f(x) = x4, f(0) = 0 ). f f(x) 0 f f(x) 0
29. T ;
f, 2 ( , ) x0. - Cf ( x0, f(x0) ( ) f ( ,x0) ( x0, ) ( x0, f(x0) Cf.
: f 2 .
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30. ;
( Fermat ) . A x0, f(x0) Cf f 2 f (x0) = 0. - f - f f . ( ).
31. ;
. . - f . ..
32. ; ;
Cf . , .
33. f - ;
- x0 ( ) ( x
0, f(x0)) f
. +
+ )x(fim)x(fim
00xxxx
ll
x = x0 Cf.
x
y
xo
Cf
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34. f - ;
x
y
Cf
., + R)x(fim =
+l (
))x(fimx
=
l y = - + ( - . )
35. f ;
+ . y = x+ Cf
+ [ ] 0)x)x(fimx
=++
l . (
[ ]x
f (x) ( x ) 0im
+ =l ).
- : y = x+ - Cf +
[ ] Rx)x(fimRx)x(f
imxx
==++
ll . ( ) .
x
y
Cf
y=+
36. ; - 2 ;
.
2 )x(Q)x(P P(x)
2 Q(x) .
. + - - .
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37. De L Hospital ;
-
.00
. -
.)
)(0)x(gim)(0)x(fim00 xxxx
==ll
)x(g)x(f
im ''
xx 0l
)x(g)x(f
im)x(g)x(f
im ''
xxxx 00==
ll
1) , . 2) , ( .
00
) 1 )
3) L Hospital . . - .
38. ;
( ) .
1
( -)
2
..
3
f , f .
4
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f , f . - . f , f , - .
5
.
6
f ( xx Cf yy.
B 7
f
8
( - )
9
f.
B 10
, , ( , ) f )
(! ) ( ) .