اتصال دالة عددية 2018 - 2019

Page 1
I
en un point continuitè
f I
0
x
I
f
0
x
00
lim ( ) ( )
xxf x f x
:
f g h x :
3
( ) 3 1f x x x  
( ) sin( )g x x
2
21
() 4
x
hx x
sin( 2)
( ) ; 2
2
(2) 1
x
f x x
x
f
gouche continuitè à –continuitè à droite
f
00
;xx
*
Ў
f
0
x
0
0
0
lim ( ) ( )
xx
xxf x f x
f
00
;xx
*
Ў
f
0
x
0
0
0
lim ( ) ( )
xx
xxf x f x
:
Ў
:
sin(3 )
( ) ; 0
( ) 2 3 0
x
f x x
x
f x x x
f
00x
:
00
sin(3 ) sin(3 )
lim lim 3 3
3
xx
xx
xx
(0) 3f
f
:
3
00
lim ( ) lim2 3 3 (0)
xx
f x x f
f
:
0
lim ( ) (0)
xf x f
f
f I
0
x
I
f
0
x
0
x
: f
Ў
:
( ) ; 1
( ) 2 3 ; 3
( ) 1 ; 3
f x x a x
f x x x
f x bx x
a b f

Page 2
continuitè sur un intervalle
f I I
f
;ab
;ab
a b
:
;ab
; ( )A a f a
; ( )B b f b
;ab
;ab
;a
;a
Ў
sinxx
cosxx
Ў
tanxx
xx
Ў
II
f g I
fg
f
fg
I
( ) 0gx
x I
1
g
f
g
I
:
f
f
D
2
( ) 3 sin(x)f x x
() 1
x
f x x x
31
() x fx x
f I g J I
( ) ( )x J g x f x   
g f J
f I g f J g J
: f
1; 
:
2
( ) ; 1
3
( ) ; 1 1
2
f x x x
x
f x x
x
f
1; 

Page 3
:
xx
0;
1; 0;  
f
1; 
2
32 x
xx
2Ў
1;1 2   Ў
f
1;1
f
(1) 1f
11
lim ( ) lim 1
xx
f x x
2
11
3
lim ( ) lim 1
2
xx
x
fx x
11
lim ( ) lim ( ) (1)
xx
f x f x f
f
f
1; 
fonction partie entière
E
Ў
n
1n x n  
x E(x)
; ( )n E n n   ў
; ( ) ( ) 1x E x x E x     Ў
; ; ( ) ( )n x E x n E x n        ўЎ
n
ў
:
n n
;1nn
n

Page 4
III image d'un intervalle par une fonction continue
f
;ab
;;f a b m M
m f
;ab
M
f
;ab
: I :
( ) 1;5fI
2
( ) 1f x x
1;2I
( ) 1:0fI
1
() 2
x
fx x
1;1I
I
fI
I
fI
;ab
( ); ( )f a f b
;ab
( ); ( )f b f a
;ab
( );lim (x)
xb
f a f
;ab
lim ( ); ( )
xbf x f a
;ab
lim (x); ( )
xaf f b
;ab
( );lim (a)
xa
f b f
;ab
lim ( );lim ( )
x a x b
f x f x
;ab
lim ( );lim ( )
x b x a
f x f x
;a
( ); lim (x)
x
f a f
;a
lim ( ); ( )
xf x f a
;a
lim ( ); lim ( )
x
xaf x f x
;a
lim ( ); lim ( )
xxa
f x f x
;b
lim ( ); (b)
xf x f
;b
(b); lim ( )
x
f f x
;b
lim ( ); lim ( )
xxb
f x f x
;b
lim ( ); lim ( )
x
xbf x f x
f I g J
()f I J
f I g J
gf
I
: :
2
( ) cos( 2 3)f x x x  
f
D
f
f f
f
D
f
f

Page 5
:
f
DЎ
2
( ) 2 3u x x x  
( ) cos( )v x x
( ) ( )f x v u x
x
Ў
u
Ў
( ) 2;u Ў
v
2,
vu
Ў
f
Ў
f I
f
I
IV théorème des valeures intermèdieres
f
,ab
k
()fa
()fb
c
;ab
()f c k
f
;ab
( ) ( ) 0f a f b
( ) 0fx
;ab
f
;ab
k f(a) f(b)
c
;ba
()f c k
f
;ab
( ) ( ) 0f a f b
( ) 0fx
a;b
:
53
31xx
0;1
:
2
11
cos
21
xx
2 ;3
f
;ab
:
( ). ( ) 0f a f b 
( ) 0fx
;ab
:
( ). 0
2
ab
f a f 
:
(b). 0
2
ab
ff
:
2
ab
a
2
ba
;2
ab
a
:
2
ab b
2
ba
;
2
ab
b
: