de riva bi liter

7/29/2019 De Riva Bi Liter

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: Drivation tude de fonctions :

Lyce My RachidLyce My RachidLyce My RachidLyce My Rachid Aguelmous Mohamed iaalouAguelmous Mohamed iaalouAguelmous Mohamed iaalouAguelmous Mohamed iaalou- 1 -

nx u( x )

(((( ))))(((( ))))r

x u x

(((( )))) (((( ))))f x g x====

(((( )))) (((( ))))f x g x

) (


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I(- 1(- Nombre drive Fonction drive

f I0x I.

f 0

x l(((( )))) (((( ))))

0

0

x x0

f x f xlim l

x x

====

l f0

x (((( ))))0f ' x.

f I I. (((( ))))f ' : x f ' x f I.

:0

h x x= = = = :(((( )))) (((( )))) (((( )))) (((( )))) (((( ))))0

0 0 0

0x x h 0

0 0

f x f x f x h f xf ' x lim limx x x x

+ + + + = == == == =

f 0x 0.

.) f : x x 0.(

f 0x 0x.2(-

1f 0

x f (((( ))))T

(((( ))))(((( ))))0 0A x , f x :(((( )))) (((( )))) (((( )))) (((( ))))0 0 0T : y f ' x x x f x= += += += +

f [[[[ [[[[1, + + + + :(((( )))) 3f x 1 x= += += += +.

(((( ))))f ' 0 (((( ))))T f .

2f

0x ) ( f

(((( ))))(((( ))))0 0A x , f x (((( ))))'d 0f x)(((( ))))'g 0f x.(

3

f 0

x(((( )))) (((( ))))

0

0

x x0

f x f xlim

x x

= = = =

(((( )))) (((( ))))

0

0

x x0

f x f xlim

x x

= = = =

f (((( ))))(((( ))))0 0A x , f x.

f

:(((( ))))

3 2f x x x x= = = =

. 1( ]]]] ]]]] [[[[ [[[[fD ,0 1,= += += += +.

2( f0 1 .


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f

0x.

(((( )))) (((( )))) (((( ))))0 0 0x f ' x x x f x + + + + f0x.

:(((( )))) (((( )))) (((( )))) (((( ))))0 0 0f x f ' x x x f x + + + + (((( ))))f x0x.

:(((( )))) (((( )))) (((( ))))0 0 0f x h f ' x h f x+ ++ ++ ++ +(((( ))))0f x h++++.

: 0

x

f ++++

:(((( ))))1

f xx

====.

1( f 1.

2( 1

0.999

1

1.002

.

3(- f f 'f

(((( ))))x a , a ]]]] [[[[, + + + +x 0]]]] [[[[, + + + +

x x]]]] [[[[, + + + +x 1]]]] [[[[, + + + +

{{{{ }}}}nx x ,n 1 ]]]] [[[[, + + + +n 1x nx ]]]] [[[[, + + + +

1

x x

]]]] [[[[ ]]]] [[[[,0 0 , + + + +2

1

x x

]]]] [[[[0,++++]]]] [[[[0

x x[[[[ [[[[0 , ++++1x

2 x]]]] [[[[0,++++

x sin x]]]] [[[[, + + + +x cos x]]]] [[[[, + + + +

x cos x]]]] [[[[, + + + +x sin x]]]] [[[[, + + + +

x tan xk ,k

2

+ + + +

2x 1 tan x++++k , k ,k

2 2

+ + + + + + + +

: :

x

3

2 cos x 1lim

x3

n

x 1

x 1lim

x 1

x4

2 cos x 2lim

2 sin x 2

.

3(- Ecriture diffrentielle

(((( ))))f x====f I :

(((( ))))dy

f ' xdx

====(((( ))))y

f ' x

====

(((( ))))dy f ' x dx==== .


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II(- f I :f g++++ffg

I:(((( ))))f g ' f ' g'+ = ++ = ++ = ++ = +(((( ))))f ' f ' ====(((( ))))fg ' f ' g fg'= += += += +.

f Ig I 1

gf

g

I:'

2

1 g'

g g

= = = =

'

2

f f ' g fg'

g g

====

.

. .

III(- Drive de la compose de deux fonctions

fg I:(((( ))))f I J.0x I.

f 0

x (((( ))))0f xg f 0x

:(((( )))) (((( )))) (((( )))) (((( ))))(((( ))))0 0 0g f ' x f ' x g' f x====.

f I (((( ))))f If I

:(((( )))) (((( )))) (((( )))) (((( ))))(((( ))))x I , g f ' x f ' x g' f x = = = =.

f :(((( )))) (((( ))))2f x cos 1 x= += += += +.

: 2u : x 1 x++++ I==== .:v : x cos x J==== .f v u==== (((( ))))u I Jf (((( )))) (((( ))))2x , f ' x 2x sin 1 x = + = + = + = +.

f Inf I:(((( ))))n n 1n , f ' nf ' f = = = =

f If I:(((( ))))

f 'f '

2 f====.

IV(- Drive de la fonction rciproque1

f I0x I.

f 0

x(((( ))))0f ' x 0 1f (((( ))))0f x

:(((( )))) (((( ))))(((( ))))(((( ))))

1

0

0

1f ' f x

f ' x

====.

f I(((( ))))(((( ))))x I , f ' x 0 1f (((( ))))f I

:(((( )))) (((( )))) (((( ))))(((( ))))

1

1

1y f I : f ' y

f ' f y

= = = =

.


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: nx x rx x.

n

x x(((( ))))n

]]]] [[[[0,++++:(((( )))) (((( ))))n

n 1n

1

x , x 'n x

++++ = = = =.

rx x(((( ))))r ]]]] [[[[0,++++:]]]] [[[[ (((( ))))r r 1x 0 , , x ' rx + = + = + = + =.

1

f :(((( )))) 3 3f x x x= += += += +.

f [[[[ [[[[0,++++ ]]]] [[[[0,++++.

) ]]]] [[[[0,++++.(:]]]] [[[[ (((( ))))(((( ))))

2

23

1x 0 , , f ' x 3 x

3 x

+ = + + = + + = + + = +.

2

f :(((( ))))2

3 3f x x x= += += += +.

f ]]]] [[[[0,++++ ]]]] [[[[0,++++) .

]]]] [[[[0,++++.(:]]]] [[[[ (((( ))))1

2 23

3

2 2x 0 , , f ' x 3 x x 3 x

3 3 x

+ = + = + + = + = + + = + = + + = + = +.

2

u I (((( ))))nx u x(((( ))))n

I:(((( ))))(((( ))))(((( ))))

(((( ))))(((( ))))n

n 1

n

u' xx I , u x '

n u x

= = = =.

u I (((( ))))(((( ))))r

x u x(((( ))))r

I:(((( ))))(((( ))))(((( )))) (((( )))) (((( ))))(((( ))))r r 1

x I , u x ' ru' x u x

= = = =.

1

f :(((( )))) 3 2f x x x x= = = = .

: f ]]]] ]]]] [[[[ [[[[fD ,0 1,= += += += +.

2x x ]]]] [[[[1, ++++]]]] [[[[,0.

f ]]]] [[[[1, ++++]]]] [[[[0:

]]]] [[[[ ]]]] [[[[ (((( ))))

(((( ))))

(((( )))) (((( ))))

2

2 23 32 2

x x ' 2 x 1

x ,0 1 , , f ' x 1 13 x x 3 x x

+ = = + = = + = = + = =

.


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2

f :(((( )))) (((( ))))2

2 3f x x 1= = = = .

: f:]]]] [[[[ ]]]] [[[[fD , 1 1,= += += += +. 2x x 1 ]]]] [[[[1, ++++]]]] [[[[1

f ]]]] [[[[1, ++++]]]] [[[[1 :

]]]] [[[[ ]]]] [[[[ (((( )))) (((( )))) (((( )))) (((( ))))(((( ))))

2 11

2 2 23 3

23

2 4 4 xx , 1 1, : f ' x x 1 ' x 1 x x 1

3 3 3 x 1

+ = = = + = = = + = = = + = = =

f [[[[ [[[[I 1,= += += += + :(((( )))) 3f x 1 x= += += += +.

1( f 1

f

.2((((( ))))f 0 1f 1(((( )))) (((( ))))1f ' 1.

V(- Fonctions primitives dune fonction

1(-

f I. fI F IF' f====.

f :(((( ))))f x x 2= += += += +.

F (((( )))) 21

F x x 2x 12

= + += + += + += + + f.

G (((( )))) 21

F x x 2x 22

= + + = + + = + + = + + f.

1

f IF I. fI I :(((( )))) (((( ))))k , x F x k + + + + .

F f I.

G fIG IG' f====G' F'====(((( ))))G F ' 0 = = = =Ik / G F k = = = =.:(((( )))) (((( ))))x I ,G x F x k = + = + = + = +.

G I :(((( )))) (((( ))))G x F x k= += += += +k G

IG' F' f = == == == =G f I.


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2f I

0x I0y .

f I GfI(((( ))))0 0G x y====.

f ]]]] [[[[I 0 ,= += += += + :(((( )))) 3 21

f x x 1x

= += += += +.

1( f I.2( G f I (((( ))))G 1 0====.

3

I I.4

fg Ik .FG fgI:F G++++ f g++++ I.kF kf I.

f :(((( )))) 2x 1, x 1f xx 4 , x 1

+ + + + ==== + > + > + > + >

.

1( f .2( f.3( G f (((( ))))G 0 0====.

f [[[[ ]]]]0, :(((( ))))f x x cos x sin x= = = = .

1((((( ))))f ' xx[[[[ ]]]]0,.2(- g [[[[ ]]]]0, :(((( ))))g x x sin x cos x= = = =

- G (((( ))))G 0 ====.

fg 0,2

:(((( )))) 2f x tan x====(((( )))) 2x

g x cos2

====

fg 0,2

.


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2(- f If II

(((( ))))x a , a (((( ))))x ax c, c+ + + + ]]]] [[[[, + + + +

x x(((( ))))2

1x x c , c

2+ + + +

]]]] [[[[, + + + +

(((( ))))nx x , n (((( ))))n 11

x x c , cn 1

+++++ + + +

++++ ]]]] [[[[

, + + + +

{{{{ }}}}(((( ))))n1

x , n 1x

(((( ))))n 1

1x x c , c

n 1

+ + + ++ + + +

+ + + + ]]]] [[[[

0,++++]]]] [[[[,0

1x

x(((( ))))x 2 x c , c+ + + + ]]]] [[[[

0,++++

{{{{ }}}}(((( ))))rx x , r 1 (((( ))))r 11x x c , cr 1

+++++ + + +

++++ ]]]] [[[[0,++++

x sin x(((( ))))x cos x c , c + + + + ]]]] [[[[, + + + +

x cos x(((( ))))x sin x c , c+ + + + ]]]] [[[[, + + + +2x 1 tan x++++(((( ))))x tan x c, c+ + + +

k , k ,k2 2

+ + + + + + + +

3(- f If I

(((( )))) (((( ))))(((( )))) (((( ))))n

x u' x u x , n (((( )))) (((( ))))n 11

x u x c , cn 1

+++++ + + +

++++

(((( ))))

(((( ))))2u' x

xu x

(((( ))))

(((( ))))1

x c , cu x

+ + + +

(((( )))) (((( ))))(((( )))) {{{{ }}}}(((( ))))r

x u' x u x , r 1 (((( ))))(((( )))) (((( ))))r 11

x u x c , cr 1

++++

+ + + + ++++

(((( ))))

(((( ))))

u' xx

u x(((( )))) (((( ))))x 2 u x c , c+ + + +

(((( )))) (((( ))))x u' x v' x++++(((( )))) (((( )))) (((( ))))x u x v x c , c+ + + + + + + +

(((( )))) (((( )))) (((( )))) (((( ))))x u' x v x v' x u x++++(((( )))) (((( )))) (((( ))))x u x v x c, c+ + + +

(((( )))) (((( )))) (((( )))) (((( ))))

(((( ))))(((( ))))2

u' x v x v' x u xx

v x

(((( ))))

(((( ))))(((( ))))

u xx c , c

v x+ + + +

(((( ))))x u' ax b ,a ,b+ + + + (((( )))) (((( ))))1

x u ax b c , ca

+ + + + + + + +

(((( ))))t cos t , , + + + + (((( ))))1

t sin t

++++

(((( ))))t sin t , , + + + + (((( ))))

1t cos t

+ + + +

u I

u I

u I


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f [[[[ [[[[0,++++ :(((( ))))(((( ))))

2

2

x 2xf x

x 1

++++====

++++.

1( ab:[[[[ [[[[ (((( ))))(((( ))))

2

bx 0 , , f x a

x 1 + = + + = + + = + + = +

++++.

2( f [[[[ [[[[0 , ++++.

3( F f [[[[ [[[[0 , ++++ :(((( ))))5

F 12

====.

f I :

(((( ))))4 2

I , f x 3x 3x x 1= = + = = + = = + = = + ]]]] [[[[ (((( )))) (((( ))))2x

I , 1 , f x x 1= == == == =

(((( )))) (((( ))))2

2 3 3f x x x 1= = = = ]]]] [[[[I 1, ,= += += += +[[[[ [[[[ (((( ))))2

xI 2 , , f x

x 1= + == + == + == + =

]]]] [[[[ (((( )))) 32 2I 0 , , f x x x x 1= + = + += + = + += + = + += + = + +]]]] [[[[ (((( )))) 4I , , f x cos x sin x= + == + == + == + =

[[[[ [[[[1 , ++++ :(((( ))))x x x 1= = = = .

1([[[[ [[[[ (((( )))) (((( ))))3

x 1 , , f x x 1 x 1 + = + + = + + = + + = + .2( [[[[ [[[[1 , ++++

3( F (((( ))))F 1 2====.

VI(- ) (

f x(((( ))))fC

(((( ))))O,i , j

.

1(- lasymptote verticale ou prallele laxe des ordonnes

x a==== (((( ))))fC )(

(((( ))))x alim f x

++++

= = = = (((( ))))x alim f x

= = = =

2(- lasymptote verticale ou prallele laxe des abscisses

y a==== (((( ))))C )((((( ))))

xlim f x a

++++

====(((( ))))xlim f x a

====.


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3(- l asymptote oblique

y ax b= += += += + (((( ))))fC++++

)(((( ))))xlim f x

++++

= = = = (((( )))) (((( ))))xlim f x ax b 0

++++

+ = + = + = + =(

)(((( ))))xlim f x

= = = = (((( )))) (((( ))))xlim f x ax b 0

+ = + = + = + =(.

(((( ))))a 0 , y ax b = + = + = + = + (((( ))))C++++

(((( ))))xlim f x

++++

= = = = (((( ))))

x

f xlim a

x++++====(((( ))))(((( ))))

xlim f x ax b

++++

= = = =.

(((( ))))x + + + +(((( ))))x .

(((( )))) (((( ))))f x ax b h x= + += + += + += + +(((( ))))xlim h x 0

==== y ax b= += += += +

(((( ))))C.

4(- Les branches paraboliques1

(((( ))))xlim f x

++++

= = = = (((( ))))

x

f xlim 0

x++++

==== (((( ))))fC

++++.

(((( ))))xlim f x

++++

= = = = (((( ))))

x

f xlim

x++++= = = = (((( ))))fC

++++.

2

(((( ))))xlim f x

++++

= = = = (((( ))))

(((( ))))x

f xlim a , a 0

x++++= = = = (((( ))))(((( ))))

xlim f x ax

++++

= = = =

(((( ))))fC y ax====++++.

(((( ))))fC

fD :

(((( ))))2

1x x

x x= += += += +

++++

(((( )))) 3x 1 x= = = =

(((( )))) 3x 2 x 1 x= + = + = + = + 31

f ( x ) x x3

= = = =


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- - :2Bac PC+SVT



x a==== (((( ))))C.(((( ))))x a

lim f x

= = = =

y a==== (((( ))))fC++++(((( ))))xlim f x a++++ ====

y a==== (((( ))))fC(((( ))))xlim f x a ====

y ax b= += += += +

(((( ))))fC++++(((( ))))(((( ))))

xlim f x ax b

++++

= = = =

(((( ))))fC

y ax====++++

(((( ))))(((( ))))x

lim f x ax++++

= = = =

(((( ))))x

f xlim a

x++++====

(((( ))))a 0

(((( ))))fC ++++(((( ))))x

f xlim

x++++= = = =

(((( ))))fC ++++(((( ))))x

f xlim 0

x++++====

(((( ))))xlim f x

++++= = = =

ax b= += += += +

(((( ))))C(((( ))))(((( ))))xlim f x ax b = = = =

(((( ))))fC y ax====

(((( ))))(((( ))))xlim f x ax

= = = =

(((( ))))x

f xlim a

x====

(((( ))))a 0

(((( ))))fC (((( ))))x

f xlim

x= = = =

(((( ))))fC (((( ))))x

f xlim 0

x====

(((( ))))xlim f x

= = = =

VI(-

1(- concavit dune courbe I.

(((( ))))C .

(((( ))))C .

2(- Point dinflexion

I0

x I.

(((( ))))(((( ))))0 0A x , f x (((( ))))C A.


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1 I

0x I.

'' I (((( ))))fC .

'' I (((( ))))fC .

''0

x 0

(((( ))))(((( ))))0 0A x , f x (((( ))))fC.

2 I

0x I.

f '0

0

x (((( ))))(((( ))))0 0A x , f x (((( ))))fC.

'

0x

0x(((( ))))C (((( ))))(((( ))))0 0A x , f x.

:(((( ))))3

1f x

x 1====

.

1(-f

D .2( .

VIII(- Axe de symtrie centre de symtrie

fD(((( ))))fC (((( ))))

2a ,b .

x a==== (((( ))))fC(((( ))))

(((( )))) (((( ))))

f f

f

x D , 2a x D

x D , f 2 a x f x

= = = =.

(((( ))))a ,b (((( ))))fC(((( ))))

(((( )))) (((( ))))

f f

f

x D , 2a x D

x D , f 2a x 2b f x

= = = = .

(((( ))))fC .

(((( ))))C.

:(((( ))))(((( ))))

3

3

x 2 x , x 2x

4 x x 2 , x 2

====

> > > >

(((( ))))fC

(((( ))))o , i , j

. x 2==== (((( ))))fC.


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IX(- 1:

f :4 3 21 1

f ( x ) x x x4 3= = = =

.1(-

fD

fD.

2(-(((( )))) (((( ))))2x , f ' x x x x 2 = = = = .

- f.3(-- (((( ))))fC .

- (((( ))))C.

4(- (((( ))))fC .

2:

f :2x 4 x

( x )x 1

++++====

++++.

1(-f

D f

D.

- (((( ))))1 , 2 f

2(-(((( ))))(((( ))))

(((( ))))

2

f 2

x 1 3x D , f ' x

x 1

+ ++ ++ ++ + = = = =

++++.

- ]]]] [[[[1 , + + + +]]]] [[[[1 .

3(- (((( ))))fC .- .- (((( ))))fC .

4(- ]]]] [[[[1 , + + + +.

- g .-(((( ))))1g xx .

3:

x :(((( ))))sin x cos x

x

1 cos 4 x

====

(((( ))))fC (((( ))))o , i , j

.

1(-D f.

2(-(((( ))))f1

x D , f x4 sin 2 x

= = = =.

- .

- E

D 0 ,2

====

.

3(-- ED 0 ,2 ==== .

- (((( ))))C [[[[ ]]]] fD


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4: n.

f 1

,3

++++ :(((( ))))

23

f ( x ) x 1 3x= + += + += + += + +.1(-:(((( ))))

xlim f x

++++

= = = =

2(- f1

3 .

-( )32 1 3x

f ' x3 1 3x

+=

+

1

,3

+ + + +

.

( )f ' x f.

3( f y x= 4( :(((( )))) (((( ))))g x f x x= = = =

g [[[[ ]]]]0 , 2.

( )f x x= [[[[ ]]]]0,2.

5( f .

5: n.

f x :31

f ( x ) x x3= = = = .

1(--f

D, f.

-:(((( ))))f 0(((( ))))xlim f x

++++

.

2(-- f .

-:]]]] [[[[ (((( ))))3 2

1 1x 0 , , f ' x 1

3 x

+ = + = + = + =

.

-(((( ))))f ' x f.

3(-- f++++.- f (((( ))))o,i , j

.

de riva bi liter

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