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AN1 – Dérivée et !iffére"tielle - #o$r et e%ercice !&'$to"omie -
Site web de votre cursus de mathématiques à l’ISTP :
http://jff-GIFA14.weebly.com
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Mathématiques – AN1 - Dérivées et Différentielles
AN1 - Dérivées et différentielles - Cours -
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Mathématiques – AN1 - Dérivées et Différentielles
1 Dérivation d&une fon7tion d&une varia8le 1+1 "ntrodu7tion Le mathématicien et phsicien an!lais Ne"ton et le mathématicien et philosophe allemand Lei#ni$ %au&uel on doit l'appellation fonction() au tournant des *+,, me et *+,,,me sicles) ont étudié le calcul des variations des fonctions ainsi &ue les propriétés des tan!entes au. cour#es/ C'est ainsi &u'est apparue la dérivation/ ne !randeur y ) e.primée en fonction d'une %ou plusieurs( autre x &u'on appellera varia#le ) n'évolue pas forcément 3 vitesse constante lors&ue sa varia#le le fait/ C'est la recherche de cette vitesse de variation &ui a donné mathémati&uement mathémati&uement la notion de dérivée de la fonction/
n définit la vitesse de variation V de de y entre entre deu. points A et 5 d'une cour#e comme étant le rapport de la variation de y par par celle de x 6 6
V =
∆ y ∆ x
Ce nom#re porte aussi le nom de taux de variation pour la fonction considérée) et de pente
pour le se!ment 7A58 %ce &ui &u i définit é!alement la tangente de l'an!le ( i, A5 ) (/ Lors&ue deu. points A et 5 sont choisis) cette vitesse V ) ainsi définie) est la vitesse moenne de
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Mathématiques – AN1 - Dérivées et Différentielles
*e nombre dérivé de la fon7tion en x est est la vitesse instantanée au 9oint A: et il dé9end de x +
ne dérivée nulle pourra donc %mais pas forcément( si!naler un sommet de la cour#e %pente nulle en ce point() c'est 3 dire un ma.imum ou un minimum pour y / Le nom#re dérivé d'une fonction en un point peut aussi ne pas e.ister ; par e.emple) en cas de non-continuité de la fonction en ce point) ou en cas de point an!uleu. %la pente de la cour#e a une certaine valeur immédiatement 3 !auche du point et une autre valeur immédiatement 3 droite(/ Le calcul des dérivées trouve de nom#reuses applications 6 < Recherche d'optimums ; < Appro.imation locale d'une fonction d'e.pression ardue par une fonction dont l'étude est connue et rapide %par e.emple 6 remplacement local d'une cour#e par sa tan!ente( 6 développements limités ) &ui sont utilisés par nos calculatrices pour calculer un sinus par e.emple ; < =tudes des mouvements mécani&ues 6 vitesses) accélérations> ; < =tudes économi&ues 6 co?t mar!inal) élasticité) etc/ ∆q dq = < ,ntensité instantanée de courant électri&ue 6 i = lim &ui est la dérivée de la ∆t →0 ∆t dt &uantité de char!e char!e électri&ue en fonction du temps ; ∆Φ < @orce =lectromotrice ,nduite % E = lim dérivée du flu. temps( ; ∆t →0 ∆t >
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Mathématiques – AN1 - Dérivées et Différentielles
1+2 Nom8re dérivé 1+2+1 *a définition du Petit *arousse+ Dérivée 6 Limite) si elle e.iste) du rapport de l'accroissement d'une fonction 3
l'accroissement correspondant correspondant de la varia#le) lors&ue ce dernier tend vers 0/ 1+2+2 *a définition mathématique+
Boit f une fonction réelle de varia#le réelle x définie sur un intervalle ouvert I de ℝ / Boit a un réel de l'intervalle I/ Lors&ue sa varia#le évolue de a 3 x) le taux de variation de f est 6
V =
f ( x ) − f ( a ) x − a
Dire &ue f est est dérivable en a ) c'est dire &ue ce tau. admet deu. limites finies lors&ue x tend vers a en lui étant inférieur et en lui étant supérieur) et &ue ces deu. limites sont é!ales/ Cette limite est alors appelée nombre dérivé de la fonction f en en a) &ue l'on note f '%a(/ 1
f ′ ( a ) = lim x → a
f ( x ) − f ( a ) x − a
L'e.pression précédente donne la valeur du nom#re dérivé de f ) pour tout réel a en le&uel elle peut tre définie/ n définit naturellement la fonction dérivée de f 6 6 fonction notée f ' ' &ui) 3 tout réel x de I) associe le nom#re f '% '% x x( %s'il e.iste(/ L'emploi de la lettre x pour la varia#le de notre fonction dérivée nous fait réécrire la limite définie au-dessus sous une formulation lé!rement différente 6
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Mathématiques – AN1 - Dérivées et Différentielles
1+2+3 "nter9rétation gra9hique de la dérivée
Les deu. notations de la définition 6 dérivée en a
Dérivée et tan!ente 6
dérivée en x
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Mathématiques – AN1 - Dérivées et Différentielles
1+2+/ Dérivée < droite: dérivée < gau7he 6
Dans cet e.emple) en x F 0) la limite 3 !auche de ∆ yA∆ x vaut –1 et la limite 3 droite vaut 1/ n peut donc tracer deu. demi-tan!entes en %0)0() mais la fonction n'est pas dériva#le en 0/
L'e.emple !raphi&ue ci-dessus est celui de la fonction d'e.pression f ( x ) = x + x2 %cour#e est définie définie sur [ −1 , + ∞[ / x + 1 / Elle est
noire() &ue l'on peut écrire aussi f ( x ) = x Etudions sa dériva#ilité en x F 0/ n a /
•
Bur ]0 , + ∞[ 6
•
Bur [− 1 , 0[ 6
f ( 0 + h ) − f ( 0 )
=
h
f ( 0 + h ) − f ( 0 ) h
f ( h) h
=
f ( h) h
= 1+ h
= − 1+ h
⇒
⇒
lim+
h→ 0
lim+
h→ 0
f ( h) h f ( h) h
= +1
= −1
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Mathématiques – AN1 - Dérivées et Différentielles
1+3 *iens entre dérivée et variations d&une fon7tion De l'interprétation !raphi&ue de la dérivée =dérivée > 9ente de la 7our8e?) on déduira aisément les propriétés suivantes 6 % I représente un intervalle de ℝ ( •
’ x5 x5 @ 0 ⇔ f est Hour tout x ∈ I) f ’ est strictement 7roissante sur I/
•
’ x5 x5 0 ⇔ f est Hour tout x ∈ I) f ’ est strictement dé7roissante sur I/
•
’ x5 Hour tout x ∈ I) f ’ est 7onstante sur I/ x5 > 0 ⇔ f est
•
’ a5 a5 > 0 ⇔ La cour#e de f admet Hour un uni&ue a ∈ I) f ’ admet un sommet % % f f %a( est un minimum ou un ma.imum( un point d’inflexion %dans ce cas) de d e pente nulle(/
Les schémas ci-dessous illustrent ce dernier point 6
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Mathématiques – AN1 - Dérivées et Différentielles
1+/ Différentielle d&une fon7tion d&une varia8le 1+/+1 *a définition du Petit *arousse+ Différentielle 6 @onction linéaire 3 la&uelle peut tre assimilée une fonction différentia#le en
un point donné %fonction &ui peut tre assimilée 3 une fonction linéaire de la varia#le &uand celle-ci tend vers le point donné(/ Cal7ul Différentiel 6 Hartie des Iathémati&ues &ui traite des propriétés locales des
fonctions) de leur comportement pour des variations infiniment petites des varia#les/ 1+/+2 *a définition mathématique+
Dire &ue la fonction f définie définie sur un se!ment ou intervalle I de ℝ est différentiable en a de et une fonction ε tels &ue 6 I) c'est dire &u'il e.iste un nom#re λ et 1
f ( a + h ) − f ( a ) = λ.h + h .ε ( h ) et
lim ε ( h ) = 0 h→0
Lors&ue h est infinitésimal) h.ε ( h ) devient né!li!ea#le devant λ h ; la fonction L telle &ue L ( h ) = λ .h est la différentielle de f en en a) et L%h( se note d f / a
1+/+3 *ien entre dérivée et différentielle
Le nom#re dérivé de f en en a est #ien entendu 6 f ′ ( a ) = lim
f ( a + h ) − f ( a )
h→0
h
est différentia8le en a en a) il vient 6 Donc) si f est 14 λ h + h ε (h ) h ε (h ) = λ + lim = λ + lim ( signe ( h ) .ε ( h ) ) = λ f ′ ( a ) = lim h →0 h→0 h →0 h h
f ( a + h )
f (a )
h f ′ ( a ) + h ε (h)
f ′ ( a ) = λ
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Mathématiques – AN1 - Dérivées et Différentielles 1+/+/ Notation différentielle de la dérivée+
Avec y = f ( x ) on note d y = df
d f = f ′ ( x ) .dx ) nous nous permettons d f ′ f x = ( ) d'écrire la dérivée sous forme de fraction 6 d x Le lien entre différentielle et dérivée étant
,l s'a!it réellement d'une fraction dont numérateur et dénominateur sont des éléments infinitésimau. %infiniment petits mathémati&ues(/ mathémati&ues(/ De faJon ima!ée nous dirons &ue d y %resp/ d f ( est la variation infiniment petite de y %resp/ f ( résultant de la variation infiniment petite d x de la varia#le x/ Hour les dérivées d'ordres supérieurs %si elles e.istent( la notation conventionnelle est la suivante) attention il ne s'a!it plus de &uotients mais d'une convention K 6
d y d dx d2 f d2 y ′ ′ • Dérivée seconde 6 f ( x ) = 2 = 2 = d x dx dx • Dérivée d'ordre n 6 %n entier naturel non nul( f
1+/+, "nter9rétation gra9hique
(n)
dn f dn y ( x) = n = n d x dx
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Mathématiques – AN1 - Dérivées et Différentielles
1+, %8tention d&e9ressions de fon7tions dérivées 1+,+1 Dérivées de fon7tions usuelles $ravail 9ersonnel 6 En utilisant les éléments de calcul des dérivées présentés en pa!es
suivantes) vous sere$ en mesure de démontrer les dérivées présentées ci-dessous/ f x5 x5
f ’ x5 x5
f f o u5 x5 x5
f f o u5’ x5 x5
%terme constant( k %terme
0
x
1
u
u'
kx
k
ku
ku'
xM
2 x
uM
2u'u
xM
u
u'uM
x
xα ) α ∈ ∈ ℝ x ln ( x )
e x
α . x x
α −1
1 2 x 1 x
e x
α
u
u
ln u
eu
α u'u
α
u′
2 u u′ u u'e
u
-1
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Mathématiques – AN1 - Dérivées et Différentielles
Dérivée d&une 7onstante 6 f 6 f x5 > k x5 > k 4 ∆ y = f ( x + h ) − f ( x ) = k − k = 0 et ce ∀ x /
Donc) pour tout x nous nous avons 6 lim h →0
f ( x ) = k
⇒
∆ y =0 ∆ x
f ′ ( x) = 0
Dérivée de la fon7tion identité 6 f 6 f x5 > x x5 > x O ∆ y = f ( x + h ) − f ( x ) = x + h − x = h et ce ∀ x / ∆ y h = lim = lim1 = 1 Donc) pour tout x nous nous avons 6 lim h →0 ∆ x h→0 h h→0
1+,+2 %9érations sur les dérivées
Les e.ercices d'autonomie vous proposent de démontrer) en appli&uant la définition de la dérivée d'une fonction) les relations suivantes) &u'il convient de #ien connatre pour calculer les dérivées des fonctions diverses &ue vous rencontrere$ en mathémati&ues) mais aussi en sciences et en techni&ues de l'in!énieur/ Boient u et v deu. fonctions dériva#les dont les dérivées respectives sont u' et v'/
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Mathématiques – AN1 - Dérivées et Différentielles
Dérivée des fon7tions 9uissan7es 6 f 6 f x5 > x n x5 > x n sera ici un entier naturel/ Ce para!raphe est aussi l'occasion de revoir un raisonnement par récurrence/ n admettra en outre &ue % uPv(' F u'Pv : uPv'/ f % x x( F xM F x/ x x) nous pouvons appli&uer l'opération de dérivation d'un produit de deu. fonctions avec u% x '% x x( F v% x x( F x et donc u'% x x( F v'% x x( F 1/ Donc 6 f '% x( F 1P x : xP1 F 2 x/ f % x x( F xQ F xM/ x x) appli&uons l'opération de dérivation d'un produit de deu. fonctions avec '% x u% x x( F xM et v% x x( F x et donc u'% x x( F 2 x et v'% x x( F 1/ Donc 6 f '% x( F 2 xP x : xMP1 F xM/
A partir de ces deu. premire étapes) nous pouvons con9ecturer une formule !énérale pour tout n 6 f ( x ) = x n ⇒ f ′ ( x ) = n.x n −1 / La démonstration par récurrence consiste 3 éta#lir l'initialisation de la proposition %est-elle vraie pour la plus petite valeur de n G( puis 3 démontrer l'implication suivante 6 si elle est vraie au ran! n) alors elle l'est au ran! n:1 6 n:1
F xn/ x x) nous pouvons appli&uer l'opération de dérivation d'un produit de deu. n n-1 fonctions avec u% x x( F x et v% x x( F x et donc u'% x x( F n/ x x %par hpothse &ue la proposition est n-1 n n vraie au ran! n( et v'% x '% x x( F 1/ Donc 6 f '% x( F n/ x x P x : x P1 F %n:1(/ x x 6 si la proposition est vraie au ran! n) alors elle l'est au ran! n:1/ Elle est récurrente/
f % x x( F x
L'initialisation %avec xM() plus la récurrence) montrent par récurrence &ue pour tout entier naturel n 6 R
( )
n
′( )
n 1
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Mathématiques – AN1 - Dérivées et Différentielles
q
p
p pq
Dérivées des 9uissan7es fra7tionnaires 6 f 6 f x5 x5 > √ x > x > x
+ous démontrere$ %aprs avoir étudié les pa!es suivantes( aussi &ue la formule s'appli&ue aussi dans le cas des e.posants fractionnaires ; ainsi on retiendra 6 p
f ( x ) = x
Har e.emple) dérivée de la racine carrée avec
f ( x ) =
f ′( x) =
⇒
q
x
p q
⇒
=
p q
p
x
q
−1
1 6 2
f ′( x) =
1 2 x
Dérivée d&une fon7tion 7om9osée 7om9osée 6 g o g o f f Boit un intervalle F du domaine de définition de f et et un intervalle G inclus dans celui de g et dans le domaine d'arrivée de f / n définit la fonction composée g o f par celle &ui) 3 x de F) associe la valeur de g s'appli&uant 3 f % x x( 6
f →u = f ( x ) ∈G g → y = g (u) x ∈ F
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Mathématiques – AN1 - Dérivées et Différentielles
(on7tions 7om9osées usuelles
@onction lo!arithme népérien
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Mathématiques – AN1 - Dérivées et Différentielles
Dérivée de la fon7tion ré7i9roque d&une fon7tion f fon7tion f Boit une fonction f #i9ective d'un intervalle intervalle I vers un intervalle J/ La récipro&ue de f est est la fonction notée f -1 ) de J vers I) telle &ue f -1 o f F F Id/
Autrement dit 6 et
-1 x( ⇔ x F f % y y( pour tout x ∈ I) y F f % x -1 pour tout y ∈ J) f % y y( F x ⇔ y F f % x x(/
Har e.emple) les fonctions carré et racine carrée sont récipro&ues) de 70 ; : ∞7 vers lui-mme/ En utilisant la notation différentielle nous avons 6 dy d x 1 1 ) donc on retiendra 6 f ′ ( x ) = et pour la récipro&ue f −1′ ( y ) = = = d x d y d y f ′ ( x ) d x T
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Mathématiques – AN1 - Dérivées et Différentielles
2 Dévelo99ements limités 2+1
"ntrodu7tion
L'idée !énérale de cette partie du chapitre est de donner les moens de calculer une valeur approchée d'une valeur f % x x( d'une fonction donnée dont on ne sait pas calculer toutes les valeurs e.actes/ Har e.emple) il n'e.iste pas de formule pour p our calculer ln% x x( ou exp% x x( ou sin% x x( ou mme √ x dans un cas !énéral %la fi!ure ci-dessous prend pour e.emple f % x x( F xM - ln x(/ Har contre) il e.iste des points A% a) f %a(( de la cour#e de telles fonctions) dont les coordonnées sont connues %dans l'e.emple !raphi&ue ci-dessous) le point A%1)1( est s?r(/ A l'aide de la connaissance d'un tel point A) on souhaite évaluer l'ordonnée f % x x( d'un autre point de la cour#e) aant une a#scisse x plus ou moins proche de a/
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Mathématiques – AN1 - Dérivées et Différentielles
2+2 Définition en a à l’ordre n est 9ustement une écriture de f % x Le développement limité de f en x( sous la forme d'un polnUme de de!ré n) additionné d'un reste 6 n
f ( x ) = P ( x ) + ( x − a ) ε ( x ) oW P% x une fonction de limite 0 &uand x tend vers a/ x( est un polnUme de de!ré n et ε une P% x x( est appelé partie ré!ulire de ce développement limité et donne 3 calculer la valeur approchée cherchée) et % x - a(nε % x x( est le reste &ui est d'autant plus fai#le &ue n est !rand %au moins 3 partir d'un certain ran!( ou &ue x est proche de a/
Vhéorme des accroissements finis 6 Boit une fonction f définie définie et continue sur un intervalle 7
b8 et dériva#le sur 8
b7/
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Mathématiques – AN1 - Dérivées et Différentielles
2+3+2 (ormules de $aBlor et dévelo99ement limité
@ormule de Valor-La!ran!e 3 l'ordre n:1 6 Bi une fonction f définie définie sur un intervalle 7 a ; x8 admet des dérivées successives f ') ') f '') '') >) f %n( continues sur cet intervalle et une dérivée d'ordre n:1 définie sur l'intervalle 8 a ; x7) alors il e.iste au moins un nom#re c ∈ 8a ; x7 tel &ue 6 f ( x ) = f ( a ) +
( x − a ) 1!
2
f ′( a) +
( x − a) 2!
f ′′ ( a ) + ... +
( x − a) n!
n +1
n
f
( n)
( x − a) ( n +1 ) f ( a) + ( c) ( n + 1) ! reste de #arane
n remar&ue &ue tous les termes de ce développement sont calcula#les) 3 l'e.ception du dernier) ce fameu. reste) pour le&uel on connat l'e.istence de c> mais pas sa valeur/ Cette formule est donc utilisée pour calculer des valeurs approchées de f % x x( lors&u'on a une connaissance e.acte de notre fonction en %pour le&uel la valeur de f et et de ses dérivées
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Mathématiques – AN1 - Dérivées et Différentielles
La formule de Valor-Xoun! est en adé&uation avec la définition d'un développement limité) aussi on retiendra 6 Boit une fonction f n fois dériva#le et dont les dérivées successives sont continues dans un intervalle &ui en!lo#e a et x/ Euel que soit x soit x) la somme
f ( a ) +
∆ x
2
f ′(a) +
( ∆ x )
f ′′ ( a ) + ... +
( ∆x )
n
f
(n)
( a)
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Mathématiques – AN1 - Dérivées et Différentielles
< Hour utiliser une valeur approchée plus précise de f % x x() on peut réaliser un développement limité de f 3 3 l'ordre 2 6 f ( x ) = f ( a ) + ( x − a ) f ′ ( a ) +
( x − a ) 2
2 2
f ′′ ( a ) + ( x − a ) .ε 2 ( x )
A l'ordre 2) la valeur approchée est une e.pression du second de!ré en x) c'est 3 dire &ue !raphi&uement on n'utilise plus une droite %la tan!ente( pour modéliser la cour#e comme 3
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Mathématiques – AN1 - Dérivées et Différentielles
2+3+3 (ormule de Ma7*aurin
Cas particulier de la formule de Valor-La!ran!e dans le cas oW a F 0 6 @ormule de IacLaurin 3 l'ordre n 6 Bi une fonction f définie définie sur l'intervalle 70 ; x8 admet des dérivées successives f ') ') f '') '') >) f %n( continues sur cet intervalle et une dérivée d'ordre n:1 en 0)
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Mathématiques – AN1 - Dérivées et Différentielles
em9le 2 6 D* 7om9let d&une fon7tion 9olBnGmiale Considérons la fonction d'e.pression f ( x ) = 4 x + 2 x2 + x + 1 /
Nous 1R o#tenons les dérivées successives suivantes 6 2 f ′ ( x ) = 12 x + 4 x + 1
f ′ ( 0 ) = 1
f ′′ ( x ) = 24 x + 4
f ′′ ( 0 ) = 4
f ′′′ ( x ) = 24
et appli&uées 3 0 6
f ′′′ ( 0 ) = 24
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Mathématiques – AN1 - Dérivées et Différentielles
2+3+/ A99li7ation de l&ordre 1 au 7al7ul 7al7 ul a99ro7hé ra9ide
Le développement limité 3 l'ordre 1 est trs utile pour éta#lir des formules de calcul approché rapide au voisina!e immédiat d'un nom#re a/ Ces formules aisées 3 mémoriser sont trs commodes pour l'estimation par calcul mental d'ordres de !randeur/ Nous reprenons f ( x ) = f ( a ) + ( x − a ) f ′ ( a ) + reste et considérons) pour x
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