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Taux de variation instantan\303\251 et interpr\303\251tation graphique\302\251 Pierre LantagneEnseignant retrait\303\251 du Coll\303\250ge de MaisonneuveLa premi\303\250re version de ce document est parue sous la version Maple 6. Ce document est la suite du document \302\253 Taux de variation moyen et interpr\303\251tation graphique \302\273. La macro-commande textplot de l'extension plots y a \303\251t\303\251 introduite et nous l'utiliserons pour bien documenter les graphiques.Bonne lecture \303\240 tous !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 Ce document Maple est ex\303\251cutable avec la version 2020.1Initialisationrestart;with(plots,display,textplot,setoptions):
setoptions(size=[300,300],axesfont=[times,roman,8],color=navy):Taux de variation instantan\303\251Saisissons \303\240 nouveau la fonction TVM cr\303\251\303\251e dans le document \302\253 Taux de variation moyen et interpr\303\251tation graphique \302\273. Les points P et Q sont les points d'une fonction f.TVM:=(P,Q)->( Q[2]-P[2] ) / (Q[1]-P[1]);Consid\303\251rons de nouveau la fonction f d\303\251finie par 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 et illustrons l'interpr\303\251tation graphique du 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.f:=x->1/(x+2):
P:=[-1/2,f(-1/2)]:
Q:=[3,f(3)]:
Courbe:=plot([x,f(x),x=-4..4]):
Points:=plot([P,Q],style=point,symbolsize=15,symbol=solidcircle,color=navy):
Texte_P:=textplot([P[1]-0.25,P[2]+0.05,cat("P",convert(P,string))],align=[below,left]):
Texte_Q:=textplot([Q[1],Q[2]+0.1,cat("Q",convert(Q,string))],align=[below,right]):
Triangle:=plot([P,[P[1],Q[2]],Q],style=line,linestyle=2,color=black):
\303\211q_S\303\251cante:=y=TVM(P,Q)*(x-P[1])+P[2]:
Droite:=plot([x,rhs(\303\211q_S\303\251cante),x=-3..4],color=khaki):
display({Courbe,Droite,Triangle,Points,Texte_P,Texte_Q},view=[-1..4,0..1.2]);En d\303\251pla\303\247ant, sur la courbe, le point Q vers le point P, on obtient, pour chaque nouvelle position du point Q, une nouvelle s\303\251cante dont on pourrait calculer la pente. Pour ainsi d\303\251placer le point Q, il faut lui donner des valeurs d'abscisses de plus en plus pr\303\250s de 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, soit la valeur d'abscisse du point P. (Notons que nous pouvons d\303\251placer sur la courbe le point Q vers le point P car la fonction f ne poss\303\250de aucune discontinuit\303\251 entre P et Q)Nous allons proc\303\251der comme suit:donnons au point Q la valeur de l'abscisse du point P plus une quantit\303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y9Rjk= que l'on fera tendre ult\303\251rieurement vers 0: soit donc QLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYmLUYjNiwtSSNtb0dGJDYvUSomdW1pbnVzMDtGJy8lJXNpemVHUSI4RicvJStleGVjdXRhYmxlR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZOLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiZRIjFGJ0Y0RjdGOi1GIzYnLUZVNiZRIjJGJ0Y0RjdGOkY0LyUnaXRhbGljR0Y5RjcvRjtRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dGVy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zfby8lKWJldmVsbGVkR0Y/LUYxNi9RIitGJ0Y0RjdGOkY9RkBGQkZERkZGSEZKRkxGTy1JI21pR0YkNidRImhGJ0Y0RmduRjdGaW4tRjE2L1EiLEYnRjRGN0Y6Rj0vRkFGOUZCRkRGRkZIRkovRk1RJjAuMGVtRicvRlBRLDAuMzMzMzMzM2VtRictRmhvNidRImZGJ0Y0RmduRjdGaW4tRiw2Ji1GIzYpLUYxNjBGM0Y0RjcvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjpGPUZARkJGREZGRkhGSkZMRk8tRlI2KC1GVTYnRldGNEY3RlxxRjotRiM2Ji1GVTYnRmZuRjRGN0ZccUY6RjRGN0Y6RltvRl1vRmBvRmJvLUYxNjBGZm9GNEY3RlxxRjpGPUZARkJGREZGRkhGSkZMRk8tRmhvNihGam9GNEZnbkY3RlxxRmluRjRGN0Y6RjRGN0Y6RjRGN0Y6RjRGN0Y6RjRGN0Y6pour de telles accroissemenst h, consid\303\251rons la suite: 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Cr\303\251ons donc un point Q variable \303\240 l'aide la liste des accroissements h ci-dessus, laquelle liste sera nomm\303\251e \316\224 (Delta).Q:=[P[1]+h,f(P[1]+h)];
Delta:=[2, 1, 2/3, 1/2, 1/3, 1/4, 1/5, 1/10, 1/20, 1/30, 1/40, 1/50, 1/100, 1/1000,1/10000,1/100000,1/1000000];Calculons maintenant tous les 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 associ\303\251s \303\240 ces diff\303\251rents points Q. Pour ce faire, cr\303\251ons, avec la macro-commande seq, la s\303\251quence de ces valeurs qui sont les pentes des s\303\251cantes passant par les points P et Q, Q se rappro\303\247ant du point P.seq(TVM(P,Q),h in Delta);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 lorsque LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictRiw2JlEiaEYnLyUlc2l6ZUdRIzEyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEpJiN4MjE5MjtGJ0YyL0Y5USdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGQy8lKXN0cmV0Y2h5R0ZDLyUqc3ltbWV0cmljR0ZDLyUobGFyZ2VvcEdGQy8lLm1vdmFibGVsaW1pdHNHRkMvJSdhY2NlbnRHRkMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZSLUklbXN1cEdGJDYlLUkjbW5HRiQ2JVEiMEYnRjJGPy1GPDYuUSIrRidGMkY/RkFGREZGRkhGSkZMRk5GUEZTLyUxc3VwZXJzY3JpcHRzaGlmdEdGZW5GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRj8=Confirmons notre intuition en \303\251valuant la limite \303\240 droite du 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 lorsque 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.P:=[-1/2,'f'(-1/2)];
Q:=[-1/2+h,'f'(-1/2+h)];
Limit(TVM(P,Q),h=0,right);
normal(%);Avec Maple, il n'aurait pas \303\251t\303\251 n\303\251cessaire de simplifier explicitement (\303\251limination du facteur h) du TVM avant d'\303\251valuer la limite, Maple le ferait \303\240 sa mani\303\250re, mais ici, on a reproduit un d\303\251veloppement manu scriptus.value();Lorsque la limite \303\240 droite existe, elle est appel\303\251e taux de variation instantan\303\251 \303\240 droite en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2L1EqJnVtaW51czA7RicvJSVzaXplR1EiOUYnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JJm1mcmFjR0YkNigtSSNtbkdGJDYmUSIxRidGL0YyRjUtRiM2Jy1GUDYmUSIyRidGL0YyRjVGLy8lJ2l0YWxpY0dGNEYyL0Y2USdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHRlIvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGam4vJSliZXZlbGxlZEdGOkYvRjJGNQ== et est not\303\251e 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 . Plus g\303\251n\303\251ralement on a la d\303\251finition suivante:D\303\211FINITIONLe taux de variation instantan\303\251 \303\240 droite en 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 d'une fonction f se d\303\251finit de la fa\303\247on suivante, lorsque la limite 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(TVI^`+`)[[x=-1/2]]=Limit(TVM(P,Q),h=0,right);
``=Limit(normal(TVM(P,Q)),h=0,right);
value(%);Rappelons que les accents graves (voir ?backquotes) sont des d\303\251limiteurs de cr\303\251ation de noms (voir ?Names). Bien que leurs usages pour la cr\303\251ation de noms de variables avec des caract\303\250res accentu\303\251es n'est pas souhaitable (c'est un peu lourd), dans ce cas-ci, leurs usages peuvent s'av\303\251rer appropri\303\251s pour une mise en forme \303\251clairante des r\303\251sultats des requ\303\252tes.
Vous savez que l'interpr\303\250te ignore les espacements et les sauts de lignes dans l'\303\251nonc\303\251 d'une requ\303\252te mais entre les deux d\303\251limiteurs de cr\303\251ation de noms, les espaces sont interpr\303\251t\303\251s comme autant de caract\303\250res blancs. Un caract\303\250re blanc est un caract\303\250re n'ayant aucun pixel noir. S'il n'y a aucun caract\303\250re entre ces deux d\303\251limiteurs, \302\253 `` \302\273, l'afficheur commencera l'affichage du r\303\251sultat par un espace tout simplement.
Lorsque le 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 existe (c'est une limite), on dit que la d\303\251riv\303\251e \303\240 droite en 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 existe. Ici, cette d\303\251riv\303\251e vaut 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. La d\303\251riv\303\251e \303\240 droite est not\303\251e plus bri\303\250vement de la mani\303\250re suivante.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkobXN1YnN1cEdGJDYnLUkmbXRleHRHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiK0YnRjUvRjlRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0ZDLyUpc3RyZXRjaHlHRkMvJSpzeW1tZXRyaWNHRkMvJShsYXJnZW9wR0ZDLyUubW92YWJsZWxpbWl0c0dGQy8lJ2FjY2VudEdGQy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlItRiM2Ji1GPDYuUSJ+RidGNUY/L0ZCRjcvRkVGNy9GR0Y3L0ZJRjcvRktGNy9GTUY3L0ZPRjcvRlFRJjAuMGVtRicvRlRGXG8tRjw2LlEiLEYnRjVGP0ZaL0ZFRjRGZm5GZ25GaG5GaW5Gam5GW28vRlRRLDAuMzMzMzMzM2VtRidGNUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0Zmby1JKG1mZW5jZWRHRiQ2JS1GIzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiZRIjFGJy8lJXNpemVHUSMxMEYnRjVGPy1GIzYnLUZicDYmUSIyRidGZXBGNUY/RmVwRjJGNUY4LyUubGluZXRoaWNrbmVzc0dGZHAvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYXEvJSliZXZlbGxlZEdGN0Y1Rj9GNUY/LUY8Ni5RIj1GJ0Y1Rj9GWkZlbkZmbkZnbkZobkZpbkZqbi9GUVEsMC4yNzc3Nzc4ZW1GJy9GVEZqcS1GPDYuUSomdW1pbnVzMDtGJ0Y1Rj9GWkZlbkZmbkZnbkZobkZpbkZqbkZQRlMtSSdtc3R5bGVHRiQ2Ji1GX3A2KC1GYnA2JlEiNEYnL0ZmcFEjMTJGJ0Y1Rj8tRmJwNiZRIjlGJ0ZnckY1Rj9GXXFGX3FGYnFGZHFGZ3JGNUY/LUkjbWlHRiQ2I1EhRidGNUY/
D\303\211FINITION
La d\303\251riv\303\251e \303\240 droite d'une fonction f se d\303\251finit de la fa\303\247on suivante, lorsque la limite 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'animation suivante montrera que si le point Q peut se rapprocher du point P par la droite, aussi pr\303\250s que l'on veut, les diff\303\251rentes s\303\251cantes tendront \303\240 se confondre avec la tangente qu'on pourrait tracer au point P du graphique de la fonction f.\303\211q_S\303\251cante:=y=TVM(P,Q)*(x-P[1])+P[2]:
Delta:=[2, 1, 2/3, 1/2, 1/3, 1/4, 1/5, 1/10, 1/20, 1/30, 1/40, 1/50, 1/100, 1/1000,1/10000,1/100000,1/1000000]:
Texte_P:=seq(textplot([P[1],P[2],"P"],align={above,right}),n=1..nops(Delta)):
Texte_Q:=seq(textplot([Q[1],Q[2],"Q"],align={above,right}),h in Delta):
S\303\251cantes:=seq(plot([[x,rhs(\303\211q_S\303\251cante),x=-2..3],[x,f(x),x=-2..3]],numpoints=100,color=[khaki,navy]),h in Delta):
Points:=seq(plot([P,Q],style=point,symbolsize=15,symbol=solidcircle,color=navy),h in Delta):
Triangle:=seq(plot([P,[P[1],Q[2]],Q],style=line,linestyle=2,color=black),h in Delta):
Accroissement:=seq(textplot([.15,.7,cat("Accroissement h = ",convert(Delta[n],string))],
align={ABOVE,RIGHT},color=red),n=1..nops(Delta)):
Pente:=seq(textplot([.15,.6,cat("Pente m = ",convert(evalf(TVM(P,Q)),string))],
align={ABOVE,RIGHT},color=red),h in Delta):
A:=display(S\303\251cantes,insequence=true):
B:=display(Points,insequence=true):
C:=display(Triangle,insequence=true):
E:=display(Accroissement,insequence=true):
F:=display(Pente,insequence=true):
G:=display(Texte_P,insequence=true):
H:=display(Texte_Q,insequence=true):
display(A,B,C,E,F,G,H,view=[-1..2,0..1.2]);Pour lancer l'animation, il faut cliquer d'abord sur le graphique pour le s\303\251lectionner : une bordure bleue atteste sa s\303\251lection. Un menu contextuel \302\253 Animation \302\273 est aussit\303\264t s\303\251lectionn\303\251 dans la barre contextuelle.Dans une d\303\251marche similaire, nous allons cr\303\251er une seconde animation pour illustrer le TVI \303\240 gauche du point P.
Puisque dans l'animation pr\303\251c\303\251dente les diff\303\251rentes valeurs d'accroissement h \303\251taient positives, les diff\303\251rents points 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 retrouvaient tous \303\240 la droite du point P. Alors, pour r\303\251aliser une animation montrant, cette-fois, un point Q se rapprochant par la gauche, aussi pr\303\250s que l'on veut du point P, il suffit de consid\303\251rer des valeurs d'accroissement h n\303\251gatives.
Cr\303\251ons, \303\240 partir de la s\303\251quence LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoJkRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRjFGMg== et avec l'aide de la macro-commande map une s\303\251quence d'accroissements n\303\251gatives. Donnez-lui le nom LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpJkRlbHRhOzJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== (Delta2). Attention, il nous faut laisser tomber la valeur d'accroissement LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjJGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ de la liste LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoJkRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRjFGMg== car l'image LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYqLUkjbW9HRiQ2LlEqJnVtaW51czA7RicvJSVzaXplR1EiOEYnL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZXLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiVRIjFGJ0ZBRkQtRiM2Jy1GaG42JVEiMkYnRkFGREZBLyUrZXhlY3V0YWJsZUdGSC8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnRkQvJS5saW5ldGhpY2tuZXNzR0Zqbi8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zpby8lKWJldmVsbGVkR0ZILUY+Ni1RIitGJ0ZERkZGSUZLRk1GT0ZRRlNGVUZYLUY+Ni1RIn5GJ0ZERkZGSUZLRk1GT0ZRRlMvRlZRJjAuMGVtRicvRllGZXAtRjk2JC1GIzYnLUY+Ni1GQEZERkZGSUZLRk1GT0ZRRlNGVUZYLUZobjYkRl9vRkRGYG9GYm9GREZERmBvRmJvRkRGREZgb0Zib0ZE (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) n'est pas sur la partie continue du graphique de f menant au point P. Alors Delta[2..nops(Delta)] pr\303\251cisera tous les \303\251l\303\251ments de la liste LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoJkRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRjFGMg== sauf le premier. La macro-commande nops donne le nombre d'op\303\251randes composant un objet. Ici, le nombre d'\303\251l\303\251ments qui composent la liste LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoJkRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRjFGMg==.Delta1:=map(x->-x,Delta[2..nops(Delta)]);Copions-collons maintenant les requ\303\252tes de l'animation pr\303\251c\303\251dente et effectuons ensuite les modifications suivantes: substituons le nom Delta par le nom Delta1. (Il y a huit substitutions)substituons P par Q et Q par P dans la structure graphique Trianglecorrigeons la position verticale du texte Accroissement \303\240 1.25 et celui de Pente \303\240 1.05modifions \303\240 la fin l'option view par view=[-2..3,0..2.5].\303\211q_S\303\251cante:=y=TVM(P,Q)*(x-P[1])+P[2]:
Texte_P:=seq(textplot([P[1],P[2],"P"],align={above,right}),n=1..nops(Delta1)):
Texte_Q:=seq(textplot([Q[1],Q[2],"Q"],align={above,right}),h in Delta1):
S\303\251cantes:=seq(plot([[x,rhs(\303\211q_S\303\251cante),x=-2..3],[x,f(x),x=-2..3]],numpoints=100,color=[khaki,navy]),h in Delta1):
Points:=seq(plot([P,Q],style=point,symbolsize=15,symbol=solidcircle,color=navy),h in Delta1):
Triangle:=seq(plot([Q,[Q[1],P[2]],P],style=line,linestyle=2,color=black),h in Delta1):
Accroissement:=seq(textplot([.15,1.25,cat("Accroissement h = ",convert(Delta1[n],string))],
align={ABOVE,RIGHT},color=red),n=1..nops(Delta1)):
Pente:=seq(textplot([.15,1.05,cat("Pente m = ",convert(evalf(TVM(P,Q)),string))],
align={ABOVE,RIGHT},color=red),h in Delta1):
A:=display(S\303\251cantes,insequence=true):
B:=display(Points,insequence=true):
C:=display(Triangle,insequence=true):
E:=display(Accroissement,insequence=true):
F:=display(Pente,insequence=true):
G:=display(Texte_P,insequence=true):
H:=display(Texte_Q,insequence=true):
display(A,B,C,E,F,G,H,view=[-2..3.5,0..2.5]);Obtenons, ici aussi, tous les 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 associ\303\251s \303\240 ces diff\303\251rents points Q.seq(TVM(P,Q),h=Delta1);Nous observons ici aussi que plus l'accroissement 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 plus le 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 se rapproche de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEjMjBGJ0YvLUYjNiQtRko2JFEjNDVGJ0YvRi8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlcvJSliZXZlbGxlZEdGNC1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRmpuRistRkc2KC1GSjYkUSI0RidGLy1GIzYkLUZKNiRRIjlGJ0YvRi9GUkZVRlhGWkYvLUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkVFZNRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYkLUkobWZlbmNlZEdGJDYmLUY2NistSSNtb0dGJDYuUSomdW1pbnVzMDtGJy8lJXNpemVHUSI5RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlctSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JVEiMUYnRkFGRC1GaG42JVEiMkYnRkFGRC8lLmxpbmV0aGlja25lc3NHRmpuLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmJvLyUpYmV2ZWxsZWRHRkgtRj42LlEiK0YnRkFGREZGRklGS0ZNRk9GUUZTRlVGWC1GLDYmUSJoRidGQUYvRjItRj42LlEoJmNvbW1hO0YnRkFGREZGL0ZKRjFGS0ZNRk9GUUZTL0ZWUSYwLjBlbUYnL0ZZUSwwLjMzMzMzMzNlbUYnRj0tRmVuNihGZ24tRjY2JEZbb0ZERl5vRmBvRmNvRmVvLUYsNiNRIUYnRkRGRC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZELyUvc3Vic2NyaXB0c2hpZnRHUSIwRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2LlEnJnJhcnI7RicvJSVzaXplR1EjMTJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGRy1GLDYtUSgmbWludXM7RidGMkY1RjgvRjtGN0Y9Rj9GQUZDL0ZGUSwwLjIyMjIyMjJlbUYnL0ZJRk8tSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEjMjBGJ0YyLUYjNiUtRlU2JFEjNDVGJ0YyLyUnaXRhbGljR0Y8L0YzUSdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYG8vJSliZXZlbGxlZEdGNy1GLDYtUSI9RidGMkY1RjhGTUY9Rj9GQUZDRkVGSC1GLDYtUSomdW1pbnVzMDtGJ0YyRjVGOEZNRj1GP0ZBRkNGTkZQLUZSNigtRlU2JFEiNEYnRjItRiM2JS1GVTYkUSI5RidGMkZnbkZpbkZbb0Zeb0Zhb0Zjb0YvRjI= lorsque 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Lorsque la limite \303\240 gauche existe, elle est appel\303\251e taux de variation instantan\303\251 \303\240 gauche en 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 et est not\303\251eLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=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. Plus g\303\251n\303\251ralement on a la d\303\251finition suivanteD\303\211FINITIONLe taux de variation instantan\303\251 \303\240 gauche en 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 d'une fonction f se d\303\251finit de la fa\303\247on suivante, lorsque la limite existeLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobXN1YnN1cEdGJDYnLUYjNiYtSSNtaUdGJDYjUSFGJy1JJm10ZXh0R0YkNiVRJFRWSUYnLyUlc2l6ZUdRIzEyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjBGOy1GIzYlRjAtSShtZmVuY2VkR0YkNictRiM2J0YwLUYxNiZRInhGJ0Y4LyUnaXRhbGljR1EldHJ1ZUYnL0Y8USdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0Y4RjsvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRlMvJSlzdHJldGNoeUdGUy8lKnN5bW1ldHJpY0dGUy8lKGxhcmdlb3BHRlMvJS5tb3ZhYmxlbGltaXRzR0ZTLyUnYWNjZW50R0ZTLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGXG8tSSVtc3ViR0YkNiVGRS1JI21uR0YkNiVRIjBGJy9GOVEiOUYnRjsvJS9zdWJzY3JpcHRzaGlmdEdGZW9GO0Y4RjsvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGOy1GIzYkLUZONi5RKiZ1bWludXMwO0YnRjhGOy9GUlEmZmFsc2VGJy9GVUZmcC9GV0ZmcC9GWUZmcC9GZW5GZnAvRmduRmZwL0ZpbkZmcC9GW29RLDAuMjIyMjIyMmVtRicvRl5vRl5xRjsvJTFzdXBlcnNjcmlwdHNoaWZ0R0Zlb0Zob0ZNLUknbXVuZGVyR0YkNiUtRiM2JS1GMTYoUSRsaW1GJ0Y4L0ZJRmZwLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y7L0krbXNlbWFudGljc0dGJFEmaW5lcnRGJ0YwRjstRiM2J0YwLUYxNiZRImhGJ0Y4RkhGSy1GTjYuUSkmI3gyMTkyO0YnRjhGO0ZRRlRGVkZYRlpGZm5GaG4vRltvUSYwLjBlbUYnL0Zeb0Zqci1JJW1zdXBHRiQ2JS1GY282JUZlb0Y4RjstRiM2JUZicEY4RjtGYHFGOy8lLGFjY2VudHVuZGVyR0ZmcC1JJm1mcmFjR0YkNigtRiM2KUYwLUYxNiZRImZGJ0Y4RkhGSy1GQTYlLUYjNiZGX28tRk42LlEiK0YnRjhGO0ZRRlRGVkZYRlpGZm5GaG5GXXFGX3FGY3JGO0Y4RjstRk42LVEoJm1pbnVzO0YnRjtGUUZURlZGWEZaRmZuRmhuRl1xRl9xRmpzLUZBNiUtRiM2JEZfb0Y7RjhGO0Y7RmNyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZgdS8lKWJldmVsbGVkR0ZmcEY7(TVI^`-`)[[x=-1/2]]=Limit(TVM(P,Q),h=0,left);
``=normal(rhs(%));
value(%);Lorsque le 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 existe (c'est une limite), on dit que la d\303\251riv\303\251e \303\240 gauche en 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 existe. Ici, cette d\303\251riv\303\251e vaut 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. La d\303\251riv\303\251e \303\240 gauche est not\303\251e plus bri\303\250vement de la mani\303\250re suivante.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
D\303\211FINITION
La d\303\251riv\303\251e \303\240 gauche d'une fonction f se d\303\251finit de la fa\303\247on suivante, lorsque la limite 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Puisque la d\303\251riv\303\251e \303\240 gauche existe, que la d\303\251riv\303\251e \303\240 droite existe et qu'elles sont \303\251gales, on dit que la fonction f est d\303\251rivable en 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 .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REMARQUE IMPORTANTE: Pour d\303\251finir la d\303\251riv\303\251e en 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 il faut absolument que l'on puisse approcher 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 sur la courbe par la droite et pas la gauche. Il faut donc que la fonction f soit continue dans tout voisinage de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjBGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdGOy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGPA==: f doit \303\252tre continue \303\240 droite de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjBGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdGOy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGPA== et \303\240 gauche de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjBGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdGOy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGPA==.C'est une condition n\303\251cessaire.Si la fonction f n'est pas continu en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYmUSIwRidGNUY3L0Y7USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZALyUlc2l6ZUdRIzEyRidGNUZB alors la fonction f n'est pas d\303\251rivable en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYmUSIwRidGNUY3L0Y7USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZALyUlc2l6ZUdRIzEyRidGNUZB.Mais ce n'est pas suffisant.Si la fonction f est continu en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYmUSIwRidGNUY3L0Y7USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZALyUlc2l6ZUdRIzEyRidGNUZB alors la fonction f n'est pas n\303\251cessairement d\303\251rivable en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYmUSIwRidGNUY3L0Y7USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZALUYvNiNRIUYnLyUlc2l6ZUdRIzEyRidGNUZB.En effet, d'une part, si l'une des deux d\303\251riv\303\251es directionnelles n'existent pas en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjBGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdGOy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGPA==, a fortiori la d\303\251riv\303\251e n'existe pas en LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjBGJy9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdGOA==. d'autre part, m\303\252me si la d\303\251riv\303\251e \303\240 gauche en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYmUSIwRidGNUY3L0Y7USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZALyUlc2l6ZUdRIzEyRidGNUZB et la d\303\251riv\303\251e \303\240 droite en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYmUSIwRidGNUY3L0Y7USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZALyUlc2l6ZUdRIzEyRidGNUZB existent toutes les deux, elles peuvent ne pas \303\252tre \303\251gales (ce sont des limites).Un exemple classique est la fonction f d\303\251finie par 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. En 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, la fonction f est continue mais pas d\303\251rivable en 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.Analysons la continuit\303\251 de f en 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.f:=x->abs(x);Limit(f(x),x=0,left);
assume(x<0);
``=Limit(f(x),x=0,left);
``=value(rhs(%));
x:='x':Limit(f(x),x=0,left);
assume(x>0);
``=Limit(f(x),x=0,left);
``=value(rhs(%));
x:='x':Puisque les deux limites directionnelles sont \303\251gales, f est continue en 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.Analysons ensuite la d\303\251rivabilit\303\251 de f en 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. P:=[0,'f'(0)]:
Q:=[0+h,'f'(0+h)]:
(TVI^`-`)[[x=0]]=Limit(TVM(P,Q),h=0,left);
assume(h<0);
``=rhs(%);
``=value(rhs(%));
h:='h':(TVI^`+`)[[x=0]]=Limit(TVM(P,Q),h=0,right);
assume(h>0);
``=rhs(%);
``=value(rhs(%));
h:='h':M\303\252me si les d\303\251riv\303\251es \303\240 gauche et \303\240 droite de f en 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 existent, elles ne sont pas \303\251gales, la fonction f n'est donc pas d\303\251rivale en 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.