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D\303\251rivation implicite\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. Le but de ce document est de transposer en Maple la d\303\251rivation implicite. On d\303\251taillera des r\303\251solutions habituelles de tangentes et de normales \303\240 des courbes d\303\251finies implicitement. Dans ce document, lorsqu'il s'agit d'une \303\251quation de deux variables, nous supposerons toujours l'existence d'au moins une fonction implicite. Bonne lecture \303\240 tous !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 Ce document Maple est ex\303\251cutable avec la version 2020.2Initialisationrestart;with(plots,implicitplot,pointplot,display,setoptions):
#with(plottools,pointplot)
setoptions(size=[400,400],scaling=constrained,axesfont=[times,roman,8],color=navy):\303\211quations implicitesIl est possible de d\303\251river implicitement avec la macro-commande diff. \303\200 l'aide de la syntaxe fonctionnelle, il faut d\303\251signer dans l'\303\251quation, laquelle des deux variables sera d\303\251sign\303\251e variable ind\303\251pendante. Par exemple, dans le cas de l'\303\251quation ci-dessous, 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 si x est d\303\251sign\303\251e variable ind\303\251pendante, soit 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 si y est d\303\251sign\303\251e variable ind\303\251pendante.Calculons LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiZRI2R5RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR0Y0LyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiZRI2R4RidGMkY1RjdGMkY1RjcvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkQvJSliZXZlbGxlZEdRJmZhbHNlRicvJSVzaXplR1EjMTJGJ0Y1L0Y4USdub3JtYWxGJw== o\303\271 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\303\211quation:=3*y(x)^2+5*x=3-5*y(x)^3;
diff(\303\211quation,x);Reste maintenant \303\240 isoler y' , c'est-\303\240-dire LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Ji1GLzYlUSNkeEYnRjJGNS8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EnTm9ybWFsRicvRjZRJ25vcm1hbEYnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZKLyUpYmV2ZWxsZWRHRj8tRi82JVEieUYnRjJGNS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYvNiVRInhGJ0YyRjVGQ0ZDRkM=. Diff(y(x),x)=solve(,diff(y(x),x));La macro-commande implicitdiff s'av\303\250re beaucoup plus souple dans le calcul des d\303\251riv\303\251es implicites et c'est la macro-commande qui sera retenue pour d\303\251river implicitement dans la suite de ce document.Macro-commande implicitdiffLa syntaxe de base est la suivante \302\253 implicitdiff(\303\211quation, var.d\303\251pendante, var.ind\303\251pendante) \302\273.Calculons de nouveau LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JI21pR0YkNiVRI2R5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYkLUYsNiVRI2R4RidGL0YyL0YzUSdub3JtYWxGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGQi8lKWJldmVsbGVkR1EmZmFsc2VGJw== avec implicitdiff. sans utiliser la syntaxe fonctionnelle pour la variable d\303\251pendante, assignez 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 \303\240 \303\211quation et calculons de nouveau LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRI2R5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JVEjZHhGJ0YyRjVGMi8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnRjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkUvJSliZXZlbGxlZEdRJmZhbHNlRicvJSVzaXplR1EjMTJGJ0Y9L0Y2USdub3JtYWxGJw==.\303\211quation:=3*y^2+5*x=3-5*y^3;
`y'`:=implicitdiff(\303\211quation,y,x);Il ne faut donc pas utiliser la notation fonctionnelle pour la variable y, celle qui est d\303\251crite implicitement par rapport \303\240 la variable x. On a donc que la macro-commande implicitdiff formule explicitement y', ce qui permet facilement d'assigner y' \303\240 un nom pour une \303\251ventuelle \303\251valuation.
\303\211valuons, par exemple, y'LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbWZyYWNHRiQ2KC1GIzYoLUkjbWlHRiQ2I1EhRictRjE2JlEiZEYnLyUlc2l6ZUdRIzEyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYxNiZRInlGJ0Y3RjpGPUY3LyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRj5RJ25vcm1hbEYnLUYjNihGMEY0LUYxNiZRInhGJ0Y3RjpGPUY3RkNGRi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGUi8lKWJldmVsbGVkR0ZFLUkobWZlbmNlZEdGJDYnLUYjNidGMC1JJW1zdWJHRiQ2JS1GIzYmRjBGN0ZDRkYtSSdtdGFibGVHRiQ2Ni1JJG10ckdGJDYmLUkkbXRkR0YkNigtRiM2KEYwLUYjNilGMEZKLUkjbW9HRiQ2LlEiPUYnRjdGRi8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGXnAvJSlzdHJldGNoeUdGXnAvJSpzeW1tZXRyaWNHRl5wLyUobGFyZ2VvcEdGXnAvJS5tb3ZhYmxlbGltaXRzR0ZecC8lJ2FjY2VudEdGXnAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZdcS1JI21uR0YkNiVRIzU1RidGN0ZGRjdGQ0ZGRjBGN0ZDRkYvJSlyb3dhbGlnbkdGMy8lLGNvbHVtbmFsaWduR0YzLyUrZ3JvdXBhbGlnbkdGMy8lKHJvd3NwYW5HRk8vJStjb2x1bW5zcGFuR0ZPRmRxRmZxRmhxLUZfbzYmLUZibzYoLUYjNihGMC1GIzYqRjBGQEZoby1GaW82LlEpJiN4MjIxMjtGJ0Y3RkZGXHBGX3BGYXBGY3BGZXBGZ3BGaXAvRlxxUSwwLjIyMjIyMjJlbUYnL0ZfcUZqci1GYXE2JVEiNEYnRjdGRkY3RkNGRkYwRjdGQ0ZGRmRxRmZxRmhxRmpxRlxyRmRxRmZxRmhxLyUmYWxpZ25HUSVheGlzRicvRmVxUSliYXNlbGluZUYnL0ZncVElbGVmdEYnL0ZpcVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjwvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0ZcdC8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0ZndC8lJmZyYW1lR0ZndC8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRkUvJS1lcXVhbGNvbHVtbnNHRkUvJS1kaXNwbGF5c3R5bGVHRkUvJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGZHQvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y3RkNGRkY3RkYvJSVvcGVuR1EifGdyRicvJSZjbG9zZUdGM0Y3RkNGRg==.
Eval('`y'`',{x=55,y=-4})=eval(`y'`,{x=55,y=-4});Pour obtenir une d\303\251riv\303\251e implicite successive d'ordre sup\303\251rieur, on proc\303\250de de la m\303\252me fa\303\247on qu'avec la macro-commande diff. Soit, par exemple , calculons 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 de l'exemple pr\303\251c\303\251dent 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. On l'obtient directement en utilisant l'op\303\251rateur de s\303\251quence $.\303\211quation:=3*y^2+5*x=3-5*y^3;
`y'''`:=implicitdiff(\303\211quation,y,x$3);Exemple 1Calculons LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiZRI2R5RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR0Y0LyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmRi5GMkY1RjcvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkEvJSliZXZlbGxlZEdRJmZhbHNlRictRi82I1EhRicvJSVzaXplR1EjMTJGJ0Y1L0Y4USdub3JtYWxGJw== si 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.\303\211quation:=exp(x*y)=x^2*y^3;
`y'`:=implicitdiff(\303\211quation,y,x);Sur la base de l'\303\251galit\303\251 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 , exprimons la d\303\251riv\303\251e y', en termes de x et y seulement. Simplifions donc y' en substituant l'expression LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYmLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSN4eUYnRjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSJeRicvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y2USdub3JtYWxGJw== par 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 dans la d\303\251riv\303\251e y'.``=subs(\303\211quation,);Il est possible de transformer encore y' en factorisant xy au num\303\251rateur et au d\303\251nominateur. Pour terminer la simplification donc, normalisons la r\303\251ponse pr\303\251c\303\251dente.normal();Exemple 2Voici un exemple classique de r\303\251solution d'un probl\303\250me de tangentes.
Soit l'ellipse d'\303\251quation 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. Calculer la pente de chacune des tangentes \303\240 cette courbe en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjNGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ et obtener leur \303\251quation respective.
Dans un m\303\252me graphique, tracer ces tangentes et la courbe. Enfin, d\303\251terminer, si possible, le point d'intersection de ces tangentes.
Solution propos\303\251eObtenons d'abord la forme canonique de l'\303\251quation de l'ellipse afin de bien mettre en \303\251vidence ses caract\303\251ristiques. C'est un travail n\303\251cessaire pour une approche judicieuse dans le trac\303\251 \303\240 faire.\303\200 l'aide de la macro-commande CompleteSquare de la sous biblioth\303\250que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JlEsd2l0aFN0dWRlbnRGJy8lJ2ZhbWlseUdRLENvdXJpZXJ+TmV3RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2Jy1GIzYoLUYsNiZRLFByZWNhbGN1bHVzRidGL0YyRjVGLy8lJXNpemVHUSMxMkYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y1Ri9GNS8lJW9wZW5HUScmbHNxYjtGJy8lJmNsb3NlR1EnJnJzcWI7RictRiw2I1EhRidGL0ZARkNGRkY1, rendons accessible cette macro-commande pour la session Maple actuellewith(Student[Precalculus],CompleteSquare);\303\211quation:=4*x^2 + 9*y^2 - 8*x + 54*y + 49 = 0;\303\211q1:=CompleteSquare(\303\211quation);\303\211q2:=\303\211q1+(36=36);\303\211q_Ellipse:=1/36*\303\211q2;L'ensemble des points du plan cart\303\251sien dont les coordonn\303\251es v\303\251rifient l'\303\251quation 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 est donc une ellipse centr\303\251e au point 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 dont l'axe principal est parall\303\250le \303\240 l'axe des x. La demi-longueur de l'axe principal vaut LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1JJm1zcXJ0R0YkNiMtSSNtbkdGJDYkUSI5RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEpJmVxdWFscztGJ0Y4LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZBLyUpc3RyZXRjaHlHRkEvJSpzeW1tZXRyaWNHRkEvJShsYXJnZW9wR0ZBLyUubW92YWJsZWxpbWl0c0dGQS8lJ2FjY2VudEdGQS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlAtRjU2JFEiM0YnRjhGOEYrLyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZBRjg= et la demi-longueur du second axe vaut LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbXNxcnRHRiQ2Iy1GIzYkLUkjbW5HRiQ2JFEiNEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y0LUkjbW9HRiQ2LVEpJmVxdWFscztGJ0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRjE2JFEiMkYnRjRGNA==, le trac\303\251 complet de cette ellipse sera obtenu avec 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 et 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
Sp\303\251cifions l'option view=[-3..5,-6..1] pour mieux visualiser l'ellipse avec les axes de coordonn\303\251es.Ellipse:=implicitplot(\303\211q_Ellipse,x=-2..4,y=-5..-1):
display(Ellipse,view=[-3..5,-6..1]);Pour calculer les pentes des tangentes, il faut obtenir d'abord LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiZRI2R5RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR0Y0LyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiZRI2R4RidGMkY1RjdGMkY1RjcvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkQvJSliZXZlbGxlZEdRJmZhbHNlRicvJSVzaXplR1EjMTJGJ0Y1L0Y4USdub3JtYWxGJw== et ensuite, faire l'\303\251valuation de la d\303\251riv\303\251e avec les coordonn\303\251es de chaque point de tangence.Obtenons d'abord la d\303\251riv\303\251e LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JI21pR0YkNiVRI2R5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYmLUYsNiVRI2R4RidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSdOb3JtYWxGJy9GM1Enbm9ybWFsRicvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkgvJSliZXZlbGxlZEdGPQ==.`y'`:=implicitdiff(\303\211quation,y,x);Pour \303\251valuer la d\303\251riv\303\251e avec les deux coordonn\303\251es de chaque point de tangence, il faut obtenir l'ordonn\303\251e de chaque point de l'ellipse ayant pour abscisse la valeur 3.Sol:=solve(\303\211quation,y);
Ordonn\303\251es:=eval(Sol,x=3);L'\303\251quation point-pente de chaque tangente est de la forme 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Obtenons la pente de chaque tangente en \303\251valuant la d\303\251riv\303\251e avec les coordonn\303\251es des points de tangence respectif.m[1]:=eval(`y'`,{x=3,y=Ordonn\303\251es[1]});
m[2]:=eval(`y'`,{x=3,y=Ordonn\303\251es[2]});La pente de la tangente passant par le point 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 est 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.
La pente de la tangente passant par le point 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 est 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Les \303\251quations de chaque tangente sont alors les suivantes.\303\211q_tangente[1]:=y=m[1]*(x-3)+Ordonn\303\251es[1];
\303\211q_tangente[2]:=y=m[2]*(x-3)+Ordonn\303\251es[2];Pour obtenir la forme 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, il suffit d'appliquer la macro-commande expand.\303\211qBis_tangente[1]:=expand(\303\211q_tangente[1]);
\303\211qBis_tangente[2]:=expand(\303\211q_tangente[2]);Superposons maintenant le trac\303\251 de l'ellipse avec ceux des deux droites tangentes.D_tangente[1]:=plot([x,rhs(\303\211q_tangente[1]),x=2..4],color=navy):
D_tangente[2]:=plot([x,rhs(\303\211q_tangente[2]),x=2..4],color=navy):
display({Ellipse,D_tangente[1],D_tangente[2]},view=[-3..5,-6..1]);Finalement, d\303\251terminons les coordonn\303\251es du point d'intersection des deux droites tangentes en r\303\251solvant le syst\303\250me suivant.solve({\303\211q_tangente[1],\303\211q_tangente[2]});Le point d'intersection des tangentes est LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYkLUYjNistSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JVEjMTFGJy8lJXNpemVHUSI5RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUY0NiVRIjJGJ0Y3RjovJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkUvJSliZXZlbGxlZEdRJmZhbHNlRictSSNtb0dGJDYuUSIsRidGN0Y6LyUmZmVuY2VHRkovJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGSi8lKnN5bW1ldHJpY0dGSi8lKGxhcmdlb3BHRkovJS5tb3ZhYmxlbGltaXRzR0ZKLyUnYWNjZW50R0ZKLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRkw2LlEqJnVtaW51czA7RidGN0Y6Rk8vRlJGSkZURlZGWEZaRmZuL0ZpblEsMC4yMjIyMjIyZW1GJy9GXG9GY28tRjQ2JVEiM0YnRjdGOkY3LyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUrZXhlY3V0YWJsZUdGSi8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjpGOkZob0ZbcEZdcEY6.D_tangente[1]:=plot([x,rhs(\303\211q_tangente[1]),x=2..11/2],color=navy):
D_tangente[2]:=plot([x,rhs(\303\211q_tangente[2]),x=2..11/2],color=navy):
Point_t:=pointplot([11/2,-3],symbol=solidcircle,symbolsize=15):
display({Point_t,Ellipse,D_tangente[1],D_tangente[2]},view=[-3..6,-6..1]);
Trouvons maintenant le point d'intersection des deux normales en ces points de tangence.La pente de la normale passant par le point 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 est 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.
La pente de la tangente passant par le point 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 est 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Les \303\251quations de chaque tangente sont alors les suivantes.\303\211q_normale[1]:=y=-1/m[1]*(x-3)+Ordonn\303\251es[1];
\303\211q_normale[2]:=y=-1/m[2]*(x-3)+Ordonn\303\251es[2];D_normale[1]:=plot([x,rhs(\303\211q_normale[1]),x=2.5..3.5],color=navy):
D_normale[2]:=plot([x,rhs(\303\211q_normale[2]),x=2.5..3.5],color=navy):
display({Ellipse,D_normale[1],D_normale[2]},view=[-3..5,-6..1]);solve({\303\211q_normale[1],\303\211q_normale[2]});Le point d'intersection des normales est 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D_normale[1]:=plot([x,rhs(\303\211q_normale[1]),x=19/9..3.5],color=navy):
D_normale[2]:=plot([x,rhs(\303\211q_normale[2]),x=19/9..3.5],color=navy):
Point_n:=pointplot([19/9,-3],symbol=solidcircle,symbolsize=15):
display({Point_n,Ellipse,D_normale[1],D_normale[2]},view=[-3..5,-6..1]);display({Ellipse,
Point_t,D_tangente[1],D_tangente[2],
Point_n,D_normale[1],D_normale[2]},
view=[-3..6,-6..1]);Exemple 3Soit l'\303\251quation 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. Tra\303\247ons le lieu d\303\251finie par cette \303\251quation.\303\211q:=y^3+y^2-5*y-x^2=-4;
Lieu:=implicitplot(\303\211q,x=-5..5,y=-5..5);Trouvons les points de ce trac\303\251 o\303\271 les tangentes sont verticales. Trouvons donc les points o\303\271 la d\303\251riv\303\251e n'existe pas.`y'`:=implicitdiff(\303\211q,y,x);La d\303\251riv\303\251e n'existe pas lorsque le d\303\251nominateur s'annule.Sol:=solve(denom()=0,{y});Obtenons les abcisses des points du trac\303\251 qui ont 1 et 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 comme ordonn\303\251es. Selon le graphique, nous devrions en trouver deux pour chaque valeur de y.Substituons la valeur de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ et LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GIzYmLUY7Ni5RKCZtaW51cztGJy8lJXNpemVHUSI5RidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjIyMjIyMjJlbUYnL0ZTRmduLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiVRIjVGJ0ZZRj4tRl1vNiVRIjNGJ0ZZRj4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmhvLyUpYmV2ZWxsZWRHRkJGWUY+RitGPkYrL0ZaUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ dans LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNidGKy1GIzYnRistRiM2KEYrLUYjNiYtSSVtc3VwR0YkNiYtRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjNGJy9GQ1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEiXkYnLUkjbW9HRiQ2LVEnJnBsdXM7RidGSS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGVy8lKXN0cmV0Y2h5R0ZXLyUqc3ltbWV0cmljR0ZXLyUobGFyZ2VvcEdGVy8lLm1vdmFibGVsaW1pdHNHRlcvJSdhY2NlbnRHRlcvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0Zgby1GOjYmRjwtRkY2JFEiMkYnRklGS0ZORkktRlI2LVEoJm1pbnVzO0YnRklGVUZYRlpGZm5GaG5Gam5GXG9GXm9GYW8tRiM2Ji1GRjYkUSI1RidGSS1GUjYtUTEmSW52aXNpYmxlVGltZXM7RidGSUZVRlhGWkZmbkZobkZqbkZcby9GX29RJjAuMGVtRicvRmJvRmRwRjxGSUYrRklGaG8tRjo2Ji1GLDYlUSJ4RidGP0ZCRmVvRktGTkZJRlEtRkY2JFEiNEYnRklGSS1GUjYtUSkmZXF1YWxzO0YnRklGVUZYRlpGZm5GaG5Gam5GXG8vRl9vUSwwLjI3Nzc3NzhlbUYnL0Zib0ZicS1GRjYkRk1GSUZJRisvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRldGSQ== puis r\303\251solvons pour x chaque \303\251quation obtenue.solve(subs(y=1,y^3-x^2+y^2-5*y+4=0),{x});
solve(subs(y=-5/3,y^3-x^2+y^2-5*y+4=0),{x});Les points recherch\303\251s sont donc 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, 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, 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 et 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.Tra\303\247ons finalement les tangentes au grahique en ces points.Tangente_1:=plot([-sqrt(849)/9,t,t=-3..-0.5],color="Niagara 16",thickness=1):
Tangente_2:=plot([-1,t,t=0..2],color="Niagara 16",thickness=1):
Tangente_3:=plot([1,t,t=0..2],color="Niagara 16",thickness=1):
Tangente_4:=plot([sqrt(849)/9,t,t=-3..-0.5],color="Niagara 16",thickness=1):
display(Lieu,Tangente_||(1..4));