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11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple P lot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 5 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 5 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 262 1 {CSTYLE " " -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 263 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 265 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 40 "Les Maths avec MAPLE V , PA S A PAS." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 29 "Commandes de calcul (rappels)" }}{SECT 1 {PARA 4 "" 0 " " {TEXT 294 20 "R\350gles fondamentales" }}{PARA 0 "" 0 "" {TEXT -1 537 "Maple est d'abord un outil \"interactif\" (on dit que c'est un la ngage 'interpr\351t\351'), ceci a de nombreux avantages (d\351tection \+ imm\351diate des erreurs, possibilit\351 de changer rapidement une don n\351e,d\351placements arbitraires dans une feuille de calcul, etc.) e t quelques inconv\351nients: comme on peut ne pas faire les calculs da ns l'ordre o\371 ils sont \351crits, il est possible de se perdre tr \350s vite. C'est pourquoi des raccourcis utiles (mais d\351conseill \351s aux utilisateurs novices) rappel\351s ci-dessous ne sont en fait jamais utilis\351s dans ce texte" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "Les calculs sont effectu\351s apr\350s ch aque \" ; \"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(10+5)^3;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%vL" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "(x+5)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&%\"xG \"\"\"\"\"&F&\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 93 "Mais il faut souvent utiliser une des fonctions de mani pulation pour avoir la forme souhait\351e" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand((x+5)^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, **$%\"xG\"\"$\"\"\"*$F%\"\"#\"#:F%\"#v\"$D\"F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "factor(x^3+15*x^2+75*x+15);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,**$%\"xG\"\"$\"\"\"*$F%\"\"#\"#:F%\"#vF*F'" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "D'autre p art, les calculs sont \"symboliques\"; pour avoir des valeurs num\351r iques, utiliser evalf:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a: =sqrt(2); a; evalf(a);evalf(a,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"aG*$\"\"##\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"##\"\" \"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5)[]4tBc8UT\"!#>" }}}{PARA 0 "" 0 "" {TEXT -1 94 "On voit que plusieurs ; sont ex\351cut\351s en m\352me temps; inve rsement, tant qu'on n'a pas tap\351 de ;" }}{PARA 0 "" 0 "" {TEXT -1 41 " les lignes continuent jusqu'au ; suivant" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "x+" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "y+" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "z;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,(%\"xG\"\"\"%\"yGF%%\"zGF%" }}}{PARA 0 "" 0 "" {TEXT -1 48 "Le dern ier calcul effectu\351 a pour abr\351viation \":" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "\"^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,(% \"xG\"\"\"%\"yGF&%\"zGF&\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 73 "Mais ra ppelons que cette abr\351viation est fortement d\351conseill\351e: en \+ effet" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "\"^2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*$,(%\"xG\"\"\"%\"yGF&%\"zGF&\"\"%" }}}{PARA 0 " " 0 "" {TEXT -1 1 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 295 11 "Affecta tion" }}{PARA 0 "" 0 "" {TEXT -1 67 "L'affectation (le fait de donner \+ une valeur) est repr\351sent\351e par :=" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x:=x1; y:=x2; x+y; y:=3; x+y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG%#x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG%# x2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#x1G\"\"\"%#x2GF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#x1G\"\"\"\"\"$F%" }}}{PARA 0 "" 0 "" {TEXT -1 40 "Supprimer une affectation se fait ainsi:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "y:='y';x+y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#x1G\"\"\"%\"yGF%" }}}{PARA 0 "" 0 "" {TEXT -1 31 "Et on ram\350ne tout \340 0 en tapant" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 10 "restart;x;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 208 "Dans ce texte, des 'restart' ont \351t\351 demand\351s au d\351but de chaque paragraphe; ne pas \+ le faire produirait peut-\352tre un texte plus \351l\351gant, mais ren drait une r\351ex\351cution \"compl\350te\" de la feuille impossible.. .." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 296 10 "Evaluation" }}{PARA 4 "" 0 "" {TEXT 303 176 "Dans certains cas, le r\351sultat semble insuffisa mment \351valu\351; nous allons voir au prochain paragraphe comment ma nipuler des formules, mais voici quelques situations fr\351quentes:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "a:=sqrt(2)+sqrt(3); evalf( a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,&*$\"\"##\"\"\"F'F)*$\" \"$F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+qVEYJ!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 155 "pour obtenir une valeur num\351rique; la pr\351ci sion (10 chiffres par d\351faut) est la valeur de Digits; mais on peut aussi ne demander qu'une \351valuation pr\351cise:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "evalf(exp(1),25);evalf(Pi); Digits:=50;eval f(Pi);Digits:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\":(Gg`BX!f%G=G= F!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"S^P*Rpr>%)G]zKQVEYQKz*e`EfTJ!#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'DigitsG\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 59 "Il existe d'autres \" \351valuateurs\", ainsi, dans les complexes" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "r:=sqrt(1+2*sqrt(2)*I);evalf(r);evalc(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG*$,&\"\"\"F'*&%\"IGF'\"\"##F'F*F*F+" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+iN@99!\"*\"\"\"%\"IG$\"+)****** ***!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$\"\"##\"\"\"F%F'%\"IGF' " }}}{PARA 0 "" 0 "" {TEXT -1 30 "ou encore l'\351valuateur bool\351en " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eq:=2+2=5;eq;evalb(eq); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&f alseG" }}}{PARA 0 "" 0 "" {TEXT -1 89 "Nous verrons plus tard l'\351va luateur \"neutre\" eval, qui donne la 'd\351finition' d'un objet :" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f:= x->1+x^2; f; eval(f); ev al(expand);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)op eratorG%&arrowGF(,&\"\"\"F-*$9$\"\"#F-F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"fG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$% )operatorG%&arrowGF&,&\"\"\"F+*$9$\"\"#F+F&F&6\"" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 280 "Dommage: cette derni \350re ligne ( proc () options builtin, remember; 92 end) nous informe que \"expand\" est \"cabl\351\" dans Maple, et que nous n'en saurons \+ pas plus long; eval(sin) ne semble gu\350re plus prometteur, mais nous verrons plus tard comment lui faire d\351voiler une partie de ses" }} {PARA 4 "" 0 "" {TEXT 297 10 "myst\350res.." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 261 48 "Techniques courantes \+ de manipulation de formules" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 265 23 "Expressions alg \351briques" }}{PARA 0 "" 0 "" {TEXT -1 76 "Les expressions usuelles s ont mises par Maple sous une forme peu simplifi\351e:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "expr:=1+( x+1)^2-5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,&!\"%\"\"\"*$,&% \"xGF'F'F'\"\"#F'" }}}{PARA 0 "" 0 "" {TEXT -1 32 "Le d\351veloppement s'obtient par :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "expand(e xpr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(!\"$\"\"\"*$%\"xG\"\"#F%F'F (" }}}{PARA 0 "" 0 "" {TEXT -1 37 "et la factorisation (rationnelle) p ar" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(expr);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"\"\"$F&F&,&F%F&!\"\"F&F &" }}}{PARA 0 "" 0 "" {TEXT -1 77 "mais il n'est pas possible de facto riser (directement) s'il y a des radicaux:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "factor(x^2-x-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,(*$%\"xG\"\"#\"\"\"F%!\"\"F(F'" }}}{SECT 1 {PARA 262 "" 0 "" {TEXT 304 30 "remarques sur la factorisation" }}{PARA 0 "" 0 "" {TEXT -1 83 "Si on connait le discriminant, par exemple, de nouvelles possibilit \351s apparaissent:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "facto r(x^2-x-1,sqrt(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(%\"xG\"\" #!\"\"\"\"\"*$\"\"&#F)F'F)F),(F&F'F(F)F*F(F)#F)\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 71 "D'autre part, la m\351thode des \"racines \351videntes \" est connu du programme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "factor(x^4+x^3+x-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\" \"!\"\"F&F&,**$F%\"\"$F&*$F%\"\"#F,F%F,F*F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 161 "Mais pour avoir directement les solutions, mieux vaut ut iliser \"solve\", et esp\351rer que le r\351sultat ne sera pas trop \" lourd\", ou donn\351 sous une forme inutilisable" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "solve(x^4-5*x^2+6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&*$\"\"##\"\"\"F$,$F#!\"\"*$\"\"$F%,$F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve (x^4-6*x^2+10);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&*$,&\"\"$\"\"\"%\"IGF&#F&\"\"#,$F#!\"\"*$,&F%F&F 'F+F(,$F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x^4+x^ 3+x-3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&\"\"\",(*$,&\"$W#F#*$\"$<%# F#\"\"#\"#7#F#\"\"$#!\"\"\"\"'*$F&#F0F.#\"\"%F.#!\"#F.F#,*F%#F#F,F2F6F 6F#*(%\"IGF#F.F*,&F%F/F2#!\"%F.F#F*,*F%F9F2F6F6F#F:#F0F+" }}}}{PARA 4 "" 0 "" {TEXT 264 23 "Passons aux fractions: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "fra:=7/6+1/x-1/(x-1)-1/(1+1/x); fra:=simplify(fr a);factor(fra);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fraG,*#\"\"(\"\" '\"\"\"*$%\"xG!\"\"F)*$,&F+F)F,F)F,F,*$,&F)F)F*F)F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fraG,$**,**$%\"xG\"\"$\"\"\"F)!#8*$F)\"\"#\"\"' !\"'F+F+F)!\"\",&F)F+F1F+F1,&F)F+F+F+F1#F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,,&%\"xG\"\"\"!\"#F'F',(*$F&\"\"#F'F&\"\")\"\"$F'F'F &!\"\",&F&F'F.F'F.,&F&F'F'F'F.#F'\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 68 "D'autres formes simplifi\351es de (fra) existent; essentiellement, on a" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "expand (fra);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**(%\"xG\"\"#,&F%\"\"\"!\"\"F(F),&F%F (F(F(F)#F(\"\"'*&F'F)F*F)#!#8F,*(F%F(F'F)F*F)F(*(F%F)F'F)F*F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 50 "qui comme on le voit, d\351veloppe le num \351rateur, et" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "convert(f ra, parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*#\"\"\"\"\"'F%*$% \"xG!\"\"F%*$,&F(F%F)F%F)F)*$,&F(F%F%F%F)F%" }}}{PARA 0 "" 0 "" {TEXT -1 145 "dont nous reparlerons (en maths) au chapitre 12. Devant une fo nction myst\351rieuse comme convert, il est temps de parler du mode d' emploi de l'aide" }}{SECT 1 {PARA 263 "" 0 "" {TEXT -1 31 "Interlude : les fichiers d'aide" }}{PARA 0 "" 0 "" {TEXT -1 107 "L'aide, sous Map le, se fait essentiellement interactivement, soit \340 la souris, soit en tapant, par exemple " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " help (convert);" }}}{PARA 0 "" 0 "" {TEXT -1 316 "mais il est recomman d\351 de continuer soi-m\352me cet exemple; de toute fa\347on, l'abond ance des informations affich\351es (en anglais) fait qu'on a int\351r \352t \340 se laisser guider par les exemples (qu'on peut au besoin co pier-coller) et \340 ne pas trop chercher \340 comprendre l'utilit\351 d'une fonction non encore \351tudi\351e en classe..." }}}{PARA 4 "" 0 "" {TEXT 266 64 "Bien entendu, toutes ces formes existent avec plusi eurs lettres:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "frb:=(x^2-a ^2)/(x^2-b^2);factor(frb); expand(frb);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$frbG*&,&*$%\"xG\"\"#\"\"\"*$%\"aGF)!\"\"F*,&F'F**$%\"bGF)F-F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&%\"xG\"\"\"%\"aG!\"\"F',&F&F 'F(F'F',&%\"bGF'F&F)F),&F,F'F&F'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*&,&*$%\"xG\"\"#\"\"\"*$%\"bGF(!\"\"F,F'F(F)*&F%F,%\"aGF(F," }}} {PARA 0 "" 0 "" {TEXT -1 7 "et m\352me" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert(frb,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*(,&*$%\"aG\"\"#!\"\"*$%\"bGF)F$F$F,F*,&F,F$%\"xGF*F*# F*F)*(F&F$F,F*,&F,F$F.F$F*F/" }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 0 "" 0 "" {TEXT 284 18 "Fonctions usuelles" }}{PARA 0 " " 0 "" {TEXT -1 150 "Les expressions \"irrationnelles\" se traitent de m\352me; encore faut-il connaitre les noms des fonctions les plus fr \351quentes; voici une liste d'exemples:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 280 45 " ---- racines carr\351es, valeur absolue, signe " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(x^2-1);abs(x+1); si gn(x^2+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$%\"xG\"\"#\"\"\"! \"\"F(#F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$absG6#,&%\"xG\"\"\"F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 155 "Il est bon de conna\356tre une autre fonction de simplif ication de Maple pour ce cas: la fonction 'radnormal' (l'option ration alized chasse les d\351nominateurs)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "a:=(1+sqrt(3))^6;b:=sqrt(2)/(sqrt(3)+sqrt(2)); radno rmal(a);radnormal(b, rationalized);a:='a':b:='b':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG*$,&\"\"\"F'*$\"\"$#F'\"\"#F'\"\"'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"bG*&\"\"##\"\"\"F&,&*$\"\"$F'F(*$F&F'F(!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"$3#\"\"\"*$\"\"$#F%\"\"#\"$?\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"##\"\"\"F%\"\"$F&F'!\"#F' " }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 281 31 " ---- logarithme, e xponentielle" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "l:=ln(4*x^2) ; expand (l); ee:=exp(1+x); expand(ee); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lG-%#lnG6#,$*$%\"xG\"\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#\"\"%\"\"\"-F%6#%\"xG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eeG-%$expG6#,&%\"xG\"\"\"F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#%\"xG\"\"\"-F%6#F(F(" }}}{PARA 264 "" 1 "" {TEXT 282 4 " " }{TEXT -1 82 "remarque: la lettre e n'est pas r\351 serv\351e; il faut , au besoin, la d\351clarer par : " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "e:=exp(1);e^x;expand(e^(x+1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eG-%$expG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#)-%$expG6#\"\"\"%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)-%$expG6#\"\"\"%\"xGF(F%F(" }}}{PARA 0 "" 0 "" {TEXT 283 80 "---- Fonctions trigonom\351triques, hyperboliques, trigo nom\351triques \"inverses\",etc." }}{PARA 0 "" 0 "" {TEXT -1 82 " La l iste compl\350te peut \352tre obtenue par help; voici celles dont nous avons besoin" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sin(Pi/4); \+ " }{TEXT -1 26 "attention \340 ne pas \351crire " }{MPLTEXT 1 0 10 "si n(pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"##\"\"\"F%F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#,$%#piG#\"\"\"\"\"%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a1:=sinh(1);cosh(1);expand(a1);convert(a1,exp);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#a1G-%%sinhG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%coshG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%sinhG6#\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#\"\"\"#F'\"\"#*$F$! \"\"#F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "arctan(1);b1:= arcsinh(1);convert(b1,ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG# \"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G-%(arcsinhG6#\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&*$\"\"##\"\"\"F(F*F*F* " }}}{PARA 0 "" 0 "" {TEXT -1 143 "Des simplifications plus sophistiqu \351es demandnt une bonne ma\356trise de expand, simplify, convert, fa ctor, etc... Ainsi, les \"formules\" de trigo:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "sin(a+b);A:=expand(sin(a+b)); factor(A);simpl ify(A);convert(A,sincos); combine(A,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#,&%\"aG\"\"\"%\"bGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,&*&-%$sinG6#%\"aG\"\"\"-%$cosG6#%\"bGF+F+*&-F-F) F+-F(F.F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"aG\"\" \"-%$cosG6#%\"bGF)F)*&-F+F'F)-F&F,F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"aG\"\"\"-%$cosG6#%\"bGF)F)*&-F+F'F)-F&F,F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"aG\"\"\"-%$cosG6#%\"bG F)F)*&-F+F'F)-F&F,F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#,&% \"aG\"\"\"%\"bGF(" }}}{PARA 0 "" 0 "" {TEXT -1 85 "Enfin! Et dans cert ains cas, l'utilisateur devra faire une partie du travail lui-m\352me " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "c:=arctan(2)+arctan(3); \+ simplify(c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG,&-%'arctanG6#\" \"#\"\"\"-F'6#\"\"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arctanG6 #\"\"#\"\"\"-F%6#\"\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand(c);factor(c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arctanG6 #\"\"#\"\"\"-F%6#\"\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arcta nG6#\"\"#\"\"\"-F%6#\"\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "d:=tan(c);simplify(d);expand(d);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"dG-%$tanG6#,&-%'arctanG6#\"\"#\"\"\"-F*6#\"\"$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$tanG6#,&-%'arctanG6#\"\"#\"\"\"-F(6#\"\"$F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(tan(x)=-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$%#PiG#!\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 13 "et pour finir" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "evalf(c-3*Pi/4);evalf(c-3*Pi /4,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!\"\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 4 "" 0 "" {TEXT 285 6 "ouf!!!" }}} {SECT 1 {PARA 0 "" 0 "" {TEXT 287 41 "R\351solution d'\351quations (et d'in\351quations)" }}{PARA 0 "" 0 "" {TEXT -1 191 "On a d\351ja vu qu elques utilisations de solve ; dans l'ensemble, ne pas lui faire une c onfiance excessive! Ainsi, si, pour des \351quations alg\351briques \" simples\", on a des r\351sultats satisfaisants:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "solve(2*x^3+5*x-7); solve(x^4+2*x^2-8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\",&#!\"\"\"\"#F#*&%\"IGF#\"#8#F#F 'F+,&F%F#F(F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&,$%\"IG\"\"#,$F$!\"#* $F%#\"\"\"F%,$F(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 112 "les solutions \+ d'\351quations plus complexes sont plus difficiles \340 lire (ou m \352me parfois fausses, voir plus loin):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(x^3=x+1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%,& *$,&\"$3\"\"\"\"*$\"#p#F'\"\"#\"#7#F'\"\"$#F'\"\"'*$F%#!\"\"F.F+,(F$#F 3F,F1F3*(%\"IGF'F.F*,&F$F/F1!\"#F'F*,(F$F5F1F3F6#F3F+" }}}{PARA 0 "" 0 "" {TEXT -1 207 "Maple introduit les abr\351viations %1 et %2 (et ne s'en sert d'ailleurs pas tr\350s bien); vous pouvez \340 pr\351sent v ous en servir aussi (mais pas n'importe comment: vous n'avez pas le dr oit d'en changer la valeur):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "### WARNING: use of labels non-interactively can give inconsistent results\nevalf(%1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bPj)o\"!#5 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "### WARNING: use of lab els non-interactively can give inconsistent results\n%1:=1;" }}{PARA 8 "" 1 "" {TEXT -1 29 "Syntax error, `:=` unexpected" }}}{PARA 0 "" 0 "" {TEXT -1 46 " Des fonctions \351tranges peuvent parfois surgir" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(exp(x)=5*x);evalf(\"); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$-%)LambertWG6##!\"\"\"\"&F(,$-F% 6$F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+=5r\"f#!#5$\"+e8kUD!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 48 "Bien s\373r, il y a des \351quation s sans solutions..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve (exp(x)=0);" }}}{PARA 0 "" 0 "" {TEXT -1 69 "A ne pas confondre avec d es \351quations que Maple ne sait pas r\351soudre!" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "solve (x^2+exp(x)=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,(-%$expG6#%#_ZG\"\"\"*$F*\"\"#F+!\"$F+" }} }{PARA 0 "" 0 "" {TEXT -1 43 "La r\351solution des in\351quations est \+ similaire" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(x^3-3*x^2 +2>0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$-%%OpenG6#,& \"\"\"F**$\"\"$#F*\"\"#!\"\"-F'6#F*-F$6$-F'6#,&F*F*F+F*%)infinityG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve((x^3-3*x^2+2)/x >=0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%-%*RealRangeG6$,$%)infinityG!\"\", &\"\"\"F**$\"\"$#F*\"\"#F(-F$6$-%%OpenG6#\"\"!F*-F$6$,&F*F*F+F*F'" }}} {PARA 0 "" 0 "" {TEXT -1 67 "Correct (si on comprend l'anglais :-)). R emarquer au passage que (-" }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 10 ") s'\351crit:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(x ^2+1>0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 116 "Quand il y a plusieurs solutions (et que la th\351orie n e permet pas d'en deviner le nombre), il faut \352tre tr\350s prudent " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eqi:=abs(x^2-4)=abs((abs (x)-1)/2);solve(eqi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqiG/-%$ab sG6#,&*$%\"xG\"\"#\"\"\"!\"%F--F'6#,&-F'6#F+#F-F,#!\"\"F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&,&#\"\"\"\"\"%F%*$\"#d#F%\"\"#F$,&#!\"\"F&F% F'F,,&F,F%*$\"#tF)F$,&F$F%F/F," }}}{PARA 0 "" 0 "" {TEXT -1 76 "Correc t, mais la prochaine va nous montrer qu'on peut l\351gitimement se m \351fier" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve (cos(5*x)= 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"#5" }}}{PARA 266 "" 1 "" {TEXT -1 153 "Il y a pourtant une infinit\351 de solutions ! C'est pire encore quand Maple choisit, de mani\350re apparemment arb itraire, d'en livrer quelques-unes seulement:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(cos(2*x)=sin(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,$%#PiG#!\"\"\"\"#,$F$#\"\"\"\"\"',$F$#\"\"&F+" }}} {PARA 0 "" 0 "" {TEXT -1 211 "D'autres probl\350mes apparaissent avec \+ plusieurs inconnues (mais l\340, c'est un peu normal); le format stand ard est solve(\{eq1,eq2,...\},\{x1,x2,...\}) (ensemble des \351quation s, suivi de l'ensemble des inconnues). Ainsi" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(\{x+a*y=b,a*x+y=2\},\{x,y\});" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<$/%\"xG,$*&,&%\"aG!\"#%\"bG\"\"\"F,,&!\"\"F,*$F )\"\"#F,F.F./%\"yG*&,&*&F)F,F+F,F,F*F,F,F-F." }}}{PARA 0 "" 0 "" {TEXT -1 152 "(on remarquera que Maple ne se pr\351occuppe pas du cas \+ a=1 ; il y a pourtant des solutions si b=2). D'autre part, savoir ce q ui est inconnu est essentiel!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(\{x+a*y=b,a*x+y=2\},\{a,b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"aG,$*&,&%\"yG\"\"\"!\"#F*F*%\"xG!\"\"F-/%\"bG*&,( *$F,\"\"#F**$F)F3F-F)F3F*F,F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 288 "Dans beaucoup des cas pr\351c\351dents, \+ pour utiliser la solution, on doit faire des affectations et du \"coup er-coller\"; Maple propose la m\351thode plus simple suivante: donner \+ un nom arbitraire \340 l'ensemble des solutions, et utiliser la comman de assign, qui change tous les '=' en ': =' . Ainsi:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "solu:=solve(\{x+a*y=b,a*x+y=2\},\{x,y\}); x;assign(solu);x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%soluG<$/%\"xG, $*&,&%\"aG!\"#%\"bG\"\"\"F.,&!\"\"F.*$F+\"\"#F.F0F0/%\"yG*&,&*&F+F.F-F .F.F,F.F.F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"aG!\"#%\"bG\"\"\"F),&!\"\"F)*$F&\"\"#F)F+ F+" }}}{PARA 0 "" 0 "" {TEXT -1 88 "Mais attention: x et y sont \340 p r\351sent fix\351s! Mieux vaut, pour continuer, les d\351saffecter" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "solve(\{x+a*y=b,a*x+y=2\},\{ x,y\});x:='x';y:='y';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/*&,&*&%\"a G\"\"\"%\"bGF)F)!\"#F)F),&!\"\"F)*$F(\"\"#F)F-F%/,$*&,&F(F+F*F)F)F,F-F -F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGF$" }}}{PARA 0 "" 0 "" {TEXT -1 84 "Quand solve v ous d\351\347oit, il reste la possibilit\351 de faire appel \340 d'aut res solveurs:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve (cos( x)=x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,&%#_ZG\"\"\"-%$ cosG6#F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fsolve(cos (x)=x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K8&3R(!#5" }}}{PARA 4 " " 0 "" {TEXT -1 4 " " }}{SECT 1 {PARA 265 "" 0 "" {TEXT 286 21 "Not e: Autres solveurs" }}{PARA 265 "" 0 "" {TEXT -1 133 "Maple permet en \+ fait de r\351soudre des \351quations de bien d'autres types : \351quat ions diophantiennes (c'est-\340-dire \340 solutions enti\350res):" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(x^2+y^2=1000);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"yGF%/%\"xG*$,&\"%+5\"\"\"*$F%\" \"#!\"\"#F+F-<$F$/F',$F(F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "isolve(x^2+y^2=1000);" }}{PARA 12 "" 1 "" {XPPMATH 20 "62<$/%\"xG \"#5/%\"yG\"#I<$/F%F)/F(F&<$/F%!#I/F(!#5<$F'/F%F1<$/F%\"#=/F(!#E<$/F%! #=/F(\"#E<$/F%F8/F(F;<$F7F:<$F$/F(F/<$F5F<<$/F%F=/F(F6<$F?FG<$F@FF<$F. F,<$F0F+<$FCF3" }}}{PARA 0 "" 0 "" {TEXT -1 41 "r\351solution num\351r ique (dans un intervalle)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fsolve(sin(x)=x/3,x=0.2..2.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +gE')yA!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Equations diff\351rentiel les (voir aussi plus loin):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqd:=diff(y(x),x$2)-4*diff(y(x),x)+3*y(x)=sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqdG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F* !\"%F,\"\"$-%$sinGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dso lve(eqd,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,*-%$c osGF&#\"\"\"\"\"&-%$sinGF&#F,\"#5*&%$_C1GF,-%$expGF&F,F,*&%$_C2GF,-F56 #,$F'\"\"$F,F," }}}{PARA 0 "" 0 "" {TEXT -1 59 "On voit que Maple intr oduit des constantes d'int\351gration..." }}{PARA 0 "" 0 "" {TEXT -1 89 "Le m\352me principe permet aussi de \"r\351soudre\" des r\351curre nces: voici la suite de Fibonacci:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "eqf:=f(n+2)=f(n+1)+f(n); solu:=(rsolve(\{eqf,f(0)=0,f (1)=1\},f(n)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqfG/-%\"fG6#,&% \"nG\"\"\"\"\"#F+,&-F'6#,&F*F+F+F+F+-F'6#F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%soluG,&*(,&\"\"\"F(*$\"\"&#F(\"\"##!\"\"F*F(),$*$,&F )F(F.F(F.F,%\"nGF(F2F.F(*(,&F.F(F)F-F(),$*$,&F)F(F(F(F.!\"#F3F(F9F.F( " }}}{PARA 0 "" 0 "" {TEXT -1 32 "ce qui semble bizarre. Pourtant:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "u:=unapply(solu,n); u(12); e valf(u(12),50); radnormal(u(12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"uGR6#%\"nG6\"6$%)operatorG%&arrowGF(,&*(,&\"\"\"F/*$\"\"&#F/\"\"##! \"\"F1F/),$*$,&F0F/F5F/F5F39$F/F9F5F/*(,&F5F/F0F4F/),$*$,&F0F/F/F/F5! \"#F:F/F@F5F/F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&\"\"\"F& *$\"\"&#F&\"\"##!\"\"F(F&,&F'F&F,F&!#8\"%'4%*&,&F,F&F'F+F&,&F'F&F&F&F. F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S/++++++++++++++++++++++S9!#Z " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$W\"" }}}{PARA 0 "" 0 "" {TEXT -1 161 "On remarquera certaines astuces utilis\351es pour obliger Mapl e \340 simplifier les expressions apparemment inexactes auquel il \351 tait parvenu; voici un contr\364le final:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "seq(radnormal(u(k)),k=0..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "60\"\"!\"\"\"F$\"\"#\"\"$\"\"&\"\")\"#8\"#@\"#M\"#b\"#*) \"$W\"\"$L#" }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 20 "Fonctions num \351riques" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {SECT 1 {PARA 257 "" 0 "" {TEXT 279 8 "D\351finiti" }{TEXT -1 3 "ons" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Les fonctions num\351riques usue lles (R->R, d\351finies par des \"formules\") sont d\351clar\351es par :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fnc:=x->x^2+3*x-1/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fncGR6#%\"xG6\"6$%)operatorG%&arr owGF(,(*$9$\"\"#\"\"\"F.\"\"$*$F.!\"\"F3F(F(6\"" }}}{PARA 0 "" 0 "" {TEXT -1 60 "(par exemple), et alors la notation fnc(x) a son sens usu el." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "fnc(3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6##\"#`\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " } {TEXT 259 45 "Attention: ne jamais changer cette \351criture: " }} {PARA 0 "" 0 "" {TEXT -1 34 "voici quelques erreurs fr\351quentes:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "fn:=x^2+3*x;fn(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fnG,&*$%\"xG\"\"#\"\"\"F'\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$-%\"xG6#\"\"\"\"\"#F(F%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g(x):=x^2+3*x;g(x);g(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"gG6#%\"xG,&*$F'\"\"#\"\"\"F'\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"xG\"\"#\"\"\"F%\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "h=x->x^2+3*x;" }}{PARA 8 "" 1 "" {TEXT -1 29 "Syntax \+ error, `->` unexpected" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "h =x^2+3*x;h(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"hG,&*$%\"xG\"\"# \"\"\"F'\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"hG6#\"\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 202 "A ce s ujet, le signe = correspond \340 de \"vraies\" \351galit\351s , comme \+ pour des \351quations, par exemple; rappelons que la fonction evalb ( \351valuateur bool\351en) dit ce qu'elle pense de l'\351galit\351 qu'o n lui propose" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "2=3; evalb( 2+2=5);evalb (2+2=4); x;evalb (2+2=x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 " " 0 "" {TEXT -1 53 "On remarque que evalb prend faux quand il ne sait \+ pas" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 36 "Quelques g\351n\351ralisat ions importantes" }}{PARA 0 "" 0 "" {TEXT -1 33 "Fonctions de plusieur s variables:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f2:=(x,y)->x ^2-x*y^3; f2(2,3);f2(3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2GR6 $%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*$9$\"\"#\"\"\"*&F/F19%\"\"$!\" \"F)F)6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "diff( f2(x,y),x);diff(f(x,y),y);diff(f2(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"#*$%\"yG\"\"$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-%\"fG6$%\"xG%\"yGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"\"%\"yG\"\"#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Calcul fonctionnel" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f3:=fnc+sin;f3(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f3G,& %$fncG\"\"\"%$sinGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$%\"xG\"\"# \"\"\"F%\"\"$*$F%!\"\"F*-%$sinG6#F%F'" }}}{PARA 0 "" 0 "" {TEXT -1 25 "Composition des fonctions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "g:=fnc@sin; " }{TEXT -1 32 "ce que nous notons f o sin... " } {MPLTEXT 1 0 5 "g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG-%\"@G6 $%$fncG%$sinG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$-%$sinG6#%\"xG\" \"#\"\"\"F%\"\"$*$F%!\"\"F-" }}}{PARA 0 "" 0 "" {TEXT -1 121 "Bijectio n r\351ciproque (le codage est assez curieux, et en fait les connaissa nces de Maple \340 ce sujet sont tr\350s d\351cevantes)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sin@@(-1);h:=(x->x^3)@@(-1);h(8);ev alf(h(8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'arcsinG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"hG-%#@@G6$R6#%\"xG6\"6$%)operatorG%&arrowGF+ *$9$\"\"$F+F+6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#--%#@@G6$R6#% \"xG6\"6$%)operatorG%&arrowGF**$9$\"\"$F*F*6\"!\"\"6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#--%#@@G6$R6#%\"xG6\"6$%)operatorG%&arrowGF** $9$\"\"$F*F*6\"!\"\"6#\"\")" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 263 15 "Etude classique" }}{PARA 0 "" 0 "" {TEXT -1 50 "La fonction fnc es t celle du paragraphe pr\351c\351dent:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "fnc(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$%\"xG\" \"#\"\"\"F%\"\"$*$F%!\"\"F*" }}}{PARA 5 "" 0 "" {TEXT -1 36 " Calculs \+ de limites et de la d\351riv\351e" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "limit(fnc(x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%*undefinedG" }}}{PARA 0 "" 0 "" {TEXT -1 52 "car la limite n'est pas la m\352me \340 droite et \340 gauche" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "limit(fnc(x),x=0,right);limit(fnc(x),x=0,left);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit(fnc(x),x=-infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%) infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Mais Maple peut d\351terminer des limites plus difficiles..." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "limit((sin(sinh(x))-sinh(si n(x)))/x^7,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"#X" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Deux poss ibilit\351s pour d\351river" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(fnc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG %&arrowGF&,(9$\"\"#\"\"$\"\"\"*$F+!\"#F.F&F&6\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 "ou" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(fnc(x ),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"\"#\"\"$\"\"\"*$F$! \"#F'" }}}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 19 "Signe de la d\351riv\351e" }}{PARA 0 "" 0 "" {TEXT -1 52 "Pour \+ \351tudier le signe de cette d\351riv\351e, factorisons:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "factor(2*x+3+1/x^2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&,(*$%\"xG\"\"$\"\"#*$F&F(F'\"\"\"F*F*F&!\"#" } }}{PARA 0 "" 0 "" {TEXT -1 113 "Des ennuis? C'est que Maple ne pense p as que vous soyez tent\351 par les valeurs suivantes (qui annulent la \+ d\351riv\351e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(2*x+ 3+1/x^2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%,(*$,&\"\"$\"\"\"*$\"\"## F'F)F)#F'F&#!\"\"F)*$F%#F-F&F,F,F',*F$#F'\"\"%F.F1F,F'*(%\"IGF'F&F*,&F $F,F.F*F'F*,*F$F1F.F1F,F'F3F," }}}{PARA 0 "" 0 "" {TEXT -1 50 "Mais s' il vous les faut absolument, essayez plut\364t" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fsolve(2*x+3+1/x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+*p]wn\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 148 "(fsolv e r\351soud les \351quations de mani\350re approch\351e; la pr\351cisi on peut \352tre am\351lior\351e, et il existe aussi une option de r \351solution dans les complexes)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Digits:=25;fsolve(2*x+3+1/x^2); fsolve(2*x^2+1,x,complex);Digi ts:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!:WE'\\&*fS!))p]wn\"!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$%\"IG$!:W%3SCva'=\"y1rq!#D,$F$$\":W%3SCva'=\"y1rqF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{PARA 259 "" 0 " " {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 15 "Trac\351 du graphe" } }{PARA 0 "" 0 "" {TEXT -1 55 "Revenons \340 notre fonction, et essayon s un premier trac\351" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "plo t(fnc);" }}{PARA 13 "" 1 "" {GLPLOT2D 284 284 284 {PLOTDATA 2 "6$-%'CU RVESG6$7bo7$$!#5\"\"!$\"1************4q!#97$$!1nmm;p0k&*!#:$\"1/C]4sN) G'F-7$$!1LL$3HF-7$$!1mm;/siq mF1$\"1)HGDmHNY#F-7$$!1****\\(y$pZiF1$\"1=*pg`l]/#F-7$$!1LLL$yaE\"eF1$ \"1,#o4(H5_;F-7$$!1mmm\">s%HaF1$\"1q2/nJ\\P8F-7$$!1******\\$*4)*\\F1$ \"1<\"Q<^x'=5F-7$$!1+++]_&\\c%F1$\"1!o*H#)3,jtF17$$!1+++]1aZTF1$\"1zL; K#y0+&F17$$!1mm;/#)[oPF1$\"1B*pB8(RhJF17$$!1LLL$=exJ$F1$\"1_i7dJlb8F17 $$!1LLLL2$f$HF1$\"1EM[;c/D:!#;7$$!1++]PYx\"\\#F1$!1#pm4mE1l)F^q7$$!1ML LL7i)4#F1$!1c`!>H\\^T\"F17$$!1****\\P'psm\"F1$!1tqOfyBA;F17$$!1****\\7 4_c7F1$!1T)=&f'p[R\"F17$$!1lmT5VBU5F1$!1Kk**GM(43\"F17$$!1:LL$3x%z#)F^ q$!1+\"onq*R0fF^q7$$!1BL$e9d;J'F^q$\"1&GIN(yHC*)!#<7$$!1ILL3s$QM%F^q$ \"1UV!*3,l(=\"F17$$!1lmT&QdDG$F^q$\"1!o4$[)*Qp@F17$$!1****\\ivF@AF^q$ \"1zI\"3Ar[)QF17$$!1n;/^wj!p\"F^q$\"13#)=t#>jV&F17$$!1MLeRx**f6F^q$\"1 6mYzI;'G)F17$$!1v;aQyxY*)Fbs$\"1i*R-&4o\"4\"F-7$$!17+D\"GyNH'Fbs$\"1;@ fjlVq:F-7$$!1!=/E]yp'\\Fbs$\"1vZp&=U')*>F-7$$!1[$eRsy.k$Fbs$\"1[(ejKyh t#F-7$$!1#F-7$$\"1Q$ e*[og!G(Fbs$!1bC+()*R6N\"F-7$$\"1NL3_Nl.5F^q$!1Go_2#GCl*F17$$\"1O$ekGR [b\"F^q$!1Y>(e%e!4%fF17$$\"1QL$3-Dg5#F^q$!12#\\%[27sSF17$$\"1TLe*['R3K F^q$!1Cc#[tk80#F17$$\"1WLLezw5VF^q$!1(3b0&o92%)F^q7$$\"1tmmmJ+IiF^q$\" 1**=\\Qs%*>lF^q7$$\"1.++v$Q#\\\")F^q$\"1,G(R$Qw\")=F17$$\"1NLLe\"*[H7F 1$\"1$oDy8lnQ%F17$$\"1++++dxd;F1$\"1Z'*Gd%H$=rF17$$\"1,++D0xw?F1$\"1() o\"R0xh+\"F-7$$\"1,+]i&p@[#F1$\"1Mr61,[?8F-7$$\"1+++vgHKHF1$\"1%Rx8)=U 0hwQ.GCDF-7$$\" 1KL$eR<*fTF1$\"1::*QbFW&HF-7$$\"1-++])Hxe%F1$\"1ygl*>[#fMF-7$$\"1mm;H! o-*\\F1$\"1$>U_!*=t'RF-7$$\"1,+]7k.6aF1$\"1$Hc:ohF`%F-7$$\"1mmm;WTAeF1 $\"1arP$G+'>^F-7$$\"1****\\i!*3`iF1$\"19\"*>0p/qdF-7$$\"1NLLL*zym'F1$ \"1@#)))*)*G9V'F-7$$\"1OLL3N1#4(F1$\"1Q\"zXEbK9(F-7$$\"1pm;HYt7vF1$\"1 \"45s\"yi%)yF-7$$\"1-+++xG**yF1$\"14EYd:+(f)F-7$$\"1qmmT6KU$)F1$\"1/+2 u99]%*F-7$$\"1OLLLbdQ()F1$\"19)[#Q*RY-\"F[w7$$\"1++]i`1h\"*F1$\"1**\\` q:*H6\"F[w7$$\"1-+]P?Wl&*F1$\"1j*f[k%*3?\"F[w7$$\"#5F*$\"1++++++*H\"F[ w-%'COLOURG6&%$RGBG$Fcal!\"\"F*F*-%%VIEWG6$;F(Fbal%(DEFAULTG" 1 2 0 1 0 2 9 0 4 2 1 0.000000 45.000000 45.000000 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Ce n'est gu\350re fameux: pour pr\351ciser la \"fen\352tre\" d'observation, il vaut mieux utili ser:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(fnc(x),x=-5..5, y=-3..7);" }{TEXT -1 12 " par exemple" }}{PARA 13 "" 1 "" {GLPLOT2D 284 284 284 {PLOTDATA 2 "6%-%'CURVESG6$7ao7$$!\"&\"\"!$\"1++++++?5!#97 $$!1LLLe%G?y%!#:$\"1G*y33F3t)F17$$!1mmT&esBf%F1$\"1k9!Rd?0`(F17$$!1LL$ 3s%3zVF1$\"1Y[a^q[niF17$$!1ML$e/$QkTF1$\"1bZhB'o!*3&F17$$!1nmT5=q]RF1$ \"1f_PF1$\"1cR(*zHl!4$F17$$!1++vo1YZNF1$\"1qLSpp )RA#F17$$!1LL3-OJNLF1$\"1t7i**y>=9F17$$!1++v$*o%Q7$F1$\"1#3x+co*pq!#;7 $$!1mmm\"RFj!HF1$\"1)QC+:fL=(!#<7$$!1LL$e4OZr#F1$!1CYFoGcgSFfn7$$!1+++ v'\\!*\\#F1$!1v)31Pwu^)Ffn7$$!1+++DwZ#G#F1$!1?,yq$3'*>\"F17$$!1+++D.xt ?F1$!1![Fm8u&Q9F17$$!1LL3-TC%)=F1$!1Q4;q'R;d\"F17$$!1mmm\"4z)e;F1$!1o# HhuS>i\"F17$$!1mmmm`'zY\"F1$!1=H$3!)exc\"F17$$!1++v=t)eC\"F1$!1'y(fSfy #Q\"F17$$!1nmm;1J\\5F1$!1vjbDC(Q4\"F17$$!1$***\\(=[jL)Ffn$!1Cc5sU\"R1' Ffn7$$!1'***\\iXg#G'Ffn$\"1:6+5QE;5Ffn7$$!1emmT&Q(RTFfn$\"1X6h#Gk]M\"F 17$$!1hm\"HdGe:$Ffn$\"1'*4Q%*Qe@BF17$$!1lm;/'=><#Ffn$\"19*R>$3#)**RF17 $$!1K$3Fpy7k\"Ffn$\"1srv<[OFcF17$$!1++D\"yQ16\"Ffn$\"1/?e\\5(Ho)F17$$! 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" 0 "" {MPLTEXT 1 0 17 "e:=f(2*a-x)+f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eG,&*&,&* $,&%\"aG\"\"#%\"xG!\"\"F+F+\"\"\"F.F.,(F*F+F,F-F-F.F-F.*&,&*$F,F+F+F.F .F.,&F,F.F-F.F-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simpli fy(e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,0*&%\"aG\"\"#%\"xG\"\" \"!\"%*$F'F(\"\"%*&F'F*F)F(F(*&F'F*F)F*F+*$F)F(F(F*F*F'!\"\"F*,(F'!\"# F)F*F*F*F1,&F)F*F1F*F1F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,0*&%\"aG\"\"#%\"xG \"\"\"!\"%*$F'F(\"\"%*&F'F*F)F(F(*&F'F*F)F*F+*$F)F(F(F*F*F'!\"\"F*,(F' !\"#F)F*F*F*F1,&F)F*F1F*F1F(" }}}{PARA 0 "" 0 "" {TEXT -1 70 "Il sembl e bien que Maple ait besoin d'\352tre aid\351: nous savons que a = 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a:=1; simplify(e);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 46 "Et donc le point C(1,4) est centre de sym\351trie" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {SECT 1 {PARA 260 "" 0 "" {TEXT -1 19 "remarque \340 ce sujet" }} {PARA 0 "" 0 "" {TEXT -1 58 "il est possible de s'en tirer si on ne co nnait pas a et b:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "a:='a'; g:=x->x^3+x^2+2*x; e1:=g(2*a-x)+g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)op eratorG%&arrowGF(,(*$9$\"\"$\"\"\"*$F.\"\"#F0F.F2F(F(6\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#e1G,,*$,&%\"aG\"\"#%\"xG!\"\"\"\"$\"\"\"*$F'F )F-F(\"\"%*$F*F,F-*$F*F)F-" }}}{PARA 0 "" 0 "" {TEXT -1 35 "et comme n ous voulons e1 constante," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(diff(e1,x)=0,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#!\"\"\"\"$ %\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 56 "a devant \352tre constante, c'e st donc que a=-1/3; en effet" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:=-1/3;simplify(e1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#! \"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!#K\"#F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 103 "D\351terminons une asymptote oblique; la m\351thode du cours est facile \340 calquer, m ais Maple propose aussi:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " asympt(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2%\"xG\"\"#F%\"\" \"*$F$!\"\"\"\"$*$F$!\"#F)*$F$!\"$F)*$F$!\"%F)*$F$!\"&F)-%\"OG6#*$F$! \"'F&" }}}{PARA 0 "" 0 "" {TEXT -1 126 "o\371 nous verrons plus tard ( ch.10) le sens des termes en 1/x^k et de la notation O(g); la partie 2 x+2 est l'asymptote cherch\351e:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{f(x),2*x+2\},x=-3..7,y=-2..14);" }}{PARA 13 "" 1 "" {GLPLOT2D 284 284 284 {PLOTDATA 2 "6&-%'CURVESG6$7cp7$$!\"$\"\"!$!1+++ +++]Z!#:7$$!1LLLe%G?y#F-$!1%ohD/#GdVF-7$$!1nmT&esBf#F-$!116SF-7$ $!1LL$3s%3zBF-$!1.Wx>Q)fk$F-7$$!1LL$e/$Qk@F-$!1&*eD%>=oF$F-7$$!1nmT5=q ]>F-$!12z%o*36=HF-7$$!1LL3_>f_ kkDF#F-7$$!1LL3-OJN8F-$!1u3mY7Db>F-7$$!1++v$*o%Q7\"F-$!1]'>o)\\Ag;F-7$ $!1jmm;RFj!*!#;$!13&oVQhjQ\"F-7$$!1JLLe4OZrFhn$!1A*Gza7!z6F-7$$!1(**** *\\n\\!*\\Fhn$!1H7!G`sO***Fhn7$$!1)*****\\ixCGFhn$!1&=&RvMxT!*Fhn7$$!1 \"******\\KqP(!#<$!1#HUy8UVT*Fhn7$$\"1pm;z*ev:\"Fhn$!1D#Gj^;7;\"F-7$$ \"1-+DJSQ%G#Fhn$!1%RFnfT8V\"F-7$$\"1NLL$347T$Fhn$!1()yE;c%4(=F-7$$\"1O LL3xxlVFhn$!1Usa0![9X#F-7$$\"1PLLLjM?`Fhn$!15[$fbhmM$F-7$$\"1pm\"HdO2V 'Fhn$!1I*>S>[*=^F-7$$\"1,+]7o7TvFhn$!1'3'Hr^[#p)F-7$$\"1M$3xcoD.)Fhn$! 1>9&eSzT;\"!#97$$\"1mm\"HK5S_)Fhn$!1V%yRLa?m\"Fhr7$$\"1L3_+7tp()Fhn$!1 5?mno4j?Fhr7$$\"1**\\7y?X:!*Fhn$!1hk+NZxmEFhr7$$\"1$3Fp^7$Q\"*Fhn$!1F1 w*Ru()4$Fhr7$$\"1m\"Hd&HVFxmgbmF hr7$$\"1\\7.27pT'*Fhn$!1EY)*)fK)zzFhr7$$\"12-)Q*)*34(*Fhn$!1S<#Q@#G=** Fhr7$$\"1m\"H2e)[w(*Fhn$!1HqTC.m-8!#87$$\"1C\"yvE()Q%)*Fhn$!1pMf_/+#)= Fjv7$$\"1$3FW&fG6**Fhn$!15o)R4H=M$Fjv7$$\"1TgFTYoy**Fhn$!1(Rf>6SMS\"!# 77$$\"1+D\"GL3Y+\"F-$\"1Z='GlQ+b'Fjv7$$\"1'R(\\,#[8,\"F-$\"1SeHpw\"Qo# Fjv7$$\"1#H#=q!)3=5F-$\"175D'[8*)p\"Fjv7$$\"1(=n)Qz#[-\"F-$\"1/ttGF\") [7Fjv7$$\"1$3_v!ycJ5F-$\"1f7>sfm4**Fhr7$$\"1zpBwwIQ5F-$\"1>Y#=7!**Q#)F hr7$$\"1v=#\\aZ]/\"F-$\"1:mq<*R'oqFhr7$$\"1rng8uy^5F-$\"1LLeR3F.iFhr7$ $\"1n;H#GF&e5F-$\"1'Q_%*3?v`&Fhr7$$\"1jl(4:n_1\"F-$\"1``X!RY&4]Fhr7$$ \"1f9m>q+s5F-$\"1b@zz=m!e%Fhr7$$\"1bjM))ouy5F-$\"1`@4>PUDUFhr7$$\"1^7. dn[&3\"F-$\"1$*R#R$GTERFhr7$$\"1U5S%\\m*)4\"F-$\"1cuM`A7^MFhr7$$\"1M3x JiW76F-$\"1u@iG5V!4$Fhr7$$\"1$z\\`OAFhr7$$\"1,+]iB0p7F-$\"1]?\"=1N)o:Fhr7$$\"1++vV&R

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L'exp\351rience mon tre que le calcul d'angle, ou plut\364t de la pente correspondante, ar ctan ((" }{XPPEDIT 18 0 "alpha+beta)" "6#,&%&alphaG\"\"\"%%betaGF%" } {TEXT -1 168 ")/2)), est difficile pour Maple, qui doit \352tre pilot \351 patiemment; il est infiniment plus facile de se procurer un vecte ur directeur de cette droite; prenant un vecteur " }{TEXT 262 9 "unita ire " }{TEXT -1 82 "sur chaque asymptote, la somme sera port\351e par \+ la bissectrice. Ainsi, le vecteur " }}{PARA 0 "" 0 "" {TEXT -1 1 "(" }{XPPEDIT 18 0 "a,b" "6$%\"aG%\"bG" }{TEXT -1 11 ") devient (" } {XPPEDIT 18 0 "a/sqrt(a^2+b^2)" "6#*&%\"aG\"\"\"-%%sqrtG6#,&*$F$\"\"#F %*$%\"bGF+F%!\"\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "b/sqrt(a^2+b^2)" "6 #*&%\"bG\"\"\"-%%sqrtG6#,&*$%\"aG\"\"#F%*$F$F,F%!\"\"" }{TEXT -1 52 ") et on obtient ici la \"somme\" de (0,1) et de (1,2)/" }{XPPEDIT 18 0 "sqrt(5)" "6#-%%sqrtG6#\"\"&" }{TEXT -1 6 ", soit" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "c:=1/sqrt(5);d:=1+2/sqrt(5);" }{TEXT -1 13 "e t une pente " }{MPLTEXT 1 0 15 "t:=expand(d/c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG,$*$\"\"&#\"\"\"\"\"##F)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG,&\"\"\"F&*$\"\"&#F&\"\"##F*F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"tG,&*$\"\"&#\"\"\"\"\"#F)F*F)" }}}{PARA 0 "" 0 " " {TEXT -1 49 "ce qui dit que notre bissectrice a pour \351quation " } {XPPEDIT 18 0 "Y=tX+b" "6#/%\"YG,&%#tXG\"\"\"%\"bGF'" }{TEXT -1 5 ", o \371 " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 20 " v\351rifie l'\351qu ation " }{XPPEDIT 18 0 "4=t+b" "6#/\"\"%,&%\"tG\"\"\"%\"bGF'" }{TEXT -1 44 ", qui affirme que C est sur la droite , d'o\371" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "plot(\{f(x),2*x+2, t*x+4-t\},x=-3.. 7,y=-2..14, color=[red, blue, green], scaling=CONSTRAINED);" }}{PARA 13 "" 1 "" {GLPLOT2D 284 284 284 {PLOTDATA 2 "6(-%'CURVESG6$7cp7$$!\"$ \"\"!$!1++++++]Z!#:7$$!1LLLe%G?y#F-$!1%ohD/#GdVF-7$$!1nmT&esBf#F-$!116 SF-7$$!1LL$3s%3zBF-$!1.Wx>Q)fk$F-7$$!1LL$e/$Qk@F-$!1&*eD%>=oF$F- 7$$!1nmT5=q]>F-$!12z%o*36=HF-7$$!1LL3_>f_kkDF#F-7$$!1LL3-OJN8F-$!1u3mY7Db>F-7$$!1++v$*o%Q7\"F-$!1]'> o)\\Ag;F-7$$!1jmm;RFj!*!#;$!13&oVQhjQ\"F-7$$!1JLLe4OZrFhn$!1A*Gza7!z6F -7$$!1(*****\\n\\!*\\Fhn$!1H7!G`sO***Fhn7$$!1)*****\\ixCGFhn$!1&=&RvMx T!*Fhn7$$!1\"******\\KqP(!#<$!1#HUy8UVT*Fhn7$$\"1pm;z*ev:\"Fhn$!1D#Gj^ ;7;\"F-7$$\"1-+DJSQ%G#Fhn$!1%RFnfT8V\"F-7$$\"1NLL$347T$Fhn$!1()yE;c%4( =F-7$$\"1OLL3xxlVFhn$!1Usa0![9X#F-7$$\"1PLLLjM?`Fhn$!15[$fbhmM$F-7$$\" 1pm\"HdO2V'Fhn$!1I*>S>[*=^F-7$$\"1,+]7o7TvFhn$!1'3'Hr^[#p)F-7$$\"1M$3x coD.)Fhn$!1>9&eSzT;\"!#97$$\"1mm\"HK5S_)Fhn$!1V%yRLa?m\"Fhr7$$\"1L3_+7 tp()Fhn$!15?mno4j?Fhr7$$\"1**\\7y?X:!*Fhn$!1hk+NZxmEFhr7$$\"1$3Fp^7$Q \"*Fhn$!1F1w*Ru()4$Fhr7$$\"1m\"Hd&HVFxmgbmFhr7$$\"1\\7.27pT'*Fhn$!1EY)*)fK)zzFhr7$$\"12-)Q*)*34(*Fhn$! 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Le graphe \351tait bien sym \351trique." }}{PARA 268 "" 1 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 267 40 "Approximations et d\351veloppements limit\351s" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 269 22 "D\351veloppements limit\351s" }}{PARA 4 "" 0 "" {TEXT 270 55 "Maple n'utilise (directement) que la formule de Taylor: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "taylor(ln(1+x),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG\"\"\"F%#!\"\"\"\"#F(#F%\"\"$F* #F'\"\"%F,#F%\"\"&F.-%\"OG6#F%\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 66 "R emarquer que le dernier terme est un \"grand O\", c'est-\340-dire que \+ " }{XPPEDIT 18 0 " h(x)=x^6f(x)" "6#/-%\"hG6#%\"xG*&F'\"\"'-%\"fG6#F' \"\"\"" }{TEXT -1 99 ", o\371 h est l'\"erreur\", et f est une fonctio n born\351e (au voisinage de 0).Pour avoir d'autres termes," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "taylor(ln(1+x),x,8);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"F%#!\"\"\"\"#F(#F%\"\"$F*#F'\"\"%F, #F%\"\"&F.#F'\"\"'F0#F%\"\"(F2-%\"OG6#F%\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 147 "(en fait, comme pour la pr\351cision d'evalf, on peut au ssi modifier la variable globale Order). La puissance du syst\350me le rend difficile \340 pi\351ger..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "taylor(ln(cos(x)/(1+sin(x))),x,15)" }{TEXT -1 25 "est obtenu en une seconde" }{MPLTEXT 1 0 1 ";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG!\"\"\"\"\"#F%\"\"'\"\"$#F%\"#C\"\"&#!#h\"%S]\" \"(#!$x#\"&wD(\"\"*#!&@0&\")+o\"*R\"#6#!&\"eT\")?.!e*\"#8-%\"OG6#F&\"# :" }}}{PARA 0 "" 0 "" {TEXT -1 97 "et ce n'est qu'en trichant qu'on pe ut l'amener \340 se tromper (ou du moins, \340 avouer son ignorance)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "taylor(exp(-1/x^2),x);" }} {PARA 8 "" 1 "" {TEXT -1 47 "Error, (in series/exp) unable to compute \+ series" }}}{PARA 0 "" 0 "" {TEXT -1 80 "En revanche, certaines formule s g\351n\351rales, pourtant simples, ne sont pas connues" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "taylor(1/(1-x),x,10);" }{TEXT -1 5 "mais " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "taylor(1/(1-x),x,n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"xG\"\"\"\"\"!F%F%F%\"\"#F%\"\"$F% \"\"%F%\"\"&F%\"\"'F%\"\"(F%\"\")F%\"\"*-%\"OG6#F%\"#5" }}{PARA 8 "" 1 "" {TEXT -1 62 "Error, wrong number (or type) of parameters in funct ion taylor" }}}{PARA 0 "" 0 "" {TEXT -1 70 "(le probl\350me vient auss i de ce que Maple ne sait pas que n est entier)" }}{PARA 0 "" 0 "" {TEXT -1 49 "Cependant, la formule g\351n\351rale pour f est connue!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "taylor(f(x),x,4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"xG-%\"fG6#\"\"!F(--%\"DG6#F&F'\" \"\",$---%#@@G6$F+\"\"#F,F'#F-F4F4,$---F26$F+\"\"$F,F'#F-\"\"'F;-%\"OG 6#F-\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 103 "Tous ces \"DL\" ne sont pa s vraiment des fonctions manipulables, mais plut\364t l'affichage de r \351sultats... " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "h:=unappl y(taylor(exp(x),x,4),x);h(x);h(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"hGR6#%\"xG6\"6$%)operatorG%&arrowGF(+-9$\"\"\"\"\"!F.F.#F.\"\"#F1# F.\"\"'\"\"$-%\"OG6#F.\"\"%F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# +-%\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#F%\"\"'\"\"$-%\"OG6#F%\"\"%" }}{PARA 8 "" 1 "" {TEXT -1 44 "Error, (in h) invalid substitution in series" } }}{PARA 0 "" 0 "" {TEXT -1 62 "Pour un vrai calcul de DL, il va falloi r \"effacer\" le terme O(" }{XPPEDIT 18 0 "x^n" "6#)%\"xG%\"nG" } {TEXT -1 113 "), ce qui demande des connaissances plus approfondies du langage de Maple; on le verra \340 la fin de ce paragraphe." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 271 28 "D\351 veloppements asymptotiques" }}{PARA 0 "" 0 "" {TEXT -1 54 "D'autres fo rmes d'approximation sont possibles, ainsi:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 57 "taylor(1/sin(x),x); series(1/sin( x),x);" }}{PARA 8 "" 1 "" {TEXT -1 53 "Error, does not have a taylor e xpansion, try series()" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG\"\" \"!\"\"#F%\"\"'F%#\"\"(\"$g$\"\"$-%\"OG6#F%\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 83 "Comme on le voit, le r\351sultat est un d\351veloppement \+ asymptotique, sur l'\351chelle des " }{XPPEDIT 18 0 "x^n" "6#)%\"xG%\" nG" }{TEXT -1 5 " (o\371 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 14 " appartient \340 " }{TEXT 272 1 "Z" }{TEXT -1 55 "); Maple peut m\352m e utiliser des \351chelles plus \351tranges:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "series(x^(x^x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG\"\"\"F%*$-%#lnG6#F$\"\"#F*,&*$F'\"\"$#F%F**$F'\"\"%F.F-, (F/#F%\"\"'*$F'\"\"&F.*$F'F3F2F0,*F4#F%\"#CF6#\"\"(F9*$F'F;#F%F0*$F'\" \")F8F5-%\"OG6#F%F3" }}}{PARA 0 "" 0 "" {TEXT -1 96 "Ce qui montre au \+ passage que, dans ce cas, l'ordre (=6) du \"DL\" n'est pas un concept \+ tr\350s clair!" }}{PARA 0 "" 0 "" {TEXT -1 37 "Si l'on n'est plus au v oisinage de 0," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DL:=taylor (ln(x),x=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DLG+/,&%\"xG\"\"\"! \"\"F(F(F(#F)\"\"#F+#F(\"\"$F-#F)\"\"%F/#F(\"\"&F1-%\"OG6#F(\"\"'" }}} {PARA 0 "" 0 "" {TEXT -1 148 "(on voit qu'il ne s'agit que d'un change ment de variable); comme pr\351c\351demment, il n'est en fait pas poss ible de manipuler cette expression: comparer" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "expand(DL); P:=x-1-(1/2)*(x-1)^2+(1/3)*( x-1)^3; expand(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/,&% \"xG\"\"\"!\"\"F&F&F&#F'\"\"#F)#F&\"\"$F+#F'\"\"%F-#F&\"\"&F/-%\"OG6#F &\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,*%\"xG\"\"\"!\"\"F'*$ ,&F&F'F(F'\"\"##F(F+*$F*\"\"$#F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,*%\"xG\"\"$#!#6\"\"'\"\"\"*$F$\"\"##!\"$F+*$F$F%#F)F%" }}}{PARA 0 "" 0 "" {TEXT -1 69 "Enfin, les d\351veloppements asymptotiques \340 l'in fini se font par asympt" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a sympt((x^3+1)/(x+2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*$%\"xG\" \"#\"\"\"F%!\"#\"\"%F'*$F%!\"\"!\"(*$F%F(\"#9*$F%!\"$!#G*$F%!\"%\"#c*$ F%!\"&!$7\"-%\"OG6#*$F%!\"'F'" }}}{PARA 0 "" 0 "" {TEXT -1 32 "qu'on p ourrait aussi obtenir par" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "series((x^3+1)/(x+2),x=infinity);" }{TEXT -1 25 "et en fait, par \"ta ylor\":" }{MPLTEXT 1 0 33 "taylor((x^3+1)/(x+2),x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*$%\"xG\"\"#\"\"\"F%!\"#\"\"%F'*$F%!\"\"! \"(*$F%F(\"#9*$F%!\"$!#G*$F%!\"%\"#c*$F%!\"&!$7\"-%\"OG6#*$F%!\"'F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*$%\"xG\"\"#\"\"\"F%!\"#\"\"%F'*$F% !\"\"!\"(*$F%F(\"#9*$F%!\"$!#G*$F%!\"%\"#c*$F%!\"&!$7\"-%\"OG6#*$F%!\" 'F'" }}}{PARA 0 "" 0 "" {TEXT 273 3 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 278 33 "(***) Manipulations d'expressions" }}{PARA 4 "" 0 "" {TEXT 290 39 "Les expressions pr\351c\351dentes ne sont pas" }{TEXT 289 1 " " }{TEXT 291 382 "des DL (\340 cause du terme en O(x^n)). Mapl e fournit la fonction de conversion convert(expr,polynom) qui donne le DL \340 notre sens, et que nous utiliserons plus loin. Mais il est im portant pour certaines applications de pouvoir \351crire ses propres f onctions de conversion (voir par exemple le paragraphe suivant), et no us allons supposer ici que nous n'avions pas trouv\351 cette option... " }}{PARA 4 "" 0 "" {TEXT 277 136 "Pour \351crire de vrais d\351velopp ements limit\351s, nous allons donc devoir p\351n\351trer un peu plus \+ profond\351ment dans les entrailles de Maple: si " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "T:=taylor(tan(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG++%\"xG\"\"\"F'#F'\"\"$F)#\"\"#\"#:\"\"&-%\"OG6#F '\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 103 "le d\351veloppement limit\351 P sera obtenu en \"tronquant\" T; pour cela d\351terminons d'abord la structure de T" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "seq(op(k,T),k=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*\"\"\"F##F#\"\"$F%#\"\"#\"#:\"\"&-%\"OG6#F#\"\"'" }}}{PARA 0 "" 0 " " {TEXT -1 213 "Est-ce clair? T est une \"somme\" ( ce que nous diraie nt op(0,T) et le manuel); nops(T) est le nombre d'objets de cette somm e, et op(k,T) en est le k-\350me objet. Combinons les (n-2) premiers \+ termes pour construire P" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " P:=sum(op(2*k-1,T)*x^(op(2*k,T)),k=1..(nops(T)/2-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,(%\"xG\"\"\"*$F&\"\"$#F'F)*$F&\"\"&#\"\"#\" #:" }}}{PARA 4 "" 0 "" {TEXT 274 56 "Cela semble fonctionner; mais la \+ nouvelle structure est:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "o p(0,P);nops(P);seq(op(k,P),k=1..nops(P));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"+G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%\"xG,$*$F#\"\"$#\"\"\"F&,$*$F#\"\"&# \"\"#\"#:" }}}{PARA 4 "" 0 "" {TEXT 275 112 " ce qui montre qu'il ne s uffisait pas de retirer les deux termes finaux de T (les codages ne so nt pas les m\352mes)" }}{PARA 4 "" 0 "" {TEXT 276 135 "Ecrivons \340 \+ pr\351sent une fonction g\351n\351rale de calcul de DL: DL(n,f) doit \+ nous donner le polyn\364me, consid\351r\351 comme une nouvelle fonctio n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DL:=proc(m,f)local T,n, P;n:=m+1;T:=taylor(f(x),x,n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "P: =sum(op(2*k-1,T)*x^(op(2*k,T)),k=1..(nops(T)/2-1)); unapply(P,x);end; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DLGR6$%\"mG%\"fG6%%\"TG%\"nG%\" PG6\"F-C&>8%,&9$\"\"\"F3F3>8$-%'taylorG6%-9%6#%\"xGF8&-%$sumG6$*&- %#opG6$,&%\"kG\"\"#!\"\"F3F5F3)F<-FD6$,$FGFHF5F3/FG;F3,&-%%nopsG6#F5#F 3FHFIF3-%(unapplyG6$F>F " 0 "" {MPLTEXT 1 0 10 "DL(6,sin);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,(9$ \"\"\"*$F+\"\"$#!\"\"\"\"'*$F+\"\"&#F,\"$?\"F&F&6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "DL(12,x->(1+x)^x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,:\"\"\"F+*$9$\"\"#F+ *$F-\"\"$#!\"\"F.*$F-\"\"%#\"\"&\"\"'*$F-F6#!\"$F4*$F-F7#\"#L\"#S*$F- \"\"(#!\"&F7*$F-\"\")#\"%f@\"%?D*$F-\"\"*#!$4#\"$S#*$F-\"#5#\"%4*)\"&! 35*$F-\"#6#!$,'\"$s'*$F-\"#7#\")j(G!=\")+%e*>F&F&6\"" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Et si nous voulons \340 pr\351sent v\351rifier nos r \350gles:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:=DL(8,sin); B :=DL(8,cos);DL(8,A/B);DL(8,tan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"AGR6#%\"xG6\"6$%)operatorG%&arrowGF(,*9$\"\"\"*$F-\"\"$#!\"\"\"\"'*$ F-\"\"&#F.\"$?\"*$F-\"\"(#F2\"%S]F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGR6#%\"xG6\"6$%)operatorG%&arrowGF(,,\"\"\"F-*$9$\"\"##!\" \"F0*$F/\"\"%#F-\"#C*$F/\"\"'#F2\"$?(*$F/\"\")#F-\"&?.%F(F(6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,*9$ \"\"\"*$F+\"\"$#F,F.*$F+\"\"&#\"\"#\"#:*$F+\"\"(#\"#<\"$:$F&F&6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,*9$ \"\"\"*$F+\"\"$#F,F.*$F+\"\"&#\"\"#\"#:*$F+\"\"(#\"#<\"$:$F&F&6\"" }}} {PARA 0 "" 0 "" {TEXT -1 56 "De m\352me, voici une fonction extrayant \+ le coefficient de " }{XPPEDIT 18 0 "x^n" "6#)%\"xG%\"nG" }{TEXT -1 21 " (s'il n'est pas nul)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "co efficient:=proc(n,f) local T,k;T:=taylor(f(x),x,n+1);op(nops(T)-3,T);e nd;" }{TEXT -1 13 "les r\350glages " }}{PARA 0 "" 0 "" {TEXT -1 56 " d es \"d\351calages\" n+1 et nops-3 se font sur un exemple..." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,coefficientGR6$%\"nG%\"fG6$%\"TG%\"kG6\"F ,C$>8$-%'taylorG6%-9%6#%\"xGF6,&9$\"\"\"F9F9-%#opG6$,&-%%nopsG6#F/F9! \"$F9F/F,F,6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "coefficie nt(6,cos);coefficient(3,exp); coefficient(6,sin);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"$?(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\" \"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"$?\"" }}}{PARA 0 "" 0 "" {TEXT -1 77 "Il est assez facile de corriger cette derni\350re er reur; \340 titre d'\351chantillon:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "coefficient:=proc(n,f) local T,k;T:=taylor(f(x),x,n+ 1);if op(nops(T)-2,T)<>n then 0 else op(nops(T)-3,T)fi;end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,coefficientGR6$%\"nG%\"fG6$%\"TG%\"kG6\"F ,C$>8$-%'taylorG6%-9%6#%\"xGF6,&9$\"\"\"F9F9@%0-%#opG6$,&-%%nopsG6#F/F 9!\"#F9F/F8\"\"!-F=6$,&F@F9!\"$F9F/F,F,6\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "coefficient(6,cos);coefficient(3,exp); coefficient( 6,sin);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"$?(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 305 32 "Repr\351sentations d'ap proximations" }}{PARA 0 "" 0 "" {TEXT -1 265 "Contrairement \340 ce qu 'on pourrait croire, la qualit\351 des approximations fournies par les DL (en dehors du voisinage de 0) n'est pas forc\351ment am\351lior \351e par le nombre de termes. Pour nous en rendre compte, \351tudions les repr\351sentations graphiques des DL de Arc tan. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "taylor(arctan(x),x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#++%\"xG\"\"\"F%#!\"\"\"\"$F(#F%\"\"&F*-%\"OG6#F% \"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 79 "Comme on l'a dit, ceci n'est pa s un polyn\364me. Voici une fonction de conversion:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "P:=n->convert(taylor(arctan(x),x,n),polynom ); P(9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGR6#%\"nG6\"6$%)opera torG%&arrowGF(-%(convertG6$-%'taylorG6%-%'arctanG6#%\"xGF59$%(polynomG F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"xG\"\"\"*$F$\"\"$#!\" \"F'*$F$\"\"&#F%F+*$F$\"\"(#F)F." }}}{PARA 0 "" 0 "" {TEXT -1 73 "Il n e reste plus qu'\340 se procurer une repr\351sentation graphique \"lis ible\":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot(\{arctan(x), seq(P(2*n+1),n=0..10)\},x=-2..2,y=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 284 284 284 {PLOTDATA 2 "60-%'CURVESG6$7\\o7$$!\"#\"\"!$\"1( zCuzCip\"!#77$$!1LL$e%G?y>!#:$\"1?()Q#>'*=V\"F-7$$!1nmm\"p0k&>F1$\"1\\ D+pPO17F-7$$!1++]P&3Y$>F1$\"1B&G-#))G95F-7$$!1LLL$Q6G\">F1$\"1n#>0Dt,^ )!#87$$!1++v3-)[(=F1$\"1r,dv`@QiFC7$$!1nm;M!\\p$=F1$\"1-41]:tTXFC7$$!1 ++Dh9H%z\"F1$\"1<]8=Q<^JFC7$$!1LLL))Qj^#y%)*!#97$$!1++Drp,B;F1$\"1ru nx8GLlF\\o7$$!1nm;C2G!e\"F1$\"1mq2=bVtUF\\o7$$!1LL$3yO5]\"F1$\"1T!#;7$$!1nmmc4`i6F1$!1g?'***F[q$!1'4OAxtEa(F[q7$$!1++++ 0\"*H\"*F[q$!1Ag;M/hFtF[q7$$!1++++83&H)F[q$!1v^T>Ic4pF[q7$$!1LLL3k(p`( F[q$!1(f0OSUaX'F[q7$$!1nmmmj^NmF[q$!1h&*fJN.eeF[q7$$!1ommm9'=(eF[q$!1v a@3**Q4`F[q7$$!1,++v#\\N)\\F[q$!1-#*oyLIBYF[q7$$!1pmmmCC(>%F[q$!11ieUa $R(RF[q7$$!1*****\\FRXL$F[q$!17)G![3f=KF[q7$$!1+++D=/8DF[q$!1%4s$=v0iC F[q7$$!1mmm;a*el\"F[q$!1Jj(>\"[+T;F[q7$$!1pmm;Wn(o)!#<$!1)=^k#f\"fm)F` u7$$!1qLLL$eV(>!#=$!1++$zwbV(>Ffu7$$\"1Mmm;f`@')F`u$\"1()>gP\"p-g)F`u7 $$\"1)****\\nZ)H;F[q$\"1U*G>kTch\"F[q7$$\"1lmm;$y*eCF[q$\"1\"R6)y396CF [q7$$\"1*******R^bJ$F[q$\"1MV)>F$\\,KF[q7$$\"1'*****\\5a`TF[q$\"1!y>S' =sORF[q7$$\"1(****\\7RV'\\F[q$\"1qhDeM!zg%F[q7$$\"1'*****\\@fkeF[q$\"1 #pp:r$)RI&F[q7$$\"1JLLL&4Nn'F[q$\"1WpRlqK%)eF[q7$$\"1*******\\,s`(F[q$ \"13P/$R%ebkF[q7$$\"1lmm\"zM)>$)F[q$\"1)p5s3bM#pF[q7$$\"1*******pfa<*F [q$\"1IXdgMQYtF[q7$$\"1HLLeg`!)**F[q$\"1_szu#HDa(F[q7$$\"1++]#G2A3\"F1 $\"1NMRYKrWrF[q7$$\"1LLL$)G[k6F1$\"1%y%4zf4h]F[q7$$\"1++]7yh]7F1$!1?(f `7(yi?F[q7$$\"1nmm')fdL8F1$!1zJ]PZM;@F17$$\"1nmm,FT=9F1$!1*y*3q')*Q3(F 17$$\"1LL$e#pa-:F1$!180%\\Ei\"))=F\\o7$$\"1nm\"HB-7a\"F1$!1odLm?RdGF\\ o7$$\"1+++Sv&)z:F1$!1qG-\"Hz^D%F\\o7$$\"1nm;%)3;C;F1$!1vUSoqf1mF\\o7$$ \"1LLLGUYo;F1$!1Up5*G6.,\"FC7$$\"1++]n'*33F1$!1*ofA( 3HH&)FC7$$\"1+]i0j\"[$>F1$!1x/9*zwf,\"F-7$$\"1++v.Uac>F1$!1F1$!1Tip#erEV\"F-7$$\"\"#F*$!1(zCuzCip\"F--%'COLOURG6&% $RGBG$\"#5!\"\"F*F*-F$6$7[o7$F($\"1(Q_8C@!e@!#67$F/$\"1*z+S?pRu\"Ff`l7 $F5$\"1?&=YZ:fS\"Ff`l7$F:$\"1d'3mfb08\"Ff`l7$F?$\"1XRp<%*)z1*F-7$FE$\" 1&))G1C2!RhF-7$FJ$\"1H*=SNP<7%F-7$FO$\"18kM%>@hg#F-7$FT$\"1cU*)*R!*)G; F-7$FY$\"1wFA$o+D+\"F-7$Fhn$\"1xx>\\L^*3'FC7$F^o$\"1&oS\"QzdcOFC7$Fco$ \"1D6u!=AJ;#FC7$Fho$\"1![&\\8vw9yF\\o7$F]p$\"1(f,zk66_#F\\o7$Fbp$\"1K^ BMZ*)>oF17$Fgp$\"1uWP&e/BA\"F17$F]q$!19`-UV/aNF[q7$Fbq$!1P0-7KeopF[q7$ Fgq$!1i1#z$=f/wF[q7$F\\r$!1VX\\>gUftF[q7$Far$!1XZX(y')*=pF[q7$Ffr$!1DS nKj!yX'F[q7$F[s$!1EH$3nn$eeF[q7$F`s$!1?@55wV4`F[q7$Fes$!1=E6zmIBYF[q7$ Fjs$!1\"oObjNR(RF[q7$F_t$!1hf:_3f=KF[q7$Fdt$!1'Q2%=v0iCF[q7$Fit$!1hj(> \"[+T;F[qF]uFcuFiu7$F_v$\"1l*G>kTch\"F[q7$Fdv$\"1Qe$)y396CF[q7$Fiv$\"1 aVtvK\\,KF[q7$F^w$\"1/a9E?sORF[q7$Fcw$\"1\\&Rdb1zg%F[q7$Fhw$\"1'Rr4ZIS I&F[q7$F]x$\"1g!HN^#p%)eF[q7$Fbx$\"1uqbw$\\zX'F[q7$Fgx$\"1xun$ftK$pF[q 7$F\\y$\"1_Q_e;**ztF[q7$Fay$\"14?%pp6Wg(F[q7$Ffy$\"1X]j7uzOqF[q7$F[z$ \"1Hot**>!4R$F[q7$F`z$!1sxB@5Xe7F17$Fez$!1$\\A3\"HEdnF17$Fjz$!1z+;pLV+ DF\\o7$F_[l$!1#>+/!pqszF\\o7$Fi[l$!1z'*=Zlt^@FC7$F^\\l$!1j7u^RF-7$F\\^l$!1\\'3IqC$)*fF-7$Fa^l$!1:TGX1f $4*F-7$Ff^l$!1Yd>i1#H8\"Ff`l7$F[_l$!1%=Yh*G&yS\"Ff`l7$F`_l$!1b%4asc^u \"Ff`l7$Fe_l$!1(Q_8C@!e@Ff`l-Fj_l6&F\\`lF*F]`lF*-F$6$7[o7$F($!1ESj\\G* Q,'F-7$F/$!1p'[)Qmtn\\F-7$F5$!1*4xFmtX4%F-7$F:$!1&H3-y@tO$F-7$F?$!1j!H O![*Gw#F-7$FE$!10p@y?F-7$FJ$!1o$3gr'=i8F-7$FO$!1_t$4#)*>J!*FC7$FT$! 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Un premier essai num\351rique (on ut ilise \"couper-coller\" pour ne pas recopier le coefficient de " } {XPPEDIT 18 0 "x^19" "6#*$%\"xG\"#>" }{TEXT -1 1 ")" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ifactor(443861162/1856156927625);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*:-%!G6#\"\"#\"\"\"-F%6#\"#JF(-F%6#\"#TF(-F% 6#\"$$GF(-F%6#\"$<'F(-F%6#\"\"$!\")-F%6#\"\"&!\"$-F%6#\"\"(!\"#-F%6#\" #6!\"\"-F%6#\"#8FD-F%6#\"#FD" }}}{PARA 0 "" 0 "" {TEXT -1 61 "Le d\351nominateur \351tait peut-\352tre \"pr\351visible\" (en eff et, 19! =" }{XPPEDIT 18 0 "2^16*3^8*5^3*7^2*11*13*17*19" "6#*2\"\"#\"# ;\"\"$\"\")\"\"&F&\"\"(F$\"#6\"\"\"\"#8F+\"#F+" }{TEXT -1 47 ") , mais le num\351rateur n'est vraiment pas clair!" }}{PARA 0 "" 0 "" {TEXT -1 80 "L'id\351e suivante est de trouver une valeur num\351rique approch\351e du coefficient de " }{XPPEDIT 18 0 "x^n" "6#)%\"xG%\"nG " }{TEXT -1 180 ". Nous allons d'abord apprendre \340 l'extraire. La m \351thode d'extraction du paragraphe (***) pr\351c\351dent serait plus facile, mais \340 d\351faut,utilisons \340 nouveau la fonction de con version:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "T:=taylor(tan(x) ,x); convert(T,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG++%\"xG\"\"\"F'#F'\"\"$F)#\"\"#\"#:\"\"&-%\"OG6#F'\"\"(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"\"\"*$F$\"\"$#F%F'*$F$\"\"&# \"\"#\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 47 "Il ne reste alors qu'\340 s \351parer le dernier terme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "T1:=taylor(tan(x),x,20);T2:=taylor(tan(x),x,19); convert(T1,polyno m)-convert(T2,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T1G+9%\" xG\"\"\"F'#F'\"\"$F)#\"\"#\"#:\"\"&#\"#<\"$:$\"\"(#\"#i\"%NG\"\"*#\"%# Q\"\"'Df:\"#6#\"&W=#\"(v53'\"#8#\"'p&H*\"*vG^Q'F,#\"(#e/k\",v)=Z&3\"F/ #\"*i6'QW\".Dw#p:c=\"#>-%\"OG6#F'\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T2G+7%\"xG\"\"\"F'#F'\"\"$F)#\"\"#\"#:\"\"&#\"#<\"$:$\"\"(#\"#i \"%NG\"\"*#\"%#Q\"\"'Df:\"#6#\"&W=#\"(v53'\"#8#\"'p&H*\"*vG^Q'F,#\"(#e /k\",v)=Z&3\"F/-%\"OG6#F'\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$% \"xG\"#>#\"*i6'QW\".Dw#p:c=" }}}{PARA 0 "" 0 "" {TEXT -1 68 "D'o\371 u ne fonction donnant le n-\350me coefficient de tan(x) (n impair):" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "coe:=proc(n)local T1,T2;T1: =taylor(tan(x),x,n+1);T2:=taylor(tan(x),x,n); (convert(T1,polynom)-con vert(T2,polynom))/x^n;end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$coeGR 6#%\"nG6$%#T1G%#T2G6\"F+C%>8$-%'taylorG6%-%$tanG6#%\"xGF5,&9$\"\"\"F8F 8>8%-F06%F2F5F7*&,&-%(convertG6$F.%(polynomGF8-F@6$F:FB!\"\"F8)F5F7FEF +F+6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "coe(5);coe(19);co e(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6##\"*i6'QW\".Dw#p:c=" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 67 "Utilisons \340 pr\351sent cet te fonction pour des estimations num\351riques:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "seq(evalf(coe(2*k+1)),k=1..10);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6,$\"+LLLLL!#5$\"+LLLL8F%$\"+(RDoR&!#6$\"+a)[p=#F* $\"+IbBj))!#7$\"+P!G@f$F/$\"+(QMeX\"F/$\"+4WF+f!#8$\"+U6H\"R#F6$\"+dz` \"p*!#9" }}}{PARA 0 "" 0 "" {TEXT -1 64 "qui tend rapidement vers 0. D ans ce cas, on a int\351r\352t \340 \351tudier " }{XPPEDIT 18 0 "u[n+1 ]/u[n]" "6#*&&%\"uG6#,&%\"nG\"\"\"F)F)F)&F%6#F(!\"\"" }{TEXT -1 11 ", \+ ici, donc" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(evalf(coe(2 *k+3)/coe(2*k+1)),k=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,$\"++++ +S!#5$\"+[!>w/%F%$\"+#e(G_SF%$\"+Cfy_SF%$\"+^0%G0%F%$\"+#fYG0%F%$\"+is %G0%F%$\"+Ot%G0%F%$\"+Xt%G0%F%$\"+Yt%G0%F%" }}}{PARA 0 "" 0 "" {TEXT -1 88 "Ce qui \"montre\" que la suite est \"\340 peu pr\350s\" g\351om \351trique, ou plus math\351matiquement que " }{XPPEDIT 18 0 "u[n]" "6 #&%\"uG6#%\"nG" }{TEXT -1 19 " est \351quivalente \340 " }{XPPEDIT 18 0 "v[n]=ak^n" "6#/&%\"vG6#%\"nG)%#akGF'" }{TEXT -1 7 ", avec " } {XPPEDIT 18 0 "k=.4052847346" "6#/%\"kG$\"+Yt%G0%!#5" }{TEXT -1 47 " \+ Reste \340 deviner ce qu'est cette constante..." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "evalf(coe(103)/coe(101),25); evalf(4/Pi^2,25); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\":z^vd3^$pXt%G0%!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\":!=bx&3^$pXt%G0%!#D" }}}{PARA 0 "" 0 "" {TEXT -1 10 "D'o\371 sort " }{XPPEDIT 18 0 "4/Pi^2" "6#*&\"\"%\"\"\"*$ %#PiG\"\"#!\"\"" }{TEXT -1 137 " ? Bonne question... En fait, deux sol utions: de la th\351orie (pas trop difficile, mais nettement niveau Sp \351, bas\351e sur l'\351tude de la s\351rie" }{XPPEDIT 18 0 "Sum(k^n* x^(2n+1),n=0..infinity" "6#-%$SumG6$*&)%\"kG%\"nG\"\"\")%\"xG,&*&\"\"# F*F)F*F*F*F*F*/F);\"\"!%)infinityG" }{TEXT -1 143 "), ou l'utilisation d'un outil Internet stup\351fiant, l'inverseur de Plouffe (http://www .lacim.uqam.ca/pi/indexf.html); quand on l'interroge sur " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 232 " (il faut lui donner une valeur appr och\351e ayant au moins 6 d\351cimales), il r\351pond (instantan\351me nt!) que cette constante n'est pas dans sa base de donn\351es, mais qu 'apr\350s quelques transformations simples, elle lui para\356t devoir \+ \352tre 4/" }{XPPEDIT 18 0 "Pi^2" "6#*$%#PiG\"\"#" }{TEXT -1 481 ". S' il n'obtient pas imm\351diatement de r\351sultat, il vous envoie un co urrier dans les minutes qui suivent, vous proposant diverses expressio ns plus complexes approchant votre valeur; s'il n'avait d\351cid\351me nt rien trouv\351, vous aurez peut-\352tre la chance d'\351changer ave c M. Plouffe en personne des informations sur votre nombre myst\351rie ux, qui sera sans doute ajout\351 \340 sa base (laquelle en comporte d \351j\340 pr\350s de 100 millions) si cela lui semble pouvoir \352tre utile \340 quelqu'un d'autre..." }}{PARA 267 "" 1 "" {TEXT -1 19 "Res te \340 d\351terminer " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 128 ". \+ La m\352me m\351thode \"num\351rique\" (ici, la th\351orie, nettement \+ plus d\351licate, demande des connaissances avanc\351es sur les s\351r ies dans " }{TEXT 292 1 "C" }{TEXT -1 9 " !) donne" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 74 "k:=4/Pi^2;evalf(coe(11)/k^5);evalf(coe(41)/k ^20); a:=evalf(coe(101)/k^50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" kG,$*$%#PiG!\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+))*4d5)!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+OZp0\")!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"aG$\"+'z%p0\")!#5" }}}{PARA 0 "" 0 "" {TEXT -1 50 "Visiblement, notre hypoth\350se se confirme; essayons" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(a*Pi);evalf(a*Pi^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+B\"zka#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1,++!)!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 78 "Cela ne prouve \351videmment rien, mais am\350ne au fort soup\347on que le co efficient " }{XPPEDIT 18 0 "u[n]" "6#&%\"uG6#%\"nG" }{TEXT -1 4 " de \+ " }{XPPEDIT 18 0 "x^(2*n+1" "6#)%\"xG,&*&\"\"#\"\"\"%\"nGF(F(F(F(" } {TEXT -1 18 " est \351quivalent \340 " }{XPPEDIT 18 0 "(8/Pi^2)(4/Pi^2 )^n=2^(2n+3)/Pi^(2n+2)" "6#/)-*&\"\")\"\"\"*$%#PiG\"\"#!\"\"6#*&\"\"%F (*$F*F+F,%\"nG*&)F+,&*&F+F(F1F(F(\"\"$F(F()F*,&*&F+F(F1F(F(F+F(F," } {TEXT -1 28 "; une derni\350re confirmation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(coe(201),20);evalf(2^203/Pi^202,20);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5Od0Gz8)f'R[!#f" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"5(\\b!Gz8)f'R[!#f" }}}{PARA 0 "" 0 "" {TEXT -1 32 "ce qui est assez satisfaisant..." }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }} }{SECT 1 {PARA 3 "" 0 "" {TEXT 268 46 "Techniques combinatoires et sui tes r\351currentes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 298 30 "Suites, sommations, r\351c urrence" }}{PARA 0 "" 0 "" {TEXT -1 57 "Maple repr\351sente les suites (finies) par la fonction seq:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "u:=seq(k^2,k=1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG6' \"\"\"\"\"%\"\"*\"#;\"#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u;u[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"\"%\"\"*\"#;\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 27 "On voit que l'indexation \+ ( " }{XPPEDIT 18 0 "u[n])" "6#&%\"uG6#%\"nG" }{TEXT -1 96 " ) se fait \+ par la notation u[n]; Maple comprend la plupart des conventions usuel les \340 ce sujet:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v[n+1] +v[n+2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"vG6#,&%\"nG\"\"\"F)F )F)&F%6#,&F(F)\"\"#F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 16 "mais pas tout es:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "u[1+3];u[n];u[Pi];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&6'\"\"\"\"\"%\"\"*\"#;\"#D6#%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&6'\"\"\"\"\"%\"\"*\"#;\"#D6#%#PiG" }}}{PARA 0 "" 0 "" {TEXT -1 197 "Les sommes et produits ont une syntaxe similaire; un petit truc a u passage pour une pr\351sentation plus agr\351able: les op\351rateur s \"inertes\", de m\352me nom que l'op\351rateur actif, mais avec une \+ majuscule" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Sum(k^2,k=1..n) =sum(k^2,k=1..n); Product(2*k+1,k=1..n)=product(2*k+1,k=1..n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*$%\"kG\"\"#/F(;\"\"\"%\"nG, **$,&F-F,F,F,\"\"$#F,F1*$F0F)#!\"\"F)F-#F,\"\"'F6F," }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%(ProductG6$,&%\"kG\"\"#\"\"\"F*/F(;F*%\"nG*()F),&F -F*F*F*F*-%&GAMMAG6#,&F-F*#\"\"$F)F*F*%#PiG#!\"\"F)" }}}{PARA 0 "" 0 " " {TEXT -1 194 "Comme toujours, on n'est pas forc\351ment plus avanc \351... Un exemple plus \351labor\351: comment faire imprimer les \"fo rmules de Bernouilli\"? Utilisons une boucle (la programmation sera ab ord\351e plus loin)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for p from 1 to 6 do Sum(k^p,k=1..n)=factor(sum(k^p,k=1..n));od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$%\"kG/F';\"\"\"%\"nG,$*&F+F* ,&F+F*F*F*F*#F*\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*$% \"kG\"\"#/F(;\"\"\"%\"nG,$*(F-F,,&F-F,F,F,F,,&F-F)F,F,F,#F,\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*$%\"kG\"\"$/F(;\"\"\"%\"nG, $*&F-\"\"#,&F-F,F,F,F0#F,\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ SumG6$*$%\"kG\"\"%/F(;\"\"\"%\"nG,$**F-F,,&F-F,F,F,F,,&F-\"\"#F,F,F,,( *$F-F2\"\"$F-F5!\"\"F,F,#F,\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $SumG6$*$%\"kG\"\"&/F(;\"\"\"%\"nG,$*(F-\"\"#,(*$F-F0F0F-F0!\"\"F,F,,& F-F,F,F,F0#F,\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*$%\"kG \"\"'/F(;\"\"\"%\"nG,$**F-F,,&F-F,F,F,F,,&F-\"\"#F,F,F,,**$F-\"\"%\"\" $*$F-F6F)F-!\"$F,F,F,#F,\"#U" }}}{PARA 0 "" 0 "" {TEXT -1 194 "Une pr \351sentation plus soign\351e encore (alignant les signes =, par exemp le) n'est pas \340 la port\351e de Maple (ou plut\364t, les efforts d \351mesur\351s n\351cessaires pour l'obtenir n'en valent gu\350re la p eine!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "Profitons-en pour montrer comment \"prouver\" ces formules par r \351currence: d\351finissons d'abord une fonction S(n) \340 l'aide de \+ l'op\351rateur unapply:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "S :=unapply(factor(sum(k^6,k=1..n)),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SGR6#%\"nG6\"6$%)operatorG%&arrowGF(,$**9$\"\"\",&F.F/F/F/F/,&F .\"\"#F/F/F/,**$F.\"\"%\"\"$*$F.F6\"\"'F.!\"$F/F/F/#F/\"#UF(F(6\"" }}} {PARA 0 "" 0 "" {TEXT -1 26 "La propri\351t\351 \340 d\351montrer (" } {XPPEDIT 18 0 "S(n)=sum(k^6,k=1..n)" "6#/-%\"SG6#%\"nG-%$sumG6$*$%\"kG \"\"'/F,;\"\"\"F'" }{TEXT -1 42 ") est vraie pour les petites valeurs \+ de n:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "1^6+2^6+3^6+4^6;S(4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%!*[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%!*[" }}}{PARA 0 "" 0 "" {TEXT -1 50 "Il suffit donc \+ de montrer que S(n)+(n+1)^6=S(n+1);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a:=S(n)+(n+1)^6; b:=S(n+1);simplify(a-b); a:='a':b :='b':" }{TEXT -1 7 "restart" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG ,&**%\"nG\"\"\",&F'F(F(F(F(,&F'\"\"#F(F(F(,**$F'\"\"%\"\"$*$F'F/\"\"'F '!\"$F(F(F(#F(\"#U*$F)F1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG,$ **,&%\"nG\"\"\"F)F)F),&F(F)\"\"#F)F),&F(F+\"\"$F)F),**$F'\"\"%F-*$F'F- \"\"'F(!\"$!\"#F)F)#F)\"#U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 299 28 "\"R\351solution\" des r\351currences" }}{PARA 0 "" 0 "" {TEXT -1 149 "La fonction rsolve de Maple (analogue \340 dsolve pour les \351qu ations diff\351rentielles) permet de \"r\351soudre\" des r\351currence s: voici la suite de Fibonacci:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "eqf:=f(n+2)=f(n+1)+f(n); solu:=rsolve(\{eqf,f(0)=0,f(1)=1\},f( n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqfG/-%\"fG6#,&%\"nG\"\"\" \"\"#F+,&-F'6#,&F*F+F+F+F+-F'6#F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%soluG,&*(,&!\"\"\"\"\"*$\"\"&#F)\"\"##F)F+F)),$*$,&F*F(F)F)F(!\"#% \"nGF)F2F(F)*(,&F(F)F*#F(F+F)),$*$,&F*F)F)F)F(F3F4F)F;F(F)" }}}{PARA 0 "" 0 "" {TEXT -1 32 "ce qui semble bizarre. Pourtant:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "u:=unapply(solu,n); u(12); evalf(u( 12),50); radnormal(u(12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6 #%\"nG6\"6$%)operatorG%&arrowGF(,&*(,&!\"\"\"\"\"*$\"\"&#F0\"\"##F0F2F 0),$*$,&F1F/F0F0F/!\"#9$F0F9F/F0*(,&F/F0F1#F/F2F0),$*$,&F1F0F0F0F/F:F; F0FBF/F0F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&!\"\"\"\"\"*$ \"\"&#F'\"\"##F'F)F',&F(F&F'F'!#8\"%'4%*&,&F&F'F(#F&F)F',&F(F'F'F'F.F/ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S/++++++++++++++++++++++S9!#Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$W\"" }}}{PARA 0 "" 0 "" {TEXT -1 155 "On remarquera les astuces utilis\351es pour obliger Maple \340 si mplifier les expressions apparemment inexactes auquel il \351tait parv enu; voici un contr\364le final:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "seq(radnormal(u(k)),k=0..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "60\"\"!\"\"\"F$\"\"#\"\"$\"\"&\"\")\"#8\"#@\"#M\"#b\"#*)\"$W\"\"$L#" }}}{PARA 0 "" 0 "" {TEXT -1 52 "Les suites arithm\351tico-g\351om\351t riques sont donn\351es par" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqa:=v(n+1)=a*v(n)+b; rsolve(eqa,v(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqaG/-%\"vG6#,&%\"nG\"\"\"F+F+,&*&%\"aGF+-F'6#F*F+F+ %\"bGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%\"vG6#\"\"!\"\"\")%\" aG%\"nGF)F)*(%\"bGF),&!\"\"F)F+F)F0F*F)F)*&F.F)F/F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 82 "On remarque que le premier terme (inconnu) est laiss \351 dans la solution dans ce cas" }}{PARA 0 "" 0 "" {TEXT -1 57 "Voyo ns comment Maple se tire de r\351currences moins faciles" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eqc:=v(n+1)=3*sqrt(v(n));rsolve(eqc ,v(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqcG/-%\"vG6#,&%\"nG\"\" \"F+F+,$*$-F'6#F*#F+\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'r solveG6$/-%\"vG6#,&%\"nG\"\"\"F,F,,$*$-F(6#F+#F,\"\"#\"\"$F/" }}} {PARA 0 "" 0 "" {TEXT -1 44 "Il va falloir l'aider; posons w(n)=ln(v(n ));" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "eqd:=w(n+1)=(w(n)/2)+ ln(3); ws:=rsolve(\{eqd,w(0)=ln(v(0))\},w(n));v(n)=simplify(exp(ws)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqdG/-%\"wG6#,&%\"nG\"\"\"F+F+, &-F'6#F*#F+\"\"#-%#lnG6#\"\"$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# wsG,(*&-%#lnG6#-%\"vG6#\"\"!\"\"\")#F.\"\"#%\"nGF.F.*&-F(6#\"\"$F.F/F. !\"#F4F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#%\"nG*&)\"\"*,&\" \"\"F,)\"\"#,$F'!\"\"F0F,)-F%6#\"\"!F-F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 300 30 "Suites d\351finies par it\351ration " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 45 "Il s'agit des suites r\351currentes de la forme " }{XPPEDIT 18 0 "u[n+1]=f(u[n])" "6#/&%\"uG6#,&%\"nG\"\"\"F)F)-%\"fG 6#&F%6#F(" }{TEXT -1 73 ". On sait que ces suites n'ont pas (le plus s ouvent) de formule explicite" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rsolve(u(n+1)=sin(u(n)),u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%'rsolveG6$/-%\"uG6#,&%\"nG\"\"\"F,F,-%$sinG6#-F(6#F+F0" }}}{PARA 0 " " 0 "" {TEXT -1 143 "Maple ne permet pas vraiment d'avoir acc\350s \+ \340 cette suite (sauf en ce qui concerne son comportement asymptotiqu e, comme on le verra plus loin);" }}{PARA 0 "" 0 "" {TEXT -1 109 "Si o n veut se procurer les premiers termes, il nous faut les d\351finir pa r une boucle; un exemple pourrait \352tre" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "u[0]:=1;for i from 0 to 9 do u[i+1]:=evalf(sin(u[i]) ); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"\"$\"+[)4ZT)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"#$\"+<9Ccu!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"$$\"+uZI%y'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"%$\"+@$=dF'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"&$\"+m*4=(e!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"'$\"+3R;Sb!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"($\"+b22h_!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\")$\"+jnq@]!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"*$\"+`NH8[!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"#5$\"+')*y&HY!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Mais l'inconv\351nient, c'est q ue " }{XPPEDIT 18 0 "u[11]" "6#&%\"uG6#\"#6" }{TEXT -1 45 " n'est pas \+ d\351fini. Cr\351ons plut\364t une fonction" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "u:=proc(n) local k, a;a:=1; for k from 1 to n do" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a:=evalf(sin(a));od;end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6#%\"nG6$%\"kG%\"aG6\"F+C$>8%\"\"\"?( 8$F/F/9$%%trueG>F.-%&evalfG6#-%$sinG6#F.F+F+6\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "u(10);u(100);u:='u':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+')*y&HY!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+! *[_)o\"!#5" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 301 14 "(***) Remarque" } {TEXT -1 1 " " }}{PARA 5 "" 0 "" {TEXT 302 134 "Ce n'est pas l'\351cri ture la plus efficace; si vous connaissez d\351j\340 la programmation r\351cursive, vous appr\351cierez sans doute la variante" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "u:=proc(n) option remember; if n=0 \+ then 1 else evalf(sin(u(n-1))) fi;end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6#%\"nG6\"6#%)rememberGF(@%/9$\"\"!\"\"\"-%&evalfG6#-%$si nG6#-F$6#,&F-F/!\"\"F/F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "u(100);u:='u':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!*[_)o\"!# 5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 269 "" 1 "" {TEXT -1 123 "Comme on l'a dit, quand Maple ne peut trouver de forme explicite, il \+ peut cependant d\351terminer un comportement asymptotique" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "T:=asympt(rsolve(u(n+1)=sin(u(n)), u(n)),n); limit(rsolve(u(n+1)=sin(u(n)),u(n)),n=infinity); limit(T,n=i nfinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,(*&\"\"$#\"\"\"\" \"#*$%\"nG!\"\"F(F)*&,&%#_CGF)*&F'F(-%#lnG6#F,F)#!\"$\"#5F)F+#F'F*F)-% \"OG6#*$F,!\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6$-%'rsol veG6$/-%\"uG6#,&%\"nG\"\"\"F/F/-%$sinG6#-F+6#F.F3/F.%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 258 "Comme on le voit, cela ne fonctionne directement que pour un appe l \340 \"asympt\" (c'est d\373 \340 la fa\347on d\351tourn\351e avec l aquelle Maple utilise les lettres muettes) .Ecrivons un petit progamme calculant directement la limite d'une suite d\351finie par it\351rati on de f :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "limsuite:=proc( f) local n,T,u;T:=asympt(rsolve(u(n+1)=f(u(n)),u(n)),n); limit(T,n=inf inity);end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)limsuiteGR6#%\"fG6%% \"nG%\"TG%\"uG6\"F,C$>8%-%'asymptG6$-%'rsolveG6$/-8&6#,&8$\"\"\"F " 0 "" {MPLTEXT 1 0 37 "limsuite(sin );limsuite(x->sqrt(1+x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"#F%*$\"\"&F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 28 "Ce qui semble raisonnable..." }}{PARA 0 "" 0 " " {TEXT -1 4 "Mais" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limsui te(cos);limsuite(sqrt);" }}{PARA 8 "" 1 "" {TEXT -1 43 "Error, (in asy mpt) unable to compute series" }}{PARA 8 "" 1 "" {TEXT -1 43 "Error, ( in asympt) unable to compute series" }}}{PARA 0 "" 0 "" {TEXT -1 87 "C ar le programme ne peut traiter que des cas \"simples\" (du moins par \+ cette approche)..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 252 "Passons \340 des repr\351sentations graphiques. La repr \351sentation \"en colima\347on\" demande de tracer des segments, ce q ue plot ne semble pas savoir faire. Une lecture attentive du fichier d 'aide montre cependant que l'option existe (\"point plots\"); testons \+ la:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot([[1,2],[2.5,1]], x=0..3,y=0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 284 284 284 {PLOTDATA 2 "6%-%'CURVESG6$7$7$$\"\"\"\"\"!$\"\"#F*7$$\"1+++++++D!#:F(-%'COLOURG 6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$%\"xG%\"yG-%%VIEWG6$;F*$\"\"$F*F @" 1 2 0 1 0 2 9 0 4 2 1 0.000000 45.000000 45.000000 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 61 "Nous voulons donc \351tablir une liste de segments de la forme (" }{XPPEDIT 18 0 "u[n],u[n+1]" "6$&%\"uG6#%\"nG &F$6#,&F&\"\"\"F*F*" }{TEXT -1 57 "), ou plus pr\351cis\351ment une li ste de segments verticaux ((" }{XPPEDIT 18 0 "u[n],u[n]" "6$&%\"uG6#% \"nG&F$6#F&" }{TEXT -1 3 "),(" }{XPPEDIT 18 0 "u[n],u[n+1]" "6$&%\"uG6 #%\"nG&F$6#,&F&\"\"\"F*F*" }{TEXT -1 47 ")) et une liste de segments h orizontaux ((" }{XPPEDIT 18 0 "u[n],u[n+1]" "6$&%\"uG6#%\"nG&F$6# ,&F&\"\"\"F*F*" }{TEXT -1 4 "), (" }{XPPEDIT 18 0 "u[n+1],u[n+1]" "6$& %\"uG6#,&%\"nG\"\"\"F(F(&F$6#,&F'F(F(F(" }{TEXT -1 460 ")). Il s'av \350re qu'il est plus facile de construire le \"plot\" de chacun de ce s segments, puis de les superposer par le programme adapt\351; le fich ier d'aide nous informe qu'il figure dans le \"package\" plots, sous l e nom de display (son appel est donc plots[display]). Il est recommand \351 de consulter les fichiers d'aide, pour comprendre les programmes \+ qui suivent Le programme de cr\351ation de ces listes n'est pas diffic ile, si on accepte de perdre un peu de place:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "creer:=proc(n,a,f)local k,u;global hliste,vlist e, plotf, xrange, yrange; u[0]:=evalf(a);for k from 0 to n-1 do u[k+1] :=evalf(f(u[k])); od;" }{TEXT -1 32 "stockage des valeurs de la suite " }{MPLTEXT 1 0 111 " for k from 0 to n-1 do hliste[k]:=plot( [[u[k],u[k+1]],[u[k+1],u[k+1]]],x=xrange,y=yrange,color=blue);" } {TEXT -1 46 "ceci est le \"plot\" du k-eme segment horizontal" } {MPLTEXT 1 0 78 "vliste[k]:=plot([[u[k],u[k]],[u[k],u[k+1]]],x=xrange, y=yrange,color=black);od;" }{TEXT -1 33 "et ceci du k-eme segment vert ical" }{MPLTEXT 1 0 41 "`c'est fait`; end;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%&creerGR6%%\"nG%\"aG%\"fG6$%\"kG%\"u G6\"F-C&>&8%6#\"\"!-%&evalfG6#9%?(8$F3\"\"\",&9$F:!\"\"F:%%trueG>&F16# ,&F9F:F:F:-F56#-9&6#&F16#F9?(F9F3F:F;F>C$>&%'hlisteGFI-%%plotG6&7$7$FH F@7$F@F@/%\"xG%'xrangeG/%\"yG%'yrangeG/%&colorG%%blueG>&%'vlisteGFI-FP 6&7$7$FHFHFSFUFX/Ffn%&blackG%+c'est~faitGF-6'FNFjn%&plotfGFWFZ6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "xrange:=0..2;yrange:=-0.2..1 .3;creer(4,1,cos);plotf:=plot(cos(x),x=xrange,y=yrange,color=red):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xrangeG;\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'yrangeG;$!\"#!\"\"$\"#8F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+c'est~faitG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plots[display]([seq(hliste[k],k=0..3),seq(vliste[k],k=0..3),plot f]);" }{TEXT -1 39 "ce qui superpose tous les \"plots\" cr\351\351s" } }{PARA 13 "" 1 "" {GLPLOT2D 284 284 284 {PLOTDATA 2 "6--%'CURVESG6$7$7 $$\"\"\"\"\"!$\"1******eI-.a!#;7$F+F+-%'COLOURG6&%$RGBGF*F*$\"*++++\"! \")-F$6$7$7$F+$\"1+++e@`v&)F-7$F:F:F/-F$6$7$7$F:$\"1+++0z*Ga'F-7$FAFAF /-F$6$7$7$FA$\"1+++(e.[$zF-7$FHFHF/-F$6$7$7$F(F(F'-F06&F2F*F*F*-F$6$7$ F.F9FO-F$6$7$F-Q'ym**F-7$$\"1LLLe0$=C\"F-$\"1>$* =%)=*H#**F-7$$\"1LLL3RBr;F-$\"16OR*ft1')*F-7$$\"1mm;zjf)4#F-$\"14ab\\; g!y*F-7$$\"1LL$e4;[\\#F-$\"1c+&F-$\"19sCSU\"\\x)F-7$$\"1+++]Z/NaF-$\"1.y>9k,f&)F-7$$\"1+++]$fC &eF-$\"1B=?bDwN$)F-7$$\"1LL$ez6:B'F-$\"1-\"=osM/7)F-7$$\"1mmm;=C#o'F-$ \"1cB*p(=B\\yF-7$$\"1mmmm#pS1(F-$\"1h5/A1*pg(F-7$$\"1++]i`A3vF-$\"1*p' H)[z7J(F-7$$\"1mmmm(y8!zF-$\"1)[6#yOZPqF-7$$\"1++]i.tK$)F-$\"122mL(oXs 'F-7$$\"1++](3zMu)F-$\"1#ejCIs\\T'F-7$$\"1nmm\"H_?<*F-$\"17****>JT!3'F -7$$\"1nm;zihl&*F-$\"1v1`dFLjdF-7$$\"1LLL3#G,***F-$\"1`O!4DF8T&F-7$$\" 1LLezw5V5!#:$\"11M%3Tx`.&F-7$$\"1++v$Q#\\\"3\"Fdv$\"1:z*G?;,q%F-7$$\"1 LL$e\"*[H7\"Fdv$\"1$R&4`BEIVF-7$$\"1+++qvxl6Fdv$\"13i;(**f.%RF-7$$\"1+ +]_qn27Fdv$\"1W`bO$=>b$F-7$$\"1++Dcp@[7Fdv$\"1bn\\8%R,<$F-7$$\"1++]2'H KH\"Fdv$\"14v'3:j,u#F-7$$\"1nmmwanL8Fdv$\"1ZTnu,0\\BF-7$$\"1+++v+'oP\" Fdv$\"1!)QKy$Gs#>F-7$$\"1LLeR<*fT\"Fdv$\"1Zaci-(=a\"F-7$$\"1+++&)Hxe9F dv$\"1u'*ow=*y6\"F-7$$\"1mm\"H!o-*\\\"Fdv$\"1*e-Gs#zqrF[o7$$\"1++DTO5T :Fdv$\"1fSphA$)oHF[o7$$\"1nmmT9C#e\"Fdv$!1HDi+l[W6F[o7$$\"1++D1*3`i\"F dv$!1**p!4]e&[aF[o7$$\"1LLL$*zym;Fdv$!1(Q.RoJWe*F[o7$$\"1LL$3N1#4&oz8F-7$$\"1nm\"HYt7v\"Fdv$!1>=$R!)*)\\z\"F-7$$\"1+++q(G**y\"Fd v$!1]+)f%*GQ<#F-7$$\"1nm;9@BM=Fdv$!1'*>t%Q$*Rg#F-7$$\"1LLL`v&Q(=Fdv$!1 .cK)=LW)HF-7$$\"1++DOl5;>Fdv$!1k:u9f)[Q$F-7$$\"1++v.Uac>Fdv$!1y&Q%G;_i PF-7$$\"\"#F*$!1C9Zl$o9;%F--F06&F2F3F*F*-%+AXESLABELSG6$%\"xG%\"yG-%%V IEWG6$;F*Ff]l;$!\"#!\"\"$\"#8Fh^l" 1 2 0 1 0 2 9 0 4 2 1 0.000000 45.000000 45.000000 0 0 "" "" "" "" "" "" "" "" "" }}}}{PARA 0 "" 0 " " {TEXT -1 75 "Il ne reste plus qu'\340 nettoyer tout cela, et \340 \+ \351crire un programme unique:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 535 "plotiter:=proc(n,a,f)local k,u;global hliste,vliste, plotf,pl otXY, xrange, yrange; u[0]:=evalf(a); for k from 0 to n-1 do u[k+1]:=evalf(f(u[k])); od; for k from 0 to n-1 do hliste[k]:=plo t([[u[k],u[k+1]],[u[k+1],u[k+1]]], x=xrange,y=yrange,color=blue); vlis te[k]:=plot([[u[k],u[k]],[u[k],u[k+1]]], x=xrange,y=yrange,color=black ); od; plotf:=plot(f(x),x=xrange,y=yrange,color=red); plotXY:=plo t(x,x=xrange,y=yrange,color=green); plots[display]([seq(hliste[k],k=0 ..n-1),seq(vliste[k], k=0..n-1),plotf,plotXY]); end; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)plotiterGR6%%\"nG%\"aG%\"fG6$%\"kG%\"uG6\"F-C(> &8%6#\"\"!-%&evalfG6#9%?(8$F3\"\"\",&9$F:!\"\"F:%%trueG>&F16#,&F9F:F:F :-F56#-9&6#&F16#F9?(F9F3F:F;F>C$>&%'hlisteGFI-%%plotG6&7$7$FHF@7$F@F@/ %\"xG%'xrangeG/%\"yG%'yrangeG/%&colorG%%blueG>&%'vlisteGFI-FP6&7$7$FHF HFSFUFX/Ffn%&blackG>%&plotfG-FP6&-FF6#FVFUFX/Ffn%$redG>%'plotXYG-FP6&F VFUFX/Ffn%&greenG-&%&plotsG6#%(displayG6#7&-%$seqG6$FM/F9;F3F;-Fgp6$Fi nFipFboFjoF-6(FNFjnFboFjoFWFZ6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plotiter(8,1,cos);" }}{PARA 13 "" 1 "" {GLPLOT2D 284 284 284 {PLOTDATA 2 "66-%'CURVESG6$7$7$$\"\"\"\"\"!$\"1******eI-.a!#;7 $F+F+-%'COLOURG6&%$RGBGF*F*$\"*++++\"!\")-F$6$7$7$F+$\"1+++e@`v&)F-7$F :F:F/-F$6$7$7$F:$\"1+++0z*Ga'F-7$FAFAF/-F$6$7$7$FA$\"1+++(e.[$zF-7$FHF HF/-F$6$7$7$FH$\"1*****pt(o8qF-7$FOFOF/-F$6$7$7$FO$\"1+++HofRwF-7$FVFV F/-F$6$7$7$FV$\"1+++]U-@sF-7$FgnFgnF/-F$6$7$7$Fgn$\"1+++=w-Q'y m**F-7$$\"1LLLe0$=C\"F-$\"1>$*=%)=*H#**F-7$$\"1LLL3RBr;F-$\"16OR*ft1') *F-7$$\"1mm;zjf)4#F-$\"14ab\\;g!y*F-7$$\"1LL$e4;[\\#F-$\"1c+&F-$\"19sCSU\"\\x)F-7$$\"1+++]Z/N aF-$\"1.y>9k,f&)F-7$$\"1+++]$fC&eF-$\"1B=?bDwN$)F-7$$\"1LL$ez6:B'F-$\" 1-\"=osM/7)F-7$$\"1mmm;=C#o'F-$\"1cB*p(=B\\yF-7$$\"1mmmm#pS1(F-$\"1h5/ A1*pg(F-7$$\"1++]i`A3vF-$\"1*p'H)[z7J(F-7$$\"1mmmm(y8!zF-$\"1)[6#yOZPq F-7$$\"1++]i.tK$)F-$\"122mL(oXs'F-7$$\"1++](3zMu)F-$\"1#ejCIs\\T'F-7$$ \"1nmm\"H_?<*F-$\"17****>JT!3'F-7$$\"1nm;zihl&*F-$\"1v1`dFLjdF-7$$\"1L LL3#G,***F-$\"1`O!4DF8T&F-7$$\"1LLezw5V5!#:$\"11M%3Tx`.&F-7$$\"1++v$Q# \\\"3\"F\\y$\"1:z*G?;,q%F-7$$\"1LL$e\"*[H7\"F\\y$\"1$R&4`BEIVF-7$$\"1+ ++qvxl6F\\y$\"13i;(**f.%RF-7$$\"1++]_qn27F\\y$\"1W`bO$=>b$F-7$$\"1++Dc 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