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0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 299 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 300 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 301 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 302 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 303 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 304 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 305 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 306 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 307 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 308 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 309 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 310 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 311 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 312 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 313 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 314 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 277 "" 0 "" {TEXT 256 55 "TP 4 Initiation a un lo giciel de calcul formel : Maple" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT -1 62 "Tapez les l ignes suivantes (ne pas oublier le point virgule !)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "3*5;" }}}{EXCHG {PARA 275 "" 0 "" {TEXT -1 39 "\" rappelle le resultat calcul precedent" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 2 "\";" }}}{EXCHG {PARA 278 "" 0 "" {TEXT -1 48 "O n peut stocker les resultats dans des variables" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "a:=3;\nb:=5;\na*b;" }}}{EXCHG {PARA 279 "" 0 " " {TEXT -1 75 "On peut utiliser des fractions, des nombres complexes, \+ des nombres reels..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "1/2 ;\n2/4;\n2+I;abs(2+I);\nI^2;\nln(2);\nPi;\n3.45; " }}}{EXCHG {PARA 280 "" 0 "" {TEXT -1 55 "On peut demander les decimales (evaluation en flottant)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" } }}{EXCHG {PARA 296 "" 0 "" {TEXT -1 41 "On peut demander encore plus d e decimales" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(Pi,20) ;" }}}{EXCHG {PARA 297 "" 0 "" {TEXT -1 122 "Un flottant, c'est une ma ntisse suivit d'un exposant (confer le cours), \non peut penetrer au c oeur de Maple pour voir ca :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "whattype(\");\nop(\"\");" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 303 "" 0 "" {TEXT -1 92 "Les tous premiers Pentium (en 19 94) avaient un bug : \nle calcul suivant ne renvoyaient pas 1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "824633702441.0*(1/8246337024 41.0);\n" }}}{EXCHG {PARA 304 "" 0 "" {TEXT -1 93 "Oups Maple serait-i l aussi bogue ?\nMais non... la precision de Maple par Defaut est donn e par" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Digits;" }}}{EXCHG {PARA 306 "" 0 "" {TEXT -1 32 "or que vaut (1/824633702441.0) ?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "1/824633702441.0;\n" }}} {EXCHG {PARA 305 "" 0 "" {TEXT -1 74 "On etait donc en dessous de la p recision, on corrige le tout et c'est bon:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "Digits:=12:\n824633702441.0*(1/824633702441.0);" }} }{EXCHG {PARA 307 "" 0 "" {TEXT -1 120 "Ouf ! Retenez bien ce type de \+ probleme, il est fondamental des que l'on manipule des nombres flottan ts (des reels, quoi)" }}{PARA 308 "" 0 "" {TEXT -1 10 "----------" }}} {EXCHG {PARA 289 "" 0 "" {TEXT -1 32 "On peut factoriser et developper " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "expand((x-1)*(x-2));\nf actor(\");" }}}{EXCHG {PARA 281 "" 0 "" {TEXT -1 28 "On peut manipuler des listes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "L:=[1,1,2]; \nL[3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "On peut manipuler des \+ ensembles" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E:=\{1,1,2\}; \nE[3];" }}}{EXCHG {PARA 282 "" 0 "" {TEXT -1 71 "Justifiez le message d'erreur que provoque la deuxieme ligne ci-dessus." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 283 "" 0 "" {TEXT -1 48 "Faiso ns maintenant quelques calculs d'integrales" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(1/x,x=u..v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 203 "Voici maintenant pourquoi on parle de trois-demi et de cinq-de mi en classe prepa : \nComparez : \"J'integre X entre la premiere et d euxieme annee\"\npuis \"j'integre X entre la deuxieme et troisieme \+ annee\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "int(x,x=1..2);\n int(x,x=2..3);" }}}{EXCHG {PARA 284 "" 0 "" {TEXT -1 51 "Les integrale s indefinies sont calculees egalement " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int(sin(x)/x,x=0..infinity);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 285 "" 0 "" {TEXT -1 59 "On peut donc fa ire des calculs de primitives et de derivees" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "int(1/x,x);\ndiff(ln(x),x);" }}}{EXCHG {PARA 286 "" 0 "" {TEXT -1 55 "On peut aussi faire des developpement en seri e deTaylor" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series(exp(x) ,x=0);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series(sin(x),x=0,8);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 287 "" 0 "" {TEXT -1 27 "On peut calculer des somme s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sum(k,k=1..n);\nfactor (\");" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sum(1/n!,n=0..infinity );" }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 137 "On peut demander de l'ai de sur une commande en utilisant le ?\nPar exemple demandons de l'aide sur la commande pour resoudre les equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "?solve" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 50 "R esolvons maintenant une equation du second degre " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "solve(x^2-x-6);" }}}{EXCHG {PARA 256 "" 0 " " {TEXT -1 131 "Bien sur Maple peut utiliser la puissance du calcul fo rmel pour donner la formule du discriminant (connu depuis les Babylonn iens) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(a*x^2+b*x+ c=0,x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 47 "De meme avec une equation du troisieme de degre" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x^3+x^2-x-1);" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 72 "La formule generale est connu so us le nom de formule de Cardan-Tartaglia" }}{PARA 0 "" 0 "" {TEXT -1 207 "(en fait Tartaglia [1500-1557] avait communique a Cardan [1501-1 576] la formule en question en faisant promettre a Cardan de garder le secret,\nCardan s'est parjure et s'est empresse de publier la formule !)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(a+b*x+c*x^2+c* x^3,x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 26 "Voici la premiere racine :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "\"[1];" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 66 "Et qu'est ce que cela donne pour une eq uation du quatrieme degre ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(a+b*x+c*x^2+c*x^3+d*x^4,x);" }}{PARA 11 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 67 "La formule est tellement ho rrible que Maple ne veut pas la donner !" }}{PARA 262 "" 0 "" {TEXT -1 37 "On va quand meme lui forcer la main :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "_EnvExplicit:=true:" }}}{EXCHG {PARA 263 "" 0 " " {TEXT -1 66 "Et on obtient la formule de Ferrari [1522-1565], l'elev e de Cardan" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(a+b*x+ c*x^2+c*x^3+d*x^4,x);" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Meme la premiere racine a une forme compl iquee :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "\"[1];" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 101 "On \+ s'est longtemps demande a quoi pouvait ressembler la formule pour une \+ equation du cinquieme degre." }}{PARA 265 "" 0 "" {TEXT -1 168 "La rep onse, trouvee par le mathematicien norvegien Niels Henrik Abel (1802-1 829) \net par le mathematicien Evariste Galois (1811-1832 mort en duel a 20 ans) c'est que :" }}{PARA 266 "" 0 "" {TEXT -1 118 "IL N'Y PAS \+ DE FORMULE POUR LES EQUATIONS DE DEGRE >4\n(cela se prouve par la theo rie des groupes, inventee par Galois)." }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 215 "De nos jours, c'est toujours un challenge de calculer nu meriquement (rapidement) les racines d'un polynome de degre eleve.\nMa ple sait le faire, on peut ainsi lui demander toutes les racines compl exes d'un polynome :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fso lve(x^5+x^4+x-3,x,complex);" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 25 " Maple sait aussi integrer" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(1/x,x=a..b);" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 67 "Les int egrales indefinies ne lui posent pas non plus de problemes :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "int(1/x,x=1..infinity);" }}} {EXCHG {PARA 270 "" 0 "" {TEXT -1 76 "Cette integrale represente l'air e sous la courbe y=1/x, tracons cette courbe" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(\{1/x,1/floor(x)\},x=1..10);" }}{PARA 13 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 161 "L'aire \+ sous la courbe en escalier est bien sur plus grande que l'aire sous la courbe continue\ndonc elle doit etre aussi infinie, est-ce que Maple \+ va le trouver ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sum(1/n, n=1..infinity);" }}}{EXCHG {PARA 272 "" 0 "" {TEXT -1 95 "En revanche, la somme des inverses des carres converge (pourquoi?) et a une valeu r remarquable" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sum(1/n^2, n=1..infinity);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 273 "" 0 "" {TEXT -1 54 "Essayez pour d'autres puissances, que remarquez-v ous ?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "Remarque : la somme con verge aussi pour des puissances meme non entieres (>1), on obtient ce que l'on appelle la fonction Zeta de Riemann." }}{PARA 0 "" 0 "" {TEXT -1 148 "On peut meme la definir (mais c'est tres complique) pour tous les nombres du plan complexe, elle joue un role TRES important e n theorie des nombres." }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 288 "" 0 "" {TEXT -1 263 "Pour les petits valeurs de x, l es deux courbes suivantes sont tres proches,\nc'est du au fait que le \+ polynome x-x^3/6 est un \"developpement limite (a l'ordre 3)\" de sin( x)\nc'est pourquoi cette approximation \"marche bien\" et est souvent \+ employee par les physiciens." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot([sin(x), x-x^3/6], x=0. .2, color=[red,blue], style=[point,line]);\n" }}}{EXCHG {PARA 290 "" 0 "" {TEXT -1 105 "Pareillement, les deux courbes ci-dessus sont tres \+ proches au voisinage de 0, mais plus loin ca se gate !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(\{tan(sin(x)),sin(tan(x))\},x= 0..2);" }}}{EXCHG {PARA 314 "" 0 "" {TEXT -1 119 "Montrer en faisant u n developpement limite a l'ordre 8 que ces deux courbes etaient en eff et proches au voisinage de 0." }}{PARA 313 "" 0 "" {TEXT -1 11 "------ -----" }}}{EXCHG {PARA 311 "" 0 "" {TEXT -1 24 "D'autres joli dessins. .." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "s := t->100/(100+(t- Pi/2)^8): r := t -> s(t)*(2-sin(7*t)-cos(30*t)/2):\nplot([r(t),t,t=-Pi /2..3/2*Pi],numpoints=2000,coords=polar,axes=none);\nplot([sin(4*x),x, x=0..2*Pi],coords=polar,thickness=3);" }}}{EXCHG {PARA 312 "" 0 "" {TEXT -1 25 "Ou meme en 3 dimensions :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot3d(abs(arccos(x+I*y)),x=-30..30,y=-30..30);" }}} {EXCHG {PARA 309 "" 0 "" {TEXT -1 74 "Le logiciel possede par ailleurs un certain nombre de packages specialises" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "?packages" }}}{EXCHG {PARA 310 "" 0 "" {TEXT -1 46 " Par exemple, pour faire de l'algebre linaire :" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 12 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 291 "" 0 "" {TEXT -1 40 "On calcule des determinants de matrice :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "det([[a,b] ,[c,d]]);\nM:=[[1,2,3,4],[-4,Pi,7,5],[3,5,6,7],[2,4,6,8]]:\nevalm(M); \ndet(M);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 292 "" 0 " " {TEXT -1 66 "Pouviez-vous prevoir que le deuxieme determinant allait donner 0 ?" }}{PARA 302 "" 0 "" {TEXT -1 21 "---------------------" } }}{EXCHG {PARA 301 "" 0 "" {TEXT -1 25 "On peut faire des tests :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "if 2>3 then print(vrai) else print(faux) fi;" }}}{EXCHG {PARA 293 "" 0 "" {TEXT -1 54 "Pour declar er une fonction, on peut faire comme suit :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=x->x^2+ln(x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 294 "" 0 "" {TEXT -1 45 "ou encore on peut utiliser une \"procedure\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f:=proc(x) \nRETURN(x^2+ln(x ))\nend;\nf(2);" }}}{EXCHG {PARA 295 "" 0 "" {TEXT -1 121 "Exercice : \+ Construisez une fonction g qui renvoie exp(x) si x>0 et x^2+1 sinon\n Tracer le graphe de la fonction sur -5..4" }}{PARA 299 "" 0 "" {TEXT -1 19 "-------------------" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 298 "" 0 "" {TEXT -1 63 "On peut calculer le i-ie me nombre premier,\nla boucle suivante " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "for i from 1 to 6 do ithprime(i) od;" }}}{EXCHG {PARA 13 "" 0 "" {TEXT -1 131 "Le mathematicien francais Pierre de Fer mat [1601-1665] avait conjecture que tous les nombres du type Fn=2^(2^ n)+1 etaient premiers." }}{PARA 300 "" 0 "" {TEXT -1 484 "Afficher les 9 premiers nombres de Fermat puis Prouvez que sa conjecture etait fau sse.\n(En fait on sait aujourd'hui apres de lourds calculs que la con jecture est fausse pour n=5...31)\nCombien de chiffre comporte F32? (R EFLECHISSEZ avant de demander n'importe quoi a Maple)\nRemarque : si o n ecrivait F40 \"a la main\", ca ferait la distance de la terre au Sol eil !\nMoralite : faire attention avant de demander a un logiciel d'af ficher certains nombres ou d'effectuer certains calculs..." }}}}{MARK "74 0 0" 38 }{VIEWOPTS 1 1 0 1 1 1803 }