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16 "a:=3;\nb:=5;\na*b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{EXCHG {PARA 279 "" 0 "" {TEXT -1 75 "On peut utiliser des fractions, des nom bres complexes, des nombres reels..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "1/2;\n2/4;\n2+I;abs(2+I);\nI^2;\nln(2);\nPi;\n3.45; \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"# \"\"\"%\"IGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"&#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$X$!\"#" }}}{EXCHG {PARA 280 "" 0 "" {TEXT -1 55 "On peut demander les decimales (evaluation en flottant)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}}{EXCHG {PARA 296 "" 0 "" {TEXT -1 41 "On peut demander encore plus de decimales" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(Pi,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5&QKz*e`EfTJ!#>" }}}{EXCHG {PARA 297 "" 0 "" {TEXT -1 122 "Un flottant, c'est une mantisse suivit d'un exposant (confer l e cours), \non peut penetrer au coeur de Maple pour voir ca :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "whattype(\");\nop(\"\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%&floatG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"5&QKz*e`EfTJ!#>" }}}{EXCHG {PARA 303 "" 0 "" {TEXT -1 92 "Les \+ tous premiers Pentium (en 1994) avaient un bug : \nle calcul suivant n e renvoyaient pas 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "8246 33702441.0*(1/824633702441.0);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+'*********!#5" }}}{EXCHG {PARA 304 "" 0 "" {TEXT -1 93 "Oups Maple \+ serait-il aussi bogue ?\nMais non... la precision de Maple par Defaut \+ est donne par" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Digits;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{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" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+H'fE@\"!#@" }}}{EXCHG {PARA 305 "" 0 "" {TEXT -1 74 "On etait donc en dessous de la precision, on corrige le tout et c' est bon:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Digits:=12:\n82 4633702441.0*(1/824633702441.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"-+++++5!#6" }}}{EXCHG {PARA 307 "" 0 "" {TEXT -1 120 "Ouf ! Retenez \+ bien ce type de probleme, il est fondamental des que l'on manipule des nombres flottants (des reels, quoi)" }}{PARA 308 "" 0 "" {TEXT -1 10 "----------" }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 32 "On peut factoris er et developper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "expand( (x-1)*(x-2));\nfactor(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$%\"x G\"\"#\"\"\"F%!\"$F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\" \"\"!\"\"F&F&,&F%F&!\"#F&F&" }}}{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];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7%\" \"\"F&\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{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];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"EG<$\"\"\"\"\"#" }}{PARA 8 "" 1 "" {TEXT -1 33 "Error, invalid subscript selector" }}}{EXCHG {PARA 282 "" 0 "" {TEXT -1 71 "Justifiez le message d'erreur que provoque la deuxieme li gne ci-dessus." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "*******\nREPON SE : \ndans un ensemble, la multiplicite du 1 n'est pas prise en compt e, on ne peut donc pas demander a E, qui n'a que 2 elements, d'affiche r son troisieme elelement.\n*******" }}}{EXCHG {PARA 283 "" 0 "" {TEXT -1 48 "Faisons maintenant quelques calculs d'integrales" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(1/x,x=u..v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#%\"vG\"\"\"-F%6#%\"uG!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 203 "Voici maintenant pourquoi on parl e de trois-demi et de cinq-demi en classe prepa : \nComparez : \"J'int egre X entre la premiere et deuxieme annee\"\npuis \"j'integre X en tre la deuxieme et troisieme annee\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "int(x,x=1..2);\nint(x,x=2..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\" \"#" }}}{EXCHG {PARA 284 "" 0 "" {TEXT -1 51 "Les integrales indefinie s sont calculees egalement " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int(sin(x)/x,x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ %#PiG#\"\"\"\"\"#" }}}{EXCHG {PARA 285 "" 0 "" {TEXT -1 59 "On peut do nc faire des calculs de primitives et de derivees" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "int(1/x,x);\ndiff(ln(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* $%\"xG!\"\"" }}}{EXCHG {PARA 286 "" 0 "" {TEXT -1 55 "On peut aussi fa ire des developpement en serie deTaylor" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series(exp(x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#+1%\"xG\"\"\"\"\"!F%\"\"\"#F%\"\"#\"\"##F%\"\"'\"\"$#F%\"#C\"\"%#F% \"$?\"\"\"&-%\"OG6#F%\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series(sin(x),x=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"xG\" \"\"\"\"\"#!\"\"\"\"'\"\"$#F%\"$?\"\"\"&#F(\"%S]\"\"(-%\"OG6#F%\"\")" }}}{EXCHG {PARA 287 "" 0 "" {TEXT -1 27 "On peut calculer des sommes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sum(k,k=1..n);\nfactor(\" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$,&%\"nG\"\"\"F'F'\"\"##F'F(F &#!\"\"F(F*F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"nG\"\"\",&F%F& F&F&F&#F&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sum(1/n!, n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"\"" }} }{EXCHG {PARA 274 "" 0 "" {TEXT -1 137 "On peut demander de l'aide sur une commande en utilisant le ?\nPar exemple demandons de l'aide sur l a commande pour resoudre les equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "?solve" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 50 "Resolv ons maintenant une equation du second degre " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(x^2-x-6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"#\"\"$" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 131 "Bien sur Maple peut utiliser la puissance du calcul formel pour donner la formule du discriminant (connu depuis les Babylonniens) :" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "solve(a*x^2+b*x+c=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#!\"&\"\"'\"\"\"*$,&\"#DF'%\"cG!#7#F'\"\"##F'F&,&F$F' F(#!\"\"F&" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 47 "De meme avec une \+ equation du troisieme de degre" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x^3+x^2-x-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"! \"\"F$" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 72 "La formule generale e st connu sous le nom de formule de Cardan-Tartaglia" }}{PARA 0 "" 0 " " {TEXT -1 207 "(en fait Tartaglia [1500-1557] avait communique a Card an [1501-1576] la formule en question en faisant promettre a Cardan d e 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 "" {XPPMATH 20 "6%,(*&%\"cG!\"\"*& ,(!#=\"\"\"F%F&*&\"\"$#F*\"\"#*&,(\"$D\"F*F%!#8*$F%F.F,F*F%F&F-F,F*F%F .#F*F,F4*&,&!#:F*F%F*F*F'#F&F,F4F8F*,*F$#F&\"\"'F5F:F8F**(%\"IGF*F,F-, &F$F4F5F8F*F-,*F$F:F5F:F8F*F<#F&F." }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 26 "Voici la premiere racine :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "\"[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"cG!\" \"*&,(!#=\"\"\"F%F&*&\"\"$#F*\"\"#*&,(\"$D\"F*F%!#8*$F%F.F,F*F%F&F-F,F *F%F.#F*F,F4*&,&!#:F*F%F*F*F'#F&F,F4F8F*" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 66 "Et qu'est ce que cela donne pour une equation du quatriem e degre ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(a+b*x+c* x^2+c*x^3+d*x^4,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,, \"\"$\"\"\"%#_ZG\"\"&*&%\"cGF(F)\"\"#F(*&F,F(F)F'F(*&%\"dGF(F)\"\"%F( " }}}{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 "" {XPPMATH 257 "6&,(*&%\"cG\" \"\"%\"dG!\"\"#F(\"\"%*(\"\"$#F&\"\"#F'F(*&,.*&F%F.,,*$F%F.\"$W\"*&F%F &F'F&!$k)F'\"%+F*$F%F,\"\")*$,6F8\"%+:*&F%F.F'F&!%+a*$F%F*!$c\"*&F%F&F 'F.!%!['*&F%F,F'F&\"#7*$F%\"\"&\"#O*$F'F,!&O2#*&F%F.F'F.\"%cM*&F'F&F%F *!$W\"*$F'F.\"&D1&F-FD#F&F,F,*(F%F&F'F&F2FP!\")*&F2#F.F,F'F&F.F5!$?\"F N\"$)GF=F9F&F2#F(F,F-#F&FD*&F'F(,$*(,4*(F%F.F2FPF/F-!#=**F%F&F'F&F2FPF /F-\"#[*(F/F-F2FTF'F&\"\"'*(F/F-F%F&F'F&!$g$*&F/F-F'F.\"$k)*(F/F-F%F.F 'F&\"#C*(F,F-F2FPF'F.\"$?(*(F,F-F2FPF%F,\"#=**F,F-F2FPF%F.F'F&!#sF&F2F WF/#F(F.F(F-FX,(F$F)F+FXFY#F(FD,(F$F)F+F[p*&F'F(,$*(,4FgnFhnFinFjnF[oF \\oF]oF^oF_oF`oFaoFboFco!$?(FeoFhnFgo\"#sF&F2FWF/FioF(F-FX,(F$F)F+F[pF ]pF[p" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Meme la premiere racine \+ a une forme compliquee :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " \"[1];" }}{PARA 11 "" 1 "" {XPPMATH 258 "6#,(*&%\"cG\"\"\"%\"dG!\"\"#F (\"\"%*(\"\"$#F&\"\"#F'F(*&,.*&F%F.,,*$F%F.\"$W\"*&F%F&F'F&!$k)F'\"%+F *$F%F,\"\")*$,6F8\"%+:*&F%F.F'F&!%+a*$F%F*!$c\"*&F%F&F'F.!%!['*&F%F,F' F&\"#7*$F%\"\"&\"#O*$F'F,!&O2#*&F%F.F'F.\"%cM*&F'F&F%F*!$W\"*$F'F.\"&D 1&F-FD#F&F,F,*(F%F&F'F&F2FP!\")*&F2#F.F,F'F&F.F5!$?\"FN\"$)GF=F9F&F2#F (F,F-#F&FD*&F'F(,$*(,4*(F%F.F2FPF/F-!#=**F%F&F'F&F2FPF/F-\"#[*(F/F-F2F TF'F&\"\"'*(F/F-F%F&F'F&!$g$*&F/F-F'F.\"$k)*(F/F-F%F.F'F&\"#C*(F,F-F2F PF'F.\"$?(*(F,F-F2FPF%F,\"#=**F,F-F2FPF%F.F'F&!#sF&F2FWF/#F(F.F(F-FX" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 101 "On s'est longtemps demande a quoi pouvait ressembler la formule pour une equation du cinquieme deg re." }}{PARA 265 "" 0 "" {TEXT -1 168 "La reponse, trouvee par le math ematicien norvegien Niels Henrik Abel (1802-1829) \net par le mathemat icien Evariste Galois (1811-1832 mort en duel a 20 ans) c'est que :" }}{PARA 266 "" 0 "" {TEXT -1 118 "IL N'Y PAS DE FORMULE POUR LES EQUAT IONS DE DEGRE >4\n(cela se prouve par la theorie des groupes, inventee par Galois)." }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 215 "De nos jours, c'est toujours un challenge de calculer numeriquement (rapidement) le s racines d'un polynome de degre eleve.\nMaple sait le faire, on peut \+ ainsi lui demander toutes les racines complexes d'un polynome :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fsolve(x^5+x^4+x-3,x,complex );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6',&$!+ " 0 "" {MPLTEXT 1 0 16 "int(1/x,x=a..b );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#%\"bG\"\"\"-F%6#%\"aG! \"\"" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 67 "Les integrales indefini es ne lui posent pas non plus de problemes :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "int(1/x,x=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 76 "C ette integrale represente l'aire sous la courbe y=1/x, tracons cette c ourbe" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(\{1/x,1/floor (x)\},x=1..10);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7W7$$\"\" \"\"\"!F(7$$\"33++v$>(3)4\"!#<$\"36hSAW\\u1\"*!#=7$$\"3%*****\\(Quh>\" F.$\"3qDZw_])*f$)F17$$\"3***\\7.`>:G\"F.$\"3A[$y*>lB.yF17$$\"3/+]7tY'o O\"F.$\"3F#)3W1K,;tF17$$\"30+v=7U%GY\"F.$\"3BIqgDz*f$oF17$$\"31++D^P#) e:F.$\"3\\BAoEM4:kF17$$\"3#******\\]Ral\"F.$\"3j6\")o%*=pSgF17$$\"3,++ ve_0_F.$\"3kC#>#e$eI9&F17$$\"3* ***\\7VsmA@F.$\"3Z@#))p*R06ZF17$$\"3*)**\\7)R&G2BF.$\"3$HYHQb(4MVF17$$ \"3-+]7ex@)\\#F.$\"3@xtL@O&G+%F17$$\"3%)**\\i&zP&)o#F.$\"3#yued#R\\>PF 17$$\"3B++]Z`I%)GF.$\"3XSHV-$RqY$F17$$\"3\"****\\P^Pn0$F.$\"3d_95(zh9F $F17$$\"3F++]#Hb3D$F.$\"3G\\ieXN6wIF17$$\"3$)****\\P,xXMF.$\"39jtxY)3@ !HF17$$\"3D++]2ngLOF.$\"34N:N4l3_FF17$$\"3&)**\\73.=/QF.$\"3S\"yB$yroG EF17$$\"3/++]<)3q+%F.$\"3q?t=^ri&\\#F17$$\"3[+++q6$)yTF.$\"3]VihUO,$R# F17$$\"3R+]789qyVF.$\"3%pc()37#y$G#F17$$\"3u*****\\W?cb%F.$\"3NmUH%o!4 &>#F17$$\"3U+]7j'G(\\ZF.$\"3IF17$$\"32+]iDt_/`F.$\"3=!)>$p5#=&)= F17$$\"3Z***\\Ppdb\\&F.$\"3H\\JDO:l>=F17$$\"3$***\\7eX)Rp&F.$\"3po,z&R Riv\"F17$$\"3J+](os:n'eF.$\"3BhFM+7`/l\"F17$$\"3Y*****\\1**fC'F.$\"385kW&*[-,;F17$$\"3t***\\itYXV'F.$\" 3@g#>8-6Tb\"F17$$\"3'***\\7.j(ph'F.$\"3JGE!3Hk7^\"F17$$\"3l***\\PBL&>o F.$\"3u]1x>gPm9F17$$\"3P*****\\kR:+(F.$\"3T3>S%Gd#G9F17$$\"3g++]P.(e>( F.$\"3wBn0jfo*Q\"F17$$\"3f**\\7GG'>P(F.$\"3fb)HL$3\\c8F17$$\"3_++]K%yW c(F.$\"3*)=?G3#o>K\"F17$$\"3R**\\781iXxF.$\"3\")=Y>J@0\"H\"F17$$\"3n** \\i&Qm\\$zF.$\"3xFR;%yW-E\"F17$$\"31++](['3?\")F.$\"3I)RJl(R^J7F17$$\" 3p)*\\7y+*QJ)F.$\"3o$[hlP1G?\"F17$$\"3,,++qfa+&)F.$\"35jnIE]Rw6F17$$\" 3>***\\(y&G9p)F.$\"3/Nt*=%)e0:\"F17$$\"3_+]7$eI2)))F.$\"3uvY6%[Lg7\"F1 7$$\"3a*****\\YzY0*F.$\"3\"e-b^K,W5\"F17$$\"3P***\\P^WSD*F.$\"305*p1f3 13\"F17$$\"3*3+++**eBV*F.$\"3/FP]u,=g5F17$$\"3L**\\78%zCi*F.$\"3$)3AZ# >L#R5F17$$\"3v**\\(o\"*[W!)*F.$\"33X$Rw8X*>5F17$$\"#5F*$\"31+++++++5F1 -%'COLOURG6&%$RGBG$Fh[l!\"\"F*F*-F$6$7_tF'7$F3F(7$F=F(7$FGF(7$FQF(7$$ \"3&**\\(o9=@[=F.F(7$FVF(7$$\"3-+voHdlm>F.F(7$$\"3#****\\()3V*))>F.F(7 $$\"3&\\i:&G\\^%*>F.F(7$$\"3w\\7Gon3+?F.$\"3++++++++]F17$$\"3duo/3'ec+ #F.Fh]l7$$\"3#)*\\7yWI7,#F.Fh]l7$$\"3))\\PMFTPA?F.Fh]l7$$\"3%***\\(o!y ^L?F.Fh]l7$$\"3v*****\\_#4y?F.Fh]l7$FenFh]l7$$\"3%***\\i?j(\\@#F.Fh]l7 $FjnFh]l7$F_oFh]l7$FdoFh]l7$$\"3D+Dcr:U'y#F.Fh]l7$FioFh]l7$$\"3/+D1*Q8 u#HF.Fh]l7$$\"3H+]iI9_qHF.Fh]l7$$\"38Dc,T%)H\")HF.Fh]l7$$\"3)*\\iS^a2# *HF.Fh]l7$$\"3ii:gcRY(*HF.Fh]l7$$\"3EvozhC&G+$F.$\"3:LLLLLLLLF17$$\"3Y (=#*p'4C3IF.Ff`l7$$\"35+v=s%HO,$F.Ff`l7$$\"3B](oH\\$=NIF.Ff`l7$F^pFf`l 7$$\"3()**\\7.kz`JF.Ff`l7$FcpFf`l7$FhpFf`l7$F]qFf`l7$$\"30+D\"y]$*)=PF .Ff`l7$FbqFf`l7$$\"3o\\(oaV()[&QF.Ff`l7$$\"3&**\\7Gc%f0RF.Ff`l7$$\"3'[ P%[E\"[4$RF.Ff`l7$$\"3A]i:!p,j&RF.Ff`l7$$\"3X(=#*>Zy*oRF.Ff`l7$$\"38D \"GQDb;)RF.Ff`l7$$\"3>%4YZk$*z)RF.Ff`l7$$\"3\"G1kc.KV*RF.Ff`l7$$\"3UJ? eE/n+SF.$\"3++++++++DF17$FgqFgcl7$$\"3#)***\\P**>H4%F.Fgcl7$F\\rFgcl7$ FarFgcl7$FfrFgcl7$F[sFgcl7$$\"3V++DEr9U[F.Fgcl7$F`sFgcl7$$\"3:v$4@$Gne \\F.Fgcl7$$\"3u]P%[2!y#)\\F.Fgcl7$$\"3>Wt_&)o!)))\\F.Fgcl7$$\"3lP4@'pL [*\\F.Fgcl7$$\"35JX*o]g3+&F.$\"36+++++++?F17$$\"3WD\"yvJ()o+&F.Fdel7$$ \"3B8`%*Q4%*=]F.Fdel7$$\"39+DJgX*4.&F.Fdel7$$\"3V]7yX!4#z]F.Fdel7$FesF del7$$\"3%3]P%Ga(f@&F.Fdel7$FjsFdel7$F_tFdel7$FdtFdel7$$\"3o****\\U,N! 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olspanG%*companionG%'concatG%%condG%)copyintoG%*crossprodG%%curlG%)def initeG%(delcolsG%(delrowsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvals G%,eigenvaluesG%-eigenvectorsG%+eigenvectsG%,entermatrixG%&equalG%,exp onentialG%'extendG%,ffgausselimG%*fibonacciG%+forwardsubG%*frobeniusG% *gausselimG%*gaussjordG%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermi teG%(hessianG%(hilbertG%+htransposeG%)ihermiteG%*indexfuncG%*innerprod G%)intbasisG%(inverseG%'ismithG%*issimilarG%'iszeroG%)jacobianG%'jorda nG%'kernelG%*laplacianG%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&mino rG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%%normG%*normalizeG%*nullspac eG%'orthogG%*permanentG%&pivotG%*potentialG%+randmatrixG%+randvectorG% %rankG%(ratformG%$rowG%'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmul G%-singularvalsG%&smithG%&stackG%*submatrixG%*subvectorG%)sumbasisG%(s wapcolG%(swaprowG%*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermo ndeG%*vecpotentG%(vectdimG%'vectorG%*wronskianG" }}}{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 "" {XPPMATH 20 "6#,&%\"dG\"\"$%\"cG!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"$\"\"%7&!\"%%#PiG\"\"(\" \"&7&F*F0\"\"'F/7&F)F+F2\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{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 0 "" 0 "" {TEXT -1 158 "*** *******\nREPONSE \noui car il y a une relation entre la premiere et la quatrieme ligne (qui le double de la premiere) donc le determinant \+ vaut 0\n**********" }}}{EXCHG {PARA 301 "" 0 "" {TEXT -1 25 "On peut f aire des tests :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "if 2>3 \+ then print(vrai) else print(faux) fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%fauxG" }}}{EXCHG {PARA 293 "" 0 "" {TEXT -1 54 "Pour declarer 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 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operato rG%&arrowGF(,&*$9$\"\"#\"\"\"-%#lnG6#F.F0F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"%\"\"\"-%#lnG6#\"\"#F%" }}}{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);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"F( F(-%'RETURNG6#,&*$9$\"\"#\"\"\"-%#lnG6#F.F0F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"%\"\"\"-%#lnG6#\"\"#F%" }}}{EXCHG {PARA 295 "" 0 "" {TEXT -1 121 "Exercice : Construisez une fonction g qui renvoie exp (x) si x>0 et x^2+1 sinon\nTracer le graphe de la fonction sur -5..4 " }}{PARA 299 "" 0 "" {TEXT -1 19 "-------------------" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "#SOLUTIONS\n\ng:=proc(x)\nif x>0 t hen RETURN(exp(x)) else RETURN(x^2+1) fi\nend;\nplot('g(x)',x=-5..4); \+ #Remarque : ne pas oublier les quotes ' et '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6#%\"xG6\"F(F(@%2\"\"!9$-%'RETURNG6#-%$expG6#F,- F.6#,&*$F,\"\"#\"\"\"F8F8F(F(" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVE SG6$7W7$$!\"&\"\"!$\"#EF*7$$!3%)****\\7c#Q![!#<$\"3$)*4J:0uwS#!#;7$$!3 =+](oKNJj%F0$\"3j6Pr&H%fYAF37$$!3s***\\([iF37$$!35+]PH;jbSF0$\"3'=p=8z9[u\"F37$$!3,+](ovK t(QF0$\"3l)e`2$4P.;F37$$!36+](=g9Fp$F0$\"32%3!48Thj9F37$$!3)***\\(=C#y ,NF0$\"3656&p)yCE8F37$$!3;+]P/AY6LF0$\"3e8/5$>yl>\"F37$$!3x****\\_Yp:J F0$\"3V4!4CvF07$$!3v****\\#H$RmBF0 $\"33*)*)y9s\")*f'F07$$!3:+](=p>e>#F0$\"3(4-E#>Ti@eF07$$!3'*****\\#=\" *H*>F0$\"30[F_`Q,s\\F07$$!3'*******H)o6#=F0$\"3>oNO2fl;VF07$$!31+](oe) H@;F0$\"3%>(G!y54'GOF07$$!3E+++b&zVW\"F0$\"3d)*>!*)HKi3$F07$$!3e**\\(o Lr-D\"F0$\"3a)>6eTyJc#F07$$!3b**\\i5WVl5F0$\"3%*RQM$[]^8#F07$$!3z#*** \\(okds)!#=$\"3d=i\"Qp*Qh[LS?Ya7F07$$!3s++v=W:gIFer$\"3l^fm]ak$4\"F07$$!3'o* *\\7tUGL\"Fer$\"3YfCY(pkx,\"F07$$\"3nu***\\77qK&!#>$\"35%3Xu]9Z0\"F07$ $\"3i%*****\\1**fCFer$\"3I&>rxP)*)y7F07$$\"3K(***\\itYXVFer$\"3scC)=%H EW:F07$$\"3f***\\7.j(phFer$\"3%\\LFip:L&=F07$$\"3Y'***\\PBL&>)Fer$\"3/ no8#HS%pAF07$$\"3P*****\\kR:+\"F0$\"3#*R;*4CqCs#F07$$\"3g++]P.(e>\"F0$ \"3OfuFKUV1LF07$$\"3f**\\7GG'>P\"F0$\"3>y6J)p#3VRF07$$\"3_++]K%yWc\"F0 $\"3F0$\"3alm(=J6Q#pF07$$\"31++](['3?@F0$\"3lv35e!e=L)F07$$\"3e**\\7y+ *QJ#F0$\"3@9(ez7p8,\"F37$$\"3,,++qfa+DF0$\"3nLgH0Z\"*=7F37$$\"33++vy&G 9p#F0$\"3'ojt_Z\"F37$$\"3_+]7$eI2)GF0$\"36oQBu#HFy\"F37$$\"3a***** \\YzY0$F0$\"3(*[`U[%Q97#F37$$\"3P***\\P^WSD$F0$\"3aeYm_g[*e#F37$$\"38+ ](=v,KM$F0$\"3*Hz!>9A'4$GF37$$\"3*3+++**eBV$F0$\"3C%[W?kc\\4$F37$$\"36 +Dc,#>u_$F0$\"3x/[k29g.MF37$$\"3L**\\78%zCi$F0$\"3e'*G)QzDIu$F37$$\"3a *****\\;kMr$F0$\"3Hs\"zoid&*4%F37$$\"3v**\\(o\"*[W!QF0$\"3Y.qiQ*\\+\\% F37$$\"3))*\\P%eWA-RF0$\"3%=W$z6lC^\\F37$$\"\"%F*$\"3iBWJ.]\")faF3-%'C OLOURG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$%\"xG%!G-%%VIEWG6$;F(F[\\l %(DEFAULTG" 2 596 596 596 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 576 548 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "On \+ voit \"bien\" le comportement polynomial a gauche et exponentiel a dr oite\n********" }}}{EXCHG {PARA 298 "" 0 "" {TEXT -1 63 "On peut calcu ler le i-ieme nombre premier,\nla boucle suivante " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "for i from 1 to 6 do ithprime(i) od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}}{EXCHG {PARA 13 "" 0 "" {TEXT -1 131 "Le mathematicien francais Pierre de Fermat [1601-1665] avait conj ecture 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 Fer mat puis Prouvez que sa conjecture etait fausse.\n(En fait on sait au jourd'hui apres de lourds calculs que la conjecture est fausse pour n= 5...31)\nCombien de chiffre comporte F32? (REFLECHISSEZ avant de deman der n'importe quoi a Maple)\nRemarque : si on ecrivait F40 \"a la main \", ca ferait la distance de la terre au Soleil !\nMoralite : faire at tention avant de demander a un logiciel d'afficher certains nombres ou d'effectuer certains calculs..." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Solution :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for i fro m 1 to 9 do print(2^(2^i)+1) od:\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$d#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&Pb'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"+(Hn\\H%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"5<;b4P2WnW=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"Hd9@ o;tB*3#z:\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"ft(4%31!\\OVY*pl%>\"G)Qv`30'['=e=.!pFM!p;)Hu=!o(pat+.k " 0 "" {MPLTEXT 1 0 56 "for i from 1 to 9 do p rint(F[i],isprime(2^(2^i)+1)) od:\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $&%\"FG6#\"\"\"%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"FG6#\"\" #%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"FG6#\"\"$%%trueG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"FG6#\"\"%%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"FG6#\"\"&%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"FG6#\"\"'%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"FG6 #\"\"(%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"FG6#\"\")%&false G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"FG6#\"\"*%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 173 "SOLUTION : le nb de chiffre c'est log \+ (du nombre) /log(10)\nil ne faut surtout PAS faire ln(F32)/ln(10) car \+ F32 est trop gros\nil faut faire ln(2^(2^32)+1) -> 2^32 * ln(2)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "2^32*ln(2)/ln(10);\nevalf(\" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%#lnG6#\"\"#\"\"\"-F&6#\"#5 !\"\"\"+'Hn\\H%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-]')R\"HH\"!\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "donc F32 a 12 milliards de chiffres !" }}}}{MARK "2 \+ 0 0" 62 }{VIEWOPTS 1 1 0 1 1 1803 }