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" }}{PARA 0 "" 0 "" {TEXT -1 82 "Attention : on n'a pas le droit de modifier les parametres pass es a la procedure :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "double:=proc(n)#calcule 2 fois n\nlocal t emp;\ntemp:=n;\nn:=2*temp;\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'doubleGf*6#%\"nG6#%%tempG6\"F*C$>8$9$>F.,$F-\"\"#F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "double(3);" }}{PARA 8 "" 1 "" {TEXT -1 53 "Error, (in double) illegal use of a formal parameter\n" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Par contre:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "redouble:=p roc(n)\nlocal temp;\ntemp:=n;\n2*temp;\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)redoubleGf*6#%\"nG6#%%tempG6\"F*C$>8$9$,$F-\"\"#F*F* F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "redouble(4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Un autre exemple, a comprendre et tester :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "g:=proc(start,n_pas)\nlocal i,temp ;\ntemp:=start;\nfor i from 1 to n_pas do\nif (temp mod 2)=1 \n then \+ temp:=3*temp+1\n else temp:=temp/2\nfi;\nprint(temp);\nod;\nend;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%&startG%&n_pasG6$%\"iG%%tem pG6\"F,C$>8%9$?(8$\"\"\"F39%%%trueGC$@%/-%$modG6$F/\"\"#F3>F/,&F/\"\"$ F3F3>F/,$F/#F3F<-%&printG6#F/F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "g(5,10);" }}{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#\"\"\"" }}{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#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "g( 10,10);" }}{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#\" \"#" }}{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#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "On constate exper imentalement que, quel que soit l'entier \"start\", apres un nombre de pas suffisamment grand on tombe sur la suite 4,2,1. On ne sait pas si c'est toujours vrai." }}}{SECT 1 {PARA 261 "" 0 "" {TEXT -1 11 "Exerc ice 1." }}{PARA 0 "" 0 "" {TEXT -1 204 "a. Ecrivez une procedure iter( x,n) qui pour tout x, 0<=x<1, itere n fois la fonction f(x)=(2*x si 2* x<1, sinon 2*x-1). Verifiez experimentalement qu'elle est \"eventuelle ment periodique\" pour x rationnel: " }}{PARA 0 "" 0 "" {TEXT -1 102 " si x=p/q, p et q entiers, il existe n0 tel que, si x0=iter(x,n0), la \+ suite iter(x0,n) est periodique." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 135 "b. Ecrivez un procedure analogue en Matl ab. Verifiez que iter(x,100)=0, quel que soit x, meme pour x=0.2, auqu el cas on devrait avoir :" }}{PARA 0 "" 0 "" {TEXT -1 74 " iter(0.2, 2)=0.4, iter(0.2,3)=0.8, iter(0.2,3)=0.6, iter(0.2,4)=0.2" }} {PARA 0 "" 0 "" {TEXT -1 37 "apres quoi cela devrait recommencer. " }} {PARA 0 "" 0 "" {TEXT -1 166 "Expliquer pourquoi, sachant que la repre sentation interne des nombres en Matlab se fait en base 2, et que les \+ fractions y sont representees par des nombres flottants." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 24 "Developpements de Taylor" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 266 14 "A une variable" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 259 96 "Pour avoir les 5 premier term es du developpement de Taylor d'une fonction d'une variable en x=0" }} {PARA 0 "" 0 "" {TEXT -1 11 " on ecrit :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "taylor(exp(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# +1%\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#F%\"\"'\"\"$#F%\"#C\"\"%#F%\"$?\"\"\" &-%\"OG6#F%F*" }}}{PARA 0 "" 0 "" {TEXT -1 59 "Mais on peut aussi en d emander plus, et en un autre point :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "taylor(exp(x),x=1,8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+5,&%\"xG\"\"\"F&!\"\"-%$expG6#F&\"\"!F(F&,$F(#F&\"\"#F.,$F(#F& \"\"'\"\"$,$F(#F&\"#C\"\"%,$F(#F&\"$?\"\"\"&,$F(#F&\"$?(F1,$F(#F&\"%S] \"\"(-%\"OGF*\"\")" }}}{PARA 5 "" 0 "" {TEXT 268 75 "La fonction dont \+ on cherche le developpement doit etre passee en parametre " }{TEXT 271 23 "sous forme d'expression" }{TEXT 272 165 ". Le dernier parametr e (ci-dessus egal a 8) est le nombre de termes calcule, en commencant \+ par le terme constant, c'est donc 1 de moins que l'ordre du developpem ent." }}{PARA 5 "" 0 "" {TEXT 270 54 "Pour en faire un vrai polynome, \+ il faut le convertir :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "t3 :=convert(taylor(exp(x),x=1,3),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3G,(-%$expG6#\"\"\"F)*&F&F),&%\"xGF)F)!\"\"F)F)*(#F)\"\"#F)F &F))F+F0F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=0.5, t3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#\"\"\"$\"*+++D'!\"* " }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT 265 10 "Exercice 2" }{TEXT 274 1 "." }}{PARA 0 "" 0 "" {TEXT -1 184 "Ecrire une procedure ayant comme parametre une fonction d'une variable f(x), une valeur a, et qui dess ine le graphe de f pres de a et de sa serie de Taylor tronquee a un or dre donne n." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 260 21 "A plusieurs var iables" }}{PARA 0 "" 0 "" {TEXT -1 117 "Voici les commandes pour calcu ler la serie de Taylor a un ordre voulu d'une fonction de deux ou plus ieurs variables :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "mtaylor (exp(x+y),[x,y]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,L\"\"\"F$%\"xGF$ %\"yGF$*&#F$\"\"#F$)F%F)F$F$*&F&F$F%F$F$*&F(F$)F&F)F$F$*&#F$\"\"'F$)F% \"\"$F$F$*(F(F$F&F$F*F$F$*(F(F$F-F$F%F$F$*&F/F$)F&F2F$F$*&#F$\"#CF$)F% \"\"%F$F$*(F/F$F&F$F1F$F$*(#F$F;F$F-F$F*F$F$*(F/F$F6F$F%F$F$*&F8F$)F&F ;F$F$*&#F$\"$?\"F$)F%\"\"&F$F$*(F8F$F&F$F:F$F$*(#F$\"#7F$F-F$F1F$F$*(F IF$F6F$F*F$F$*(F8F$FAF$F%F$F$*&FCF$)F&FFF$F$" }}}{PARA 0 "" 0 "" {TEXT -1 57 "Par defaut, la serie est tronquee a partir de l'ordre 5. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "mtaylor(sin(x^2+y^2),[x, y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\"\"F(*$)%\"y GF'F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "On peut demander plus de termes si l'on veut :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "mtaylor(sin(x^2+y^2),[x,y],7 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"\"#\"\"\"F(*$)%\"yGF 'F(F(*&#F(\"\"'F(*$)F&F.F(F(!\"\"*&#F(F'F(*&F*F()F&\"\"%F(F(F1*&#F(F'F (*&)F+F6F(F%F(F(F1*&#F(F.F(*$)F+F.F(F(F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Ou encore :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "mtaylor(exp(x^2+y^2),[x,y],10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,@\"\"\"F$*$)%\"xG\"\"#F$F$*$)%\"yGF(F$F$*&# F$F(F$)F'\"\"%F$F$*&F*F$F&F$F$*&F-F$)F+F/F$F$*&#F$\"\"'F$)F'F5F$F$*(F- F$F*F$F.F$F$*(F-F$F2F$F&F$F$*&F4F$)F+F5F$F$*&#F$\"#CF$)F'\"\")F$F$*(F4 F$F*F$F6F$F$*(#F$F/F$F2F$F.F$F$*(F4F$F:F$F&F$F$*&F " 0 "" {MPLTEXT 1 0 17 "f:=y^2-x*(x-1)^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*$)%\"yG\"\"#\"\"\"F**&%\" xGF*),&F,F*F*!\"\"F)F*F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(\{diff(f,x)=0,diff(f,y)=0\});" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$<$/%\"yG\"\"!/%\"xG#\"\"\"\"\"$<$F$/F(F*" }}}{EXCHG {PARA 0 "> 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Faites ?plot3d[options] pour en savoir plus. L'option la plus impor tante est \"orientation=[a,b]\"; cela determine le point de vue par l es angles d'Euler theta et phi. On peut les modifier interactivement a l'aide de la souris ou a partir du menu, " }{TEXT 273 31 "apres avoir selectionne l'image" }{TEXT -1 26 " en cliquant sur celle-ci." }}} {SECT 1 {PARA 20 "" 0 "" {TEXT 276 0 "" }{TEXT 277 0 "" }{TEXT 278 9 " Question." }}{PARA 0 "" 0 "" {TEXT -1 30 "C'est quoi les angles d'Eule r?" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT 269 10 "Exercice 3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 177 "Trouver a l'aide de Maple les poi nts critiques de la fonction y^2-x^2+x^4; trouver une representation \+ du graphe de cette fonction permettant de comprendre les points critiq ues." }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 262 12 "La Hessienne" }}{PARA 0 "" 0 "" {TEXT -1 170 "La hessienne d'une fonction en un point critiq ue (a,b) est la forme quadratique constituee par les termes d'ordre ex actement 2 du developpement de Taylor au point (a,b) :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "hess:=proc(f,a,b)\n1/2*subs(x=a,y= b,diff(f,x,x))*x^2+subs(x=a,y=b,diff(f,x,y))*x*y+\n1/2*subs(x=a,y=b,di ff(f,y,y))*y^2;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%hessGf*6%% \"fG%\"aG%\"bG6\"F*F*,(*&-%%subsG6%/%\"xG9%/%\"yG9&-%%diffG6%9$F1F1\" \"\")F1\"\"#F:#F:F<*(-F.6%F0F3-F76%F9F1F4F:F1F:F4F:F:*(F=F:-F.6%F0F3-F 76%F9F4F4F:)F4F " 0 "" {MPLTEXT 1 0 9 "print(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"yG\"\"#\"\"\"F (*&%\"xGF(),&F*F(F(!\"\"F'F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "hess(f,1,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\" #\"\"\"!\"\"*$)%\"yGF'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "hess(f,1/3,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"# \"\"\"F(*$)%\"yGF'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "c ontourplot(hess(f,1/3,0),x=-1..1,y=-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 257 257 257 {PLOTDATA 2 "6+-%'CURVESG657$7$$!3S+++++++#*!#=$ !3)om;zWWWk*F*7$F+F(7$F-7$$!\"\"\"\"!$!39\"44f@@@\"))F*7$7$F0$\"3.!44f @@@\"))F*7$$!3$QyMav#\\9)*F*$\"3,#yMav#\\9!*F*7$7$$!35pm\"zWWWk*F*$\"3 1(************>*F*F97$F?7$$!3?M$eRAAAU*F*$\"3)>LeRAAAU*F*7$7$$!3G***** ********>*F*$\"3)om;zWWWk*F*FE7$7$F3F0F'7$FK7$$!3B%yMav#\\9!*F*$\"3h\" yMav#\\9)*F*7$7$$!3Z%44f@@@\"))F*$\"2c***************!#*F*F]v7$7$$!3'oOOhUUUU)F*FB7$$!33]V!Q?%f6%) F*$\"3NYV!Q?%f6#*F*7$7$Fhq$\"3\"HLeRAAAA*F*F\\w7$Ffu7$F^rF\\r7$7$FdrFb rFfw7$7$$!3!>++++++S)F*Fcw7$$!3VZXqwvvvzF*$\"3>UXqwvvv&*F*7$7$Fdr$\"3V **\\i!*)))))))*F*F]x7$7$FhrF0Fhw7$Fcx7$$!3D.+]Pz]OvF*$\"3s)***\\Pz]O** F*7$7$$!3Gsm;a=&=X(F*$\"2y***************FhnFix7$7$F\\sF07$$\"3#***** \\Pz]OvF*F_s7$7$FgsFesFfy7$7$Fgs$\"3l,]i!*)))))))*F*7$$\"3++ +++++'F*F^_l7$7$$!3qkmmmmmm)*F*Fg_l7$F0$!3zr&G9dG9x&F*7$7$F0$\"3zr&G9d G9x&F*7$$!3:6Uot%*y:**F*$\"3#)4Uot%*y:fF*7$7$$!3.ommmmmm)*F*$\"3m)**** *********fF*Fd`l7$7$$!3#pmmmmmm')*F*F]al7$$!3u+++++++'*F*$\"3-******** *****R'F*7$7$$!3qNLLLLLL$*F*$\"3E)************z'F*Fcal7$Fial7$$!3(yZ!> w/>w#*F*$\"3tv/>w/>woF*7$7$F($\"3IyxxxxxxpF*F_bl7$7$Fhq$!3P++++++?zF*7 $$!37*344444p)F*Fdr7$F\\clFe^l7$7$$!3],++++++#*F*Ffbl7$$!3E,+++++?*)F* $\"3t)***********>tF*7$7$$!3Y#444444p)F*FgsFdcl7$Fjcl7$$!3_`4Q_4Q_&)F* $\"3q^4Q_4Q_xF*7$7$Fhq$\"3E************>zF*F^dl7$7$FdrF]cl7$FjblFhq7$F idlFibl7$7$Fhq$\"3:)***********>zF*7$$!3p,+++++g\")F*$\"3O)*********** f\")F*7$7$$!3f-+++++?zF*$\"3O'************R)F*F_el7$Feel7$$!3#R&4Q_4Q_ xF*$\"3I^4Q_4Q_&)F*7$7$Fdr$\"37*344444p)F*F[fl7$7$F[_lFi^l7$Ff^lF(7$Ff flFhdl7$Fafl7$$!31-+++++?tF*$\"3/************>*)F*7$7$$!3j\"yxxxxx(pF* FBFifl7$F_gl7$$!31z/>w/>woF*$\"3Vt/>w/>w#*F*7$7$F[_l$\"3EJLLLLLL$*F*Fc gl7$Fefl7$Fa_lF__l7$7$Fg_lFe_lF]hl7$Figl7$$!3Y.++++++kF*$\"3U(******** ****f*F*7$7$$!3)3+++++++'F*$\"3qkmmmmmm)*F*Fahl7$7$F^`lF07$Fg_lF[`l7$7 $Fg_lFjhl7$$!3:8Uot%*y:fF*$\"3#y?%ot%*y:**F*7$7$$!3Nx&G9dG9x&F*FbzFail 7$7$Fb`lF07$Fg`lFe`l7$7$F]alF[alF\\jl7$7$$\"3a(*************fF*$\"3#pm mmmmm')*F*7$$\"3Nx&G9dG9x&F*Ffn7$7$F]alFaal7$FfalFdal7$7$F\\blFjalFjjl 7$7$F]alFcjl7$$\"3q&)***********R'F*$\"3_3++++++'*F*7$7$F\\bl$\"3[LLLL LLL$*F*F_[m7$F\\[m7$FbblF`bl7$7$FfblF(Fi[m7$7$FfblFbcl7$$\"3%)******** ****>tF*Fecl7$7$FgsF[dlF^\\m7$7$Fgs$\"3c$444444p)F*7$$\"3j\"yxxxxx(pF* FB7$7$$\"3_!yxxxxx(pF*FBFe[m7$Fb\\m7$$\"3f]4Q_4Q_xF*F_dl7$7$FedlFhqF_] m7$7$F]elFhq7$FbelF`el7$7$FhelFfelFf]m7$7$Fjt$\"3f-+++++?zF*7$F[^mFjt7 $F]^mFd\\m7$Fh]m7$$\"3>]4Q_4Q_&)F*F\\fl7$7$FbflFdrF`^m7$Fd^m7$F\\glFjf l7$7$FBF`glFf^m7$7$FBFh\\m7$Fe\\mFgs7$F[_mFj]m7$Fh^m7$$\"3au/>w/>w#*F* Fdgl7$7$FjglF[_lF^_m7$Fb_m7$FdhlFbhl7$7$FjhlFhhlFd_m7$7$FjhlFg_l7$Fdil Fbil7$7$FbzFhilFi_m7$7$FfnFfjl7$FcjlFajl7$7$FcjlF]al7$Fb[mF`[m7$7$Ff[m F\\blFa`m7$Fc`m7$FBF\\]m-Fhp6&FjpF[q#\"\"&F^qF2-F$6[r7$7$F($!3Y++]ibbb ^F*7$$!3n)**\\7yxxd*F*$!3o,++++++WF*7$7$Faam$!3C-++++++WF*7$$!33G9d`!> w%)*F*$!3+u&Gk%4Q_PF*7$7$$!30LLe9666**F*$!33-++++++OF*Fiam7$F_bm7$F0$! 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"Etude locale aux voi sinage des points critiques" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 329 "On va faire tracer les niveaux de f et de sa hessienne, convenablement t ranslates, pour juger de l'approximation. Afin de pouvoir representer \+ les 2 dessins sur un meme graphique, on assigne d'abord chaque dessin \+ a une variable, puis on utilise la procedure \"display(..)\" qui repre sente sur un meme graphique une liste de dessins." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "ff2:=subs(x=x-1/3,hess(f,1/3,0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ff2G,&*$),&%\"xG\"\"\"#F*\"\"$!\"\"\"\"#F *F**$)%\"yGF.F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "p1:=c ontourplot(f,x=1/3-1..1/3+1,y=-1..1,color=blue):p2:=contourplot(ff2,x= 1/3-1..1/3+1,y=-1..1,color=red):display([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 329 329 329 {PLOTDATA 2 "63-%'CURVESG6.7$7$$!32]P4#*zwbf!#=$ !\"\"\"\"!7$$!3VI]TD\"3w,'F*$!3gR;DT&e!\\)*F*7$7$$!3%o9;d!\\_$G'F*$!3S +++++++#*F*F.7$F47$$!3z4#o0VKU`'F*$!3jg%)4OUVK&)F*7$7$$!3NVL?V(oRe'F*$ !3!3++++++S)F*F:7$F@7$$!3ipmmmmmmmF*$!3v;f<#yxx:)F*7$7$FG$\"3(*=f<#yxx :)F*7$$!3MCmcFo,0mF*$\"3=_***3;]$Q$)F*7$7$$!3YWL?V(oRe'F*$\"3Y(******* *****R)F*FO7$FU7$$!3!e;B%*\\I\")Q'F*$\"3C$\\cF$QY@*)F*7$7$$!3&z9;d!\\_ $G'F*$\"31(************>*F*Fen7$F[o7$$!3b5^]LXPihF*$\"3[O%Qo'yq&\\*F*7 $7$$!3=^P4#*zwbfF*$\"2m***************!#qiF*$!3sJ]sdLZ'>(F*7$7$$!3[j#\\F^=?R'F*$!3g,+++ +++oF*F]r7$7$$!3Oi#\\F^=?R'F*Ffr7$$!3aDA))RkU*e'F*$!33YWyE-CxgF*7$7$$! 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$Fgbal7$$\"1+++++D$4\"F\\x$\"14+++++vMF27$7$FicalFegqF]dal7$Fcdal7$$\" 11Zw6%Hb4\"F\\x$\"1?%HN#)eq/%F27$7$F_calFjjsFedal7$7$F_calF^dq7$F_d`l$ \"13++++++YF27$7$Fc^u$\"1=++++++]F2F^ealFcdx-%+AXESLABELSG6$%\"xG%\"yG " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C urve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 275 10 "Exercice 4" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 142 "Faites une etude semblable pour les poin ts critiques des fonctions g(x,y)=y^2-(x-1)*x^2*(x+1), u(x,y)=x^3-y^3+ x^2-y^2, h(x,y)=x^3+y^3+x*y-x-y." }}{PARA 0 "" 0 "" {TEXT -1 306 "Pour gagner du temps (?), ecrivez une procedure qui admet comme parametres une fonction f(x,y) et un point singulier (a,b) de f et qui, apres av oir verifie que (a,b) est bien un point singulier, trace les contours \+ de f et de sa hessienne translatee en (a,b), sur un rectangle centre e n (a-r,b-r)x(a+r,b+r)." }}{PARA 0 "" 0 "" {TEXT -1 74 "Testez cette pr ocedure sur d'autres fonctions de votre choix. Par exemple:" }}{PARA 0 "" 0 "" {TEXT -1 47 " f1(x,y)=y^2+x^4-x^2 ; f2(x,y)=(x^2+y^2)^2-8xy; " }}}}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }