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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT 256 45 "Etude local des fonctions a 1 et 2 variables." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 16 "Solutions et \+
sug" }{TEXT 257 0 "" }{TEXT -1 8 "gestions" }}{PARA 260 "" 0 "" {TEXT 
-1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 24 "Version du 10 avril 2002" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 263 "" 0 "" {TEXT -1 
11 "Exercice 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 124 "iter:=proc(x,n)\nlocal i, temp;\ntemp:=x;\nfo
r i from 1 to n do\ntemp:=2*temp;\nif temp>=1 then temp:=temp-1 fi;\no
d;\ntemp;\nend;\n\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%iterGf*6$%\"
xG%\"nG6$%\"iG%%tempG6\"F,C%>8%9$?(8$\"\"\"F39%%%trueGC$>F/,$F/\"\"#@$
1F3F/>F/,&F/F3F3!\"\"F/F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "iter(0.2,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\")!\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "iter(0.2,11);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"\"'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "iter(0.2,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "iter(0.2,13);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"%!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "iter(0.2,14);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#$\"\")!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Cela a l'air d'et
re preriodique. Verifions encore:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "for i from 1 to 20 do iter(0.2,i) od;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"'!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"\"%!\"\"" }}{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#$\"\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"'!
\"\"" }}{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#$\"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "
On peut aussi modifier la procedure iter pour qu'elle produise une lis
te d'iteres:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "iter_liste
:=proc(a,n)\nlocal i,L,temp;\nL:=[];\ntemp:=a;\nfor i from 1 to n do\n
temp:=2*temp;\nif temp>=1 then temp:=temp-1 fi;\nL:=[op(L),temp];\nod;
\nL;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+iter_listeGf*6$%\"aG%
\"nG6%%\"iG%\"LG%%tempG6\"F-C&>8%7\">8&9$?(8$\"\"\"F79%%%trueGC%>F3,$F
3\"\"#@$1F7F3>F3,&F3F7F7!\"\">F07$-%#opG6#F0F3F0F-F-F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "iter_liste(3/7,10);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7,#\"\"'\"\"(#\"\"&F&#\"\"$F&F$F'F)F$F'F)F$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "iter_liste(5/13,20);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#76#\"#5\"#8#\"\"(F&#\"\"\"F&#\"\"#F&#
\"\"%F&#\"\")F&#\"\"$F&#\"\"'F&#\"#7F&#\"#6F&#\"\"*F&#\"\"&F&F$F'F)F+F
-F/F1F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Cela a bien l'air d'et
re periodique!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "iter_list
e(3/56,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,#\"\"$\"#G#F%\"#9#F%
\"\"(#\"\"'F*#\"\"&F*F)F+F-F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
57 "Ici, la suite devient periodique apres la 3eme iteration." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "iter_liste(sqrt(2.0)/2,100);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`q$\"*iN@9%!\"*$\"*CrUG)F&$\"*[U&
olF&$\"*'\\3PJF&$\"*#*pTF'F&$\"*%)R$[DF&$\"*ozm4&F&$\")OfL>F&$\")s=nQF
&$\")WPMxF&$\"*)[(oa\"F&$\"*w\\P4$F&$\"*_*\\(='F&$\"*/**\\P#F&$\"*3)**
\\ZF&$\"*;'***\\*F&$\"*K#*****)F&$\"*k%)***zF&$\"*Gp***fF&$\"*cQ***>F&
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\"*_$o'\\$F&$\"*/nL*pF&$\"*3Mn)RF&$\"*;oM(zF&$\"*KOp%fF&$\"*ksQ*=F&$\"
*GXxy$F&$\"*c!\\vvF&$\"*7\")4:&F&$\")Ci>IF&$\")[CRgF&$\"*'*[y?\"F&$\"*
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om/YF&$\"*wL$4#*F&$\"*_n'=%)F&$\"*/Nt$oF&$\"*3qYn$F&$\"*;S$\\tF&$\"*K!
o)p%F&$\"*kgtR*F&$\"*G@Zz)F&$\"*cU%*e(F&$\"*7&))y^F&$\")CqxNF&$\")[Sbr
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\"*;7wq\"F&$\"*KC_T$F&$\"*k[/$oF&$\"*G(*3m$F&$\"*c%z@tF&$\"*7*eVYF&$\"
*CyrG*F&$\"*[cVd)F&$\"*'Hr[rF&$\"*#fU(H%F&$\"*%=&[f)F&$\"*o.(*=(F&$\"*
O2%zVF&$\"*s9)e()F&$\"*WHw^(F&$\"*))e_.&F&$\"(w<0(F&$\")_N59F&$\")/r?G
F&$\")3UTcF&$\"*;%GG6F&$\"*KolD#F&$\"*kOJ^%F&$\"*Gti-*F&$\"*cYD0)F&" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Cela n'a pas tellement l'air d'
etre periodique! On a meme l'impression que les valeurs se baladnet un
 peu partout dans [0,1[, et c'est ce qui se passe en effet." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "============ " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Si on essaie de faire une
 procedure semblable en Matlab, voici ce que cela donne :" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "On ecrit la focntion dans un fichier :" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "function iter(x,n)\ntemp=x;\nfo
r i=1:n\n\011temp=2*temp;\n\011if temp>=1 temp=temp-1;\n\011\011end\ne
nd\ntemp\n\n\011Voici le resultat de l'execution dans la ligne de comm
ande :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
207 "\310iter(0.2,2);\n\ntemp =\n\n    0.8000\n\n\310iter(0.2,3);\n\nt
emp =\n\n    0.6000\n\n\310iter(0.2,4);\n\ntemp =\n\n    0.2000\n\n
\310iter(0.2,8);\n\ntemp =\n\n    0.2000\n\n\310iter(0.2,16);\n\ntemp \+
=\n\n    0.2000\n\n\310iter(0.2,100);\n\ntemp =\n\n     0" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 458 "En fait, quelle que soit la valeur de de
part, on trouve 0 au bout d'un moment! La raison en est que pour repre
senter x, 0 <= x < 1, en base 2, on ecrit x sous la forme sous la form
e d'une somme finie de termes de la forme a(n)/(2^n), avec a(n)=0 ou 1
, n entier positif; la suite des a(n) donne les chiffres a droite de l
a virgule dans la representation binaire de x (exemple: x=0.675 en bas
e 10; x=1/2+1/8, donc x s'ecrit 0.101 en base 2). Or  2^n*(1/2^n)=1." 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 261 "" 0 "" {TEXT 
-1 11 "Exercice 2." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "f est la fon
ction, representee " }{TEXT 258 28 "sous forme d'expression en x" }
{TEXT -1 88 ", a est le point ou on la developpe, n l'ordre, et on des
siners le graphe sur [a-h,a+h]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "montre:=proc(f,a,h,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "lo
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or(f,x=a,n+1),polynom);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot([f,
approx],x=a-h..a+h,color=[red,green],style=[line,point]);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'mo
ntreGf*6&%\"fG%\"aG%\"hG%\"nG6#%'approxG6\"F-C$>8$-%(convertG6$-%'tayl
orG6%9$/%\"xG9%,&9'\"\"\"F=F=%(polynomG-%%plotG6&7$F7F0/F9;,&F:F=9&!\"
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radplot3dG%-implicitplotG%/implicitplot3dG%(inequalG%-listcontplotG%/l
istcontplot3dG%0listdensityplotG%)listplotG%+listplot3dG%+loglogplotG%
(logplotG%+matrixplotG%(odeplotG%'paretoG%*pointplotG%,pointplot3dG%*p
olarplotG%,polygonplotG%.polygonplot3dG%4polyhedra_supportedG%.polyhed
raplotG%'replotG%*rootlocusG%,semilogplotG%+setoptionsG%-setoptions3dG
%+spacecurveG%1sparsematrixplotG%+sphereplotG%)surfdataG%)textplotG%+t
extplot3dG%)tubeplotG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "On rappe
lle la procedure qui calcule la hessienne de f au point (a.b):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "hess:=proc(f,a,b)\n1/2*subs
(x=a,y=b,diff(f,x,x))*x^2+subs(x=a,y=b,diff(f,x,y))*x*y+\n1/2*subs(x=a
,y=b,diff(f,y,y))*y^2;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%hes
sGf*6%%\"fG%\"aG%\"bG6\"F*F*,(*&-%%subsG6%/%\"xG9%/%\"yG9&-%%diffG6%9$
F1F1\"\"\")F1\"\"#F:#F:F<*(-F.6%F0F3-F76%F9F1F4F:F1F:F4F:F:*(F=F:-F.6%
F0F3-F76%F9F4F4F:)F4F<F:F:F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
71 "et voici une procedure qui affiche les contours de f et de sa hess
ienne" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 407 "montre:=proc(f,a,
b,r)\n#presume que f est a deux variables nommees x et y\nlocal h,th,p
lot1,plot2;\nif (abs(subs(x=a,diff(f,x)))<0.01) and (abs(subs(x=a,diff
(f,x)))<0.01)\nthen \nh:=hess(f,a,b);\nth:=subs(\{x=x-a,y=y-b\},h);\np
lot1:=contourplot(th,x=a-r..a+r,y=b-r..b+r,color=red);\nplot2:=contour
plot(f,x=a-r..a+r,y=b-r..b+r,color=green);\ndisplay([plot1,plot2]);\ne
lse print(`(a,b) n'est pas point critique`)\nfi;\nend;" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%'montreGf*6&%\"fG%\"aG%\"bG%\"rG6&%\"hG%#thG%&pl
ot1G%&plot2G6\"F0@%2-%$absG6#-%%subsG6$/%\"xG9%-%%diffG6$9$F:$\"\"\"!
\"#C'>8$-%%hessG6%F?F;9&>8%-F76$<$/%\"yG,&FPFAFI!\"\"/F:,&F:FAF;FRFE>8
&-%,contourplotG6&FK/F:;,&F;FA9'FR,&F;FAFgnFA/FP;,&FIFAFgnFR,&FIFAFgnF
A/%&colorG%$redG>8'-FX6&F?FZFin/F^o%&greenG-%(displayG6#7$FVFao-%&prin
tG6#%?(a,b)~n'est~pas~point~critiqueGF0F0F0" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 163 "crit:=proc(f)\nlocal sols;\n#procedure pour cherc
her les points citiques\n#presume que f est a deux variables nommees x
 et y\nsolve(\{diff(f,x),diff(f,y)\},\{x,y\});\nend;\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%critGf*6#%\"fG6#%%solsG6\"F*-%&solveG6$<$-%%dif
fG6$9$%\"xG-F06$F2%\"yG<$F3F6F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "f:=y^2-(x-1)*x^2*(x+1);s:=crit(f);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fG,&*$)%\"yG\"\"#\"\"\"F**(,&%\"xGF*F*!\"\"F*)F-F
)F*,&F-F*F*F*F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG6$<$/%\"xG
\"\"!/%\"yGF)<$F*/F(-%'RootOfG6$,&*$)%#_ZG\"\"#\"\"\"F5F6!\"\"/%&label
G%$_L7G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "allvalues(op(s[2
])[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"xG,$*$-%%sqrtG6#\"\"#\"
\"\"#F+F*/F$,$F&#!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 
"montre(f,0,0,0.2);" }}{PARA 13 "" 1 "" {GLPLOT2D 386 289 289 
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