{VERSION 3 0 "IBM INTEL NT" "3.0" }
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{SECT 0 {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 89 "Esculier 16 8 02 9h48
 zip 250 maple 20020724_exoRMS.mws=01_328.mws version 16 08 02 15h40" 
}}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 56 "Exercice  propos\351 par Vidia
ni le 24/07/2002 sur UPS-math" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 468 
"L'exercice 328 de la rms a pour \351nonc\351 :\nSoit (P) la parabole \+
y^2=2px p>0 ; Si  M(x,0) avec x\\geq 0, montrer qu'il\nexiste un uniqu
e point M'(x',0) avec x'>x tel que le carr\351 de diagonale MM'\nait s
es deux autres sommets sur la parabole.\nExpliciter f : x---> x' ;\nSo
it alors x_0\\geq 0 et la r\351currence x(n+1)=f(x_n) ; montrer que x_
n tend\nvers plus l'infini et donner un d\351veloppement asymtpotique \+
\340 deux termes de\n(x_n) ;\n\nOn trouve f(x)=x+2p+2*\\sqrt\{2px+p^2
\} ..... etc" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 143 "Si l'on veu
t des carr\351s, supprimer les unconstrained, et mettre constrained da
ns le display, mais l'\351chelle cause un \351crasement de la parabole
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1687 "restart:\n eq:=y^2-2
*p*x:\n  y3:=(-x+x1)/2; x3:=(x+x1)/2 ;\n  y4:=(x-x1)/2 : x4:=(x+x1)/2 \+
:\n eq3:=simplify(x3*2*p-y3^2);eq4:=simplify(x4*2*p-y4^2):\n sol3:=sol
ve(\{eq3\},\{x1\}) :\n res:=[allvalues(sol3[1])];\n assign(res[1]); x1
;\n f:=proc(x)  2*p+x+2*sqrt(p^2+2*p*x) end;\n\n# ********** ordonn
\351es en progression arith. *********************************\n ordon
nee:=(f(x)-x)/2;\n difference:=subs(x=f(x),ordonnee)-ordonnee;\n # ***
**  Maple ne simplifie pas (il faut l'aider un peu !)\n assume(X>0,p>0
):\n simplify(subs(x=(X^2-p^2)/2/p,difference));\n# ********* \347a fa
it bien 2p, c'est \351vident dixit Douillet ! ********************\n a
ssume(p>0):\nseq(simplify((f@@n)(p*(2*a-1/2))/p+1/2-2*a-4*n*sqrt(a)),n
=1..10);\n p:='p':\n u:=proc(n,u0,p)\n 2*p*n^2+u0+2*n*sqrt(2*u0*p+p^2)
\n end;\n\n p:=1/2: u00:=1:\n printf(\"suite des u_n par applications \+
successives de f :\\n%a\\n\\n\",\n [seq((f@@n)(u00),n=1..12)]);\n prin
tf(\"suite des u_n par la formule de r\351currence propos\351e /\n:\\n
%a\\n\\n\",[seq(u(n,u00,p),n=1..12)]);\n printf(\"suite des diff\351re
nces des ordonn\351es :\\n%a\\n\\n\",\n[seq(simplify(subs(x=u(n,u00,p)
,difference)),n=1..12)]);\n\n with(plots):\n nb:=10;  p:=1: u0:=10: \n
# ***** valeurs \340 changer selon r\351sultats d\351sir\351s*******\n
 dep:=(f(u0)-u0)/2:\n\nquad:=plot([seq(evalf([[0,dep+i*2*p],[(u(i,u0,p
)+u(i+1,u0,p))/2,dep+i*2*p]])\n,i=1..nb)],\n       color=green,linesty
le=6):\n carre:=display([seq(display(polygonplot([\n      seq(evalf([[
u(n,u0,p),0],subs(x=u(n,u0,p),x1=u(n+1,u0,p),[x3,y3]),\n[u(n+1,u0,p),0
],subs(x=u(n,u0,p),x1=u(n+1,u0,p),[x4,y4])]),n=1..i)],color=blue,style
=line))\n         ,i=1..nb)],insequence=true):\n yf:=subs(x=u(nb,u0,p)
,x1=u(n+1,u0,p),y3):\n para:=plot([t^2/2/p,t,t=-yf..yf],color=red,scal
ing=constrained):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{MPLTEXT 1 0 142 "display([quad,carre,para],scaling=unc
onstrained,title=\"Animation des\ncarr\351s\");\n # ******* avec const
rained c'est evidemment tr\350s aplati ******" }{TEXT -1 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#y3G,&%\"xG#!\"\"\"\"#%#x1G#\"\"\"F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G,&%\"xG#\"\"\"\"\"#%#x1GF'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G,,*&%\"pG\"\"\"%\"xGF(F(*&F'\"
\"\"%#x1GF(F(*$)F)\"\"#F+#!\"\"\"\"%*&F)F+F,F+#F(F/*$)F,F/F+F0" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG7#<#/%#x1G,(%\"pG\"\"#%\"xG\"\"
\"*$-%%sqrtG6#,&*$)F*F+\"\"\"F-*&F*F-F,F-F+F5F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(%\"pG\"\"#%\"xG\"\"\"*$-%%sqrtG6#,&*$)F$F%\"\"\"F'*&F
$F'F&F'F%F/F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"F(F(
,(%\"pG\"\"#9$\"\"\"-%%sqrtG6#,&*$)F*F+\"\"\"F-*&F*F-F,F-F+F+F(F(F(" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)ordonneeG,&%\"pG\"\"\"*$-%%sqrtG6#
,&*$)F&\"\"#\"\"\"F'*&F&F'%\"xGF'F/F0F'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%+differenceG,&*$-%%sqrtG6#,&*$)%\"pG\"\"#\"\"\"\"\"\"*&F-F/,(F
-F.%\"xGF0*$-F(6#,&F+F0*&F-F0F3F0F.F/F.F0F.F/F0F4!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$%#p|irG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,
\"\"#\"\")\"#=\"#K\"#]\"#s\"#)*\"$G\"\"$i\"\"$+#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"uGR6%%\"nG%#u0G%\"pG6\"F*F*,(*&9&\"\"\")9$\"\"#\"\"
\"F19%F.*&F0F.-%%sqrtG6#,&*&F3F.F-F2F1*$)F-F1F2F.F.F1F*F*F*" }}{PARA 
6 "" 1 "" {TEXT -1 49 "suite des u_n par applications successives de f
 :" }}{PARA 6 "" 1 "" {TEXT -1 170 "[2+5^(1/2), 5+2*5^(1/2), 10+3*5^(1
/2), 17+4*5^(1/2), 26+5*5^(1/2), 37+6*5^(1/2), 50+7*5^(1/2), 65+8*5^(1
/2), 82+9*5^(1/2), 101+10*5^(1/2), 122+11*5^(1/2), 145+12*5^(1/2)]" }}
{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 53 "suite des
 u_n par la formule de r\351currence propos\351e /" }}{PARA 6 "" 1 "" 
{TEXT -1 1 ":" }}{PARA 6 "" 1 "" {TEXT -1 170 "[2+5^(1/2), 5+2*5^(1/2)
, 10+3*5^(1/2), 17+4*5^(1/2), 26+5*5^(1/2), 37+6*5^(1/2), 50+7*5^(1/2)
, 65+8*5^(1/2), 82+9*5^(1/2), 101+10*5^(1/2), 122+11*5^(1/2), 145+12*5
^(1/2)]" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 
37 "suite des diff\351rences des ordonn\351es :" }}{PARA 6 "" 1 "" 
{TEXT -1 36 "[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]" }}{PARA 6 "" 1 "" 
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nbG\"#5" }}{PARA 
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 762 "# ****
*** autre possibilit\351***********************************\ncarre:= d
isplay([seq(display(\{\n         polygonplot([ seq(evalf([ [0,dep+n*2*
p], [(u(n,u0,p)+u(n+1,u0,p))/2,dep+n*2*p] ]),\n                       \+
n=1..i)],linestyle=2, color=green),\n         polygonplot( [seq(evalf(
[[u(n,u0,p),0],subs(x=u(n,u0,p),x1=u(n+1,u0,p),[x3,y3]),\n[u(n+1,u0,p)
,0],subs(x=u(n,u0,p),x1=u(n+1,u0,p),[x4,y4])]),n=1..i)], color=blue,st
yle=patch,linestyle=0)\}) ,i=1..nb)],insequence=true):\n# ****** on pe
ut tracer avec style =line\n yf:=subs(x=u(nb,u0,p),x1=u(n+1,u0,p),y3):
\n para:=plot([t^2/2/p,t,t=-yf..yf],color=red):\n display([carre,para]
,scaling=unconstrained,title=\"Animation des\ncarr\351s : biele parabo
lique !\");\n # ******* avec constrained c'est evidemment tr\350s apla
ti ******\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" 
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