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"" 0 261 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 "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 41 "Corrig\351 des exercice s Maple Ensam-PT 2000" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 0 "" {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT 259 0 "" }{TEXT 257 11 "Ex ercice 79" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg): " }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->1/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-9$ !\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "AM:=vector(2 ,[x-a,y-f(a)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AMG-%'vectorG6#7 $,&%\"xG\"\"\"%\"aG!\"\",&%\"yGF+*&\"\"\"F1F,!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "AB:=vector(2,[b-a,f(b)-f(a)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ABG-%'vectorG6#7$,&%\"bG\"\"\"%\"aG!\"\", &*&\"\"\"F0F*!\"\"F+*&F0F0F,F1F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "a:=2*b:M:=map(eval,concat(AB,AM));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"MG-%'matrixG6#7$7$,$%\"bG!\"\",&%\"xG\"\"\"F+!\"# 7$,$*&\"\"\"F4F+!\"\"#F/\"\"#,&%\"yGF/F3#F,F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,( *&%\"yG\"\"\")%\"bG\"\"#\"\"\"F+F*!\"$%\"xGF(F,F*!\"\"#!\"\"F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "P:=numer(det(M));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,(*&%\"yG\"\"\")%\"bG\"\"#\"\"\"!\"#F* \"\"$%\"xG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "L" }{TEXT 313 0 "" }{TEXT -1 42 "a famille de droites (AB) est donn\351e par :" }} {PARA 0 "> " 0 "" {XPPEDIT 19 1 "D_b-x-2*b^2*y+3*b = 0;" "6#/,*%$D_bG \"\"\"%\"xG!\"\"*(\"\"#F&*$%\"bG\"\"#F&%\"yGF&F(*&\"\"$F&F,F&F&\"\"!" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 251 "L'enveloppe de cette fa mille de droites sobtient en r\351solvant le syst\350me en (x,y) des \351quations des droitres D_b et D'_b, ou dans notre cas, en expriman t que l'\351quation du second degr\351 en b repr\351sent\351e par D_b \+ a une racine double. D'o\371 ce qui suit :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Delta:=coeff(P,b,1)^2-4*coeff(P,b,2)*coeff(P,b,0);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*%$D_bG\"\"\"%\"xG!\"\"*&%\"yGF&)% \"bG\"\"#\"\"\"!\"#F,\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%& DeltaG,&\"\"*\"\"\"*&%\"yGF'%\"xGF'!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "L'enveloppe de la famille de droites (AB) est l'hyperbol e d' \351quation :" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "9-8*xy = 0;" "6 #/,&\"\"*\"\"\"*&\"\")F&%#xyGF&!\"\"\"\"!" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&\"\"*\"\"\"%#xyG!\")\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "implicitplot(Delta,x=-10..10,y=-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%'CURVESG6_o7$7$$!1************* >*!#:$!1Al&p3EGA\"!#;7$$!1')*********\\P*F*$!1-++++++7F-7$F.7$$!#5\"\" !$!1++++++D6F-7$F'7$$!1MLLLL$3<*F*$!1nmmmm;H7F-7$7$$!1*************R)F *$!1:dG9dGR8F-F;7$FA7$$!1,+++v=<#)F*$!1++++D\"GQ\"F-7$7$$!1+++++++wF*$ !1PZ*y:j-[\"F-FG7$FM7$$!1eG9dGk>sF*$!1Vr&G9d.e\"F-7$7$$!1************* z'F*$!1$)eqk&F*$!1R:YQ:Yj@F-7$$!1&***** *****\\i&F*$!1-++++++?F-7$FjoF_o7$Feo7$$!1NLLLL$e%[F*$!1mmmmm;aBF-7$7$ $!1*************R%F*$!1#=====ob#F-Fap7$7$$!1,++++++OF*$!1++++++DJF-7$$ !1Sr&G9dy,%F*$!1-++++++GF-7$FbqFgp7$7$$!1+++++++GF*$!1Vr&G9dy,%F-7$$!1 )**********\\7$F*$!1-++++++OF-7$F^rF]q7$7$$!\"#F7$!1,+++++DcF-7$$!1Q:Y Q:Yj@F*$!1-++++++_F-7$Fjr7$$!1lmmmm;aBF*$!1QLLLL$e%[F-7$7$$!1!=====ob# F*$!1-++++++WF-F`s7$7$$!1\"=====ob#F*FisFiq7$7$$!1+++++++7F*$!1,+++++v $*F-7$$F,F*$!1+++++++#*F-7$Fet7$$F?F*$!1LLLLL$3<*F-7$7$$!19dG9dGR8F*$! 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F.,(F(F)F6F.F7F.F.#!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "S3:=sum(S[k]^3,k=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S3G, **$),&\"\"\"F)*$-%%sqrtG6#,&\"#6F)*$-F,6#\"\"&\"\"\"\"\"(F4#F)\"\"#\" \"$F4F)*$),&F)F)F*#!\"\"F7F8F4F)*$),&F)F)*$-F,6#,&F/F)F0!\"(F4F6F8F4F) *$),&F)F)FAF " 0 "" {MPLTEXT 1 0 13 "simp lify(S3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 0 "" } {TEXT 263 0 "" }{TEXT 264 11 "Exercice 82" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "restart:u:=n->cos(Pi*(n^3+a*n^2+b*n)^(1/3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6#%\"nG6\"6$%)operatorG%&arrowG F(-%$cosG6#*&%#PiG\"\"\"),(*$)9$\"\"$\"\"\"F1*&%\"aGF1)F6\"\"#F8F1*&% \"bGF1F6F1F1#F1F7F8F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "En f actorisant n^3, on obtient :" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "u_n = c os(n*Pi(1+a/n+b/(n^2))^(1/3));" "6#/%$u_nG-%$cosG6#*&%\"nG\"\"\")-%#Pi G6#,(\"\"\"F**&%\"aGF*F)!\"\"F**&%\"bGF**$F)\"\"#F3F**&\"\"\"F*\"\"$F3 F*" }}{PARA 0 "" 0 "" {TEXT -1 34 "D'o\371 le d\351veloppement g\351n \351ralis\351 :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "inter:=series((1 +a/n+b/n^2)^(1/3),n=infinity,3);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%&interG,0\"\"\"F&*&%\"aG\"\"\"%\"nG! \"\"#F&\"\"$*&,&%\"bGF,*$)F(\"\"#F)#!\"\"\"\"*F)*$)F*\"\"#F)F+F&*&,&*& F(F&F0F&#!\"#F6*$)F(F-F)#\"\"&\"#\")F)*$)F*\"\"$F)F+F&*&,(*$)F0F3F)F4* &F2F)F0F)#FB\"#F*$)F(\"\"%F)#!#5\"$V#F)*$)F*\"\"%F)F+F&*&,(*&F(F)FJF)F L*&F@F)F0F)#!#SFS*$)F(FBF)#\"#A\"$H(F)*$)F*\"\"&F)F+F&-%\"OG6#*&F)F)*$ )F*\"\"'F)F+F&" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "u_n = (-1)^n* cos(a*Pi/3+(b/3-a^2/9)*Pi/n+O(1/(n^2)));" "6#/%$u_nG*&),$\"\"\"!\"\"% \"nG\"\"\"-%$cosG6#,(*(%\"aGF+%#PiGF+\"\"$F)F+*(,&*&%\"bGF+\"\"$F)F+*& F1\"\"#\"\"*F)F)F+F2F+F*F)F+-%\"OG6#*&\"\"\"F+*$F*\"\"#F)F+F+" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "u_n tend \+ vers " }{XPPEDIT 18 0 "cos(Pi*a/3);" "6#-%$cosG6#*(%#PiG\"\"\"%\"aGF( \"\"$!\"\"" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 75 "u_n ne ten d pas vers 0 - et alors la s\351rie diverge- si a est diff\351rent de \+ " }{XPPEDIT 18 0 "3/2+3*k;" "6#,&*&\"\"$\"\"\"\"\"#!\"\"F&*&\"\"$F&% \"kGF&F&" }}{PARA 0 "" 0 "" {TEXT -1 17 "Dans le cas o\371 " } {XPPEDIT 18 0 "a = 3/2+3*k;" "6#/%\"aG,&*&\"\"$\"\"\"\"\"#!\"\"F(*&\" \"$F(%\"kGF(F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 323 "alors u_n tend vers 0. C'est une condition n\351cessaire de convergence de la s\351rie. Alors, il est facile de voir que la s \351rie de t.g u_n converge comme la somme de deux s\351ries convergen tes, la premi\350re de terme g\351n\351ral v_n converge gr\342ce au th \351or\350me sp\351cial des s\351ries altern\351es, la deuxi\350me de \+ t.g w_n converge absolument : " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "v_n = (-1)^n*K/n;" "6#/%$v_nG*(),$\"\"\"!\"\"%\"nG\"\"\"%\"KGF+F*F)" }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "w_n = O(1/(n^2));" "6#/%$w_nG-%\"OG6#*& \"\"\"\"\"\"*$%\"nG\"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 298 0 "" }{TEXT 299 0 "" } {TEXT 300 11 "Exercice 83" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "restart:f:=x->if x=1 then 1 \nelif type(x,even) then \+ x/2 else 3*x+1 fi:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "mapro c:=proc(u0,n)\na:=u0:for k to n do\na:=f(a) od:\na\nend:" }}{PARA 7 " " 1 "" {TEXT -1 41 "Warning, `a` is implicitly declared local" }} {PARA 7 "" 1 "" {TEXT -1 41 "Warning, `k` is implicitly declared local " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "conjec:=proc(u0)\nfor k from 0 while maproc(u0,k)<>1 do od:\nk\nend:" }}{PARA 7 "" 1 "" {TEXT -1 41 "Warning, `k` is implicitly declared local" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "conjec(57);time(conjec(125647987451 ));conjec(125647987451);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#K" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$!e!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$\\#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ma proc(125647987451,248);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 265 0 "" }{TEXT 266 11 "Exercice 84" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=matrix(3,3,[a,b,c,b,a,b,c ,b,a]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%%\"aG %\"bG%\"cG7%F+F*F+7%F,F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 182 "On d\351finit une matrice inconnue M et on va r\351soudre le syst\350me \+ form\351 par l'\351galit\351 \340 0 des 9 coefficients de la matrice C =AM-MA, o\371 les inconnues sont les coefficients m_(i,j) de M. " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "M:=matrix(3,3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"MG-%&arrayG6%;\"\"\"\"\"$F(7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "C:=evalm(A&*M-M&*A):eq:=\{seq(seq(C[i,j] ,j=1..3),i=1..3)\}:inc:=[seq(seq(M[i,j],j=1..3),i=1..3)]:X:=genmatrix( eq,inc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG-%'matrixG6#7+7+\"\" !F*%\"bG,$%\"cG!\"\",$F+F.F*F*F*F+7+F-F*F*F+F*F*F*F/F,7+F*F-F*F*F+F*F/ F*F/7+F*F*F-F*F*F+F,F/F*7+F*F/F,F+F*F*F-F*F*7+F/F*F/F*F+F*F*F-F*7+F,F/ F*F*F*F+F*F*F-7+F+F*F*F*F/F,F+F*F*7+F*F+F*F/F*F/F*F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "kernel(X);L:=[op(kernel(X))];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%-%'vectorG6#7+\"\"\"\"\"!F)F)F(F)F)F )F(-F%6#7+F)F(F)F(,$*&%\"cG\"\"\"%\"bG!\"\"!\"\"F(F)F(F)-F%6#7+F)F)F(F )F(F)F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7%-%'vectorG6#7+ \"\"\"\"\"!!\"\"F+F+F+F,F+F*-F'6#7+F+F+F*F+F*F+F*F+F+-F'6#7+F+F**&%\"c G\"\"\"%\"bG!\"\"F*F+F*F3F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Evidemment, on est ici dans le cas o\371 b n'est pas nul. L'espace ve ctoriel des matrices commutant ave A est de dimension 3." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 " for k to nops(kernel(X)) do M.k:=matrix(3,3 ,op(k,L)) od :base:=seq(eval(M.k),k=1..nops(kernel(X)));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%baseG6%-%'matrixG6#7%7%\"\"\"\"\"!!\"\"7%F,F, F,7%F-F,F+-F'6#7%7%F,F,F+7%F,F+F,7%F+F,F,-F'6#7%7%F,F+*&%\"cG\"\"\"%\" bG!\"\"7%F+F,F+7%F:F+F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Une au tre m\351thode moins sophistiqu\351e :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "M:=matrix(3,3):C:=evalm(A&*M-M&*A):eq:=\{seq(seq(C[i,j],j=1.. 3),i=1..3)\}:inc:=\{seq(seq(M[i,j],j=1..3),i=1..3)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "S:=solve(eq,inc);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG<+/&%\"MG6$\"\"\"F*&F(6$\"\"$F-/&F(6$F*F-&F(6$ F-F*/&F(6$F*\"\"#&F(6$F-F6/&F(6$F6F6*&,(*&%\"cGF*F7F*!\"\"*&%\"bGF*F1F *F**&FB\"\"\"F+F*F*FDFB!\"\"/F1F1/F7F7/F+F+/&F(6$F6F-F7/&F(6$F6F*F7" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "assign(S);M;map(eval,M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"MG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%&%\"MG6$\"\"$F+&F)6$F+\"\"#&F)6$F+\"\"\"7%F,*&,( *&%\"cGF1F,F1!\"\"*&%\"bGF1F/F1F1*&F9\"\"\"F(F1F1F;F9!\"\"F,7%&%\"?GF0 &F?F-&F?F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "Maple traite ici l e cas o\371 b n'est pas \351gal \340 0.On voi, dans ce cas, que que l 'ensemble des matrices M telles que AM-MA=0 est un espace vectoriel de dimension 3." }}{PARA 0 "" 0 "" {TEXT -1 12 "Et si b= 0 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "b:=0:A:=map(eval,A);M:=matrix(3,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%%\"aG\"\"!% \"cG7%F+F*F+7%F,F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%&array G6%;\"\"\"\"\"$F(7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "C: =evalm(A&*M-M&*A):eq:=\{seq(seq(C[i,j],j=1..3),i=1..3)\}:inc:=\{seq(se q(M[i,j],j=1..3),i=1..3)\}:S:=solve(eq,inc):assign(S);map(eval,M);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%&%\"MG6$\"\"$F+\"\"!&F )6$F+\"\"\"7%F,&%\"?G6$\"\"#F4F,7%&F2F.F,&F2F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "ou alors :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "M: =matrix(3,3):C:=evalm(A&*M-M&*A):eq:=\{seq(seq(C[i,j],j=1..3),i=1..3) \}:inc:=[seq(seq(M[i,j],j=1..3),i=1..3)]:X:=genmatrix(eq,inc):for k to nops(kernel(X)) do M.k:=matrix(3,3,op(k,[op(kernel(X))])) od :base:=s eq(eval(M.k),k=1..nops(kernel(X)));" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Dans tous les cas, le sous-espace vectoriel des matrices \+ commutant avec A est de dimension 3." }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%baseG6%-%'matrixG6#7%7%\"\"!F+\"\"\"7%F+F+F+7%F,F+F+-F'6#7%F.F-F* -F'6#7%F-7%F+F,F+F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 11 "Exercice 85" }{TEXT -1 0 "" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "restart:f:=unapply( ((x^3+a*x-3)*exp(1/x))/(1+x),(a ,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"aG%\"xG6\"6$%)ope ratorG%&arrowGF)*&*&,(*$)9%\"\"$\"\"\"\"\"\"*&9$F5F2F5F5!\"$F5F5-%$exp G6#*&F4F4F2!\"\"F5F4,&F5F5F2F5F=F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(f(2,x),x=-10..10,y=-100..100);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7\\r7$$!#5\"\"!$\" 12U$=`)\\G5!#87$$!1nmm;p0k&*!#:$\"12pr&Q#zL%*!#97$$!1LL$3s%HaF1$ \"1#pQgpeeE$F47$$!1******\\$*4)*\\F1$\"1k@1^LvAGF47$$!1+++]_&\\c%F1$\" 1RCV&G%z;CF47$$!1+++]1aZTF1$\"12#f(omWx \"F47$$!1LLL$=exJ$F1$\"1ShhV\")=t9F47$$!1LLLL2$f$HF1$\"1)o.\"[0'eD\"F4 7$$!1++]PYx\"\\#F1$\"1mVvZ\"QD0\"F47$$!1nmTNz>&H#F1$\"13y.5myG)*F17$$! 1MLLL7i)4#F1$\"1$y2LIZ?H*F17$$!1+]PMLy!*>F1$\"1I]eYR'G3*F17$$!1mmTNa%H )=F1$\"1e***yBX6&*)F17$$!1K$ek`2^x\"F1$\"1+29^MA>*)F17$$!1****\\P'psm \"F1$\"1(f\\Z;JQ-*F17$$!1**\\(o/Efh\"F1$\"1O;>D?uQ\"*F17$$!1***\\iX#ek :F1$\"1k-*e\">F4$*F17$$!1**\\il)QK^\"F1$\"1A:Kv#o/b*F17$$!1*****\\F&*= Y\"F1$\"1l1H(3oS))*F17$$!1***\\P43#f8F1$\"1tvNnmv(4\"F47$$!1****\\74_c 7F1$\"1hJE\"QR'=8F47$$!1m\"z>E\\H?\"F1$\"1<3#yLFN`\"F47$$!1K$e9hx$\\6F 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7$F]x$\"15#RsDxqk&F/7$Fbx$\"19Vhf'fgp&F/7$Fgx$\"1x8fys.PdF/7$F\\y$\"1# eTrAA$odF/7$Fay$\"11U$zz>qz&F/7$Ffy$\"1:)yMk(\\;eF/7$F[z$\"1#*om&*p6Je F/7$F`z$\"1[Ua'z?%ReF/Fdz-Fhz6&FjzF*F*F]dl-%+AXESLABELSG6$Q\"x6\"%!G-% *AXESTICKSG6$7%/F)%*-infinityG/F*%\"0G/Ffz%)infinityGF\\^m-%%VIEWG6$;F )Ffz%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "series(f(a,x),x=-infinity,2 ),series(f(a,x),x=+infinity,2);" }}{PARA 0 "" 0 "" {TEXT -1 89 "On lit ainsi les deux asymptotes et la position de la courbe par rapport aau x asymptotes:" }}{PARA 0 "" 0 "" {TEXT -1 58 "pour x tendant vers +in fini, l'asymptote a pour \351quation " }{XPPEDIT 18 0 "y = -2*a*exp(Pi /2)*x-exp(pi/2);" "6#/%\"yG,&**\"\"#\"\"\"%\"aGF(-%$expG6#*&%#PiGF(\" \"#!\"\"F(%\"xGF(F0-F+6#*&%#piGF(\"\"#F0F0" }}{PARA 0 "" 0 "" {TEXT -1 61 "et pour x tendant vers -infini, l'asymptote a pour \351quation " }{XPPEDIT 18 0 "y = 2*a*exp(Pi/2)*x+exp(pi/2);" "6#/%\"yG,&**\"\"# \"\"\"%\"aGF(-%$expG6#*&%#PiGF(\"\"#!\"\"F(%\"xGF(F(-F+6#*&%#piGF(\"\" #F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,(*(%\"aG\"\"\"-%$expG6#,$%#P iG#F&\"\"#F&%\"xGF&!\"#F'!\"\"-%\"OG6#,$*&\"\"\"F6F.!\"\"F0F&,(F$F-F'F &-F26#F5F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 312 11 "Exercice 87" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "restart:eq:=diff(y(x),x)-y(x)*tan(x)+cos(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG,(-%%diffG6$-%\"yG6#%\"xGF,\"\" \"*&F)F--%$tanGF+F-!\"\"-%$cosGF+F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Evidemment on r\351soud s\351par\351ment sur chaque intervalle \+ ouvert du type :" }}{PARA 0 "" 0 "" {TEXT -1 6 "I_k=] " }{XPPEDIT 18 0 "-Pi/2+k*Pi,Pi/2+k*Pi;" "6$,&*&%#PiG\"\"\"\"\"#!\"\"F(*&%\"kGF&F%F&F &,&*&F%F&\"\"#F(F&*&F*F&F%F&F&" }{TEXT -1 3 " [." }}{PARA 0 "" 0 "" {TEXT -1 52 "On d\351termine la solution sur l'intervalle ouvert ] " }{XPPEDIT 18 0 "-Pi/2,pi/2;" "6$,$*&%#PiG\"\"\"\"\"#!\"\"F(*&%#piGF&\" \"#F(" }{TEXT -1 23 " [ qui s'annule en 0 :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dsolve(\{eq,y(0)=0\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&,&-%$sinG6#,$F'\"\"#!\"\"F'!\"#\"\"\" -%$cosGF&!\"\"#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "On \+ r\351soud l'\351quation sur chaque I_k :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "S:=dsolve(eq,y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assign(S):y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&,(-%$sinG6#,$%\"xG\"\"#!\"\"F*!\"#%$_C1G\"\"%\"\"\"-%$cosG6#F*!\"\"# \"\"\"F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Chaque solution sur ] [ est du type pr\351c\351dent. pou r pouvoir la prolonger en " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\" \"\"#!\"\"" }{TEXT -1 47 " , il faut qu'elle ait une limite finie en \+ " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "limit(numer(y(x)),x=Pi/2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&%#PiG!\"\"%$_C1G\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Le d\351nominateur tend vers 0 en . donc, pour qu'il y a it limite finie en " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\" \"" }{TEXT -1 41 " , il faut et cela ne suffit pas que " } {XPPEDIT 18 0 "_C1 = Pi/4;" "6#/%$_C1G*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 3 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y:=x->1/4*( -sin(2*x)- 2*x+Pi)/cos(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGR6#%\"xG6\"6$ %)operatorG%&arrowGF(,$*&,(-%$sinG6#,$9$\"\"#!\"\"F3!\"#%#PiG\"\"\"\" \"\"-%$cosG6#F3!\"\"#F8\"\"%F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "limit(y(x),x=Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Mais en " }{XPPEDIT 18 0 "-Pi/2;" "6#,$*&%#PiG\"\"\"\"\"#!\"\"F(" }{TEXT -1 11 " , on a \+ :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(y(x),x=-Pi/2);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Ainsi, il n'y a pas de solution sur R. Mais, on peut trouver une solution sur ] " }{XPPEDIT 18 0 "-Pi /2,3*Pi/2;" "6$,$*&%#PiG\"\"\"\"\"#!\"\"F(*(\"\"$F&F%F&\"\"#F(" } {TEXT -1 19 " [ en posant y( " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\" \"\"\"\"#!\"\"" }{TEXT -1 38 ") =0 et quel que soit x diff\351rent de \+ " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 11 " \+ , y(x)=" }{XPPEDIT 18 0 "(-sin(2*x)-2*x+Pi)/(4*cos(x));" "6#*&,(-%$s inG6#*&\"\"#\"\"\"%\"xGF*!\"\"*&\"\"#F*F+F*F,%#PiGF*F**&\"\"%F*-%$cosG 6#F+F*F," }{MPLTEXT 1 0 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*unde finedG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 310 0 "" } {TEXT 311 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 301 0 "" }{TEXT 302 0 "" }{TEXT 303 11 "Exercice 88" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P:=x->x^3+a*x^2+a*x+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGR6 #%\"xG6\"6$%)operatorG%&arrowGF(,**$)9$\"\"$\"\"\"\"\"\"*&%\"aGF2)F/\" \"#F1F2*&F4F1F/F2F2F2F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 " On utilise les relations reliant les racines " }{XPPEDIT 18 0 "x_i,i \+ = 1 .. 4;" "6$%$x_iG/%\"iG;\"\"\"\"\"%" }{TEXT -1 22 " et les coeffici ents :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "eq:=\{x.1+x.2=1,x.1+x.2+x .3=-a,x.1*x.2+x.1*x.3+x.2*x.3=a,x.1*x.2*x.3=-1\}:inc:=\{seq(x.i,i=1..3 ),a\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "S:=solve(eq,inc); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG6%<&/%#x2G\"\"#/%#x3G#\"\"\" F)/%\"aG#!\"$F)/%#x1G!\"\"<&/F(F4F*F./F3F)<&/F+F4/F(-%'RootOfG6#,(*$)% #_ZGF)\"\"\"F-FAF4F-F-/F/\"\"!/F3,&F;F4F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a:=0:factor(P(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"x G\"\"\"F&F&F&,(*$)F%\"\"#\"\"\"F&F%!\"\"F&F&F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:=-3/2: factor(P(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\"xG\"\"\"!\"# F'F',&F&\"\"#!\"\"F'F',&F&F'F'F'F'#F'F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 329 0 "" }{TEXT 330 0 "" }{TEXT 331 11 "Exercice 89" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "restart:f:=x->(a*x^4+b*x^2+c)/((x-1)^3*(x+2)^5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&,(*&%\"aG \"\"\")9$\"\"%\"\"\"F0*&%\"bGF0)F2\"\"#F4F0%\"cGF0F4*&),&F2F0!\"\"F0\" \"$F4),&F2F0F8F0\"\"&F4!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "On convertit la fraction en \351l\351ments simples." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "convert(f(x),parfrac,x);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#,2*&,(%\"aG\"\"$%\"cG\"\"&%\"bG!\"#\"\"\",&%\"xG\"\" \"!\"\"F/!\"\"#F/\"$H(*&,(F*F/F&\"\"(F(!\"&F,*$)F-\"\"#F,F1F2*&,(F*F/F &F/F(F/F,*$)F-\"\"$F,F1#F/\"$V#*&F%F,,&F.F/\"\"#F/F1#F0F3*&,(F*F7F&\"# ;F(\"#5F,*$)FC\"\"#F,F1FE*&,(F*F0F&\"\")F(FDF,*$)FC\"\"$F,F1#F0\"#\")* &,&F&FHF(F0F,*$)FC\"\"%F,F1#F/\"#F*&,(F*\"\"%F&FHF(F/F,*$)FC\"\"&F,F1# F0Fen" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "La condition demand\351e est donc : -2b+3a+5c=0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 11 "Exercice 90" }}{PARA 0 "" 0 "" {TEXT -1 173 "On \351crit l'\351quation d'une sph\350re de centre A(a,b,c), \+ de rayon |c|, ce qui exprime qu'elle est tangente au plan z=0. Et on e xprime qu'elle passe par les trois points donn\351s :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "restart:f:=(x,y,z)->(x-a)^2+(y-b)^2+(z-c)^2-c^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6%%\"xG%\"yG%\"zG6\"6$%)ope ratorG%&arrowGF*,**$),&9$\"\"\"%\"aG!\"\"\"\"#\"\"\"F3*$),&9%F3%\"bGF5 F6F7F3*$),&9&F3%\"cGF5F6F7F3*$)FAF6F7F5F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eq:=\{f(1,-2,1),f(-3,1,2),f(2,2,2)\}:inc:=\{a,b, c\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "S:=solve(eq,inc):as sign(S);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a,b,c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,&#!\"\"\"\"&\"\"\"-%'RootOfG6#,(%#_ZG\"$-$* $)F,\"\"#\"\"\"\"#EF'F'F$F(,&#\"#;F&F'F(#!#>F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "allvalues(a);allvalues(b); allvalues(c);" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"# D\"#E\"\"\"*$-%%sqrtG6#\"$6*\"\"\"#!\"\"F&,&F$F'F(#F'F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$,&#!$^\"\"#E\"\"\"*$-%%sqrtG6#\"$6*\"\"\"#\"\"&F &,&F$F'F(#!\"&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"$d'\"#E\"\"\" *$-%%sqrtG6#\"$6*\"\"\"#!#>F&,&F$F'F(#\"#>F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "a:=25/26-1/26*sqrt(911):b:=-151/26+5/26*sqrt(91 1):c:=657/26-19/26*sqrt(911):f(x,y,z);expand(f(1,-2,1)),expand(f(-3,1, 2)),expand(f(2,2,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$),(%\"xG \"\"\"#!#D\"#EF(*$-%%sqrtG6#\"$6*\"\"\"#F(F+\"\"#F1F(*$),(%\"yGF(#\"$^ \"F+F(F,#!\"&F+F3F1F(*$),(%\"zGF(#!$d'F+F(F,#\"#>F+F3F1F(*$),&#\"$d'F+ F(F,#!#>F+F3F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!F#F#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "a:=25/26+1/26*sqrt(911):b:= -151/26-5/26*sqrt(911):c:=657/26+19/26*sqrt(911):f(x,y,z);expand(f(1,- 2,1)),expand(f(-3,1,2)),expand(f(2,2,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$),(%\"xG\"\"\"#!#D\"#EF(*$-%%sqrtG6#\"$6*\"\"\"#!\" \"F+\"\"#F1F(*$),(%\"yGF(#\"$^\"F+F(F,#\"\"&F+F4F1F(*$),(%\"zGF(#!$d'F +F(F,#!#>F+F4F1F(*$),&#\"$d'F+F(F,#\"#>F+F4F1F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "On t este les diff\351rentes valeurs de a,b,c et on s'aper\347oit qu'il exi ste deux sph\350res solutions, correspondant \340 a=" }{XPPEDIT 18 0 " 25/26-1/26*sqrt(911);" "6#,&*&\"#D\"\"\"\"#E!\"\"F&*(\"\"\"F&\"#EF(-%% sqrtG6#\"$6*F&F(" }}{PARA 0 "" 0 "" {TEXT -1 4 "b= " }{XPPEDIT 18 0 " -151/26+5/26*sqrt(911);" "6#,&*&\"$^\"\"\"\"\"#E!\"\"F(*(\"\"&F&\"#EF( -%%sqrtG6#\"$6*F&F&" }{TEXT -1 11 " , c= " }{XPPEDIT 18 0 "657/26 -19/26*sqrt(911);" "6#,&*&\"$d'\"\"\"\"#E!\"\"F&*(\"#>F&\"#EF(-%%sqrtG 6#\"$6*F&F(" }{TEXT -1 11 " et a= " }{XPPEDIT 18 0 "25/26+1/26*sqr t(911);" "6#,&*&\"#D\"\"\"\"#E!\"\"F&*(\"\"\"F&\"#EF(-%%sqrtG6#\"$6*F& F&" }{TEXT -1 8 " , b= " }{XPPEDIT 18 0 "-151/26-5/26*sqrt(911);" "6 #,&*&\"$^\"\"\"\"\"#E!\"\"F(*(\"\"&F&\"#EF(-%%sqrtG6#\"$6*F&F(" } {TEXT -1 9 " , c= " }{XPPEDIT 18 0 "657/26+19/26*sqrt(911);" "6#,&* &\"$d'\"\"\"\"#E!\"\"F&*(\"#>F&\"#EF(-%%sqrtG6#\"$6*F&F&" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 314 0 "" }{TEXT 315 0 "" }{TEXT 316 11 "Exercice 91" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "resta rt:A:=(n,t)->t^n+1/t^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGR6$% \"nG%\"tG6\"6$%)operatorG%&arrowGF),&)9%9$\"\"\"*&\"\"\"F3F.!\"\"F1F)F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "simplify(A(n,t)-A(1, t)*A(n-1,t)+A(n-2,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "A(0,t),A(1,t);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"\"#,&%\"tG\"\"\"*&\"\"\"F(F%!\"\"F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "P:=proc(n)\noption remember; if n=0 then 2 elif n=1 then x else expand(P(n-1)*P(1)-P(n-2)) fi\nend; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGR6#%\"nG6\"6#%)rememberGF(@' /9$\"\"!\"\"#/F-\"\"\"%\"xG-%'expandG6#,&*&-F$6#,&F-F1!\"\"F1F1-F$6#F1 F1F1-F$6#,&F-F1!\"#F1F;F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "P(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "P(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &*$)%\"xG\"\"$\"\"\"\"\"\"F&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "L:=NULL:n:=10:for k to n do\nP.k:=unapply(P(k),x):L:= L,combine(P.k(2*cos(x)),trig) od :L;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6,,$-%$cosG6#%\"xG\"\"#,$-F%6#,$F'F(F(,$-F%6#,$F'\"\"$F(,$-F%6#,$F'\" \"%F(,$-F%6#,$F'\"\"&F(,$-F%6#,$F'\"\"'F(,$-F%6#,$F'\"\"(F(,$-F%6#,$F' \"\")F(,$-F%6#,$F'\"\"*F(,$-F%6#,$F'\"#5F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "for k to n do cos(k*x)=1/2* P.k(2*cos(x)) od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"xGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$%\"xG\"\"#,&*$)-F%6#F(F)\"\"\"F)!\"\"\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$%\"xG\"\"$,&*$)-F%6#F( F)\"\"\"\"\"%F-!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$%\" xG\"\"%,(*$)-F%6#F(F)\"\"\"\"\")*$)F-\"\"#F/!\")\"\"\"F5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$cosG6#,$%\"xG\"\"&,(*$)-F%6#F(F)\"\"\"\"#;*$ )F-\"\"$F/!#?F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$%\"xG \"\"',**$)-F%6#F(F)\"\"\"\"#K*$)F-\"\"%F/!#[*$)F-\"\"#F/\"#=!\"\"\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$%\"xG\"\"(,**$)-F%6# F(F)\"\"\"\"#k*$)F-\"\"&F/!$7\"*$)F-\"\"$F/\"#cF-!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$%\"xG\"\"),,*$)-F%6#F(F)\"\"\"\"$G\"*$ )F-\"\"'F/!$c#*$)F-\"\"%F/\"$g\"*$)F-\"\"#F/!#K\"\"\"F=" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$cosG6#,$%\"xG\"\"*,,*$)-F%6#F(F)\"\"\"\"$c#* $)F-\"\"(F/!$w&*$)F-\"\"&F/\"$K%*$)F-\"\"$F/!$?\"F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$%\"xG\"#5,.*$)-F%6#F(F)\"\"\"\"$7&*$)F -\"\")F/!%!G\"*$)F-\"\"'F/\"%?6*$)F-\"\"%F/!$+%*$)F-\"\"#F/\"#]!\"\"\" \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 317 0 "" }{TEXT 318 0 "" }{TEXT 319 11 "Exercice 92" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "restart:f:=x->(a*x^5+x^4+b*x^3+c*x^ 2+x+d)/(x*(x^2+1)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"x G6\"6$%)operatorG%&arrowGF(*&,.*&%\"aG\"\"\")9$\"\"&\"\"\"F0*$)F2\"\"% F4F0*&%\"bGF0)F2\"\"$F4F0*&%\"cGF0)F2\"\"#F4F0F2F0%\"dGF0F4*&F2\"\"\") ,&*$F>F4F0F0F0\"\"#F4!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(f(x),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,2*&% \"aG\"\"\"%\"xGF&F&*&%\"dGF&-%#lnG6#F'F&F&-F+6#,&*$)F'\"\"#\"\"\"F&F&F &#F&F2*&F-F&F)F3#!\"\"F2*&-%'arctanGF,F&%\"bGF&F4*&F9F3F%F3#!\"$F2*&,* *&,(F%F2F;!\"#F2F&F&F'F3F&F2F&F)F2%\"cGFCF3F/!\"\"#F&\"\"%F9F4" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Il y a vraisemblablement erreur d' \351nonc\351." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 320 0 "" }{TEXT 321 0 "" }{TEXT 322 11 "Exercice 93" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "restart:P:=x->x^4+a*x^3+sqrt(3)*x^2+b*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGR6#%\"xG6\"6$%)operatorG%&arrowGF(,**$)9$ \"\"%\"\"\"\"\"\"*&%\"aGF2)F/\"\"$F1F2*&-%%sqrtG6#F6F2)F/\"\"#F1F2*&% \"bGF2F/F2F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P(1+2* I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,!\"(\"\"\"%\"IG!#C*&,&!#6F%F& !\"#F%%\"aGF%F%*&,&!\"$F%F&\"\"%F%-%%sqrtG6#\"\"$\"\"\"F%*&,&F%F%F&\" \"#F%%\"bGF%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "eq:=\{eva lc(Re(P(1+2*I))),evalc(Im(P(1+2*I)))\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG<$,*!\"(\"\"\"*$-%%sqrtG6#\"\"$\"\"\"!\"$%\"aG!#6%\"bGF(,* !#CF(F)\"\"%F0!\"#F2\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "inc:=\{a,b\}:S:=solve(eq,inc):assign(S):a;b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"#F%*$-%%sqrtG6#\"\"$\"\"\"#!\"\"F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"#D\"\"#\"\"\"*$-%%sqrtG6#\"\"$\" \"\"#!\"&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(P(x),x );" }}{PARA 0 "" 0 "" {TEXT -1 86 "Evidemment, on retrouve que 0 est r acine et que le conjugu\351 de 1+2i est aussi racine !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!,&\"\"\"F%%\"IG\"\"#,&F%F%F&!\"#,&*$-%%sqrtG6 #\"\"$\"\"\"#F%F'#!\"&F'F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 0 "" }{TEXT 269 0 "" }{TEXT 270 11 "Exercice 94" }}{PARA 0 "" 0 "" {TEXT -1 137 "On r\351soud le s yst\350me form\351 par les \351quations de la drote D_t et de la droit e D'_t, obtenue en d\351rivant par rapport \340 t les coefficients :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "restart:f:=t->x*sin(t)-y*cos(t)-( sin(t))^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "inc:=\{x,y\}: eq:=\{f(t),diff(f(t),t)\}:S:=solve(eq,inc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "assign(S):x,y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ ,$*&-%$sinG6#%\"tG\"\"\",&!\"#F)*$)F%\"\"#\"\"\"F)F)!\"\"*&F-F/-%$cosG F'F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "Nous avons ainsi une rep r\351sentation param\351trique de l'enveloppe, qui est visiblement sym \351trique par rapport \340 Ox,Oy et donc O. On trace l'enveloppe :" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot([x,y,t=0..2*Pi],title=`Envelo ppe`);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CUR VESG6$7eq7$\"\"!F(7$$\"1Gl8!#;$\"1b>?P%34n%!#=7$$\"1&H&>\"z2^q#F, $\"1z\\E21`Y=!#<7$$\"1Vy;iY.JQF,$\"1\"Ge[O<)RPF57$$\"1p-weF+/\\F,$\"1n eTAP@3iF57$$\"1&)4oZC\\IgF,$\"1()Qw_`)3c*F57$$\"1XU$Q:_h0(F,$\"1,i]1Io P8F,7$$\"1FF_0gRtzF,$\"1AQJ[)>4v\"F,7$$\"1lL>8I]l()F,$\"1(>Jm+HS<#F,7$ $\"1nr>UtEC%*F,$\"1rAyI[&\\e#F,7$$\"1zsX#4!o_**F,$\"1Cc/mm*e'HF,7$$\"1 1MoD.F c\"y5F[o$\"1:XW2C%4s$F,7$$\"1usM1D5(3\"F[o$\"1y=G(fE#GQF,7$$\"1<.%>2 eOs5F[o$\"1k$R5-Z'eNF,7$$\"1n+aJI-f5F[o$\"1)=4*H)[EE$F,7$$\"1\\hSr(pR/ \"F[o$\"14#oi'[beGF,7$$\"1sNL'R&HH5F[o$\"1]#)4^yKkBF,7$$\"1Fk(H(4!z,\" F[o$\"1+&o;6(*e'=F,7$$\"1WvnC7%)35F[o$\"1*yV,,)z?8F,7$$\"1L&GSLRA+\"F[ o$\"1Y9Ui?+\"o'F57$$\"1#Ho#y,++5F[o$!1/m8Xw1rf!#>7$$\"18_&[&yL-5F[o$!1 ?uY]#ye#oF57$$\"1E!>,?l!45F[o$!1e\\_1)*=P8F,7$$\"1TM+TL9>5F[o$!1@%3V21 w#>F,7$$\"1P]vp*z=.\"F[o$!1nqfn!*)3Y#F,7$$\"1AW%eK#zW5F[o$!1AX4Ks#H)GF ,7$$\"1')H?+w#z0\"F[o$!1x9\")**e&fB$F,7$$\"1\"*3:iYBs5F[o$!1_?)z*p)fb$ F,7$$\"1F#33JIJ3\"F[o$!1G&eLZ4.w$F,7$$\"1-x)3;!>'3\"F[o$!1qw9*op5\"QF, 7$$\"1Li>G0%QF,7$$\"1gm%3%[l)3\"F[o$!149ca%**)[QF,7 $$\"1-hn'fSx3\"F[o$!1-:rI&fl$QF,7$$\"1B)[m#o.z5F[o$!17%e))zt4t$F,7$$\" 19w`*GQ(f5F[o$!1^>$e$*Qj_$F,7$$\"1#eDcNGE.\"F[o$!1&QGxp0OF$F,7$$\"1Eya \\St_**F,$!1pnp\"[Qf'HF,7$$\"1r[Xw;z=%*F,$!112)f%QF\"e#F,7$$\"1!)4=3,0 _()F,$!1Rx'))RZi;#F,7$$\"1dEzeqt%*zF,$!1Jp\")>9Xh8F,$!1G+KI*e:O%F/7$$!1UjX_FlvEF,$!1*oi\")flh!=F57$$!1D.'HB Kf\"QF,$!1f2788\")4PF57$$!1aSv([%=-\\F,$!18)en?`M?'F57$$!1;ZpaiS)*fF,$ !1]<&[(ez`%*F57$$!1?6,>P!)**pF,$!1I0=)oRYJ\"F,7$$!1guj/Ou@zF,$!1at'z39 cs\"F,7$$!1g^xP3l>()F,$!1ut4NzfZ@F,7$$!10$e!Gy#[P*F,$!1203!H'*=b#F,7$$ !1(zsqZFY!**F,$!1X0nM6#*GHF,7$$!1qj([UD+.\"F[o$!19X_3;%3D$F,7$$!1IG%ow S'e5F[o$!1M(>&zKX:NF,7$$!1#Hx)QGiy5F[o$!1Lz%=fYis$F,7$$!1o&)*HSuw3\"F[ o$!1-[ABqoNQF,7$$!1r$y]hd')3\"F[o$!1S^\\m%Q*[QF,7$$!1!)*HK$*zz3\"F[o$! 10BoL=%*QQF,7$$!1**RYa)Ge3\"F[o$!1U[N%*)f_!QF,7$$!17;?rnR#3\"F[o$!1AS! o5fwu$F,7$$!1'f9r?#or5F[o$!1d3)zmGZa$F,7$$!1\\\"\\o&*>z0\"F[o$!1e\\%)) GodB$F,7$$!1J145tNW5F[o$!1ekio[2qGF,7$$!1![$\\V*p5.\"F[o$!1Z)*pPB9JCF, 7$$!1#\\Y%3\"y\"=5F[o$!1FHl68')z=F,7$$!1Sw][NdIF/7$$!1i\\xJV(> +\"F[o$\"1Q^s1G]uiF57$$!1G'>]V=\"35F[o$\"1^[:D)pjE\"F,7$$!1Tf*z9nw,\"F [o$\"12XxNH-a=F,7$$!1=]7rM#*H5F[o$\"1M#\\'4AC)Q#F,7$$!1G#fyN(\\W5F[o$ \"17nu\"f@U(GF,7$$!1;\"R`j'Qf5F[o$\"1&e'QsRUrKF,7$$!11Jhz;Ws5F[o$\"1_0 \"*[P=gNF,7$$!1;N*4KzD3\"F[o$\"1fWf4N#3v$F,7$$!1E2A`-4'3\"F[o$\"1eN8E' o%4QF,7$$!1[ZTC?<)3\"F[o$\"1ATqI:#=%QF,7$$!1W$oIG.')3\"F[o$\"1mOiY(z\" [QF,7$$!1(>t%H*or3\"F[o$\"1Z)z1a'3HQF,7$$!147;ZN+y5F[o$\"1-f'*[9@>PF,7 $$!1,BdieAf5F[o$\"1$GhxI[7_$F,7$$!1#*\\G)eIC.\"F[o$\"1\\hS0\"o=F$F,7$$ !1Fc!*)pns&**F,$\"1,ShzVXpHF,7$$!1HsC:mL3%*F,$\"1x;EC\"fUd#F,7$$!1%=SH ->$>()F,$\"1MI>-wSZ@F,7$$!15B_'z?a)zF,$\"1m*)p[%[ov\"F,7$$!1_[8Rm#[9(F ,$\"1(o(QjeYu8F,7$$!1SbtNaQRhF,$\"1(4jF\"GkH**F57$$!1#*)R#*HaN.&F,$\"1 .7lm2i_lF57$$!1I&pB\"eq&*QF,$\"1FxBx+wpQF57$$!1AcASjs'p#F,$\"1X?&\\,)* \\$=F57$$!1X5=(*4(4O\"F,$\"1a%=.%*)QTYF/7$$\"1\\E7Gu#3k\"!#C$\"1p*)>Am yIn!#M-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%&TITLEG6#%*EnveloppeG-%+AXESLA BELSG6$%!GFa]m-%%VIEWG6$%(DEFAULTGFe]m" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "On m\350 ne l'\351tude demand\351e :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "x:=u napply(x,t):y:=unapply(y,t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "factor(simplify(diff(x(t),t))),factor(simplify(diff(y(t),t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&-%$cosG6#%\"tG\"\"\",&!\"\"F(*$)F$ \"\"#\"\"\"\"\"$F(*&-%$sinGF&F(F)F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "Les points stationnaires correspondent aux param\350tres t ann ulant \340 la fois x' et y'. On obtient le point stationnaire dd coord onn\351es positives, les trois autres autres s'obtiennent par les sym \351tries annonc\351es : " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x(arcc os(1/sqrt(3))),y(arccos(1/sqrt(3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$,$*$-%%sqrtG6#\"\"'\"\"\"#\"\"%\"\"*,$*$-F&6#\"\"$F)#\"\"#F," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 323 0 "" }{TEXT 324 0 "" }{TEXT 325 11 "Exercice 95" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 99 "A:=matrix(3,3,[1-2*a*u,-2*a*v,-2*a*w,-2*b*u,1- 2*b*v,-2*b*w,-2*c*u,-2*c*v,1-2*c*w]);Id:=diag(1,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%,&\"\"\"F+*&%\"aGF+%\"uGF+! \"#,$*&F-\"\"\"%\"vGF+F/,$*&F-F2%\"wGF+F/7%,$*&%\"bGF+F.F2F/,&F+F+*&F: F2F3F2F/,$*&F:F2F6F2F/7%,$*&%\"cGF+F.F2F/,$*&FBF2F3F2F/,&F+F+*&FBF2F6F 2F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IdG-%'matrixG6#7%7%\"\"\"\" \"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 325 "Bien s \373r, on suppose que a,b,c ne sont pas tous nuls, de m\352me que u,v, w. Dans ce cas, sans calcul, on voit que 1 est valeur propre et que le SEP associ\351 est le plan ux+vy+wz=0. Or la trace de A vaut 3-2(au+b v+cw)=1. Comme 1 est au moins valeur propre double, car dimSEP(1)=2, l a troisi\350me racine du polyn\364me caract\351ristique " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 26 " v\351rifie Tr(A)= 1= 1+1 \+ + " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 10 " et donc " } {XPPEDIT 18 0 "lambda = -1;" "6#/%'lambdaG,$\"\"\"!\"\"" }{TEXT -1 310 " . Ce qui signifie que A a deux valeurs propres r\351elles 1, d' ordre de multiplicit\351 2, le sous-espace propre associ\351 \351tant \+ le plan ux+vy+wz=0 l'autre valeur propre est -1, simple, le sous-esp ace propre associ\351 \351tant la droite ker(A+Id) d'\351quations - si elles sont bien ind\351pendantes- (2-2au)x -2avy-2awz=0 ; " }}{PARA 0 "" 0 "" {TEXT -1 51 " -2bu x +(2-2bv)y -2bwz=0, dirig\351e par le ve cteur :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "A1:=crossprod(vector(3, [2-2*a*u,-2*a*v,-2*a*w]),vector(3,[-2*b*u,2-2*b*v,-2*b*w]));map(simpli fy,A1,\{a*u+b*v+c*w=1\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%' vectorG6#7%,&**%\"aG\"\"\"%\"vGF,%\"bGF,%\"wGF,\"\"%*(F+\"\"\"F/F2,&\" \"#F,*&F.F2F-F2!\"#F,F4,&**F+F2F/F2F.F2%\"uGF,F0*(,&F4F,*&F+F2F9F2F6F, F.F2F/F2F4,&*&F;F2F3F2F,**F+F2F-F2F.F2F9F2!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$*&%\"aG\"\"\"%\"wGF*\"\"%,$*&%\"bGF*F+ \"\"\"F,,$*&%\"cGF*F+F0F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "Com me on voit, on suppose que w n'est pas nul. Si ce n'est pas le cas, c' est u ou v. Dans tous les cas, SEP(-1) est la droite dirig\351e par le vecteur A(a,b,c) ." }}{PARA 0 "" 0 "" {TEXT -1 142 "L'application ass oci\351e \340 A est la sym\351trie par rapport au plan SEP(1), d'\351q uation ux+vy+wz=0, parall\350lement \340 la droite dirig\351e par A(a ,b,c)." }}{PARA 0 "" 0 "" {TEXT -1 161 "Pour avoir une isom\351trie, n \351cessairement ces sous-espaces propres sont orthogonaux, ce qui \+ \351quivaut \340 ce que les vecteurs U(u,v,w) et A(a,b,c) sont colin \351aires." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 304 0 "" }{TEXT 305 0 "" }{TEXT 306 11 "Exercice \+ 96" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "Attention, d'une se ssion Maple \340 une autre, les r\351sultats affich\351s varient et la solution est donc \340 adapter en fonction." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warn ing, new definition for trace" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 " On suppose naturellement que a,b,c ne sont pas tous nuls." }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 59 "A:=matrix(3,3,[a-b-c,2*a,2*a,2*b,b-a-c,2*b,2 *c,2*c,c-a-b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7 %7%,(%\"aG\"\"\"%\"bG!\"\"%\"cGF.,$F+\"\"#F07%,$F-F1,(F-F,F+F.F/F.F37% ,$F/F1F6,(F/F,F+F.F-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v p:=eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vpG6$7%,(%\"aG !\"\"%\"bGF)%\"cGF)\"\"#<$-%'vectorG6#7%\"\"!\"\"\"F)-F/6#7%F3F2F)7%,( F(F3F*F3F+F3F3<#-F/6#7%*&F(\"\"\"F*!\"\"F3*&F+F>F*F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "Maple fait comme si a, ou b ou c, selon les se ssions n'est pas nul. En fait, il est facile de voir que le vecteur (a ,b,c) est propre pour a+b+c." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "P:= concat(vector([a,b,c]),vector([-1, 0, 1]), vector([-1, 1, 0]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7%7%%\"aG!\"\"F+7%% \"bG\"\"!\"\"\"7%%\"cGF/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG!\"\"%\"bGF%%\"cG F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "P est inversible, i.e il existe une base de vecteurs propres, \+ autrement dit A est diagonalisable si et seulement si det(P) n'est pas nul. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Donc " }{TEXT 339 68 "P est diag onalisable ssi elle est inversible, i.e ssi a+b+c non nul." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 340 1 " " }{TEXT -1 38 " On se place dans ce cas dans la suite." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " Delta:=evalm(inverse(P)&*A&*P);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&DeltaG-%'matrixG6#7%7%,(*&,(*&,(%\"aG\"\"\"%\"bG!\"\"%\"cGF2 \"\"\",(F/F0F1F0F3F0!\"\"F0*&F1F4F5F6\"\"#*&F3F4F5F6F8F0F/F0F0*&,(*&F/ F4F5F6F8*&,(F1F0F/F2F3F2F4F5F6F0F9F8F0F1F0F0*&,(FF0F4F5F6F2FNF8F0F1F4F0*&,(FSFDFLFD*& *&FPF4FBF0F4F5F6F0F0F3F4F0,*FJF0FNFDFSFDFYF0,*FJF0FLF8FSFDFUF27%,(*&,( *&*&F1F4F.F4F4F5F6F2*&*&,&F/F0F3F0F0F1F4F4F5F6F8FLFDF0F/F4F0*&,(*&*&F1 F4F/F4F4F5F6FD*&*&F_oF4F>F4F4F5F6F0FLFDF0F1F4F0*&,(FboFDF]oF8*&*&F1F4F BF4F4F5F6F2F0F3F4F0,*F[oF0FLF8FboFDFhoF2,*F[oF0F]oFDFboFDFdoF0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Delta:=map(simplify,Delta); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG-%'matrixG6#7%7%,(%\"aG\" \"\"%\"bGF,%\"cGF,\"\"!F/7%F/,(F+!\"\"F-F2F.F2F/7%F/F/F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Le calcul des puissances de A s'obtient a lors facilement : " }{TEXT -1 1 " " }{XPPEDIT 18 0 "A^n;" "6#)%\"AG%\" nG" }{TEXT -1 0 "" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "P*Delta^n*P^(-1) ;" "6#*(%\"PG\"\"\")%&DeltaG%\"nGF%)F$,$\"\"\"!\"\"F%" }{TEXT -1 2 " \+ " }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Delta_puis_n:=m ap(x->x^n,Delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-Delta_puis_nG- %'matrixG6#7%7%),(%\"aG\"\"\"%\"bGF-%\"cGF-%\"nG\"\"!F17%F1),(F,!\"\"F .F5F/F5F0F17%F1F1F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "A_pu is_n:=evalm(P&*Delta_puis_n&*inverse(P));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)A_puis_nG-%'matrixG6#7%7%,(*&*&%\"aG\"\"\"),(F-F.%\" bGF.%\"cGF.%\"nGF.\"\"\"F0!\"\"F.*&*&),(F-!\"\"F1F:F2F:F3F.F2F.F4F0F5F .*&*&F8F4F1F.F4F0F5F.,(F+F.F6F.*&*&F8F4,&F-F.F2F.F.F4F0F5F:,(F+F.*&*&F 8F4,&F-F.F1F.F.F4F0F5F:F;F.7%,&*&*&F1F4F/F4F4F0F5F.F;F:,&FGF.F>F.FF7%, &*&*&F2F4F/F4F4F0F5F.F6F:FK,&FLF.FBF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A_puis_n:=map(normal,A_puis_n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)A_puis_nG-%'matrixG6#7%7%*&,(*&%\"aG\"\"\"),(F-F.%\" bGF.%\"cGF.%\"nGF.F.*&),(F-!\"\"F1F7F2F7F3F.F2F.F.*&F5\"\"\"F1F.F.F9F0 !\"\"*&*&F-F9,&F/F.F5F7F.F9F0F:F;7%*&*&F1F9F=F9F9F0F:*&,(*&F1F9F/F9F.* &F5F9F-F9F.F4F.F9F0F:F?7%*&*&F2F9F=F9F9F0F:FF*&,(*&F2F9F/F9F.FDF.F8F.F 9F0F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 295 0 " " }{TEXT 296 0 "" }{TEXT 297 11 "Exercice 97" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart:eq:=x*diff(y(x),x)-(x-1)*y(x)-exp(x):S:=dsolv e(eq,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG/-%\"yG6#%\"xG,&- %$expGF(\"\"\"*&*&F+F-%$_C1GF-\"\"\"F)!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "assign(S):y(x);f:=unapply(y(x),(x,_C1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG\"\"\"*&*&F$F(%$_C1GF( \"\"\"F'!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%$_C1 G6\"6$%)operatorG%&arrowGF),&-%$expG6#9$\"\"\"*&*&F.F29%F2\"\"\"F1!\" \"F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{seq(f( x,k),k=-2..2)\},x=-5..5,y=-5..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6$7ao7$$!\"&\"\"!$\"1`'>()zDJV*!#=7$$ !1LLLe%G?y%!#:$\"1&3$zv\\L)=\"!#<7$$!1mmT&esBf%F1$\"1^eY0Q*RX\"F47$$!1 LL$3s%3zVF1$\"1]X))f.EE=F47$$!1ML$e/$QkTF1$\"1v*)=q_A+BF47$$!1nmT5=q]R F1$\"1oc()4R=)*GF47$$!1LL3_>f_PF1$\"1'o$RYc&ef$F47$$!1++vo1YZNF1$\"1&) y!*oGL.XF47$$!1LL3-OJNLF1$\"1YF/S(y_p&F47$$!1++v$*o%Q7$F1$\"1MP?t*4]@( F47$$!1mmm\"RFj!HF1$\"1bt#[Cx,B*F47$$!1LL$e4OZr#F1$\"1)ygO/)4]6!#;7$$! 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GFhs7$F[_l$\"1U[bUVanMFhs7$F`_l$\"1D#e(*3p#zUFhs7$Fe_l$\"1actMto\">&Fh s7$Fj_l$\"1`pQd-1zkFhs7$$\"1,+voTAqUF1$\"1qZ?Lqw`rFhs7$F_`l$\"1*G+MUN( )*yFhs7$$\"1M$eRA5\\Z%F1$\"1bS=\"*=ny()Fhs7$Fd`l$\"1JJP8_jc(*Fhs7$Ff^m $\"1Vn05aXz5F\\v7$Fi`l$\"1WfNl()G%>\"F\\v7$$\"1+D1k2/P[F1$\"1og%G_c4E \"F\\v7$F^_m$\"1qiQ " 0 "" {MPLTEXT 1 0 19 "se ries(f(x,k),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG%\"kG!\"\" ,&\"\"\"F(F%F(\"\"!,&F(F(F%#F(\"\"#\"\"\",&F+F(F%#F(\"\"'\"\"#,&F/F(F% #F(\"#C\"\"$,&F3F(F%#F(\"$?\"\"\"%-%\"OG6#F(\"\"&" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 80 "Il y a une seule fonction solution sur R, correspo ndant \340 k= 0 : la fonction y= " }{XPPEDIT 18 0 "e^x;" "6#)%\"eG%\"x G" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 292 0 "" }{TEXT 293 0 "" }{TEXT 294 11 "Exercice \+ 98" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "k1:=1/(sqrt(3))*vector(3,[1 ,-1,1]):i1:=1/(sqrt(2))*vector(3,[1,1,0]):j1:=crossprod(k1,i1):P:=conc at(i1,j1,k1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Aprim:=mat rix(3,3,[cos(Pi/3), -sin(Pi/3),0,sin(Pi/3),cos(Pi/3),0,0,0,1]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A:=evalm(P&*Aprim&*transpose (P));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%#\"\"# \"\"$#!\"#F,#!\"\"F,7%#\"\"\"F,F*F-7%F*F2F*" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 95 "On a construit une base orthonorm\351e privil\351gi\351 e dans laquelle la rotation s'exprime simplement." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 289 0 "" } {TEXT 290 0 "" }{TEXT 291 11 "Exercice 99" }}{PARA 0 "" 0 "" {TEXT -1 15 "cf exercice 89." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 286 0 "" }{TEXT 287 0 "" }{TEXT 288 12 "Exercice 100" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:wi th(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition fo r norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for tra ce" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A:=matrix(4,4,[0,0,0, 1,0,0,0,0,0,b,a,0,1,0,0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG -%'matrixG6#7&7&\"\"!F*F*\"\"\"7&F*F*F*F*7&F*%\"bG%\"aGF*7&F+F*F*F*" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "vp:=eigenvects(A);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vpG6&7%%\"aG\"\"\"<#-%'vectorG6#7& \"\"!F.F(F.7%F(F(<#-F+6#7&F(F.F.F(7%F.F(<#-F+6#7&F.,$*&F'\"\"\"%\"bG! \"\"!\"\"F(F.7%F>F(<#-F+6#7&F>F.F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P:=concat(seq(op( vp[k][3]), k=1..nops([vp])));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7&7&\"\"!\"\"\"F*!\" \"7&F*F*,$*&%\"aG\"\"\"%\"bG!\"\"F,F*7&F+F*F+F*7&F*F+F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"aG\"\"\"%\"bG!\"\"!\"#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 36 "Maple fait comme si b n'est pas nul." }}{PARA 0 "" 0 " " {TEXT 326 6 "Cas 1 " }{TEXT -1 13 ": b non nul :" }}{PARA 0 "" 0 "" {TEXT 327 7 "Cas 1-1" }{TEXT -1 26 " : b non nul et a non nul." }} {PARA 0 "" 0 "" {TEXT -1 142 "Le d\351terminant de P est non nul : il \+ existe une base de vecteurs propres et donc A est diagonalisable. On p roc\350de dans ce cas \340 la r\351duction :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Delta:=evalm(inverse(P)&*A&*P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG-%'matrixG6#7&7&\"\"!F*F*F*7&F*!\"\"F*F*7&F*F* \"\"\"F*7&F*F*F*%\"aG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 328 8 "Cas 1-2 \+ " }{TEXT -1 19 ": b non nul et a=0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a:=0:A1:=map(eval,A);eigenvects(A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'matrixG6#7&7&\"\"!F*F*\"\"\"7&F*F*F*F*7&F*%\"bGF*F*7&F+ F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%\"\"!\"\"#<#-%'vectorG6#7& F$F$\"\"\"F$7%F+F+<#-F(6#7&F+F$F$F+7%!\"\"F+<#-F(6#7&F2F$F$F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 223 "Comme on le voit, A n'est pas di agonalisable, car la valeur propre 0 est d'ordre de multiplicit\351 2, alors que la dimension du sous-espace propre associ\351 est \351gale \+ \340 1, donc strictement inf\351rieure \340 l'ordre de multiplicit\351 ." }}{PARA 0 "" 0 "" {TEXT 338 5 "Cas 2" }{TEXT -1 7 " : b=0." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "unassign('a'):a;b:=0:A2:=map(eval,A );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"aG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'matrixG6#7&7&\"\"!F*F*\"\"\"7&F*F*F*F*7&F*F*% \"aGF*7&F+F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vp2:=ei genvects(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vp2G6&7%\"\"\"F'<# -%'vectorG6#7&F'\"\"!F-F'7%F-F'<#-F*6#7&F-F'F-F-7%!\"\"F'<#-F*6#7&F4F- F-F'7%%\"aGF'<#-F*6#7&F-F-F'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "P:=concat(seq(op( vp2[k][3]), k=1..nops([vp])));det(P);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7&7&\"\"\"\"\"!!\"\" F+7&F+F*F+F+7&F+F+F+F*7&F*F+F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#! \"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Comme le d\351terminant de P n'est pas nul, il existe une base de vecteurs propres et A est diagonalisable." }}{PARA 262 "" 0 "" {TEXT -1 82 "Conclusion : Si b non nul, A est diagonalisable si et seu lement si a est non nul. " }}{PARA 263 "" 0 "" {TEXT -1 28 "Si b=0, Ae st diagonalisable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 332 0 "" }{TEXT 333 0 "" }{TEXT 334 12 "Exercice \+ 101" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "resta rt:u:=unapply(2*sqrt(n)-sum(1/sqrt(k),k=1..n),n):u(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$-%%sqrtG6#%\"nG\"\"\"\"\"#-%$sumG6$*&F)F)*$-F& 6#%\"kGF)!\"\"/F2;\"\"\"F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "v:=unapply(2*sqrt(n+1)-sum(1/sqrt(k),k=1..n),n):v(n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$-%%sqrtG6#,&%\"nG\"\"\"F*F*\"\"\" \"\"#-%$sumG6$*&F+F+*$-F&6#%\"kGF+!\"\"/F4;F*F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "limit(u(n)-v(n),n=infinity);deltau:=norm al(2*sqrt(n+1)-2*sqrt(n)-1/sqrt(n+1));deltav:=normal(2*sqrt(n+2)-2*sqr t(n+1)-1/sqrt(n+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'deltauG*&,(%\"nG\"\"#\"\"\"F)*&-%%s qrtG6#F'\"\"\"-F,6#,&F'F)F)F)F.!\"#F.*$-F,6#F1F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'deltavG,$*&,(*&-%%sqrtG6#,&%\"nG\"\"\"\"\"#F.\" \"\"-F*6#,&F-F.F.F.F0!\"#F-F/\"\"$F.F0*$-F*6#F3F0!\"\"!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "expand((2*n+1)^2-4*n*(n+1)); expand(4*(n+2)*(n+1)-(2*n+3)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{PARA 260 "" 1 "" {TEXT -1 110 "Ainsi, la suite (u_n) est croissante, la suite (v_n) est d\351croissante et (u_n-v_n) a pour limite 0 en +infiny." }}{PARA 0 " " 0 "" {TEXT -1 37 "Les deux suites sont bien adjacentes." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "m:=100:points_u:=pointplot(\{seq([l,u(l)],l=1. .m)\},color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "point s_v:=pointplot(\{seq([l,v(l)],l=1..m)\},color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(\{points_u,points_v\});" }}{PARA 13 "" 1 "" {GLPLOT2D 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l$\"+CrUG=F,7$Fgcl$\"+[J!)H:F,7$F`]l$\"+FSS8:F,7$F3$\"+kj*)p:F,7$F[jl$ \"+N[*pv\"F,7$Fd^l$\"+dn\"G^\"F,7$F[el$\"+zF9H:F,7$Fdfl$\"+Sm(y_\"F,7$ F`s$\"+/'etc\"F,7$F]cl$\"+G.`7:F,7$Fit$\"+]Ha:j&G;F,7$Fcgl$\"+]?5E:F,7$Fhgl$\"+S3(*Q;F, 7$FL$\"+'=&y>;F,7$Fbq$\"+Bl(ob\"F,7$Fbhl$\"+24aD:F,7$Fgq$\"+Qose:F,7$F ]hl$\"+yQ*\\_\"F,7$Fen$\"+!o^Ah\"F,7$Fhp$\"+Bo7b:F,7$Fail$\"+1/YC:F,7$ Fcp$\"+c*oMb\"F,7$Fjn$\"+')Rp0;F,7$Fghl$\"+A*RR_\"F,7$F^p$\"+e^*=b\"F, 7$Fjx$\"+r\"=**f\"F,7$Ffil$\"+7>VB:F,7$F\\il$\"+ue$H_\"F,7$Fcbl$\"+&Hr >^\"F,7$Fadl$\"+H%)p6:F,7$F\\dl$\"+T)H9^\"F,7$Fbcl$\"+Ga;6:F,7$Ffdl$\" +v]!4^\"F,7$Feel$\"+*p[1^\"F,7$F_fl$\"+whR5:F,7$Fifl$\"*SZ,^\"Fjjl-Fej l6&FgjlFhjlF)F)" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }} }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Maple conna\356t une expression \+ de la limite cherch\351e :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "l:=limit(u(n),n=+infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"lG,$-%%ZetaG6##\"\"\"\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4XNg9 !\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 335 0 "" }{TEXT 336 0 "" }{TEXT 337 12 "Exercice 102" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}{PARA 7 " " 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 " " {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=matrix(3,3,[1,a,b,1,c,d,1,e,f]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"%\"aG%\"bG7%F*% \"cG%\"dG7%F*%\"eG%\"fG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "On d \351finit la matrice P de passage de la base canonique de " } {XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 104 " \340 la base (u, v,w) ; on v\351rifie bien qu'il s'agit d'une base en montrant que le d \351terminant est non nul." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "u:=ve ctor(3,[1,1,1]):v:=vector(3,[1,0,-1]):w:=vector(3,[1,-1,1]):P:=concat( u,v,w);det(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7% 7%\"\"\"F*F*7%F*\"\"!!\"\"7%F*F-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# !\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Il faut que Delta= P^(-1 )AP soit une matrice diagonale, ce qui am\350ne \340 r\351soudre un \+ syst\350me o\371 les inconnues sont a,b,c,d,e,f." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Delta:=evalm(inverse(P)&*A&*P):Delta1:=map(eval,Delta );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Delta1G-%'matrixG6#7%7%,0\"\" \"F+%\"aG#F+\"\"%%\"cG#F+\"\"#%\"eGF-%\"bGF-%\"dGF0%\"fGF-,*F+F+F3#!\" \"F.F4#F8F1F5F7,0F+F+F,F7F/F9F2F7F3F-F4F0F5F-7%,*F,F0F2F9F3F0F5F9,&F3F 9F5F0,*F,F9F2F0F3F0F5F97%,.F,F-F/F9F2F-F3F-F4F9F5F-,(F3F7F4F0F5F7,.F,F 7F/F0F2F7F3F-F4F9F5F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "fo r i to 3 do Delta1[i,i]:=0 od:Delta1:=map(eval,Delta1);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'Delta1G-%'matrixG6#7%7%\"\"!,*\"\"\"F,%\"bG#! \"\"\"\"%%\"dG#F/\"\"#%\"fGF.,0F,F,%\"aGF.%\"cGF2%\"eGF.F-#F,F0F1#F,F3 F4F97%,*F6F:F8F2F-F:F4F2F*,*F6F2F8F:F-F:F4F27%,.F6F9F7F2F8F9F-F9F1F2F4 F9,(F-F.F1F:F4F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "eq:= \{seq(seq(Delta1[i,j],j=1..3),i=1..3)\}:inc:=\{a,b,c,d,e,f\}:S:=solve( eq,inc):assign(S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "'a'=a ,'b'=b,'c'=c,'d'=d,'e'=e,'f'=f;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(/% \"aG\"\"#/%\"bG\"\"\"/%\"cGF%/%\"dGF(/%\"eGF%/%\"fGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "On v\351rifie que pour ces valeurs, A est bien diagonalisable." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "A:=map(eva l,A);Delta:=map(eval,Delta);evalm(A-P&*Delta&*inverse(P));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"#F*F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG-%'matrixG6#7%7%\"\"%\"\"!F+7 %F+F+F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!F(F( F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 242 "Une autre m\351thode est d'exprimer que les vecteurs u et Au s ont colin\351aires, de m\352me v et Av, w et Aw. Pour exprimer que u e t Au sont colin\351aires, on exprimera que leur produit vectoriel est \+ nul. On passe ancore par la r\351solution d'un syst\350me." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "restart:with(linalg):A:=matrix(3,3,[1,a,b, 1,c,d,1,e,f]);u:=vector(3,[1,1,1]):v:=vector(3,[1,0,-1]):w:=vector(3,[ 1,-1,1]):C:=concat(crossprod(u,evalm(A&*u)),crossprod(v,evalm(A&*v)),c rossprod(w,evalm(A&*w)));" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new defi nition for trace" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6# 7%7%\"\"\"%\"aG%\"bG7%F*%\"cG%\"dG7%F*%\"eG%\"fG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7%7%,*%\"eG\"\"\"%\"fGF,%\"cG!\"\"%\" dGF/,&F,F,F0F/,,!\"#F,F+F,F-F/F.F,F0F/7%,*%\"aGF,%\"bGF,F+F/F-F/,(F3F, F7F,F-F,,*F6F/F7F,F+F,F-F/7%,*F.F,F0F,F6F/F7F/F1,,\"\"#F,F.F/F0F,F6F/F 7F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "inc:=\{a,b,c,d,e,f\} :eq:=\{seq(seq(C[i,j],j=1..3),i=1..3)\}:S:=solve(eq,inc):assign(S):map (eval,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\" \"#F(F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "P:=concat(u,v, w):evalm(inverse(P)&*A&*P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matr ixG6#7%7%\"\"%\"\"!F)7%F)F)F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 280 0 "" }{TEXT 281 0 "" } {TEXT 282 12 "Exercice 103" }}{PARA 0 "" 0 "" {TEXT -1 45 "Encore une \+ fois, on doit r\351soudre un syst\350me." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "restart:f:=x->sin(x)-(a*x+b*x^3+c*x^5)/(1+d*x^2+e*x^4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%& arrowGF(,&-%$sinG6#9$\"\"\"*&,(*&%\"aGF1F0F1F1*&%\"bGF1)F0\"\"$\"\"\"F 1*&%\"cGF1)F0\"\"&F:F1F:,(F1F1*&%\"dGF1)F0\"\"#F:F1*&%\"eGF1)F0\"\"%F: F1!\"\"!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s:=se ries(f(x),x=0,10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"sG+/%\"xG,& \"\"\"F(%\"aG!\"\"\"\"\",(#F*\"\"'F(%\"bGF**&F)F(%\"dGF(F(\"\"$,*%\"cG F**&F)\"\"\"%\"eGF(F(*&,&F/F*F0F(F(F1F6F*#F(\"$?\"F(\"\"&,(*&F9F6F7F6F **&,*F4F*F5F(*&F1F6F/F(F(*&F)F6)F1\"\"#F6F*F(F1F6F*#F*\"%S]F(\"\"(,(#F (\"'!)GOF(*&F@F6F7F6F**&,,*&F7F6F/F6F(*(F7F6F)F6F1F6!\"#*&F1F6F4F(F(*& FCF6F/F6F**&F)F6)F1\"\"$F6F(F(F1F6F*\"\"*-%\"OG6#F(\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p:=convert(s,polynom);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pG,,*&,&\"\"\"F(%\"aG!\"\"F(%\"xGF(F(*&,(#F *\"\"'F(%\"bGF**&F)F(%\"dGF(F(F()F+\"\"$\"\"\"F(*&,*%\"cGF**&F)F5%\"eG F(F(*&,&F0F*F1F(F(F2F5F*#F(\"$?\"F(F()F+\"\"&F5F(*&,(*&F " 0 "" {MPLTEXT 1 0 83 "eq:=\{seq(coeff(p,x^k),k=1..degree(p))\}: inc:=\{a,b,c,d,e\}:S:=solve(eq,inc):assign(S):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "'a'=a,'b'=b,'c'=c,'d'=d,'e'=e;'f(x)'=f(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'/%\"aG\"\"\"/%\"bG#!#`\"$'R/%\"cG#\"$^ &\"'?j;/%\"dG#\"#8F*/%\"eG#\"\"&\"&)36" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%$sinGF&\"\"\"*&,(F'F+*$)F'\"\"$\"\"\"#!#`\"$'R *$)F'\"\"&F1#\"$^&\"'?j;F1,(F+F+*$)F'\"\"#F1#\"#8F4*$)F'\"\"%F1#F7\"&) 36!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series(f(x) ,x=0,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"xG#!#6\"*+)GsX\"#6-% \"OG6#\"\"\"\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 283 0 "" }{TEXT 284 0 "" }{TEXT 285 12 "Ex ercice 104" }}{PARA 0 "" 0 "" {TEXT -1 58 "Comme le coefficient de y' \+ : x(x-1) ne s'annule pas sur ] " }{XPPEDIT 18 0 "1,infinity;" "6$\"\" \"%)infinityG" }{TEXT -1 86 " [, l'ensemble des solutions de l'\351qua tion diff\351rentielle propos\351e est du type y=f_0+ " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 136 " f_1, o\371 f_0 o\371 f_0 est une solution particuli\350re de l'\351quation et f_1 une solution non nulle de l'\351quation sans second membre associ\351e." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "restart:eq:=x*(x-1)*diff(y(x),x)+(x-1)*y(x)-x^ 2+1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "S:=dsolve(eq,y(x)); assign(S):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG/-%\"yG6#%\"xG,(F) #\"\"\"\"\"#F,F,*&%$_C1G\"\"\"F)!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y:=unapply(y(x),(_C1,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGR6$%$_C1G%\"xG6\"6$%)operatorG%&arrowGF),(9%#\"\" \"\"\"#F0F0*&9$\"\"\"F.!\"\"F0F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "a:=plot(y(0,x),x=-10..10,color=black):b:=plot(y(-10, x),x=-10..10,y=-10..10,color=red):c:=plot(y(10,x),x=-10..10,y=-10..10, color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(a,b,c);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$ $!#5\"\"!$!\"%F*7$$!1nmm;p0k&*!#:$!1LLLe%G?y$F07$$!1LL$3f_FF07$$!1,+]P8# \\4(F0$!1++vo1YZDF07$$!1mm;/siqmF0$!1LL3-OJNBF07$$!1****\\(y$pZiF0$!1+ +v$*o%Q7#F07$$!1LLL$yaE\"eF0$!1mmm\"RFj!>F07$$!1mmm\">s%HaF0$!1LL$e4OZ r\"F07$$!1******\\$*4)*\\F0$!1+++v'\\!*\\\"F07$$!1+++]_&\\c%F0$!1+++Dw Z#G\"F07$$!1+++]1aZTF0$!1+++D.xt5F07$$!1mm;/#)[oPF0$!1JL$3-TC%))!#;7$$ !1LLL$=exJ$F0$!1lmm;4z)e'Fcp7$$!1LLLL2$f$HF0$!1jmmmOlzYFcp7$$!1++]PYx \"\\#F0$!1****\\(=t)eCFcp7$$!1MLLL7i)4#F0$!1\"ommmh5$\\!#<7$$!1****\\P 'psm\"F0$\"12+]7=lj;Fcp7$$!1****\\74_c7F0$\"1/+]PaRY29F07$$\"1NLLe\"*[H7F0$\"1nm;zXu9;F07$$\"1++++dxd;F0$\"1+++ ]y))G=F07$$\"1,++D0xw?F0$\"1++]i_QQ?F07$$\"1,+]i&p@[#F0$\"1,+D\"y%3TAF 07$$\"1+++vgHKHF0$\"1++]P![hY#F07$$\"1lmmmZvOLF0$\"1LLL$Qx$oEF07$$\"1, ++]2goPF0$\"1+++v.I%)GF07$$\"1KL$eR<*fTF0$\"1mm\"zpe*zIF07$$\"1-++])Hx e%F0$\"1,++D\\'QH$F07$$\"1mm;H!o-*\\F0$\"1LLe9S8&\\$F07$$\"1,+]7k.6aF0 $\"1,+D1#=bq$F07$$\"1mmm;WTAeF0$\"1LLL3s?6RF07$$\"1****\\i!*3`iF0$\"1+ +DJXaETF07$$\"1NLLL*zym'F0$\"1ommm*RRL%F07$$\"1OLL3N1#4(F0$\"1om;a<.YX F07$$\"1pm;HYt7vF0$\"1NLe9tOcZF07$$\"1-+++xG**yF0$\"1,++]Qk\\\\F07$$\" 1qmmT6KU$)F0$\"1NL$3dg6<&F07$$\"1OLLLbdQ()F0$\"1ommmxGp`F07$$\"1++]i`1 h\"*F0$\"1++D\"oK0e&F07$$\"1-+]P?Wl&*F0$\"1,+v=5s#y&F07$$\"#5F*$\"\"'F *-%'COLOURG6&%$RGBGF*F*F*-F$6$7ao7$F($!\"$F*7$F.$!1p!)[gqWOFF07$F4$!1A @4^2h.DF07$F9$!1g))y6OHPAF07$F>$!1;,Q-]sj>F07$FC$!1G#[)))Q5&o\"F07$FH$ !1nl;l&z,U\"F07$FM$!1VT7s<+Q6F07$FR$!1yYPR>/i$)Fcp7$FW$!1!p!>aGcK_Fcp7 $Ffn$!1/HDp4Vf=Fcp7$F[o$\"1cJFubjq7Fcp7$F`o$\"1F`2T(3r,&Fcp7$Feo$\"1v> rHNC\"3*Fcp7$Fjo$\"1r&4E1(HP8F07$F_p$\"1XG=(*)R$pkhF07$F_r$\"1lK1)GA-L)F07$Fdr$\"1.@Rlht!3f%Fg_l7$$!1n;/^wj!p\"Fcp$\"1-VdB[Z1 gFg_l7$$!1MLeRx**f6Fcp$\"16`3lk!\\r)Fg_l7$$!1v;aQyxY*)Fhq$\"1Qn_'\\ts7 \"!#87$$!17+D\"GyNH'Fhq$\"1\")fCGlg)f\"Fial7$$!1!=/E]yp'\\Fhq$\"1/I_j! 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On recherche par des f ormules faciles \340 \351tablir, quand on connait son cours, les formu les donnant les coordonn\351es (Y(t),Y(t)) du centre de courbure I(t) \+ en O \340 la courbe (C_t)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "rest art:y:=x->t*x*exp(x);yprim:=unapply(diff(y(x),x),x);ysec:=unapply(diff (yprim(x),x),x);yprim(0):=yprim(0);ysec(0):=ysec(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGR6#%\"xG6\"6$%)operatorG%&arrowGF(*(%\"tG\"\" \"9$F.-%$expG6#F/F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yprimG R6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&%\"tG\"\"\"-%$expG6#9$F/F/*(F.\" \"\"F3F/F0F5F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ysecGR6#%\" xG6\"6$%)operatorG%&arrowGF(,&*&%\"tG\"\"\"-%$expG6#9$F/\"\"#*(F.\"\" \"F3F/F0F6F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&yprimG6#\"\" !%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%%ysecG6#\"\"!,$%\"tG\"\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "X:=t->-(yprim(0)/ysec(0 ))*(1+yprim(0)^2):X(t):=X(t);Y:=t->(1+yprim(0)^2)/ysec(0):Y(t):=Y(t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"XG6#%\"tG,&#!\"\"\"\"#\"\"\"* $)F'F+\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"YG6#%\"tG,$*&,& \"\"\"F+*$)F'\"\"#\"\"\"F+F/F'!\"\"#F+F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 224 "La courbe d\351crite par I(t) est sym\351trique par rapp ort \340 OX : son \351tude manuelle serait facile. 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On d\351finit la projection orthogonale sur R_3[X] p_de_f de la \+ fonction f propos\351e, en exprimant que f-p_de_F est orthogonale \340 R_3[X], donc \340 1,x,x^2,x^3, ce qui conduit \340 r\351soudre un sys t\350me." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ps:=unapply(int(f(t)*g(t),t= -1..1),(f,g));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#psGR6$%\"fG%\"gG6 \"6$%)operatorG%&arrowGF)-%$intG6$*&-9$6#%\"tG\"\"\"-9%F3F5/F4;!\"\"F5 F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "norme:=unapply(sq rt(ps(f,f)),f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&normeGR6#%\"fG6 \"6$%)operatorG%&arrowGF(*$-%%sqrtG6#-%$intG6$*$)-9$6#%\"tG\"\"#\"\"\" /F8;!\"\"\"\"\"F:F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " f:=x->12*x^7+5*x^6+x+1:p_de_f:=(a,b,c,d,x)->a+b*x+c*x^2+d*x^3:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "for k from 1 to 4 do e.k:=un apply(x^(k-1),x) od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "g:= unapply(f(x)-p_de_f(a,b,c,d,x),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eq:=\{seq(ps(g,e.k),k=1..4)\}:inc:=\{a,b,c,d\}:S:=sol ve(eq,inc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "assign(S):'a '=a,'b'=b,'c'=c,'d'=d;'p_de_f(x)'=p_de_f(a,b,c,d,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/%\"aG#\"#6\"#@/%\"bG#!#HF&/%\"cG#\"#D\"\"(/%\"dG#\" $S\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'p_de_fG6#%\"xG,*#\"#6\"# @\"\"\"F'#!#HF**$)F'\"\"#\"\"\"#\"#D\"\"(*$)F'\"\"$F2#\"$S\"F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "On peut calculer la distance de f \+ \340 R_3[X] :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "simplify(norme(g)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%%sqrtG6#\"&-9*\"\"\"-F&6#\" #lF)#\"\")\"&:]\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "a:=plo t(f(x),x=-1..1,color=blue):\nb:=plot(11/21-29/11*x+25/7*x^2+140/11*x^3 ,x=-1..1,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wit h(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(a,b); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$ 7W7$$!\"\"\"\"!$!\"(F*7$$!1nmm;p0k&*!#;$!1C1y`\\R8\\!#:7$$!1LL$30y*>8nAF07$$!1LLL$yaE\"eF0$\"1Yu@#Q?bU$F07$$!1nmm\">s %HaF0$\"1o!ehx?B=%F07$$!1+++]$*4)*\\F0$\"1$p=KCij%[F07$$!1+++]_&\\c%F0 $\"1[X2TNz\"R&F07$$!1+++]1aZTF0$\"19WPPwi`eF07$$!1nm;/#)[oPF0$\"1xS\"p L(>XiF07$$!1MLL$=exJ$F0$\"12eT2%Gep'F07$$!1MLLL2$f$HF0$\"1#*p7Qx_tqF07 $$!1++]PYx\"\\#F0$\"1t8M*>OI^(F07$$!1MLLL7i)4#F0$\"1gTo9))\\.zF07$$!1+ +]P'psm\"F0$\"1D?A3YPL$)F07$$!1++]74_c7F0$\"1l#py_;Ou)F07$$!1JLL$3x%z# )FW$\"1YtF(>l?<*F07$$!1MLL3s$QM%FW$\"1_#\\+e;cc*F07$$!1^omm;zr)*!#>$\" 1LLL3#G,***F07$$\"1(Rso;F37$$\"1****\\7k.6aF0$\"1NgrZ+fH=F37$$\"1mmm;WTAeF0$\"1tCot1D\\ ?F37$$\"1****\\i!*3`iF0$\"1F,\\q2!GP#F37$$\"1MLLL*zym'F0$\"1O+nu@X4GF3 7$$\"1LLL3N1#4(F0$\"1Znk%)pKGMF37$$\"1mm;HYt7vF0$\"1vV+))H@rUF37$$\"1* ******p(G**yF0$\"1g>HhOs2`F37$$\"1mmmT6KU$)F0$\"1f%ei'H$R*oF37$$\"1++] P$[/a)F0$\"1gq.F+>rxF37$$\"1LLLLbdQ()F0$\"1IDt&fi(p()F37$$\"1nm\"zW?) \\*)F0$\"1jpD4Oz$)**F37$$\"1++]i`1h\"*F0$\"1'*pWWe+P6!#97$$\"1++++PDj$ *F0$\"1S9V\"p#o(G\"F[[l7$$\"1++]P?Wl&*F0$\"1^5&H?9zX\"F[[l7$$\"1++v=5s #y*F0$\"15<7&QW]m\"F[[l7$$\"\"\"F*$\"#>F*-%'COLOURG6&%$RGBGF*F*$\"*+++ +\"!\")-F$6$7S7$F($!1&*4n&*4n&*fF37$F.$!1eb')\\2?A[F37$F5$!1(388)QF.RF 37$F:$!1Olu_w\"z(HF37$F?$!1h:4#Q+i:#F37$FD$!1\"yX=hf& RIm)F07$FN$!1DOE\\_\"R`$F07$FS$\"1%pO>T')[Q*FW7$FY$\"1i^hyy#=h%F07$Fhn $\"1DG&>!H!Qj(F07$F]o$\"1Bhm\\xi4(*F07$Fbo$\"1WB=1KdW6F37$Fgo$\"13-)R! y\"3E\"F37$F\\p$\"1vKQzwcB8F37$Fap$\"1([a')*ePV8F37$Ffp$\"1c_$)fK\"oK \"F37$F[q$\"1L?s!*pe$G\"F37$F`q$\"16qr)fsb?\"F37$Feq$\"1$=F9QSn6\"F37$ Fjq$\"1\\1;RSl.5F37$F_r$\"1#))p;BC@'))F07$Fdr$\"1M^#4d_Mf(F07$Fir$\"1N /sofCSkF07$F^s$\"1+Mh_b:k_F07$Fds$\"1!=*p!>#=yTF07$Fis$\"18Z7zur&R$F07 $F^t$\"1E!zA4IJx#F07$Fct$\"1\\&Hf*)[*GCF07$Fht$\"1!*RAP5JVCF07$F]u$\"1 Hx*yp= (F07$Fav$\"1ek.U!\\Lh*F07$Ffv$\"1+`))***R\\H\"F37$F[w$\"16tX!y:#z;F37$ F`w$\"1^YGGNNf@F37$Few$\"1W(p\"yno6FF37$Fjw$\"1P)ejR!f$Q$F37$F_x$\"1Sh X8j*o7%F37$Fdx$\"1m\\2*R!R!*\\F37$Fix$\"1&)p]H6lbfF37$F^y$\"1U7A+89VpF 37$Fcy$\"1_[J-(e\"*>)F37$F]z$\"1Kd??v;S%*F37$Fgz$\"1R#y'4E7*3\"F[[l7$F b[l$\"1L2WF&*)3C\"F[[l7$F\\\\l$\"1=Zh=Zh=9F[[l-Fa\\l6&Fc\\lFd\\lF*F*-% +AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F\\\\l%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 59 "___________ ________________________________________________" }}}}{MARK "122 2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }