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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Exercice 5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 327 "La fonction d\351finie par l'int\351gran
de est continue sur tout intervalle ferm\351 born\351 [x,x^2]. f est d
onc bien d\351finie et d\351rivable sur R.\nOn commence par tracer le \+
graphe de f (on pourra calcul\351e des valeur approch\351e de f).\nOn \+
poursuit l'\351tude de mani\350re classique: \351tude aux bornes et du
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0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "f:=x->Int(1/sqrt(1+t^4),t=x..x^2);" }}{PARA 11 "" 1 "
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11 "" 1 "" {XPPMATH 20 "6#*,^$#!\"\"\"\"##\"\"\"F'F),&**,&F)F)*&^#F&F)
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$$\"0++!Q4%zn$F-$\"0++!eP=y>F]cl7$$\"0++I>$[<PF-$\"0++?&G)\\'>F]cl7$$
\"0+++z8yv$F-$\"0++!)*[j^>F]cl7$$\"0++qaj0!QF-$\"0++?H;w$>F]cl7$$\"0++
!*)3wRQF-$\"0++5j\")[#>F]cl7$$\"0++qp`%yQF-$\"0++SlBC\">F]cl7$$\"0++]n
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%%VIEWG6$;F)Fjin;$!\"&!\"\"Fjin" 1 2 0 1 10 0 2 6 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
51 "Etude des limites de f en -infinity et en +infinity" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(f(x),x=infinity);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%&limitG6$-%$IntG6$*&\"\"\"F**$,&F*F**$)%\"
tG\"\"%F*F*#F*\"\"#!\"\"/F/;%\"xG*$)F6F2F*/F6%)infinityG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 106 "Cel\340 ne fonctionne pas, on encadre l'
int\351grande par 0 \340 gauche et 1/t^2 \340 droite au voisinage de +
infinity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "limit(int(1/t^2
,t=x..x^2),x=infinity);" }}{PARA 7 "" 1 "" {TEXT -1 110 "Warning, unab
le to determine if 0 is between x and x^2; try to use assumptions or u
se the AllSolutions option\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(f(x),x=-infinity)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6$-%$IntG6$*&\"\"\"F**$,
&F*F**$)%\"tG\"\"%F*F*#F*\"\"#!\"\"/F/;%\"xG*$)F6F2F*/F6,$%)infinityGF
3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Un argument d'int\351grabil
it\351 montre que cette limite existe et est donn\351e par l'int\351gr
ale g\351n\351ralis\351e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 
"limite_moins_infini:=int(1/sqrt(1+t^4),t=-infinity..infinity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4limite_moins_infiniG,$*&#\"\"\"\"\"
#F(-%%BetaG6$#F(\"\"%F-F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "evalf(limite_moins_infini);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+a$\\\"3P!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
25 "Etude de la d\351riv\351e de f." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "derivee:=diff(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%(deriveeG,&*(\"\"#\"\"\"%\"xGF(,&F(F(*$)F)\"\")F(F(#!\"\"F'F(*
&F(F(*$,&F(F(*$)F)\"\"%F(F(#F(F'F/F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "derivee:=simplify(derivee);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(deriveeG*(,&*(\"\"#\"\"\"%\"xGF),&F)F)*$)F*\"\"%F)F)
#F)F(F)*$,&F)F)*$)F*\"\")F)F)F/!\"\"F)F+#F5F(F1F6" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 27 "numerateur:=numer(derivee);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%+numerateurG,&*(\"\"#\"\"\"%\"xGF(,&F(F(*$)F)\"
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"" {MPLTEXT 1 0 52 "sol:=solve(op(1,numerateur)^2=op(2,numerateur)^2,x
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG6**$-%'RootOfG6$,*\"\"\"F
+*$)%#_ZG\"\"%F+F+*&F/F+F.F+!\"\"*&F/F+)F.\"\"$F+F1/%&indexGF+#F+\"\"#
,$F&F1*$-F(6$F*/F6F8F7,$F:F1*$-F(6$F*/F6F4F7,$F?F1*$-F(6$F*/F6F/F7,$FD
F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solCalc = [allvalues(
sol)];" }}{PARA 8 "" 1 "" {TEXT -1 152 "Error, invalid input: too many
 and/or wrong type of arguments passed to allvalues; first unused argu
ment is -RootOf(1+_Z^4-4*_Z-4*_Z^3,index = 1)^(1/2)\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "solSort:=select(z-> Im(Z) =0 and is
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