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{SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 202 10 "Exercice 3" }{TEXT 203 
0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 47 "restart:Digits:=
30:with(plots):with(plottools):" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 204 36 "fI:=t->4/(1+t^2):aI:=0:bI:=1:NI:=20:"
 }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 39 
"fJ:=t->exp(-t*t):aJ:=-10:bJ:=10:NJ:=20:" }{MPLTEXT 1 204 0 "" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 "" 
0 "" {TEXT 205 11 "Question a)" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 ">
 " 0 "" {MPLTEXT 1 204 26 "Rectangle_d:=proc(f,a,b,N)" }{MPLTEXT 1 
204 0 "" }{MPLTEXT 1 204 11 "\nlocal i,S;" }{MPLTEXT 1 204 0 "" }
{MPLTEXT 1 204 6 "\nS:=0;" }{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 21 "\n
for i from 1 to N do" }{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 22 "\n S:=S
+f(a+i*(b-a)/N);" }{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 4 "\nod;" }
{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 14 "\nS:=S*(b-a)/N;" }{MPLTEXT 1 
204 0 "" }{MPLTEXT 1 204 5 "\nend:" }{MPLTEXT 1 204 0 "" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{TEXT 206 11 "Question b)" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 ""
 0 "" {TEXT 207 21 "Calcul approch\351 de I " }{TEXT 203 5 "avec " }
{TEXT 207 13 "20 Rectangles" }{TEXT 203 3 " o\371" }{TEXT 203 0 "" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 22 "I=Int(fI(t),t=aI..bI);" 
}{MPLTEXT 1 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#/^#\"\"\"-I$In
tG6$I*protectedGF)I(_syslibG6\"6$,$*$,&F%F%*$I\"tGF+\"\"#F%!\"\"\"\"%/
F1;\"\"!F%" }{TEXT 208 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 
204 47 "I_(approch\351):=evalf(Rectangle_d(fI,aI,bI,NI)); " }{MPLTEXT 
1 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>-I#I_G6\"6#I)approch|dy
GF&$\"?4S2/ZMYGT&p)f<\"4$!#H" }{TEXT 208 0 "" }}}{EXCHG {PARA 201 "> "
 0 "" {MPLTEXT 1 204 37 " erreur:=evalf(abs(I_(approch\351)-Pi));" }
{MPLTEXT 1 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>I'erreurG6\"$
\"=>VEg:]xkcjmmT]!#H" }{TEXT 208 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 207 21 "Calcul a
pproch\351 de J " }{TEXT 203 5 "avec " }{TEXT 207 13 "20 Rectangles" }
{TEXT 203 3 " o\371" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 204 27 "J=(Int(fJ(t),t=aJ..bJ))**2;" }{MPLTEXT 1 204 0 "" }
}{PARA 203 "" 1 "" {XPPMATH 20 "6#/I\"JG6\"*$-I$IntG6$I*protectedGF*I(
_syslibGF%6$-I$expGF)6#,$*$I\"tGF%\"\"#!\"\"/F2;!#5\"#5F3" }{TEXT 208 
0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 52 "J_(approch\351):
=evalf((Rectangle_d(fJ,aJ,bJ,NJ))**2); " }{MPLTEXT 1 204 0 "" }}{PARA 
203 "" 1 "" {XPPMATH 20 "6#>-I#J_G6\"6#I)approch|dyGF&$\"?P&)f9V\"RjkN
*fECUJ!#H" }{TEXT 208 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 
204 37 " erreur:=evalf(abs(J_(approch\351)-Pi));" }{MPLTEXT 1 204 0 ""
 }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>I'erreurG6\"$\";4-E]!o+J&eM1+l!#H
" }{TEXT 208 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}
}{EXCHG {PARA 201 "> " 0 "" {TEXT 206 11 "Question c)" }{MPLTEXT 1 
204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 24 "Points_Rd:=p
roc(f,a,b,N)" }{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 11 "\nlocal i,S;" }
{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 9 "\nS:=NULL;" }{MPLTEXT 1 204 0 "
" }{MPLTEXT 1 204 58 "\nfor i from 1 to N do S:=S,[i/N,Rectangle_d(f,a
,b,i)]; od;" }{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 5 "\n[S];" }
{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 5 "\nend;" }{MPLTEXT 1 204 0 "" }}
{PARA 204 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 
"6#>I*Points_RdG6\"f*6&I\"fGF%I\"aGF%I\"bGF%I\"NGF%6$I\"iGF%I\"SGF%F%F
%C%>8%I%NULLGI*protectedGF3?(8$\"\"\"F69'I%trueGF3>F16$F17$*&F5F6F7!\"
\"-I,Rectangle_dGF%6&9$9%9&F57#F1F%F%F%" }{TEXT 208 0 "" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 204 30 "plot(Points_Rg(fI,aI,bI,50)); "
 }{TEXT 209 33 "Trac\351 des points obtenus pour le " }{TEXT 210 20 "c
alcul approch\351 de I" }{MPLTEXT 1 204 0 "" }}{PARA 205 "" 1 "" 
{GLPLOT2D 400 400 400 {PLOTDATA 2 "6%-%'CURVESG6$7T7$$\"3/+++++++?!#>$
\"\"%\"\"!7$$\"35+++++++SF*$\"33+++++++O!#<7$$\"3y**************fF*$\"
3cc-Tc-TcMF37$$\"3=+++++++!)F*$\"3WN#)eqk<\"Q$F37$$\"3/+++++++5!#=$\"3
!)*)4\"Q6E\\L$F37$$\"3%**************>\"FA$\"3&)Hz8L(HOI$F37$$\"39++++
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$\"3-+++++++AFA$\"36;0:NV7JKF37$$\"3!**************R#FA$\"33%=!=!eoPA$
F37$$\"33+++++++EFA$\"3oT+jP&Hv@$F37$$\"3E+++++++GFA$\"3s>e9&)3<7KF37$
$\"3))**************HFA$\"312**oz&=v?$F37$$\"31+++++++KFA$\"3L*QT?hTM?
$F37$$\"3C+++++++MFA$\"3u:]\"=[R)*>$F37$$\"3')*************f$FA$\"3e.:
\"f!Qj'>$F37$$\"3-+++++++QFA$\"3A`A1_Dw$>$F37$$\"3A+++++++SFA$\"3]GT&p
)f<\">$F37$$\"3%)*************>%FA$\"3m<EGsP$))=$F37$$\"3-+++++++WFA$
\"3keZWYGq'=$F37$$\"3=+++++++YFA$\"3)yQ6Y&ev%=$F37$$\"3#)*************
z%FA$\"3!3*\\To*pH=$F37$$\"3++++++++]FA$\"3y`7$p)fK\"=$F37$$\"3;++++++
+_FA$\"3[$GMPk2)zJF37$$\"3M+++++++aFA$\"3oPl%p1,%yJF37$$\"3a+++++++cFA
$\"3u#=uUN%4xJF37$$\"3g*************z&FA$\"35O,WNs(e<$F37$$\"3w*******
*******fFA$\"33j1u,3uuJF37$$\"3%**************>'FA$\"3wpPl(GxO<$F37$$
\"39+++++++kFA$\"3+u\\<$*)zE<$F37$$\"3I+++++++mFA$\"3#)onGQEurJF37$$\"
3[+++++++oFA$\"3'QxzHCg3<$F37$$\"3a**************pFA$\"3QK0uF!G+<$F37$
$\"3u*************>(FA$\"3GPaaI=CpJF37$$\"3!**************R(FA$\"3)pg_
s$z\\oJF37$$\"33+++++++wFA$\"3kl[$H-$znJF37$$\"3E+++++++yFA$\"3')G()>-
T7nJF37$$\"3U+++++++!)FA$\"3]5O#p[)[mJF37$$\"3]*************>)FA$\"3-8
(f,v$)e;$F37$$\"3o*************R)FA$\"37RB;&p2`;$F37$$\"3')***********
**f)FA$\"3)G#)H(G$eZ;$F37$$\"3-+++++++))FA$\"3YpI;QQBkJF37$$\"3A++++++
+!*FA$\"350bGrDtjJF37$$\"3S+++++++#*FA$\"37U%f!>IDjJF37$$\"3Y*********
****R*FA$\"3QxNT+QziJF37$$\"3k*************f*FA$\"3WD.'*[ONiJF37$$\"3#
)*************z*FA$\"3#>$RL,9$>;$F37$$\"\"\"F-$\"3[`K#p)f_hJF3-%&COLOR
G6&%$RGBG$\"#5!\"\"$F-Fe[lFf[l-%+AXESLABELSG6$Q!6\"Fj[l-%%VIEWG6$;$\"1
6++++++SF*$\"&'>5!\"%;$\"2%=<m1lvWJ!#;$\"2_`h-[pn,%Fh\\l" 1 2 2 0 10 
1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}{TEXT 211 0 "" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 204 30 "plot(Points_Rg(fJ,aJ,bJ,50)); "
 }{TEXT 209 33 "Trac\351 des points obtenus pour le " }{TEXT 210 20 "c
alcul approch\351 de J" }{MPLTEXT 1 204 0 "" }}{PARA 205 "" 1 "" 
{GLPLOT2D 400 400 400 {PLOTDATA 2 "6%-%'CURVESG6$7T7$$\"3/+++++++?!#>$
\"3kt;/_>:Su!#g7$$\"35+++++++SF*$\"#5\"\"!7$$\"3y**************fF*$\"3
]$>/L!=r#*>!#@7$$\"3=+++++++!)F*$\"37z)Q,++++&!#<7$$\"3/+++++++5!#=$\"
3;`()466Dl9FC7$$\"3%**************>\"FC$\"3\\)\\B*oHVLLF?7$$\"39++++++
+9FC$\"3?(yN(['\\TU(FC7$$\"3-+++++++;FC$\"3Sy0vqAl4DF?7$$\"3#*********
*****z\"FC$\"39U%z'=SA$H\"F?7$$\"35+++++++?FC$\"3qOcp0IEt?F?7$$\"3-+++
++++AFC$\"3'3xrg^=Mf\"F?7$$\"3!**************R#FC$\"3ij<ek:(R(=F?7$$\"
33+++++++EFC$\"3.B(f(p@n<<F?7$$\"3E+++++++GFC$\"3t7$>\")*Hf+=F?7$$\"3)
)**************HFC$\"3up[,Gfpe<F?7$$\"31+++++++KFC$\"3'fQDhbc)y<F?7$$
\"3C+++++++MFC$\"3'GsnPh<'p<F?7$$\"3')*************f$FC$\"31x'3vt\\Ox
\"F?7$$\"3-+++++++QFC$\"3WcK(**)Q(>x\"F?7$$\"3A+++++++SFC$\"35_m#[?PEx
\"F?7$$\"3%)*************>%FC$\"3-^&))*yrQs<F?7$$\"3-+++++++WFC$\"3'ol
&4FpZs<F?7$$\"3=+++++++YFC$\"3gcKX[iWs<F?7$$\"3#)*************z%FC$\"3
;Ub$\\BcCx\"F?7$$\"3++++++++]FC$\"3An=IRJXs<F?7$$\"3;+++++++_FC$\"3w;J
3`SXs<F?7$$\"3M+++++++aFC$\"3Mm&Hiz`Cx\"F?7$$\"3a+++++++cFC$\"3oc)z\\'
QXs<F?7$$\"3g*************z&FC$\"3;KqXZQXs<F?7$$\"3w**************fFC$
\"3aw(4<&QXs<F?7$$\"3%**************>'FC$\"3[wps]QXs<F?7$$\"39+++++++k
FC$\"3;UK%4&QXs<F?7$$\"3I+++++++mFC$\"3=Gz*3&QXs<F?7$$\"3[+++++++oFC$
\"3%*op!4&QXs<F?7$$\"3a**************pFC$\"3?^_!4&QXs<F?7$$\"3u*******
******>(FC$\"31ib!4&QXs<F?7$$\"3!**************R(FC$\"3y3b!4&QXs<F?7$$
\"33+++++++wFC$\"3A<b!4&QXs<F?7$$\"3E+++++++yFC$\"3W:b!4&QXs<F?7$$\"3U
+++++++!)FC$\"35;b!4&QXs<F?7$$\"3]*************>)FC$\"3)e^04&QXs<F?7$$
\"3o*************R)FCFbx7$$\"3')*************f)FCF]x7$$\"3-+++++++))FC
Fbx7$$\"3A+++++++!*FCFbx7$$\"3S+++++++#*FC$\"3K;b!4&QXs<F?7$$\"3Y*****
********R*FCFbx7$$\"3k*************f*FCF]x7$$\"3#)*************z*FCF]x
7$$\"\"\"F3Fcy-%&COLORG6&%$RGBG$F2!\"\"$F3FfzFgz-%+AXESLABELSG6$Q!6\"F
[[l-%%VIEWG6$;$\"16++++++SF*$\"&'>5!\"%;$!\"#Ffz$\"2-++++++-\"!#:" 1 
2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}{TEXT 211 0 "" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> \+
" 0 "" {TEXT 206 11 "Question d)" }{MPLTEXT 1 204 0 "" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 204 23 "Suite_Rd:=proc(f,a,b,N)" }
{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 23 "\nlocal i,S,suite,delta;" }
{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 24 "\ndelta:=(b-a)/N;S:=NULL;" }
{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 23 "\nfor i from 0 to N-1 do" }
{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 135 "\n S:=S,polygon([[a+i*delta,0
],[a+(i+1)*delta,0],[a+(i+1)*delta,f(a+(i+1)*delta)],[a+i*delta,f(a+(i
+1)*delta)]],color=pink,thickness=2);" }{MPLTEXT 1 204 0 "" }{MPLTEXT 
1 204 4 "\nod;" }{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 3 "\nS;" }
{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 5 "\nend:" }{MPLTEXT 1 204 0 "" }}
}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 200 ""
 0 "" {TEXT 207 50 "Trac\351 des rectangels correspondant \340 l'int\3
51grale I" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 
204 71 "RdI:=Suite_Rd(fI,aI,bI,NI):GI:=plot(fI,aI..bI,color=black,thic
kness=3):" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 204 23 "plots[display](RdI,GI);" }{MPLTEXT 1 204 0 "" }}
{PARA 205 "" 1 "" {GLPLOT2D 758 400 400 {PLOTDATA 2 "69-%)POLYGONSG6%7
&7$$\"\"!F)F(7$$\"?++++++++++++++]!#JF(7$F+$\"?*[/#=F\"\\.'elP\\-!*R!#
H7$F(F/-%&COLORG6&%$RGBG$\"#5!\"\"$\"2gTk>\">THv!#<$\"1=AFNWygz!#;-%*T
HICKNESSG6#\"\"#-F$6%7&F*7$$\"?++++++++++++++5!#IF(7$FH$\"?SgRgRgRgRgR
gRgRF17$F+FLF3F@-F$6%7&FG7$$\"?++++++++++++++:FJF(7$FS$\"?y()\\W-5^*z(
4S/)>\"RF17$FHFVF3F@-F$6%7&FR7$$\"?++++++++++++++?FJF(7$Fgn$\"?:YQ:YQ:
YQ:YQ:YQF17$FSFjnF3F@-F$6%7&Ffn7$$\"?++++++++++++++DFJF(7$Fao$\"?HN#)e
qk<THN#)eqkPF17$FgnFdoF3F@-F$6%7&F`o7$$\"?++++++++++++++IFJF(7$F[p$\"?
5@`Qi[$=?U1xC(pOF17$FaoF^pF3F@-F$6%7&Fjo7$$\"?++++++++++++++NFJF(7$Fep
$\"?AZ7c_OkRy_(QuMc$F17$F[pFhpF3F@-F$6%7&Fdp7$$\"?++++++++++++++SFJF(7
$F_q$\"?M5$z8C<b'*o?'eF[MF17$FepFbqF3F@-F$6%7&F^q7$$\"?++++++++++++++X
FJF(7$Fiq$\"?SELSELSELSELSELF17$F_qF\\rF3F@-F$6%7&Fhq7$$F,FJF(7$Fcr$\"
?++++++++++++++KF17$FiqFerF3F@-F$6%7&Fbr7$$\"?++++++++++++++bFJF(7$F
\\s$\"?Hpf4]f!*o@Zus,rIF17$FcrF_sF3F@-F$6%7&F[s7$$\"?++++++++++++++gFJ
F(7$Ffs$\"?#)eqk<THN#)eqk<THF17$F\\sFisF3F@-F$6%7&Fes7$$\"?+++++++++++
+++lFJF(7$F`t$\"?ZG-B,(H*f6'3z]>\"GF17$FfsFctF3F@-F$6%7&F_t7$$\"?+++++
+++++++++qFJF(7$Fjt$\"?-`HWm\\uh#*QePc%o#F17$F`tF]uF3F@-F$6%7&Fit7$$\"
?++++++++++++++vFJF(7$Fdu$\"?+++++++++++++gDF17$FjtFguF3F@-F$6%7&Fcu7$
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