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{SECT 0 {SECT 1 {PARA 4 "" 0 "" {TEXT 256 11 "Exercice 26" }}{PARA 0 "
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->(gamma+2*arccosh(x/2))/(sqrt(x^2-4));" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f_1Gf*6#%\"xG6\"6$%)operatorG%
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{XPPMATH 20 "6#>%$f_2Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&%%betaG\"
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&#F/F1F/9$F/F/F/F/F/-%%sqrtG6#,&*$)F8F1F/F/\"\"%!\"\"F@F(F(F(" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "f_1;" "
6#%$f_1G" }{TEXT -1 17 " est solution de " }{XPPEDIT 18 0 "G;" "6#%\"G
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"" {XPPMATH 20 "6#>%\"aG,(*(\"\"#\"\"\")%#t|irGF'F(*&F(F(*(,&F'!\"\"F*
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{TEXT -1 17 " est solution de " }{XPPEDIT 18 0 "G;" "6#%\"GG" }{TEXT 
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "3) " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f:=x->piecew
ise(x<(-2),f_1(x),-2<x and x<2,f_2(x),x>2,f_3(x));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewise
G6(29$!\"#-%$f_1G6#F032F1F02F0\"\"#-%$f_2GF42F8F0-%$f_3GF4F(F(F(" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "limit(f(x),x=-2,left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%'sig
numG6#%&alphaG\"\"\"%)infinityGF(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "limit(f(x),x=-2,right);" }{TEXT -1 0 "" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*&-%'signumG6#,&*&\"\"#!\"\"%%betaG\"\"\"F,%#PiG
F,F,%)infinityGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Pour que " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 16 " soit de classe " }
{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 51 " 1 en -2 elle doit \352tre \+
continue ; on suppose donc " }{XPPEDIT 18 0 "alpha = 0;" "6#/%&alphaG
\"\"!" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "beta = -2*Pi;" "6#/%%betaG,$
*&\"\"#\"\"\"%#PiGF(!\"\"" }{TEXT -1 3 ".  " }{MPLTEXT 1 0 1 "\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "alpha:=0:beta:=-2*Pi;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG,$*&\"\"#\"\"\"%#PiGF(!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f:=x->piecewise(x<(-2),f_1
(x),-2<x and x<2,f_2(x),x>2,f_3(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6(29$!\"#-%$f_
1G6#F032F1F02F0\"\"#-%$f_2GF42F8F0-%$f_3GF4F(F(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "limit(f(x),x=-2,left);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "limit(f(x),x=-2,right);" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "f;" "6#%\"fG" }{TEXT -1 35 " est prolongeable par contiuit\351 e
n " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT -1 30 "  (on pose qu'elle vau
t -1 en " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT -1 1 ")" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 72 "limit((f_1(-2+h)-(-1))/h,h=0,left);limit((f_2(-2+h)-(-1))/h,h=
0,right); " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 51 "  est d\351riv
able en -2 et son nombre d\351riv\351 est -1/6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##!\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"
\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "limit(diff(f_1(x),x)
,x=-2,left);limit(diff(f_2(x),x),x=-2,right); " }{TEXT -1 14 "la d\351
riv\351e de " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 75 "  est contin
ue en -2 puisque sa limite en -2 co\357ncide avec sa valeur en -2." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##!\"\"\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 8 "Lorsque " }{XPPEDIT 18 0 "alpha = 0;" "6#/%&alph
aG\"\"!" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "beta = -2*Pi;" "6#/%%betaG
,$*&\"\"#\"\"\"%#PiGF(!\"\"" }{TEXT -1 15 ", la fonction  " }{XPPEDIT 
18 0 "f;" "6#%\"fG" }{TEXT -1 22 "  est une solution de " }{XPPEDIT 
18 0 "G;" "6#%\"GG" }{TEXT -1 19 " sur l'intervalle ]" }{XPPEDIT 18 0 
"-infinity;" "6#,$%)infinityG!\"\"" }{TEXT -1 6 ";2[ . " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(f
(x),x=2,left); " }{TEXT -1 15 "ces valeurs de " }{XPPEDIT 18 0 "alpha;
" "6#%&alphaG" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }
{TEXT -1 40 " provoquent une discontinuit\351 de f en 2." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "limit(f(x),x=2,right);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&-%'signumG6#%#k3G\"\"\"%)infinityGF(" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "On montre de la m\352me f
a\347on que " }{XPPEDIT 18 0 "beta = 0;" "6#/%%betaG\"\"!" }{TEXT -1 
4 " et " }{XPPEDIT 18 0 "gamma = 0;" "6#/%&gammaG\"\"!" }{TEXT -1 15 "
 permettent \340  " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 21 "  d'
\352tre solution de " }{XPPEDIT 18 0 "G;" "6#%\"GG" }{TEXT -1 23 " sur
 l'intervalle ]-2;+" }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }
{TEXT -1 73 "[ et que cette solution est maximale. Mais il n'existe pa
s de valeurs de " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 2 ", \+
" }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "
gamma;" "6#%&gammaG" }{TEXT -1 18 " pour lesquelles  " }{XPPEDIT 18 0 
"f;" "6#%\"fG" }{TEXT -1 18 "  est solution de " }{XPPEDIT 18 0 "G;" "
6#%\"GG" }{TEXT -1 5 " sur " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 
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