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\newcommand{\TITLE}{Fonction exponentielle}
\author{Y. Morel}
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\begin{document}
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\noindent
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La fonction exponentielle est l'unique fonction définie et dérivable
sur $\R$ solution de
l'équation $f'=f$ et telle que $f(0)=1$.
\vspd
On la note $x\mapsto \exp(x)=e^x$.
\vspd
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\item {\bf\ul{Règles de calcul}}\
Ce sont les règles
usuelles de calcul sur les puissances:
\hspace{3.2cm}$e^0=1$\ ,\ $e^1=e\simeq 2,718$
\hspace{3.2cm}$e^{a+b}=e^a\,e^b$\ , \
$e^{-a}=\dfrac{1}{e^a}$\ , \
$e^{a-b}=\dfrac{e^a}{e^b}$\ , \
$\lp e^a\rp^b=e^{a\,b}$
\vspd
\item {\bf\ul{Dérivée}} \
$\exp'=\exp$
et pour toute fonction $u$ dérivable $\lp e^u\rp'=u'\,e^u$
\vspd
\item {\bf\ul{Limites}}\
$\dsp\bullet\lim_{x\to-\infty}e^x=0$,
donc la droite d'équation $y=0$ (l'axe des ordonnées)
\hspace{4.2cm}est une
asymptote en $-\infty$.
\hspace{1.55cm}$\dsp\bullet\lim_{x\to+\infty}e^x=+\infty$
$\bullet$ Croissances comparées:
$\dsp\lim_{x\to+\infty}\dfrac{e^x}{x^n}=+\infty$
\ et \
$\dsp\lim_{x\to-\infty}x^n e^x=0$
$\bullet$ Taux d'accroissement en $0$:
$\dsp\lim_{x\to0}\dfrac{e^x-1}{x}=1$.
\vspd
\item {\bf\ul{Le logarithme et l'exponentielle}} sont des fonctions
réciproques:
$\bullet$
Pour tous $x$ et $y$, $x>0$,
$e^{\ln(x)}=x$\ ,\ $\ln\lp e^y\rp=y$
\ et\
$y=\ln(x)\iff e^y=x$
\vspd
\item {\bf\ul{Composition avec l'exponentielle:}}
$f=e^{u}$, donc $f'=u'e^{u}$
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