% dérivation
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\lfoot{\small \textit{Lycée Joachim du Bellay}}
\cfoot{\small \textit{Mathématiques, prépa ECE1}}
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%%%     Exercice 465
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%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer les dérivées des fonctions suivantes (sans donner de justification concernant l'existence de cette dérivée)~:
\begin{eqnarray*}
a(x)&=&\ln (1+x^{2})\\
b(x)&=&\dfrac{e^{2x}}{x^{2}-1}\\
c(x)&=&\exp (x+1/x)\\
d(x)&=&\sqrt{x^{2}+x+1}\\
e(x)&=&\frac{-x+2}{x+1}\\
f(x)&=&\frac{x^2+2x+2}{3-x}
\end{eqnarray*}

\end{exercice}


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%%%     Exercice 471
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\begin{exercice}
\nopagebreak[5]\'Etudier la continuité et la dérivabilité de la fonction $f$ définie sur $\bf R$ par :
\[f(x)=x^2-|x|\]
\end{exercice}


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%%%     Exercice 467
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\begin{exercice}
\nopagebreak[5]\'Etudier la dérivabilité de la fonction suivante~:
\[a:x\mapsto \left\{ 
\begin{array}{l}
x\ln x \quad\textrm{ si } x>0\\ 
0\quad\textrm{ si } x=0
\end{array}
\right.\] 
\end{exercice}


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%%%     Exercice 468
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\begin{exercice}
\nopagebreak[5]\'Etudier la dérivabilité de la fonction suivante~:
\[b:x\mapsto \left\{
\begin{array}{l}
x^{2}\ln x \quad\textrm{ si } x>0 \\ 
0\quad\textrm{ si }x=0
\end{array}
\right.\]

\end{exercice}


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%%%     Exercice 466
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\begin{exercice}
\nopagebreak[5]\'Etudier la dérivabilité de la fonction $f:x\mapsto x\sqrt{x-x^{2}}$.
\end{exercice}


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%%%     Exercice 472
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soit :
\[f(x)=\frac{\sqrt{1+x}-1}{x}\]
\'Etudier la continuité (ainsi que les éventuels prolongements) et la dérivabilité de $f$. 
\end{exercice}


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%%%     Exercice 473
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soit $f$ la fonction définie sur $\mathbf R^*$ par :
\[f(x)=\frac{1}{x}\]
Calculer $f^{(n)}$ pour tout $n\in\mathbf N$ (indice~: conjecturer une formule puis la prouver par récurrence).
\end{exercice}


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%%%     Exercice 474
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soit $f$ la fonction définie sur $\mathbf R_+^*$ par :
\[f(x)=\sqrt x\]
Montrer que, pour tout $n\in\mathbf N^*$~:
\[\forall x\in\mathbf R_+^*,\quad f^{(n)}(x)=\dfrac{(-1)^{n+1}\sqrt x}{2^n x^n}\]
\end{exercice}


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%%%     Exercice 475
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soit $f$ la fonction définie sur $\bf R$ par :
\[f(x)=\frac{1}{2}x|x|\]
Démontrer que $f$ est de classe $\mathcal C^1$ mais qu'elle n'est pas de classe $\mathcal C^2$.
\end{exercice}


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%%%     Exercice 476
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]\'Etudier la fonction définie sur $\bf R$ par :
\[f(x)=x^3-3x^2+x+1\]
\end{exercice}


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%%%     Exercice 477
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Combien de racines admet le polynôme $x^4+x-1$~?
\end{exercice}


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%%%     Exercice 478
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Etudier la fraction rationnelle :
\[f(x)=\frac{x}{x^2+3x+2}\]
\end{exercice}


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%%%     Exercice 479
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Trouver tous les extremums locaux de la fraction rationnelle définie par :
\[f(x)=\frac{x}{1+x^2}\]
\end{exercice}


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%%%     Exercice 480
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Trouver tous les extremums locaux du polynôme défini par :
\[f(x)=x^4+4x^3+6x^2+8x+1\]
\end{exercice}


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%%%     Exercice 183
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Etudier la fonction suivante~:
\[f(x)=\frac{x^4+2x-3}{x^3-1}\]
\end{exercice}


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%%%     Exercice 184
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Etudier la fonction suivante~:
\[f(x)=(x^2+3x-4)^{\frac{1}{2}}\]
\end{exercice}


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%%%     Exercice 185
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Etudier la fonction suivante~:
\[f(x)=(x^2-x-2)^{-\frac{1}{2}}\]
\end{exercice}


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%%%     Exercice 188
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Etudier la fonction suivante~:
\[f(x)=\frac{e^{2x}}{x^2-1}\]
\end{exercice}


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%%%     Exercice 189
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Etudier la fonction suivante~:
\[f(x)=\exp(x+x^{-1})\]
\end{exercice}


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%%%     Exercice 190
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Etudier la fonction suivante~:
\[f(x)=\sqrt{x^2+x+1}\]
\end{exercice}


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%%%     Exercice 195
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Démontrer que~:
\[\forall x\ge 1,\; \ln(x)\le x-1\]
\end{exercice}


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%%%     Exercice 196
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Démontrer que~:
\[\forall x>0,\; \frac{2}{x}+\frac{x}{5}\ge 2\sqrt{10}\]
\end{exercice}


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%%%     Exercice 197
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Démontrer que~:
\[\forall x\in ]0,e^{-1}[,\; x\ln(x^{-1})<e^{-1}\]
\end{exercice}


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%%%     Exercice 481
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $p$ un réel positif fixé.
Démontrer que parmi les rectangles de périmètre $p$, c'est le carré de côté $p/4$ qui a une aire maximale (on considérera un rectangle de périmètre $p$ et dont un côté est de longueur $x$, on exprimera son aire $a(x)$ puis on étudiera la fonction $a$).
\end{exercice}


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%%%     Exercice 482
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Chercher le ou les points de la parabole d'équation $y=x^2$ qui sont les plus proches du point $(6,3)$.
\end{exercice}


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%%%     Exercice 483
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Encadrer les nombres suivants à l'aide du théorème des accroissements finis :
\begin{eqnarray*}
A&=&\sqrt{10001}-100\\
B&=&\frac{1}{0,99}-1\\
C&=&\ln(1,01)\\
\end{eqnarray*}
\end{exercice}


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%%%     Exercice 484
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\begin{enumerate}
\item Démontrer que pour tout entier $n\ge 1$~:
\[\frac{1}{n+1}\le \ln(n+1)-\ln(n)\le \frac{1}{n}\]
\item En déduire que $\sum\limits_{k=1}^n\frac{1}{k}\ge \ln(n+1)$.
\item Déterminer $\lim\limits_{n\to +\infty} \sum\limits_{k=1}^n \frac{1}{k}$.
\end{enumerate}
\end{exercice}


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%%%     Exercice 22
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]% ESC 2001 (T)

Soit $f$ la fonction définie pour tout $x$ réel par~:
\[f(x)=\left(1+x+\frac{x^{2}}{2}\right)e^{-x}\]
On désigne par $\mathcal C_f$ sa courbe représentative dans un repère du plan.
\begin{enumerate}
\item Déterminer les limites de $f$ en $-\infty$ et en $+\infty$.
\item Etudier les branches infinies de $\mathcal C_f$.
\item Déterminer la fonction dérivée $f'$ de $f$ et dresser le
tableau de variations de~$f$.
\item Déterminer une équation de la tangente à $\mathcal C_f$ au point d'abscisse~0. Tracer cette tangente dans un repère ortho\-gonal d'un\-ités 2 cm sur l'axe des abscisses et 10 cm sur l'axe des ordonnées.
\item Donner l'allure de $\mathcal C_f$ et ses asymptotes éventuelles dans ce même
repère. On donne les valeurs suivantes~: $f(-1)\approx 1,36$; $f(1)\approx 0,92$; $f(2)\approx 0,68$; $f(5)\approx 0,12$.
\end{enumerate}
\end{exercice}


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%%%     Exercice 470
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Etudier la convexité des fonctions suivantes~:
\begin{eqnarray*}
f(x)&=&6x^{5}-15x^{4}+10x^{3}+1\\
g(x)&=&\dfrac{x^{2}}{x+1}\\
h(x)&=&\ln \left( \dfrac{x-1}{x+1}\right)
\end{eqnarray*}

\end{exercice}

\label{fin}
\end{document}