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\begin{document}
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\begin{center}
\textbf{\Large \decofourleft~BTS Conception de produits industriels~\decofourright\\
Session 2010}
\end{center}

\textbf{EXERCICE 1 \hfill 10 points}

\begin{center}
\textbf{\emph{Les parties A et B de cet exercice peuvent être traitées de façon indépendante.}}
\end{center}

\emph{A.	Résolution d'une équation différentielle}

\medskip

On considère l'équation différentielle

\[(E):\  2y^{\prime\prime} + 2y^{\prime}+ y = \left(5x^2 +22x + 31\right)\text{e}^x\]

où $y$ est une fonction de la variable réelle $x$, définie et deux fois dérivable sur $\R$, $y'$ la fonction dérivée de $y$ et $y^{\prime\prime}$ sa fonction dérivée seconde.

\medskip

\begin{enumerate}
	\item Déterminer les solutions de l'équation différentielle $(E_0) :\ 2y^{\prime\prime} + 2y^{\prime} + y = 0$.
	\item Montrer que la fonction $g$ définie sur $\R$ par $g(x)= \left(x^2+2x+3\right)\text{e}^x$ est une solution particulière de l'équation $(E)$.
	\item En déduire l'ensemble des solutions de l'équation différentielle $(E)$.
	\item Déterminer la solution particulière $f$ de l'équation différentielle $(E)$ qui vérifie les conditions initiales $f(0) = 3$ et $f^{\prime}(0) = 5$.
\end{enumerate}


\emph{B.	Étude d'une fonction}

\medskip

Soit $f$ la fonction définie sur $\R$ par $f(x) = \left(x^2 + 2x + 3\right)\text{e}^x$.

On désigne par $C$ la courbe représentative de la fonction $f$ dans un repère orthonormal \Oij.

\medskip

\begin{enumerate}
\item Démontrer que, pour tout $x$ de $\R$,
	
\[f'(x) = \left(x^2 + 4x + 5\right)\text{e}^x.\]

\item Étudier le signe de $f'(x)$ lorsque $x$ varie dans $\R$.
\item 
	\begin{enumerate}
		\item Déterminer $\lim_{x\to+\infty} f(x)$.
		\item Déterminer $\lim_{x\to-\infty} f(x)$. Que peut-on en déduire pour la courbe $C$?
	\end{enumerate}
\item Établir le tableau de variation de $f$ sur $\R$.
\item 
	\begin{enumerate}
		\item Démontrer que le développement limité à l'ordre 2, au voisinage de 0, de la fonction $f$ est :
	
$f(x) = 3 + 5x+ \frac{9}{2}x^2 + x^2\varepsilon(x)$ avec $\lim_{x\to0} \varepsilon(x) = 0$.
		\item En déduire une équation de la tangente $T$ à la courbe $C$ au point d'abscisse 0.
		\item Étudier la position relative de $C$ et $T$ au voisinage du point d'abscisse 0.
	\end{enumerate}
\end{enumerate}

\vspace{0,5cm}

\textbf{EXERCICE 2\hfill 3 points}

\medskip

La courbe $C$ ci-dessous est la représentation graphique dans un repère orthogonal de la fonction $f$ définie sur $[- 1, 1]$ par $f(x) = \frac{1}{5}\left(\text{e}^x+ \text{e}^{-x}\right)$.

%Courbe
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%		CCCCC		OOOOO		U   U		RRRR		BBBB		EEEEE
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%		C				O		O		U   U		RRRR		BBBB		EEE
%		C				O		O		U   U		R R			B   B		E
%		CCCCC		OOOOO		UUUUU		R  R		BBBB		EEEEE

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On considère le solide de révolution engendré par la rotation de la courbe $C$ autour de l'axe des abscisses.

On désigne par $V$ le volume, en unités de volume, de ce solide.

On admet que $V=\displaystyle\int_{-1}^1 \pi{\left[f(x)\right]}^2\text{d}x$.
\begin{enumerate}
	\item Vérifier que: $V=\displaystyle\int_{-1}^1\dfrac{\pi}{25}\left(2+\text{e}^{2x} + \text{e}^{-2x}\right)\text{d}x$.
	\item Démontrer que :  $V=\dfrac{\pi}{25}\left(4 + \text{e}^2-\text{e}^{-2}\right)$.
	\item Donner la valeur approchée arrondie à $10^{-2}$ de $V$.
\end{enumerate}
\emph{Le solide obtenu ci-dessus est le modèle d'un élément de mobilier urbain.}

\newpage

\textbf{EXERCICE 3}\hfill (7 points)

\medskip

Le plan est muni d'un repère orthonormal \Oij{} où l'unité graphique est 4 ~centimètres.

On souhaite construire la courbe de Bézier $C$ définie par les points de définition suivants donnés par leurs coordonnées :
$A_0(0~;~0)$; $A_1(0~;~2)$; $A_2\left(3~;~\frac{3}{4}\right)$.

On rappelle que la courbe de Bézier définie par les points de définition $A_i (0 \leqslant i \leqslant n)$ est l'ensemble des points $M(t)$ tels que, pour tout $t$ de l'intervalle $[0~ ;~1]$ :

\[\overrightarrow{OM(t)}= \sum_0^n B_{i,n}(t)~ \overrightarrow{OA_i}~~~~~ \text{où } B_{i,n}(t) = C_n^i~ t^i~ {(1 -t)}^{n-i}.\]

\begin{enumerate}
	\item Développer, réduire et ordonner les polynômes $B_{i,2}(t)$ avec $0 \leqslant i \leqslant 2$.
	\item On note $\left(f(t), g(t)\right)$ les coordonnées du point $M(t)$ de la courbe $C$.
	
Démontrer qu'un système d'équations paramétriques de la courbe $C$ est :
	
$\left\{	
\begin{minipage}{4cm}{
$x= f (t) =3t^2$\\
$y=g(t)=4t-\frac{13}{4}t^2$}
\end{minipage}\right.$
 où $t$ appartient à l'intervalle $[0~;~1]$.
\item Étudier les variations de $f$ et $g$ sur $[0~;~1]$ et rassembler les résultats dans un tableau unique.
\item Déterminer un vecteur directeur de la tangente à la courbe $C$ :
	\begin{enumerate}
		\item au point $A_0$,
		\item au point $A_2$,
		\item au point $M\left(\frac{8}{13}\right)$.
	\end{enumerate}\label{tangentes}
\item La figure est à réaliser sur une feuille de papier millimétré.
	Construire les tangentes définies au \ref{tangentes} et la courbe $C$. Que constate-t-on ?
\end{enumerate}
\end{document}