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\begin{center}\section*{\textcolor{red}{Spécialité de Terminale : devoir sur feuille \no 2}}\end{center}


\subsection{}%TES Amérique du Nord juin 2017
Une grande université, en pleine croissance d'effectifs, accueillait \np{27500}~étudiants en septembre 2016.

Le président de l'université est inquiet car il sait que, malgré une gestion optimale des locaux et une répartition des étudiants sur les divers sites de son université, il ne pourra pas accueillir plus de \np{33000}~étudiants.

Une étude statistique lui permet d'élaborer un modèle de prévisions selon lequel, chaque année :

\setlength\parindent{8mm}
\begin{itemize}
\item[$\bullet~~$]$150$ étudiants démissionnent en cours d'année universitaire (entre le 1\up{er} septembre et le 30 juin) ;
\item[$\bullet~~$]les effectifs constatés à la rentrée de septembre connaissent une augmentation de 4\,\% par rapport à ceux du mois de juin qui précède.
\end{itemize}
\setlength\parindent{0mm}

\smallskip

Pour tout entier naturel $n$, on note $u_n$ le nombre d'étudiants estimé selon ce modèle à la rentrée de septembre $2016 + n$, on a donc $u_0 = \np{27500}$.\index{suite}

\medskip

\begin{enumerate}
\item 
	\begin{enumerate}
		\item Estimer le nombre d'étudiants en juin 2017.
		\item Estimer le nombre d'étudiants à la rentrée de septembre 2017.
	\end{enumerate}
\item  Justifier que, pour tout entier naturel $n$, on a $u_{n+1} = 1,04u_n - 156$.
\item  Recopier et compléter les lignes L5, L6, L7 et L9 de l'algorithme suivant afin qu'il donne l'année à partir de laquelle le nombre d'étudiants à accueillir dépassera la capacité maximale de l'établissement.\index{algorithme}
	
\begin{center}
\begin{tabularx}{0.85\linewidth}{l l l XX}
L1& Variables : &$n$ est un nombre entier naturel\\
L2& 			&$U$ est un nombre réel\\
L3& Traitement :&$n$ prend la valeur 0\\
L4& 			&$U$ prend la valeur \np{27500}\\
L5& 			&Tant que $U \leqslant \ldots\ldots$ faire\\
L6& 			& \hspace*{1cm} $n$ prend la valeur \dotfill\\
L7& 			& \hspace*{1cm} $U$ prend la valeur \dotfill.\\
L8& 			&Fin Tant que\\
L9& Sortie: &Afficher \dotfill\\
\end{tabularx}
\end{center}
\item 
	\begin{enumerate}
		\item On fait fonctionner cet algorithme pas à pas.
		
Recopier le tableau suivant et le compléter en ajoutant le nombre nécessaire de colonnes ; on arrondira les valeurs de $U$ à l'unité.
		
\begin{center}
\begin{tabularx}{\linewidth}{|*{3}{X|}m{5cm}|}\hline		
&Initialisation &Étape 1 &\ldots\\ \hline
Valeur de $n$& 0 &\ldots&\\ \hline
Valeur de $U$& \np{27500}&\ldots&\\ \hline
\end{tabularx}
\end{center}
		\item Donner la valeur affichée en sortie de cet algorithme.
	\end{enumerate}
\item  On cherche à calculer explicitement le terme général $u_n$ en fonction de $n$.
	
Pour cela, on note $\left(v_n\right)$ la suite définie, pour tout entier naturel $n$, par $v_n = u_n - \np{3900}$.
	\begin{enumerate}
		\item Montrer que $\left(v_n\right)$ est une suite géométrique dont on précisera la raison et le premier terme.\index{suite géométrique}
		\item En déduire que, pour tout entier naturel $n$, \: $u_n = \np{23600} \times 1,04^n + \np{3900}$.
		\item Déterminer la limite de la suite $\left(u_n\right)$ et en donner une interprétation dans le contexte de l'exercice.
	\end{enumerate}
\end{enumerate}


\begin{multicols}{2}
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\subsection{}
On considère les suites $\left(u_n\right)$ et $\left(v_n\right)$ définies pour tout entier nturel $n$ par :
\[\begin{cases}u_0=1\\u_{n+1}=\dfrac{3u_n+2v_n}{5}\end{cases}\text{ et }\begin{cases}v_0=2\\v_{n+1}=\dfrac{2u_n+3v_n}{5}\end{cases}.\]

\begin{enumerate}
\item Calculer $u_1$ , $v_1$ , $u_2$ et $v_2$.

\item On considère la suite $\left(d_n\right)$ définie pour tout entier naturel $n$ par $d_n=v_n-u_n$.
\begin{enumerate}
\item Montrer que la suite $\left(d_n\right)$ est une suite géométrique dont on donnera sa raison et son premier terme.

\item En déduire l’expression de $d_n$ en fonction de $n$.
\end{enumerate}

\item On considère la suite $\left(s_n\right)$ définie pour tout entier naturel $n$ par $s_n=u_n+v_n$.
\begin{enumerate}
\item Calculer $s_0$ , $s_1$ et $s_2$ . Que peut-on conjecturer ?

\item Montrer que, pour tout $n\in\mathbb{N}$ , $s_{n+1}=s_n$. Qu’en déduit-on ?
\end{enumerate}

\item En déduire une expression de $u_n$ et $v_n$ en fonction de $n$.

\item Déterminer, en fonction de $n\in\mathbb{N}$ :
\begin{enumerate}
\item $T_n=u_0+u_1+\cdots+u_n$.

\item $W_n=v_0+v_1+\cdots+v_n$.
\end{enumerate}
\end{enumerate}


\end{multicols}


\subsection{\textcolor{blue}{D'après le livre CQFD 1\iere{}}}%page 78
Dans un lointain futur, sur la planète Thêta, vivent Horion et Apolline au sein d’une colonie d’humains, gouvernée avec bienveillance par une I.A. Mais tout n’est pas si paisible sur la planète Thêta\dots
\og{}Le tapis est mité\fg{} constate Horion.\og{}Tu pourrais au moins
cacher ta joie\fg{} s’emporte Apolline qui adorait son tapis laineux.\\
Le tapis carré de 1 m de côté, ravissant, selon les goûts d’Apolline, hideux selon les critères d’Horion, est effectivement attaqué par des mites.
\og{}Il s’agit de mites de l’espèce Waclaw\fg{}, croit bon de préciser l’I.A. à qui personne n’avait rien demandé, exaspérant davantage Apolline.\\
Les mites ont leur stratégie ; le premier jour, elles s’attaquent à la partie centrale du tapis : si on partage le tapis
carré en 9 carrés de même aire, les mites mangent le carré central. Le second jour, elles s’attaquent de la même
manière aux 8 carrés autour du carré manquant, et ainsi de suite pour les autres jours.\\
(Remarque : la figure \og{}limite\fg{} qu'on obtient ainsi d'appelle le \og{}tapis de Sierpinski\fg{}, du nom du mathématicien polonais Waclaw Franciszek Sierpinski 1882-1961)



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\end{multicols}

\end{center}

Horion, ravi, commente : \og{}Au final, cela fera un joli tapis en dentelle, et dans quelques jours elles feront des troussi petits qu’on ne les verra plus. \fg{}.\\
Apolline lui répond, furieuse : \og{}Le tapis va disparaître complètement ! \fg{}. Horion n’en espérait pas tant !\\
Étudions ces deux affirmations ;\\
Pour tout entier $n\geqslant$, on désigne par $c_n$ la longueur, en m, du côté d’un trou carré qui apparaît au $n$-ième jour et $A_n$ l’aire, en m$^2$, de la surface totale mangée par les mites après $n$ jours.
\begin{enumerate}
\item Montrer que $\left(c_n\right)$ est une suite géométrique et, pour tout entier $n\geqslant$, exprimer $c_n$ en fonction de $n$.

\item Quelle est la valeur de $A_1$ ?\\
Vérifier ensuite que $A_2=\dfrac{17}{81}$

\item Chaque jour, les mites mangent $\dfrac{1}{9}$ de la partie du tapis restante.\\
Montrer que, pour tout entier $n\geqslant$, on a $A_{n+1}=\dfrac{8}{9}A_n+\dfrac{1}{9}$.

\item 
\begin{enumerate}
\item Pour tout entier $n\geqslant$, on pose $B_n= A_n-1$. Montrer que $\left(B_n\right)$ est une suite géométrique.

\item Pour tout entier $n\geqslant$, en déduire l’expression de $B_n$, puis de $A_n$ en fonction de $n$.
\end{enumerate}

\item 
\begin{enumerate}
\item À partir de combien de jours la surface du tapis sera-t-elle mangée à 99\:\% au moins ?

\item Sachant qu’à une distance supérieure à 1 m, un oeil humain normalement constitué ne peut distinguer de détail sur un tapis inférieur à 0,5 mm, au bout de combien de jours ne verra-t-on plus les nouveaux trous formés ?
\end{enumerate}

\item Que penser de chacune des affirmations de Horion et d’Apolline ?
\end{enumerate}

\subsection{}
Résoudre dans $\mathbb{R}$ :



\begin{multicols}{2}
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\begin{enumerate}[a)]
\item $\mathrm{e}^{x^2+6x+5}\geqslant 1$

\item $\mathrm{e}^{-x^2-3x+5}\geqslant \mathrm{e}$.
\end{enumerate}


\end{multicols}

\begin{multicols}{2}
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\subsection{}

\begin{enumerate}
\item On définit sur $\mathbb{R}$ la fonction $g : x \mapsto  x^2\mathrm{e}^{x} - 1$.
\begin{enumerate}
\item Déterminer une expression de la dérivée de $g$.

\item Donner le tableau de signes de cette dérivée sur $\mathbb{R}$.

\item En déduire le tableau de variations de $g$ sur $\mathbb{R}$.

\item En déduire que l'équation $g(x)=0$ admet une solution unique dans $\mathbb{R}$.

\item En déduire le tableau de signes de g(x) sur $\mathbb{R}$.
\end{enumerate}

\item On considère la fonction $f : x \mapsto  \mathrm{e}^{x} + \dfrac{1}{x}$ définie et dérivable sur $\mathbb{R}^*$.
\begin{enumerate}
\item Expliquer pourquoi la fonction $f$ n’est pas définie en 0.

\item Déterminer une expression de la dérivée de $f$.

\item Donner le tableau de signes de cette dérivée sur $\mathbb{R}^*$.

\item En déduire le tableau de variations de f sur $\mathbb{R}^*$.

\end{enumerate}

\end{enumerate}


\end{multicols}
\label{fin}
\end{document}