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% --- Basic Settings ---
% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{2BAC.PC/SVT}
\def\examtitle{Devoir 03 - S01, Par R. MOUNCIF}
\def\schoolname{Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{visit ~~\wsite~~ for more!}}
\def\province{Direction provinciale\newline de Beni Mellal}
\def\logo{\includegraphics[width=\linewidth]{images/logo-men.png}}
\def\wsite{\href{https://www.mosaid.xyz}{\textcolor{magenta}{\texttt{www.mosaid.xyz}}}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}


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% Custom theorem environments (uncomment if needed)
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% Additional packages (uncomment if needed)
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\begin{document}



\vspace*{-0.75cm}
% Exercise 1 (originally 1)
\printexo{1}{}{
On considère la fonction numérique \(f\) définie sur \(]-\infty; 3[\) par :
~\(
f(x) = \frac{x}{\sqrt{3-x}} \)~
 et soit  ~$(C_f)$~ sa courbe représentative dans un repère orthonormé ~$ (O; \vec{i}; \vec{j})$~.
\begin{enumerate}[tight]
    \item Calculer :
    ~\(
    \lim_{x \to 3^-} f(x), \text{ puis interpréter le résultat graphiquement.}
    \)~

    \item Montrer que :
    ~\(
    \lim_{x \to -\infty} f(x) = -\infty \text{ et que : } \lim_{x \to -\infty} \frac{f(x)}{x} = 0,
    \)~
    puis en déduire la branche infinie de \((C_f)\) au voisinage de \(-\infty\).

    \item Montrer que :
    ~\(
    \forall x \in ]-\infty; 3[ : f'(x) = \frac{6-x}{2\sqrt{3-x}(3-x)},
    \)~
    puis justifier que \(f\) est strictement croissante sur \(]-\infty; 3[\).

    \item
    \begin{enumerate}[tight]
        \item Montrer que :
        ~\(
        \forall x \in ]-\infty; 3[ : f(x) - x = \frac{x(x-2)}{\sqrt{3-x}(1+\sqrt{3-x})}.
        \)~
        \item En déduire la position relative de \((C_f)\) et la droite (D) d’équation \(y = x\).
    \end{enumerate}

    \item Déterminer l’équation de la tangente \((T')\) à la courbe \((C_f)\) au point d’abscisse 2.

    \item Construire les droites (D), \((T')\) et la courbe \((C_f)\) dans le repère \((O; \vec{i}; \vec{j})\).

    \item
    \begin{enumerate}[tight]
        \item Montrer que \(f\) admet une fonction réciproque \(f^{-1}\) définie sur un intervalle \(J\) à déterminer.
        \item Montrer que \(f^{-1}\) est dérivable en 0, puis montrer que :
        ~\(
        (f^{-1})'(0) = \sqrt{3}.
        \)~
        \item Déterminer l’équation de la tangente \((T'')\) à la courbe \((C_{f^{-1}})\) au point d’abscisse 0.
        \item Construire, dans le même repère \((O; \vec{i}; \vec{j})\), la courbe \((C_{f^{-1}})\). (Utiliser deux couleurs différentes).
    \end{enumerate}
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
On considère la suite numérique \((u_n)\) définie par :
~\(
u_0 = 5 \text{ et } u_{n+1} = \frac{7u_n+4}{2u_n+5} \text{ pour tout } n \in \mathbb{N}.
\)~
\begin{enumerate}[tight]
    \item
    \begin{enumerate}[tight]
        \item Montrer que : \(\forall n \in \mathbb{N} : u_n \geq 2\).
        \item Montrer que : \(\forall n \in \mathbb{N} : u_{n+1} - u_n = \frac{-2(u_n-2)(u_n+1)}{2u_n+5}\).
        \item En déduire que la suite \((u_n)\) est décroissante puis que : \(\forall n \in \mathbb{N} : 2 \leq u_n \leq 5\).
        \item En déduire que la suite \((u_n)\) est convergente.
    \end{enumerate}

    \item
    \begin{enumerate}[tight]
        \item Montrer que : \(\forall n \in \mathbb{N} : |u_{n+1} - 2| \leq \frac{1}{3}(u_n - 2)\).
        \item En déduire que : \(\forall n \in \mathbb{N} : |u_n - 2| \leq 3\left(\frac{1}{3}\right)^n\).
        \item Calculer \(\lim u_n\).
    \end{enumerate}

    \item On pose : \(\forall n \in \mathbb{N} : v_n = \frac{u_n-2}{u_n+1}\).
    \begin{enumerate}[tight]
        \item Montrer que \((v_n)\) est une suite géométrique de raison \(q = \frac{1}{3}\) et déterminer son premier terme.
        \item Calculer \(v_n\) en fonction de \(n\) et en déduire que :
        ~\(
        \forall n \in \mathbb{N} : u_n = \frac{2 + \frac{1}{2}(\frac{1}{3})^n}{1 - \frac{1}{2}(\frac{1}{3})^n}.
        \)~
        \item Retrouver de nouveau : \(\lim u_n\).
    \end{enumerate}

    \item On considère la somme :
    ~\(
    S_n = \sum_{i=0}^n v_i = v_0 + v_1 + v_2 + \cdots + v_n.
    \)~
    Montrer que :
    ~\(
    S_n = \frac{3}{4} - \frac{1}{4}\left(\frac{1}{3}\right)^n.
    \)~

    \item Calculer en fonction de \(n\), le produit :
    ~\(
    P_n = \prod_{i=1}^n v_i = v_1 \times v_2 \times v_3 \times \cdots \times v_n.
    \)~

    \item On pose : \(\forall n \in \mathbb{N} : w_n = \sqrt{7 + u_n}\), calculer \(\lim_{n \to \infty} w_n\).
\end{enumerate}
}





% Bottom message
\vspace*{-0.5cm}
\begin{center}
  \normalsize{%
    \RL{\arabicfont
    ﴿وَمَا خَلَقْنَا ٱلسَّمَـٰوَٰتِ وَٱلْأَرْضَ وَمَا بَيْنَهُمَآ إِلَّا بِٱلْحَقِّ ۗ وَإِنَّ ٱلسَّاعَةَ لَـَٔاتِيَةٌ ۖ فَٱصْفَحِ ٱلصَّفْحَ ٱلْجَمِيلَ﴾ (الحجر 85)
    }%
  }%
\end{center}

\end{document}