series exercices : Limite et dérivation

📅 February 01, 2024   |   👁️ Views: 1.07K   |   ❓ 45 questions


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\usepackage{amsmath,amsfonts,amssymb}
\usepackage{enumitem}
\usepackage{multirow}
\usepackage{xcolor}
\usepackage[ddmmyyyy]{datetime}

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\begin{document}
\thispagestyle{empty}
\begin{center}
    \begin{tabularx}{\textwidth}{@{} CCC @{}}
        %\toprule
            \multirow{2}{*}{\parbox{\linewidth}{Prof MOSAID \newline \mylink }}
            & Serie - Limites et dérivation & 1BAC-SE \\
        \bottomrule
    \end{tabularx}
\end{center}
\textbf{\underline{Exercice 1:}}\\
\noindent
\begin{tabular}{@{}p{0.01\textwidth}|p{0.30\textwidth}p{0.30\textwidth}p{0.38\textwidth}}
    & Calculez les limites suivantes : & &\\
    &\begin{enumerate}[topsep=0pt, partopsep=0pt, parsep=0pt, itemsep=0pt]
       \item  $\displaystyle \lim_{x \to 2^-} (3x^3 - 2x^2 + 5x - 1)$
       \item  $\displaystyle \lim_{x \to 1^+} \frac{x^2 - 3x + 2}{x - 1}$
       \item  $\displaystyle \lim_{x \to +\infty} \frac{\sqrt{x} - 3}{x}$
       \item  $\displaystyle \lim_{x \to \pi^-} \frac{\sin(x)}{x}$
       \item  $\displaystyle \lim_{x \to -1^+} (x^4 - 2x^2 + 3)$
       \item  $\displaystyle \lim_{x \to 2^-} \frac{x^2 + x - 6}{x^2 - 4}$
       \item  $\displaystyle \lim_{x \to \frac{\pi}{2}^-} \tan(x)$
    \end{enumerate}
    &\begin{enumerate}[topsep=0pt, partopsep=0pt, parsep=0pt, itemsep=0pt, start=8]
        \item  $\displaystyle \lim_{x \to -4^+} \frac{\sqrt{x^2 + 3x + 4} - 2}{x + 4}$
        \item  $\displaystyle \lim_{x \to 3^+} \frac{x^3 - 2x^2 + 3x - 1}{x^2 - 9}$
        \item  $\displaystyle \lim_{x \to 0} \frac{\sin(2x)}{x}$
        \item  $\displaystyle \lim_{x \to \infty} \frac{\tan(x)}{x}$
        \item  $\displaystyle \lim_{x \to -2^-} \frac{x^3 - 3x^2 - 2x - 2}{x^2 + 4x + 4}$
        \item  $\displaystyle \lim_{x \to 1^+} (x^3 - 3x^2 + 2x - 1)$
        \item  $\displaystyle \lim_{x \to 5^+} \frac{\sqrt{x^2 - 25}}{x - 5}$
    \end{enumerate}
    &\begin{enumerate}[topsep=0pt, partopsep=0pt, parsep=0pt, itemsep=0pt, start=15]
       \item  $\displaystyle \lim_{x \to -3^+} (2x^3 + 5x^2 - 4)$
       \item  $\displaystyle \lim_{x \to -\infty} \frac{2x^2 - 5x}{x}$
       \item  $\displaystyle \lim_{x \to \frac{\pi}{4}^-} \frac{\sin(x)}{x}$
       \item  $\displaystyle \lim_{x \to \infty} \frac{3x^2 - 4x + 1}{x^2 + 2x + 1}$
       \item  $\displaystyle \lim_{x \to -\infty} (2x^3 - 5x^2 + 3x - 4)$
    \end{enumerate} \\
\end{tabular}
\\
\textbf{\underline{Exercice 2:}}\\
\noindent
\begin{tabular}{@{}p{0.01\textwidth}|p{0.98\textwidth}}
    &Determiner l'équation de la tangente puis le nombre dérivé en \(x_0\):\\
    &\hspace*{0.5cm}1.\(\quad\)$x_0 = 1$\hspace*{0.3cm};\hspace*{0.5cm} $f(x) = 2x^3 - 5x^2 + 3x - 4$.\\
    &\hspace*{0.5cm}2.\(\quad\)$x_0 = 2$\hspace*{0.3cm};\hspace*{0.5cm} $f(x) = \frac{1}{x}$.\\
    &\hspace*{0.5cm}3.\(\quad\)$x_0 = 9$\hspace*{0.3cm};\hspace*{0.5cm} $f(x) = \sqrt{x}$ .\\
    &\hspace*{0.5cm}4.\(\quad\)$x_0 = \frac{\pi}{4}$\hspace*{0.3cm};\hspace*{0.5cm} $f(x) = \sin(x)$.\\
\end{tabular}
\\
\\
\textbf{\underline{Exercice 3:}}\\
\noindent
\begin{tabular}{@{}p{0.01\textwidth}|p{0.48\textwidth}p{0.50\textwidth}}
    & Calculez le nombre dérivé à droite et à gauche& pour les fonctions suivantes: \\
    &\hspace*{0.5cm}1.\(\quad\)$x_0 = 0$\hspace*{0.3cm};\hspace*{0.5cm}
    \( f(x) = \begin{cases} x^2 & \text{si } x < 0 \\ 2x + 1 & \text{si } x \geq 0 \end{cases} \).
    &\hspace*{0.5cm}2.\(\quad\)$x_0 = 2$\hspace*{0.3cm};\hspace*{0.5cm}
     \( g(x) = \begin{cases} -x & \text{si } x < 2 \\ x^2 & \text{si } x \geq 2 \end{cases} \).\\
    &\hspace*{0.5cm}3.\(\quad\)$x_1 = 1$\hspace*{0.3cm};\hspace*{0.5cm}
    \( h(x) = \begin{cases} \sqrt{2x} & \text{si } x < 1 \\ 3x^2 & \text{si } x \geq 1 \end{cases} \).
    &\hspace*{0.5cm}4.\(\quad\)$x_0 = -1$\hspace*{0.3cm};\hspace*{0.5cm}
    \( k(x) = \begin{cases} x^3 & \text{si } x < -1 \\ \frac{1}{x} & \text{si } x \geq -1 \end{cases} \).\\
\end{tabular}
\\
\textbf{\underline{Exercice 4:}}\\
\noindent\begin{tabular}{@{}p{0.01\textwidth}|p{0.30\textwidth}p{0.30\textwidth}p{0.38\textwidth}}
    & Calculez \(f'(x)\) pour les &fonctions suivantes:\hspace*{2.5cm}& \\
    &\begin{enumerate}[topsep=0pt, partopsep=0pt, parsep=0pt, itemsep=3pt]
        \item $f(x) = 3x^2 - 4x + 2$
        \item $f(x) = \frac{1}{2}x^3 - 2x^2 + 3x$
        \item $f(x) = \sqrt{x} + 2x$
        \item $f(x) = x^2 + \sqrt{x} - \frac{1}{x}$
        \item $f(x) = \sin(x) + \cos(x)$
        \item $f(x) = \frac{x^2 + 1}{x}$
        \item $f(x) = \tan(x)$
    \end{enumerate}
    &\begin{enumerate}[topsep=0pt, partopsep=0pt, parsep=0pt, itemsep=3pt, start=8]
        \item $f(x) = \frac{\cos(x)}{x^2}$
        \item $f(x) = x \cdot \sqrt{x+1}$
        \item $f(x) = \sqrt{(x^2 + 1)(x+2)}$
        \item $f(x) = x^2 \cdot \sqrt{x}$
        \item $f(x) = \frac{x^3}{\sqrt{x}}$
        \item $f(x) = \frac{\sqrt{x}}{x + 1}$
        \item $f(x) = \sin(x) \cdot \cos(x)$
    \end{enumerate}
    &\begin{enumerate}[topsep=0pt, partopsep=0pt, parsep=0pt, itemsep=3pt, start=15]
        \item $f(x) = \frac{\tan(x)}{\sqrt{x}}$
        \item $f(x) = \sqrt{x} \cdot \sqrt{x^3}$
        \item $f(x) = \frac{\sqrt{2x+1}}{x^2}$
        \item $f(x) = \frac{\cos(x)}{\sqrt{x}}$
        \item $f(x) = x^2 \cdot \frac{x+2}{\sqrt{x+2}}$
        \item $f(x) = \frac{\sqrt{x^3}}{x + 1}$
    \end{enumerate} \\
\end{tabular}
\\
\textbf{\underline{Exercice 5:}}\\
\noindent\begin{tabular}{@{}p{0.01\textwidth}|p{0.48\textwidth}p{0.48\textwidth}}
    & Etudier les variations des fonctions suivantes : & \\
    &\begin{enumerate}[topsep=0pt, partopsep=0pt, parsep=0pt, itemsep=3pt, start=1]
        \item \( f(x) = 3x^3 - 4x^2 - 9x + 2 \)
        \item \( g(x) = \frac{2x^2 - 5x}{x - 3} \)
        \item \( h(x) = 4x^3 - 6x^2 - 15x + 3 \)
    \end{enumerate}
    &\begin{enumerate}[topsep=0pt, partopsep=0pt, parsep=0pt, itemsep=3pt, start=4]
        \item \( k(x) = \frac{x^2 - 3x}{x + 1} \)
        \item \( m(x) = 5x^3 - 2x^2 - 12x + 4 \)
        \item \( n(x) = \frac{3x^2 - 6}{x - 2} \)
    \end{enumerate} \\
\end{tabular}
\\
\\
\textcolor{white}{.}\hfill \underline{MOSAID le \today}\\
\textcolor{white}{.}\hfill \mylink
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Practice each exercise, then check your answers against the provided solutions. Repeat until you master the concepts.

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The course "Dérivation" covers key concepts of maths for 1-bac-science. Designed to help students master the curriculum.

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