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mode 100644 index 0000000..9b7eae9 --- /dev/null +++ b/TES/Logarithme/Etude_fonction/2E_variation.tex @@ -0,0 +1,23 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\title{Étude des variations} +\tribe{Terminale TESL} +\date{Mai 2020} + +\pagestyle{empty} +\geometry{left=10mm,right=10mm, top=10mm} + +\renewcommand{\baselinestretch}{0.8} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, +} + +\begin{document} + +\input{banque.tex} +\printcollection{banque} + +\end{document} diff --git a/TES/Logarithme/Etude_fonction/banque.tex b/TES/Logarithme/Etude_fonction/banque.tex index 4e45524..915a8b8 100644 --- a/TES/Logarithme/Etude_fonction/banque.tex +++ b/TES/Logarithme/Etude_fonction/banque.tex @@ -35,4 +35,97 @@ \end{enumerate} \end{multicols} \end{exercise} + +\begin{exercise}[subtitle={Étude de fonction}, step={2}, topics={Logarithme}] + On considère la fonction $f$ définie sur $\intFF{1}{11}$ par + \[ + f(x) = -0.5x^2 + 2x + 15\ln(x) + \] + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est + \[ + f'(x) = \frac{-x^2 + 2x + 15}{x} + \] + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \item Montrer que l'équation $f(x) = 0$ admet une unique solution, $\alpha$, sur $\intFF{1}{11}$. + \item Donner une valeur approchée de $\alpha$. + \item En déduire le tableau de signe de $f$. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Étude de fonction}, step={2}, topics={Logarithme}] + On considère la fonction $f$ définie sur $\intFO{0}{+\infty}$ par + \[ + f(x) = \frac{1 + \ln(x)}{x} + \] + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est + \[ + f'(x) = \frac{-\ln(x)}{x^2} + \] + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \item Déterminer le minimum de la fonction $f$. + \item En déduire le tableau de signe de $f$. + \end{enumerate} +\end{exercise} + + +\begin{exercise}[subtitle={Recherche par dichotomie}, step={2}, topics={Logarithme}] + On considère la fonction $f$ définie sur $\intFF{1}{5}$ par + \[ + f(x) = 3x -10 + 4\ln(x) + \] + \begin{enumerate} + \item + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est + \[ + f'(x) = \frac{3x + 4}{x} + \] + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \item Montrer que l'équation $f(x) = 0$ admet une unique solution, $\alpha$, sur $\intFF{1}{5}$. + \end{enumerate} + \item On souhaite trouver un encadrement de $\alpha$ par la méthode de dichotomie. + + Pour cela, on propose l'algorithme suivant: + \begin{center} + \begin{minipage}{0.5\textwidth} + \begin{algorithm}[H] + \SetAlgoLined + $a \leftarrow 1$ \; + $b \leftarrow 5$ \; + \Tq{$b-a \leq 0.01$}{ + $m \leftarrow \dfrac{b+a}{2}$ \; + \eSi{f(m) > 0}{ + $a \leftarrow m$\; + }{ + $b \leftarrow m$\; + } + } + \Retour{$a, b$} + \end{algorithm} + \end{minipage} + \end{center} + \begin{enumerate} + \item En vous aidant du tableau ci-dessous (vous pouvez ajouter des lignes si nécessaire) exécuter l'algorithme pour trouver un encadrement d'amplitude 0.01 de $\alpha$. + + \begin{center} + \begin{tabular}{|*{5}{p{2cm}|}} + \hline + $a$ & $b$ & $(b-a) \leq 0.01$ & $m$ & $f(m) > 0$ \\ + \hline + & & & & \\ + \hline + & & & & \\ + \hline + & & & & \\ + \hline + \end{tabular} + \end{center} + \item Expliquer le fonctionnement de cet algorithme en quelques phrases. + \end{enumerate} + \end{enumerate} +\end{exercise} + + \collectexercisesstop{banque} diff --git a/TES/Logarithme/Etude_fonction/index.rst b/TES/Logarithme/Etude_fonction/index.rst index 2359159..be2acbc 100644 --- a/TES/Logarithme/Etude_fonction/index.rst +++ b/TES/Logarithme/Etude_fonction/index.rst @@ -2,7 +2,7 @@ #################################################################### :date: 2020-05-05 -:modified: 2020-05-05 +:modified: 2020-05-06 :authors: Bertrand Benjamin :category: TESL :tags: Logarithme @@ -27,8 +27,19 @@ Cours sur la représentation graphique du logarithme et les formules de dérivat Étape 2: Dérivation et étude de variations ========================================== +.. image:: 2E_variation.pdf + :height: 200px + :alt: Exerices techniques d'étude de signe de fonctions + + Étape 3: Calculs d'aires ======================== Étape 4: Annales Bac ==================== + +- Polynésie Juin 2019 ex 4 +- Métro Sept 2019 Ex 3 +- Liban mai 2018 Ex 4 -> dérivation d'un quotient +- Métropole 2017 Ex 4 -> Loi de Benford (Plus de nombre qui commencent par 1 que par 9) +- Polynésie sept 2017 Ex1 -> avec équation de tangente