From 6da20efdd66845314a2e5789d76b5dcc60246d2a Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Tue, 23 Mar 2021 18:40:21 +0100 Subject: [PATCH] Feat: 2E et 2B sur ln --- .../08_Logarithme_Neperien/2B_fonction_ln.pdf | Bin 0 -> 76468 bytes .../08_Logarithme_Neperien/2B_fonction_ln.tex | 72 ++++++++++++++++++ .../2E_etude_fonction.pdf | Bin 0 -> 77888 bytes .../2E_etude_fonction.tex | 22 ++++++ .../08_Logarithme_Neperien/exercises.tex | 44 +++++++++++ TST_sti2d/08_Logarithme_Neperien/index.rst | 15 +++- 6 files changed, 151 insertions(+), 2 deletions(-) create mode 100644 TST_sti2d/08_Logarithme_Neperien/2B_fonction_ln.pdf create mode 100644 TST_sti2d/08_Logarithme_Neperien/2B_fonction_ln.tex create mode 100644 TST_sti2d/08_Logarithme_Neperien/2E_etude_fonction.pdf create mode 100644 TST_sti2d/08_Logarithme_Neperien/2E_etude_fonction.tex diff --git a/TST_sti2d/08_Logarithme_Neperien/2B_fonction_ln.pdf b/TST_sti2d/08_Logarithme_Neperien/2B_fonction_ln.pdf new file mode 100644 index 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logarithme} + +\begin{definition}[ Fonction logarithme népérien ] +La \textbf{fonction logarithme} notée $\ln$ est définie sur $\R^{+*}=\intOO{0}{+\infty}$ par $\ln :x \mapsto ln(x)$. + + \begin{minipage}{0.5\textwidth} + \begin{itemize} + \item Elle est continue et dérivable sur $\R^{+*}$ + \item Elle est négative sur $\intOO{0}{1}$ + \item Elle est positive sur $\intOO{1}{+\infty}$ + \item $\ln(1) = 0$ et $\ln(e) = 1$ + \end{itemize} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=5]{$x$/1,$f(x)$/2}% + {$0$, $+\infty$}% + \tkzTabVar{D-/$-\infty$, +/$+\infty$}% + \end{tikzpicture} + \end{minipage} + \hfill + \begin{minipage}{0.4\textwidth} + \begin{tikzpicture}[yscale=0.8, xscale=1] + \tkzInit[xmin=0,xmax=6,xstep=1, + ymin=-3,ymax=3,ystep=1] + \tkzGrid + \tkzAxeXY[up space=0.5,right space=.5] + \tkzFct[domain = 0.01:6, line width=1pt]{log(x)} + \tkzText[draw,fill = brown!20](5,-2.5){$f(x)=\ln(x)$} + \end{tikzpicture} + \end{minipage} + +\end{definition} + + +\begin{propriete}[(admise) Dérivée ] +La dérivée de la fonction logarithme est la fonction inverse +\[ + \forall x \in \R \qquad \ln'(x) = \frac{1}{x} +\] + +On en déduit, pour tout $x > 0$: +\begin{itemize} + \item $\ln'(x) = \frac{1}{x}$ et $\frac{1}{x} > 0$ alors la fonction logarithme est \dotfill +\end{itemize} +\end{propriete} + +\paragraph{Exemple de calcul} + +On souhaite étudier les variations de $f(x) = 5x + \ln(x)$ +\begin{itemize} + \item Valeur de $x$ possibles - ensemble de définition. + \item Démontrons que la dérivée de $f(x)$ est égale à $f'(x) = \frac{5x + 1}{x}$ + \item Étudions le signe de $f'(x)$ puis les variations de $f(x)$. +\end{itemize} +\afaire{} + + + + + +\end{document} diff --git a/TST_sti2d/08_Logarithme_Neperien/2E_etude_fonction.pdf b/TST_sti2d/08_Logarithme_Neperien/2E_etude_fonction.pdf new file mode 100644 index 0000000000000000000000000000000000000000..45ded16b23fe270ace9a0fdf13bb6f8b941fb120 GIT binary patch literal 77888 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b/TST_sti2d/08_Logarithme_Neperien/2E_etude_fonction.tex new file mode 100644 index 0000000..3830af5 --- /dev/null +++ b/TST_sti2d/08_Logarithme_Neperien/2E_etude_fonction.tex @@ -0,0 +1,22 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Logarithme Népérien - Exercices} +\date{mars 2021} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, +} + +\begin{document} +\setcounter{exercises}{3} + +\input{exercises.tex} +\printcollection{banque} +\vfill +\printcollection{banque} +\vfill + +\end{document} diff --git a/TST_sti2d/08_Logarithme_Neperien/exercises.tex b/TST_sti2d/08_Logarithme_Neperien/exercises.tex index 8643bd2..664ad6b 100644 --- a/TST_sti2d/08_Logarithme_Neperien/exercises.tex +++ b/TST_sti2d/08_Logarithme_Neperien/exercises.tex @@ -50,5 +50,49 @@ \end{multicols} \end{exercise} +\begin{exercise}[subtitle={Éléments remarquables du logarithme}, step={2}, topics={Logarithme}] + \begin{enumerate} + \item Tracer l'allure de la courbe représentative du logarithme. + \item Repérer les éléments remarquables de cette représentation graphique. + \item Tracer le tableau de signe de $\ln$. + \item Tracer le tableau de variation de $\ln$. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Dériver les fonctions}, step={2}, topics={Logarithme}] + Dériver les fonctions suivantes puis mettre sous une forme pratique pour l'étude de signe. + \begin{multicols}{3} + \begin{enumerate} + \item $f(x) = x-2-\ln(x)$ + \item $f(x) = 2x^2 - 2x + 4\ln(x)$ + \item $f(x) = x\ln(x)$ + \item $f(x) = (x+1)\ln(x)$ + \item $f(x) = (\ln(x) + 1)^2$ + \item(*) $f(x) = \frac{1 + \ln(x)}{x}$ + \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) = 10x - 15\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{10x-15}{x}$. + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Étude de fonction - Bis}, 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 Étude du numérateur de $f'(x)$: $N(x) = -x^2 + 2x + 15$ + \begin{enumerate} + \item Démontrer que $x=5$ et $x=-3$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} \collectexercisesstop{banque} diff --git a/TST_sti2d/08_Logarithme_Neperien/index.rst b/TST_sti2d/08_Logarithme_Neperien/index.rst index 8702040..6f94620 100644 --- a/TST_sti2d/08_Logarithme_Neperien/index.rst +++ b/TST_sti2d/08_Logarithme_Neperien/index.rst @@ -2,7 +2,7 @@ Logarithme Népérien ################### :date: 2021-03-18 -:modified: 2021-03-18 +:modified: 2021-03-23 :authors: Benjamin Bertrand :tags: Analyse, Logarithme :category: TST_sti2d @@ -30,9 +30,20 @@ Les élèves n'ont pas à faire tous les calculs, uniquement les premiers de cha Étape 2: Dérivation du logarithme ================================= +Exercices: Étude de variations de fonction utilisant le logarithme. + +.. image:: ./2E_etude_fonction.pdf + :height: 200px + :alt: Exercices autour de la dérivation de ln + +Le bilan est à recopier après avoir fait l'exercice 1. + Cours: Définition de la fonction logarithme népérien, graph, dérivée, signe -Exercices: Étude de variations de fonction utilisant le logarithme. +.. image:: ./2B_fonction_ln.pdf + :height: 200px + :alt: Étude de la fonction ln + Étape 3: Primitive de la fonction inverse =========================================