From cef32d3530903c8807a23625049936d7568117e7 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Thu, 18 Mar 2021 10:22:21 +0100 Subject: [PATCH] =?UTF-8?q?Feat:=20D=C3=A9but=20du=20chapitre=20sur=20le?= =?UTF-8?q?=20logarithme=20n=C3=A9p=C3=A9rien?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../1B_relations_fonctionnelles.pdf | Bin 0 -> 80085 bytes .../1B_relations_fonctionnelles.tex | 77 ++++++++++++++++++ .../1E_manip_equations.tex | 18 ++++ .../08_Logarithme_Neperien/exercises.tex | 10 +++ TST_sti2d/08_Logarithme_Neperien/index.rst | 41 ++++++++++ 5 files changed, 146 insertions(+) create mode 100644 TST_sti2d/08_Logarithme_Neperien/1B_relations_fonctionnelles.pdf create mode 100644 TST_sti2d/08_Logarithme_Neperien/1B_relations_fonctionnelles.tex create mode 100644 TST_sti2d/08_Logarithme_Neperien/1E_manip_equations.tex create mode 100644 TST_sti2d/08_Logarithme_Neperien/exercises.tex create mode 100644 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zV$criWhwxg!~YwBApFQwFlBKU(bx|xQv(!PgD*+?c&hqX3ptQF=7OWj)-Nv$GMO%V z_=T-zdg+Bl9aXfmtI1V1vxnhT4Hg_%^NfGlJkL{1-aOBH@U*+gwYyo@x+*kk3Ha*R k=lG*)NfZA2V|H*f06V&VcvTSCIa%4+5Xi|z 0$, il existe un unique nombre $b$ tel que $\e^b = a$. + \medskip + + $b$ est appelé \textbf{logarithme népérien} de $a$ et est noté $\ln(a)$. On peut alors noter + \[ + e^b = a \qquad \equiv \qquad b = \ln(a) + \] +\end{definition} + +\begin{propriete} + \begin{itemize} + \item Soit $a$ un nombre réel alors $\ln(\e^a) = a$. + \item Soit $a$ un nombre réel strictement positif alors $\e^{\ln(a)} = a$. + \item Valeurs particulières + \[ + \ln(1) = 0 \qquad \ln(\e) = 1 + \] + \end{itemize} +\end{propriete} + +\paragraph{Exemples}% + +\begin{itemize} + \item Résolution de l'équation $e^{2x - 1} = 2$ + \\[2cm] + \item Résolution de l'équation $\ln(2x + 1) = -2$ + \\[2cm] +\end{itemize} +\afaire{Résoudre ces équations} + +\section{Relations fonctionnelles} + +\begin{propriete}{Relations fonctionnelles} +Soient $a$ et $b$ deux nombres réels strictement positifs et $n$ un entier naturel. +\begin{align*} + \log(a \times b) &= \log(a) + \log(b)\\ + \log(a^n) &= n\log(a) \\ + \log\left( \frac{a}{b} \right) &= \log(a) - \log(b) \\ + \log\left( \frac{1}{a} \right) &= - \log(a) \\ +\end{align*} +\end{propriete} + +\paragraph{Exemple}% + +Soit $f(x) = 10 + 2\ln\left(\frac{5}{4\times x}\right)$. Montrons que l'on peut écrire +\[ + f(x) = 10 + 2\ln(1,25) - 2\ln(x) +\] +\afaire{Démontrer l'égalité} + + + + + + + + + +\end{document} diff --git a/TST_sti2d/08_Logarithme_Neperien/1E_manip_equations.tex b/TST_sti2d/08_Logarithme_Neperien/1E_manip_equations.tex new file mode 100644 index 0000000..2d0a21b --- /dev/null +++ b/TST_sti2d/08_Logarithme_Neperien/1E_manip_equations.tex @@ -0,0 +1,18 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Logarithme Népérien - Cours} +\date{mars 2021} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=1, +} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} + +\end{document} \ No newline at end of file diff --git a/TST_sti2d/08_Logarithme_Neperien/exercises.tex b/TST_sti2d/08_Logarithme_Neperien/exercises.tex new file mode 100644 index 0000000..76b53b6 --- /dev/null +++ b/TST_sti2d/08_Logarithme_Neperien/exercises.tex @@ -0,0 +1,10 @@ +\collectexercises{banque} +\begin{exercise}[subtitle={<++>}, step={1}, origin={<++>}, topics={Logarithme Népérien}, tags={analyse, logarithme}] + <++> +\end{exercise} + +\begin{solution} + <++> +\end{solution} + +\collectexercisesstop{banque} \ No newline at end of file diff --git a/TST_sti2d/08_Logarithme_Neperien/index.rst b/TST_sti2d/08_Logarithme_Neperien/index.rst new file mode 100644 index 0000000..115143f --- /dev/null +++ b/TST_sti2d/08_Logarithme_Neperien/index.rst @@ -0,0 +1,41 @@ +Logarithme Népérien +################### + +:date: 2021-03-18 +:modified: 2021-03-18 +:authors: Benjamin Bertrand +:tags: Analyse, Logarithme +:category: TST_sti2d +:summary: Étude de la fonction logarithme + +Dans cette séquence, le cours sera donné en amont car il y a peu de choses à découvrir. + +Étape 1: Logarithme décimal et logarithme népérien +================================================== + +Cours: Définition du logarithme népérien, lien avec le logarithme décimal et formules + +.. image:: ./1B_relations_fonctionnelles.pdf + :height: 200px + :alt: Cours sur le logarithme népérien - relation fonctionnelles et définition + +Exercices: reprises des exercices déjà faits en tronc commun avec le logarithme décimal adapté au logarithme népérien. + +Étape 2: Dérivation du logarithme +================================= + +Cours: Définition de la fonction logarithme népérien, graph, dérivée, signe + +Exercices: Étude de variations de fonction utilisant le logarithme. + +Étape 3: Primitive de la fonction inverse +========================================= + +Cours: Primitive de la fonction inverse et point sur la primitive de ln + +Exercices: calculs de primitive et d'aires avec la fonction inverse. + +Étape 4: Annales de l'ancien bac sti2d +====================================== + +