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a/TST/12_Fonction_inverse/2E_manip_inverse_derivation.tex b/TST/12_Fonction_inverse/2E_manip_inverse_derivation.tex new file mode 100644 index 0000000..e8a143b --- /dev/null +++ b/TST/12_Fonction_inverse/2E_manip_inverse_derivation.tex @@ -0,0 +1,23 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Fonction inverse - exercices } +\date{Mai 2021} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, +} + +\setlength\columnsep{10pt} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} +\vfill +\printcollection{banque} +\vfill + +\end{document} diff --git a/TST/12_Fonction_inverse/exercises.tex b/TST/12_Fonction_inverse/exercises.tex index 683e797..14d1e4e 100644 --- a/TST/12_Fonction_inverse/exercises.tex +++ b/TST/12_Fonction_inverse/exercises.tex @@ -1,10 +1,41 @@ \collectexercises{banque} -\begin{exercise}[subtitle={<++>}, step={1}, origin={<++>}, topics={Fonction inverse}, tags={fonctions inverse}] - <++> +\begin{exercise}[subtitle={"factorisation"}, step={2}, origin={Création}, topics={Fonction inverse}, tags={fonctions inverse}] + Démontrer les égalités suivantes + \begin{multicols}{3} + \begin{enumerate} + \item $x + 1 + \dfrac{1}{x} = \dfrac{x^2 + x + 1}{x}$ + \item $x + 1 + \dfrac{-1}{x^2} = \dfrac{x^3 + x^2 - 1}{x^2}$ + + \item $2x - 5 + \dfrac{5}{x^2} = \dfrac{2x^3 -5 x^2 + 5}{x^2}$ + \item $\dfrac{3}{x} + 2x + 1= \dfrac{2x^2 + x + 3}{x}$ + + \item $1 - \dfrac{121}{x^2} = \dfrac{(x-11)(x+11)}{x^2}$ + \item $9 - \dfrac{1}{x^2} = \dfrac{(3x - 1)(3x+1)}{x^2}$ + \end{enumerate} + \end{multicols} \end{exercise} -\begin{solution} - <++> -\end{solution} - -\collectexercisesstop{banque} \ No newline at end of file +\begin{exercise}[subtitle={Dérivation}, step={2}, origin={Création}, topics={Fonction inverse}, tags={fonctions inverse}] + \begin{enumerate} + \item Dériver les fonctions suivantes + \begin{multicols}{4} + \begin{enumerate} + \item $f(x) = x - 6 + \dfrac{4}{x}$ + \item $g(x) = 2x + 4 + \dfrac{8}{x}$ + \item $h(x) = x + 2 + \dfrac{1}{x}$ + \item $i(x) = 3x + 40 + \dfrac{2700}{x}$ + \end{enumerate} + \end{multicols} + \item En réutilisant les fonctions ci-dessus démontrer que l'on peut mettre leur dérivée sous la forme suivante + \begin{multicols}{2} + \begin{enumerate} + \item $f'(x) = \dfrac{(x-2)(x+2)}{x^2}$ + \item $g'(x) = \dfrac{(2x-2)(x+1)}{x^2}$ + \item $h'(x) = \dfrac{(x-1)(x+1)}{x^2}$ + \item $i'(x) = \dfrac{3(x-30)(x+30)}{x^2}$ + \end{enumerate} + \end{multicols} + \item Pour chacune des fonctions, étudier le signe de leur dérivée puis en déduire leurs variations. + \end{enumerate} +\end{exercise} +\collectexercisesstop{banque} diff --git a/TST/12_Fonction_inverse/index.rst b/TST/12_Fonction_inverse/index.rst index 5f1b653..f7c8007 100644 --- a/TST/12_Fonction_inverse/index.rst +++ b/TST/12_Fonction_inverse/index.rst @@ -2,7 +2,7 @@ Fonction inverse ################ :date: 2021-05-06 -:modified: 2021-05-17 +:modified: 2021-05-18 :authors: Benjamin Bertrand :tags: Fonctions inverse :category: TST @@ -25,9 +25,21 @@ Bilan: nécessité d'étudier la fonction 1/x pour l'étude du coût moyen :height: 200px :alt: Définition et cadre de la fonction inverse -Étape 2: Bastonage sur des exercices types. -=========================================== +Étape 2: manipulation de la fonction inverse et dérivation +========================================================== + +Après la lecture du cours et un exemple au tableau de "factorisation" d'un polynôme avec une fonction inverse, les élèves s'entrainent. On introduira ensuite la dérivée. Ou alors on le fait dans l'autre sens...! + +.. image:: ./2E_manip_inverse_derivation.pdf + :height: 200px + :alt: Dérivation et factorisation avec la fonction inverse + +Bilan: Dérivée de la fonction inverse .. image:: ./2B_derivation.pdf :height: 200px :alt: Dérivée de la fonction inverse + +Étape 3: Bastonage sur des exercices types. +=========================================== +