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mode 100644 index 0000000..6d979b9 --- /dev/null +++ b/TST_sti2d/05_Fonction_Exponentielle/2E_composee.tex @@ -0,0 +1,23 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Fonction Expronentielle - dérivation et étude de signe} +\date{décembre 2020} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, +} + +\begin{document} + +\setcounter{exercise}{3} + +\input{exercises.tex} +\printcollection{banque} +\vfill +\printcollection{banque} +\vfill + +\end{document} diff --git a/TST_sti2d/05_Fonction_Exponentielle/exercises.tex b/TST_sti2d/05_Fonction_Exponentielle/exercises.tex index 3898322..de0145e 100644 --- a/TST_sti2d/05_Fonction_Exponentielle/exercises.tex +++ b/TST_sti2d/05_Fonction_Exponentielle/exercises.tex @@ -35,4 +35,38 @@ \end{multicols} \end{exercise} +\begin{exercise}[subtitle={Étude de fonctions}, step={2}, origin={Création}, topics={Fonction Expronentielle}, tags={Analyse, exponentielle}] + Calculer la dérivée, étudier son signe et en déduire les variations de la fonction initiale. + \begin{multicols}{3} + \begin{enumerate} + \item $f(x) = e^{-3x}$ , $I = \R$ + \item $g(x) = 100e^{-0.5x + 1}$ , $I=\R$ + \item $h(x) = e^{-x^2}$ , $I = \R$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Décroissance radioactive}, step={2}, origin={Création}, topics={Fonction Expronentielle}, tags={Analyse, exponentielle}] + La loi de décroissance radioactive est décrite par la formule suivant où $t$ représente le temps en $s$, $N(t)$ la quantité d'éléments radioactifs et $\tau$ le temps de demi-vie en $s^{-1}$: $N(t) = N_0 \times e^{-\frac{t}{\tau}}$ + + On fixe $\tau = 2$. + \begin{enumerate} + \item Calculer $N'(t)$ la dérivée de $N(t)$. + \item Étudier le signe de $N'(t)$ et en déduire les variations de $N(t)$. + \item Tracer l'allure de la courbe représentative de $N(t)$. + \item Que peut-on dire de la quantité d'éléments radioactifs après un long moment? + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Charge d'une batterie}, step={2}, origin={Inspiration de l'annal Antille septembre 2019}, topics={Fonction Expronentielle}, tags={Analyse, exponentielle}] + On souhaite charger une batterie de 22kWh. Le profil de charge est décrit par le fonction $c(t) = 22 - 22e^{-0.55t}$ où $t$ décrit le temps en heure. + \begin{enumerate} + \item Calculer et interpréter $c(0)$. + \item Calculer $C'(t)$ la dérivée de $C(t)$. + \item Étudier le signe de $C'(t)$ et en déduire les variations de $C(t)$. + \item Tracer l'allure de la représentation graphique de $C(t)$. + \item Est-il possible de charger entièrement la batterie? + \end{enumerate} +\end{exercise} + \collectexercisesstop{banque} diff --git a/TST_sti2d/05_Fonction_Exponentielle/index.rst b/TST_sti2d/05_Fonction_Exponentielle/index.rst index d51a777..c061051 100644 --- a/TST_sti2d/05_Fonction_Exponentielle/index.rst +++ b/TST_sti2d/05_Fonction_Exponentielle/index.rst @@ -2,7 +2,7 @@ Fonction Exponentielle ###################### :date: 2020-12-07 -:modified: 2020-12-07 +:modified: 2021-01-02 :authors: Benjamin Bertrand :tags: Analyse, Exponentielle :category: TST_sti2d @@ -24,4 +24,22 @@ Bilan/cours à recopier à la maison :alt: Cours sur la fonction exp +Étape 2: Composée avec exponentielle +==================================== + +À travers un exemple, on explique comment calculer la dérivée d'une fonction composée avec l'exponentielle. Puis les élèves vont s'exercer sur des exercices techniques puis d'application de l'exponentielle. + +.. image:: ./2E_composee.pdf + :height: 200px + :alt: Exercices sur la dérivation de fonctions composées avec l'exponentielle. + +Cours sur le dérivation de fonctions composées avec exponentielle. + +.. image:: ./2B_composee.pdf + :height: 200px + :alt: Cours sur le dérivation de fonctions composées avec exponentielle. + +Étape 3: intégration de l'exponentielle +======================================= +