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0000000..e3b1910 --- /dev/null +++ b/TST_sti2d/07_Equation_differentielle/4E_mise_situation.tex @@ -0,0 +1,19 @@ +\documentclass[a4paper, 10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Équation differentielle - Cours} +\date{février 2021} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=4, +} + +\begin{document} +\setcounter{exercise}{4} + +\input{exercises.tex} +\printcollection{banque} + +\end{document} diff --git a/TST_sti2d/07_Equation_differentielle/exercises.tex b/TST_sti2d/07_Equation_differentielle/exercises.tex index 9acf5d8..496fb8c 100644 --- a/TST_sti2d/07_Equation_differentielle/exercises.tex +++ b/TST_sti2d/07_Equation_differentielle/exercises.tex @@ -80,4 +80,64 @@ \end{multicols} \end{exercise} +\begin{exercise}[subtitle={Résolution d'équations différentielles}, step={4}, origin={Création}, topics={Équation differentielle}, tags={analyse, exponentiel, dérivation}] + On veut étudier la chute d'un objet dû à l'attraction terrestre. On lâché l'objet à $t=0$ avec une vitesse nulle. + \begin{enumerate} + \item On suppose que que l'objet ne subit pas de frottement. Il est donc uniquement soumis à l'attraction terrestre et donc le bilan des forces donne l'équation différentielle suivante + \[ + \frac{dv}{dt} = 9,81 + \] + où $v(t)$ est la vitesse en fonction du temps. + \begin{enumerate} + \item Déterminer l'expression de la vitesse en fonction du temps. + \item Tracer l'allure de la courbe représentative de la vitesse. Que peut-on dire de la vitesse quand le temps devient très grand? + \end{enumerate} + \item On ajoute une force de résistance dû aux frottements avec l'aire. Le bilan des force donne l'équation différentielle suivante + \[ + \frac{dv}{dt} = -0.5v(t) + 9,81 + \] + \begin{enumerate} + \item Déterminer la solution générale de l'équation différentielle. + \item En prenant en compte, les conditions initiales, déterminer la solution unique de l'équation. + \item Tracer l'allure de la courbe représentative de la vitesse. Que peut-on dire de la vitesse quand le temps devient très grand? + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Résolution d'équations différentielles}, step={4}, origin={Annale}, topics={Équation differentielle}, tags={analyse, exponentiel, dérivation}] + L'octane est un hydrocarbure qui entre dans la composition de l'essence. + + Lorsqu'on chauffe un mélange d'octane et de solvant dans une cuve, une réaction chimique transforme progressivement l'octane en un carburant plus performant, appelé iso-octane. + + La concentration d'octane, en moles par litre, dans la cuve est modélisée par une fonction $f$ du temps $t$, exprimé en minutes. On admet que cette fonction $f$, définie et dérivable sur l'intervalle $[0~;~+\infty[$, est une solution, sur cet intervalle, de l'équation différentielle suivante: + + \[(E)~:~y'+0,12y=0,003.\] + + À l'instant $t = 0$, la concentration d'octane dans la cuve est de $0,5$~mole par litre (mol.L$^{-1}$). + + \begin{enumerate} + \item + \begin{enumerate} + \item Déterminer la solution générale de l'équation différentielle $(E)$. + \item Donner $f(0)$. + \item Vérifier que la fonction $f$ est définie sur $[0~;~+\infty[$ par $f(t) = 0,475\e^{-0,12t}+0,025$. + \end{enumerate} + \item + \begin{enumerate} + \item Calculer la fonction dérivée de la fonction $f$ sur l'intervalle $[0~;~+\infty[$. + \item Étudier le sens de variation de la fonction $f$ sur l'intervalle $[0~;~+\infty[$. + \item Interpréter cette réponse dans le contexte de l'exercice. + \end{enumerate} + \item Calculer, en justifiant votre réponse, à la minute près, le temps nécessaire pour obtenirune concentration en octane dans la cuve de $0,25$ mole par litre. + \item + \begin{enumerate} + \item Par une lecture graphique, déterminer $\ds \lim_{t\to +\infty} f(t)$. + + Interpréter le résultat dans le contexte. + + \item Le processus de transformation de l'octane en iso-octane est arrêté au bout d'une heure. Expliquer ce choix. + \end{enumerate} + \end{enumerate} +\end{exercise} + \collectexercisesstop{banque} diff --git a/TST_sti2d/07_Equation_differentielle/index.rst b/TST_sti2d/07_Equation_differentielle/index.rst index 38b786f..7405e77 100644 --- a/TST_sti2d/07_Equation_differentielle/index.rst +++ b/TST_sti2d/07_Equation_differentielle/index.rst @@ -2,7 +2,7 @@ ####################### :date: 2021-02-07 -:modified: 2021-02-10 +:modified: 2021-02-11 :authors: Benjamin Bertrand :tags: Analyse, Exponentiel, Dérivation :category: TST_sti2d @@ -60,4 +60,8 @@ Bilan: trouver une solution particulière Exercices techniques de résolution. Puis problèmes de mise en situation (bilan des forces - chute libre avec frottement ...). +.. image:: ./4E_mise_situation.pdf + :height: 200px + :alt: Problèmes utilisant les équations différentielles + Bilan: vidéo de correction par un élève?