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+\DeclareExerciseCollection{banque} +\xsimsetup{ + step={5}, +} + +\begin{document} + +Dans les exercices suivants, vous aurez un moment besoin de calculer l'équation d'une tangente. Je vous rappelle que l'équation de la tangente à la courbe représentative de $f$ au point d'abscisse $a$ se calcule avec la formule suivante +\[ + y = f'(a)(x-a) + f(a) + \] + +\input{banque.tex} +\printcollection{banque} + +\end{document} diff --git a/Tsti2d/Analyse/Operation_limites/banque.tex b/Tsti2d/Analyse/Operation_limites/banque.tex index ee673da..c8b9ab5 100644 --- a/Tsti2d/Analyse/Operation_limites/banque.tex +++ b/Tsti2d/Analyse/Operation_limites/banque.tex @@ -195,4 +195,173 @@ \item Compléter le tableau de variations en y ajoutant les limites et les valeurs remarquables. \end{enumerate} \end{exercise} + +\begin{exercise}[subtitle={Vitesse de rotation}, step={5}, topics={Limite}] + % Polynésie Sept 2018 Ex 4 + On considère la fonction $w$ définie pour tout réel positif $t$ par : + + \[w(t) = 4 \text{e}^{-200t} + 146.\] + + On note $C$ la courbe représentative de la fonction $w$ dans un repère orthonormé. + + \medskip + + \begin{enumerate} + \item + \begin{enumerate} + \item Calculer $w(0)$. + \item Déterminer la limite de la fonction $w$ lorsque $t$ tend vers $+ \infty$ et interpréter graphiquement cette limite. + \end{enumerate} + \item On note $w'$ la fonction dérivée de la fonction $w$ sur l'intervalle + $[0~;~+ \infty[$. + \begin{enumerate} + \item Pour tout réel positif $t$, calculer $w'(t)$. + \item Étudier le signe de $w'$ sur l'intervalle $[0~;~+ \infty[$. + \item Dresser le tableau de variation de la fonction $w$ sur l'intervalle $[0~;~+ \infty[$. + \item Déterminer une équation de la tangente à la courbe $C$ au point d'abscisse 0 . + \end{enumerate} + \end{enumerate} + + \bigskip + + \textbf{Partie B} + + \medskip + + On étudie l'évolution de la vitesse d'un moteur dont la vitesse de rotation à vide est de + $150$~rad.s$^{-1}$. + + On s'intéresse à une phase particulière appelée phase d'embrayage. + + Durant cette phase, la vitesse de rotation du moteur, exprimée en rad.s$^{-1}$, est modélisée par une fonction solution de l'équation différentielle $(E)$ : + + \[\dfrac{1}{200}y' + y = 146\] + + où $y$ désigne une fonction dérivable de la variable réelle $t$ positive et exprimée en seconde. + + \medskip + + \begin{enumerate} + \item + \begin{enumerate} + \item Résoudre cette équation différentielle. + \item Vérifier que la fonction $w$ étudiée dans la \textbf{partie A} est la fonction solution de l'équation différentielle $(E)$ vérifiant la condition initiale $w(0) = 150$. + \end{enumerate} + \item Interpréter, dans le contexte de l'exercice, la limite de $w(t)$ lorsque $t$ tend vers $+ \infty$ ainsi que le sens de variation de la fonction $w$, déterminés dans la \textbf{partie A}. + \item On considère que la vitesse de rotation du moteur, exprimée en rad.s$^{-1}$, est stabilisée + + lorsque la quantité $\dfrac{w(t)-146}{146}$ est inférieure à $0,01$. + + Calculer le temps mis par le moteur pour stabiliser sa vitesse. On donnera la valeur + exacte puis la valeur arrondie au millième de seconde. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Carbon 14}, step={5}, topics={Limite}] + % Nouvelle Calédone Novembre 2018 Ex1 + \emph{Les trois parties de cet exercice peuvent être traitées de façon indépendante. } + + \smallskip + + Les organismes vivants contiennent naturellement du carbone 14 (élément radioactif) provenant des rayons cosmiques, qui est constamment renouvelé et qui se maintient à la valeur de 15,3 unités. + + À leur mort, l'assimilation cesse et le carbone 14 présent se désintègre. + + \smallskip + + On note $f(t)$ la concentration en carbone 14 présent dans un organisme à l'instant $t$ après sa mort (t exprimé en milliers d'années). + + \bigskip + + \textbf{Partie A :} + + \medskip + + On admet que $f$ est une solution sur $[0~;~+\infty[$ de l'équation différentielle : + + \[ y' = - 0,124y \qquad (E).\] + + \smallskip + + \begin{enumerate} + \item Résoudre l'équation différentielle $(E)$. + \item Déterminer la solution $f$ de $(E)$ vérifiant la condition initiale $f(0) = 15,3$. + \end{enumerate} + + \bigskip + + \textbf{Partie B :} + + \medskip + + On admet que la fonction $f$ est définie par $f(t) = 15,3\text{e}^{-0,124t}$ sur + $[0~;~+\infty[$. + + \medskip + + \begin{enumerate} + \item Déterminer les variations de $f$ sur $[0~;~+\infty[$. + \item Déterminer la limite de $f$ au voisinage de l'infini. + + Interpréter ce résultat dans le contexte de l'énoncé. + \end{enumerate} + + \bigskip + + \textbf{Partie C :} + + \medskip + + On rappelle que la fonction $f$ donnée dans la partie B donne la concentration en carbone 14 dans un organisme après sa mort en fonction de $t$ (en milliers d'années). + + \medskip + + \begin{enumerate} + \item Des archéologues ont trouvé des fragments d'os présentant une concentration en carbone 14 égale à $7,27$~unités. + + Justifier que l'on peut estimer l'âge de ces fragments d'os à \np{6000}~ans. + \item Lorsque la concentration en carbone 14 d'un organisme devient inférieure à $0,3$\,\% de sa valeur initiale on ne peut pas dater raisonnablement à l'aide du carbone 14. + + Déterminer l'âge à partir duquel un organisme ne peut plus être daté au carbone 14. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Octane}, step={5}, topics={Limite}] + % Métropole Mai 2019 Ex 3 + 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 Calculer, en justifiant votre réponse, $\ds \lim_{t\to +\infty} f(t)$. + + 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/Tsti2d/Analyse/Operation_limites/index.rst b/Tsti2d/Analyse/Operation_limites/index.rst index 49ec29d..e66a8dd 100644 --- a/Tsti2d/Analyse/Operation_limites/index.rst +++ b/Tsti2d/Analyse/Operation_limites/index.rst @@ -2,7 +2,7 @@ Opération sur les limites de fonctions pour l'année 2019-2020 en terminale STI ################################################################################ :date: 2020-05-03 -:modified: 2020-05-08 +:modified: 2020-05-23 :authors: Bertrand Benjamin :category: Tsti2d :tags: Fonctions, Limites @@ -62,6 +62,16 @@ Cours: limite d'une fonction composée avec exponentielle ou logarithme :height: 200px :alt: Limites d'une fonction composée avec exponentielle ou logarithme +Exercices + +.. image:: 4E_composee.pdf + :height: 200px + :alt: Exercices sur les limites de fonctions contenant du log et de l'exp Étape 6: Annales ================ + +.. image:: 5E_annales.pdf + :height: 200px + :alt: Exercices d'annales avec des questions sur les limites. +