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+\date{Mai 2020} + +\pagestyle{empty} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step={5}, +} + +\begin{document} + +\input{banque.tex} +\printcollection{banque} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: + diff --git a/1ST/Fonctions_reelle/PolyDeg3/banque.tex b/1ST/Fonctions_reelle/PolyDeg3/banque.tex index 891999a..7d25c59 100644 --- a/1ST/Fonctions_reelle/PolyDeg3/banque.tex +++ b/1ST/Fonctions_reelle/PolyDeg3/banque.tex @@ -148,4 +148,55 @@ \item En déduire le nombre de masque que l'entreprise doit produire pour gagner de l'argent. \end{enumerate} \end{exercise} + +\begin{exercise}[subtitle={Lot}, step={5}, topics={Polynôme degré 3}] + % Inspiré de T1CMATH00290 + Un artisan produit et vend des sachets de viennoiseries. En notant, $x$ le nombre de sachets de viennoiseries ses coûts sont calculables avec la formule suivante: + \[ + C(x) = x^3 - 120x^2 + 10x + \] + \begin{enumerate} + \item Calculer le coût de production pour 75 sachets. + \item Chaque sachet est vendu 10\euro. + \begin{enumerate} + \item Justifier que le bénéfice se calcule alors avec la formule suivante: + \[ + B(x) = x^3 - 120x^2 + \] + \item Tracer l'allure de la courbe représentative de $B(x)$, conjecturer puis démontrer les racines du polynômes. + \item Démontrer que $B(x)$ peut s'écrire + \[ + B(x) = x^2(x-120) + \] + \item Étudier le signe de $B(x)$. + \item En déduire la production maximal avant que l'artisan commence à perdre de l'argent. + \end{enumerate} + \item Recherche du maximum des bénéfices. + \begin{enumerate} + \item Déterminer $B'(x)$ la dérivée de $B(x)$. + \item Montrer que l'on peut écrire + \[ + B'(x) = 3x(x-80) + \] + \item Étudier le signe de $B'(x)$ et en déduire les variations de $B(x)$. + \item En déduire le nombre de sachet que l'artisan doit produire pour maximiser ses bénéfices. + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Population de bactéries}, step={5}, topics={Polynôme degré 3}] + \textit{Cet exercice est un problème ouvert. C'est à dire qu'il y a de nombreuses façon d'y apporter une réponse qui pourra être plus ou moins précise. C'est à vous de choisir les outils qui vous semblent les plus pertinents puis de détailler votre démarche - qui est aussi importante que le résultat final.} + + La population de bactéries dans une solution est modélisée par la fonction suivante + \[ + f(x) = -0,01t^3 + 4t^2 + 2t + \] + où $t$ représente le temps en heure depuis le début de l'expérience. + + + Déterminer quand la population de bactéries va s'éteindre et quand elle aura atteint son maximum. + +\end{exercise} + + \collectexercisesstop{banque} diff --git a/1ST/Fonctions_reelle/PolyDeg3/index.rst b/1ST/Fonctions_reelle/PolyDeg3/index.rst index b5080ba..9d35088 100644 --- a/1ST/Fonctions_reelle/PolyDeg3/index.rst +++ b/1ST/Fonctions_reelle/PolyDeg3/index.rst @@ -2,7 +2,7 @@ Polynômes du 3e degré pour l'année 2019-2020 avec les premières technologiq ############################################################################### :date: 2020-04-10 -:modified: 2020-05-14 +:modified: 2020-05-23 :authors: Bertrand Benjamin :tags: Fonctions, Polynômes :category: 1techno @@ -62,4 +62,10 @@ Exercices sur les racines et la factorisation :height: 200px :alt: Exercices sur l'étude de signe de polynômes +Étape 5: Type E3C +================= + +.. image:: 5E_E3C.pdf + :height: 200px + :alt: Exercices type E3C diff --git a/1ST/semaines.rst b/1ST/semaines.rst index efa92ed..5d3c3e6 100644 --- a/1ST/semaines.rst +++ b/1ST/semaines.rst @@ -63,7 +63,23 @@ S22 - Questions flashs (3x5min) S22 - Polynômes de degré 3 -------------------------- -Exercices type E3C +Exercice 3 sur la racine double. + +.. image:: ./Fonctions_reelle/PolyDeg3/3E_racine_facto.pdf + :height: 200px + :alt: Racine et factorisation de polynômes de degré 3 + +Exercice 2: étude de signe + +.. image:: ./Fonctions_reelle/PolyDeg3/4E_etude_sgn.pdf + :height: 200px + :alt: Racine et factorisation de polynômes de degré 3 + +Exercices type E3C et problèmes ouverts: + +.. image:: ./Fonctions_reelle/PolyDeg3/5E_E3C.pdf + :height: 200px + :alt: Exercice type E3C et problème ouvert. S22 - Programmation -------------------