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Import work from year 2014-2015

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Benjamin Bertrand
2017-06-16 09:48:07 +03:00
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\documentclass{/media/documents/Cours/Prof/Enseignements/Archive/2014-2015/tools/style/classConn}
% Title Page
\title{}
\author{}
\date{}
\begin{document}
\begin{multicols}{2}
Nom - Prénom - Classe:
\section{Connaissance}
\begin{enumerate}
\item Donner la formule du discriminant
\vfill
\begin{eqnarray*}
\Delta & = & \parbox{3cm}{\dotfill}
\end{eqnarray*}
\vfill
\item Combien y a-t-il de solution à l'équation $ax^2 + bx + c = 0$ quand $\Delta = 0$?
\\[0.5cm]
.\dotfill
\\[0.5cm]
\vfill
\item On suppose que $\Delta = 0$ donner les formules pour les deux solutions
\vfill
\begin{eqnarray*}
x_1 & = & \parbox{1cm}{\dotfill} \\[1cm]
x_2 & = & \parbox{1cm}{\dotfill}
\end{eqnarray*}
\vfill
\item Faire le calcul suivant: On donne $a = -1$, $b = 1$ et $c = -2$
\vfill
\begin{eqnarray*}
A = a^2 - 2a - c & = & \parbox{1cm}{\dotfill}
\end{eqnarray*}
\vfill
\end{enumerate}
\columnbreak
Nom - Prénom - Classe
\section{Connaissance}
\begin{enumerate}
\item Donner la formule du discriminant
\vfill
\begin{eqnarray*}
\Delta & = & \parbox{3cm}{\dotfill}
\end{eqnarray*}
\vfill
\item Combien y a-t-il de solution à l'équation $ax^2 + bx + c = 0$ quand $\Delta > 0$?
\\[0.5cm]
.\dotfill
\\[0.5cm]
\item On suppose que $\Delta = 0$ donner les formules pour calculer la solution
\vfill
\begin{eqnarray*}
x_1 & = & \parbox{1cm}{\dotfill}
\end{eqnarray*}
\vfill
\item Faire le calcul suivant: On donne $a = -2$, $b = 1$ et $c = -1$
\vfill
\begin{eqnarray*}
A = a^2 - 2a - c & = & \parbox{1cm}{\dotfill}
\end{eqnarray*}
\vfill
\end{enumerate}
\end{multicols}
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "master"
%%% End:

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\documentclass[a4paper,10pt, table]{/media/documents/Cours/Prof/Enseignements/Archive/2014-2015/tools/style/classCours}
\usepackage{/media/documents/Cours/Prof/Enseignements/Archive/2014-2015/2014_2015}
% Title Page
\titre{Polynôme du 2nd degré}
% \seconde \premiereS \PSTMG \TSTMG
\classe{\PSTMG}
\date{Novembre 2014}
\begin{document}
\maketitle
\section{Polynôme du 2nd degré}
\begin{Def}
$f$ est un polynôme du 2nd degré quand elle est la forme
\begin{eqnarray*}
f(x) & = & ax^2 + bx + c
\end{eqnarray*}
avec $a$, $b$ et $c$ des rééls tels que $a \neq 0$.
\end{Def}
\begin{Ex}
Les coûts de productions d'une entreprise se calcul à partir de la fonction suivante
\begin{eqnarray*}
f(x) & = & x^2 + 5x + 125
\end{eqnarray*}
$f(x)$ est une fonction polynôme du 2nd degré et $a = 1$, $b = 5$ et $c = 125$.
Pour calculer les coûts pour 10 ordinateurs, on remplace les $x$ par 10 dans l'expression de $f$
\begin{eqnarray*}
f(10) & = & 10^2 + 5 \times 10 + 125
\end{eqnarray*}
On dessine le tableau fait en TP et on note comment faire calculer par un ordinateur.
\end{Ex}
\begin{Prop}
Soit $f(x) = ax^2 + bx + c$ alors la courbe représentative de $f$ est une \textbf{parabole} de la forme \textit{2graphiques en fonction du signe de $a$}.
\end{Prop}
\section{Équation du 2nd degré}
\begin{Def}
$f(x) = ax^2 + bx + c$ alors on définit
\begin{eqnarray*}
\Delta & = & b^2 - 4ac
\end{eqnarray*}
$\Delta$ sera le nombre qui déterminera le nombre de solution à une équation.
\end{Def}
\begin{Ex}
3 exemples en fonction du signe de $\Delta$
\end{Ex}
\section{$\Delta$ et les polynômes du 2nd degré}
\begin{Prop}
Tableau de signe et graphique en fonction de $\Delta$ et $a$
\end{Prop}
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "master"
%%% End:

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Notes sur le cours autour des polynômes du 2nd degré pour les 1stmg
###################################################################
:date: 2015-07-01
:modified: 2015-07-01
:tags: Fonctions,Cours
:category: 1er_STMG
:authors: Benjamin Bertrand
:summary: Pas de résumé, note créée automatiquement parce que je ne l'avais pas bien fait...
`Lien vers resume.pdf <resume.pdf>`_
`Lien vers 2nd_deg.tex <2nd_deg.tex>`_
`Lien vers 2nd_deg.pdf <2nd_deg.pdf>`_

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\documentclass[a4paper,10pt, table]{/media/documents/Cours/Prof/Enseignements/Archive/2014-2015/tools/style/classDS}
\usepackage{/media/documents/Cours/Prof/Enseignements/Archive/2014-2015/2014_2015}
\usepackage{tkz-tab}
% Title Page
\titre{Production et bénéfice}
% \seconde \premiereS \PSTMG \TSTMG
\classe{\PSTMG}
\date{11 décembre 2014}
%\duree{1 heure}
%\sujet{%{{infos.subj%}}}
% DS DSCorr DM DMCorr Corr
\typedoc{Corr}
\printanswers
\begin{document}
\maketitle
Une entreprise produit et vend des chaises de bureau. Chacune de ces chaises sont vendus 56\euro d'unité.
Pour un nombre de chaises $q$ produite, le coût de production est donnée par la fonction suivante
\begin{eqnarray*}
C(q) & = & 0,1q^2 + 10q + 450
\end{eqnarray*}
On supposera que toutes les chaises produites sont vendue.
\begin{solution}
On peut extraire de ce texte d'information 3 éléments importants:
\begin{itemize}
\item Une chaise est vendue 56\euro.
\item $q$ : nombre de chaise produite
\item Le coût: $C(q) = 0,1q^2 + 10q + 450$
\end{itemize}
\end{solution}
\begin{enumerate}
\item
\begin{enumerate}
\item Calculer les coûts fixes (quand la production est nulle).
\begin{solution}
On veut calculer $C(q)$ pour $q = 0$ (production nulle)
\begin{eqnarray*}
C(0) & = & 0,1\times 0^2 + 10 \times 0 + 450 \\
C(0) & = & 450
\end{eqnarray*}
Les coûts fixes sont dont de 450.
\end{solution}
\item Calculer le coût de fabrication de 10 chaises.
\begin{solution}
Coût de production pour 10 chaises
\begin{eqnarray*}
C(10) & = & 0,1\times 10^2 + 10 \times 10 + 450 = 560
\end{eqnarray*}
La production de 10 chaises coûtera 560.
\end{solution}
\item Determiner la quantitié de chaises pour que les coûts soient égaux à 690\euro.
\begin{solution}
On cherche $q$ tel que $C(q) = 690$.
\begin{eqnarray*}
0,1q^2 + 10q + 450 & = & 690 \\
0,1q^2 + 10q + 450 - 690 &=& 690 \\
0,1q^2 + 10q - 240 &=& 0
\end{eqnarray*}
On reconnait une équation du second degré avec
\begin{eqnarray*}
a = 0,1 \qquad b = 10 \qquad c = -240
\end{eqnarray*}
On calcule le discriminant
\begin{eqnarray*}
\Delta & = & b^2 - 4ac = 10^2 - 4\times 0,1 \times (-240) = 196
\end{eqnarray*}
$\Delta$ est positif, il y a donc deux solutions.
\begin{eqnarray*}
x_1 & = & \frac{-b - \sqrt{\Delta=}}{2a} = \frac{-10 - \sqrt{196}}{2\times0,1} = -120 \\
x_2 & = & \frac{-b + \sqrt{\Delta=}}{2a} = \frac{-10 + \sqrt{196}}{2\times0,1} = 20 \\
\end{eqnarray*}
La solution $x_1 = -120$ est impossible car on ne peut pas produire un nombre de chaise négatif.
Donc pour que les coûts soient égaux à 690\euro, il faut produire 20 chaises.
\end{solution}
\end{enumerate}
\item Exprimer les recettes $R(q)$ en fonction de $q$.
\begin{solution}
Comme chaque chaise est vendue 56\euro, les recettes se calculent en multipliant le nombre de chaise vendues par 56. Donc
\begin{eqnarray*}
R(q) & = & 56q
\end{eqnarray*}
\end{solution}
\item
\begin{enumerate}
\item Montrer que les bénéfices s'exprime par
\begin{eqnarray*}
B(q) & = & -0,1q^2 + 46q - 450
\end{eqnarray*}
\begin{solution}
On rappelle que
\begin{center}
\ovalbox{Bénéfice = Recette - Coût}
\end{center}
Donc
\begin{eqnarray*}
B(q) & = & R(q) - C(q) \\
B(q) &=& 56q - (0,1q^2 + 10q + 450) \\
&& \mbox{Un "-" devant une parenthèse change }\\
&& \mbox{le signe de ce qui est à l'interieur}\\
B(q) &=& 56q - 0,1q^2 - 10q - 450 \\
B(q) &=& -0,1q^2 + 46q - 450
\end{eqnarray*}
\end{solution}
\item Si l'entreprise produit et vend 14 chaises, fait-elle du bénéfice?
\begin{solution}
Bénéfice pour la production et la vente de 14 chaises
\begin{eqnarray*}
B(14) & = & -0,1\times 14^2 + 46\times14 - 56 \\
B(14) &=& 174,4
\end{eqnarray*}
L'entreprise fait 174,4\euro de bénéfices.
\end{solution}
\item Tracer le tableau de signe de $B(q)$.
\begin{solution}
Tableau de signe de $B(q) = -0,1q^2 + 46q - 450$.
\begin{eqnarray*}
a = -0,1 \qquad b = 46 \qquad c = -450
\end{eqnarray*}
Calcul du discriminant
\begin{eqnarray*}
\Delta & = & b^2 - 4ac = 46^2 - 4\times (-0,1) \times (-450)\\
\Delta &=& 1936
\end{eqnarray*}
$\Delta$ est positif il y a donc deux racines
\begin{eqnarray*}
x_1 & = & \frac{-b - \sqrt{\Delta=}}{2a} = \frac{-46 - \sqrt{1936}}{2\times(-0,1)} = 450 \\
x_2 & = & \frac{-b + \sqrt{\Delta=}}{2a} = \frac{-46 + \sqrt{1936}}{2\times(-0,1)} = 10 \\
\end{eqnarray*}
\begin{center}
\begin{tikzpicture}
\tkzTabInit{$q$ / 1 ,$B(q)$ /1 }%
{$-\infty$ , $10$ , $450$, $+\infty$ }%
\tkzTabLine{ ,- ,z ,+ ,z ,-, }
\end{tikzpicture}
\end{center}
\end{solution}
\item Combien de chaise au minimum doit-elle produire pour faire du bénéfice? Au maximum?
\begin{solution}
Pour faire des bénéfices (là où il y a des + dans le tableau), l'entreprise doit produire au minimum 10 chaises et au maximum 450.
\end{solution}
\end{enumerate}
\end{enumerate}
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "master"
%%% End:

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Notes sur des exercices autour des polynômes du 2nd degré
#########################################################
:date: 2015-07-01
:modified: 2015-07-01
:tags: Fonctions,Exo
:category: 1er_STMG
:authors: Benjamin Bertrand
:summary: Pas de résumé, note créée automatiquement parce que je ne l'avais pas bien fait...
`Lien vers production.pdf <production.pdf>`_
`Lien vers production.tex <production.tex>`_
`Lien vers Exo_corr.pdf <Exo_corr.pdf>`_
`Lien vers Exo_corr.tex <Exo_corr.tex>`_
`Lien vers benef.ods <benef.ods>`_

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\documentclass[a4paper,10pt,xcolor=table]{/media/documents/Cours/Prof/Enseignements/Archive/2014-2015/tools/style/classPres}
\usepackage{/media/documents/Cours/Prof/Enseignements/Archive/2014-2015/2014_2015}
\author{}
\title{}
\date{}
\begin{document}
\begin{frame}{Production et bénéfices}
Une entreprise produit et vend des chaises de bureau. Chacune de ces chaises sont vendus 56\euro d'unité.
Pour un nombre de chaises $q$ produite, le coût de production est donnée par la fonction suivante
\begin{eqnarray*}
C(q) & = & 0,1q^2 + 10q + 450
\end{eqnarray*}
On supposera que toutes les chaises produites sont vendue.
\begin{enumerate}
\item
\begin{enumerate}
\item Calculer les coûts fixes (quand la production est nulle).
\item Calculer le coût de fabrication de 10 chaises.
\item Determiner la quantitié de chaises pour que les coûts soient égaux à 690\euro.
\end{enumerate}
\item Exprimer les recettes $R(q)$ en fonction de $q$.
\item
\begin{enumerate}
\item Montrer que les bénéfices s'exprime par
\begin{eqnarray*}
B(q) & = & -0,1q^2 + 46q - 450
\end{eqnarray*}
\item Si l'entreprise produit et vend 14 chaises, fait-elle du bénéfice?
\item Tracer le tableau de signe de $B(q)$.
\item Combien de chaise au minimum doit-elle produire pour faire du bénéfice? Au maximum?
\end{enumerate}
\end{enumerate}
\end{frame}
\end{document}