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left] node [above] {$+t_m$} node [below] {$\times (1+t_m)$} (termB.north) ; + \path[->] (termB.north) edge [bend left] node [above] {$+t_m$} node [below] {$\times (1+t_m)$} (termC.north) ; + \path[->] (termC.north) edge [bend left] node [above] {$+t_m$} node [below] {$\times (1+t_m)$} (termD.north) ; + + \path[->] (termA.south) edge [bend right] node [above] {$+t_m$} node [below] {$\times (1+T)$} (termD.south); + \end{tikzpicture} + \end{minipage} +\end{definition} + +\medskip +\hline + +\input{exercises.tex} +\printcollection{banque} + +\vfill + +\setcounter{section}{2} +\section{Taux d'évolution moyen} + +\begin{definition}[Taux d'évolution moyen] + \begin{minipage}{0.45\linewidth} + On note $t_m$ le taux d'évolution et $T$ le taux d'évolution global et $n$ le nombre de 'sous-évolutions'. Alors + \[ + 1 + T = (1 + t_m)^n \qquad \qquad 1 + t_m = (1 + T)^{\frac{1}{n}} + \] + + \end{minipage} + \hfill + \begin{minipage}{0.5\linewidth} + \begin{tikzpicture}[ + roundnode/.style={circle, draw=highlightbg, fill=green!5, very thick, minimum size=3mm}, + node distance=2cm and 2cm + ] + %Nodes + \node[roundnode] (termA) {\makebox[0.5cm]{}}; + \node[roundnode] (termB) [right=of termA] {\makebox[0.5cm]{}}; + \node[roundnode] (termC) [right=of termB] {\makebox[0.5cm]{}}; + \node[roundnode] (termD) [right=of termC] {\makebox[0.5cm]{}}; + + %Lines + \path[->] (termA.north) edge [bend left] node [above] {$+t_m$} node [below] {$\times (1+t_m)$} (termB.north) ; + \path[->] (termB.north) edge [bend left] node [above] {$+t_m$} node [below] {$\times (1+t_m)$} (termC.north) ; + \path[->] (termC.north) edge [bend left] node [above] {$+t_m$} node [below] {$\times (1+t_m)$} (termD.north) ; + + \path[->] (termA.south) edge [bend right] node [above] {$+t_m$} node [below] {$\times (1+T)$} (termD.south); + \end{tikzpicture} + \end{minipage} +\end{definition} + +\medskip +\hline + +\printcollection{banque} + +\end{document} diff --git a/TST/06_Prolongement_geometrique_vers_exponentiel/exercises.tex b/TST/06_Prolongement_geometrique_vers_exponentiel/exercises.tex index d3de511..b97e5c3 100644 --- a/TST/06_Prolongement_geometrique_vers_exponentiel/exercises.tex +++ b/TST/06_Prolongement_geometrique_vers_exponentiel/exercises.tex @@ -116,6 +116,28 @@ \end{enumerate} \end{exercise} +\begin{exercise}[subtitle={Applications directes}, step={4}, origin={Création}, topics={Prolongement géométrique vers exponentiel}, tags={exponentiel, suite, programmation}] + Les questions suivantes peuvent être traité de façon indépendantes. + \begin{enumerate} + \item Le chiffre d'affaire d'une entreprise a augmenté de 20\% en 10ans. Quel a été le taux d'évolution moyen annuel? + \item On sait qu’entre 2002 et 2009, le nombre d’internautes en Chine est passé de 60 millions à 385 millions. + \begin{enumerate} + \item Quel est le taux d'évolution global du nombre d'internautes? + \item Calculer le taux d’évolution moyen annuel du nombre d’internautes en Chine entre 2002 et 2009. + \end{enumerate} + \item On étudie les abonnements à un grand quotidien de 2011 à 2015. En 2011, il y avait 620 214 abonnés. En 2015, il y en avait 555 239. + \begin{enumerate} + \item Quel est le taux d'évolution global du nombre d'abonnement? + \item Calculer le taux d’évolution annuel moyen du nombre d’abonnés entre 2011 et 2015. + \end{enumerate} + \item Une quantité a connait un taux d'évolution moyen de 2\% par ans. + \begin{enumerate} + \item Cette quantité vaut 234\euro en 2010. Combien va-t-elle valoir en 2015? + \item Quel est le taux d'évolution global entre 2010 et 2015? + \item Quel est le taux d'évolution global entre 2010 et 2020? + \end{enumerate} + \end{enumerate} +\end{exercise} \collectexercisesstop{banque} diff --git a/TST/06_Prolongement_geometrique_vers_exponentiel/index.rst b/TST/06_Prolongement_geometrique_vers_exponentiel/index.rst index a9c74e5..b7112a2 100644 --- a/TST/06_Prolongement_geometrique_vers_exponentiel/index.rst +++ b/TST/06_Prolongement_geometrique_vers_exponentiel/index.rst @@ -2,7 +2,7 @@ Prolongement géométrique vers exponentiel ######################################### :date: 2020-10-15 -:modified: 2020-12-03 +:modified: 2020-12-17 :authors: Benjamin Bertrand :tags: Exponentiel, Suite, Programmation :category: TST @@ -44,5 +44,19 @@ Bilan: représentation graphique en fonctions des paramètres Étape 3: Manipulations techniques de l'exponentielle ==================================================== -Exercices techniques avec l'exponentielle +.. image:: ./3E_expressions.pdf + :height: 200px + :alt: Exercice de manipulation techniques des puissances. + + +Étape 4: Taux d'évolution moyen +=============================== + +.. image:: ./4E_moyennes.pdf + :height: 200px + :alt: Cours et exercices sur le taux d'évolution moyen. + + + +