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zFd9GvUn?JDEuVlVAFVwf&nqA6EuRR=02wtQgyb7aI3qwRBgir%P%|T#AtM|zRq~H- zk67wKRSiXIm$&vft%856nyMb5O--eb&zhFXt6y77ZPMi0n&P9mjjb@OJPCri10oJE zKn56O4Gh!r3aZ`)8){QEV{l>0x&m$QqZlZ(gqNPy+!;$~xqrJxW~5QqI=4f3`d literal 0 HcmV?d00001 diff --git a/TST/07_Logarithme_et_equation_puissance/3E_manip.tex b/TST/07_Logarithme_et_equation_puissance/3E_manip.tex new file mode 100644 index 0000000..6cc5ea7 --- /dev/null +++ b/TST/07_Logarithme_et_equation_puissance/3E_manip.tex @@ -0,0 +1,21 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Logarithme et équation puissance - Cours} +\date{Janvier 2021} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=3, +} +\setlength{\columnseprule}{0pt} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} +\vfill +\printcollection{banque} + +\end{document} diff --git a/TST/07_Logarithme_et_equation_puissance/exercises.tex b/TST/07_Logarithme_et_equation_puissance/exercises.tex index b6d6c32..324ccd7 100644 --- a/TST/07_Logarithme_et_equation_puissance/exercises.tex +++ b/TST/07_Logarithme_et_equation_puissance/exercises.tex @@ -146,4 +146,52 @@ \end{enumerate} \end{exercise} +\begin{exercise}[subtitle={Manipulation d'expressions}, step={3}, origin={Création}, topics={Logarithme et equation puissance}, tags={logarithme, fonctions}] + Simplifier les calculs suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \log(2) + \log(3)$ + \item $B = \log(9) - \log(3)$ + \item $C = \log(2) + \log(0.5)$ + \item $D = \log(2^3) + \log(2^4)$ + \item $E = \log(2\times 3^2) - \log(6)$ + \item $F = -\log(2) + \log(5)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Simplification}, step={3}, origin={Création}, topics={Logarithme et equation puissance}, tags={logarithme, fonctions}] + Simplifier les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = \log(10^x^2)$ + \item $B = 10^{\log(x^2+1)}$ + \item $C = 10^{3\log(5)}$ + \item $D = \log(10^{4x}\times 10^{-x})$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Population de renards}, step={3}, origin={Création}, topics={Logarithme et equation puissance}, tags={logarithme, fonctions}] + Dans un parc régional, on étudie une espèce de renards. Cette population était de \np{1240} renards à la fin de l'année 2016. + + Les études ont montré que cette population diminue de 15\% par an. + + Pour compenser cette diminution, le parc décide d'introduire chaque année 30 renards. + + On modélise alors la population de renard par la suite $(u_n)$ définie par la relation de récurrence suivante \\$u_{n+1} = 0.85u_n +30$. + \begin{enumerate} + \item Calculer $u_1$ et $u_2$ + \item Est-ce que la suite $(u_n)$ est géométrique? + \end{enumerate} + On veut chercher une formule explicite pour cette suite $(u_n)$. Pour cela, on passe par une suite annexe $(v_n)$ définie par $v_n = u_n - 200$ + \begin{enumerate} + \setcounter{enumi}{2} + \item Calculer $v_0$ et $v_1$ + \item La suite $(v_n)$ est géométrique de raison $0,85$. Exprimer $v_n$ en fonction de $n$. + \item Démontrer que $u_n = 1040\times 0.85^n + 200$ + \item Par le calcul, déterminer quand la population va atteindre 500 individus. + \end{enumerate} +\end{exercise} + \collectexercisesstop{banque}