From 565f281f4cb3eac038cddcdb11abb4d61450c9db Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Mon, 17 Jan 2022 09:03:02 +0100 Subject: [PATCH] Feat: Deux exercices sur les tableaux pour DS5 --- 2nd/Evaluations/DS_2022-01-18/exercises.tex | 102 ++++++++++++++++++++ 2nd/Evaluations/DS_2022-01-18/sujet.pdf | Bin 0 -> 33443 bytes 2nd/Evaluations/DS_2022-01-18/sujet.tex | 31 ++++++ 3 files changed, 133 insertions(+) create mode 100644 2nd/Evaluations/DS_2022-01-18/exercises.tex create mode 100644 2nd/Evaluations/DS_2022-01-18/sujet.pdf create mode 100644 2nd/Evaluations/DS_2022-01-18/sujet.tex diff --git a/2nd/Evaluations/DS_2022-01-18/exercises.tex b/2nd/Evaluations/DS_2022-01-18/exercises.tex new file mode 100644 index 0000000..f4b743b --- /dev/null +++ b/2nd/Evaluations/DS_2022-01-18/exercises.tex @@ -0,0 +1,102 @@ +\begin{exercise}[subtitle={Cducosto}, step={1}, origin={Création?}, topics={Fonctions}, tags={Tableau de signes, Tableau de variations, inéquations}, points=5] + L'entreprise Cducosto produit des outils de bricolages. + \begin{enumerate} + \item Leur premier produit est un marteau. Voici les tableaux décrivant le signe et les variations des bénéfices (notés $B(x)$) en fonction du nombre de marteau qu'elle produit et vend. + + \begin{center} + \begin{tikzpicture} + \tkzTabInit[]{$x$/1,Signes de $B(x)$/2}{0, 30, 120, 150} + \tkzTabLine{, -, z, +, z, -,} + \end{tikzpicture} + + \begin{tikzpicture}[baseline=(a.north)] + \tkzTabInit[]{$ x $/1, Variations de $ B(x) $/2}{0, 75, 150} + \tkzTabVar{ -/-175, +/100, -/-175} + \end{tikzpicture} + \end{center} + + \begin{enumerate} + \item Si l'entreprise produit 10 marteaux, fait-elle des bénéfices? + \item Sur quel intervalle doit-elle produire pour que ses bénéfices soient positifs? + \item Quelle quantité de marteaux doit-elle produire pour faire un maximum de bénéfices? + \end{enumerate} + + \item Leur deuxième produit est une visseuse automatique. Le bénéfice liés à cet outil est donné par la fonction suivante: + \begin{eqnarray*} + f:x & \mapsto & 2x - 3 + \end{eqnarray*} + \begin{enumerate} + \item Tracer et démontrer le tableau de signes de cette fonction. + \item À partir de combien de visseuses l'entreprise fait-elle du bénéfice? + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{center} + \begin{tikzpicture}[xscale=1] + \begin{axis}[ + xscale=2, + axis lines = center, + %grid = both, + xlabel = {Quantité}, + %xtick={0, 20, ..., 150}, + xtick distance=10, + ylabel = {Bénéfices}, + ytick distance=50, + ymax=150, + grid=major + ] + \addplot[domain=0:150,samples=40, color=red, very thick]{-0.05*x*x + 7.5*x - 180}; + \end{axis} + \end{tikzpicture} + \end{center} + \begin{enumerate} + \item Pour que les bénéfices soient positifs , il faut que la production reste sur l'intervalle $\intFF{3}{120}$ + \end{enumerate} + \item + \begin{enumerate} + \item + On cherche là où la fonction $f$ est positive + \begin{eqnarray*} + f(x) & > &0\\ + 2x - 3 & > & 0 \\ + 2x & > & 3 \\ + && \mbox{2 est positif, on ne change}\\ + && \mbox{le sens de l'inégalité}\\ + x &>& \frac{3}{2} = 1,5 + \end{eqnarray*} + Donc $f(x)$ est positive quand $x$ est supérieur à 1.5. + + \begin{tikzpicture} + \tkzTabInit[]{$x$/1,$f(x)$/1}{0, {1,5}, $+\infty$} + \tkzTabLine{ ,-, z, +,} + \end{tikzpicture} + \item + À partir de 2 visseuses l'entreprise fait des bénéfices (là où dans le tableau au dessus il y a un +) + + \end{enumerate} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Tableaux}, step={1}, origin={Création?}, topics={Fonctions}, tags={Tableau de signes, Tableau de variations}, points=5] + \begin{enumerate} + \item Tracer le tableau de signes puis le tableau de variation de la fonction suivante + \begin{center} + \begin{tikzpicture}[yscale=0.5] + \tkzInit[xmin=-5,xmax=5,xstep=1, + ymin=-5,ymax=4,ystep=1] + \tkzGrid + \tkzAxeXY + \draw[very thick, color=red] plot [smooth,tension=0.5, mark=*] coordinates{(-4, -4) (-3.5, -3) (-3, 0) (-2, 1) (-1, 0) (0, -3) (1, -1) (2, -3) (2.5,0) (3, 2) (4, 3)}; + \draw (4,3) node[above right] {$\mathcal{C}_f$}; + \end{tikzpicture} + \end{center} + \item En utilisant la calculatrice tracer le tableau de signe puis le tableau de variation de la fonction + \[ + g(x) = x^3 + x^2 - 2x + \] + \end{enumerate} +\end{exercise} diff --git a/2nd/Evaluations/DS_2022-01-18/sujet.pdf b/2nd/Evaluations/DS_2022-01-18/sujet.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fc8c10cf99e8f126fd793f41c481165c66e3b42f GIT binary patch literal 33443 zcmce;1CTAD1*6 zElrG^@#&OY44waDMAXjF2A}S){eN7+fKMkVXy=Zv_2;Mb_zd*_x#{53$rw00TKqN6 z@SkV?WB+Gd(ZtEl#nH&b37_T9km8@IEUcYP9RD1x4V+DcO^ob}|Hzd#u{Cox$7f){ z=jFx!uMd=yv!jWD4V3%(BK(Xz`OGd50Qe{14U~zk@&7vW=i6WR&-Q2Lf1e)g|1v!| z{;hg2{AIy^jX@ax0>Qt}A`FcGI!*q)*kbtCfs64kbN~Bn!uS^i{x#@gWcv4N@|T!wHI~fMu4lvv>9}zSHaX@-+I>PB&1m36%isECtWc0|4NqS2GHgvz>&PE=yuE 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