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+\pagestyle{empty} +\geometry{left=10mm,right=10mm, top=10mm} + +\renewcommand{\baselinestretch}{0.8} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=4, + %solution/print=true, +} + +\begin{document} + +\maketitle + +\input{banque.tex} +\printcollection{banque} + +\end{document} diff --git a/TES/Logarithme/Etude_fonction/banque.tex b/TES/Logarithme/Etude_fonction/banque.tex index bc19af3..1b9e6eb 100644 --- a/TES/Logarithme/Etude_fonction/banque.tex +++ b/TES/Logarithme/Etude_fonction/banque.tex @@ -167,4 +167,368 @@ \end{enumerate} \end{enumerate} \end{exercise} + +\begin{exercise}[subtitle={Coût de fabrication}, step={4}, topics={Logarithme}] + % Polynésie Juin 2019 Ex 4 + \begin{center}\textbf{Les deux parties de cet exercice sont indépendantes}\end{center} + + \textbf{Partie A} + + \medskip + + Une entreprise produit chaque année entre $100$ et $900$ pneus pour tracteurs. + + On considère la fonction $f$ définie sur l'intervalle [1~;~9] par + + \[f(x ) = 0,5 x^2 - 7x + 14 + 6\ln (x).\] + + On admet que la fonction $f$ modélise le coût moyen annuel de fabrication d'un pneu, exprimé en centaines d'euros, pour $x$ centaines de pneus produits. + + \medskip + + \begin{enumerate} + \item La fonction $f$ est dérivable sur l'intervalle [1~;~9] et on note $f'$ sa fonction dérivée. + + Démontrer que pour tout réel $x$ de l'intervalle [1~;~9] on a : $f'(x)= \dfrac{x^2 -7 x + 6}{x}$. + \item + \begin{enumerate} + \item Justifier les variations suivantes de la fonction $f$ sur l'intervalle [1~;~9] : + + \begin{center} + \begin{tikzpicture}[baseline=(a.north)] + \tkzTabInit[lgt=3,espcl=3]{$x$/1,Variations de\\ $f(x)$/2}{1, 6, 9} + \tkzTabVar{+/ , -/ , +/ } + \end{tikzpicture} + \end{center} + + \item Justifier que, sur l'intervalle [1~;~9], l'équation $f(x) = 5$ admet une unique solution $\alpha$. + \item Donner un encadrement au centième près de $\alpha$. + \item On considère l'algorithme ci-dessous: + + \begin{center} + \begin{tabularx}{0.5\linewidth}{|X|}\hline + $X \gets 1$\\ + $Y \gets 7,5$\\ + Tant que $Y > 5$\\ + \hspace{12mm}$X \gets X + 0,01$\\ + \hspace{12mm}$Y \gets 0,5X^2 - 7X + 14 + 6*\ln (X)$\\ + Fin Tantque\\ \hline + \end{tabularx} + \end{center} + + À la fin de l'exécution de l'algorithme, quelle valeur numérique contient la variable $X$? + \end{enumerate} + \item Pour quelle quantité de pneus, le coût moyen annuel de fabrication d'un pneu est-il minimal ? + À combien s'élève-t-il ? + \end{enumerate} + + \bigskip + + \textbf{Partie B} + + \medskip + + + Cette même entreprise envisage la fabrication de semoirs (gros matériel agricole). + On admet que la fonction $g$ définie sur l'intervalle [0~;~100] par + + \[g (x) = 2x -1 + \text{e}^{0,05x}\] + + modélise le coût de fabrication, exprimé en centaines d'euros, de $x$ semoirs. + + \medskip + + \begin{enumerate} + \item Donner une primitive $G$ de la fonction $g$ sur l'intervalle [0~;~100]. + \item Calculer la valeur moyenne de la fonction $g$ sur l'intervalle [0~;~100]. + \item Interpréter ce résultat dans le contexte de l'exercice. + \end{enumerate} +\end{exercise} + +\begin{solution} + \textbf{Partie A} + + \medskip + + Une entreprise produit chaque année entre $100$ et $900$ pneus pour tracteurs. + + On considère la fonction $f$ définie sur l'intervalle $\left [1~;~9\strut\right ]$ par + $f(x ) = 0,5 x^2 - 7x + 14 + 6\ln (x)$. + + On admet que la fonction $f$ modélise le coût moyen annuel de fabrication d'un pneu, exprimé en centaines d'euros, pour $x$ centaines de pneus produits. + + \medskip + + \begin{enumerate} + \item% La fonction $f$ est dérivable sur l'intervalle $\left [1~;~9\strut\right ]$ et on note $f'$ sa fonction dérivée. + Pour tout réel $x$ de l'intervalle $\left [1~;~9\strut\right ]$ on a + $f(x ) = 0,5 x^2 - 7x + 14 + 6\ln (x)$ donc\\ + $f'(x)= 0,5\times 2x - 7 + 6\times \dfrac{1}{x} + = x-7-\dfrac{6}{x} + = \dfrac{x^2-7x+6}{x}$. + + %Démontrer que pour tout réel $x$ de l'intervalle $\left [1~;~9\strut\right ]$ on a : $f'(x)= \dfrac{x^2 -7 x + 6}{x}$. + \item + \begin{enumerate} + \item On va justifier les variations suivantes de la fonction $f$ sur l'intervalle [1~;~9] : + + \begin{center} + {\renewcommand{\arraystretch}{1.1} + \psset{nodesep=3pt,arrowsize=2pt 3} % paramètres + \def\esp{\hspace*{1.5cm}}% pour modifier la largeur du tableau + \def\hauteur{0pt}% mettre au moins 20pt pour augmenter la hauteur + $\begin{array}{|c| *4{c} c|} + \hline + x & 1 & \esp & 6 & \esp & 9 \\ + \hline + & \Rnode{max1}{\phantom{0}} & & & & \Rnode{max2}{\phantom{0}} \\ + \text{Variations de } f & & & & & \rule{0pt}{\hauteur} \\ + & & & \Rnode{min}{\phantom{0}} & & \rule{0pt}{\hauteur} + \ncline{->}{max1}{min} \ncline{->}{min}{max2} + %\rput*(-3.7,0.65){\Rnode{zero}{\blue 0}} + %\rput(-3.7,1.7){\Rnode{alpha}{\blue \alpha}} + %\ncline[linestyle=dotted, linecolor=blue]{alpha}{zero} + %\rput*(-1.3,0.65){\Rnode{zero2}{\red 0}} + %\rput(-1.3,1.7){\Rnode{beta}{\red \beta}} + %\ncline[linestyle=dotted, linecolor=red]{beta}{zero2} + \\ + \hline + \end{array}$ + } + \end{center} + + Sur $\left [1~;~9\strut\right ]$, $f'(x)$ est du signe de $x^2-7x+6$. + + $\Delta = 7^2 - 4\times 1\times 6 = 49-24=25=5^2$ donc le trinôme $x^2-7x+6$ admet deux racines:\\[3pt] + $x'=\dfrac{-b-\sqrt{\Delta}}{2a} = \dfrac{7-5}{2}=1$ et $x''=\dfrac{-b+\sqrt{\Delta}}{2a}=\dfrac{7+5}{2}=6$. + + \smallskip + + On en déduit le signe du trinôme (positif à l'extérieur des racines) donc de $f'(x)$: + + \begin{center} + \renewcommand{\arraystretch}{1.5} + \def\esp{\hspace*{1.2cm}} + $\begin{array}{|c | *{5}{c} |} + \hline + x & 1 & \esp & 6 & \esp & 9 \\ + \hline + f'(x) & & \pmb{-} & \vline\hspace{-2.7pt}{0} & \pmb{+} & \\ + \hline + \end{array}$ + \end{center} + + Cela justifie les variations de $f$. + + \item% Justifier que, sur l'intervalle [1~;~9], l'équation $f(x) = 5$ admet une unique solution $\alpha$. + On complète le tableau de variations de $f$: + $f(1)=7,5$, $f(6)\approx 0,75$ et $f(9)\approx 4,7$ + + \begin{center} + {\renewcommand{\arraystretch}{1.3} + \psset{nodesep=3pt,arrowsize=2pt 3} % paramètres + \def\esp{\hspace*{1.5cm}}% pour modifier la largeur du tableau + \def\hauteur{0pt}% mettre au moins 20pt pour augmenter la hauteur + $\begin{array}{|c| *4{c} c|} + \hline + x & 1 & \esp & 6 & \esp & 9 \\ + \hline + & \Rnode{max1}{7,5} & & & & \Rnode{max2}{\approx 4,7} \\ + \text{Variations de } f & & & & & \rule{0pt}{\hauteur} \\ + & & & \Rnode{min}{\approx 0,75} & & \rule{0pt}{\hauteur} + \ncline{->}{max1}{min} \ncline{->}{min}{max2} + \rput*(-4.5,0.65){\Rnode{zero}{\blue 5}} + \rput(-4.5,1.7){\Rnode{alpha}{\blue \alpha}} + \ncline[linestyle=dotted, linecolor=blue]{alpha}{zero} + \\ + \hline + \end{array}$ + } + \end{center} + + On en déduit que sur $\left [1~;~9\strut\right ]$, l'équation $f(x)=5$ admet une solution unique $\alpha$. + + + \item %Donner un encadrement au centième près de $\alpha$. + $\left. + \begin{array}{l} + f(2)\approx 6,2 > 5\\ + f(3)\approx 4,1 < 5 + \end{array} + \right\rbrace + \Rightarrow + \alpha \in \left [ 2~;\,3\strut\right ]$ + \hfill + $\left. + \begin{array}{l} + f(2,5)\approx 5,1 > 5\\ + f(2,6)\approx 4,9 < 5 + \end{array} + \right\rbrace + \Rightarrow + \alpha \in \left [ 2,5~;\, 2,6 \strut\right ]$ + + $\left. + \begin{array}{l} + f(2,55)\approx 5,018 > 5\\ + f(2,56)\approx 4,997 < 5 + \end{array} + \right\rbrace + \Rightarrow + \alpha \in \left [ 2,55~;\, 2,56 \strut\right ]$ + + \item On considère l'algorithme ci-dessous: + + \begin{center} + \begin{tabularx}{0.5\linewidth}{|X|}\hline + $X \gets 1$\\ + $Y \gets 7,5$\\ + Tant que $Y > 5$\\ + \hspace{12mm}$X \gets X + 0,01$\\ + \hspace{12mm}$Y \gets 0,5X^2 - 7X + 14 + 6*\ln (X)$\\ + Fin Tantque\\ \hline + \end{tabularx} + \end{center} + + À la fin de l'exécution de l'algorithme, la variable $X$ contient la valeur $2,56$, première valeur au centième pour laquelle $Y>5$. + \end{enumerate} + \item Le coût moyen annuel de fabrication d'un pneu est minimal quand la fonction $f$ atteint son minimum c'est-à-dire pour $x=6$; c'est donc pour la fabrication de 600 pneus que le coût moyen annuel de fabrication d'un pneu est minimal. Ce coût est, en euro, de $f(6)\times 100 \approx 75$. + %À combien s'élève-t-il ? + \end{enumerate} + + \bigskip + + \textbf{Partie B} + + \medskip + + Cette même entreprise envisage la fabrication de semoirs (gros matériel agricole). + + On admet que la fonction $g$ définie sur l'intervalle $\left [0~;~100\strut\right ]$ par + $g (x) = 2x -1 + \e^{0,05x}$ + modélise le coût de fabrication, exprimé en centaines d'euros, de $x$ semoirs. + + \medskip + + \begin{enumerate} + \item %Donner une primitive $G$ de la fonction $g$ sur l'intervalle $\left [0~;~100\strut\right ]$. + Sur l'intervalle $\left [0~;~100\strut\right ]$, la fonction $g$ a pour primitive la fonction $G$ définie par\\ + $G(x)=x^2 - x + \dfrac{\e^{0,05x}}{0,05} = x^2-x +20\e^{0,05x}$. + + \item La valeur moyenne $m$ de la fonction $g$ sur l'intervalle $\left [0~;~100\strut\right ]$ est: + + $m=\dfrac{1}{100-0} \ds\int_{0}^{100} g(x) \d x = \dfrac{1}{100} \left [G(100) - G(0) \strut\right ] + =\dfrac{1}{100}\left [ \left ( \np{9900} + 20\e^{5}\right ) - \left ( 20\right ) \right ] + = \np{9880} +20\e^{5}\\[3pt] + \phantom{m} + \approx \np{128,46}$ + + \item %Interpréter ce résultat dans le contexte de l'exercice. + Le coût moyen d'un semoir est donc, en euro, $128,46 \times 100 = \np{12846}$. + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonction}, step={4}, topics={Logarithme}] + % Métropole Septembre 2019 Ex 3 + La courbe $\mathcal{C}_f$ ci-dessous est la courbe représentative d'une fonction $f$ définie et deux + fois dérivable sur l'intervalle [1,1~;~8]. + + \begin{center} + \begin{tikzpicture}[yscale=0.5, xscale=1.5, domain=1:8] + \tkzInit[xmin=0,xmax=9,xstep=1, + ymin=0,ymax=12,ystep=1] + \tkzGrid + \tkzAxeXY[up space=0.5,right space=.5] + \clip (1.1, 0) rectangle (9,12); + \tkzFct[line width=1pt]{(2.*x-1.)/log(x)} + \end{tikzpicture} + \end{center} + + \textbf{Les parties A et B peuvent être traitées de manière indépendante.} + + \bigskip + + \textbf{Partie A : étude graphique} + + \medskip + + \begin{enumerate} + \item Donner une valeur approchée du minimum de la fonction $f$ sur l'intervalle + [1,1~;~8] + \item Quel est le signe de $f'(5)$ ? Justifier. + \item Encadrer l'intégrale $\displaystyle\int_2^4 f(x)\:\text{2}4 f(x)\:\text{d}x$ par deux entiers consécutifs. + \item La fonction $f$ est-elle convexe sur [1,1~;~3] ? Justifier. + \end{enumerate} + + \bigskip + + \textbf{Partie B : étude analytique} + + \medskip + + On admet que $f$ est la fonction définie sur l'intervalle [1,1~;~8] par + + \[f(x) = \dfrac{2x - 1}{\ln (x)}.\] + + \smallskip + + \begin{enumerate} + \item Montrer que, pour tout réel $x$ de l'intervalle [1,1~;~8], on a : + + \[f'(x) = \dfrac{2\ln (x) - 2 + \frac{1}{x}}{(\ln (x))^2}\] + + \item Soit $h$ la fonction définie sur [1,1~;~8] par : $h(x) = 2\ln (x) - 2 + \frac{1}{x}$. + \begin{enumerate} + \item Soit $h'$ la fonction dérivée de $h$ sur l'intervalle [1,1; 8]. + + Montrer que, pour tout réel $x$ de l'intervalle [1,1~;~8], + + \[h'(x) = \dfrac{2x - 1}{x^2}.\] + + \item En déduire les variations de la fonction $h$ sur l'intervalle [1,1~;~8]. + \item Montrer que l'équation $h(x) = 0$ admet une unique solution $\alpha$ sur + l'intervalle [1,1~;~8]. Donner un encadrement de $\alpha$ par deux entiers + consécutifs. + \end{enumerate} + \item Déduire des résultats précédents le signe de $h(x)$ sur l'intervalle [1,1~;~8]. + \item À l'aide des questions précédentes, donner les variations de [ sur [1,1~;~8]. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Loi de Benfort}, step={4}, topics={Logarithme}] + % Métropole Juin 2017 Ex 4 + Dans cet exercice, on considère le premier chiffre des entiers naturels non nuls, en écriture décimale. Par exemple, le premier chiffre de \np{2017} est 2 et le premier chiffre de 95 est 9. + + Dans certaines circonstances, le premier chiffre d'un nombre aléatoire non nul peut être modélisé par une variable aléatoire $X$ telle que pour tout entier $c$ compris entre 1 et 9, + + \[P(X = c) = \dfrac{\ln (c + 1) - \ln (c)}{\ln(10)}.\] + + Cette loi est appelée loi de Benford. + + \medskip + + \begin{enumerate} + \item Que vaut $P(X = 1)$ ? + \item On souhaite examiner si la loi de Benford est un modèle valide dans deux cas particuliers. + \begin{enumerate} + \item \textbf{Premier cas} + + Un fichier statistique de l'INSEE indique la population des communes en France au 1\ier{} janvier 2016 (champ: France métropolitaine et départements d'outre-mer de la Guadeloupe, de la Guyane, de la Martinique et de la Réunion). + + À partir de ce fichier, on constate qu'il y a \np{36677} communes habitées. Parmi elles, il y a \np{11094} communes dont la population est un nombre qui commence par le chiffre 1. + + Cette observation vous semble-t-elle compatible avec l'affirmation : \og{}le premier chiffre de la population des communes en France au 1 er janvier 2016 suit la loi de Benford \fg{} ? + \item \textbf{Deuxième cas} + + Pour chaque candidat au baccalauréat de la session 2017, on considère sa taille en centimètres. + + On désigne par $X$ la variable aléatoire égale au premier chiffre de la taille en centimètres d'un candidat pris au hasard. + + La loi de Benford vous semble-t-elle une loi adaptée pour $X$ ? + \end{enumerate} + \end{enumerate} +\end{exercise} + + + \collectexercisesstop{banque}