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z8N?#V&H~^RB>A61KHHHqv$yy}FxXlCgPlq8&-&dK^oc1jOWJ+{7vg3n4yI=Rto~g< zpTLJ5yvG{hTnk&U5^mV_;|7Jjp_W@!R)Zc-Sc*n$!90xgf|KoCd!Ww zqSOq=O%~y)6egfE#7Z?nSjt652SS|!k)*$gAQk3epX3tZ8Dik&!DZw@H5-BP<{`8j z5qag}yyOx?lR#+U29u`f=6#PbN|~3cKuf7WsZdHO!LMLVd6cbiO38t#fJ^C7tdPdK zT(p+qytm`5%J8hqP!ecu%1{%qX~+m7TnQ0a34`DcCg)ZcnC=ide35?#V;*D<`S1UQ a3&_YB}, step={1}, origin={<++>}, topics={Logarithme et equation puissance}, tags={logairthme, fonctions}] - <++> +\begin{exercise}[subtitle={Étude graphique}, step={1}, origin={Création}, topics={Logarithme et equation puissance}, tags={logairthme, fonctions}] + \begin{minipage}{0.6\linewidth} + \begin{enumerate} + \item On note $f(x) = 10^x$. Laquelle des fonctions tracées sur le graphique à droite correspond à la représentation graphique de $f(x)$. + \item Reconnaître les formules des autres fonctions puissances représentée sur le graphique. + \item Résoudre graphiquement les équations suivantes + \[ + f(x) = 20 \qquad \qquad 10^x = 100 \qquad \qquad 10^x = 80 + \] + \item Résoudre graphiquement $f(x) \geq 50$. + \end{enumerate} + \end{minipage} + \begin{minipage}{0.6\linewidth} + \begin{tikzpicture}[yscale=0.5, xscale=1.5] + \tkzInit[xmin=-2,xmax=2,xstep=1, + ymin=0,ymax=100,ystep=10] + \tkzGrid + \tkzAxeXY + \tkzFct[domain = -3:2, line width=1pt]{10**x} + \tkzFct[domain = -3:2,color=blue,very thick]{15**x} + \tkzFct[domain = -3:2,color=red,very thick]{0.1**x} + \tkzFct[domain = -3:2,color=green,very thick]{40**x} + \tkzFct[domain = -3:2,color=gray,very thick]{0.2**x} + \end{tikzpicture} + \end{minipage} \end{exercise} -\begin{solution} - <++> -\end{solution} +\begin{exercise}[subtitle={Économie d'échelle}, step={1}, origin={Création}, topics={Logarithme et equation puissance}, tags={logairthme, fonctions}] + Une usine produit des pièces pour les voitures. Produire en grande quantité permet de réduire les coûts de production, c'est \textbf{une économie d'échelle}. On modélise le prix unitaire (pour produire une pièce) par la fonction $f(x) = 200\times 10^{-0.1x}$ où $x$ représente la quantité produite par l'usine en une journée. Cette fonction est représenter ci-dessous. -\collectexercisesstop{banque} \ No newline at end of file + \begin{center} + \begin{tikzpicture}[yscale=0.4, xscale=0.8] + \tkzInit[xmin=0,xmax=200,xstep=10, + ymin=0,ymax=200,ystep=20] + \tkzGrid + \tkzDrawX[label={\textit{Quantité produite}},above=10pt] + \tkzLabelX + \tkzDrawY[label={\textit{Prix unitaire (en \euro)}}, right=10pt] + \tkzLabelY + \tkzFct[domain = 0:200, line width=1pt]{200*10**(-0.1*x)} + \end{tikzpicture} + \end{center} + \begin{enumerate} + \item Vous utiliserez le graphique pour répondre aux questions suivantes + \begin{enumerate} + \item Quel est le coût unitaire pour une production de 10 pièces? Combien cela va-t-il coûter au total? + \item Combien de pièces doit-on produire pour que le coût unitaire soit environ égal à 100\euro? + \item Combien de pièces doit-on produire pour que le coût unitaire soit inférieur à 40\euro? + \item Résoudre l'inéquation $f(x) \geq 80$. + \item (sti2d) Si l'on produit une infinité de prièce. Quel va être le prix unitaire de celles-ci? + \end{enumerate} + \item Vous justifierez vos réponses aux questions suivantes avec un calcul + \begin{enumerate} + \item Quel est le coût unitaire pour une production de 20 pièces? Combien cela va-t-il coûter au total? + \item Quel est le coût unitaire pour une production de 170 pièces? Combien cela va-t-il coûter au total? + \item (*) Combien de pièces doit-on produire pour que le coût unitaire soit inférieur à 10\euro? + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Stockage de données}, step={1}, origin={Création}, topics={Logarithme et equation puissance}, tags={logairthme, fonctions}] + En informatique, un \textbf{bit} est représenté par un 1 ou un 0. C'est l'unité de base mesurer le poids d'une information numérique: 1bit peut décrire 2 choses, 2bits peut décrire 4 choses, 3bits 8 ... Si on note $x$ le nombre de bits, alors le nombre d'information différentes qu'il est possible de décrire est donné par la fonction $f(x) = 2^x$. + \begin{enumerate} + \item Décrire la fonction $f(x)$. Quel type de fonction reconnaît-on? + \item Combien de d'informations peut-on décrire avec 8bits (c'est un octet)? + \item Combien de d'informations peut-on décrire avec 128bits? + \item Combien de bit doit-on utiliser pour décrire \np{1000000} information différentes? + \end{enumerate} +\end{exercise} + +\collectexercisesstop{banque} diff --git a/TST/07_Logarithme_et_equation_puissance/index.rst b/TST/07_Logarithme_et_equation_puissance/index.rst index 66e8930..ab3e405 100644 --- a/TST/07_Logarithme_et_equation_puissance/index.rst +++ b/TST/07_Logarithme_et_equation_puissance/index.rst @@ -2,7 +2,7 @@ Logarithme et équation puissance ################################ :date: 2020-12-17 -:modified: 2020-12-17 +:modified: 2021-01-01 :authors: Benjamin Bertrand :tags: Logairthme, Fonctions :category: TST @@ -13,6 +13,10 @@ Logarithme et équation puissance Situations utilisant les fonctions puissances où l'on calcule des valeurs et où l'on utilise le tableur de la calculatrice pour connaître un seuil. +.. image:: ./1E_equations.pdf + :height: 200px + :alt: Exercices sur les fonctions puissances et le résolution d'équations graphique + Bilan: On pose l'inéquation du seuil et on introduit la fonction log décimal Étape 2: Premières équations/inéquations