From 414f4ed2d330ac105847163b924fc9d7e186a3d3 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Thu, 3 Dec 2020 14:16:18 +0100 Subject: [PATCH] Feat: Bilan 1 sur les fonctions puissances --- .../1B.tex | 14 ---- .../1B_prologement.pdf | Bin 0 -> 76211 bytes .../1B_prologement.tex | 76 ++++++++++++++++++ .../index.rst | 21 +++-- 4 files changed, 92 insertions(+), 19 deletions(-) delete mode 100644 TST/06_Prolongement_geometrique_vers_exponentiel/1B.tex create mode 100644 TST/06_Prolongement_geometrique_vers_exponentiel/1B_prologement.pdf create mode 100644 TST/06_Prolongement_geometrique_vers_exponentiel/1B_prologement.tex diff --git a/TST/06_Prolongement_geometrique_vers_exponentiel/1B.tex b/TST/06_Prolongement_geometrique_vers_exponentiel/1B.tex deleted file mode 100644 index ae1093d..0000000 --- a/TST/06_Prolongement_geometrique_vers_exponentiel/1B.tex +++ /dev/null @@ -1,14 +0,0 @@ -\documentclass[a4paper,10pt]{article} -\usepackage{myXsim} - -\author{Benjamin Bertrand} -\title{Prolongement géométrique vers exponentiel - Cours} -\date{octobre 2020} - -\pagestyle{empty} - -\begin{document} - -\maketitle - -\end{document} \ No newline at end of file diff --git a/TST/06_Prolongement_geometrique_vers_exponentiel/1B_prologement.pdf b/TST/06_Prolongement_geometrique_vers_exponentiel/1B_prologement.pdf new file mode 100644 index 0000000000000000000000000000000000000000..de0424e45e8d97e8530fce10d1379a45cc7b7a25 GIT binary patch literal 76211 zcmce-V~{S*wk`a$R@=61+csC*wr$&X_iEd=d$n!bw!e3ud(Msf<3`-?&)yYL8CjXL zGHS$_H7j%0NKyq6F(|7j;Oq>W<|7lTzqL(qTHFGv6U}Pou@8h3QEUcYP90};ftPPw^L`;nAj7^~U z_@JDe9Zd{upxoCN;HMQRrgwn>;9o$uP$ss<|K0OX?O%`1{?EYw83-KzE(o0eF$j$R zBIn;i1LMCa|IhHi$n@!&=u)9cD`0tO`8|HF_d6@2*!ih69Hg3#D>^!=+en4Z$2r7{j+T~WpB_&6vly8R zmkgV`vbJVY;SVyg**ZGWBCD_OZo;NM4{Xu;@=UXA+Ol|S)TGt<@@Rqc;nHqKWnxvi z)t@JpFDTfktWBbUH49g3__V=7YdKf;u74G? z;$F;Mey_kr5<6|l~r33LR{|GH#$sF6c^%;+GAFyE`+^VVv_c5 zPql80&D)^}+Wj(XNPh!ye<&=0Bu?hoW&EoQHK1c6z+Dy+_ELAzo8~WGG3VJL%o^|5 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b/TST/06_Prolongement_geometrique_vers_exponentiel/1B_prologement.tex @@ -0,0 +1,76 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Prolongement géométrique vers exponentiel - Cours} +\date{décembre 2020} +\tribe{TST} + +\pagestyle{empty} + +\begin{document} + +\maketitle + +\section{Fonctions puissances / exponentielles} + +On peut prolonger une suite géométrique de sorte à ce que l'on puisse calculer sa valeur pour des valeurs de $n$ négative ou à virgule. On a ainsi transformé une suite en une fonction. + +\begin{definition} + Soit $a$ un nombre réel positif. + + La fonction \textit{puissance} ou \textit{exponentielle} de base $a$ est la fonction + \[ + x \mapsto a^x + \] + Cette fonction est définie sur $\R$. +\end{definition} + +\paragraph{Exemples}% +\begin{itemize} + \item Soit $f(x) = 2^x$ la fonction puissance de base 2. + \[ + f(3) = ... \qquad \qquad f(-1) = ... \qquad \qquad f(0,5) = ... + \] + \item Soit $g(x) = 10^x$ la fonction puissance de base 10. + \[ + g(1) = ... \qquad \qquad g(0) = ... \qquad \qquad g(-5) = ... \qquad \qquad g(2,2) = ... + \] + \item Soit $h(x) = ...$ la fonction puissance de base 1,5. + \item Soit $i(x) = 0.5^x$ la fonction puissance de base 0.5. +\end{itemize} + +\afaire{compléter les exemples} + +\begin{propriete} + Soit $a$ un nombre réel positif et $f(x) = a^x$ la fonction puissance de base $a$. Alors + \[ + f(0) = a^0 = 1 \qquad \qquad f(1) = a^1 = a + \] + Soit $x$ et $y$ 2 nombres réels + \[ + a^x \times a^y = a^{x+y} \qquad \qquad + a^{-x} = \frac{1}{a^{x}} \qquad \qquad + \frac{a^x}{a^y} = a^{x-y} \qquad \qquad + (a^x)^y = a^{x\timesy} + \] +\end{propriete} + +\paragraph{Exemples}% +\begin{itemize} + \item Simplification des expressions + \[ + \frac{10^2\times 10^3}{10^10} = \qquad \qquad \qquad (2^3\times2^5)^3 = + \] + \item Réduction d'expressions + \[ + (1+2^x)(1-2^x) = + \] + \item Factorisation + \[ + 3\times 10^x + (2x-1)10^x = + \] +\end{itemize} +\afaire{compléter les exemples} + +\end{document} diff --git a/TST/06_Prolongement_geometrique_vers_exponentiel/index.rst b/TST/06_Prolongement_geometrique_vers_exponentiel/index.rst index 137b360..c828eb9 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: 2020-10-15 +:modified: 2020-12-03 :authors: Benjamin Bertrand :tags: Exponentiel, Suite, Programmation :category: TST @@ -11,14 +11,25 @@ Prolongement géométrique vers exponentiel Étape 1: Prolongement ===================== -Compléter les trous d'une suite géométrique et revenir en arrière. +Activité intro: Compléter les trous d'une suite géométrique et revenir en arrière. + +Cours: Définition des fonctions exponentielles et règles de calculs + +.. image:: ./1B_prologement.pdf + :height: 200px + :alt: Cours sur les fonctions puissances Étape 2: Tracer les courbes =========================== Tracer les courbes de fonctions exponentielle. C'est l'occasion d'utiliser la calculatrice pour calculer des valeurs. Le but étant de conclure sur la croissance des exponentielles en fonction de la forme de la formule. -Étape 3: programmation combler les trous -======================================== +On donne 8 fonctions exponentielles et exponentielles multipliées par un réel. Les élèves en groupe tracent un graphique précisément puis donne l'allure des autres en se partageant le travail. Ils cherchent ensuite à établir une règle pour la croissance. + +Cours: représentation graphique en fonctions des paramètres + +Étape 3: Manipulations techniques de l'exponentielle +==================================================== + +Exercices techniques avec l'exponentielle -Calculer les valeurs décimales d'une exponentielle à partie d'un algo et de programmation.