From 2bbad86df57fda0e993571b480a3542ad884990a Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Sat, 2 Jan 2021 10:33:35 +0100 Subject: [PATCH] =?UTF-8?q?Feat:=20=C3=A9tape=201=20sur=20le=20log=20pour?= =?UTF-8?q?=20les=20TST?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../1B.tex | 14 ---- .../1B_logarithme.pdf | Bin 0 -> 85119 bytes .../1B_logarithme.tex | 68 ++++++++++++++++++ .../1E_equations.pdf | Bin 98277 -> 98249 bytes .../exercises.tex | 7 +- .../index.rst | 13 +++- 6 files changed, 83 insertions(+), 19 deletions(-) delete mode 100644 TST/07_Logarithme_et_equation_puissance/1B.tex create mode 100644 TST/07_Logarithme_et_equation_puissance/1B_logarithme.pdf create mode 100644 TST/07_Logarithme_et_equation_puissance/1B_logarithme.tex diff --git a/TST/07_Logarithme_et_equation_puissance/1B.tex b/TST/07_Logarithme_et_equation_puissance/1B.tex deleted file mode 100644 index cf73961..0000000 --- a/TST/07_Logarithme_et_equation_puissance/1B.tex +++ /dev/null @@ -1,14 +0,0 @@ -\documentclass[a4paper,10pt]{article} -\usepackage{myXsim} - -\author{Benjamin Bertrand} -\title{Logarithme et equation puissance - Cours} -\date{décembre 2020} - -\pagestyle{empty} - -\begin{document} - -\maketitle - -\end{document} \ No newline at end of file diff --git a/TST/07_Logarithme_et_equation_puissance/1B_logarithme.pdf b/TST/07_Logarithme_et_equation_puissance/1B_logarithme.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d0a494d8a9732f9775f16d42928c36ea7c2871ee GIT binary patch literal 85119 zcmce-bFlF4(j~fW+k3y;Hs5XAwr$(CZQHhO+qSLyJ2Ugm{chD%-TCL7R3()>-FYf2 ztJA6Om5|5@iO?|6vOtl{Ee)?hF%!@e*cwDY@Gj6VR#28dw+`IziDXIvY6s zj}Z}D2Wu!g0RdY#0!{jV{2A!~$EpoQ_b zqph=pp|K+Y%ReIuP;^qpHl|Ky1Pm+$|7-bY7IP~nV+R5{5i5NsV z)y3af4ApMju^UvG9_QYL8s>p}-l&qf?s2W#%;(0t^yyro;`KPhZ^?+?PuO?moK<@=0PMaQSpW4nuuR!R*aw~4sE0os>H5=t(J6Jp#GQryzg z5~)*HS^iQ*9Hmq+uZht*5b>#&Iq 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On appelle cela un \textbf{seuil}. + +Par exemple, dans l'exercice \textit{économie d'échelle}, le coût unitaire est donné par la fonction $f(x) = 200\times10^{-0.1x}$ et l'on se demande à partir de quelle quantité produite, le coût unitaire passe en dessous de 10\euro. Cette question de seuil se traduit par l'inéquation suivante +\[ + f(x) = 200\times 10^{-0.1x} \leq 10 +\] + +Pour résoudre cette inéquation, on peut envisager 3 méthodes +\begin{itemize} + \item \textbf{Tâtonnement}: en calculant successivement des de $f(x)$ et en essayant de s'approcher de 10. Cette méthode peut être rendu efficace en utilisant la calculatrice ou le tableur. + \item \textbf{Algorithme}: en programmant un algorithme puis en faisant trouver le résultat par un ordinateur. On étudiera cette méthode plus tard. + \item \textbf{Résolution exacte}: en résolvant de manière exacte l'inéquation. Pour cela, on a besoin d'une nouvelle fonction, le logarithme. +\end{itemize} + +\section{Fonction logarithme} + +Il existe plusieurs fonction logarithme, nous en étudierons une seule: le logarithme décimal. + +\begin{definition}{Fonction logarithme décimal} + Pour tout nombre réel $a > 0$, il existe un unique nombre $b$ tel que $10^b = a$. + \medskip + + $b$ est appelé \textbf{logarithme décimal} de $a$ et est noté $\log(a)$. On peut alors noter + \[ + e^b = a \qquad \equiv \qquad b = \log(a) + \] + \medskip + + La fonction \textbf{logarithme décimal}, notée $\log$, est la fonction qui à tout $x$ réel \textbf{strictement positif} associe $\log(x)$ +\end{definition} + +\begin{propriete} + \begin{itemize} + \item Soit $a$ un nombre réel alors $\log(10^a) = a$. + \item Soit $a$ un nombre réel strictement positif alors $10^{\log(a)} = a$. + \item Valeurs particulières + \[ + \log(1) = 0 \qquad \log(10) = 1 + \] + \end{itemize} +\end{propriete} + +\paragraph{Exemple} + +Résolution de l'inéquation +\[ + 200\times 10^{-0.1x} \leq 10 +\] +\afaire{résoudre cette inéquation} + + + +\end{document} diff --git a/TST/07_Logarithme_et_equation_puissance/1E_equations.pdf b/TST/07_Logarithme_et_equation_puissance/1E_equations.pdf index 4dfc4afd71513accdb7be87b2e4b94cdf11339a4..47a8107292c4bc3dacc392fd6a9167f9ca9d3328 100644 GIT binary patch delta 19764 zcmXt9b8sci+l_79I2+r%NjA34&BnH!+}QTU#>Td7+s4KlZTRK=RegWV)Kov`JkNC3 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