From 0ec87c145b734c0c601779abd544d396ec1749f8 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Fri, 8 May 2020 11:28:40 +0200 Subject: [PATCH] Feat: Suite du chapitre sur les limites pour les Tsti2d --- .../Operation_limites/2E_puissances.pdf | Bin 64992 -> 64992 bytes .../3E_fraction_rationnelle.pdf | Bin 68592 -> 68592 bytes .../Analyse/Operation_limites/4B_composee.pdf | Bin 0 -> 81796 bytes .../Analyse/Operation_limites/4B_composee.tex | 61 +++++++++++++ .../Analyse/Operation_limites/4E_composee.pdf | Bin 0 -> 103380 bytes .../Analyse/Operation_limites/4E_composee.tex | 21 +++++ Tsti2d/Analyse/Operation_limites/banque.tex | 85 ++++++++++++++++++ Tsti2d/Analyse/Operation_limites/index.rst | 12 ++- Tsti2d/semaines.rst | 22 +++-- 9 files changed, 192 insertions(+), 9 deletions(-) create mode 100644 Tsti2d/Analyse/Operation_limites/4B_composee.pdf create mode 100644 Tsti2d/Analyse/Operation_limites/4B_composee.tex create mode 100644 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+\section{Composée} + +\subsection*{Propriété - composée avec l'exponentielle} +Soit $u(x)$ une fonction, +\begin{itemize} + \item Si $\ds \lim u(x) = +\infty$ alors $\lim e^{u(x)} = +\infty$. + \item Si $\ds \lim u(x) = -\infty$ alors $\lim e^{u(x)} = 0$. + \item Si $\ds \lim u(x) = a$ alors $\lim e^{u(x)} = e^a$. +\end{itemize} + +\subsubsection*{Remarque} +Ici les limites ne sont pas précisées car elles n'influencent pas sur le résultat. + +\subsubsection*{Exemples}% +Limites suivantes +\[ + \lim_{x \rightarrow +\infty} e^{2x + 1} = +\] +\[ + \lim_{x \rightarrow -\infty} e^{-x + 1}= +\] +\[ + \lim_{x \rightarrow -\infty} e^{\frac{1}{x}}= +\] +\afaire{} + +\subsection*{Propriété - composée avec le logarithme} +Soit $u(x)$ une fonction strictement positive, +\begin{itemize} + \item Si $\ds \lim u(x) = +\infty$ alors $\lim \ln(u(x)) = +\infty$. + \item Si $\ds \lim u(x) = 0$ alors $\lim \ln(u(x)) = -\infty$. + \item Si $\ds \lim u(x) = a$ alors $\lim \ln(u(x)) = \ln(a)$. +\end{itemize} + +\subsubsection*{Remarque} +Ici les limites ne sont pas précisées car elles n'influencent pas sur le résultat. + +\subsubsection*{Exemples}% +Limites suivantes +\[ + \lim_{x \rightarrow +\infty} \ln(2x + 1) = +\] +\[ + \lim_{x \rightarrow 1} \ln(-x + 1)= +\] +\[ + \lim_{x \rightarrow 0} \ln(2x + 2)= +\] +\afaire{} + + +\end{document} diff --git a/Tsti2d/Analyse/Operation_limites/4E_composee.pdf b/Tsti2d/Analyse/Operation_limites/4E_composee.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d0eb10c15b4fbb6bdd69a5730e80252eb509027b GIT binary patch literal 103380 zcmce+bFeVavMso6+qP}n*0*ijwr$(q-?nYrwr$Vv+;{K9OuUJh_s=;U(H#|4Srxr{ zWoKm8DpCazFJhVZw1J`*60&nA&|>(|7j;Oq>W<|LIYJqL(qTHFGv6U}Pou@9UpcEUcYP90};ftPPw^L`;nAj7^~U_@JDe z9Zd{upxieW;b#;mX7+#pz`p_SpiFFy|9j-0*1ukz{hx*ZGY~lbT@X0`V-OhsMb5v4 z2F8C;{-5E2k?G$C;y>q1jQ=)oV)_@y{~0Ej{zdw~#hsX#|8pSz6$Jl1AlUwMApYgK z{}vF;|N7hh$DRH=d^0hzF#WIi{r?}nuQhe-u-VakUhC3*DnW!{JlL#wNXPIbWr_JoWA`%9wY2OoTk> zsArlMfaqh3QRi8Z{4$COV 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+\geometry{left=10mm,right=10mm, top=10mm} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step={4}, +} + +\begin{document} + +\input{banque.tex} +\printcollection{banque} + +\end{document} diff --git a/Tsti2d/Analyse/Operation_limites/banque.tex b/Tsti2d/Analyse/Operation_limites/banque.tex index 82eed0a..ee673da 100644 --- a/Tsti2d/Analyse/Operation_limites/banque.tex +++ b/Tsti2d/Analyse/Operation_limites/banque.tex @@ -110,4 +110,89 @@ \item En vous aidant de la calculatrice, tracer l'allure de la courbe de $f$ et noter les asymptotes. \end{enumerate} \end{exercise} + +\begin{exercise}[subtitle={Composée avec une exponentielle}, step={4}, topics={Limite}] + Retrouver les limites suivantes + \begin{multicols}{3} + \begin{enumerate} + \item $\ds \lim_{x \rightarrow +\infty} e^{2x + 1} = $ + \item $\ds \lim_{x \rightarrow +\infty} e^{-4x - 10} = $ + + \item $\ds \lim_{x \rightarrow -\infty} e^{2x^3 + 2x - 1}$ + \item $\ds \lim_{x \rightarrow -\infty} e^{\frac{3}{x}}$ + + \item $\ds \lim_{x \rightarrow 3} e^{5x + 2}= $ + \item $\ds \lim_{x \rightarrow 1+} e^{\frac{1}{x-1}}= $ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Solution d'équations différentielles}, step={4}, topics={Limite}] + \begin{enumerate} + \item On souhaite étudier la solution de l'équation différentielle + \[ + \begin{cases} + y' = -2y\\ + y(0) = 10 + \end{cases} + \] + \begin{enumerate} + \item Déterminer la solution de cette équations. + \item Déterminer la limite en $+\infty$ de la solution. + \end{enumerate} + \item On souhaite étudier la solution de l'équation différentielle + \[ + \begin{cases} + y' = 10y\\ + y(0) = 1 + \end{cases} + \] + \begin{enumerate} + \item Déterminer la solution de cette équations. + \item Déterminer la limite en $+\infty$ de la solution. + \end{enumerate} + \item On souhaite étudier la solution de l'équation différentielle + \[ + \begin{cases} + y' = -2y + 10\\ + y(0) = 3 + \end{cases} + \] + \begin{enumerate} + \item Déterminer la solution de cette équations. + \item Déterminer la limite en $+\infty$ de la solution. + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Composée avec un Logarithme}, step={4}, topics={Limite}] + Retrouver les limites suivantes + \begin{multicols}{3} + \begin{enumerate} + \item $\ds \lim_{x \rightarrow +\infty} \ln(2x + 1) = $ + \item $\ds \lim_{x \rightarrow 1+} \ln(x-1) = $ + + \item $\ds \lim_{x \rightarrow 0} x + \ln(x)= $ + \item $\ds \lim_{x \rightarrow +\infty} x + \ln(x)= $ + + \item $\ds \lim_{x \rightarrow +\infty} \ln\frac{2x+1}{x-1}$ + \item $\ds \lim_{x \rightarrow -\infty} \ln\frac{-5x^2 + 2}{10x^2 + x + 1}$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Composée avec un Logarithme}, step={4}, topics={Limite}] + Soit $f$ la fonction définie sur $\intOO{0}{+\infty}$ pas + \[ + f(x) = 1 + 2\frac{\ln x}{x} + \] + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est + \[ + f'(x) = \frac{2 - 2\ln x}{x^2} + \] + \item Étudier le signe de $f'(x)$ et en déduire les variations de $f$. + \item Compléter le tableau de variations en y ajoutant les limites et les valeurs remarquables. + \end{enumerate} +\end{exercise} \collectexercisesstop{banque} diff --git a/Tsti2d/Analyse/Operation_limites/index.rst b/Tsti2d/Analyse/Operation_limites/index.rst index 591b1b4..49ec29d 100644 --- a/Tsti2d/Analyse/Operation_limites/index.rst +++ b/Tsti2d/Analyse/Operation_limites/index.rst @@ -53,11 +53,15 @@ Exercices :height: 200px :alt: Exercices sur les limites d'une fraction rationnelle -Étape 4: Limite avec l'exponentielle -==================================== +Étape 4: Composée avec l'exponentielle et logarithme +==================================================== + +Cours: limite d'une fonction composée avec exponentielle ou logarithme + +.. image:: 4B_composee.pdf + :height: 200px + :alt: Limites d'une fonction composée avec exponentielle ou logarithme -Étape 5: Limite avec le logarithme -================================== Étape 6: Annales ================ diff --git a/Tsti2d/semaines.rst b/Tsti2d/semaines.rst index fde4ac2..873a21b 100644 --- a/Tsti2d/semaines.rst +++ b/Tsti2d/semaines.rst @@ -35,9 +35,9 @@ S20 - Programme Visio (2h) On commencera exactement à l'heure précisée donc essayez de vous connecter au moins 5min avant pour résoudre les éventuels problèmes techniques (je donnerai le lien 15 min avant). On commencera directement par la correction des questions flashs alors faites-les avant! -- Mardi 15h15-15h45: présentation de la semaine -- Mercredi 11h15-11h45: -- Jeudi 15h45-16h45: +- Mardi 15h15-15h45: présentation de la semaine, cours sur les limites des polynômes +- Mercredi 11h15-11h45: Q/R sur les exercices d'annales et introduction des limites des fractions rationnelles +- Jeudi 15h45-16h45: Q/R exercices limites polynômes et fraction rationnelles, début composée. S20 - Questions flashs (3x5min) ------------------------------- @@ -76,13 +76,25 @@ S20 - Opérations et limites (2h15) Cours: limite d'une fraction rationnelle -.. image:: 3B_fraction_rationnelle.pdf +.. image:: ./Analyse/Operation_limites/3B_fraction_rationnelle.pdf :height: 200px :alt: Limites d'une fraction rationnelle Exercices: limite d'une fraction rationnelle -.. image:: 3E_fraction_rationnelle.pdf +.. image:: ./Analyse/Operation_limites/3E_fraction_rationnelle.pdf + :height: 200px + :alt: Exercices sur les limites d'une fraction rationnelle + +Cours: limite d'une fonction composée avec exponentielle ou logarithme + +.. image:: ./Analyse/Operation_limites/4B_composee.pdf + :height: 200px + :alt: Limites d'une fraction rationnelle + +Exercice 1(uniquement) + +.. image:: ./Analyse/Operation_limites/4E_composee.pdf :height: 200px :alt: Exercices sur les limites d'une fraction rationnelle