From fdd6be3e95ea8d4f180fb3ad155e1ebc0ebd6406 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Fri, 8 May 2020 09:48:25 +0200 Subject: [PATCH] Feat: limite d'une fraction rationnelle --- .../Operation_limites/2E_puissances.pdf | Bin 64992 -> 64992 bytes .../3B_fraction_rationnelle.pdf | Bin 0 -> 67537 bytes .../3B_fraction_rationnelle.tex | 41 ++++++++++++++++++ .../3E_fraction_rationnelle.pdf | Bin 0 -> 68592 bytes .../3E_fraction_rationnelle.tex | 21 +++++++++ Tsti2d/Analyse/Operation_limites/banque.tex | 32 ++++++++++++++ Tsti2d/Analyse/Operation_limites/index.rst | 23 ++++++++-- Tsti2d/semaines.rst | 30 +++++++++++-- 8 files changed, 140 insertions(+), 7 deletions(-) create mode 100644 Tsti2d/Analyse/Operation_limites/3B_fraction_rationnelle.pdf create mode 100644 Tsti2d/Analyse/Operation_limites/3B_fraction_rationnelle.tex create mode 100644 Tsti2d/Analyse/Operation_limites/3E_fraction_rationnelle.pdf create mode 100644 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en $+\infty$ ou $-\infty$ d'une fraction rationnelle est la même que le quotient des fonctions monômes de plus haut degré de son numérateur et de son dénominateur: +\[ + \lim_{x \rightarrow +\infty} \frac{ax^n + bx^{n-1} + ... + c}{a'x^m + b'x^{m-1} + ... + c'} = \lim_{x\rightarrow +\infty} \frac{ax^n}{a'x^m} +\] +\[ + \lim_{x \rightarrow -\infty} \frac{ax^n + bx^{n-1} + ... + c}{a'x^m + b'x^{m-1} + ... + c'} = \lim_{x\rightarrow -\infty} \frac{ax^n}{a'x^m} +\] + + +\subsubsection*{Exemples}% +Limites suivantes +\[ + \lim_{x \rightarrow +\infty} \frac{2x^3 + 3}{x + 1} = +\] +\[ + \lim_{x \rightarrow -\infty} \frac{2x - 4}{x^2 + 1} = +\] +\afaire{} + +\subsubsection*{Remarque} +Cette propriété n'est valable que pour les limites en $+\infty$ et $-\infty$. +\[ + \lim_{x \rightarrow 0+} \frac{2x^2 + 2}{x + 3} = +\] +\[ + \lim_{x \rightarrow 0+} \frac{2x^2 + 2}{x} = +\] +\afaire{} +\end{document} diff --git a/Tsti2d/Analyse/Operation_limites/3E_fraction_rationnelle.pdf 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Exercices} +\tribe{Terminale Tsti2d} +\date{Mai 2020} + +\pagestyle{empty} +\geometry{left=10mm,right=10mm, top=10mm} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step={3}, +} + +\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 7772e1d..82eed0a 100644 --- a/Tsti2d/Analyse/Operation_limites/banque.tex +++ b/Tsti2d/Analyse/Operation_limites/banque.tex @@ -78,4 +78,36 @@ \end{multicols} \end{exercise} +\begin{exercise}[subtitle={Limites de fractions rationnelles}, step={3}, topics={Limite}] + Retrouver les limites suivantes + \begin{multicols}{3} + \begin{enumerate} + \item $\ds \lim_{x \rightarrow +\infty} \frac{x + 1}{x^2 - 3} = $ + \item $\ds \lim_{x \rightarrow +\infty} \frac{4x^3 - 12}{2x^2 - 3} = $ + \item $\ds \lim_{x \rightarrow +\infty} \frac{2x + 1}{5x - 3} = $ + + \item $\ds \lim_{x \rightarrow -\infty} \frac{0.4x+1}{0.2x^2 - 3} = $ + \item $\ds \lim_{x \rightarrow -\infty} \frac{x^3 + 3x^2 +1}{x^5 + 2x - 3} = $ + \item $\ds \lim_{x \rightarrow -\infty} \frac{x+2x^2}{3x^2 - 3} = $ + + \item $\ds \lim_{x \rightarrow 0+} \frac{x+1}{x^2 - 3} = $ + \item $\ds \lim_{x \rightarrow 4+} \frac{x+1}{x^2 - 3} = $ + \item $\ds \lim_{x \rightarrow 1-} \frac{x+1}{x - 1} = $ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Étude d'une fonction rationnelle}, step={3}, topics={Limite}] + On souhaite étudier la fonction $f(x)$ définie sur $\intOO{-\infty}{1}\cup\intOO{1}{+\infty}$ par + \[ + f(x) = \frac{x^2 + x - 1}{x - 1} + \] + \begin{enumerate} + \item Déterminer la valeur interdite de $f$. + \item Calculer la dérivé de $f$. + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \item Compléter le tableau de variation en y ajoutant les limites que vous justifierez. + \item En vous aidant de la calculatrice, tracer l'allure de la courbe de $f$ et noter les asymptotes. + \end{enumerate} +\end{exercise} \collectexercisesstop{banque} diff --git a/Tsti2d/Analyse/Operation_limites/index.rst b/Tsti2d/Analyse/Operation_limites/index.rst index e29ed3e..591b1b4 100644 --- a/Tsti2d/Analyse/Operation_limites/index.rst +++ b/Tsti2d/Analyse/Operation_limites/index.rst @@ -2,7 +2,7 @@ Opération sur les limites de fonctions pour l'année 2019-2020 en terminale STI ################################################################################ :date: 2020-05-03 -:modified: 2020-05-07 +:modified: 2020-05-08 :authors: Bertrand Benjamin :category: Tsti2d :tags: Fonctions, Limites @@ -38,11 +38,26 @@ Exercices :height: 200px :alt: Exercice sur les limites de fonctions avec les puissances. -Étape 3: Limite avec l'exponentielle +Étape 3: Limite d'une fraction rationnelle +========================================== + +Cours: limite d'une fraction rationnelle + +.. image:: 3B_fraction_rationnelle.pdf + :height: 200px + :alt: Limites d'une fraction rationnelle + +Exercices + +.. image:: 3E_fraction_rationnelle.pdf + :height: 200px + :alt: Exercices sur les limites d'une fraction rationnelle + +Étape 4: Limite avec l'exponentielle ==================================== -Étape 4: Limite avec le logarithme +Étape 5: Limite avec le logarithme ================================== -Étape 5: Annales +Étape 6: Annales ================ diff --git a/Tsti2d/semaines.rst b/Tsti2d/semaines.rst index 12c0eea..fde4ac2 100644 --- a/Tsti2d/semaines.rst +++ b/Tsti2d/semaines.rst @@ -53,14 +53,38 @@ S20 - Annales loi exponentielles (30min) Deux exercices complets du bac de l'année dernière. Seules les parties B sont sur la loi exponentielle. Je vous devez à les faire en entier (sauf la partie C de l'exercice 1 qui est sur un chapitre que l'on a pas encore fait). -.. image:: 4E_annales.pdf +.. image:: ./Probabilite/Loi_exponentielle/4E_annales.pdf :height: 200px :alt: Exercices d'annales sur la loi exponentielle **À rendre: exercice 1 partie A et B OU exercice 2 partie B** -S20 - Opérations et limites ---------------------------- +S20 - Opérations et limites (2h15) +---------------------------------- + +- Cours: Limites des fonctions puissances et des polynômes + +.. image:: ./Analyse/Operation_limites/1B_puissances.pdf + :height: 200px + :alt: Limites des fonctions puissances. + +- Exercices: limites des fonctions polynômes et puissance + +.. image:: ./Analyse/Operation_limites/2E_puissances.pdf + :height: 200px + :alt: Exercices sur les limites des fonctions puissances et polynômes. + +Cours: limite d'une fraction rationnelle + +.. image:: 3B_fraction_rationnelle.pdf + :height: 200px + :alt: Limites d'une fraction rationnelle + +Exercices: limite d'une fraction rationnelle + +.. image:: 3E_fraction_rationnelle.pdf + :height: 200px + :alt: Exercices sur les limites d'une fraction rationnelle Semaine 19: 4/05 -> 8/05 ========================