Feat: 1E pour les TST_sti2d sur les limites
This commit is contained in:
parent
82ab0ebf62
commit
73f318d6dc
Binary file not shown.
|
|
@ -0,0 +1,166 @@
|
|||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\author{Benjamin Bertrand}
|
||||
\title{Limites de fonctions - Cours}
|
||||
\date{avril 2021}
|
||||
|
||||
\pagestyle{empty}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
\section{Tableaux de variations et limites des fonctions de référence}
|
||||
|
||||
\begin{itemize}
|
||||
\item Fonction carré $x\mapsto x^2$
|
||||
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
\begin{tikzpicture}[yscale=.5, xscale=.8]
|
||||
\tkzInit[xmin=-4,xmax=4,xstep=1,
|
||||
ymin=0,ymax=10,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY[up space=0.5,right space=.5]
|
||||
\tkzFct[domain = -5:5, line width=1pt]{x**2}
|
||||
\tkzText[draw,fill = brown!20](2.5,1){$f(x)=x^2$}
|
||||
\end{tikzpicture}
|
||||
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{tikzpicture}
|
||||
\tkzTabInit[lgt=2,espcl=3]{$x$/1,$f(x)$/3}%
|
||||
{$-\infty$, $0$, $+\infty$}%
|
||||
\tkzTabVar{+/$+\infty$, -/0, +/$+\infty$}%
|
||||
\end{tikzpicture}
|
||||
|
||||
Limites
|
||||
\[
|
||||
\lim_{x\rightarrow-\infty} x^2 = +\infty \qquad
|
||||
\lim_{x\rightarrow+\infty} x^2 = +\infty
|
||||
\]
|
||||
\end{minipage}
|
||||
|
||||
\item Fonction cube $x\mapsto x^3$
|
||||
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
\begin{tikzpicture}[yscale=0.5, xscale=1]
|
||||
\tkzInit[xmin=-3,xmax=3,xstep=1,
|
||||
ymin=-10,ymax=10,ystep=2]
|
||||
\tkzGrid
|
||||
\tkzAxeXY[up space=0.5,right space=.5]
|
||||
\tkzFct[domain = -3:3, line width=1pt]{x**3}
|
||||
\tkzText[draw,fill = brown!20](2,-8){$f(x)=x^3$}
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{tikzpicture}
|
||||
\tkzTabInit[lgt=2,espcl=5]{$x$/1,$f(x)$/3}%
|
||||
{$-\infty$, $+\infty$}%
|
||||
\tkzTabVar{-/$-\infty$, +/$+\infty$}%
|
||||
\tkzTabVal{1}{2}{0.5}{0}{0}
|
||||
\end{tikzpicture}
|
||||
|
||||
Limites
|
||||
\[
|
||||
\lim_{x\rightarrow-\infty} x^3 = -\infty \qquad
|
||||
\lim_{x\rightarrow+\infty} x^3 = +\infty
|
||||
\]
|
||||
\end{minipage}
|
||||
|
||||
\item Fonction inverse $x \mapsto \dfrac{1}{x}$
|
||||
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
\begin{tikzpicture}[yscale=.5, xscale=.8]
|
||||
\tkzInit[xmin=-4,xmax=4,xstep=1,
|
||||
ymin=-5,ymax=5,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY[up space=0.5,right space=.5]
|
||||
\tkzFct[domain = -5:-0.01, line width=1pt]{1/x}
|
||||
\tkzFct[domain = 0.01:5, line width=1pt]{1/x}
|
||||
\tkzText[draw,fill = brown!20](3,-4){$f(x)=\frac{1}{x}$}
|
||||
\tkzHLine[color=red,style=solid,line width=1.2pt]{0}
|
||||
\tkzVLine[color=green,style=solid,line width=1.2pt]{0}
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{tikzpicture}
|
||||
\tkzTabInit[lgt=1.5,espcl=3]{$x$ /1,$f(x)$ /3}
|
||||
{$-\infty$,$0$,$+\infty$}%
|
||||
\tkzTabVar{+/
|
||||
$0$ / ,-D+/ $-\infty$ / $+\infty$ , -/ $0$ /}
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
|
||||
Limites
|
||||
\[
|
||||
\lim_{x\rightarrow-\infty} \frac{1}{x} = 0 \qquad
|
||||
\lim_{\substack{x\rightarrow 0 \\ <}} \frac{1}{x} = -\infty \qquad
|
||||
\lim_{\substack{x\rightarrow 0 \\ >}} \frac{1}{x} = +\infty \qquad
|
||||
\lim_{x\rightarrow+\infty} \frac{1}{x} = 0
|
||||
\]
|
||||
|
||||
\pagebreak
|
||||
\item Fonction exponentielle $x\mapsto e^x$
|
||||
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
\begin{tikzpicture}[yscale=1, xscale=.8]
|
||||
\tkzInit[xmin=-5,xmax=2,xstep=1,
|
||||
ymin=0,ymax=5,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY[up space=0.5,right space=.5]
|
||||
\tkzFct[domain = -5:2, line width=1pt]{exp(x)}
|
||||
\tkzText[draw,fill = brown!20](2,0.5){$f(x)=\text{e}^{x}$}
|
||||
\tkzHLine[color=red,style=solid,line width=1.2pt]{0}
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{tikzpicture}
|
||||
\tkzTabInit[lgt=2,espcl=5]{$x$/1,$f(x)$/3}%
|
||||
{$-\infty$, $+\infty$}%
|
||||
\tkzTabVar{-/$0$, +/$+\infty$}%
|
||||
\end{tikzpicture}
|
||||
|
||||
Limites
|
||||
\[
|
||||
\lim_{x\rightarrow-\infty} e^x = 0 \qquad
|
||||
\lim_{x\rightarrow+\infty} e^x = +\infty
|
||||
\]
|
||||
\end{minipage}
|
||||
|
||||
|
||||
\item Fonction logarithme népérien $x \mapsto \ln{x}$
|
||||
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
\begin{tikzpicture}[yscale=0.8, xscale=1]
|
||||
\tkzInit[xmin=0,xmax=6,xstep=1,
|
||||
ymin=-3,ymax=3,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY[up space=0.5,right space=.5]
|
||||
\tkzFct[domain = 0.01:6, line width=1pt]{log(x)}
|
||||
\tkzText[draw,fill = brown!20](5,-2.5){$f(x)=\ln(x)$}
|
||||
\tkzVLine[color=green,style=solid,line width=1.2pt]{0}
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{tikzpicture}
|
||||
\tkzTabInit[lgt=2,espcl=5]{$x$/1,$f(x)$/3}%
|
||||
{$0$, $+\infty$}%
|
||||
\tkzTabVar{D-/$-\infty$, +/$+\infty$}%
|
||||
\end{tikzpicture}
|
||||
|
||||
Limites
|
||||
\[
|
||||
\lim_{\substack{x\rightarrow 0\\ > }} \ln{x} = -\infty \qquad
|
||||
\lim_{x\rightarrow+\infty} \ln{x} = +\infty
|
||||
\]
|
||||
\end{minipage}
|
||||
|
||||
\end{itemize}
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
|
|
@ -0,0 +1,20 @@
|
|||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\author{Benjamin Bertrand}
|
||||
\title{Limites de fonctions - Cours}
|
||||
\date{avril 2021}
|
||||
|
||||
\DeclareExerciseCollection{banque}
|
||||
\xsimsetup{
|
||||
step=1,
|
||||
}
|
||||
|
||||
\pagestyle{empty}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\input{exercises.tex}
|
||||
\printcollection{banque}
|
||||
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,129 @@
|
|||
\collectexercises{banque}
|
||||
\begin{exercise}[subtitle={Limites de fonctions}, step={1}, origin={Création}, topics={Limites de fonctions}, tags={Fonctions, limites}]
|
||||
\begin{tikzpicture}[yscale=.5, xscale=.8]
|
||||
\tkzInit[xmin=-5,xmax=5,xstep=1,
|
||||
ymin=0,ymax=10,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY
|
||||
\tkzFct[domain = -5:5, line width=1pt]{x**2}
|
||||
\tkzText[draw,fill = brown!20](3,1){$f(x)=x^2$}
|
||||
\end{tikzpicture}
|
||||
\hfill
|
||||
\begin{tikzpicture}[yscale=0.5, xscale=1]
|
||||
\tkzInit[xmin=-4,xmax=4,xstep=1,
|
||||
ymin=-10,ymax=10,ystep=2]
|
||||
\tkzGrid
|
||||
\tkzAxeXY
|
||||
\tkzFct[domain = -5:5, line width=1pt]{x**3}
|
||||
\tkzText[draw,fill = brown!20](1,-2){$f(x)=x^3$}
|
||||
\end{tikzpicture}
|
||||
|
||||
\begin{tikzpicture}[yscale=1, xscale=.8]
|
||||
\tkzInit[xmin=-5,xmax=5,xstep=1,
|
||||
ymin=0,ymax=5,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY
|
||||
\tkzFct[domain = -5:5, line width=1pt]{exp(x)}
|
||||
\tkzText[draw,fill = brown!20](2,1){$f(x)=\text{e}^{x}$}
|
||||
\end{tikzpicture}
|
||||
\hfill
|
||||
\begin{tikzpicture}[yscale=1, xscale=1.5]
|
||||
\tkzInit[xmin=0,xmax=5,xstep=1,
|
||||
ymin=-3,ymax=3,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY
|
||||
\tkzFct[domain = 0.01:5, line width=1pt]{log(x)}
|
||||
\tkzText[draw,fill = brown!20](2,2){$f(x)=\ln(x)$}
|
||||
\end{tikzpicture}
|
||||
|
||||
\begin{tikzpicture}[yscale=1.5, xscale=1]
|
||||
\tkzInit[xmin=-2,xmax=7,xstep=1,
|
||||
ymin=-2,ymax=2,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY
|
||||
\tkzFct[domain = -2:8, line width=1pt]{1 - exp(-x)}
|
||||
\tkzText[draw,fill = brown!20](1,1.5){$f(x)=1-e^{-x}$}
|
||||
\end{tikzpicture}
|
||||
\hfill
|
||||
\begin{tikzpicture}[yscale=.5, xscale=.8]
|
||||
\tkzInit[xmin=-5,xmax=5,xstep=1,
|
||||
ymin=-5,ymax=5,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY
|
||||
\tkzFct[domain = -5:-0.01, line width=1pt]{1/x}
|
||||
\tkzFct[domain = 0.01:5, line width=1pt]{1/x}
|
||||
\tkzText[draw,fill = brown!20](-2,2){$f(x)=\frac{1}{x}$}
|
||||
\end{tikzpicture}
|
||||
|
||||
\begin{tikzpicture}[yscale=0.5, xscale=.8]
|
||||
\tkzInit[xmin=-5,xmax=5,xstep=1,
|
||||
ymin=-1,ymax=10,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY
|
||||
\tkzFct[domain = -5:-0.01, line width=1pt]{1/x**2}
|
||||
\tkzFct[domain = 0.01:5, line width=1pt]{1/x**2}
|
||||
\tkzText[draw,fill = brown!20](3,3){$f(x)=\frac{1}{x^2}$}
|
||||
\end{tikzpicture}
|
||||
\hfill
|
||||
\begin{tikzpicture}[yscale=1.5, xscale=.8]
|
||||
\tkzInit[xmin=-5,xmax=5,xstep=1,
|
||||
ymin=-2,ymax=2,ystep=1]
|
||||
\tkzGrid
|
||||
\tkzAxeXY
|
||||
\tkzFct[domain = -5:5, line width=1pt]{cos(x)}
|
||||
\tkzText[draw,fill = brown!20](3,1){$f(x)=\cos{x}$}
|
||||
\end{tikzpicture}
|
||||
|
||||
À l'aide des graphiques ci-dessus, déterminer graphiquement les quantités suivantes
|
||||
|
||||
\begin{multicols}{3}
|
||||
\begin{enumerate}
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item $\ds \lim_{x\rightarrow +\infty} x^2 = $
|
||||
\item $\ds \lim_{x\rightarrow -\infty} x^2 = $
|
||||
\end{enumerate}
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item $\ds \lim_{x\rightarrow +\infty} x^3 = $
|
||||
\item $\ds \lim_{x\rightarrow -\infty} x^3 = $
|
||||
\end{enumerate}
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item $\ds \lim_{x\rightarrow +\infty} e^x = $
|
||||
\item $\ds \lim_{x\rightarrow -\infty} e^x = $
|
||||
\end{enumerate}
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item $\ds \lim_{x\rightarrow +\infty} \ln(x) = $
|
||||
\item $\ds \lim_{x\rightarrow 0} \ln(x) = $
|
||||
\end{enumerate}
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item $\ds \lim_{x\rightarrow +\infty} 1-e^{-x} = $
|
||||
\item $\ds \lim_{x\rightarrow -\infty} 1-e^{-x} = $
|
||||
\end{enumerate}
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item $\ds \lim_{x\rightarrow -\infty} \frac{1}{x} = $
|
||||
\item $\ds \lim_{\substack{x\rightarrow 0 \\ <}} \frac{1}{x} = $
|
||||
\item $\ds \lim_{\substack{x\rightarrow 0 \\ >}} \frac{1}{x} = $
|
||||
\item $\ds \lim_{x\rightarrow +\infty} \frac{1}{x} = $
|
||||
\end{enumerate}
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item $\ds \lim_{x\rightarrow -\infty} \frac{1}{x^2} = $
|
||||
\item $\ds \lim_{\substack{x\rightarrow 0 \\ <}} \frac{1}{x^2} = $
|
||||
\item $\ds \lim_{\substack{x\rightarrow 0 \\ >}} \frac{1}{x^2} = $
|
||||
\item $\ds \lim_{x\rightarrow +\infty} \frac{1}{x^2} = $
|
||||
\end{enumerate}
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item $\ds \lim_{x\rightarrow +\infty} \cos(x) = $
|
||||
\item $\ds \lim_{x\rightarrow -\infty} \cos(x) = $
|
||||
\end{enumerate}
|
||||
\end{enumerate}
|
||||
\end{multicols}
|
||||
\end{exercise}
|
||||
|
||||
\collectexercisesstop{banque}
|
||||
|
|
@ -0,0 +1,36 @@
|
|||
Limites de fonctions
|
||||
####################
|
||||
|
||||
:date: 2021-04-22
|
||||
:modified: 2021-04-22
|
||||
:authors: Benjamin Bertrand
|
||||
:tags: Fonctions, Limites
|
||||
:category: TST_sti2d
|
||||
:summary: Découverte et calculs de limites de fonctions
|
||||
|
||||
Étape 1: Découverte graphique des limites à connaître
|
||||
=====================================================
|
||||
|
||||
(à distance)
|
||||
|
||||
À partir de graphiques lire les valeurs des limites.
|
||||
|
||||
.. image:: ./1E_fonction_reference.pdf
|
||||
:height: 200px
|
||||
:alt: Déduire des graphiques les limites des fonctions de références
|
||||
|
||||
Bilan: Tableau de variation et limites des fonctions de références
|
||||
|
||||
.. image:: ./1B_fonction_reference.pdf
|
||||
:height: 200px
|
||||
:alt: Cours sur les tableau de variation et limites des fonctions de références
|
||||
|
||||
Étape 2: limites de polynômes
|
||||
=============================
|
||||
|
||||
Établir les règles de simplifications des limites avec les polynômes. Début du calcul formel de limites.
|
||||
|
||||
Étape 3: Croissances comparés avec l'exponentielle
|
||||
==================================================
|
||||
|
||||
|
||||
|
|
@ -2,7 +2,7 @@ Terminale technologique spécialité sti2d
|
|||
########################################
|
||||
|
||||
:date: 2020-08-21
|
||||
:modified: 2021-03-18
|
||||
:modified: 2021-04-22
|
||||
:authors: Bertrand Benjamin
|
||||
:category: TST_sti2d
|
||||
:tags: Progression
|
||||
|
|
@ -38,12 +38,11 @@ Période 4 (Février mars avril - 7 semaines)
|
|||
|
||||
- `Équation différentielle linéaire et affine <./07_Equation_differentielle>`_
|
||||
- `Étude fonction logarithme népérien <./08_Logarithme_Neperien>`_
|
||||
- Limite de fonctions
|
||||
|
||||
Période 5 (Mai juin - 10 semaines)
|
||||
==================================
|
||||
|
||||
- Limites avec l'exponentielle et les polynômes
|
||||
- `Limite de fonctions <./09_Limites_de_fonctions>`_
|
||||
- Propriété de l'intégrales
|
||||
- Composition de fonctions
|
||||
- Formule de duplication du sin et du cos
|
||||
|
|
|
|||
Loading…
Reference in New Issue