From 4dd754b1b19c149d1b32c1dd448ac64fd5da4ca1 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Tue, 25 Apr 2023 11:11:18 +0200 Subject: [PATCH] Feat(2nd): exercices et cours sur les intervalles --- .../1B_Intervalles.pdf | Bin 0 -> 81000 bytes .../2B_Ensemble_nombres_reels.pdf | Bin 0 -> 50170 bytes .../exercises.tex | 181 ++++++++++++++++++ 2nd/15_Intervalles_et_nombres_reels/index.rst | 62 ++++++ .../plan_de_travail.pdf | Bin 0 -> 53314 bytes .../plan_de_travail.tex | 67 +++++++ 6 files changed, 310 insertions(+) create mode 100644 2nd/15_Intervalles_et_nombres_reels/1B_Intervalles.pdf create mode 100644 2nd/15_Intervalles_et_nombres_reels/2B_Ensemble_nombres_reels.pdf create mode 100644 2nd/15_Intervalles_et_nombres_reels/exercises.tex create mode 100644 2nd/15_Intervalles_et_nombres_reels/index.rst create mode 100644 2nd/15_Intervalles_et_nombres_reels/plan_de_travail.pdf create mode 100644 2nd/15_Intervalles_et_nombres_reels/plan_de_travail.tex diff --git 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step={1}, origin={Camille}, topics={ Intervalles et nombres réels }, tags={ Inéquation, Intervalle, Nombres }, mode={\searchMode}] + \begin{minipage}{0.5\linewidth} + On a représenté ci-contre la fonction $f$. + \begin{enumerate} + \item Résoudre l'inéquation $f(x) \geq 1$ + \begin{enumerate} + \item d'écrire l'ensemble des solutions sous forme d'un intervalle. + \item Recopier et compléter la phrase suivante + \begin{center} + $f(x)$ est plus petit ou égal à 1 lorsque $x$ est plus grand que ... et plus petit que ... + \end{center} + \item Recopier et compléter la phrase suivante + \begin{center} + $\ldots \leq x \leq \ldots$ si et seulement si $f(x) \leq 1$ + \end{center} + \item Représenter les solutions de l'inéquation sur un axe gradué. + + \begin{tikzpicture}[xscale=0.7] + \draw[gray](-5.5,0)grid(4.5,0); + \draw[-stealth]|-(4.5,0)node[above]{$x$}; + \foreach \x in {-5,...,4} \draw (\x,-.1) -- (\x,0); + \end{tikzpicture} + \end{enumerate} + \end{enumerate} + \end{minipage} + \hfill + \begin{minipage}{0.4\linewidth} + \begin{tikzpicture}[xscale=1.3] + \tkzInit[xmin=-2.5,xmax=2.5,xstep=1, + ymin=-1.3,ymax=4.2,ystep=1] + \tkzGrid[sub] + \tkzAxeXY + \tkzFct[domain = -3:3,color=red,very thick]% + {x**2 - 1}; + \end{tikzpicture} + \end{minipage} + \begin{enumerate} + \setcounter{enumi}{1} + \item Reprendre les questions précédentes avec l'inéquation $f(x) > 1$. + \item Quels sont les différences entres les solutions de l'inéquation $f(x) \geq 1$ et $f(x) > 1$? + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Représentation d'intervalles}, step={1}, origin={Camille}, topics={ Intervalles et nombres réels }, tags={ Inéquation, Intervalle, Nombres }, mode={\trainMode}] + Compléter le tableau suivant + + \newcommand{\Raxe}{% + \begin{tikzpicture}[xscale=0.7] + \draw[gray](-5.5,0)grid(4.5,0); + \draw[-stealth]|-(4.5,0)node[above]{$x$}; + \foreach \x in {-5,...,4} \draw (\x,-.1) -- (\x,0); + \end{tikzpicture} + } + + \begin{tabular}{|p{6cm}|c|c|c|} + \hline + En français & Inégalité & sur la droite & Notation \\ + \hline + Ensemble des réels strictement plus grand que -1 & & \Raxe & \\ + \hline + & $-2 \leq x \leq$ 1 & \Raxe & \\ + \hline + & $1 \leq x < 3$ & \Raxe & \\ + \hline + & & \Raxe & $x\in \intOO{2}{5}$\\ + \hline + & & \Raxe & $x\in \intFO{2}{+\infty}$\\ + \hline + & & + \begin{tikzpicture}[xscale=0.7] + \draw[gray](-5.5,0)grid(4.5,0); + \draw[-stealth]|-(4.5,0)node[above]{$x$}; + \foreach \x in {-5,...,4} \draw (\x,-.1)node[below]{\x} -- (\x,0); + \draw[very thick, color=red](-5.5, 0) -- (3, 0) node {\large \textbf [}; + \end{tikzpicture} + & \\ + \hline + \end{tabular} +\end{exercise} + +\begin{exercise}[subtitle={Inéquations}, step={1}, origin={Classique}, topics={ Intervalles et nombres réels }, tags={ Inéquation, Intervalle, Nombres }, mode={\trainMode}] + Résoudre les inéquations suivantes et donner la réponse sous forme d'un axe gradué et d'un intervalle. + \begin{multicols}{4} + \begin{enumerate} + \item $4x + 5 > 0$ + \item $-4x + 5 \geq 5$ + \item $0.3x + 4 \leq 0.1x$ + \item $-8x + 5 < 7$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Union et intersection}, step={2}, origin={Ma tête}, topics={ Intervalles et nombres réels }, tags={ Inéquation, Intervalle, Nombres }, mode={\trainMode}] + Représenter les intervalles suivants sur l'axe des réels puis si c'est possible, proposer une écriture plus simple. + \begin{multicols}{4} + \begin{enumerate} + \item $\intFF{2}{5} \cap \intFO{3}{8}$ + \item $\intOF{-\infty}{3} \cap \intFO{-4}{3}$ + \item $\intFF{-2}{4} \cup \intOO{3}{7}$ + \item $\intFF{-3}{0} \cup \intFO{3}{+\infty}$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Inéquation graphique le retour!}, step={2}, origin={Ma tête}, topics={ Intervalles et nombres réels }, tags={ Inéquation, Intervalle, Nombres }, mode={\trainMode}] + \begin{minipage}{0.4\linewidth} + Sur le graphique ci-contre, on a tracé les représentations de 3 fonctions $f$, $g$ et $h$. + + Résoudre les inéquations suivantes en utilisant le graphique, vous donnerez les solutions sous forme d'intervalles. + \begin{enumerate} + \item $f(x) < 3$ + \item $f(x) \geq 0$ + \item $g(x) > 0$ + \item $g(x) \leq 1$ + \item $h(x) < f(x)$ + \item $h(x) \geq -2$ + \end{enumerate} + \end{minipage} + \hfill + \begin{minipage}{0.5\linewidth} + \begin{tikzpicture}[xscale=1.5, yscale=0.8] + \tkzInit[xmin=-3,xmax=3,xstep=1, + ymin=-3,ymax=4,ystep=1] + \tkzGrid + \tkzAxeXY + \tkzFct[domain = -3:3,color=red,very thick]{x**2 - 1}; + \tkzText(-1.8, 3.2){$\mathcal{C}_f$}; + + \tkzFct[domain = -3:3,color=green,very thick]{0.5*x+1}; + \tkzText(2.5, 1.8){$\mathcal{C}_g$}; + + \tkzFct[domain = -3:3,color=blue,very thick]{exp(x)-2}; + \tkzText(-2.5, -1.5){$\mathcal{C}_h$}; + \end{tikzpicture} + \end{minipage} +\end{exercise} + +\begin{exercise}[subtitle={Droite des réels}, step={3}, origin={Ma tête}, topics={ Intervalles et nombres réels }, tags={ Inéquation, Intervalle, Nombres }, mode={\trainMode}] + On a tracé un axe des nombres réels.za + + \begin{tikzpicture} + \draw[gray](0,0)grid(18.5,0); + \draw[-stealth]|-(18.5,0)node[above]{$x$}; + \foreach \x in {0,...,18} \draw (\x,-.1) -- (\x,0); + \draw (8, 0) node[below] {0}; + \draw (14, 0) node[below] {1}; + \draw (10, 0) node{$\bullet$} node[above] {B}; + \draw (1, 0) node{$\bullet$} node[above] {A}; + \end{tikzpicture} + + \begin{multicols}{2} + \begin{enumerate} + \item Représenter les nombres suivants sur cette droite : + \[ + -1 \qquad \frac{1}{6} \qquad \frac{-2}{3} \qquad \frac{3}{2} \qquad \frac{7}{6} \qquad \frac{4}{3} + \] + \item A quel nombre peut-on associer les points $A$ et $B$ ? + \item (*) Tracer l'ensemble des points à une distance strictement inférieur à $\dfrac{1}{2}$ du point $B$. Décrire cet ensemble sous forme d'un intervalle. + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Appartenance}, step={3}, origin={Ma tête}, topics={ Intervalles et nombres réels }, tags={ Inéquation, Intervalle, Nombres }, mode={\trainMode}] + Compléter à l'aide du signe $\in$ ou $\not \in$. + \begin{multicols}{4} + \begin{enumerate}[label={\alph*)}] + \item $2 \ldots \intOO{-1}{3}$ + \item $\dfrac{1}{3} \ldots \intFO{1}{3}$ + + \item $2 \ldots \intOO{-2}{2}$ + \item $0 \ldots \intFO{0}{+\infty}$ + + \item $100 \ldots \intOO{-\infty}{1}$ + \item $\dfrac{1}{10} \ldots \intFO{0.01}{0.2}$ + + \item $-1 \ldots \intOO{-1}{0}$ + \item $\dfrac{-3}{3} \ldots \intFF{-1}{3}$ + \end{enumerate} + \end{multicols} + +\end{exercise} diff --git a/2nd/15_Intervalles_et_nombres_reels/index.rst b/2nd/15_Intervalles_et_nombres_reels/index.rst new file mode 100644 index 0000000..74e4fa0 --- /dev/null +++ b/2nd/15_Intervalles_et_nombres_reels/index.rst @@ -0,0 +1,62 @@ +Intervalles et nombres réels +############################ + +:date: 2023-04-25 +:modified: 2023-04-25 +:authors: Benjamin Bertrand +:tags: Inéquation, Intervalle, Nombres +:category: 2nd +:summary: Ensemble de nombres réels représenté sous forme d'un intervalle. + + +Éléments du programme +===================== + +Capacités attendues en fin de chapitre + +- Associer à chaque point de la droite graduée un unique nombre réel et réciproquement. +- Représenter un intervalle de la droite numérique. +- Déterminer si un nombre réel appartient à un intervalle donné. +- Modéliser un problème par une inéquation. +- Résoudre une inéquation du premier degré. + +Plan de travail +=============== + +.. image:: ./Plan de travail.pdf + :height: 200px + :alt: Plan de travail + +.. image:: ./solutions.pdf + :height: 200px + :alt: Solution des exercices + + + +Étape 1: Inéquation graphique et intervalle +------------------------------------------- + +Cours + +.. image:: ./1B_Intervalles.pdf + :height: 200px + :alt: cours sur les intervalles + + +Étape 2: Union et intersection d'intervalles +-------------------------------------------- + +Étape 3: Nombres réels et droites des réels +------------------------------------------- + +Cours + +.. image:: ./2B_Ensemble_nombres_reels.pdf + :height: 200px + :alt: cours sur la droite de nombres réels + + +Remarques +--------- + +On travaillera les semaines qui suivront la séquence avec des inéquations résolues avec un tableau de signes en questions flashs diff --git a/2nd/15_Intervalles_et_nombres_reels/plan_de_travail.pdf b/2nd/15_Intervalles_et_nombres_reels/plan_de_travail.pdf new file mode 100644 index 0000000000000000000000000000000000000000..94ecdf87ad15bad31dd51127abc1105cfd43a53c GIT binary patch literal 53314 zcmce+V~{3M*CkrEx@_Yq+qP}nwr$(CZQHi(sxGUGJ^jwyduJkU#QgidjL1BhduQ%* zR_+x$u@0Gnh!`y+9V-mk%Ld>Uu$FNVk%;4Y;W?fUl~(7a~BIjMpi;TKEnTZ zfpK{t5MSo^z=l8oAU73IO$n%4wjX(X8 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+\author{Benjamin Bertrand} +\title{Intervalles de nombres réels - Plan de travail} +\tribe{2nd} +\date{} + +\pagestyle{empty} + +\DeclareExerciseCollection{banque} +\xsimsetup{ +} + +\begin{document} +\maketitle + +\begin{multicols}{2} + Savoir-faire de la séquence + \begin{itemize} + \item Associer à chaque point de la droite graduée un unique nombre réel et réciproquement. + \item Représenter un intervalle de la droite numérique. + \item Déterminer si un nombre réel appartient à un intervalle donné. + \item Modéliser un problème par une inéquation. + \item Résoudre une inéquation du premier degré. + \end{itemize} + + Ordre des étapes à respecter + + \begin{center} + \Ovalbox{ + \begin{tikzpicture} + \node (E2) [right of=E1] {1}; + \node (E3) [above right of=E2] {2}; + \node (E4) [below right of=E2] {3}; + + \path[->] (E2) edge (E3); + \path[->] (E2) edge (E4); + \end{tikzpicture} + } + \end{center} +\end{multicols} + + +\section{Les intervalles} + +\listsectionexercises + +\section{Union et intersection d'intervalles} + +\listsectionexercises + +\section{Droite des réels} + +\listsectionexercises + +\bigskip +\legendMode +\bigskip + +\bigskip + +\input{exercises.tex} +\printcollection{banque} + + +\end{document}