From a9f2e61ed95cffe3521f94d64f477f77a4b7a981 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Sun, 22 Nov 2020 10:39:29 +0100 Subject: [PATCH] =?UTF-8?q?Feat:=20exercices=20de=20calculs=20d'int=C3=A9g?= =?UTF-8?q?rales?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../2E_primitives_integrales.pdf | Bin 0 -> 93328 bytes .../2E_primitives_integrales.tex | 18 +++ .../04_Integrale_et_Primitives/exercises.tex | 110 ++++++++++++++++++ .../04_Integrale_et_Primitives/index.rst | 4 + 4 files changed, 132 insertions(+) create mode 100644 TST_sti2d/04_Integrale_et_Primitives/2E_primitives_integrales.pdf create mode 100644 TST_sti2d/04_Integrale_et_Primitives/2E_primitives_integrales.tex diff --git a/TST_sti2d/04_Integrale_et_Primitives/2E_primitives_integrales.pdf b/TST_sti2d/04_Integrale_et_Primitives/2E_primitives_integrales.pdf new file mode 100644 index 0000000000000000000000000000000000000000..97090bdac54616a97be85e8cb60cf1aa65bed8b2 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+1,18 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Integrale et Primitives - Cours} +\date{novembre 2020} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, +} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} + +\end{document} diff --git a/TST_sti2d/04_Integrale_et_Primitives/exercises.tex b/TST_sti2d/04_Integrale_et_Primitives/exercises.tex index 31e1cfb..25e7f37 100644 --- a/TST_sti2d/04_Integrale_et_Primitives/exercises.tex +++ b/TST_sti2d/04_Integrale_et_Primitives/exercises.tex @@ -71,4 +71,114 @@ f(x) = x \qquad g(x) = 2 \qquad h(x) = x^2 \qquad i(x) = x^3 \qquad j(x) = x^n \qquad k(x) = \dfrac{1}{x^2} \qquad l(x) = \cos(x) \] \end{exercise} + +\begin{exercise}[subtitle={Calculs de primitives }, step={2}, origin={Création}, topics={Integrale et Primitives}, tags={Intégrale, Primitive, physique}] + \begin{enumerate} + \item Calculer les primitives de fonctions suivantes + \begin{multicols}{3} + \begin{enumerate} + \item $f(x) = 2x + 1$ + \item $g(t) = t^2-2t +2$ + + \item $h(x) = 2x(4x+1)$ + \item $i(x) = x + 1 + \frac{1}{x^2}$ + + \item $j(x) = 3x - \cos(x)$ + \item $k(x) = x^{10} + \sin(x)$ + \end{enumerate} + \end{multicols} + \item Calculer les primitives de fonctions suivantes en respectant les contraintes + \begin{multicols}{2} + \begin{enumerate} + \item $f(x) = 2x + 1$ et $F(0) = 5$ + \item $g(t) = t^2-2t +2$ et $G(10) = 0$ + \end{enumerate} + \end{multicols} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Calculs d'intégrales}, step={2}, origin={Création}, topics={Integrale et Primitives}, tags={Intégrale, Primitive, physique}] + Calculer les valeurs suivantes + \begin{multicols}{3} + \begin{enumerate} + \item $\ds A = \int_1^2 9x^2 - 2x + 2\; dx$ + \item $\ds B = \int_3^4 5x^3 + 2x^2 + 1\; dx$ + \item $\ds C = \int_{\frac{\pi}{2}}^{\pi} \cos(x) \; dx$ + \end{enumerate} + \end{multicols} +\end{exercise} + +% \begin{exercise}[subtitle={Relation de Chasles}, step={2}, origin={Création}, topics={Integrale et Primitives}, tags={Intégrale, Primitive, physique}] +% On souhaite calculer plusieurs intégrales de la fonction $f(x) =3x^2 + 4x - 1$ +% \begin{enumerate} +% \item Calculer un primitive de $f$. +% \item Représenter graphiquement les quantités suivantes puis les calcules. +% \[ +% \int_{1}^2 f(x) \;dx \qquad +% \int_{2}^3 f(x) \;dx \qquad +% \] +% \item Représenter graphiquement la quantité $\ds \int_{1}^3 f(x) \;dx$ et déduire sa valeur à partir de la questions précédente +% \item (*) Quelle formule peut-on conjecturer des deux questions précédentes? (si vous êtes pas trompé, cette formule s'appelle la relation de Chasles). +% \end{enumerate} +% \end{exercise} + + +\begin{exercise}[subtitle={Propriétés de l'intégrales}, step={2}, origin={Création}, topics={Integrale et Primitives}, tags={Intégrale, Primitive, physique}] + Dans cet exercice, le calcul de plusieurs intégrales devrait vous permettre d'intuiter les propriétés de l'intégrale (du même type de la relation de Chasles dans le premier exercice). + + \noindent + \begin{minipage}{0.5\textwidth} + Pour cela, on va s'intéresser aux deux fonctions suivantes (représentée ci-contre) + + \[ + f(x) = 2x - 4 \qquad \qquad g(x) = x^2 - 6x + 11 + \] + \end{minipage} + \begin{minipage}{0.5\textwidth} + \begin{tikzpicture}[yscale=0.3, xscale=1.2, domain=-1:6] + \tkzInit[xmin=-1,xmax=6,xstep=1, ymin=-4,ymax=7,ystep=1] + \tkzGrid + \tkzAxeXY[up space=0.5,right space=.2] + \tkzFct{2*x-4} + \tkzFct{(x-3)*(x-3)+2} + %\draw plot[id=g] function {(x-3)*(x-3)+2} node[right] {$g(x)$}; + \end{tikzpicture} + \end{minipage} + \begin{enumerate} + \item Influence du signe de la fonction + \begin{enumerate} + \item Calculer les quantités suivantes + \[ + \int_{1}^2 f(x) \;dx \qquad \qquad \int_3^4 g(x) \;dx + \] + \item Quel est le signe de $f(x)$ sur $\intFF{1}{2}$ puis sur $\intFF{3}{4}$? + \item Que peut-on conjecturer sur le lien entre le signe de la fonction et le signe de l'intégrale? + \end{enumerate} + \item Croissance de l'intégrale Pour les questions qui suivent on définira + \[ + h(x) = f(x) - g(x) + \] + \begin{enumerate} + \item Démontrer que $h(x) = -(x-3)(x-5)$ puis étudier le signe de $h(x)$. + \item Calculer les quantités suivantes + \[ + \int_3^5 h(x) \;dx + \] + \item En déduire, la comparaison des quantités suivantes + \[ + \int_3^5 f(x) \;dx \qquad \qquad \int_3^5 g(x) \;dx + \] + \item Que peut-on conjecturer de la questions (a) et (c)? + \end{enumerate} + \item Aire entre deux courbes. + \begin{enumerate} + \item Représenter sur le graphique la quantité + \[ + \int_3^5 f(x) \;dx - \int_3^5 g(x) \;dx + \] + \item En déduire, une méthode pour calculer l'aire contenue entre 2 courbes. + \end{enumerate} + \end{enumerate} +\end{exercise} + \collectexercisesstop{banque} diff --git a/TST_sti2d/04_Integrale_et_Primitives/index.rst b/TST_sti2d/04_Integrale_et_Primitives/index.rst index 41a8083..dc8ba61 100644 --- a/TST_sti2d/04_Integrale_et_Primitives/index.rst +++ b/TST_sti2d/04_Integrale_et_Primitives/index.rst @@ -38,6 +38,10 @@ Cours: Formulaire des primitives Calculs techniques de primitives puis d'intégrales. +.. image:: ./2E_primitives_integrales.pdf + :height: 200px + :alt: Exercices sur les calculs de primitives et d'intégrales + Étape 3: Problèmes avec des intégrales ======================================