From cc0f917df10e601655aafbf7fe73ae639c18a81d Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Thu, 3 Sep 2020 10:22:56 +0200 Subject: [PATCH] =?UTF-8?q?Feat:=20Aire=20sous=20la=20courbe=20en=20premie?= =?UTF-8?q?r=20chapitre=20et=20premi=C3=A8re=20=C3=A9tape=20faite?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../1B_integrale_aire.pdf | Bin 0 -> 77043 bytes .../1B_integrale_aire.tex | 118 ++++++++++++++++++ .../01_Aire_sous_la_courbe/1E_batteries.pdf | Bin 0 -> 26893 bytes .../01_Aire_sous_la_courbe/1E_batteries.tex | 23 ++++ .../01_Aire_sous_la_courbe/exercises.tex | 51 ++++++++ TST_sti2d/01_Aire_sous_la_courbe/index.rst | 38 ++++++ .../1B_vitesse.pdf | Bin .../1B_vitesse.tex | 0 .../1E_vitesse.pdf | Bin .../1E_vitesse.tex | 0 .../2B_derivee.pdf | Bin .../2B_derivee.tex | 0 .../2E_derivation.pdf | Bin .../2E_derivation.tex | 0 .../3E_problemes.pdf | Bin .../3E_problemes.tex | 0 .../4B_trigo.pdf | Bin .../4B_trigo.tex | 0 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b/TST_sti2d/01_Aire_sous_la_courbe/1B_integrale_aire.tex @@ -0,0 +1,118 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Aire sous la courbe - Cours} +\date{septembre 2020} + +\pagestyle{empty} + +\begin{document} + +\maketitle + +\section{Formules d'aire} + +\subsection*{Aire d'un triangle} + +\begin{center} + \begin{minipage}{0.2\textwidth} + \begin{tikzpicture}[scale=0.8] + \draw (0, 0) -- (1, 2) -- (3, 0) -- node [midway, below] {$b$} cycle; + \draw (1, 0) -- node [midway, right] {$h$} (1, 2); + \end{tikzpicture} + \end{minipage} + \begin{minipage}{0.2\textwidth} + \begin{tikzpicture}[scale=0.8] + \draw (0, 0) -- node [midway, left] {$h$} (0, 2) -- (3, 0) -- node [midway, below] {$b$} cycle; + \end{tikzpicture} + \end{minipage} + \begin{minipage}{0.2\textwidth} + \[ + Aire = \frac{b \times h}{2} + \] + \end{minipage} +\end{center} + +\section{Intégrale} + +Dans l'exercice sur la charge des batteries, on a vu que pour calculer l'énergie totale captée sur une période de temps, il fallait calculer une aire. Cette aire est appelée \textbf{intégrale}. + +\subsection*{Définition} + +\begin{minipage}{0.6\textwidth} +Soit $f$ une fonction continue et positive sur un intervalle $\intFF{a}{b}$ et $\mathcal{C}_f$ sa représentation graphique. + +\textbf{L'intégrale de $f$ sur $\intFF{a}{b}$} est l'aire situé entre l'axe des abscisses, la courbe $\mathcal{C}_f$ et les droites d'équations $x=a$ et $x=b$ (zone hachurée sur le graphique). + +On note cette quantité: +\[ + \int_a^b f(x) dx +\] +\end{minipage} +\begin{minipage}{0.3\textwidth} + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=-3,xmax=4,ymin=-2,ymax=4] + \tkzDrawXY + \tkzFct[domain = -2.15:3.2]{(2+\x)*exp(-\x)} + \tkzDrawArea[pattern=north west lines,domain =-1:2] + %tkzRep + \draw (-1,0) node [below] {a}; + \draw (2,0) node [below] {b}; + \end{tikzpicture} +\end{minipage} + +\subsection*{Exemples} + +\begin{itemize} + \item Énergie captée entre 14h et 20h par "l'électricité". + + \begin{tikzpicture}[scale=0.8] + \draw (0, 3) node [above] {Puissance (W)}; + \draw (0, 1) node [left] {100}; + \draw (12, 0) node [above right] {Heure}; + \draw (3, 0) node [below] {6}; + \draw (6, 0) node [below] {12}; + \draw (9, 0) node [below] {18}; + \draw (12, 0) node [below] {24}; + \draw (12, 0) node [above right] {Heure}; + \draw[very thin, gray, xstep=0.5] (0,0) grid (12,3); + \draw[->, very thick] (-0.5,0) -- (12.5,0); + \draw[->, very thick] (0,-0.5) -- (0,3.2); + + \draw[very thick, color=red] plot coordinates{(0,1) (3,1) (3,2) (6,2) (6,1) (9,1) (9,2) (12,2) (12,1)}; + \end{tikzpicture} + + Cette quantité correspond à l'aire sous la courbe entre 14 et 20. C'est une intégrale + \[ + \int_{14}^{20} P(t) dt = + \] + \afaire{Hachurer zone correspondant et calculer cette quantité} + + \item Énergie captée entre 11h et 20h par "le solaire". + + \begin{tikzpicture}[scale=0.8] + \draw (0, 3) node [above] {Puissance (W)}; + \draw (0, 1) node [left] {100}; + \draw (12, 0) node [above right] {Heure}; + \draw (3, 0) node [below] {6}; + \draw (6, 0) node [below] {12}; + \draw (9, 0) node [below] {18}; + \draw (12, 0) node [below] {24}; + \draw (12, 0) node [above right] {Heure}; + \draw[very thin, gray, xstep=0.5] (0,0) grid (12,3); + \draw[->, very thick] (-0.5,0) -- (12.5,0); + \draw[->, very thick] (0,-0.5) -- (0,3.2); + + \draw[very thick, color=red] plot coordinates{(0, 0) (3,0) (5.5,3) (7,3) (10,0) (12,0) }; + \end{tikzpicture} + + Cette quantité correspond à l'aire sous la courbe entre 11 et 20. C'est une intégrale + \[ + \int_{11}^{20} P(t) dt = + \] + \afaire{Hachurer zone correspondant et calculer cette quantité} +\end{itemize} + + +\end{document} diff --git a/TST_sti2d/01_Aire_sous_la_courbe/1E_batteries.pdf b/TST_sti2d/01_Aire_sous_la_courbe/1E_batteries.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4ea31c7f8bacb328695223789cd5069209b216db GIT binary patch literal 26893 zcmce7V~{9OkLcL8ZQHhO+qUhwW83x}+cS4;+qV7gx4U2MR=roX@6Y#jRi{o)(x*F> zbSgPHAyp6&qh+LHg(96@9A1TDAz&b|H?o4_;en!8@pLdHpx2N$vNAPxfudJ-HFEin z88Le&TPS)VA$t!3ZHB-6j12#A>q620gYq9wF+$PH8oD@H{^el&%kh`~5BQfu$<*22 z)yde@nSk~0j1m;RjH#WuivF7yD_aV^bR%(2m0$sHpa z6G~Xqz?77up}snK3GzV5msn3o%9P}cqy^;ZY6Fjh-aV9q@Ct|E%1U@9FD)d&L8J*S zzBR;1!`T{GxF+|{o$~9+{(7y7k5JN)8k;@RkHDM^6$*2pffY?tKVABKDTyB4wkn>a#gpn zUb0@4N%-q|%hk10*PlmaS*L1-Qc)wBWwqT4OwVgS{K>C(GxSL5<-^Xub(8aeQ&nnA zx}5AmFbuS0HWTGm)W~oVQbgZbihb$)+O1n_C01h^w@V}vwdrc7a=@UYe_IpDk*dU} z*0xMDV${C!srJ3#VQm~0&Rc(*#`o)fkm+ekcAHctZuOAOM;Bdv3D2xXI{-mRNPrQ` 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z7-*?8Cn(somXUgfk_Msii7_hEhs1kVx6}FfC`FTbU!Bxf&CxeCQR}Fvux9Ut`OH^% z{kI&~V3ybrpWMDZi9SxbCi)HE#;GXp|C1^H_a(~z*4R-rF@>g=ur)UMm!u9G^OsVN zl7+ho0ozxqqxO%64if?6Kk7LE0Pu@E{9nor61JuQf`6r*UosC8!US48oUEK|98652 zOahFetQ-uC%px3Yf{Y@Z%nX7;oT5UUyafNV$=4|V)^uR}^8LTftxzYp(R$=j0(UxY zZ^8q_e_VT`NFq, very thick] (-0.5,0) -- (12.5,0); + \draw[->, very thick] (0,-0.5) -- (0,3.2); + + \draw[very thick, color=red] plot coordinates{(0,1) (3,1) (3,2) (6,2) (6,1) (9,1) (9,2) (12,2) (12,1)}; + \end{tikzpicture} + + \item \textbf{Solaire} en prenant compte la variation de l'ensoleillement la capacité varie comme ci-dessous + + \begin{tikzpicture}[scale=0.8] + \draw (0, 3) node [above] {Puissance (W)}; + \draw (0, 1) node [left] {100}; + \draw (12, 0) node [above right] {Heure}; + \draw (3, 0) node [below] {6}; + \draw (6, 0) node [below] {12}; + \draw (9, 0) node [below] {18}; + \draw (12, 0) node [below] {24}; + \draw (12, 0) node [above right] {Heure}; + \draw[very thin, gray, xstep=0.5] (0,0) grid (12,3); + \draw[->, very thick] (-0.5,0) -- (12.5,0); + \draw[->, very thick] (0,-0.5) -- (0,3.2); + + \draw[very thick, color=red] plot coordinates{(0, 0) (3,0) (5.5,3) (7,3) (10,0) (12,0) }; + \end{tikzpicture} + \end{itemize} + + \begin{enumerate} + \item Combien de batteries pourront être rechargées entre 14h et 20h avec chacune de ses 3 solutions? + \item Quels sont les solutions qui permettent de recharger tout le parc de batteries sur une journée? + \end{enumerate} +\end{exercise} + + +\collectexercisesstop{banque} diff --git a/TST_sti2d/01_Aire_sous_la_courbe/index.rst b/TST_sti2d/01_Aire_sous_la_courbe/index.rst new file mode 100644 index 0000000..3efdc1d --- /dev/null +++ b/TST_sti2d/01_Aire_sous_la_courbe/index.rst @@ -0,0 +1,38 @@ +Aire sous la courbe +################### + +:date: 2020-08-14 +:modified: 2020-08-14 +:authors: Benjamin Bertrand +:tags: Intégrale, Analyse +:category: TST_sti2d +:summary: Calcul d'intégrales en se basant sur le calcul de l'aire sous la courbe. + +Étape 1: Puissance pour charger des batteries +============================================= + +Activité autour du stockage d'énergie dans des batteries. Les élèves ont trois sources (constante, créneau et en "bosse"). Ils devront venir au calcul d'aire sous la courbe pour obtenir l'énergie captée. + +.. image:: ./1E_batteries.pdf + :height: 200px + :alt: Calcule d'aire sous la courbe pour sommer une énergie. + +Cours: Formules de calculs d'airs et notation intégrales. + +.. image:: ./1B_integrale_aire.pdf + :height: 200px + :alt: Bilan sur la définition de l'intégrale. + +Étape 2: Calculs théoriques +=========================== + +Même activité que précédemment mais avec des fonctions mathématiques. Le but est d'apprivoiser la notation intégrales puis de construire des formules de calculs d'intégrales pour les fonctions constantes, linéaires et affines. + +Cours: les formules trouvées. + +Étape 3: Approximation par la méthode des rectangles +==================================================== + +Dans cette étape, on reprend la situation initiale avec la puissance. Mais cette fois ci, les courbes sont plus réalistes. Les élèves vont devoir donc aller chercher à approximer l'aire sous la courbe avec des formes plus simples. + +Cours: la méthode d'approximation par les rectangles. diff --git a/TST_sti2d/01_Derivation/1B_vitesse.pdf b/TST_sti2d/02_Derivation/1B_vitesse.pdf similarity index 100% rename from TST_sti2d/01_Derivation/1B_vitesse.pdf rename to TST_sti2d/02_Derivation/1B_vitesse.pdf diff --git a/TST_sti2d/01_Derivation/1B_vitesse.tex b/TST_sti2d/02_Derivation/1B_vitesse.tex similarity index 100% rename from TST_sti2d/01_Derivation/1B_vitesse.tex rename to TST_sti2d/02_Derivation/1B_vitesse.tex diff --git a/TST_sti2d/01_Derivation/1E_vitesse.pdf b/TST_sti2d/02_Derivation/1E_vitesse.pdf similarity index 100% rename from TST_sti2d/01_Derivation/1E_vitesse.pdf rename to TST_sti2d/02_Derivation/1E_vitesse.pdf diff --git a/TST_sti2d/01_Derivation/1E_vitesse.tex b/TST_sti2d/02_Derivation/1E_vitesse.tex similarity index 100% rename from TST_sti2d/01_Derivation/1E_vitesse.tex rename to TST_sti2d/02_Derivation/1E_vitesse.tex diff --git a/TST_sti2d/01_Derivation/2B_derivee.pdf b/TST_sti2d/02_Derivation/2B_derivee.pdf similarity index 100% rename from TST_sti2d/01_Derivation/2B_derivee.pdf rename to TST_sti2d/02_Derivation/2B_derivee.pdf diff --git a/TST_sti2d/01_Derivation/2B_derivee.tex b/TST_sti2d/02_Derivation/2B_derivee.tex similarity index 100% rename from TST_sti2d/01_Derivation/2B_derivee.tex rename to TST_sti2d/02_Derivation/2B_derivee.tex diff --git a/TST_sti2d/01_Derivation/2E_derivation.pdf b/TST_sti2d/02_Derivation/2E_derivation.pdf similarity index 100% rename from TST_sti2d/01_Derivation/2E_derivation.pdf rename to TST_sti2d/02_Derivation/2E_derivation.pdf diff --git a/TST_sti2d/01_Derivation/2E_derivation.tex b/TST_sti2d/02_Derivation/2E_derivation.tex similarity index 100% rename from TST_sti2d/01_Derivation/2E_derivation.tex rename to TST_sti2d/02_Derivation/2E_derivation.tex diff --git a/TST_sti2d/01_Derivation/3E_problemes.pdf b/TST_sti2d/02_Derivation/3E_problemes.pdf similarity index 100% rename from TST_sti2d/01_Derivation/3E_problemes.pdf rename to TST_sti2d/02_Derivation/3E_problemes.pdf diff --git a/TST_sti2d/01_Derivation/3E_problemes.tex b/TST_sti2d/02_Derivation/3E_problemes.tex similarity index 100% rename from TST_sti2d/01_Derivation/3E_problemes.tex rename to TST_sti2d/02_Derivation/3E_problemes.tex diff --git a/TST_sti2d/01_Derivation/4B_trigo.pdf b/TST_sti2d/02_Derivation/4B_trigo.pdf similarity index 100% rename from TST_sti2d/01_Derivation/4B_trigo.pdf rename to TST_sti2d/02_Derivation/4B_trigo.pdf diff --git a/TST_sti2d/01_Derivation/4B_trigo.tex b/TST_sti2d/02_Derivation/4B_trigo.tex similarity index 100% rename from TST_sti2d/01_Derivation/4B_trigo.tex rename to TST_sti2d/02_Derivation/4B_trigo.tex diff --git a/TST_sti2d/01_Derivation/4E_trigo.pdf b/TST_sti2d/02_Derivation/4E_trigo.pdf similarity index 100% rename from TST_sti2d/01_Derivation/4E_trigo.pdf rename to TST_sti2d/02_Derivation/4E_trigo.pdf diff --git a/TST_sti2d/01_Derivation/4E_trigo.tex b/TST_sti2d/02_Derivation/4E_trigo.tex similarity index 100% rename from TST_sti2d/01_Derivation/4E_trigo.tex rename to TST_sti2d/02_Derivation/4E_trigo.tex diff --git a/TST_sti2d/01_Derivation/Vitesse instantanée.ggb b/TST_sti2d/02_Derivation/Vitesse instantanée.ggb similarity index 100% rename from TST_sti2d/01_Derivation/Vitesse instantanée.ggb rename to TST_sti2d/02_Derivation/Vitesse instantanée.ggb diff --git a/TST_sti2d/01_Derivation/exercises.tex b/TST_sti2d/02_Derivation/exercises.tex similarity index 100% rename from TST_sti2d/01_Derivation/exercises.tex rename to TST_sti2d/02_Derivation/exercises.tex diff --git a/TST_sti2d/01_Derivation/index.rst b/TST_sti2d/02_Derivation/index.rst similarity index 100% rename from TST_sti2d/01_Derivation/index.rst rename to TST_sti2d/02_Derivation/index.rst