From 087d3df807adfb4ac2dafa0e5ef3c007ab054362 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Thu, 10 Nov 2022 16:03:54 +0100 Subject: [PATCH] =?UTF-8?q?Feat:=20Premi=C3=A8re=20grosse=20moiti=C3=A9=20?= =?UTF-8?q?du=20chapitre=20sur=20le=20nombre=20d=C3=A9riv=C3=A9e?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 1ST/03_Nombre_derive_et_tangente/1B.tex | 14 - .../1B_taux_accroissement.pdf | Bin 0 -> 28929 bytes .../1B_taux_accroissement.tex | 71 +++++ .../2B_tangente.pdf | Bin 0 -> 22065 bytes .../2B_tangente.tex | 47 +++ .../exercises.tex | 298 +++++++++++++++++- 1ST/03_Nombre_derive_et_tangente/index.rst | 13 +- .../plan_de_travail.pdf | Bin 24054 -> 47868 bytes .../plan_de_travail.tex | 13 +- .../solutions.pdf | Bin 0 -> 25538 bytes .../solutions.tex | 2 +- 11 files changed, 428 insertions(+), 30 deletions(-) delete mode 100644 1ST/03_Nombre_derive_et_tangente/1B.tex create mode 100644 1ST/03_Nombre_derive_et_tangente/1B_taux_accroissement.pdf create mode 100644 1ST/03_Nombre_derive_et_tangente/1B_taux_accroissement.tex create mode 100644 1ST/03_Nombre_derive_et_tangente/2B_tangente.pdf create mode 100644 1ST/03_Nombre_derive_et_tangente/2B_tangente.tex create mode 100644 1ST/03_Nombre_derive_et_tangente/solutions.pdf diff --git a/1ST/03_Nombre_derive_et_tangente/1B.tex b/1ST/03_Nombre_derive_et_tangente/1B.tex deleted file mode 100644 index 1f65126..0000000 --- a/1ST/03_Nombre_derive_et_tangente/1B.tex +++ /dev/null @@ -1,14 +0,0 @@ -\documentclass[a4paper,10pt]{article} -\usepackage{myXsim} - -\author{Benjamin Bertrand} -\title{Nombre dérivé et tangente - Cours} -\date{novembre 2022} - -\pagestyle{empty} - -\begin{document} - -\maketitle - -\end{document} diff --git a/1ST/03_Nombre_derive_et_tangente/1B_taux_accroissement.pdf b/1ST/03_Nombre_derive_et_tangente/1B_taux_accroissement.pdf new file mode 100644 index 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coller le plus possible à la courbe en ce point. +\end{minipage} + \hfill + \begin{minipage}{0.45\linewidth} + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid= both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + ymin = -2, + ] + \addplot[domain=0:4,samples=20, color=red, very thick]{(x-3)^2-1}; + \addplot[mark=*, very thick, only marks] coordinates {(1,3) (3,-1)}; + \end{axis} + \end{tikzpicture} + \end{minipage} +\end{definition} + +\afaire{Tracer les tangentes aux points} + +\end{document} diff --git a/1ST/03_Nombre_derive_et_tangente/exercises.tex b/1ST/03_Nombre_derive_et_tangente/exercises.tex index 0727733..0f66e72 100644 --- a/1ST/03_Nombre_derive_et_tangente/exercises.tex +++ b/1ST/03_Nombre_derive_et_tangente/exercises.tex @@ -2,9 +2,9 @@ On souhaite évaluer la situation financière d'une entreprise. Pour cela, nous avons les chiffres d'affaires de quelques années \begin{center} - \begin{tabular}{|p{5cm}|*{5}{c|}} + \begin{tabular}{|p{7cm}|*{5}{c|}} \hline - Année & 1980 & 1995 & 2000 & 2010 & 2020 \\ + Année & 1980 & 1995 & 2000 & 2008 & 2020 \\ \hline Chiffre d'affaires (en milliers d'euros) & 10 & 18 & 29 & 45 & 50 \\ \hline @@ -13,14 +13,302 @@ \begin{enumerate} \item Tracer un repère et y placer les points pour représenter graphiquement le tableau. - \item Sur quel période, la progression du chiffre d'affaires a été le plus rapide? + \item Sur quel période, la progression du chiffre d'affaires a été le plus rapide ? + \item Trouver une façon de vérifier ce classement grâce au graphique. + \item Trouver un calcul qui permet de justifier ce classement sans ambigüités. + \end{enumerate} +\end{exercise} + +\begin{annexe} + \begin{tabular}{|p{4cm}|*{4}{p{3cm}|}} + \hline + Période & 1980-1990 & 1990-2000 & 2000-2008 & 2008-2020 \\ + \hline + Écart horizontal & & & & \\ + \hline + Écart vertical & & & & \\ + \hline + Rapport (vertical sur horizontal) & & & & \\ + \hline + \end{tabular} + +\end{annexe} + + +\begin{exercise}[subtitle={Vitesse moyenne d'une balle}, step={1}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\trainMode}] + \noindent + \begin{minipage}{0.55\linewidth} + On lance une balle et on décrit la hauteur ($h$ en m) en fonction du temps ($t$ en secondes) dans le graphique ci-contre + \begin{enumerate} + \item Quelle est la hauteur de la balle après 5 s ? + \item Calculer le taux de variation de la hauteur entre $t=0$ et $t=4$. + \item Calculer la vitesse moyenne entre $t=2$ et $t=10$. + \item Calculer la vitesse moyenne entre $t=10$ et $t=16$. + \item (*) Comment peut on deviner avant de faire le calcul le signe du taux de variation (ou de la vitesse moyenne)? + \end{enumerate} + \end{minipage} + \hfill + \begin{minipage}{0.4\linewidth} + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid = both, + xlabel = {$t$ en s}, + xtick distance=2, + ylabel = {$h$ en m}, + ytick distance=1, + ] + \addplot[domain=0:20,samples=20, color=red, very thick]{-0.1*x^2+2*x}; + \end{axis} + \end{tikzpicture} + \end{minipage} +\end{exercise} + +\begin{exercise}[subtitle={Taux de variations}, step={1}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\trainMode}] + \begin{enumerate} + \item Calculer le taux de variation de la fonction $f(x) = 3x + 1$ entre $x = 1$ et $x = 5$. + \item Calculer le taux de variation de la fonction $f(x) = -2x + 10$ entre $x = -3$ et $x = 4$. + \item Calculer le taux de variation de la fonction $f(x) = x^2 + x + 1$ entre $x = 5$ et $x = 10$. + \item Calculer le taux de variation de la fonction $f(x) = x^2 - 5$ entre $x = -3$ et $x = 3$. \end{enumerate} \end{exercise} \begin{solution} - + \begin{enumerate} + \item On veut calculer + \[ + \frac{f(5) - f(1)}{5 - 1} + \] + Pour cela, il faut tout d'abord calculer $f(5)$ et $f(1)$ + \[ + f(5) = 3\times 5 + 1 = 15 + 1 = 16 + \qquad \qquad + f(1) = 3\times 1 + 1 = 3 + 1 = 4 + \] + Donc le taux de variation est + \[ + \frac{f(5) - f(1)}{5 - 1} = \frac{16 - 4}{5-1} = \frac{12}{4} = 3 + \] + On peut interpréter cela en disant qu'en moyenne entre 1 et 5, la fonction a augmenté de 3 par unité. + \item + \[ + f(4) = -2\times 4 + 10 = 2 \qquad \qquad + f(-3) = -2\times (-3) + 10 = 16 + \] + \[ + \frac{f(5) - f(-3)}{5 - (-3)} = \frac{2 - 16}{5 + 3} = \frac{-14}{8} = \frac{-7}{4} + \] + \item + \[ + f(10) = 10^2 + 10 + 1 = 111 + \qquad \qquad + f(5) = 5^2 + 5 + 1 = 31 + \] + \[ + \frac{f(10) - f(5)}{10 - 5} = \frac{111 - 31}{10 - 5} = \frac{142}{5} = 28.4 + \] + \item + \[ + f(3) = 3^2 - 5 = 4 + \qquad \qquad + f(-3) = (-3)^2 - 5 = 4 + \] + \[ + \frac{f(3) - f(-3)}{3 - (-3)} = \frac{4 - 4}{6} = 0 + \] + \end{enumerate} \end{solution} +% Tangente et taux de variation +\begin{exercise}[subtitle={Tangente}, step={2}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\searchMode}] + Dans cet exercice, nous allons étudier comment se comporte le taux d'accroissement et la corde quand on fixe un point et que l'on fait se rapprocher l'autre point. L'étude de ce comportement mènera au concept de tangente. + \begin{enumerate} + \item Avec une fonction représentée par le graphique ci-dessous. -\begin{exercise}[subtitle={Résultats d'une entreprise}, step={1}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\searchMode}] + \begin{minipage}{0.45\linewidth} + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid= both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + ymin = 0, + ymax = 11, + ] + \addplot[domain=0:5,samples=20, color=red, very thick]{(x-3)^2 + 1}; + \end{axis} + \end{tikzpicture} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{enumerate} + \item On fixe le point $A$ qui est sur la courbe à l'abscisse 1. Repérer ce point sur le graphique. Quelle est la valeur de $f(1)$? + \item Représenter la corde entre $A$ et le point d'abscisse 5. Calculer le taux de variations entre 1 et 5. + \item Représenter la corde entre $A$ et le point d'abscisse 4. Calculer le taux de variations entre 1 et 4. + \item Faire la même chose pour l'abscisse 3, 2 puis 1,5. + \end{enumerate} + \end{minipage} + + \begin{enumerate} + \setcounter{enumi}{4} + \item Que peut-on observer sur les cordes? + \end{enumerate} + + \item Avec une fonction représentée par la formule: $f(x) = (x-3)^2 + 1$ + \begin{enumerate} + \item Calculer le taux de variation entre 1 et 2. + \item Calculer le taux de variation entre 1 et 1,5. + \item Calculer le taux de variation entre 1 et 1,25. + \item Calculer le taux de variation entre 1 et 0. + \item Calculer le taux de variation entre 1 et 0,5. + \item Calculer le taux de variation entre 1 et 0,75. + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Tracer des tangentes}, step={2}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\trainMode}] + Tracer les tangentes aux points marqués sur les graphiques + + \begin{minipage}{0.5\linewidth} + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid= both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + ymin = 0, + ymax = 11, + ] + \addplot[domain=0:5,samples=20, color=red, very thick]{(x-3)^2 + 1}; + \addplot [black, mark=*, very thick, only marks] coordinates {(3,1)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(1,5)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(4,2)}; + \end{axis} + \end{tikzpicture} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid= both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + ymin = -6, + ymax = 6, + ] + \addplot[domain=-2:2,samples=50, color=red, very thick]{sin(deg(x*pi/2))*5}; + \addplot [black, mark=*, very thick, only marks] coordinates {(-1,-5)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(1.5,5*sin(deg(1.5*pi/2)))}; + \addplot [black, mark=*, very thick, only marks] coordinates {(0,0)}; + \end{axis} + \end{tikzpicture} + \end{minipage} +\end{exercise} + + +\begin{exercise}[subtitle={Tracer une courbe}, step={2}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\trainMode}] + \begin{multicols}{2} + \begin{enumerate} + \item Tracer une courbe passant par les points. + + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid= both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + ymin = -6, + ymax = 6, + ] + \addplot [black, mark=*, very thick, only marks] coordinates {(-2,-3)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(-1,-5)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(0,0)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(1.5,5)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(2,2)}; + \end{axis} + \end{tikzpicture} + + \item Tracer une courbe passant par les points en respectant les tangentes. + + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid= both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + ymin = -6, + ymax = 6, + ] + \addplot [black, mark=*, very thick, only marks] coordinates {(-2,-3)}; + \addplot [mark=, very thick] coordinates {(-2,-3) (-1.8, -3.5)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(-1,-5)}; + \addplot [mark=, very thick] coordinates {(-1.2,-5) (-0.8, -5)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(0,0)}; + \addplot [mark=, very thick] coordinates {(-0.2,0) (0.2, 0)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(1.5,5)}; + \addplot [mark=, very thick] coordinates {(1.3, 4.8) (1.7, 5.2)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(2,2)}; + \addplot [mark=, very thick] coordinates {(1.8, 2) (2, 2)}; + \end{axis} + \end{tikzpicture} + \end{enumerate} + \end{multicols} +\end{exercise} + +% Nombre dérivé et tangente + + +\begin{exercise}[subtitle={Lire le nombre dérivé}, step={3}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\trainMode}] + Sur les courbes suivantes, tracer les tangentes aux points puis lire graphiquement le nombre dérivé. + Tracer les tangentes aux points marqués sur les graphiques + + \begin{minipage}{0.5\linewidth} + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid= both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + ] + \addplot[domain=-4:2,samples=20, color=red, very thick]{0.1*(x+1)^3 + 1}; + \addplot [black, mark=*, very thick, only marks] coordinates {(3,1)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(1,5)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(4,2)}; + \end{axis} + \end{tikzpicture} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid= both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + ymin = -6, + ymax = 6, + ] + \addplot[domain=-2:2,samples=50, color=red, very thick]{sin(deg(x*pi/2))*5}; + \addplot [black, mark=*, very thick, only marks] coordinates {(-1,-5)}; + \addplot [black, mark=*, very thick, only marks] coordinates {(1.5,5*sin(deg(1.5*pi/2)))}; + \addplot [black, mark=*, very thick, only marks] coordinates {(0,0)}; + \end{axis} + \end{tikzpicture} + \end{minipage} + +\end{exercise} + +\begin{exercise}[subtitle={Tracer la courbe avec les nombres dérivés}, step={3}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\trainMode}] \end{exercise} diff --git a/1ST/03_Nombre_derive_et_tangente/index.rst b/1ST/03_Nombre_derive_et_tangente/index.rst index fe41a8f..cd6c50e 100644 --- a/1ST/03_Nombre_derive_et_tangente/index.rst +++ b/1ST/03_Nombre_derive_et_tangente/index.rst @@ -2,7 +2,7 @@ Nombre dérivé et tangente ######################### :date: 2022-11-09 -:modified: 2022-11-09 +:modified: 2022-11-10 :authors: Benjamin Bertrand :tags: Dérivation :category: 1ST @@ -43,6 +43,13 @@ Commentaires Progression =========== +Plan de travail + +.. image:: ./plan_de_travail.pdf + :height: 200px + :alt: Plan de travail + + Étape 1: Taux d'accroissement ------------------------------ @@ -57,9 +64,9 @@ Bilan: calculs et classement des périodes. Définition du taux d'accroissement Trois exercices où les élèves trouvent des images, tracent des droites et calculent le taux d'accroissement. À chaque fois, on part d'un seul point puis on se rapproche de plus en plus. -Un exercice se base sur la lecture graphique, le suivant sur une formule et le dernier sur une programme qui calcul des images. +Un exercice se base sur la lecture graphique, le suivant sur une formule et le dernier sur un programme qui calcul des images. -Bilan: Notion de tangente. +Bilan: Notion de tangente et taux de variations d'une fonction Étape 3: Les droites -------------------- diff --git a/1ST/03_Nombre_derive_et_tangente/plan_de_travail.pdf b/1ST/03_Nombre_derive_et_tangente/plan_de_travail.pdf index 9b3f8859d2f9c5f83ec54afc48ce48fdbbb46eab..621d9e713e0986a01dd34fa01e8f46f5a637a3a1 100644 GIT binary patch literal 47868 zcmce6V~}ml(&pQ?PTRI^+qP}nwr$%sPTRI^+udjS+;1kniI|Cr`E&1%*lVTgsg=93 zGHX{YVmToZYI+)GNaD$*zQv){*}-8*COkSkTLTM7E-pM;C3ibxJX$qb0}EqACp=n3 zX9K5yNknWNtnq07UjJo5k4GyYVC#mb`O}mRkDl(ozS?-S()vyg=6~h%|E2lY^+&E? z>}c!kU})@!$NVEw_<=IFax!-Kxm)Qw84DR3+8X`zD`jkB>STsT&y2^zgZIBakd96c 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