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zTk%4+9(voHnTZQ9lI@e?FYoN({Rp;DgPpcDP&m9txt?4@-!&Ni=~i(@zPlj4o1p{P zOj#rL>5Hx++SC(je>K?UXz_5iP0vKWoR1W1p>5csk+Q{gGr=BdO7&71MGntFU~#K}gVoR2#92lhf-HiBWMQ~S z2-t+sq{G;+96XfcR!FSqHwy-6iEpvh<- zv|vACUFR<Ff+cNy4g9pHGRb-Y9pZ!ih7 z`-ADeu#5XDrrY#%=o7<1*V&@CZxfdHPJF596_<5$EraJ8M%@Vhm|IjKUd?+@)S=AO z!TcFB`lgAkWmIHxpx^sT%C_^;Q&$`9{sl&+Yg1MuVz rf0-`cF+0b1P;zX7%Ktuxb96FraB};aiNUZib8@o7kdcYWi^KeXLh*XB diff --git a/1ST/05_Fonction_derivee/1E_fonction_derivee.tex b/1ST/05_Fonction_derivee/1E_fonction_derivee.tex deleted file mode 100644 index e38fa69..0000000 --- a/1ST/05_Fonction_derivee/1E_fonction_derivee.tex +++ /dev/null @@ -1,23 +0,0 @@ -\documentclass[a4paper,10pt]{article} -\usepackage{myXsim} -\usepackage{tikz} -\usepackage{pgfplots} - -\author{Benjamin Bertrand} -\title{Fonction derivé - Exercices} -\date{Janvier 2023} - -\DeclareExerciseCollection[step=1]{banque} -\xsimsetup{collect} - -\pagestyle{empty} - - -\begin{document} -\input{exercises.tex} - -\printcollection{banque} -\vfill -\printcollection{banque} - -\end{document} diff --git a/1ST/05_Fonction_derivee/1_techniques.tex b/1ST/05_Fonction_derivee/1_techniques.tex new file mode 100644 index 0000000..ecf52c9 --- /dev/null +++ b/1ST/05_Fonction_derivee/1_techniques.tex @@ -0,0 +1,309 @@ +\begin{exercise}[subtitle={Calculs de dérivée}, step={1}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\trainMode}] + Calculer les fonctions dérivées des fonctions suivantes + + \begin{multicols}{3} + \begin{enumerate} + \item $f(x) = - 6x - 7$ + \item $g(x) = 10x + 3$ + \item $h(x) = - 4x - 3$ + \item $i(x) = - 10x - 7$ + \item $j(x) = - 8x - 4$ + \item $k(x) = - 3x^{2} + 8x - 1$ + \item $l(x) = - 10x + 6$ + \item $m(x) = 10x^{2} + 5x + 6$ + \item $n(x) = - 10x^{2} + 6x - 2$ + \item $o(x) = 5x^{2}$ + \item $p(x) = - 9x^{2} + 4x$ + \item $q(x) = - 2x^{2} - 4$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{multicols}{2} + \begin{enumerate} + \item $f(x) = - 6x - 7$ + + \[ + f'(x) = - 6 + \] + \item $g(x) = 10x + 3$ + + \[ + g'(x) = 10 + \] + \item $h(x) = - 4x - 3$ + + \[ + h'(x) = - 4 + \] + \item $i(x) = - 10x - 7$ + + \[ + i'(x) = - 10 + \] + \item $j(x) = - 8x - 4$ + + \[ + j'(x) = - 8 + \] + \item $k(x) = - 3x^{2} + 8x - 1$ + + \[ + k'(x) = - 6x + 8 + \] + \item $l(x) = - 10x + 6$ + + \[ + l'(x) = - 10 + \] + \item $m(x) = 10x^{2} + 5x + 6$ + + \[ + m'(x) = 20x + 5 + \] + \item $n(x) = - 10x^{2} + 6x - 2$ + + \[ + n'(x) = - 20x + 6 + \] + \item $o(x) = 5x^{2}$ + + \[ + o'(x) = 10x + \] + \item $p(x) = - 9x^{2} + 4x$ + + \[ + p'(x) = - 18x + 4 + \] + \item $q(x) = - 2x^{2} - 4$ + + \[ + q'(x) = - 4x + \] + \end{enumerate} + \end{multicols} +\end{solution} + +\begin{exercise}[subtitle={Fonction affines - technique}, step={2}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\trainMode}] + Reprendre l'exercice précédent pour les fonctions suivantes: + + \begin{multicols}{2} + \begin{enumerate} + \item $f(x) = - 8x + 5$ + \item $g(x) = - 9x - 6$ + \item $h(x) = - 2x + 8$ + \item $i(x) = - 5x - 4$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item Étude de la fonction $f(x) = - 8x + 5$ + \begin{itemize} + \item Fonction dérivée : $f'(x) = - 8$ + \item Comme $- 8 < 0$ la fonction est décroissante + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=10]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{\hspace{2cm}, \hspace{2cm}}% + \tkzTabLine{,-,}% + \tkzTabVar{+/ ,-/ }% + \end{tikzpicture} + \end{center} + \end{itemize} + \item Étude de la fonction $g(x) = - 9x - 6$ + \begin{itemize} + \item Fonction dérivée : $g'(x) = - 9$ + \item Comme $- 9 < 0$ la fonction est décroissante + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=10]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{\hspace{2cm}, \hspace{2cm}}% + \tkzTabLine{,-,}% + \tkzTabVar{+/ ,-/ }% + \end{tikzpicture} + \end{center} + \end{itemize} + \item Étude de la fonction $h(x) = - 2x + 8$ + \begin{itemize} + \item Fonction dérivée : $h'(x) = - 2$ + \item Comme $- 2 < 0$ la fonction est décroissante + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=10]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{\hspace{2cm}, \hspace{2cm}}% + \tkzTabLine{,-,}% + \tkzTabVar{+/ ,-/ }% + \end{tikzpicture} + \end{center} + \end{itemize} + \item Étude de la fonction $i(x) = - 5x - 4$ + \begin{itemize} + \item Fonction dérivée : $i'(x) = - 5$ + \item Comme $- 5 < 0$ la fonction est décroissante + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=10]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{\hspace{2cm}, \hspace{2cm}}% + \tkzTabLine{,-,}% + \tkzTabVar{+/ ,-/ }% + \end{tikzpicture} + \end{center} + \end{itemize} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Fonction affines - technique}, step={2}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\trainMode}] + Reprendre l'exercice précédent pour les fonctions suivantes : + + \begin{multicols}{2} + \begin{enumerate} + \item $f(x) = 3x^{2} + 10x - 3$ + \item $g(x) = 4x^{2} + 2x - 2$ + \item $h(x) = - 4x^{2} + 2x - 7$ + \item $i(x) = - 9x^{2} + 9x - 9$ + \item $j(x) = - x^{2} + 8x + 4$ + \item $k(x) = 6x^{2} + 9x + 9$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item Étude de la fonction $f(x) = 3x^{2} + 10x - 3$ + \begin{itemize} + \item Fonction dérivée : $f'(x) = 6x + 10$ + \item On résout l'inéquation $f'(x) \geq 0$ pour déterminer quand la fonction $f'$ est positive. + \begin{align*} + f(x) & \geq 0 \\ + 6x + 10 & \geq 0 \\ + 6x + 10 + - 10 &\geq 0 + - 10 \\ + 6x &\geq - 10 \\ + \frac{6x}{6} &\geq \frac{- 10}{6} \\ + x &\geq \dfrac{- 5}{3} \\ + \end{align*} + Donc $f(x)$ est positif quand $x$ est plus \textbf{grand} que $\dfrac{- 5}{3}$ + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=4]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{, $\dfrac{- 5}{3}$ ,}% + \tkzTabLine{, -, z, +, } + \tkzTabVar{+/ ,-/$f(\dfrac{- 5}{3}) = \dfrac{- 102}{9}$ , +/}% + \end{tikzpicture} + \end{center} + \end{itemize} + \item Étude de la fonction $g(x) = 4x^{2} + 2x - 2$ + \begin{itemize} + \item Fonction dérivée : $g'(x) = 8x + 2$ + \item On résout l'inéquation $g'(x) \geq 0$ pour déterminer quand la fonction $g'$ est positive. + \begin{align*} + g(x) & \geq 0 \\ + 8x + 2 & \geq 0 \\ + 8x + 2 + - 2 &\geq 0 + - 2 \\ + 8x &\geq - 2 \\ + \frac{8x}{8} &\geq \frac{- 2}{8} \\ + x &\geq \dfrac{- 1}{4} \\ + \end{align*} + Donc $g(x)$ est positif quand $x$ est plus \textbf{grand} que $\dfrac{- 1}{4}$ + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=4]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{, $\dfrac{- 1}{4}$ ,}% + \tkzTabLine{, -, z, +, } + \tkzTabVar{+/ ,-/$f(\dfrac{- 1}{4}) = \dfrac{- 36}{16}$ , +/}% + \end{tikzpicture} + \end{center} + \end{itemize} + \item Étude de la fonction $h(x) = - 4x^{2} + 2x - 7$ + \begin{itemize} + \item Fonction dérivée : $h'(x) = - 8x + 2$ + \item On résout l'inéquation $h'(x) \geq 0$ pour déterminer quand la fonction $h'$ est positive. + \begin{align*} + h(x) & \geq 0 \\ + - 8x + 2 & \geq 0 \\ + - 8x + 2 + - 2 &\geq 0 + - 2 \\ + - 8x &\geq - 2 \\ + \frac{- 8x}{- 8} &\leq \frac{- 2}{- 8} \\ + x &\leq \dfrac{1}{4} \\ + \end{align*} + Donc $h(x)$ est positif quand $x$ est plus \textbf{petit} que $\dfrac{1}{4}$ + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=4]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{, $\dfrac{1}{4}$ ,}% + \tkzTabLine{, +, z, -, } + \tkzTabVar{-/ ,+/$f(\dfrac{1}{4}) = \dfrac{- 108}{16}$ , -/}% + \end{tikzpicture} + \end{center} + \end{itemize} + \item Étude de la fonction $i(x) = - 9x^{2} + 9x - 9$ + \begin{itemize} + \item Fonction dérivée : $i'(x) = - 18x + 9$ + \item On résout l'inéquation $i'(x) \geq 0$ pour déterminer quand la fonction $i'$ est positive. + \begin{align*} + i(x) & \geq 0 \\ + - 18x + 9 & \geq 0 \\ + - 18x + 9 + - 9 &\geq 0 + - 9 \\ + - 18x &\geq - 9 \\ + \frac{- 18x}{- 18} &\leq \frac{- 9}{- 18} \\ + x &\leq \dfrac{1}{2} \\ + \end{align*} + Donc $i(x)$ est positif quand $x$ est plus \textbf{petit} que $\dfrac{1}{2}$ + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=4]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{, $\dfrac{1}{2}$ ,}% + \tkzTabLine{, +, z, -, } + \tkzTabVar{-/ ,+/$f(\dfrac{1}{2}) = \dfrac{- 27}{4}$ , -/}% + \end{tikzpicture} + \end{center} + \end{itemize} + \item Étude de la fonction $j(x) = - x^{2} + 8x + 4$ + \begin{itemize} + \item Fonction dérivée : $j'(x) = - 2x + 8$ + \item On résout l'inéquation $j'(x) \geq 0$ pour déterminer quand la fonction $j'$ est positive. + \begin{align*} + j(x) & \geq 0 \\ + - 2x + 8 & \geq 0 \\ + - 2x + 8 + - 8 &\geq 0 + - 8 \\ + - 2x &\geq - 8 \\ + \frac{- 2x}{- 2} &\leq \frac{- 8}{- 2} \\ + x &\leq 4 \\ + \end{align*} + Donc $j(x)$ est positif quand $x$ est plus \textbf{petit} que $4$ + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=4]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{, $4$ ,}% + \tkzTabLine{, +, z, -, } + \tkzTabVar{-/ ,+/$f(4) = 20$ , -/}% + \end{tikzpicture} + \end{center} + \end{itemize} + \item Étude de la fonction $k(x) = 6x^{2} + 9x + 9$ + \begin{itemize} + \item Fonction dérivée : $k'(x) = 12x + 9$ + \item On résout l'inéquation $k'(x) \geq 0$ pour déterminer quand la fonction $k'$ est positive. + \begin{align*} + k(x) & \geq 0 \\ + 12x + 9 & \geq 0 \\ + 12x + 9 + - 9 &\geq 0 + - 9 \\ + 12x &\geq - 9 \\ + \frac{12x}{12} &\geq \frac{- 9}{12} \\ + x &\geq \dfrac{- 3}{4} \\ + \end{align*} + Donc $k(x)$ est positif quand $x$ est plus \textbf{grand} que $\dfrac{- 3}{4}$ + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=4]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{, $\dfrac{- 3}{4}$ ,}% + \tkzTabLine{, -, z, +, } + \tkzTabVar{+/ ,-/$f(\dfrac{- 3}{4}) = \dfrac{90}{16}$ , +/}% + \end{tikzpicture} + \end{center} + \end{itemize} + \end{enumerate} +\end{solution} diff --git a/1ST/05_Fonction_derivee/bopytex_config.py b/1ST/05_Fonction_derivee/bopytex_config.py new file mode 100644 index 0000000..599b6b5 --- /dev/null +++ b/1ST/05_Fonction_derivee/bopytex_config.py @@ -0,0 +1,12 @@ +# bopytex_config.py +from mapytex.calculus.random import expression as random_expression +from mapytex import render +import random + +random.seed(0) # Controlling the seed allows to make subject reproductible + +render.set_render("tex") + +direct_access = { + "random_expression": random_expression, +} diff --git a/1ST/05_Fonction_derivee/exercises.tex b/1ST/05_Fonction_derivee/exercises.tex index e700c0a..de4b5a5 100644 --- a/1ST/05_Fonction_derivee/exercises.tex +++ b/1ST/05_Fonction_derivee/exercises.tex @@ -1,15 +1,21 @@ -\begin{exercise}[subtitle={Construction de la fonction derivée}, step={1}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }] +\begin{exercise}[subtitle={Construction de la fonction dérivée}, step={1}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\searchMode}] Pour chacun des graphiques ci-dessous compléter les tableaux pour trouver les nombres dérivés. \begin{enumerate} \item ~ \begin{minipage}{0.4\textwidth} - \begin{tikzpicture}[yscale=.45, xscale=1] - \tkzInit[xmin=-3,xmax=3,xstep=1, - ymin=-5,ymax=5,ystep=1] - \tkzGrid - \tkzAxeXY[up space=0.5,right space=.5] - \tkzFct[domain = -3:3, line width=1pt]{-x**2} + \begin{tikzpicture}[] + \begin{axis}[ + axis lines = center, + grid = both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + legend pos = north west, + ] + \addplot[domain=-3:3,samples=80, color=red, very thick]{-x^2}; + \end{axis} \end{tikzpicture} \end{minipage} \hfill @@ -34,12 +40,18 @@ \item ~ \begin{minipage}{0.4\textwidth} - \begin{tikzpicture}[yscale=.35, xscale=1] - \tkzInit[xmin=-3,xmax=3,xstep=1, - ymin=-7,ymax=7,ystep=1] - \tkzGrid - \tkzAxeXY[up space=0.5,right space=.5] - \tkzFct[domain = -3:3, line width=1pt]{0.5*x**2 - 2} + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid = both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + legend pos = north west, + ] + \addplot[domain=-3:3,samples=80, color=red, very thick]{3*x}; + \end{axis} \end{tikzpicture} \end{minipage} \hfill @@ -60,11 +72,131 @@ \hline \end{tabular} \end{minipage} - \item Pour les deux fonctions précédentes, à partir des valeurs déjà trouvées, ne pourrait-on pas trouver une formule qui pourrait calculer tous les nombres dérivés de ces fonctions? \\ Combien vaudrait dans chacun des cas $f'(10)$? $f'(0,5)$? + \item ~ + + \begin{minipage}{0.4\textwidth} + \begin{tikzpicture} + \begin{axis}[ + axis lines = center, + grid = both, + xlabel = {$x$}, + xtick distance=1, + ylabel = {$f(x)$}, + ytick distance=1, + legend pos = north west, + ] + \addplot[domain=-3:3,samples=80, color=red, very thick]{0.5*(x-1)^2-2}; + \end{axis} + \end{tikzpicture} + \end{minipage} + \hfill + \begin{minipage}{0.5\textwidth} + \begin{tabular}{|m{2cm}|c|} + \hline + x & Nombre dérivé $f'(x)$\\ + \hline + -2 & \\ + \hline + -1 & \\ + \hline + 0 & \\ + \hline + 1 & \\ + \hline + 2 & \\ + \hline + \end{tabular} + \end{minipage} + + \item Pour chacune des fonctions précédentes, à partir des valeurs déjà trouvées, ne pourrait-on pas trouver une formule qui pourrait calculer tous les nombres dérivés de ces fonctions ? \\ Combien vaudrait dans chacun des cas $f'(10)$ ? $f'(0,5)$ ? + \end{enumerate} +\pagebreak +\end{exercise} + +\begin{exercise}[subtitle={Utilisation de la fonction dérivée}, step={1}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\trainMode}] + Ci-dessous, vous trouverez des couples de fonction avec leur dérivée. + + \begin{center} + \begin{tabular}{cp{2cm}c} + $f(x) = 5x^3 - x^2 + 0.3x + 1$ & & $g(x) = 0.3x^5 - 3x^2 + 5x + 1$ \\ + $f'(x) = 15x^2 - 2x + 0.3$ & & $g'(x) = 1.5x^4 - 6x + 5$ + \end{tabular} + \end{center} + + \begin{enumerate} + \item Déterminer le nombre dérivé de la fonction $f$ au point d'abscisse $x=2$ + \item Que peut-on dire sur la croissance de la fonction $f$ autour du point d'abscisse $x=2$? + \item Déterminer le nombre dérivé de la fonction $g$ au point d'abscisse $x=5$ + \item Que peut-on dire sur la croissance de la fonction $g$ autour du point d'abscisse $x=5$? + \item Que peut-on dire sur la croissance de la fonction $f$ autour du point d'abscisse $x=1$? + \item Que peut-on dire sur la croissance de la fonction $g$ autour du point d'abscisse $x=4$? + \item Que peut-on dire sur la croissance de la fonction $f$ autour du point d'abscisse $x=111$? + \item Vérifier vos résultats en traçant les fonctions $f$ et $g$ sur votre calculatrice. \end{enumerate} \end{exercise} -\begin{exercise}[subtitle={Gestion hôtelière}, step={1}, origin={???}, topics={ Fonction dérivée }, tags={ Dérivation }] +\begin{exercise}[subtitle={Calcul de la fonction dérivée}, step={1}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\groupMode}] + \begin{center} + \begin{tabular}{l|l|l|l|l} + $f(x) = 2x + 1$ & $g(x) = 3$ & $h(x) = 5x + 1$ & $i(x) = x^2 + x + 1$ & $j(x) = 3x^2 - 10x - 100$\\ + $f'(x) = 2$ & $g'(x) = 0$ & $h'(x) = 5$ & $i'(x) = 2x + 1$ & $j'(x) = 6x - 10$ + \end{tabular} + \end{center} + + En observant les couples fonctions et dérivées précédentes, déterminer les fonctions dérivées suivantes + \begin{center} + \begin{tabular}{l|l|l|l|l} + $f(x) = 4$ & $g(x) = 3x+2$ & $h(x) = -7x + 19$ & $i(x) = x^2 + 3x + 9$ & $j(x) = 4x^2 - x - 100$\\ + $f'(x) = ...$ & $g'(x) = ...$ & $h'(x) = ... $ & $i'(x) = ...$ & $j'(x) = ...$ + \end{tabular} + \end{center} + + Expliquer votre méthode pour déterminer ces dérivées. +\end{exercise} + + +% ------ +% Fonction de degré 1 + +\begin{exercise}[subtitle={Fonction affines}, step={2}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\searchMode}] + On définit la fonction $f(x) = 5x - 10$ dont on veut étudier les variations. + \begin{enumerate} + \item Calculer $f'(x)$ la fonction dérivée de $f(x)$. + \item Quel est le signe de $f'(x)$? Que peut-on déduire sur la croissance de $f$? + \item Recopier et compléter le tableau suivant + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=10]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{\hspace{5cm}, \hspace{5cm}}% + \tkzTabLine{,,}% + \tkzTabVar{,}% + \end{tikzpicture} + \end{center} + \end{enumerate} +\end{exercise} + +% ------ +% Fonction de degré 2 + +\begin{exercise}[subtitle={Fonction polynôme}, step={2}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\searchMode}] + On définit la fonction $f(x) = 3x^2 - 2x + 10$ dont on veut étudier les variations. + \begin{enumerate} + \item Calculer $f'(x)$ la fonction dérivée de $f(x)$. + \item Quel est le signe de $f'(x)$? + \item Recopier et compléter le tableau suivant + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=10]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{\hspace{5cm}, \hspace{5cm}}% + \tkzTabLine{,,}% + \tkzTabVar{,}% + \end{tikzpicture} + \end{center} + \end{enumerate} +\end{exercise} + +% ------ +% Mise en situations + +\begin{exercise}[subtitle={Gestion hôtelière}, step={3}, origin={???}, topics={ Fonction dérivée }, tags={ Dérivation }] Le nombre d'offre "séjour exclusif" vendues peut être modélisé par la fonction suivante $N(x) = -0.6x + 219$ où $x$ désigne le prix de vente en euro. \begin{enumerate} \item On se place dans le cas où le prix de vente est de 150\euro. @@ -84,7 +216,7 @@ \end{enumerate} \end{exercise} -\begin{exercise}[subtitle={Crème de beauté}, step={1}, origin={???}, topics={ Fonction dérivée }, tags={ Dérivation }] +\begin{exercise}[subtitle={Crème de beauté}, step={3}, origin={???}, topics={ Fonction dérivée }, tags={ Dérivation }] Une entreprise fabrique des flacons de crème de beauté. Cette entreprise peut fabriquer jusqu'à 60 flacons par jour. \begin{enumerate} \item Chaque flacon est vendu 250\euro. On note $R(x)$ les recettes des ventes journalière des flacons où $x$ désigne le nombre de flacon produit. Déterminer l'expression de $R$ en fonction de $x$. diff --git a/1ST/05_Fonction_derivee/index.rst b/1ST/05_Fonction_derivee/index.rst index c848f9a..b357af3 100644 --- a/1ST/05_Fonction_derivee/index.rst +++ b/1ST/05_Fonction_derivee/index.rst @@ -2,7 +2,7 @@ Fonction dérivée ################ :date: 2023-01-04 -:modified: 2023-01-04 +:modified: 2023-01-05 :authors: Benjamin Bertrand :tags: Dérivation :category: 1ST @@ -31,13 +31,35 @@ Capacités attendues Progression =========== +Plan de travail +--------------- + +.. image:: ./plan_de_travail.pdf + :height: 200px + :alt: Plan de travail + +Solutions + +.. image:: ./solutions.pdf + :height: 200px + :alt: Solution des exercices techniques + + + + Étape 1: Découverte de la fonction dérivée ------------------------------------------ -À partir de graphique, les élèves tracent les tangentes et détermine les nombres dérivées. Ils doivent ensuite "deviner" la transformation de x vers le nombre dérivé. +À partir de graphique, les élèves tracent les tangentes et détermine les nombres dérivés. Ils doivent ensuite "deviner" la transformation de x vers le nombre dérivé. + +Ensuite, ils utilisent des fonctions dérivées pour calculer les nombres dérivés et connaître la croissance des fonctions. + +Enfin, en groupe, ils vont devoir chercher une méthode pour calculer des fonctions dérivées et produire un bilan. 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a/1ST/05_Fonction_derivee/plan_de_travail.tex +++ b/1ST/05_Fonction_derivee/plan_de_travail.tex @@ -1,5 +1,7 @@ \documentclass[a4paper,12pt]{article} \usepackage{myXsim} +\usepackage{pgfplots} +\pgfplotsset{compat=1.18} \author{Benjamin Bertrand} \title{Fonction dérivée - Plan de travail} @@ -22,22 +24,31 @@ Savoir-faire de la séquence \begin{itemize} - \item + \item Interpréter géométriquement le nombre dérivé comme coefficient directeur de la tangente + \item Calculer la dérivée d’une fonction polynôme de degré inférieur ou égal à deux. + \item Déterminer le sens de variation d'un polynôme de degré inférieur ou égal à 2. \end{itemize} \bigskip -Ordre des étapes à respecter - - -\section{} +\section{Découverte de la fonction dérivée} \listsectionexercises +\section{Utilisation de la fonction dérivée} + +\listsectionexercises + +\section{Étude de variation des fonctions} + +\listsectionexercises + + \pagebreak \input{exercises.tex} +\input{1_techniques.tex} \printcollection{banque} 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a/1ST/05_Fonction_derivee/tpl_techniques.tex b/1ST/05_Fonction_derivee/tpl_techniques.tex new file mode 100644 index 0000000..d5b7a18 --- /dev/null +++ b/1ST/05_Fonction_derivee/tpl_techniques.tex @@ -0,0 +1,182 @@ +\begin{exercise}[subtitle={Calculs de dérivée}, step={1}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\trainMode}] + Calculer les fonctions dérivées des fonctions suivantes + \Block{ + set fonctions = { + "f": random_expression("{a}x + {b}", [],), + "g": random_expression("{a}x + {b}", [],), + + "h": random_expression("{a}x + {b}", [],), + "i": random_expression("{a}x + {b}", [],), + + "j": random_expression("{a}x + {b}", [],), + "k": random_expression("{a}x^2 + {b}x + {c}", [],), + + "l": random_expression("{a}x + {b}", [],), + "m": random_expression("{a}x^2 + {b}x + {c}", [],), + + "n": random_expression("{a}x^2 + {b}x + {c}", [],), + "o": random_expression("{a}x^2", [],), + + "p": random_expression("{a}x^2 + {b}x", [],), + "q": random_expression("{a}x^2 + {c}", [],), + } + } + \begin{multicols}{3} + \begin{enumerate} + %- for name, f in fonctions.items() + \item $\Var{name}(x) = \Var{f}$ + %- endfor + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{multicols}{2} + \begin{enumerate} + %- for name, f in fonctions.items() + \item $\Var{name}(x) = \Var{f}$ + + \[ + \Var{name}'(x) = \Var{f.differentiate()} + \] + %- endfor + \end{enumerate} + \end{multicols} +\end{solution} + +\begin{exercise}[subtitle={Fonction affines - technique}, step={2}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\trainMode}] + Reprendre l'exercice précédent pour les fonctions suivantes: + \Block{ + set functions = { + "f": random_expression("{a}x + {b}", [],), + "g": random_expression("{a}x + {b}", [],), + + "h": random_expression("{a}x + {b}", [],), + "i": random_expression("{a}x + {b}", [],), + } + } + \begin{multicols}{2} + \begin{enumerate} + %- for name, f in functions.items() + \item $\Var{name}(x) = \Var{f}$ + %- endfor + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + %- for name, f in functions.items() + \item Étude de la fonction $\Var{name}(x) = \Var{f}$ + %- set f1 = f.differentiate() + \begin{itemize} + \item Fonction dérivée : $\Var{name}'(x) = \Var{f1}$ + %- if f1 > 0 + \item Comme $\Var{f1} > 0$ la fonction est croissante + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=10]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{\hspace{2cm}, \hspace{2cm}}% + \tkzTabLine{,+,}% + \tkzTabVar{-/ ,+/ }% + \end{tikzpicture} + \end{center} + %- elif f1 < 0 + \item Comme $\Var{f1} < 0$ la fonction est décroissante + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=10]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{\hspace{2cm}, \hspace{2cm}}% + \tkzTabLine{,-,}% + \tkzTabVar{+/ ,-/ }% + \end{tikzpicture} + \end{center} + %- endif + \end{itemize} + %- endfor + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Fonction affines - technique}, step={2}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\trainMode}] + Reprendre l'exercice précédent pour les fonctions suivantes : + \Block{ + set functions = { + "f": random_expression("{a}x^2 + {b}x + {c}", ["a>0"],), + "g": random_expression("{a}x^2 + {b}x + {c}", ["a>0"],), + + "h": random_expression("{a}x^2 + {b}x + {c}", [],), + "i": random_expression("{a}x^2 + {b}x + {c}", [],), + + "j": random_expression("{a}x^2 + {b}x + {c}", [],), + "k": random_expression("{a}x^2 + {b}x + {c}", [],), + } + } + \begin{multicols}{2} + \begin{enumerate} + %- for name, f in functions.items() + \item $\Var{name}(x) = \Var{f}$ + %- endfor + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + %- for name, f in functions.items() + \item Étude de la fonction $\Var{name}(x) = \Var{f}$ + %- set f1 = f.differentiate() + \begin{itemize} + \item Fonction dérivée : $\Var{name}'(x) = \Var{f1}$ + %- if f1[1] > 0 + \item On résout l'inéquation $\Var{name}'(x) \geq 0$ pour déterminer quand la fonction $\Var{name}'$ est positive. + %- set cst = -f1[0] + %- set coef = f1[1] + %- set racine = cst / coef + \begin{align*} + \Var{name}(x) & \geq 0 \\ + \Var{f1} & \geq 0 \\ + \Var{f1} + \Var{cst} &\geq 0 + \Var{cst} \\ + \Var{f1 + cst} &\geq \Var{0 + cst} \\ + \frac{\Var{f1 + cst}}{\Var{coef}} &\geq \frac{\Var{cst}}{\Var{coef}} \\ + x &\geq \Var{racine.simplify()} \\ + \end{align*} + Donc $\Var{name}(x)$ est positif quand $x$ est plus \textbf{grand} que $\Var{racine.simplify()}$ + %- set racine = racine.simplify() + %- set img_racine = f(racine) + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=4]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{, $\Var{racine}$ ,}% + \tkzTabLine{, -, z, +, } + \tkzTabVar{+/ ,-/$f(\Var{racine}) = \Var{img_racine}$ , +/}% + \end{tikzpicture} + \end{center} + %- elif f1[1] < 0 + \item On résout l'inéquation $\Var{name}'(x) \geq 0$ pour déterminer quand la fonction $\Var{name}'$ est positive. + %- set cst = -f1[0] + %- set coef = f1[1] + %- set racine = cst / coef + \begin{align*} + \Var{name}(x) & \geq 0 \\ + \Var{f1} & \geq 0 \\ + \Var{f1} + \Var{cst} &\geq 0 + \Var{cst} \\ + \Var{f1 + cst} &\geq \Var{0 + cst} \\ + \frac{\Var{f1 + cst}}{\Var{coef}} &\leq \frac{\Var{cst}}{\Var{coef}} \\ + x &\leq \Var{racine.simplify()} \\ + \end{align*} + Donc $\Var{name}(x)$ est positif quand $x$ est plus \textbf{petit} que $\Var{racine.simplify()}$ + %- set racine = racine.simplify() + %- set img_racine = f(racine) + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=3,espcl=4]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{, $\Var{racine}$ ,}% + \tkzTabLine{, +, z, -, } + \tkzTabVar{-/ ,+/$f(\Var{racine}) = \Var{img_racine}$ , -/}% + \end{tikzpicture} + \end{center} + %- endif + \end{itemize} + %- endfor + \end{enumerate} +\end{solution}