From 561a44b7191a01f0fe2ab681ae85318de36270d1 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Tue, 3 Jan 2023 18:56:12 +0100 Subject: [PATCH] Feat(2nd): fin des exercices et solution pour le chapitre sur les tableaux de fonctions --- .../1_exercises_inequation.tex | 212 ++++++++++++++++++ .../bopytex_config.py | 12 + .../exercises.tex | 40 ---- .../plan_de_travail.pdf | Bin 49363 -> 48892 bytes .../plan_de_travail.tex | 1 + .../solutions.pdf | Bin 37461 -> 50081 bytes .../solutions.tex | 3 +- .../tpl_exercises_inequation.tex | 117 ++++++++++ 8 files changed, 343 insertions(+), 42 deletions(-) create mode 100644 2nd/08_Tableaux_representant_une_fonction/1_exercises_inequation.tex create mode 100644 2nd/08_Tableaux_representant_une_fonction/bopytex_config.py create mode 100644 2nd/08_Tableaux_representant_une_fonction/tpl_exercises_inequation.tex diff --git a/2nd/08_Tableaux_representant_une_fonction/1_exercises_inequation.tex b/2nd/08_Tableaux_representant_une_fonction/1_exercises_inequation.tex new file mode 100644 index 0000000..30fc5ea --- /dev/null +++ b/2nd/08_Tableaux_representant_une_fonction/1_exercises_inequation.tex @@ -0,0 +1,212 @@ +\begin{exercise}[subtitle={Inéquation et tableau de signes}, step={4}, origin={Création}, topics={Inéquations}, tags={ Statistiques, Fractions }, mode={\searchMode}] + Tracer le tableau de signes des fonctions suivantes sans tracer le graphique. Une fois le tableau de signes terminé, vous vérifierez votre tableau avec la calculatrice. + + + \begin{multicols}{2} + \begin{enumerate} + \item $f(x) = 6x + 2$ + \item $g(x) = 9x + 10$ + \item $h(x) = 6x + 8$ + \item $i(x) = - 8x - 4$ + \item $j(x) = 8x - 1$ + \item $k(x) = 6x - 3$ + \item $m(x) = \dfrac{9}{- 4} \times x + \dfrac{- 9}{2}$ + \end{enumerate} + \end{multicols} + +\end{exercise} + +\begin{solution} + \begin{multicols}{2} + \begin{enumerate} + \item $f(x) = 6x + 2$ + Pour déterminer les valeurs de $x$ pour lesquelles $f(x)$ est positive, il faut résoudre l'inéquation + + \begin{align*} + f(x) & \geq 0 \\ + 6x + 2 & \geq 0 \\ + 6x + 2 + - 2 &\geq 0 + - 2 \\ + 6x &\geq - 2 \\ + \frac{6x}{6} &\geq \frac{- 2}{6} \\ + x &\geq \dfrac{- 1}{3} \\ + \end{align*} + + Donc $f(x)$ est positif quand $x$ est supérieur à $\dfrac{- 1}{3}$. On en déduit le tableau de signe + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ x $/1,$ f(x) $/1}{, $\dfrac{- 1}{3}$ ,} + \tkzTabLine{, -, z, +, } + \end{tikzpicture} + \end{center} + + \item $g(x) = 9x + 10$ + Pour déterminer les valeurs de $x$ pour lesquelles $f(x)$ est positive, il faut résoudre l'inéquation + + \begin{align*} + g(x) & \geq 0 \\ + 9x + 10 & \geq 0 \\ + 9x + 10 + - 10 &\geq 0 + - 10 \\ + 9x &\geq - 10 \\ + \frac{9x}{9} &\geq \frac{- 10}{9} \\ + x &\geq \dfrac{- 10}{9} \\ + \end{align*} + + Donc $g(x)$ est positif quand $x$ est supérieur à $\dfrac{- 10}{9}$. On en déduit le tableau de signe + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ x $/1,$ g(x) $/1}{, $\dfrac{- 10}{9}$ ,} + \tkzTabLine{, -, z, +, } + \end{tikzpicture} + \end{center} + + \item $h(x) = 6x + 8$ + Pour déterminer les valeurs de $x$ pour lesquelles $f(x)$ est positive, il faut résoudre l'inéquation + + \begin{align*} + h(x) & \geq 0 \\ + 6x + 8 & \geq 0 \\ + 6x + 8 + - 8 &\geq 0 + - 8 \\ + 6x &\geq - 8 \\ + \frac{6x}{6} &\geq \frac{- 8}{6} \\ + x &\geq \dfrac{- 4}{3} \\ + \end{align*} + + Donc $h(x)$ est positif quand $x$ est supérieur à $\dfrac{- 4}{3}$. On en déduit le tableau de signe + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ x $/1,$ h(x) $/1}{, $\dfrac{- 4}{3}$ ,} + \tkzTabLine{, -, z, +, } + \end{tikzpicture} + \end{center} + + \item $i(x) = - 8x - 4$ + Pour déterminer les valeurs de $x$ pour lesquelles $f(x)$ est positive, il faut résoudre l'inéquation + + \begin{align*} + i(x) & \geq 0 \\ + - 8x - 4 & \geq 0 \\ + - 8x - 4 + 4 &\geq 0 + 4 \\ + - 8x &\geq 4 \\ + \frac{- 8x}{- 8} &\leq \frac{4}{- 8} \\ + x &\leq \dfrac{1}{- 2} \\ + \end{align*} + + Donc $i(x)$ est positif quand $x$ est inférieur à $\dfrac{1}{- 2}$. On en déduit le tableau de signe + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ x $/1,$ i(x) $/1}{, $\dfrac{1}{- 2}$ ,} + \tkzTabLine{, +, z, -, } + \end{tikzpicture} + \end{center} + \item $j(x) = 8x - 1$ + Pour déterminer les valeurs de $x$ pour lesquelles $f(x)$ est positive, il faut résoudre l'inéquation + + \begin{align*} + j(x) & \geq 0 \\ + 8x - 1 & \geq 0 \\ + 8x - 1 + 1 &\geq 0 + 1 \\ + 8x &\geq 1 \\ + \frac{8x}{8} &\geq \frac{1}{8} \\ + x &\geq \dfrac{1}{8} \\ + \end{align*} + + Donc $j(x)$ est positif quand $x$ est supérieur à $\dfrac{1}{8}$. On en déduit le tableau de signe + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ x $/1,$ j(x) $/1}{, $\dfrac{1}{8}$ ,} + \tkzTabLine{, -, z, +, } + \end{tikzpicture} + \end{center} + + \item $k(x) = 6x - 3$ + Pour déterminer les valeurs de $x$ pour lesquelles $f(x)$ est positive, il faut résoudre l'inéquation + + \begin{align*} + k(x) & \geq 0 \\ + 6x - 3 & \geq 0 \\ + 6x - 3 + 3 &\geq 0 + 3 \\ + 6x &\geq 3 \\ + \frac{6x}{6} &\geq \frac{3}{6} \\ + x &\geq \dfrac{1}{2} \\ + \end{align*} + + Donc $k(x)$ est positif quand $x$ est supérieur à $\dfrac{1}{2}$. On en déduit le tableau de signe + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ x $/1,$ k(x) $/1}{, $\dfrac{1}{2}$ ,} + \tkzTabLine{, -, z, +, } + \end{tikzpicture} + \end{center} + + \item $m(x) = \dfrac{9}{- 4} \times x + \dfrac{- 9}{2}$ + Pour déterminer les valeurs de $x$ pour lesquelles $f(x)$ est positive, il faut résoudre l'inéquation + + \begin{align*} + m(x) & \geq 0 \\ + \dfrac{9}{- 4} \times x + \dfrac{- 9}{2} & \geq 0 \\ + \dfrac{9}{- 4} \times x + \dfrac{- 9}{2} + \dfrac{9}{2} &\geq 0 + \dfrac{9}{2} \\ + \dfrac{9}{- 4} \times x + \dfrac{0}{2} &\geq \dfrac{9}{2} \\ + \frac{\dfrac{9}{- 4} \times x + \dfrac{0}{2}}{\dfrac{9}{- 4}} &\leq \frac{\dfrac{9}{2}}{\dfrac{9}{- 4}} \\ + x &\leq - 2 \\ + \end{align*} + + Donc $m(x)$ est positif quand $x$ est inférieur à $- 2$. On en déduit le tableau de signe + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ x $/1,$ m(x) $/1}{, $- 2$ ,} + \tkzTabLine{, +, z, -, } + \end{tikzpicture} + \end{center} + \end{enumerate} + \end{multicols} +\end{solution} + + +\begin{exercise}[subtitle={Tableau de signes et produits}, step={4}, origin={Création}, topics={Inéquations}, tags={ Statistiques, Fractions }, mode={\searchMode}] + Tracer le tableau de signes des fonctions suivantes sans tracer le graphique. Une fois le tableau de signes terminé, vous vérifierez votre tableau avec la calculatrice. + + \begin{multicols}{2} + \begin{enumerate} + \item $f(x) = (5x + 5)(3x + 7)$ + \item $g(x) = (9x + 10)(4x + 5)$ + \item $h(x) = (- 3x - 9)(4x + 4)$ + \item $i(x) = (- 2x - 10)(5x - 4)$ + \end{enumerate} + \end{multicols} + +\end{exercise} + +\begin{solution} + Cette correction n'explique pas le raisonnement, mais donne uniquement les réponses. Les valeurs sont arrondis à $10^{-2}$ mais il est plus pertinent de garder les valeurs exactes. + \begin{enumerate} + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=3]{$ x $/1,$ f(x) $/1}{, $-2.33$ ,$-1$, } + \tkzTabLine{, +, z, -, z, +, } + \end{tikzpicture} + \end{center} + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=3]{$ x $/1,$ g(x) $/1}{, $-1.25$ ,$-1.11$, } + \tkzTabLine{, +, z, -, z, +, } + \end{tikzpicture} + \end{center} + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=3]{$ x $/1,$ h(x) $/1}{, $-3$ ,$-1$, } + \tkzTabLine{, -, z, +, z, -, } + \end{tikzpicture} + \end{center} + \item + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=3]{$ x $/1,$ i(x) $/1}{, $-5$ ,$0.80$, } + \tkzTabLine{, +, z, -, z, +, } + \end{tikzpicture} + \end{center} + \end{enumerate} + +\end{solution} diff --git a/2nd/08_Tableaux_representant_une_fonction/bopytex_config.py b/2nd/08_Tableaux_representant_une_fonction/bopytex_config.py new file mode 100644 index 0000000..599b6b5 --- /dev/null +++ b/2nd/08_Tableaux_representant_une_fonction/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/2nd/08_Tableaux_representant_une_fonction/exercises.tex b/2nd/08_Tableaux_representant_une_fonction/exercises.tex index 28ee9ac..95e1ef2 100644 --- a/2nd/08_Tableaux_representant_une_fonction/exercises.tex +++ b/2nd/08_Tableaux_representant_une_fonction/exercises.tex @@ -189,13 +189,6 @@ \end{enumerate} \end{solution} -\begin{exercise}[subtitle={Vrai-Faux}, step={2}, origin={Création}, topics={Fonctions}, tags={Tableaux de signes, Tableaux de variations}, mode={\trainMode}] -\end{exercise} - -\begin{solution} - -\end{solution} - \begin{exercise}[subtitle={Tracer un graphique à partir de tableaux}, step={3}, origin={Création}, topics={Fonctions}, tags={Tableaux de signes, Tableaux de variations}, mode={\trainMode}] Tracer des graphiques qui correspondent aux tableaux suivants @@ -300,36 +293,3 @@ Vous devez inventer le tableau de signes d'une fonction $f$ et le tableau de variations d'une fonction $g$. Puis vous inventerez 6 propositions vraies ou fausses. Enfin vous proposerez un correction de votre exercice. \end{exercise} - -\begin{exercise}[subtitle={Tableaux de signes}, step={4}, origin={Création}, topics={Fonctions}, tags={Tableaux de signes, Tableaux de variations}, mode={\trainMode}] - Tracer le tableau de signes des fonctions suivantes sans tracer le graphique. Une fois le tableau de signes terminé, vous vérifierez votre tableau avec la calculatrice. - - \begin{tasks}(3) - \task $f(x) = 2x$ - \task $g(x) = 5x$ - \task $h(x) = x + 2$ - - \task $i(x) = x - 5$ - \task $j(x) = x - 1$ - \task $k(x) = 2x + 4$ - - \task $l(x) = 6x - 12$ - \task $m(x) = -2x + 6$ - \task $n(x) = -5x - 10$ - \end{tasks} -\end{exercise} - - -\begin{exercise}[subtitle={Tableaux de signes et produit}, step={4}, origin={Création}, topics={Fonctions}, tags={Tableaux de signes, Tableaux de variations}. mode={\trainMode}] - Tracer le tableau de signes des fonctions suivantes sans tracer le graphique. Une fois le tableau de signes terminé, vous vérifierez votre tableau avec la calculatrice. - - \begin{tasks}(3) - \task $f(x) = (x + 1)(x - 1)$ - \task $g(x) = (x - 2)(x - 5)$ - \task $h(x) = 2x (x - 1)$ - - \task $i(x) = (2x + 6)(3x - 12)$ - \task $j(x) = (x - 1)(-5x + 10)$ - \task $j(x) = (x + 1)(-x + 2)$ - \end{tasks} -\end{exercise} diff --git a/2nd/08_Tableaux_representant_une_fonction/plan_de_travail.pdf b/2nd/08_Tableaux_representant_une_fonction/plan_de_travail.pdf index 298fa95b80a7d85533acf0d2a6d1f8e072957c84..68f4acf6b4cfea1b82a39d6e7725a1635ac86c48 100644 GIT binary patch delta 39781 zcmZ^KV{{o+qR8~ZDV3iG_jq$x%a)bzCYjVKfTwfeY$$} z>N>mXboF8&`1=P~eIyjX%)|0cQ1ILTJ>Xnioz0Bw;5^qabY&ejIZ=A9>gI@<2lpJG zL82i!Z+L|r?EL7w;W8hUu%#G3-UxfhRjTx48y8DK66nd537og@5)$ddcc6KE`Nv+A z10K{4f6ck<5#qel(4M^uQ5W@Br!Fq zvch>ojSfJ7q}Or03(Za_&}q-;SA(8mC~hIb9^Gq0r2v<~vSTqzD%T(aB=Y3N!QVS! zv*E;5S|yE{o>Gv8gL6^Ys#Yt9Iw@jGS1z` zW!;0FF}I3!A)ygPmEF(wtBc~z7O}#>EQ*rYYTnA~Nl1F#fg{<(D 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On en déduit le tableau de signe + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ x $/1,$ \Var{name}(x) $/1}{, $\Var{racine}$ ,} + \tkzTabLine{, -, z, +, } + \end{tikzpicture} + \end{center} + + %- else + \frac{\Var{function + cst}}{\Var{coef}} &\leq \frac{\Var{cst}}{\Var{coef}} \\ + x &\leq \Var{racine.simplify()} \\ + \end{align*} + + %- set racine = racine.simplify() + Donc $\Var{name}(x)$ est positif quand $x$ est inférieur à $\Var{racine}$. On en déduit le tableau de signe + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=1]{$ x $/1,$ \Var{name}(x) $/1}{, $\Var{racine}$ ,} + \tkzTabLine{, +, z, -, } + \end{tikzpicture} + \end{center} + %- endif + %- endfor + \end{enumerate} + \end{multicols} +\end{solution} + + +\begin{exercise}[subtitle={Tableau de signes et produits}, step={4}, origin={Création}, topics={Inéquations}, tags={ Statistiques, Fractions }, mode={\searchMode}] + Tracer le tableau de signes des fonctions suivantes sans tracer le graphique. Une fois le tableau de signes terminé, vous vérifierez votre tableau avec la calculatrice. + \Block{ + set fonctions = { + "f": random_expression("({a}x+{b})({c}x+{d})", ["a!=c"], global_config = {"min_max": (1, 10)}), + "g": random_expression("({a}x+{b})({c}x+{d})", ["a!=c"], global_config = {"min_max": (1, 10)}), + "h": random_expression("({a}x+{b})({c}x+{d})", ["a!=c"]), + "i": random_expression("({a}x+{b})({c}x+{d})", ["a!=c"]), + } + } + \begin{multicols}{2} + \begin{enumerate} + %- for (name, function) in fonctions.items() + \item $\Var{name}(x) = \Var{function}$ + %- endfor + \end{enumerate} + \end{multicols} + +\end{exercise} + +\begin{solution} + Cette correction n'explique pas le raisonnement, mais donne uniquement les réponses. Les valeurs sont arrondis à $10^{-2}$ mais il est plus pertinent de garder les valeurs exactes. + \begin{enumerate} + %- for (name, function) in fonctions.items() + \item + %- set f = function.simplify() + \begin{center} + \begin{tikzpicture} + \tkzTabInit[lgt=2,espcl=3]{$ x $/1,$ \Var{name}(x) $/1}{, $\Var{f.roots[0].raw | round(2) }$ ,$\Var{f.roots[1].raw | round(2)}$, } + %- if f[0] > 0 + \tkzTabLine{, +, z, -, z, +, } + %- else + \tkzTabLine{, -, z, +, z, -, } + %- endif + \end{tikzpicture} + \end{center} + %-endfor + \end{enumerate} + +\end{solution}