From 4d002dae30b277e00c534184eab2fecf13cac88e Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Sun, 10 May 2020 12:40:01 +0200 Subject: [PATCH] Feat: poly deg 3 pour les 1ST --- .../PolyDeg3/3B_forme_facto.pdf | Bin 66286 -> 66286 bytes .../PolyDeg3/3E_racine_facto.pdf | Bin 7558 -> 51037 bytes .../PolyDeg3/4E_etude_sgn.pdf | Bin 0 -> 51478 bytes .../PolyDeg3/4E_etude_sgn.tex | 26 ++++++ 1ST/Fonctions_reelle/PolyDeg3/banque.tex | 80 +++++++++++++++++- 1ST/Fonctions_reelle/PolyDeg3/index.rst | 10 ++- 1ST/semaines.rst | 22 ++++- 7 files changed, 132 insertions(+), 6 deletions(-) create mode 100644 1ST/Fonctions_reelle/PolyDeg3/4E_etude_sgn.pdf create mode 100644 1ST/Fonctions_reelle/PolyDeg3/4E_etude_sgn.tex diff --git a/1ST/Fonctions_reelle/PolyDeg3/3B_forme_facto.pdf b/1ST/Fonctions_reelle/PolyDeg3/3B_forme_facto.pdf index 0548f6d439bb1325911471cb7785d1cbca2b5e16..58923aca8858cea26146a78e365b4558fc42758e 100644 GIT binary patch delta 110 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+\DeclareExerciseCollection{banque} +\xsimsetup{ + step={4}, +} + +\begin{document} + +\input{banque.tex} +\printcollection{banque} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: + diff --git a/1ST/Fonctions_reelle/PolyDeg3/banque.tex b/1ST/Fonctions_reelle/PolyDeg3/banque.tex index cbd1bc3..891999a 100644 --- a/1ST/Fonctions_reelle/PolyDeg3/banque.tex +++ b/1ST/Fonctions_reelle/PolyDeg3/banque.tex @@ -68,8 +68,84 @@ \end{enumerate} \end{exercise} -\begin{exercise}[subtitle={Racines}, step={3}, topics={Polynôme degré 3}] - +\begin{exercise}[subtitle={Marche suivante: degré 3}, step={3}, topics={Polynôme degré 3}] + \begin{enumerate} + \item Soit $P(x) = 3x^3 - 6x^2 + 9$ un polynôme de degré 3. + \begin{enumerate} + \item Expliquer pourquoi $P$ est un polynôme de degré 3. + \item Démontrer que $x=1$, $x=2$ et $x=-1$ sont des racines de $P$. + \item En vous inspirant du travail fait sur les polynômes de degré 2, proposer une forme factorisée de $P(x)$. + \item Développer cette forme factorisée pour retrouver l'expression de $P(x)$ initiale. + \end{enumerate} + \item Soit $Q(x) = 5(x-2)(x+1)(x+2)$ une fonction. + \begin{enumerate} + \item Développer l'expression de $Q(x)$ pour vérifier que c'est une fonction polynôme de degré 3. + \item Conjecturer 3 racines de $Q(x)$ puis démontrer qu'elles sont bien des racines. + \end{enumerate} + \end{enumerate} \end{exercise} +\begin{exercise}[subtitle={Racines et factorisation}, step={3}, topics={Polynôme degré 3}] + Soit $P(x) = 2x^3 + 8x^2 + 2x - 12$ une fonction polynôme de degré 3. + \begin{enumerate} + \item Tracer la courbe représentative de $P(x)$ et conjecturer les valeurs des racines de $P(x)$. + \item Parmi les valeurs suivantes lesquelles sont des racines de $P(x)$. + \[ + -3 \qquad + -2 \qquad + -1 \qquad + 0 \qquad + 1 \qquad + 2 \qquad + 3 + \] + \item Proposer une forme factorisée pour $P(x)$ et vérifier la en développant l'expression. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={La racine double!}, step={3}, topics={Polynôme degré 3}] + Soit $P(x) = 2x^3 + 2x^2 -10x - 6$ une fonction polynôme de degré 3. + \begin{enumerate} + \item Tracer la courbe représentative de $P(x)$ et conjecturer les valeurs des racines de $P(x)$. + \item Parmi les valeurs suivantes lesquelles sont des racines de $P(x)$ (toutes les racines se trouvent parmi ces valeurs). + \[ + -3 \qquad + -2 \qquad + -1 \qquad + 0 \qquad + 1 \qquad + 2 \qquad + 3 + \] + \item Proposer une forme factorisée pour $P(x)$ et vérifier la en développant l'expression. + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Étude de signe}, step={4}, topics={Polynôme degré 3}] + Pour chacune des fonctions suivantes, réaliser le tableau de signe. + \begin{multicols}{2} + \begin{enumerate} + \item $f(x) = 3(x-1)(x-10)(x+2)$ + \item $g(x) = 2(x-2)(x+3)(x+2)$ + \item $h(x) = -3(x-1)(x-10)(x+2)$ + \item $i(x) = -2(x-1)(x+1)(x+2)$ + \item $j(x) = -3(x-1)(x-10)^2$ + \item(*) $k(x) = (2x-1)(-x-10)(x+2)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Étude des profits}, step={4}, topics={Polynôme degré 3}] + Un usine produit chaque jours entre 0 et 50 milles masques. Une étude statistique a montré que les bénéfices pouvaient être modélisés par la fonction suivante: + \[ + f(x) = x^3 - 96x^2+2489,25x - \np{10171,25} + \] + \begin{enumerate} + \item Démontrer que $f(x) = (x-5)(x-39,5)(x-51,5)$. + \item En déduire les racines de $f$. + \item Étudier le signe de $f(x)$. + \item En déduire le nombre de masque que l'entreprise doit produire pour gagner de l'argent. + \end{enumerate} +\end{exercise} \collectexercisesstop{banque} diff --git a/1ST/Fonctions_reelle/PolyDeg3/index.rst b/1ST/Fonctions_reelle/PolyDeg3/index.rst index a632402..872e3cf 100644 --- a/1ST/Fonctions_reelle/PolyDeg3/index.rst +++ b/1ST/Fonctions_reelle/PolyDeg3/index.rst @@ -2,7 +2,7 @@ Polynômes du 3e degré pour l'année 2019-2020 avec les premières technologiq ############################################################################### :date: 2020-04-10 -:modified: 2020-05-08 +:modified: 2020-05-10 :authors: Bertrand Benjamin :tags: Fonctions, Polynômes :category: 1techno @@ -47,8 +47,14 @@ Cours: :height: 200px :alt: Bilan sur la forme factorisée d'un polynôme de degré 3. -Exercices: +Exercices sur les racines et la factorisation .. image:: 3E_racine_facto.pdf :height: 200px :alt: Exercices sur les racines et la factorisation. + +Exercices sur les racines et la factorisation + +.. image:: 4E_etude_signe.pdf + :height: 200px + :alt: Exercices sur l'étude de signe de polynômes diff --git a/1ST/semaines.rst b/1ST/semaines.rst index e9d0c93..54f0133 100644 --- a/1ST/semaines.rst +++ b/1ST/semaines.rst @@ -60,8 +60,26 @@ S20 - Questions flashs (3x5min) :height: 200px :alt: Questions flash pour vendredi 15 mai -S20 - Polynômes de degré 3 --------------------------- +S20 - Polynômes de degré 3 (45min) +---------------------------------- + +- Exercice 1 + +.. image:: ./1ST/Fonctions_reelle/PolyDeg3/3E_racine_facto.pdf + :height: 200px + :alt: Racine et factorisation de polynômes de degré 3 + +- Cours: racines et factorisation des polynômes de degré 3 + +.. image:: ./1ST/Fonctions_reelle/PolyDeg3/3B_forme_facto.pdf + :height: 200px + :alt: Cours sur les polynômes de degré 3 + +- Suite des exercices. + +.. image:: ./1ST/Fonctions_reelle/PolyDeg3/3E_racine_facto.pdf + :height: 200px + :alt: Racine et factorisation de polynômes de degré 3 S20 - Programmation (45min) ---------------------------