From 2e61e281ec83566066bc7590d1cde19f55cdde88 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Fri, 19 May 2023 10:22:17 +0200 Subject: [PATCH] =?UTF-8?q?Feat(1ST):=20chapitre=20sur=20les=20polynomes?= =?UTF-8?q?=20de=20degr=C3=A9=203?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 1ST/09_Polynomes_de_degre_3/1B.tex | 14 -- 1ST/09_Polynomes_de_degre_3/1B_polyDeg3.pdf | Bin 0 -> 33104 bytes 1ST/09_Polynomes_de_degre_3/1B_polyDeg3.tex | 64 +++++++ .../2B_forme_facto.pdf | Bin 0 -> 36425 bytes .../2B_forme_facto.tex | 90 ++++++++++ .../4B_racine_cubique.pdf | Bin 0 -> 30017 bytes .../4B_racine_cubique.tex | 54 ++++++ 1ST/09_Polynomes_de_degre_3/exercises.tex | 163 +++++++++++++++++- 1ST/09_Polynomes_de_degre_3/index.rst | 50 +++++- .../plan_de_travail.pdf | Bin 1613 -> 48320 bytes .../plan_de_travail.tex | 15 +- 11 files changed, 425 insertions(+), 25 deletions(-) delete mode 100644 1ST/09_Polynomes_de_degre_3/1B.tex create mode 100644 1ST/09_Polynomes_de_degre_3/1B_polyDeg3.pdf create mode 100644 1ST/09_Polynomes_de_degre_3/1B_polyDeg3.tex create mode 100644 1ST/09_Polynomes_de_degre_3/2B_forme_facto.pdf create mode 100644 1ST/09_Polynomes_de_degre_3/2B_forme_facto.tex create mode 100644 1ST/09_Polynomes_de_degre_3/4B_racine_cubique.pdf create mode 100644 1ST/09_Polynomes_de_degre_3/4B_racine_cubique.tex diff --git a/1ST/09_Polynomes_de_degre_3/1B.tex b/1ST/09_Polynomes_de_degre_3/1B.tex deleted file mode 100644 index 6286d80..0000000 --- a/1ST/09_Polynomes_de_degre_3/1B.tex +++ /dev/null @@ -1,14 +0,0 @@ -\documentclass[a4paper,10pt]{article} -\usepackage{myXsim} - -\author{Benjamin Bertrand} -\title{Polynomes de degré 3 - Cours} -\date{mai 2023} - -\pagestyle{empty} - -\begin{document} - -\maketitle - -\end{document} diff --git a/1ST/09_Polynomes_de_degre_3/1B_polyDeg3.pdf b/1ST/09_Polynomes_de_degre_3/1B_polyDeg3.pdf new file mode 100644 index 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Alors + + \begin{tabular}{cc} + \begin{tikzpicture}[yscale=.4, xscale=0.8] + \tkzInit[xmin=-5,xmax=5,xstep=1, + ymin=-5,ymax=10,ystep=1] + \tkzAxeXY + \tkzFct[domain = -5:5, line width=1pt]{0.5*x*x*x+1} + \tkzText[draw,fill = brown!20](2.5,-2){$f(x)=0.5x^3+1$} + \end{tikzpicture} + & + \begin{tikzpicture}[yscale=.4, xscale=0.8] + \tkzInit[xmin=-5,xmax=5,xstep=1, + ymin=-5,ymax=10,ystep=1] + \tkzAxeXY + \tkzFct[domain = -5:5, line width=1pt]{-0.1*x*x*x+3} + \tkzText[draw,fill = brown!20](2.5,6){$f(x)=-0.1x^3+3$} + \end{tikzpicture} \\ + Si $a > 0$, $f$ est croissante sur $\R$. & + Si $a < 0$, $f$ est décroissante sur $\R$. + \end{tabular} + +On remarque que comme pour les autres types de polynômes, la valeur de $b$ peut se lire sur l'axe des ordonnées. +\end{propriete} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/1ST/09_Polynomes_de_degre_3/2B_forme_facto.pdf b/1ST/09_Polynomes_de_degre_3/2B_forme_facto.pdf new file mode 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\textbf{racines} du polynôme. + \end{minipage} + \hfill + \begin{minipage}{0.5\linewidth} + Forme graphique + + \begin{tikzpicture}[yscale=.4, xscale=0.8] + \tkzInit[xmin=-5,xmax=5,xstep=1, + ymin=-5,ymax=10,ystep=1] + \tkzAxeXY + \tkzFct[domain = -5:5, line width=1pt]{0.5*(x+2)*(x-1)*(x-2)} + \tkzText[draw,fill = brown!20](2.5,-2){$f(x)=0.5(x+2)(x-1)(x-2)$} + \end{tikzpicture} + \end{minipage} +\end{propriete} + + +\subsubsection*{Exemple}% +Montrons que $-1$, $-2$ et $1$ sont des racines de +\[ + P(x) = 2x^3 + 3x^2 - 3x - 2 +\] +On en déduit la forme factorisée de $P(x)$ +\afaire{} + + +\begin{definition}[Racines doubles et triples] + \begin{multicols}{2} + \begin{itemize} + \item On appelle \textbf{racine double} une racine qui apparait 2 fois dans la forme factorisé. On a alors dans le cas où $x_1$ est une racine double + \[ + P(x) = a(x-x_1)(x-x_1)(x-x_3) = a(x-x_1)^2(x-x_3) + \] + + \begin{tikzpicture}[yscale=.4, xscale=0.8] + \tkzInit[xmin=-5,xmax=5,xstep=1, + ymin=-5,ymax=10,ystep=1] + \tkzAxeXY + \tkzFct[domain = -5:5, line width=1pt]{0.5*(x+2)*(x-1)**2} + \tkzText[draw,fill = brown!20](2.5,-2){$f(x)=0.5(x+2)(x-1)^2$} + \end{tikzpicture} + \item On appelle \textbf{racine triple} une racine qui apparait 3 fois dans la forme factorisé. On a alors dans le cas où $x_1$ est une racine triple + \[ + P(x) = a(x-x_1)(x-x_1)(x-x_1) = a(x-x_1)^3 + \] + + \begin{tikzpicture}[yscale=.4, xscale=0.8] + \tkzInit[xmin=-5,xmax=5,xstep=1, + ymin=-5,ymax=10,ystep=1] + \tkzAxeXY + \tkzFct[domain = -5:5, line width=1pt]{-0.5*(x-1)**3} + \tkzText[draw,fill = brown!20](2.5,2){$f(x)=-0.5(x-1)^3$} + \end{tikzpicture} + \end{itemize} + \end{multicols} +\end{definition} + +\subsection*{Méthode: étude de signe}% + +Étudions le signe de $P(x) = 2(x+1)(x+2)(x-1)$ + +\afaire{} + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/1ST/09_Polynomes_de_degre_3/4B_racine_cubique.pdf b/1ST/09_Polynomes_de_degre_3/4B_racine_cubique.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f68c8a268eb6ee89235be6ecd53a660fea841f7c GIT binary patch literal 30017 zcmce-18}6>_U|9tPA0Z(+vwQF#5QJP+qP}nww+AuOzb=F)p`GQ&pq|4`>%7ls-N!K 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z$0ybclpd(&k@l3XjN7Rl&SqhpL9W6YY!i{7xdly+X|_b?+n$#Rn^J8c2SQ>LsJ8^VtKAKbYCOAtRQl0vzK_;oF}!3k#Y9bxJ~9;yzp>C_R9B zn+0l3KRN=^W8hHghxuf`o({k(R@xSq*JgA_v3igyUEaWCPwvyxa`4E#W^ErHni5+| zlxmTzq6cp=x?YpPZcCJUTs_)a|LLxuHFNE%6qp;`f~Y>~FyZ`rrz3PmsP0*88@#a% zt&8LTKQkEcbMpVzT=>tMw6>~+B?*&^qq)U@8f1dgUa$kEnvIVI33!!B>p$EvS-}pO z|M+48fo>$+|7G_l<7f#a;r=Hj0=u)xh=b$!MLET}x!IXTSU5SjMY+J~3WuZ^t0<3% z1P70#2s<~Y;QuBBJkI}1GlZ4xU&SZ>J3qjh6jFpz!G*a_ht>Rj829artTL~SrmWZ2 z#MGiKh?KN`c1gOoe~e9AdsPpY*A6vbYNnC@m_8tV^kCdqc-H&cPBoRG0F{#a1?t8w z`=GQ~3h0;)DT_)YCt(|#F1Ow&e3DlD_sfj(N6RI2z1ijFGd{!%6B8xg=r2rjv5LE^ z3&x)XS!4BIWc5rVi@sv5CIy8yOd3PN2^l|{0NKr@GNK*(tYJMa?YEN!Y*=i9N0GSo zSG$=2b0*I)C4oEc_P%5WJbP`hu||E7ja}6KJiB*@S&V(*kr!61N$OLF!EwMH5Xw8G iiW_}, topics={ Polynomes de degré 3 }, tags={ Fonction }] +\begin{exercise}[subtitle={Identification}, step={1}, origin={Ma tête}, topics={ Polynômes de degré 3 }, tags={ Fonction }, mode={\trainMode}] Identifier le degré des polynômes ci-dessous ainsi que les coefficients (quand ils ne sont pas factorisés). \begin{tasks}(3) @@ -19,3 +19,164 @@ \task $f(x) = 3(x+2)(x-1)(x-2)$ \end{tasks} \end{exercise} + +\begin{exercise}[subtitle={Représentation graphique}, step={1}, origin={Ma tête}, topics={ Polynômes de degré 3 }, tags={ Fonction }, mode={\groupMode}] + On souhaite étudier les représentations graphiques des fonctions suivantes : + \begin{multicols}{3} + \begin{enumerate}[label={$\alph*(x) = $}, wide] + \item $2x^3$ + \item $5x^3 + 1$ + \item $2(x - 1)(x - 4)(x+2)$ + \item $-2x^3$ + \item $-2(x + 3)(x - 1)(x+2)$ + \item $5x^3 - 3$ + \item $2x^3 + 3$ + \item $2(x - 2)(x - 4)(x+1)$ + \item $-0.5x^3$ + \item $0.5x^3$ + \item $2x^3 - 1$ + \item $-2(x + 1)(x - 4)x$ + \end{enumerate} + \end{multicols} + + \begin{enumerate} + \item Regroupe les fonctions sur des critères de forme de la formule. + \item Pour chaque fonction, en vous aidant de la calculatrice, tracer l'allure du graphique. On ne demande pas un tracé précis, mais une forme générale qui respecte la position par rapport aux axes. + \item Faire une conjecture sur le lien entre la forme de la fonction et la forme du graphique associé. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Factorisation et racines}, step={2}, origin={Ma tête}, topics={ Polynômes de degré 3 }, tags={ Fonction }, mode={\trainMode}] + \begin{enumerate} + \item Soit $f(x) = 2x^3 - 4x^2 - 2x + 4$ + \begin{enumerate} + \item Quel est le degré de ce polynôme ? + \item Par les nombres suivants, lesquels sont des racines de $f(x)$ ? + \[ + -3 \qquad + -2 \qquad + -1 \qquad + 0 \qquad + 1 \qquad + 2 \qquad + 3 + \] + \item Conjecturer une forme factorisée de $f(x)$ (on ne vous demande pas de la prouver). + \item Tracer l'allure de la fonction $f(x)$. + \end{enumerate} + \item Mêmes questions pour la fonction $g(x) = 2x^3 -4x^2 -18x + 36$ + \item Mêmes questions pour la fonction $h(x) = 2x^3 - 8x^2 + 8x$ + \item Combien de racines semblent avoir les polynômes de degré 3 ? + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Étude de signe}, step={2}, topics={Polynôme degré 3}, mode={\trainMode}] + 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={2}, topics={Polynôme degré 3}, mode={\trainMode}] + Une usine produit chaque jour 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 Expliquer que l'on peut mettre $f(x)$ sous la forme $f(x) = (x-5)(x-39,5)(x-51,5)$ en calculant les racines. + \item Étudier le signe de $f(x)$. + \item En déduire le nombre de masques que l'entreprise doit produire pour gagner de l'argent. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Lot}, step={3}, origin={T1CMATH00290}, topics={Polynôme degré 3}, mode={\trainMode}] + Un artisan produit et vend des sachets de viennoiseries. En notant, $x$ le nombre de sachets de viennoiseries ses coûts sont calculables avec la formule suivante : + \[ + C(x) = x^3 - 120x^2 + 10x + \] + \begin{enumerate} + \item Calculer le coût de production pour 75 sachets. + \item Chaque sachet est vendu 10\euro. + \begin{enumerate} + \item Justifier que le bénéfice se calcule alors avec la formule suivante: + \[ + B(x) = x^3 - 120x^2 + \] + \item Tracer l'allure de la courbe représentative de $B(x)$, conjecturer puis démontrer les racines du polynômes. + \item Démontrer que $B(x)$ peut s'écrire + \[ + B(x) = x^2(x-120) + \] + \item Étudier le signe de $B(x)$. + \item En déduire la production maximal avant que l'artisan commence à perdre de l'argent. + \end{enumerate} + \item Recherche du maximum des bénéfices. + \begin{enumerate} + \item Déterminer $B'(x)$ la dérivée de $B(x)$. + \item Montrer que l'on peut écrire + \[ + B'(x) = 3x(x-80) + \] + \item Étudier le signe de $B'(x)$ et en déduire les variations de $B(x)$. + \item En déduire le nombre de sachet que l'artisan doit produire pour maximiser ses bénéfices. + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Population de bactéries}, step={3}, topics={Polynôme degré 3}, mode={\searchMode}] + \textit{Cet exercice est un problème ouvert. C'est-à-dire qu'il y a de nombreuses façons d'y apporter une réponse qui pourra être plus ou moins précise. C'est à vous de choisir les outils qui vous semblent les plus pertinents puis de détailler votre démarche - qui est aussi importante que le résultat final.} + + La population de bactéries dans une solution est modélisée par la fonction suivante + \[ + f(x) = -0,01t^3 + 4t^2 + 2t + \] + où $t$ représente le temps en heure depuis le début de l'expérience. + + + Déterminer quand la population de bactéries va s'éteindre et quand elle aura atteint son maximum. +\end{exercise} + + +\begin{exercise}[subtitle={Solution des équations $x^3=k$}, step={4}, topics={Polynôme degré 3}, mode={\searchMode}] + Dans cet exercice, nous allons chercher à résoudre les équations du type $x^3=k$. Pour cela, nous allons porter une attention particulière à la fonction $f(x) = x^3$. + \begin{enumerate} + \item Tracer la courbe représentative de $f(x) = x^3$ avec $x$ allant de $-3$ à $3$ en cherchant à être le plus précis possible. + \end{enumerate} + Les questions suivantes se répondent en utilisant le graphique. + \begin{enumerate} + \setcounter{enumi}{1} + \item Résoudre graphiquement l'équation $x^3 = 8$. + \item Même question pour $x^3 = -8$. + \item Même question pour $x^3 = 4$. + \item Même question pour $x^3 = -2$. + \item Même question pour $x^3 = 0$. + \item De manière générale, combien l'équation $x^3 = k$ a-t-elle de solutions ? + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Équations cubiques}, step={4}, topics={Polynôme degré 3}, mode={\trainMode}] + Résoudre les équations suivantes (les questions avec (*) sont plus compliquées - mais pas impossibles !) + + \begin{multicols}{3} + \begin{enumerate} + \item $x^3 = 8$ + \item $x^3 = 27$ + \item $x^3 = 64$ + + \item $x^3 = -27$ + \item $x^3 = 10$ + \item $x^3 = -5$ + + \item (*) $2x^3 = 16$ + \item (*) $-4x^3 = 40$ + \item (*) $3x^3 + 1 = 8$ + \end{enumerate} + \end{multicols} +\end{exercise} diff --git a/1ST/09_Polynomes_de_degre_3/index.rst b/1ST/09_Polynomes_de_degre_3/index.rst index 390e551..53f91a6 100644 --- a/1ST/09_Polynomes_de_degre_3/index.rst +++ b/1ST/09_Polynomes_de_degre_3/index.rst @@ -2,7 +2,7 @@ Polynomes de degré 3 #################### :date: 2023-05-18 -:modified: 2023-05-18 +:modified: 2023-05-19 :authors: Benjamin Bertrand :tags: Fonction :category: 1ST @@ -15,14 +15,50 @@ Polynomes de degré 3 Contenus -------- -Capacités attendues -------------------- +Fonctions polynômes de degré 3: + +- représentations graphiques des fonctions: ax3, x ax3 + b ; +- racines et signe d’un polynôme de degré 3 de la forme a(x - x1)(x - x2)(x - x3) ; +- équation x3 = c ; racine cubique d’un nombre réel positif ; notations -Commentaires ------------- Progression =========== -Étape 1: --------- +Plan de travail: + +.. image:: ./plan_de_travail.pdf + :height: 200px + :alt: Plan de travail + +Étape 1: Forme algébrique et forme graphique +-------------------------------------------- + +Bilan: Sur les formes algébriques et la forme graphique + +.. image:: ./1B_polyDeg3.pdf + :height: 200px + :alt: Bilan sur les formes + + +Étape 2: Racine et signe +------------------------ + +Bilan: sur la forme factorisée et l'étude de signes + +.. image:: ./2B_forme_facto.pdf + :height: 200px + :alt: Bilan sur la forme factorisée. + + +Étape 3: Problèmes +------------------ + +Étape 4: Équations cubiques +--------------------------- + +Bilan: racine cubique + +.. image:: ./4B_racine_cubique.pdf + :height: 200px + :alt: Bilan sur la racine cubique diff --git a/1ST/09_Polynomes_de_degre_3/plan_de_travail.pdf b/1ST/09_Polynomes_de_degre_3/plan_de_travail.pdf index d9e414d2695d7c27a9ed82f1722ed1ef75b726b8..b7af6dfd174f069cf3f6417888b4a36c66b78122 100644 GIT binary patch literal 48320 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