From 99dea014f052449e04ccbb747019f290795b124d Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Tue, 7 Mar 2023 15:43:47 +0100 Subject: [PATCH] Feat(1ST): fin du cours sur les polynomes de deg 2 --- 1ST/07_Polynome_du_2nd_degre/1B.tex | 14 -- 1ST/07_Polynome_du_2nd_degre/1B_polyDeg2.pdf | Bin 0 -> 27395 bytes 1ST/07_Polynome_du_2nd_degre/1B_polyDeg2.tex | 68 ++++++ 1ST/07_Polynome_du_2nd_degre/2B_cas_part.pdf | Bin 0 -> 38983 bytes 1ST/07_Polynome_du_2nd_degre/2B_cas_part.tex | 117 +++++++++ .../3B_racines_facto.pdf | Bin 0 -> 30207 bytes .../3B_racines_facto.tex | 67 +++++ 1ST/07_Polynome_du_2nd_degre/4B_parabole.pdf | Bin 0 -> 28815 bytes 1ST/07_Polynome_du_2nd_degre/4B_parabole.tex | 56 +++++ 1ST/07_Polynome_du_2nd_degre/exercises.tex | 229 +++++++++++++++++- 1ST/07_Polynome_du_2nd_degre/index.rst | 28 ++- .../plan_de_travail.pdf | Bin 0 -> 188937 bytes .../plan_de_travail.tex | 33 +-- 13 files changed, 580 insertions(+), 32 deletions(-) delete mode 100644 1ST/07_Polynome_du_2nd_degre/1B.tex create mode 100644 1ST/07_Polynome_du_2nd_degre/1B_polyDeg2.pdf create mode 100644 1ST/07_Polynome_du_2nd_degre/1B_polyDeg2.tex create mode 100644 1ST/07_Polynome_du_2nd_degre/2B_cas_part.pdf create mode 100644 1ST/07_Polynome_du_2nd_degre/2B_cas_part.tex create mode 100644 1ST/07_Polynome_du_2nd_degre/3B_racines_facto.pdf create mode 100644 1ST/07_Polynome_du_2nd_degre/3B_racines_facto.tex create mode 100644 1ST/07_Polynome_du_2nd_degre/4B_parabole.pdf create mode 100644 1ST/07_Polynome_du_2nd_degre/4B_parabole.tex create mode 100644 1ST/07_Polynome_du_2nd_degre/plan_de_travail.pdf diff --git a/1ST/07_Polynome_du_2nd_degre/1B.tex b/1ST/07_Polynome_du_2nd_degre/1B.tex deleted file mode 100644 index 16fad82..0000000 --- a/1ST/07_Polynome_du_2nd_degre/1B.tex +++ /dev/null @@ -1,14 +0,0 @@ -\documentclass[a4paper,10pt]{article} -\usepackage{myXsim} - -\author{Benjamin Bertrand} -\title{Polynome du 2nd degré - Cours} -\date{mars 2023} - 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+\section{Polynôme de degré 2} + +Dans l'exercice sur le volume d'une boite, on a abouti à l'étude de la fonction suivante +\[ + V(x) = +\] +\afaire{Écrire la formule factorisée puis développée qui permet de calculer le volume} +Pour étudier les variations et trouver le maximum, il a fallut dériver $V$ +\[ + V'(x) = +\] +\afaire{Dériver la fonction $V$} + +À cause du "$^2$", on ne peut pas trouver où la tangente est horizontale car on ne sait pas résoudre $V'(x) = 0$. + +C'est ce type de fonction que l'on va étudier dans ce chapitre. + +\begin{definition}[Polynome du 2nd degré] + +On appelle \textbf{fonction polynôme du second degré} tout fonction $f$ définie sur $\R$ par +\[ + f(x) = ax^2 + bx + c +\] +où $a$, $b$ et $c$ sont des nombres réels et $a$ n'est pas nul. + +\bigskip + +\noindent +On appelle l'expression algébrique $ax^2 + bx + c$ \textbf{trinôme du second degré}. + +\end{definition} + +\subsubsection*{Exemples} + \begin{multicols}{2} + \begin{enumerate} + \item $f(x) = 3x^2 - 10x + 2$ + \vspace{1cm} + \item $f(x) = 3 + 4x^2 - x$ + \vspace{1cm} + \item $f(x) = 3x^3 - 10x$ + \vspace{1cm} + \item $f(x) = - 10x + 2$ + \vspace{1cm} + \item $f(x) = 3x^2$ + \vspace{1cm} + \item $f(x) = (2x+1)(x-1)$ + \vspace{1cm} + \end{enumerate} + \end{multicols} +\afaire{Parmi les fonction ci-dessus, lesquelles sont des fonctions polynôme du second degré? Quand elles le sont, préciser les valeurs de $a$, $b$ et $c$.} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/1ST/07_Polynome_du_2nd_degre/2B_cas_part.pdf b/1ST/07_Polynome_du_2nd_degre/2B_cas_part.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b79242dc3c3393cc734c58d42bc46c3f83741a66 GIT binary patch literal 38983 zcmce-b8u!|vpzhrZQHgpv2EL#*tVTaZ0n9~+qOLuTa$d5=ll+;zEkJF_q}W5uI{yZ z?cROut5)?|O{O3sM#n_Y22D1()VDaaIy*QF%}U5fXlG;v&C5&3pyFX~O30ugZ)9a^ z>`cg@>|*5nk3`JQ(T0%W@AaP+OoR+VLU!(i+FxB637HuGC2Sm4`)-yulo-}XHyYVV>^?tab-+x&7Ca>nb-*V`3e8? z2kqqSXliH!?Y_Q%Fs(p2y$k#e;t$X@w5hGh|I>W^`}_PHUzY!UesKQF{NVbx`oZ)! zg8!O;F#QF=zb_(8%>O!1{=L{@`qzPr`ER)YeKBGF3xR(Px|mu1y`KD~$$yb2?EhX* z{(|{m;|c5kMA!dC@my>iO#e;t|4s21J`>ePVy;Iz`VPLkvp2rqLO_*N6M<%s(RTy( z5@wh1CBV-mKYeS^?C=xA`lPp-qvi5cJnS}i&FqI_{}4V!BF26GL5%Zz7wmk0J7Dn`iCc|AqK(0LFx{{=PL$L(Z}siA;|Z#2aNLT8V4bb zMoqfGm2f{|e 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100644 index 0000000..6976dec --- /dev/null +++ b/1ST/07_Polynome_du_2nd_degre/2B_cas_part.tex @@ -0,0 +1,117 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\title{Polynômes du 2e degré - Cours} +\tribe{1ST} +\date{Mars 2023} + +\pagestyle{empty} + +\begin{document} + +\setcounter{section}{1} +\section{Représentation graphique} + + +\begin{definition}[Parabole] + Soit $f(x) = ax^2 + bx + c$ un polynôme du second degré. + + La représentation graphique de $f$ s'appelle une \textbf{parabole}. + + \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]{x*x-x+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]{-x*x-x+3} + \end{tikzpicture} \\ + Cas où $a > 0$ les branches sont orientées vers le haut. & + Cas où $a < 0$ les branches sont orientées vers le bas. + \end{tabular} + +\end{definition} +\section{Fonctions particulières} + +Dans le programme de première ST, seules 3 formes de polynômes du 2nd degré sont à savoir étudier et reconnaître. + +\begin{center} + \Large + \textbf{$x \mapsto ax^2$} +\end{center} + +\begin{tabular}{cc} + \begin{tikzpicture}[yscale=.4, xscale=2] + \tkzInit[xmin=-2,xmax=2,xstep=1, + ymin=-5,ymax=10,ystep=1] + \tkzAxeXY + \end{tikzpicture} + & + \begin{tikzpicture}[yscale=.4, xscale=1.3] + \tkzInit[xmin=-3,xmax=3,xstep=1, + ymin=-10,ymax=5,ystep=1] + \tkzAxeXY + \end{tikzpicture} \\ + f(x) = 2x^2 & + f(x) = -3x^2 & +\end{tabular} + +\pagebreak + +\begin{center} + \Large + \textbf{$x \mapsto ax^2 + b$} +\end{center} + +\begin{tabular}{cc} + \begin{tikzpicture}[yscale=.4, xscale=2] + \tkzInit[xmin=-2,xmax=2,xstep=1, + ymin=-5,ymax=10,ystep=1] + \tkzAxeXY + \end{tikzpicture} + & + \begin{tikzpicture}[yscale=.4, xscale=1.3] + \tkzInit[xmin=-3,xmax=3,xstep=1, + ymin=-10,ymax=5,ystep=1] + \tkzAxeXY + \end{tikzpicture} \\ + f(x) = 2x^2 + 2 & + f(x) = -3x^2 + 4& +\end{tabular} + + +\vspace{3cm} +\begin{center} + \Large + \textbf{$x \mapsto a(x - x_1)(x - x_2)$} +\end{center} +\begin{tabular}{cc} + \begin{tikzpicture}[yscale=.4, xscale=2] + \tkzInit[xmin=-2,xmax=2,xstep=1, + ymin=-5,ymax=10,ystep=1] + \tkzAxeXY + \end{tikzpicture} + & + \begin{tikzpicture}[yscale=.4, xscale=1.3] + \tkzInit[xmin=-3,xmax=3,xstep=1, + ymin=-10,ymax=5,ystep=1] + \tkzAxeXY + \end{tikzpicture} \\ + f(x) = 2(x-2)(x+1) & + f(x) = -3(x+2)(x-1)& +\end{tabular} + + +\afaire{Tracer l'allure des fonctions sur les graphiques} +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/1ST/07_Polynome_du_2nd_degre/3B_racines_facto.pdf b/1ST/07_Polynome_du_2nd_degre/3B_racines_facto.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c0383c2d1384c7cfaba2e37261ec98c4a0776de1 GIT binary patch literal 30207 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factorisée.} + +\begin{definition}[Racine] + On appelle \textbf{racine} d'un polynôme $f(x)$ une valeur de $x$ telle que $f(x) = 0$. +\end{definition} + +\paragraph{Exemple} +$3$ est une racine de $f(x) = x^2-2x-3$ car +\[ + f(3) = 3^2 - 2\times3 -3 = 9 - 6 - 3 = 0 +\] + +\begin{propriete} + Une racine d'un polynôme correspond à l'abscisse d'un point d'intersection entre la courbe représentative du polynôme et l'axe des abscisses. +\end{propriete} + +\paragraph{Exemple:} On a vu que $3$ est une racine de $f(x) = x^2-2x-3$. On peut aussi le "voir" sur un graphique, car la courbe coupe l'axe des abscisses en $x=3$. + +\begin{minipage}{0.5\linewidth} + \begin{tikzpicture}[yscale=.4, xscale=1] + \tkzInit[xmin=-4,xmax=4,xstep=1, + ymin=-5,ymax=10,ystep=1] + \tkzAxeXY + \tkzFct[domain = -5:5, line width=1pt]{x*x -2*x - 3} + \end{tikzpicture} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \afaire{Trouver sur le graphique une autre racine puis démontrer que c'est bien une racine} +\end{minipage} + +\begin{propriete} + Soit $f(x) = ax^2 + bx + c$ un polynôme du 2nd degré. Alors + \begin{itemize} + \item S'il a 2 racines $x_1$ et $x_2$ alors on peut le factoriser et on a + \[ + f(x) = a(x-x_1)(x-x_2) + \] + \item S'il a 1 racine $x_1$ alors on peut le factoriser et on a + \[ + f(x) = a(x-x_1)^2 + \] + \item S'il n'a pas de racine, alors on ne peut pas le factoriser. + \end{itemize} +\end{propriete} + +\paragraph{Exemple} +\afaire{Proposer une factorisation de $f(x) = 2x^2-4x-6$} + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/1ST/07_Polynome_du_2nd_degre/4B_parabole.pdf b/1ST/07_Polynome_du_2nd_degre/4B_parabole.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c741b3a6c65808bb2d3a08080bf4a7b94dac1b8a GIT binary patch literal 28815 zcmce-1CT9U@-N!9ZQC|Z+qUgKZQJfXZQHhO+s0|z);r()-+l8%+=+=7_r}bQsM-}P zS7la4=3aZ{@0Ud?FDy#SNXH6AI=R%hIJ7!DI1I%?z(8PYXbHu`LqM70){Q{x&#epuj12#C=n&A$7&tjt{58(_ zKazi(|BNe|INCZp7@0T{u>KiR{G-ak+R4P>&$qRKlZmj2k*)C`yV52$W=`e=jI0EF 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+\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\title{Polynômes du 2e degré - Cours} +\tribe{1ST} +\date{Mars 2020} + +\pagestyle{empty} + +\begin{document} + +\setcounter{section}{3} +\section{Éléments géométriques à reconnaitre sur une parabole} + +\begin{propriete} + Soit $f(x)$ un polynôme avec une ou deux racines nommées $x_1$ et $x_2$. On sait que l'on a + \[ + f(x) = x(x-x_1)(x-x_2) + \] + Alors la parabole représentative de $f$ a les caractéristiques suivantes: + \begin{itemize} + \item L'axe de symétrie de la parabole a pour équation $y = \dfrac{x_1+x_2}{2}$ + \item Le sommet de la parabole a pour abscisse $\dfrac{x_1+x_2}{2}$ + \end{itemize} +\end{propriete} + + +\paragraph{Exemple} + +Soit $f(x) = -3(x-2)(x+4)$ sa courbe représentative a été tracée ci-dessous. + +\begin{minipage}{0.4\linewidth} + \begin{tikzpicture}[yscale=.4, xscale=0.8] + \tkzInit[xmin=-6,xmax=4,xstep=1, + ymin=-5,ymax=30,ystep=2] + \tkzGrid + \tkzAxeXY + \tkzFct[domain = -6:4, line width=1pt]{-3*(x-2)*(x+4)} +\end{tikzpicture} +\end{minipage} +\begin{minipage}{0.6\linewidth} + \afaire{ + \begin{enumerate} + \item Déterminer et tracer l'axe de symétrie. + \item Calculer les coordonnées du sommet de la parabole. + \end{enumerate} + } +\end{minipage} + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/1ST/07_Polynome_du_2nd_degre/exercises.tex b/1ST/07_Polynome_du_2nd_degre/exercises.tex index 5521032..e8b4baf 100644 --- a/1ST/07_Polynome_du_2nd_degre/exercises.tex +++ b/1ST/07_Polynome_du_2nd_degre/exercises.tex @@ -1,7 +1,234 @@ -\begin{exercise}[subtitle={Volume d'une boite}, step={1}, origin={classique}, topics={ Polynome du 2nd degré }, tags={ Tableau de signes, variation, dérivation }] +\begin{exercise}[subtitle={Volume d'une boite}, step={1}, origin={classique}, topics={ Polynome du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\searchMode}] On dispose d'une feuille cartonnée pour construire des boites sans couvercle. \includegraphics[scale=0.3]{./fig/boite} Où doit-on plier les bords pour avoir une boite la plus grande possible ? \end{exercise} + + +\begin{exercise}[subtitle={Identification des coefficients}, step={2}, origin={classique}, topics={ Polynome du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\trainMode}] + Identifier les polynômes du 2nd et déterminer les coefficients $a$, $b$ et $c$. + \begin{tasks}(3) + \task $f(x) = 7x^2 + 2x + 0.2$ + \task $f(x) = -2x^2 - 8x + 2$ + \task $f(x) = 3x^3 - 10x - 2$ + + \task $f(x) = 6 + 4x^2 - 5x$ + \task $f(x) = -8 - 3x^2 + x$ + \task $f(x) = 3 + 4x^2 - x^3$ + + \task $f(x) = - 10x + 2$ + \task $f(x) = - 10x^2 + 0.25$ + \task $f(x) = -5x^2 + x$ + + \task $f(x) = (5x-2)(3x-1)$ + \task $f(x) = (2x+1)(0.1x-10)$ + \task (*)$f(x) = 3(x+2)(x-1)$ + \end{tasks} +\end{exercise} + +\begin{exercise}[subtitle={Représentation graphique}, step={2}, origin={classique}, topics={ Polynôme du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\searchMode}] + On souhaite étudier les représentations graphiques des fonctions suivantes : + \begin{multicols}{3} + \begin{enumerate}[label={$\alph*(x) = $}, wide] + \item $2x^2$ + \item $5x^2 + 1$ + \item $2(x - 1)(x - 4)$ + \item $-2x^2$ + \item $-2(x + 3)(x - 1)$ + \item $5x^2 - 3$ + \item $2x^2 + 3$ + \item $2(x - 2)(x - 4)$ + \item $-0.5x^2$ + \item $0.5x^2$ + \item $2x^2 - 1$ + \item $-2(x + 1)(x - 4)$ + \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={Racines}, step={3}, origin={classique}, topics={ Polynôme du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\trainMode}] + Les phrases suivantes sont-elles justes ou fausses? Justifier + \begin{enumerate} + \item La valeur $x=-1$ est une racine du polynôme $f(x) = 3^2-2x-3$. + \item La valeur $x=3$ est une racine du polynôme $g(x) = 5(x-3)(x+1)$. + \item La valeur $x=4$ est une racine du polynôme $h(x) = 2x^2-2x-24$. + \item La valeur $x=-3$ est une racine du polynôme $h(x) = 2x^2-2x-24$. + \item Les valeurs $x=-10$ et $x=2$ sont deux racines du polynôme $i(x) = x^2+8x-20$. + \item Les valeurs $x=-10$ et $x=2$ sont deux racines du polynôme $j(x) = (x+10)(x-2)$. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Racines et factorisation}, step={3}, origin={classique}, topics={ Polynôme du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\searchMode}] + \begin{enumerate} + \item Soient 2 fonctions polynômes du 2nd degré + \[ + f(x) = 5x^2 - 26x + 5 \qquad g(x) = 5(x-5)(x-0.2) + \] + \begin{enumerate} + \item Démontrer que $x=5$ et $x=0.2$ sont 2 racines de $f$ + \item Démontrer que $x=5$ et $x=0.2$ sont 2 racines de $g$ + \item Démontrer que $f(x) = g(x)$ pour toutes valeurs de $x$ réelles. + \item Tracer la représentation graphique de $f$. Que ce passe-t-il pour les valeurs $x=5$et $x=0.2$? + \end{enumerate} + \item Soit $h$ une fonction polynôme du 2nd degré + \[ + h(x) = x^2 + 2x - 15 + \] + \begin{enumerate} + \item Tracer la représentation graphique de $f$. Conjecturer (lire sur le graphique) les valeurs des 2 racines. + \item En vous inspirant de ce qui a été fait avant, conjecturer une forme factorisée de $f$. Démontrer que cette forme factorisée convient. + \end{enumerate} + \item Proposer une méthode pour factoriser un polynôme du 2nd degré. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Factoriser}, step={3}, origin={classique}, topics={ Polynôme du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\trainMode}] + Dans cet exercice, on souhaite factoriser des polynômes du 2nd degré. + \begin{enumerate} + \item On veut factoriser $f(x) = 3x^2 - 9x -30$. + \begin{enumerate} + \item Démontrer que 5 est une racine de $f$. + \item Parmi les nombres suivants, quels sont ceux qui sont des racines de $f$. + \[ + -3\qquad -2 \qquad -1 \qquad 1 \qquad 2 + \] + \item Démontrer que $f(x)$ est égal à $3(x+2)(x-5)$. + \end{enumerate} + \item On veut factoriser $g(x) = 2x^2 - 6x + 4$. + \begin{enumerate} + \item Tracer la courbe représentative de $f$ et trouver les racines de $g$ + \item Proposer une factorisation de $g$ en se basant sur les racines. + \item Démontrer que cette factorisation est juste par un calcul. + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Etude de signes}, step={4}, origin={classique}, topics={ Polynôme du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\trainMode}] + Tracer le tableau de signe des fonctions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $f(x) = 3(x-2)(x+1)$ + \item $g(x) = 5(x+6)(x+2)$ + \item $h(x) = -2(x-5)(x-1)$ + \item $i(x) = -0.1(x-0.2)(x+10)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Et la boite dans tout ça?}, step={4}, origin={classique}, topics={ Polynôme du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\trainMode}] + On a maintenant tous les outils pour terminer et résoudre l'exercice de la boite. On rappelle que l'on souhaiter trouver le maximum de la fonction + \[ + V(x) = x(20-2x)(20-2x) = 4x^3 - 80x^2 + 400x + \] + On avait alors dérivé $V$ et trouvé + \[ + V'(x) = 12x^2 - 160x + 400 + \] + On s'était arrêté là car on ne savait pas résoudre $V'(x)=0$. + \begin{enumerate} + \item Démontrer que $x=10$ et $x=\frac{10}{3}$ sont deux racines de $V'(x)$. + \item Démontrer que $V'(x) = 12(x-10)(x-\dfrac{10}{3})$ + \item Tracer le tableau de signes de $V'(x)$ pour $x$ variant entre 0 et 10. + \item En déduire le tableau de variations de $V(x)$ pour $x$ variant entre 0 et 10. + \item Pour quelle valeur de $x$, le volume de la boite est-il maximal? + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Etude de fonction}, step={5}, origin={classique}, topics={ Polynôme du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\trainMode}] + On définit la fonction $f$ sur $\R$ par + \[ + f(x) = -0,1x^2 -0,3x +1,8 + \] + \begin{enumerate} + \item Calculer l'image de 3 et interpréter le résultat. + \item Démontrer que -6 est une racine de $f$. + \item Démontrer que l'on a $f(x) = -0,1(x-3)(x+6)$. + \item Tracer le tableau de signe de $f$. + \item Dériver la fonction $f$. + \item En déduire le tableau de variations de $f$. + \item Déterminer les coordonnées du sommet de la représentation graphique de $f$. + \item Tracer l'allure de la représentation graphique. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Technique}, step={5}, origin={classique}, topics={ Polynôme du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\trainMode}] + \begin{enumerate} + \item On considère la fonction polynôme $f$ définie sur $\R$ par $f(x) = -3(x+1)(x-5)$ et $(P)$ la parabole représentant cette fonction. + \begin{enumerate} + \item Donner les 2 racines de $f$ + \item Développer $f$ + \item Déterminer les coordonnées du sommet $S$ de la parabole $(P)$. + \item Dresser le tableau de signe de $f$. + \item Parmi les représentations graphiques ci-dessous laquelle correspond à $(P)$? Justifier. + + \hspace{-2cm} + \begin{tabular}{ccc} + \begin{tikzpicture}[yscale=.4, xscale=0.6] + \tkzInit[xmin=-2,xmax=7,xstep=1, + ymin=-5,ymax=30,ystep=2] + \tkzGrid + \tkzAxeXY + \tkzFct[domain = -2:7, line width=1pt]{-3*(x+1)*(x-5)} + \end{tikzpicture} + & + \begin{tikzpicture}[yscale=.4, xscale=0.6] + \tkzInit[xmin=-2,xmax=7,xstep=1, + ymin=-30,ymax=5,ystep=2] + \tkzGrid + \tkzAxeXY + \tkzFct[domain = -2:7, line width=1pt]{3*(x+1)*(x-5)} + \end{tikzpicture} + & + \begin{tikzpicture}[yscale=.5, xscale=0.6] + \tkzInit[xmin=-2,xmax=7,xstep=1, + ymin=-5,ymax=20,ystep=2] + \tkzGrid + \tkzAxeXY + \tkzFct[domain = -2:7, line width=1pt]{-3*(x+1)*(x-4)} + \end{tikzpicture} + \\ + courbe 1 & Courbe 2 & Courbe 3 + \end{tabular} + \end{enumerate} + \item Résoudre l'équation $f(x) < 15$ + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Fruits en conserves}, step={5}, origin={classique}, topics={ Polynôme du 2nd degré }, tags={ Tableau de signes, variation, dérivation }, mode={\trainMode}] + Une entreprise commercialise des fruits en conserve. Elle en produit entre 0 et 13 tonnes par mois et vend l'intégralité de sa production. + + On note $x$ la production en tonne de fruits et on définit : + \begin{itemize} + \item La fonction $C(x)$ qui modélise les coûts de production + \[ + C(x) = x^3 - 15 x^2 + 75x + \] + \item La fonction $R(x)$ qui modélise les recettes + \[ + R(x) = 36,75x + \] + \end{itemize} + \begin{enumerate} + \item Déterminer les coûts puis les recettes pour une production de 8,5tonnes. + \item La fonction $B(x)$ modélise les bénéfices de l'entreprise. C'est à dire la différence entre les recettes et les coûts + \[ + B(x) = R(x) - C(x) + \] + L'entreprise fait-elle des bénéfices quand elle produit 8,5tonnes? + \item Démontrer que $B(x) = -x^3 +15x^2 - 38,25x$ + \item On note $B'$ la dérivée de $B$. Démontrer que $B'(x) = -3x^2 + 30x - 38,25$. + \item Démontrer que $x=8,5$ et $x=1,5$ sont deux racines de $B'(x)$. + \item En déduire une forme factorisée de $B'(x)$. + \item En déduire le tableau de signe de $B'(x)$ + \item En déduire le tableau de variation de $B(x)$ + \item Pour quelle quantité de fruit produit, l'entreprise fait-elle un maximum de profit? + \end{enumerate} +\end{exercise} diff --git a/1ST/07_Polynome_du_2nd_degre/index.rst b/1ST/07_Polynome_du_2nd_degre/index.rst index 68dada9..9582656 100644 --- a/1ST/07_Polynome_du_2nd_degre/index.rst +++ b/1ST/07_Polynome_du_2nd_degre/index.rst @@ -2,7 +2,7 @@ Polynome du 2nd degré ##################### :date: 2023-03-06 -:modified: 2023-03-06 +:modified: 2023-03-07 :authors: Benjamin Bertrand :tags: Tableau de signes, Variation, Dérivation :category: 1ST @@ -27,8 +27,32 @@ Progression Étape 1: Tache complexe ----------------------- +Problème du volume maximum d'une boite faite à partir d'une feuille de carton. Le travail est à faire en groupe avec rendu. On invitera les élèves à modéliser le volume par une fonction et à se rendre compte qu'on ne peut pas étudier le signe de la dérivé. Donc pour optimiser, ils pourront utiliser le tableur. + +Bilan: pre bilan de la tache complexe et definition des polynômes de degré 2. + Étape 2: Étude de signe de polynomes de degré 2 ----------------------------------------------- -Étape 3: Étude de variations de polynomes de degré 3 +Identification de polynomes du 2nd degré et début du lien que l'on peut faire entre la forme de la formule et la forme du graphique associé. + +Bilan: représentation graphique des polynomes du 2nd degré + +Étape 3: Racine d'un polynôme +----------------------------- + +On donne la définition d'une racine et c'est au tour des élèves de vérifier si tel nombre est racine d'un polynôme. Ils découvriront ensuite le lien entre les racines et la forme factorisée. + +Bilan: Définition des racines et lien avec la forme factorisée. + +Étape 4: Sommet de la parabole +------------------------------ + +L'étude des variations des polynomes de degré 2 permet de calculer le sommet de la parabole. + +Bilan: Coordonnées du sommet de la parabole + +Étape 5: Étude de variations de polynômes de degré 3 ---------------------------------------------------- + +Exercices complets mettant en scène des polynomes de degrés 3 et leur étude de variations. diff --git a/1ST/07_Polynome_du_2nd_degre/plan_de_travail.pdf b/1ST/07_Polynome_du_2nd_degre/plan_de_travail.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7c1b727bcca23bbf881fb3393d59f25ecf0afced GIT binary patch literal 188937 zcmeFYWpo@%vM#!9LCa!hwwRg0VrFKxn3*jmi^*cQn3-G5U@_s)_wQ= zcyG=8I*RV<>ddO@jEsmc>Ge(BW5DDH@1T1<0EEN z^>i>JX4F(LwlXtuA!bx@HFo*CMcm%WmYDI+_U|K@i5Z23?LCNfz^5`1Gc*0;tw+o# 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