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zdr&f}69Rj8o62erTd4{|?sgLl#0i-LL6n15)L)G`aIMivd8tBaQfUDY6Y&$t6FW^m z1|*p}_wq1si(vW`#Xfq=K(mz)K1dNzUd=GMl~76>A(%#C*tL8-c0zc(cz;rtety^r z6aEW0a06Ur>Yrp1JLtd+0tq8D)@VO`v=IE=44mbO(pXpPb@q3|@uM&SssXl{0s4^v z{!%_94D}FCM@zAjr)g!Clee+8(&6nY{Ziuic}TTsO0B&PO@(${Q$^-WlUe{ZZZ8pT pA5(TORd#u5?uNed)VKgX1npbeF%J+P3>za0D>NCIn1VR;{{R?=&C~z@ diff --git a/TST_sti2d/DS/DS_20_11_12/DS_20_11_12.tex b/TST_sti2d/DS/DS_20_11_12/DS_20_11_12.tex index ae719f4..223ba93 100644 --- a/TST_sti2d/DS/DS_20_11_12/DS_20_11_12.tex +++ b/TST_sti2d/DS/DS_20_11_12/DS_20_11_12.tex @@ -33,7 +33,20 @@ Le barème est donné à titre indicatif, il pourra être modifié. \item Quelle est la valeur exacte de $\cos(\dfrac{-2\pi}{3})$? Justifier votre réponse. \end{enumerate} \end{exercise} +\vfill +\begin{exercise}[subtitle={Citerne}, points=6] + Soit $f$ la fonction définie sur $\R$ par + \[ f(t) = 10t^3 - 4t^3 + 2.5t^2 - 6t + 10\] + \begin{enumerate} + \item Calculer $f'(t)$ puis en déduire que $f'(t) = (2t^2+1)(5t-6)$. + \item Étudier le signe de $f'(t)$ et en déduire les variations de $f(t)$. + \item La fonction $f$ a-t-elle un maximum? Un minimum? Quelle est alors sa valeur? + \end{enumerate} +\end{exercise} +\vfill + +\pagebreak \begin{exercise}[subtitle={Complexes}, points=7] On note i le nombre complexe de module 1 et d'argument $\dfrac{\pi}{2}$ qui vérifie $i^2 = -1$. @@ -71,43 +84,6 @@ Le barème est donné à titre indicatif, il pourra être modifié. \end{enumerate} \end{exercise} -\begin{exercise}[subtitle={Citerne}, points=6] - - \begin{minipage}{0.6\textwidth} - On se propose de fabriquer avec le moins de tôle possible une citerne fermée en forme de parallélépipède rectangle dont le volume intérieur doit être de $12m^3$. La longueur est aussi fixée à $3m$ par le cahier des charges. - - On peut donc faire varier uniquement la largeur (notée $x$) et la hauteur (notée $h$) de la cuve. - \end{minipage} - \hfill - \begin{minipage}{0.3\textwidth} - \includegraphics[scale=0.8]{./fig/citerne} - \end{minipage} - - \begin{enumerate} - \item Expliquer pourquoi quand la largeur $x$ change, la hauteur $h$ doit elle aussi changer pour respecter les contraintes. - \item Démontrer que l'on doit avoir $h = \dfrac{4}{x}$. - \item On note $S(x)$ l'aire totale de la citerne (c'est à dire la somme des aires des six faces). Montrer que l'on peut écrire - \[ - S(x) = 6x + 8 + \frac{24}{x} - \] - \item Démontrer que - \[ - S'(x) = \frac{6x^2-24}{x^2} - \] - \item Démontrer que - \[ - S'(x) = \frac{6(x-2)(x+2)}{x^2} - \] - \item En déduire que le tableau de variation de $S(x)$ sur $\intOF{0}{10}$ est . - - \begin{tikzpicture}[baseline=(a.north)] - \tkzTabInit[lgt=3,espcl=3]{$x$/1, $S(x)$/2}{$0$, $2$, $10$} - \tkzTabVar{D+/ , -/ , +/ } - \end{tikzpicture} - \item Déterminer les valeurs de $x$ et $h$ correspondant à une utilisation minimal de tôle. Quel sera alors la surface de tôle utilisé? - \end{enumerate} -\end{exercise} - \end{document} %%% Local Variables: