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100508d..b06652e 100644 --- a/TST_sti2d/DS/DS_20_10_08/DS_20_10_08.tex +++ b/TST_sti2d/DS/DS_20_10_08/DS_20_10_08.tex @@ -14,14 +14,23 @@ Le barème est donné à titre indicatif, il pourra être modifié. -Une part importante de la note sera dédiée à la rédaction, aux explications et à l'utilisation des notations mathématiques. - \begin{exercise}[subtitle={Automatismes}, points=6] Dans cet exerice les questions sont indépendantes. \begin{enumerate} \begin{multicols}{2} \item Calculer la valeur de l'intégrale suivante. - \item Donner un encadrement de l'intégrale suivante. + \[ + \int_2^8 0.1x + 3 \; dx + \] + \columnbreak + \item Donner un encadrement de l'intégrale entre 1 et 4. + + \begin{tikzpicture}[scale=1, yscale=0.4] + \tkzInit[xmin=-0.1,xmax=5,ymax=5] + \tkzGrid + \tkzAxeXY + \tkzFct[color=red, very thick]{4*sin(0.5*\x)} + \end{tikzpicture} \end{multicols} \begin{multicols}{2} \item Soit $f(x) = 5x^6 + \dfrac{1}{2}x^2 - \dfrac{x^3}{2} + 10$, calculer @@ -32,7 +41,7 @@ Une part importante de la note sera dédiée à la rédaction, aux explications \begin{multicols}{2} \item Calculer la valeur de $\cos(\vec{OI};\vec{OA})$? - \begin{tikzpicture}[scale=3] + \begin{tikzpicture}[scale=1.5] \cercleTrigo \foreach \x in {0,30,...,360} { % dots at each point @@ -44,7 +53,7 @@ Une part importante de la note sera dédiée à la rédaction, aux explications \end{tikzpicture} \item Calculer la valeur de $\sin(\vec{OI};\vec{OA})$? - \begin{tikzpicture}[scale=3] + \begin{tikzpicture}[scale=1.5] \cercleTrigo \foreach \x in {0,30,...,360} { % dots at each point @@ -58,14 +67,15 @@ Une part importante de la note sera dédiée à la rédaction, aux explications \end{enumerate} \end{exercise} -\begin{exercise}[subtitle={Vitesse}, points=4] - On lance une fusée hydrolique en l'air verticalement à $t = 0$. La hauteur de la fusée est modélisée par le fonction $z(t) = ...$ où $t$ est en seconde et $z(t)$ en m. Cette fonction est représentée dans le graphique. +\begin{exercise}[subtitle={Vitesse}, points=3] + On lance une fusée hydrolique en l'air verticalement à $t = 0$. La hauteur de la fusée est modélisée par le fonction $z(t) = -0,49x^2 + 6x$ où $t$ est en seconde et $z(t)$ en m. Cette fonction est représentée dans le graphique. - - \begin{tikzpicture}[baseline=(current bounding box.south), xscale=0.5, yscale=0.4] + \noindent + \begin{minipage}{0.4\textwidth} + \begin{tikzpicture}[baseline=(current bounding box.south), xscale=0.5, yscale=0.35] \tkzInit[xmin=0,xmax=14,xstep=1, - ymin=0,ymax=200,ystep=20] + ymin=0,ymax=20,ystep=2] \tkzGrid \tkzDrawX[label={$t (s)$},above=0pt] \tkzDrawY[label={$Hauteur (m)$}, right=2pt ] @@ -73,19 +83,28 @@ Une part importante de la note sera dédiée à la rédaction, aux explications \tkzLabelY \tkzFct[color=red,very thick,% domain=0:12.3 - ]{-4.9*\x**2+60*\x}; + ]{-0.49*\x**2+6*\x}; \tkzFct[color=red,very thick,% domain=12.3:14 ]{0}; - \end{tikzpicture} - - + \end{tikzpicture} + \end{minipage} + \begin{minipage}{0.6\textwidth} \begin{enumerate} \item Calculer la vitesse moyenne de la fusée entre 5s et 10s. Expliquer à quoi cette valeur correspond sur le graphique. \item Quelle est la vitesse instantanée de la fusée après 15s de vol? \item Déterminer la valeur de $t$ telle que la vitesse de la fusée est nulle. À quel moment cela correspond-il dans la trajectoire de la fusée? \end{enumerate} + + \end{minipage} +\end{exercise} +\begin{exercise}[subtitle={Démonstration}, points=1] + Soit $g(x) = 5x$. On veut connaître la dérivée de $g(x)$ au point $x$. + \begin{enumerate} + \item Calculer $\dfrac{\Delta g}{\Delta x}$ en $x_1 = x$ et $x_2 = x +h$ + \item En rendant $h$ très petit (proche de 0) déterminer $\dfrac{dg}{dx}$. + \end{enumerate} \end{exercise}