From 85507d5d68a947b5aa03252a12215c0fb966f43d Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Wed, 19 May 2021 08:23:38 +0200 Subject: [PATCH] =?UTF-8?q?Feat:=205e=20pour=20les=20maths=20compl=C3=A9me?= =?UTF-8?q?ntaires?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../03_Logarithme/5B_fonction_ln.pdf | Bin 0 -> 68740 bytes .../03_Logarithme/5B_fonction_ln.tex | 73 +++++++++++ .../03_Logarithme/5E_fonction_ln.pdf | Bin 0 -> 76322 bytes .../03_Logarithme/5E_fonction_ln.tex | 19 +++ Complementaire/03_Logarithme/exercises.tex | 56 +++++++++ Complementaire/03_Logarithme/index.rst | 14 ++- Complementaire/DM/2105_DM1/01_2105_DM1.tex | 0 Complementaire/DM/2105_DM1/all_2105_DM1.pdf | Bin 0 -> 59445 bytes Complementaire/DM/2105_DM1/tpl_2105_DM1.tex | 25 ++++ Complementaire/DM/2105_DM1/tpl_bassin.tex | 55 ++++++++ .../DM/2105_DM1/tpl_optimisation.tex | 119 ++++++++++++++++++ 11 files changed, 360 insertions(+), 1 deletion(-) create mode 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z`qWm9DtK!z4JY*1)2%#b?=IIocXeyy)qwEN#v55z-kFwJPhrbmwaneT$Crq5iat8~ zOD-H8A38yr3nrg)jI~#EOL4p1T5=IQ&H1`n5@VxgFAeynD^0_FIe)5nc%gH92%{QaqGudN zDP^#9elJEm{A^ayZJyFzynK{=@nKcP<@>vZ-!bU#=|6J$n}zs``p`qEafA0RETTQR ztjC3=I~(@C!HO;t)Tb27vdd3HZL3im-q*f>Z z581XfjOEe~&a6rWvbgla6`~Clj1@p)LHYS53KkYX{Xrn*Y558kpdK-ZY{jN zPKuwmbi&yQN7K(+q-dwSJXfr- 0$: +\begin{itemize} + \item $\ln'(x) = \dfrac{1}{x}$ et $\dfrac{1}{x} > 0$ alors la fonction logarithme est \dotfill + \item $\ln''(x) = \makebox[2cm]{\dotfill}$ et $\makebox[2cm]{\dotfill}$ alors la fonction logarithme est \dotfill +\end{itemize} + +\subsection*{Exemples de calculs} + +Calcul de la dérivée de $f(x) = 2x + 1 - 4\ln(x)$ +\afaire{} + +Calcul de la dérivée de $f(x) = (2x+1)\ln(x)$ +\afaire{} + +Calcul de la dérivée de $f(x) = \dfrac{2x+1}{\ln(x)}$ +\afaire{} + +\end{document} diff --git a/Complementaire/03_Logarithme/5E_fonction_ln.pdf b/Complementaire/03_Logarithme/5E_fonction_ln.pdf new file mode 100644 index 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zCK4iW%f^k${=^a!NJP{@yS)MTm;)t(=OaWTh>x&=N(Y4N2Eyb45w?KP@j!T7AmY_8 zsPJEK!}9)~YQ8E;hi~7iQg^G{=45VkbO;Og3aezMr3UAd4q7)XmS*lR{}?=$JT9`c z^Lw0hQfhXxFLy95ck=S2V0YQno6CB)f~fue`?5Pg^z0!n@1zY1Cp#xMCkiE{h@2?O Fe*gzK5extT literal 0 HcmV?d00001 diff --git a/Complementaire/03_Logarithme/5E_fonction_ln.tex b/Complementaire/03_Logarithme/5E_fonction_ln.tex new file mode 100644 index 0000000..df66ad1 --- /dev/null +++ b/Complementaire/03_Logarithme/5E_fonction_ln.tex @@ -0,0 +1,19 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Logarithme - Cours} +\date{Mai 2021} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=5, +} + +\begin{document} + +\input{exercises.tex} + +\printcollection{banque} +\vfill +\end{document} diff --git a/Complementaire/03_Logarithme/exercises.tex b/Complementaire/03_Logarithme/exercises.tex index bec6df3..7bc043f 100644 --- a/Complementaire/03_Logarithme/exercises.tex +++ b/Complementaire/03_Logarithme/exercises.tex @@ -210,4 +210,60 @@ \end{multicols} \end{exercise} +\begin{exercise}[subtitle={Représentation graphique de $\ln$}, step={5}, origin={Création}, topics={Logarithme Népérien}, tags={analyse, logarithme}] + \begin{enumerate} + \item Tracer l'allure de la courbe représentative du logarithme. + \item Repérer les éléments remarquables de cette représentation graphique. + \item Tracer le tableau de signe de $\ln$. + \item Tracer le tableau de variation de $\ln$. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Dériver les fonctions}, step={5}, topics={Logarithme}] + Dériver les fonctions suivantes puis mettre sous une forme pratique pour l'étude de signe. + \begin{multicols}{3} + \begin{enumerate} + \item $f(x) = x-2-\ln(x)$ + \item $f(x) = 2x^2 - 2x + 4\ln(x)$ + \item $f(x) = x\ln(x)$ + \item $f(x) = (x+1)\ln(x)$ + \item $f(x) = (\ln(x) + 1)^2$ + \item(*) $f(x) = \frac{1 + \ln(x)}{x}$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Étude de fonction}, step={5}, topics={Logarithme}] + On considère la fonction $f$ définie sur $\intFF{1}{11}$ par + \[ + f(x) = -0.5x^2 + 2x + 15\ln(x) + \] + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est + \[ + f'(x) = \frac{-x^2 + 2x + 15}{x} + \] + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \item Montrer que l'équation $f(x) = 0$ admet une unique solution, $\alpha$, sur $\intFF{1}{11}$. + \item Donner une valeur approchée de $\alpha$. + \item En déduire le tableau de signe de $f$. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Étude de fonction}, step={5}, topics={Logarithme}] + On considère la fonction $f$ définie sur $\intFO{0}{+\infty}$ par + \[ + f(x) = \frac{1 + \ln(x)}{x} + \] + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est + \[ + f'(x) = \frac{-\ln(x)}{x^2} + \] + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \item Déterminer le minimum de la fonction $f$. + \item En déduire le tableau de signe de $f$. + \end{enumerate} +\end{exercise} + \collectexercisesstop{banque} diff --git a/Complementaire/03_Logarithme/index.rst b/Complementaire/03_Logarithme/index.rst index d669aa2..9c39ab9 100644 --- a/Complementaire/03_Logarithme/index.rst +++ b/Complementaire/03_Logarithme/index.rst @@ -2,7 +2,7 @@ Logarithme ########## :date: 2021-04-25 -:modified: 2021-05-05 +:modified: 2021-05-19 :authors: Benjamin Bertrand :tags: Exponentielle, Logarithme :category: Complementaire @@ -77,3 +77,15 @@ Bilan: Autres relations fonctionnelles et résolutions d'(in)équations Étape 5: Étude de la fonction ln ================================ + +Avec la calculatrice, les élèves découvrent ln comme une fonction. Puis on donne la formule de la dérivée et on étudier les variations + +.. image:: ./5E_fonction_ln.pdf + :height: 200px + :alt: Étude de fonctions avec ln + +Bilan: éléments remarquables du logarithme et dérivée + +.. image:: ./5B_fonction_ln.pdf + :height: 200px + :alt: éléments remarquables et dérivée diff --git a/Complementaire/DM/2105_DM1/01_2105_DM1.tex b/Complementaire/DM/2105_DM1/01_2105_DM1.tex new file mode 100644 index 0000000..e69de29 diff --git a/Complementaire/DM/2105_DM1/all_2105_DM1.pdf b/Complementaire/DM/2105_DM1/all_2105_DM1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..abd1d6cc428c85d8f913339f41c1a547051e21c6 GIT binary patch literal 59445 zcma&MW00mn(>2((?Y^gNPTRI^+qP{__q1)>wr$%scb@NipN-wvjo7WIxZ-3+RYq1; z{Wx)*B$XEyrDddJfgzn;99o58126#W3@u@}xnby~O>E7a%>k@Tfd3gV^r9Bl&L)lk zdQocwXA@x)BRgXg7+zi&Cuc_!0~;9kb!KG-AVDE;9Z()nAtNZ0y!gxsl~I#YGa#^H zBye%-a1v!gW&R!J72!V89BL#ec(nO|z-urIph7uJbr=&{J-~X3nls)WC z0QBl|hL$Eq&M@>!E{4wk%Oh&%X!9?xf8YPLf)R#ZURgw)?q8xNvIaIL01A3n14nvi z8+&?tV^d258(McsCusp`dM7&JqF7dCOVFfvgT6V&-v!vA{zv-Y1Q;_fV_;OjA|G|tLfPeFU```W6K8-Hfd1cm0MIL&xH|*>lh^;9KCwBin$(S2WP zX0?n9kMCt+L=*tx_58CUwNsr}mqE+Rlcn+WMp_GRZ(r_7lv)RWr+`E{)+&t?Ik{gR z@Dnj!z-f7>Pv4RlG!zu%^9v2#)#;Kcp3}0j-%du>xWX==*P2SW33f9WnLtzM>PtR28@9PqJP8^J4VqD?$a z#!wqX)aS+gHGkI#R^kO_6_g*B)TspWB$msCw5?) z6>EFxqy3@oDv1q}YiT|1PNdxrg5rV#x?d;lX| zSya8H7scHUSWr4dgop;`4KgvmI#Q%L60G@KAK6j<;&#JL1}HQH zCnBtl!(ahV2>43pCQBEp8uJO#1cPg;qFb7K`HKOU^b;;75nWeUTQ$Og&U$AYtX)}t z#?`LCR4`6xqN+vzWLnn=WmP5^N-}cA(JQX?SpV@zqI8RO`ogGu@NFiT4eA$Y~wI_@;_Z@d`1 zUsK)|@}=;$uXUq&B5B3dr3IFg7^=urWrx1G6Ob5r$y#jZWDh%tLzgAO+>A}7BzRDa zgK$i(_R@8&2_JlZdeu?Pp~OL;v=X#)qG+%>*oN3{i{2hMBjVaA7+KCd@{^6O-8dwC zIjeD-IcX!CC$QK>M1od_(C!8EH^~YH^e~3&$`sX0uuQ`$773|$Zz9+;s&OXK{}v-M zk#XkeIbg)P@8kHgbvm!@3W8cIzMJ|Ph_}r?_V*c4?vzM^H%OFcc8D8N;wp2Dw_B)a zINaajte};=l8@$Tho;DtT4D_X4hYZ&{-%BA z-~ytV8I!M8vVd4zUqO}|y7!)^%We7vBG93a9;p#5ltlKr_M#fA*&8NtQ&#)Qpw{?U zx|Nb4@CV4G<*E>#gnKj7${bD0oVH8`_8WtWLJV4X%++NY@j`i|UN(iQpM4fbF^N@; z3lBX(|J05S^@tzlOormEzQa6lFx)R#n%}O{(scHxHxFadFd;n6L2;!=j75}z0UQiX zNQY5^*-yGu7S%>16zmi&)Bk%(11{Q0D>jKIGYuv4A^_@>f8g8(p-(5Zo@ZI-0R+RM z;KCNA9lT+J@R)<(*`LjuhbYS&uld5oDOB=7$lv>OmNzW5W1VdUl3XqLC};b07rA}h zfAP=9^I~!~SDPRvv>!MY80$|>1}w5IGCSR@6qi8a1KzdV-0wopklnay)C1Cky|;om z@n44Z$=}C2q zqc@XIsN+tDM64{G?Y7!)5C`LYCfYgZ0g=h0*7|y&fEiRCn^mF<)wihZ^p{5RnzUQL zPRFM%1lA*gO#=%c^dg|&ZSBW`I2;*&t#}8)%ZybhUK4$yARVF*)2NdpDexOwJbNIa z2MIHt0d&k0U`0|iv0iddKfMa*Fw)KEKeH?PYgXN^QfJAh1qbP4pF+qL<9${l<>l?3 zr)LPHN7UN`v*(nB=r)H|OpfLorcAmHOj6NSa>&1bvzU0|Ad?p!W!fyoMH!ZNJhfDp zeK;(+kW;N8Te%9lhQ`tEC4b!jM+^0@A0*LV$sO#c*Yv_jXWZ9Z=Shk7$jXcGsCvF0 zf?ySBEA;tvN>vOHJverXK{eR2VIqA(kCyBZR#RrE38%NP5l=Zph6@EhPhAZ=cF1)N zxHgotVe(hV*;n#4Z=I?W7{xc}tZc0|$(GC=Ud9%pydF$vl^r16vf;PL$!Kq?0`H#qeFcC6g>@Chf~ea8Pi^0>OWHOsqY=i+=tWCGQj z{(m8R|6wx!ck0OqVEoO@{$C2h1Yl)nVEM20j|2S|0@&Et|A(Xge>0RAH*n=-n+3l4 zqC)Wo3c%F-)~~H!i4*m`iXE49`13LIS&5Qv;*u~>D1iXVUwOICP~H`BGaubEA3r${ zJ}10pH&&*;%qx>DJ#Ew5m6Pod-1p54xZyC9=tBcDu*pCamKJ4qpuqgO0D2?}{?Tzu zC=m{!?`70_93doeib<7yKBPz}D5#KS$^d!uT#QNd-0N$&T}-gsDDm4UNuU5AV1WFm z9RThZusnzdzYVZvCy+}THJHR`IaH7raqcq$#UrmDJ@BPI0#HwDE5(bQn|~)P7%+H{ zHn1ij8FUc5ix}$=j=?`54=MSDPq0>sBUj!YnwM8jPEI!-HW4$lHg#_o!d*}S0m$dy z8E^epL3~r-W+B@6zmtchMqwD7{fT~qA>h7=b{!Zje}| z79h^MNGm

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pour le jeudi 27 mai} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\Block{include "./tpl_optimisation.tex"} +\Block{include "./tpl_bassin.tex"} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: + diff --git a/Complementaire/DM/2105_DM1/tpl_bassin.tex b/Complementaire/DM/2105_DM1/tpl_bassin.tex new file mode 100644 index 0000000..1b304d2 --- /dev/null +++ b/Complementaire/DM/2105_DM1/tpl_bassin.tex @@ -0,0 +1,55 @@ +\begin{exercise}[subtitle={Bassin}] + %- set Vinit = randint(1, 10)*100000 + %- set tx = round((random()+1)/2, 1) + Le clinker est un constituant du ciment qui résulte de la cuisson d'un mélange composé de calcaire et d'argile. La fabrication du clinker nécessite des fours à très haute température qui libèrent dans l'air une grande quantité de dioxyde de carbone (CO$_2$). + + Dans une cimenterie, la fabrication du clinker s'effectue de 7 h 30 à 20 h, dans une pièce de volume \np{\Var{Vinit}}~dm$^3$. + + À 20 h, après une journée de travail, le taux volumique de CO$_2$ dans la pièce est de \Var{tx}\,\%. + \begin{enumerate} + %- set v20 = int(Vinit*tx/100) + \item Justifier que le volume de CO$_2$ présent dans cette pièce à 20 h est de \np{\Var{v20}}~dm$^3$ . + %- set q = round(random()/10, 2) + %- set c = randint(20, 60)*10 + %- set v0 = int(v20 - c) + %- set t = sympy.symbols("t") + %- set V = v0*exp(- q*t) + c + %- set Vp = V.diff() + \item On modélise le volume de CO$_2$ présent dans la pièce par une fonction du temps $t$ écoulé après 20h (exprimé en minutes) qui pour formule $V(t) = V_0e^{-\Var{q}t} + \Var{c}$ + \begin{enumerate} + \item Démontrer que $V_0$ est égale à \np{\Var{v0}}. + %- set decal = randint(1, 4) + \item Quel sera, au dm$^3$ près, le volume de CO$_2$ dans cette pièce à \Var{20+decal} h ? + \item Démontrer que $V'(t) = \Var{latex(Vp)}$. + \item Étudier le signe de $V'(t)$ puis en déduire le sens de variation de $V(t)$. + \item Que peut-on dire du volume de CO$_2$ quand $t$ devient grand? + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item Volume à 20h: $\Var{Vinit}\times \Var{tx/100} = \Var{v20}$ + \item + \begin{enumerate} + \item $t=0$ correspond à 20h. + + Donc $V(0) = \Var{v20} = V_0e^{-\Var{q}\times 0} + \Var{c} = V_0 + \Var{c}$ + + Donc $V_0 = \Var{v20} - \Var{c} = \Var{v0}$ + \item Il faut calculer $V(t)$ pour $t = \Var{decal}$ donc + \[ + V(\Var{decal}) = \Var{round(V.subs(t, str(decal)), 2)} + \] + \item Pas de correction pour cette question. + \item Pas de correction pour cette question. + \item Pas de correction pour cette question. + \end{enumerate} + \end{enumerate} +\end{solution} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: + diff --git a/Complementaire/DM/2105_DM1/tpl_optimisation.tex b/Complementaire/DM/2105_DM1/tpl_optimisation.tex new file mode 100644 index 0000000..ff43c25 --- /dev/null +++ b/Complementaire/DM/2105_DM1/tpl_optimisation.tex @@ -0,0 +1,119 @@ +\begin{exercise}[subtitle={Optimisation de matière}] + %- set Vl = Integer.random("{a}", min_value=2, max_value=10) + %- set l = Integer.random("{a}", min_value=2, max_value=5) + %- set V = Vl*l + %- set Snum = Expression.from_str(str(l*2)+"*x^2 +" + str(Vl*2) + "*x +" + str(V*2)) + %- set dSnum = Snum.differentiate()*"x" - Snum + \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 $\Var{V}m^3$. La longueur est aussi fixée à $\Var{l}m$ 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} + \begin{tikzpicture} + \pgfmathsetmacro{\cubex}{3} + \pgfmathsetmacro{\cubey}{1} + \pgfmathsetmacro{\cubez}{2} + \draw[black,fill=gray] (0,0,0) -- ++(-\cubex,0,0) -- ++(0,-\cubey,0) node [midway, left] {$h$} -- ++(\cubex,0,0) node [midway, below] {$x$} -- cycle; + \draw[black,fill=gray] (0,0,0) -- ++(0,0,-\cubez) -- ++(0,-\cubey,0) -- ++(0,0,\cubez) node [midway, right] {$\Var{l}m$} -- cycle; + \draw[black,fill=gray] (0,0,0) -- ++(-\cubex,0,0) -- ++(0,0,-\cubez) -- ++(\cubex,0,0) -- cycle; + \end{tikzpicture} + \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 $h = \dfrac{\Var{Vl}}{x}$. + \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 + \[ + S(x) = \Var{2*l}x + \Var{2*Vl} + \frac{\Var{2*V}}{x} + \] + \item Démontrer que + \[ + S(x) = \frac{\Var{Snum}}{x} + \] + \item Démontrer que + \[ + S'(x) = \frac{\Var{dSnum}}{x^2} + \] + \item En déduire le tableau de variation de $S(x)$ sur $\intOF{0}{10}$. + \item Déterminer les valeurs de $x$ et $h$ correspondant à une utilisation minimal de tôle. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item Le volume étant fixe si l'on fait varier $x$, $h$ doit aussi varier. + \begin{itemize} + \item Si $x = 2$ alors conserver un volume de $V=\Var{V}$, $h$ doit être égale à $\Var{Vl/2}$ + \item Si $x = 3$ alors conserver un volume de $V=\Var{V}$, $h$ doit être égale à $\Var{Vl/3}$ + \end{itemize} + \item Pour calculer le volume, on a + \begin{eqnarray*} + V &=& h\times x \times \Var{l} \\ + \Var{V} &=& h\times x \times \Var{l} \\ + x &=& \frac{\Var{V}}{h\times \Var{l}} = \frac{\Var{Vl}}{h} + \end{eqnarray*} + \item Pour calculer la surface totale, on ajoute la surface de chaque face. On a donc le calcul suivant + \begin{eqnarray*} + S(x) &=& x\times h \times 2 + x\times\Var{l}\times2 + h\times \Var{l}\times 2\\ + S(x) &=& x\times \frac{\Var{Vl}}{x} \times 2 + x\times\Var{l}\times2 + \frac{\Var{Vl}}{x}\times \Var{l}\times 2\\ + S(x) &=& \Var{2*l}x + \Var{2*Vl} + \frac{\Var{2*V}}{x} + \end{eqnarray*} + \item Pour trouver cette nouvelle forme, on met chaque élément sur le même dénominateur + \begin{eqnarray*} + S(x) &=& \Var{2*l}x + \Var{2*Vl} + \frac{\Var{2*V}}{x}\\ + S(x) &=& \frac{\Var{2*l}x\times x}{x} + \frac{\Var{2*Vl}\times x}{x} + \frac{\Var{2*V}}{x}\\ + S(x) &=& \frac{\Var{Snum}}{x} + \end{eqnarray*} + \item On retrouve la formule $\frac{u}{v}$ à dériver + \[ + u(x) = \Var{Snum} \Rightarrow u'(x) = \Var{Snum.differentiate()} + \] + \[ + v(x) = x \Rightarrow v'(x) = 1 + \] + Donc au numérateur on obtient + \begin{eqnarray*} + u'(x)\times v(x) - u(x)\times v'(x) &=& (\Var{Snum.differentiate()})\times x - (\Var{Snum})\times 1\\ + &=& \Var{dSnum} + \end{eqnarray*} + Donc + \[ + S'(x) = \frac{\Var{dSnum}}{x^2} + \] + \item Tableau de variations de $S$ + + \begin{itemize} + \item Valeur interdite: $x^2 = 0 \equiv x = 0$ + \item Signe de $\Var{dSnum}$: c'est un polynôme du 2e degré + \[ + \Delta = \Var{dSnum.delta} > 0 + \] + Il y a donc 2 racines + \[ + x_1 = \Var{dSnum.roots[0]} \qquad + x_2 = \Var{dSnum.roots[1]} + \] + Et on sait que $\Var{dSnum}$ est du signe de $a$ donc positif en dehors des racines + \item Le dénominateur $x^2$ est toujours positif. + \item Tableau de variations + + \begin{tikzpicture}[baseline=(a.north)] + \tkzTabInit[lgt=3,espcl=3]{$x$/1,$\Var{dSnum}$/1, $x^2$/1, $S'$/1, $S$/2}{$0$, $\Var{dSnum.roots[0]}$, $10$} + \tkzTabLine{d,-, z, +, } + \tkzTabLine{d,+, , +, } + \tkzTabLine{d,-, z, +, } + \tkzTabVar{D+/ , -/ , +/ } + \end{tikzpicture} + + \end{itemize} + \item On a donc une surface minimal pour $x=\Var{dSnum.roots[1]}$ et $h = \Var{Vl*dSnum.roots[1]}$. + \end{enumerate} +\end{solution} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: +