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-
-\subsection{Courbes de \tkzname{Van der Waals}}
-
-\bigskip
-Soient $v$ le volume d'une masse fluide et $p$ sa pression.
-$b$ et $k$ sont deux nombres réels strictement positifs. On souhaite étudier une formule exprimant la dépendance de ces variables proposée par Van~der~Waals.
-\[
- p(v)= \frac{-3}{v^2} + \dfrac{3k}{v-b}
-\]
-
-définie sur l'intervalle $I=\big]b~;~+\infty\big]$
-
-\subsubsection{Tableau de variations}
-\begin{center}
-
- \begin{tkzexample}[]
- \begin{tikzpicture}
- \tkzTab%
- { $v$ /1,%
- $g'(v)$ /1,%
- $g(v)$ /3%
- }%
- { $b$ ,%
- $3b$ ,%
- $+\infty$%
- }%
- {0,$+$,$0$,$-$,t}
- {-/ $0$ /,%
- +/$\dfrac{8}{27b}$ /,%
- -/ $0$ /}%
- \end{tikzpicture}
-\end{tkzexample}
-\end{center}
-
-
-
-\newpage
-
-\subsubsection{ Première courbe avec $ b=1$}
- Quelques courbes pour $r\leq\ v \leq\ 6$
-
-\medskip
-\begin{center}
- \begin{tkzexample}[]
- \begin{tikzpicture}[xscale=2,yscale=2.5]
- \tkzInit[xmin=0,xmax=6,ymax=0.5,ystep=0.1]
- \tkzDrawX[label=$v$]
- \tkzDrawY[label=$g(v)$]
- \tkzGrid(0,0)(6,0.5)
- \tkzFct[color = red,domain =1:6]{(2*(x-1)*(x-1))/(x*x*x)}
- \tkzDrawTangentLine[color=blue,draw](3)
- \tkzDefPointByFct(1)
- \tkzText[draw, fill = brown!30](4,0.1){$g(v)=2\dfrac{(v-1)^2}{v^3}$}
- \end{tikzpicture}
- \end{tkzexample}
-\end{center}
-
-
-\newpage
-\subsubsection{ Deuxième courbe $b=1/3$ }
-
-
-\medskip
-\begin{center}
- \begin{tkzexample}[]
- \begin{tikzpicture}[scale=1.2]
- \tkzInit[xmin=0,xmax=2,xstep=0.2,ymax=1,ystep=0.1]
- \tkzAxeXY
- \tkzGrid(0,0)(2,1)
- \tkzFct[color = red,domain =1/3:2]{(2*(\x-1./3)*(\x-1./3))/(\x*\x*\x)}
- \tkzDrawTangentLine[draw,color=blue,kr=.5,kl=.5](1)
- \tkzDefPointByFct(1)
- \tkzText[draw,fill = brown!30](1.2,0.3)%
- {$g(v)=2\dfrac{\left(v-\dfrac{1}{3}\right)^2}{v^3}$}
- \end{tikzpicture}
- \end{tkzexample}
-\end{center}
-
-
-\newpage
-\subsubsection{ Troisième courbe $ b=32/27$ }
-
-
-\medskip
-\begin{center}
- \begin{tkzexample}[]
- \begin{tikzpicture}[scale=1.2]
- \tkzInit[xmin=0,xmax=10,ymax=.35,ystep=0.05];
- \tkzAxeXY
- \tkzGrid(0,0)(10,.35)
- \tkzFct[color = red,
- domain =1.185:10]{(2*(\x-32./27)*(\x-32./27))/(\x*\x*\x)}
- \tkzDrawTangentLine[draw,color=blue,kr=2,kl=2](3.555)
- \tkzText[draw,fill = brown!30](5,0.3)%
- {$g(v)=2\dfrac{\left(v-\dfrac{32}{27}\right)^2}{v^3}$}
- \end{tikzpicture}
- \end{tkzexample}
-\end{center}
-
- \newpage
-\subsection{Valeurs critiques}
-\subsubsection{Courbes de \tkzname{Van der Walls} }
-%<–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––>
-
-
-\begin{tkzexample}[]
-\begin{tikzpicture}[scale=4]
- \tkzInit[xmax=3,ymax=2];
- \tkzAxeXY
- \tkzGrid(0,0)(3,2)
- \tkzFct[color = red,domain =1/3:3]{0.125*(3*\x-1)+0.375*(3*\x-1)/(\x*\x)}
- \tkzDefPointByFct[draw](2)
- \tkzDefPointByFct[draw](3)
- \tkzDrawTangentLine[draw,color=blue](1)
- \tkzFct[color = green,domain =1/3:3]{0.125*(3*x-1)}
- \tkzSetUpPoint[size=8,fill=orange]
- \tkzDefPointByFct[draw](3)
- \tkzDefPointByFct[draw](1/3)
- \tkzDefPoint(1,1){f}
- \tkzDrawPoint(f)
- \tkzText[draw,fill = white,text=red](1,1.5)%
-{$f(x)=\dfrac{1}{8}(3x-1)+\dfrac{3}{8}\left(\dfrac{3x-1}{x^2}\right)$}
-\tkzText[draw,fill = white,text=green](2,0.4){$g(x) = \dfrac{3x-1}{8}$}
-\end{tikzpicture}
-\end{tkzexample}
-
-\newpage
-\subsubsection{Courbes de \tkzname{Van der Walls} (suite)}
-
-
-\begin{tkzexample}[]
-\begin{tikzpicture}[xscale=4,yscale=1.5]
- \tkzInit[xmin=0,xmax=3,ymax=3,ymin=-4]
- \tkzGrid(0,-4)(3,3)
- \tkzAxeXY
- \tkzClip
- \tkzVLine[color=red,style=dashed]{1/3}
- \tkzFct[color=red,domain = 0.35:3]{-3/(x*x) +4/(3*x-1)}
- \tkzFct[color=blue,domain = 0.35:3]{-3/(x*x) +27/(4*(3*x-1))}
- \tkzFct[color=orange,domain = 0.35:3]{-3/(x*x) +8/(3*x-1)}
- \tkzFct[color=green,domain = 0.35:3]{-3/(x*x) +7/(3*x-1)}
- \tkzText[draw,fill = white,text=brown](2,-2)%
- {$f(x)=-\dfrac{3}{x^2}+\dfrac{8\alpha}{3x-1}$ \hspace{.5cm}%
- avec $\alpha \in%
- \left\{\dfrac{1}{2}~;~\dfrac{27}{32}~;~\dfrac{7}{8}~;~1\right\}$}
-\end{tikzpicture}
-\end{tkzexample}
-
- \endinput \ No newline at end of file