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authorKarl Berry <karl@freefriends.org>2020-04-12 21:33:18 +0000
committerKarl Berry <karl@freefriends.org>2020-04-12 21:33:18 +0000
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tree71d747ce98db7c12c482309e3c43805064502153 /Master/texmf-dist/doc/latex/tkz-fct/TKZdoc-fct-VDW.tex
parent459ff31e0bc16b6adf838e7f1e69050496341559 (diff)
tkz-fct (12apr20)
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+%!TEX root = /Users/ego/Boulot/TKZ/tkz-fct/doc-fr/TKZdoc-fct-main.tex
+\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 \emph{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 \emph{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 \emph{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=Maroon](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