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diff --git a/Master/texmf-dist/doc/latex/tkz-fct/TKZdoc-fct-VDW.tex b/Master/texmf-dist/doc/latex/tkz-fct/TKZdoc-fct-VDW.tex new file mode 100644 index 00000000000..189e2d04278 --- /dev/null +++ b/Master/texmf-dist/doc/latex/tkz-fct/TKZdoc-fct-VDW.tex @@ -0,0 +1,154 @@ +%!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
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