\documentclass[12pt,a4paper,portrait,notitlepage]{report}
\usepackage[utf8]{inputenc}
\usepackage[french]{babel}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{ifthen}
\usepackage[top=0.5cm,bottom=2.5cm,left=1.5cm,right=1.5cm]{geometry}
\author{Alain Busser\\IREMI 974}
\usepackage{tikz}
\usetikzlibrary{automata,shapes,arrows,shadings}
\tikzset{
v/.style={initial text={\tikz{\node[draw,fill,color=green] {};}},
circle,draw,ultra thick,minimum width=1.4cm},
a/.style={rectangle,draw,ultra thick,minimum width=1.4cm,minimum height=1.4cm},
rouge/.style={->,>=stealth',ultra thick,color=red},
bleu/.style={->,>=stealth',ultra thick,color=blue},
arc/.style={->,>=stealth',ultra thick}
}

\begin{document}

\begin{center}
\begin{LARGE}
\textbf{Dérivée de la fonction cube}
\end{LARGE}
\end{center}

Le cube de 

\begin{center}
%$\left(
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v,accepting] (a) at (-1,-1) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
%\right)^{3}$
\end{center}

est le produit de

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v,accepting] (a) at (-1,-1) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v] (a) at (1,1) {};
\node[v] (b) at (2,0) {};
\node[v,accepting] (c) at (0,0) {};
\node[v,initial] (d) at (-1,1) {};
\draw[bleu] (a) to (b);
\draw[bleu,bend left=15] (a) to (c);
\draw[rouge,bend right=15] (a) to (c);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[bleu] (d) to (c);
\draw[rouge] (d) to (a);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}


par 


\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v,accepting] (a) at (-1,-1) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

soit

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v,accepting] (a) at (-1,-1) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,-1) {};
\node[v] (c) at (2,0) {};
\node[v] (b) at (1,1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[bleu] (c) to (o);
\draw[rouge] (c) to (d);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (o);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (1,1.5) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v] (c) at (1,0) {}; 
\node[v] (d) at (2,0) {}; 
\node[v,accepting] (e) at (3,0) {}; 
\draw[bleu, bend left] (o) to (e);
\draw[rouge, bend right] (o) to (a);
\draw[rouge] (o) to (d);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=15] (c) to (d);
\draw[rouge, bend right=15] (c) to (d);
\draw[bleu, bend left=15] (d) to (e);
\draw[rouge, bend right=15] (d) to (e);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\draw[bleu, bend left=45] (b) to (d);
\draw[rouge, bend right=45] (b) to (d);
\draw[bleu, bend left=45] (c) to (e);
\draw[rouge, bend right=45] (c) to (e);
\draw[bleu, bend left=45] (a) to (d);
\draw[rouge, bend right=45] (a) to (d);
\draw[bleu, bend left=45] (b) to (e);
\draw[rouge, bend right=45] (b) to (e);
\draw[bleu, bend left=45] (a) to (e);
\draw[rouge, bend right=45] (a) to (e);
\end{tikzpicture}
\end{center}

Tous ces termes sont infinitésimaux, sauf le premier

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v,accepting] (a) at (-1,-1) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\end{center}

Ainsi, si on ajoute au jeu ci-dessus le terme

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v,accepting] (a) at (-1,-1) {}; 
\draw[rouge] (o) to (a);
\end{tikzpicture}
\end{center}

qui neutralise

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v,accepting] (a) at (-1,-1) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\end{center}

il restera la somme infinitésimale

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,-1) {};
\node[v] (c) at (2,0) {};
\node[v] (b) at (1,1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[bleu] (c) to (o);
\draw[rouge] (c) to (d);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (o);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (1,1.5) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v] (c) at (1,0) {}; 
\node[v] (d) at (2,0) {}; 
\node[v,accepting] (e) at (3,0) {}; 
\draw[bleu, bend left] (o) to (e);
\draw[rouge, bend right] (o) to (a);
\draw[rouge] (o) to (d);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=15] (c) to (d);
\draw[rouge, bend right=15] (c) to (d);
\draw[bleu, bend left=15] (d) to (e);
\draw[rouge, bend right=15] (d) to (e);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\draw[bleu, bend left=45] (b) to (d);
\draw[rouge, bend right=45] (b) to (d);
\draw[bleu, bend left=45] (c) to (e);
\draw[rouge, bend right=45] (c) to (e);
\draw[bleu, bend left=45] (a) to (d);
\draw[rouge, bend right=45] (a) to (d);
\draw[bleu, bend left=45] (b) to (e);
\draw[rouge, bend right=45] (b) to (e);
\draw[bleu, bend left=45] (a) to (e);
\draw[rouge, bend right=45] (a) to (e);
\end{tikzpicture}
\end{center}

C'est un jeu positif. Il existe donc une stratégie gagnante pour les bleus. Laquelle~?

Maintenant si on multiplie

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v] (a) at (-1,-1) {}; 
\node[v] (b) at (-1,-2) {}; 
\node[v,accepting] (c) at (-1,-3) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,-1) {};
\node[v] (c) at (2,0) {};
\node[v] (b) at (1,1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[bleu] (c) to (o);
\draw[rouge] (c) to (d);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (o);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

par

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (2,0) {};
\node[v,initial] (a) at (1,1) {}; 
\node[v] (b) at (0,0) {}; 
\draw[bleu] (a) to (o);
\draw[rouge] (a) to (b);
\draw[bleu,bend left=15] (b) to (o);
\draw[rouge,bend right=15] (b) to (o);
\end{tikzpicture}
\end{center}

on obtient la même somme que ci-avant (le jeu infinitésimal)~:

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,-1) {};
\node[v] (c) at (2,0) {};
\node[v] (b) at (1,1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[bleu] (c) to (o);
\draw[rouge] (c) to (d);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (o);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (1,1.5) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v] (c) at (1,0) {}; 
\node[v] (d) at (2,0) {}; 
\node[v,accepting] (e) at (3,0) {}; 
\draw[bleu, bend left] (o) to (e);
\draw[rouge, bend right] (o) to (a);
\draw[rouge] (o) to (d);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=15] (c) to (d);
\draw[rouge, bend right=15] (c) to (d);
\draw[bleu, bend left=15] (d) to (e);
\draw[rouge, bend right=15] (d) to (e);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\draw[bleu, bend left=45] (b) to (d);
\draw[rouge, bend right=45] (b) to (d);
\draw[bleu, bend left=45] (c) to (e);
\draw[rouge, bend right=45] (c) to (e);
\draw[bleu, bend left=45] (a) to (d);
\draw[rouge, bend right=45] (a) to (d);
\draw[bleu, bend left=45] (b) to (e);
\draw[rouge, bend right=45] (b) to (e);
\draw[bleu, bend left=45] (a) to (e);
\draw[rouge, bend right=45] (a) to (e);
\end{tikzpicture}
\end{center}


Ce qui signifie que le taux d'accroissement de la fonction $x \mapsto x^{3}$ en

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v,accepting] (a) at (-1,-1) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v] (a) at (-1,-1) {}; 
\node[v] (b) at (-1,-2) {}; 
\node[v,accepting] (c) at (-1,-3) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}
\begin{tikzpicture}[scale=2]
\node[v,accepting] (o) at (0,0) {};
\node[v] (d) at (1,-1) {};
\node[v] (c) at (2,0) {};
\node[v] (b) at (1,1) {};
\node[v,initial] (a) at (-1,1) {};
\draw[bleu,bend left=15] (d) to (o);
\draw[rouge,bend right=15] (d) to (o);
\draw[bleu] (c) to (o);
\draw[rouge] (c) to (d);
\draw[bleu,bend left=15] (b) to (c);
\draw[rouge,bend right=15] (b) to (c);
\draw[rouge] (a) to (b);
\draw[bleu] (a) to (o);
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,1) {};
\node[v] (a) at (-1,0) {}; 
\node[v] (b) at (0,0) {}; 
\node[v,accepting] (c) at (1,0) {}; 
\draw[bleu, bend left] (o) to (c);
\draw[rouge, bend right] (o) to (a);
\draw[bleu, bend left=15] (a) to (b);
\draw[rouge, bend right=15] (a) to (b);
\draw[bleu, bend left=15] (b) to (c);
\draw[rouge, bend right=15] (b) to (c);
\draw[bleu, bend left=45] (a) to (c);
\draw[rouge, bend right=45] (a) to (c);
\end{tikzpicture}
\end{center}

Mais deux des termes sont infinitésimaux. Donc le nombre dérivé de $x \mapsto x^{3}$ en 

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v,accepting] (a) at (1,0) {}; 
\draw[bleu] (o) to (a);
\end{tikzpicture}
\end{center}

est

\begin{center}
\begin{tikzpicture}[scale=2]
\node[v,initial] (o) at (0,0) {};
\node[v] (a) at (1,0) {}; 
\node[v] (b) at (2,0) {}; 
\node[v,accepting] (c) at (3,0) {}; 
\draw[bleu] (o) to (a);
\draw[bleu] (a) to (b);
\draw[bleu] (b) to (c);
\end{tikzpicture}
\end{center}


\end{document}


