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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 144603248533929--
%%--Machines mathématiques--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
Les symboles ci-dessous d\'{e}signent des machines \'{e}l\'{e}mentaires qui
servent :

%%machine1
%% \begin{pspicture}(0,.8)(3.2,2.7)
%% \psset{dimen=middle,linewidth=.5pt}
%% \pswedge*(1,2){0.25}{270}{90}
%% \psline(0.6,1.75)(1,1.75)
%% \psline(0.6,2.25)(1,2.25)
%% \psline(1.25,2)(1.65,2)
%% \rput[Br](0.4,2.25){$b$}
%% \rput[Br](0.4,1.75){$a$}
%% \rput[Bl](1.85,2){$a+b$}
%% \rput[B](1.5,1){à additionner}
%% \end{pspicture}
%%machine2
%% \begin{pspicture}(0,.8)(3,2.7)
%% \psset{dimen=middle,linewidth=.5pt}
%% \pswedge(1,2){0.25}{270}{90}
%% \psline(0.6,1.75)(1,1.75)
%% \psline(0.6,2.25)(1,2.25)
%% \psline(1.25,2)(1.65,2)
%% \rput[Br](0.4,2.25){$b$}
%% \rput[Br](0.4,1.75){$a$}
%% \rput[Bl](1.85,2){$ab$}
%% \rput[B](1.2,1){à multiplier}
%% \end{pspicture}
%%machine3
%% \begin{pspicture}(0,.8)(3,2.7)
%% \psset{dimen=middle,linewidth=.5pt}
%% \pspolygon(1,1.75)(1,2.25)(1.45,2)
%% \psline(0.6,2)(1,2)
%% \psline(1.45,2)(1.85,2)
%% \rput[Br](0.4,2){$a$}
%% \rput[Bl](2.05,2){$a^2$}
%% \rput[B](1.3,1){à élever au carré}
%% \end{pspicture}
%

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%cropbottom "0";  filename 'Include/machine1.eps';file-properties "XNPEU";}}}%
%BeginExpansion
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height=0.6384in,
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{Include/machine1.eps}%
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%EndExpansion
&
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{Include/machine3.eps}%
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\]


\begin{enumerate}
\item On consid\`{e}re les machines suivantes:

%%machine4
%% \begin{pspicture}(0,-0.5)(3.1,2)
%% \psset{dimen=middle,linewidth=.5pt}
%% \pswedge(0.9,1.25){0.25}{270}{90}
%% \psline(0.5,1)(0.9,1)
%% \psline(0.5,1.5)(0.9,1.5)
%% \psline(1.15,1.25)(1.55,1.25)
%% \rput[Br](0.3,1){$-3$}
%% \rput[Br](0.3,1.5){$4$}
%% \rput[Br](0.3,0.5){$5$}
%% \psline(.5,0.5)(1.55,0.5)(1.55,.75)
%% \pswedge*(1.95,1){0.25}{270}{90}
%% \psline(1.55,1.25)(1.95,1.25)
%% \psline(1.55,.75)(1.95,.75)
%% \psline(2.2,1)(2.6,1)
%% \rput[Bl](2.8,1){$A$}
%% \end{pspicture}
%%machine5
%% \begin{pspicture}(0,-0.5)(5,2.4)
%% \psset{dimen=middle,linewidth=.5pt}
%% \rput[Br](0.3,0){$5$}
%% \rput[Br](0.3,2){$-2$}
%% \psline(0.5,0)(2.8,0)
%% \psline(0.5,2)(2.8,2)
%% \pspolygon(1,.25)(1,.75)(1.45,.5)
%% \pspolygon(1,1.25)(1,1.75)(1.45,1.5)
%% \pswedge(2.8,0.25){0.25}{270}{90}
%% \pswedge(2.8,1.75){0.25}{270}{90}
%% \psline(0.6,0)(0.6,.5)(1,.5)
%% \psline(0.6,2)(0.6,1.5)(1,1.5)
%% \psline(1.45,.5)(1.85,.5)(2.4,1.5)(2.8,1.5)
%% \psline(1.45,1.5)(1.85,1.5)(2.4,.5)(2.8,.5)
%% \psline(3.05,0.25)(3.45,0.25)(3.45,0.75)(3.85,0.75)
%% \psline(3.05,1.75)(3.45,1.75)(3.45,1.25)(3.85,1.25)
%% \pswedge*(3.85,1){0.25}{270}{90}
%% \psline(4.15,1)(4.55,1)
%% \rput[Bl](4.75,1){$B$}
%% \rput[B](2.5,-0.5){Machine $T$}
%% \end{pspicture}
%

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{Include/machine4.eps}%
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& \hspace{2cm} &
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%{\special{ language "Scientific Word";  type "GRAPHIC";
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%original-height 1.0585in;  cropleft "0";  croptop "1";  cropright "1";
%cropbottom "0";  filename 'Include/machine5.eps';file-properties "XNPEU";}}}%
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{Include/machine5.eps}%
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\end{tabular}
\
\]


\begin{enumerate}
\item Calculer les r\'{e}sultats $A$ et $B$.

\item Calculer le r\'{e}sultat de la machine $T$ si on y introduit
$\displaystyle{\frac{1}{2}}$ et $\displaystyle{\frac{1}{3}}$ ?

\item Quel est le r\'{e}sultat fourni si on y introduit $a$ et $b$ ?
\end{enumerate}

\item Comparer $A$ et $B$ dans les deux cas suivant :

\begin{itemize}
\item \textit{Premier cas:}

%%machine6
%% \begin{pspicture}(0,-0.8)(4.5,1.2)
%% \psset{dimen=middle,linewidth=.5pt}
%% \rput[Br](0.3,.75){$x$}
%% \rput[Br](0.3,-0.25){$-1$}
%% \rput[Br](0.3,-0.75){$2$}
%% \psline(0.5,.75)(2.05,.75)
%% \psline(1,.75)(1,.25)(1.4,.25)
%% \psline(0.5,-.25)(1.4,-.25)
%% \pswedge(1.4,0){0.25}{270}{90}
%% \psline(1.65,0)(2.05,0)
%% \pspolygon(2.05,.5)(2.05,1)(2.5,.75)
%% \pswedge*(2.05,-0.25){0.25}{270}{90}
%% \psline(0.5,-.75)(1.65,-.75)(1.65,-.5)(2.05,-.5)
%% \psline(2.5,.75)(2.9,.75)(2.9,.25)(3.3,.25)
%% \psline(2.3,-.25)(3.3,-.25)
%% \pswedge*(3.3,0){0.25}{270}{90}
%% \psline(3.55,0)(3.95,0)
%% \rput[Bl](4.15,0){$A$}
%% \end{pspicture}
%%machine7
%% \begin{pspicture}(0,-0.8)(4.5,1.2)
%% \psset{dimen=middle,linewidth=.5pt}
%% \rput[Br](0.3,.75){$-2$}
%% \rput[Br](0.3,0.){$x$}
%% \rput[Br](0.3,-0.75){$1$}
%% \psline(.5,.75)(1.4,.75)
%% \psline(.5,0)(1,0)
%% \psline(.5,-.75)(1.4,-.75)
%% \psline(1.4,.25)(1,.24)(1,-.25)(1.4,-.25)
%% \pswedge*(1.4,0.5){0.25}{270}{90}
%% \pswedge*(1.4,-0.5){0.25}{270}{90}
%% \psline(1.65,0.5)(2.05,0.5)(2.05,0.25)(2.45,0.25)
%% \psline(1.65,-0.5)(2.05,-0.5)(2.05,-0.25)(2.45,-0.25)
%% \pswedge(2.45,0){0.25}{270}{90}
%% \psline(2.7,0)(3.1,0)
%% \rput[Bl](3.3,0){$B$}
%% \end{pspicture}
%

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\]


\item \textit{Second cas:}

%%machine8
%% \begin{pspicture}(0,-0.7)(3,0.7)
%% \psset{dimen=middle,linewidth=.5pt}
%% \rput[Br](0.3,.25){$a$}
%% \rput[Br](0.3,-0.25){$b$}
%% \psline(0.5,.25)(0.9,.25)
%% \psline(0.5,-.25)(0.9,-.25)
%% \pswedge*(.9,0){0.25}{270}{90}
%% \psline(1.15,0)(1.55,0)
%% \pspolygon(1.55,-.25)(1.55,.25)(2,0)
%% \psline(2,0)(2.4,0)
%% \rput[Bl](2.6,0){$A$}
%% \end{pspicture}
%%machine9
%% \begin{pspicture}(0,-0.7)(3,0.7)
%% \psset{dimen=middle,linewidth=.5pt}
%% \rput[Br](0.3,.25){$a$}
%% \rput[Br](0.3,-0.25){$b$}
%% \psline(0.5,.25)(0.9,.25)
%% \psline(0.5,-.25)(0.9,-.25)
%% \pspolygon(0.9,0)(0.9,.5)(1.35,.25)
%% \pspolygon(0.9,0)(0.9,-.5)(1.35,-.25)
%% \psline(1.35,.25)(1.75,.25)
%% \psline(1.35,-.25)(1.75,-.25)
%% \pswedge*(1.75,0){0.25}{270}{90}
%% \psline(2,0)(2.4,0)
%% \rput[Bl](2.6,0){$B$}
%% \end{pspicture}
%

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& \hspace{2cm} &
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\end{itemize}

\item Dessiner la machine qui donne $(a+2b)^{2}$.

\item Calculer $A(x)$ et $B(x)$, puis comparer $A(x)$ et $B(x)$.

%%machine10
%% \begin{pspicture}(0,0)(7,3.6)
%% \psset{dimen=middle,linewidth=.5pt}
%% \rput[Br](0.3,3.5){$x$}
%% \rput[Br](0.3,2.5){$2$}
%% \rput[Br](0.3,1.5){$5$}
%% \rput[Br](0.3,0.5){$-7$}
%% \rput[Br](0.3,0){$8$}
%% \psline(.5,3.5)(1.5,3.5)
%% \psline(1,3.5)(1,1)
%% \pspolygon(1.5,3.25)(1.5,3.75)(1.95,3.5)
%% \psline(1.95,3.5)(2.35,3.5)
%% \psline(1,3)(2.35,3)
%% \pswedge(2.35,3.25){0.25}{270}{90}
%% \psline(2.6,3.25)(3,3.25)(3,3)(3.4,3)
%% \psline(.5,2.5)(3.4,2.5)
%% \pswedge(3.4,2.75){0.25}{270}{90}
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%% \psline(1,2)(1.5,2)
%% \pspolygon(1.5,1.75)(1.5,2.25)(1.95,2)
%% \psline(1.95,2)(2.35,2)
%% \psline(0.5,1.5)(2.35,1.5)
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%%machine11
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%% \rput[Br](0.3,0){$8$}
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%% \end{pspicture}
%

\[%
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\]


On constate que la deuxi\`{e}me machine est plus \og\'{e}conomique\fg que la
premi\`{e}re. Elle est bas\'{e}e sur un proc\'{e}d\'{e} publi\'{e} par le
math\'{e}maticien anglais H\"{o}rner en 1819. Une m\'{e}thode analogue a
\'{e}t\'{e} utlis\'{e}e par Newton 150 ans auparavant. Les math\'{e}maticiens
chinois utilisaient un proc\'{e}d\'{e} analogue issu de la pratique du boulier.

\item Construire la machine de H\"{o}rner associ\'{e} \`{a} : $C(x)=3x^{2}%
-4x+3$.
\end{enumerate}
\end{exercise}


\end{document}