Source Latex
de la correction du devoir
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
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{\pagestyle{empty}}%
{%
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\vspace*{-.5cm}
\ct{\bf\LARGE{Corrig\'e du devoir de math\'ematiques}}
\bgex
\begin{enumerate}[a)]
\item
\bgmp[t]{6.cm}
$\dsp h:x\mapsto \sqrt{(x-3)(5-x)}$. \
$x$ ne doit pas prendre de valeurs telles que $(x-3)(5-x)<0$,
et donc, d'apr\`es le tableau de signes,
\ul{$\mathcal{D}_h=[3;5]$}.
\enmp
\bgmp[t]{10cm}
\[
\begin{tabular}[t]{|c|ccccccc|}\hline
$x$&$-\infty$ & & $3$ & & $5$ & & $+\infty$ \\\hline
$x-3$ & & $-$ & \zb & $+$ & $|$ & $+$ &\\\hline
$5-x$ & & $+$ & $|$ & $+$ & \zb & $-$ &\\\hline
$(x-3)(5-x)$ & & $-$ & \zb & $+$ & \zb & $-$ & \\\hline
\end{tabular}
\]
\enmp
\item $\dsp g:x\mapsto \frac{\sqrt{3x-6}}{(x+3)(2x-5)}$. \
$x$ ne doit pas prendre de valeurs telles que $3x-6<0$ et
$(x+3)(2x-5)=0$,
soit $x<2$ et $x=-3$ et $x=\frac{5}{2}$.
Ainsi, \ul{$\mathcal{D}_g=[2;+\infty[\ \setminus \la\frac{5}{2}\ra$}.
\end{enumerate}
\enex
\bgex
Soit $M(x;y)\in\mathcal{C}_f\cap\mathcal{C}_g$,
alors on a, $y=f(x)=g(x)$.
\[f(x)=g(x)\iff x^2+3x+4=-3x+4\iff x^2+6x=0\iff x(x+6)=0
\iff \Bigl( x=0 \text{ ou } x=-6\Bigr)\]
Ainsi ces courbes ont deux points d'intersection:
$M_1(0;4)$ et $M_2(-6;22)$.
\enex
\bgex
L'expression $f(x)$ est d\'efinie pour des valeurs de $x$ telles que
$x^2-3\not=0$, soit $x\not=-\sqrt{3}$ et $x\not=-\sqrt{3}$.
Ainsi, l'ensemble de d\'efinition de $f$ est
$\mathcal{D}_f=\R\setminus\la -\sqrt{3};\sqrt{3}\ra
=]-\infty;-\sqrt{3}[\cup]-\sqrt{3};\sqrt{3}[\cup]\sqrt{3};+\infty[$.
\vspd
Soit $u:x\mapsto x^2$ la fonction carr\'e.
\[
\begin{tabular}{|c|cccccccccc|}\hline
$x$ & $-\infty$ && $-\sqrt{3}$ && $0$ && $\sqrt{3}$ && $+\infty$& \\\hline
\raisebox{0.2cm}[0.8cm]{$u$} &&
\psline{->}(-0.6,0.6)(0.6,0.3)&
\rput(0.,0.3){$3$}&&
\psline{->}(-1.1,0.3)(-0.1,0.)&
\rput(-0.5,-0.){$0$}&
\psline{->}(-1.1,0.)(-0.1,0.3)
\rput(0.,0.3){$3$}&
\psline{->}(-0.5,0.3)(0.7,0.6)&&
\\\hline
\raisebox{0.2cm}[0.8cm]{$u-3$}&&
\psline{->}(-0.6,0.6)(0.6,0.3)&
\rput(0.,0.3){$0$}&&
\psline{->}(-1.1,0.3)(-0.2,0.)&
\rput(-0.5,-0.){$-3$}&
\psline{->}(-1.,0.)(-0.1,0.3)
\rput(0.,0.3){$0$}&
\psline{->}(-0.5,0.3)(0.7,0.6)&&
\\\hline
\raisebox{0.2cm}[0.8cm]{$\dfrac{1}{u-3}$}&&
\psline{->}(-0.8,0.)(0.6,0.6)&
\psline(0,0.8)(0,-0.1)\psline(0.06,0.8)(0.06,-0.1)&&
\psline{->}(-1,0.)(-0.3,0.5)&
\rput(-0.6,0.5){$-\frac{1}{3}$}&
\psline{->}(-0.9,0.5)(-0.3,0.)
\psline(0,0.8)(0,-0.1)\psline(0.06,0.8)(0.06,-0.1)&&
\psline{->}(-0.8,0.6)(0.6,0.)&
\\\hline
\end{tabular}
\]
\enex
\bgex
\bgit
\item[a)] La fonction $-2u$ a un sens de variation contraire \`a celui
de $u$, puis $f=-2u+1$ a le m\^eme sens de variation que $-2u$:
\[
\begin{tabular}{|c|ccccccc|}\hline
$x$ & $0$ && $1$ && $2$ &&$+\infty$ \\\hline
&&&$2$&&&&\\
$-2u$ && \psline{->}(-0.3,-0.3)(0.3,0.4)
&&\psline{->}(-0.3,0.5)(0.3,0.2)&0&
\psline{->}(-0.3,0.)(0.3,-0.3)&\\
&0&&&&&&\\\hline
&&&$3$&&&&\\
$f=-2u+1$ && \psline{->}(-0.3,-0.3)(0.3,0.4)
&&\psline{->}(-0.3,0.5)(0.3,0.2)&1&
\psline{->}(-0.3,0.)(0.3,-0.3)&\\
&1&&&&&&\\\hline
\end{tabular}
\]
\vspd
\item[b)] La fonction $\sqrt{u}$ est d\'efinie lorsque $u(x)\geqslant0$,
donc pour $x\in[2\,;\,+\infty[$, et a sur cete intervalle le m\^eme
sens que $u$.
La fonction $-\dfrac{1}{2}\sqrt{u}$ a ensuite un sens de variation
contraire \`a celui de $\sqrt{u}$, puis
$-\dfrac{1}{2}\sqrt{u}+3$ a le m\^eme sens de variation que
$-\dfrac{1}{2}\sqrt{u}$.
\[
\begin{tabular}{|c|ccc|}\hline
$x$ & $2$ &&$+\infty$ \\\hline
&&&\\
$\sqrt{u}$ && \psline{->}(-0.3,-0.3)(0.6,0.5)&\\
&0&&\\\hline
&0&&\\
$-\dfrac{1}{2}\sqrt{u}$ && \psline{->}(-0.3,0.5)(0.6,-0.3)&\\
&&&\\\hline
&3&&\\
$-\dfrac{1}{2}\sqrt{u}+3$ && \psline{->}(-0.3,0.5)(0.6,-0.3)&\\
&&&\\\hline
\end{tabular}
\]
\vspd
\item[c)] La fonction $h=\dfrac{1}{u}$ est d\'efinie lorsque
$u(x)\not=0$, donc pour $x\in]0\,;\,2[\cup]2\,;\,+\infty[$, et a sur
cet intervalle, un sens de variation contraire \`a celui de $u$:
\[
\begin{tabular}{|c|ccccccc|}\hline
$x$ & $0$ && $1$ && $2$ &&$+\infty$ \\\hline
&&&$-1$&&&&\\
$\dfrac{1}{u}$ &
\psline(0,-0.6)(0,0.8)\psline(-0.08,-0.6)(-0.08,0.8)
& \psline{->}(-0.3,-0.3)(0.3,0.5)
&&\psline{->}(-0.3,0.6)(0.3,0.2)&
\psline(0,-0.6)(0,0.8)\psline(0.08,-0.6)(0.08,0.8)
&
\psline{->}(-0.1,0.)(0.5,-0.4)&
\\
&&&&&&&\\\hline
\end{tabular}
\]
\enit
\enex
\label{LastPage}
\end{document}
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