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\begin{document}

\parindent=0pt

\bf{Question}

\quad Sketch the contours for each of the following functions
$f:\br^2\to\br$\ :
\medskip
\begin{center}
\begin{tabular}{lllll}$f(x_1,x_2)=\hspace{0.5cm}
$&(a)&$x_1^2$&(d)&$x_1^2+x_2$\\
&(b)&$x_1^2-2x_1x_2+x_2^2$&(e)&$x_1^2-2x_1x_2$\\
&(c)&$x_1^2-2x_1x_2+2x_2^2$&(f)&$\sin x_1\sin x_2$
\end{tabular}
\end{center}


\bf{Answer}

\begin{description}
\item{(a)}
$$ \ $$
\begin{center}
\epsfig{file=342-1A-1.eps, width=40mm}
\end{center}

\begin{eqnarray*}
x^2 = c, \ \ \ x & = & \pm \sqrt{c} \ \ \ (c >0)\\ & = & 0 \ \ \
(c=0)
\end{eqnarray*}

\item{(b)}
$$ \ $$
\begin{center}
\epsfig{file=342-1A-2.eps, width=40mm}
\end{center}

$$x^2-2xy+y^2 = (x-y)^2 = c$$

\item{(c)}
$$ \ $$
\begin{center}
\epsfig{file=342-1A-3.eps, width=40mm}

Ellipses
\end{center}

\begin{eqnarray*}
x-2xy+2y^2 & = & (x-y)^2+y^2\\ & = & u^2 + v^2 = c
\end{eqnarray*}
In co-ordinates $x=u+v$, $y=v$.

\item{(d)}
$$ \ $$
\begin{center}
\epsfig{file=342-1A-4.eps, width=40mm}

Parabolas
\end{center}

$$y=-x^2+c$$

\item{(e)}
$$ \ $$
\begin{center}
\epsfig{file=342-1A-1.eps, width=40mm}
\end{center}

$$x^2-2xy = x(x-2y) = c$$ $c=0$: Pair of lines

$c \ne 0$: Hyperbolae

\item{(f)}
$$ \ $$
\begin{center}
\epsfig{file=342-1A-1.eps, width=70mm}
\end{center}

$c=0$: lines $x=n\pi$, $y=m\pi$ ($m,n \in \textbf{Z}$).

$c=1$: max points *

$c=-1$: min points $\bullet$

empty when $|c|>1$
\end{description}




\end{document}