<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\usepackage{epsfig}
\newcommand\ds{\displaystyle}
\setlength{\textwidth}{6.5in} \setlength{\textheight}{9in}
\setlength{\topmargin}{-.25in} \setlength{\oddsidemargin}{-.25in}
\newcommand\br{\textbf{R}}
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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}
$&amp;(a)&amp;$x_1^2$&amp;(d)&amp;$x_1^2+x_2$\\
&amp;(b)&amp;$x_1^2-2x_1x_2+x_2^2$&amp;(e)&amp;$x_1^2-2x_1x_2$\\
&amp;(c)&amp;$x_1^2-2x_1x_2+2x_2^2$&amp;(f)&amp;$\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 &amp; = &amp; \pm \sqrt{c} \ \ \ (c &gt;0)\\ &amp; = &amp; 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 &amp; = &amp; (x-y)^2+y^2\\ &amp; = &amp; 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 $|c|&gt;1$
\end{description}




\end{document}
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