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\begin{document}

\title{Mathematics 2D, Fall 1997}
\author{Marcel Oliver}
\date{Midterm Exam}

\maketitle

\begin{enumerate}

\item Determine whether or not the function
\begin{equation*}
  f(x,y) =
  \begin{cases}
    \dfrac{x^2 - y^2}{x^2 + y^2} & \text{when } x \neq 0 \text{ or }
       y \neq 0 \\
    0 & \text{otherwise}
  \end{cases}
\end{equation*}
is continuous.
\hfill (10)

\item The cone with equation
\begin{equation*}
  z^2 = x^2 + y^2 
\end{equation*}
and the plane with equation
\begin{equation*}
  z = \tfrac12 \, x - \tfrac12 \, y + 1
\end{equation*}
intersect in an ellipse.  Use the method of Lagrange multipliers to
find the highest point (above the $xy$-plane) on this ellipse.
\hfill (20)

\item
\begin{enumerate}
\item Suppose you are standing at the point $(1,1,\tfrac34)$ in a
valley that has the shape of the graph of
\begin{equation*}
  z = 1 - \frac1{x^2+y^2+2}.
\end{equation*}
In what horizontal direction should you go to maintain a constant
altitude?

\item Find an equation for the plane tangent to the surface from part
(a) at the given point.
\end{enumerate}
\hfill(15+10)

\item
\begin{enumerate}
\item Let $\boldsymbol{r}=\boldsymbol{r}(t)$ be a parametric curve on
the surface defined through
\begin{equation*}
  F (\boldsymbol{x}) = 0.
\end{equation*}
Show that $\boldsymbol\nabla F$ and $\tfrac{d \boldsymbol{r}}{d t}$
are perpendicular.

\item Suppose that $w = f(u)$ and $u = x - c t$.  Show that
\begin{equation*}
  \frac{\partial w}{\partial t}
  + c \frac{\partial w}{\partial x} = 0.
\end{equation*}
\end{enumerate}
\hfill(10+10)

\item
\begin{enumerate}
\item Show that the function
\begin{equation*}
  g(x,y) = y^3 - 3 x^2 y \tag{*}
\end{equation*}
does not have a local extremum.

\item Locate the extrema of (*) when $x$ and $y$ are constrained to
the the unit disc $x^2 + y^2 \leq 1$.
\end{enumerate}
\hfill (10+15)

\end{enumerate}

\end{document}
