\documentclass[12pt]{article}
\author{William Conley}
\title{Math 32A - Midterm 1}
\date{Sunday, 16 November, 2003}

\usepackage{amsfonts}
\usepackage{amsmath}

\begin{document}
\pagestyle{myheadings}
\markright{Math 32A - Midterm 1}

\begin{enumerate}
  \item (10 points) Suppose that a swan (not a duck!!) is swimming in the circle 
    described by the path 
    $$ \vec{c}(t) = (cos t, sin t) \qquad t\in\mathbb{R}^+ $$
    and that the water temperature is given by 
    $$ T(x,y) = x^2e^y \qquad x,y\in\mathbb{R}. $$
    \begin{enumerate}
      \item Find $DT(x,y)$ and explain what this represents.
      \item Find $D(T\circ c)(t)$ and explain what this represents.
    \end{enumerate}

  \item (10 points) Determine whether the following statements are true or 
    false. If a statement is true, indicate how you could show it. If a 
    statement is false, provide a counterexample. 

    \begin{enumerate}
      \item If $f(x,y,z) = x^4 + xy + z^3$ then $\nabla f(1,0,1)$ is perpendicular 
        to the surface $f=2$ at the point $(1,0,1)$. 

        \vspace{1ex}

        This statement is (circle one): \hspace{1.5cm} True \hspace{1.5cm} False \\
        Explanation or counterexample: 

        \vspace{1cm}

      \item If $f(x,y) = ln y$ then $\nabla f(x,y) = \frac{1}{y}$. 

        \vspace{1ex}

        This statement is (circle one): \hspace{1.5cm} True \hspace{1.5cm} False \\
        Explanation or counterexample: 

        \vspace{1cm}

    \end{enumerate}

  \item (10 points) Find the absolute maxima and minima of the function 
    $$ f(x,y) = 5x^2 - 2y^2 + 2 $$
    on the disk $x^2 + y^2 \le 1$

  \item (10 points) Pictured are a contour map of f and a (dashed) curve with 
    equation $g(x,y) = 8$. Estimate the maximum and minimum values of f subject to 
    the constraint $g(x,y) = 8$. Explain your choices. 

\end{enumerate}

\end{document}