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\begin{document}
\title{Engineering and Science Mathematics I \\ Standard Track}
\author{Midterm I}
\date{October 9, 2002}
\maketitle

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%% \sin(\alpha + \beta) = \sin \alpha \, \cos \beta + \cos \alpha \,
%% \sin \beta \\
%% \sin(\alpha - \beta) = \sin \alpha \, \cos \beta - \cos \alpha \,
%% \sin \beta \\
%% \cos(\alpha + \beta) = \cos \alpha \, \cos \beta - \sin \alpha \,
%% \sin \beta \\
%% \cos(\alpha - \beta) = \cos \alpha \, \cos \beta + \sin \alpha \,
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\begin{enumerate}

\item Compute the following limits.
\begin{enumerate}
\item $\displaystyle \lim_{z \to 3} \frac{2 - \sqrt{1+z}}{3-z}$
\item $\displaystyle \lim_{r \to -1} \frac{r^2-1}{2r^2+3r+1}$
\item $\displaystyle \lim_{w \to 0} \frac{w}{\sin \sqrt w}$
\end{enumerate}
\hfill(5+5+5)

\item The following functions are discontinuous at a point.  Can you
remove the discontinuity?  Explain.

\begin{enumerate}
\item $\displaystyle g(y) = y^2 \, \cos \frac1y$
\item $\displaystyle h(\theta) = \frac{\theta}{|\theta|}$
\item $\displaystyle \psi(r) = \frac{r}{1-\cos r}$
\end{enumerate}
\hfill(5+5+5)

\item Differentiate the following functions.
\begin{enumerate}
\item $\displaystyle \ell(t) = \frac{(\sin t)^2}{\cos t}$
\item $\displaystyle m(p) = e^{-\ln p}$
\item $\displaystyle n(q) = \frac{\sin (q^2)}{\cos q}$
\end{enumerate}
\hfill(5+5+5)

\item Find all points $(x, y)$ on the graph of $x^{2/3} + y^{2/3} = 8$
where lines tangent to the graph have slope $1$.
\hfill (10)

\item Use the definition of the derivative to show that $\cos' \theta
= -\sin \theta$.
\hfill (10)

\item Find the global minimum and maximum values, if they exist, of
the function
\[
  f(x) = \frac{x^2+1}{(x-1)^2}
\]
\hfill (10)

\item A square sheet of cardboard of width $W$ will be made into a box
by cutting equal-sized squares from each corner and folding up the
four edges. What will be the dimensions of the box with largest
volume?
\hfill (10)

\item A farmer owns a $10\,$km long stretch of land between two
parallel rivers that are $2\,$km apart.  What is the area of the
largest rectangular corral he can enclose with
\begin{enumerate}
\item $2\,$km of fencing,
\item $5\,$km of fencing,
\end{enumerate}
assuming that no fence is needed along the river.
\hfill (10+5)

\end{enumerate}
\end{document}
