\documentclass[12pt]{article}

\usepackage{amsmath}

\def\a{a}
\def\b{b}
\def\x{x}
\def\y{y}
\def\z{z}
\def\f{f}
\def\g{g}
\def\h{h}

\begin{document}

\title{Mathematics 2A, Section D, Fall 1996}
\author{Marcel Oliver}
\date{Midterm Exam I}

\maketitle

\begin{enumerate}

\item Compute the derivative of each of the following functions:
\begin{enumerate}

\item $p(x) = \a \, x^{100} + \b$, 

\item $\f(\x) = \sin^2 (\x) + \cos^2(\x)$,

\item $\g(\y) = \sin (3 \, \cos \y)$,

\item $\h(\z) = \dfrac{\a + \b \, \z}{\a - \b \, \z}$,

\end{enumerate}
where $\a$ and $\b$ do not depend on $\z$. \hfill (5+5+10+10)

\item Consider the function
\begin{equation*}
  \h(\z) = \frac{\z^{\textstyle \frac{4}{3}}}{{\z}}.
\end{equation*}
\begin{enumerate}

\item Find the domain and the range of $\h$.

\item For which value of $\z$ is $\h$ discontinuous?  Is the discontinuity
removable? Explain your answer.

\item Compute the derivative of $\h$.

\item For which value of $\z$ is $\h'$ discontinuous?  Is the discontinuity
removable? Explain your answer.

\item Sketch the graphs of $\h$ and $\h'$.  You may use the same coordinate
system provided you label the graphs clearly.

\end{enumerate}
\hfill (5+5+5+5+10)

\item Let $\displaystyle f(x) = {\sqrt{x}}$.
\begin{enumerate}

\item Compute $f'(x)$ by using the power rule.

\item Compute $f'(x)$ by using the definition of the derivative as the
limit of a difference quotient.

\end{enumerate}
\hfill (10+20)


\item In the lands of Pythagoria people prefer triangular shapes over
rectangular ones. So a farmer plans to enclose an isosceles triangular
pasture adjacent to a straight river. He needs to enclose an area of 100
square meters, wants to use as little fence as possible, but does not know
how much fence he should buy. Solve the farmer's problem, assuming that no
fencing is needed along the river. \hfill (30)

\end{enumerate}
\end{document}


\item In class we have seen that
\begin{equation*}
  \lim_{\x \to 0} \frac{\sin\x}{\x} = 1.
\end{equation*}
Use this fact to compute
\begin{equation*}
  \lim_{\x \to 0} \frac{\x}{\sin \x}.
\end{equation*}
State the limit law(s) you are using, and say why they are applicable.  Do
\emph{not} use the theorem of l'Hospital.

