\documentclass[11pt]{amsart}
\parindent=0pt

% The next four lines override the settings in the amsart style file.
\voffset=-0.8in    % Moves text up
\hoffset=-1.2in    % Moves text left
\textwidth=6.8in   % Sets new page text width
\textheight=9.3in  % Sets new page text height

\usepackage{amssymb}
\usepackage{graphicx} % For including EPS files

\input{isomath}
\input{mathenv}

%% ================================================================
\begin{document}

\thispagestyle{empty} % Removes page numbering for this page only
\centerline{\textbf{\large Exam \#2 $\cdot$ Thursday, October 9, 2008}}
\vspace*{.2in}
MATH 124 $\cdot$ Calculus I $\cdot$ Section 26 $\cdot$ Fall 2008  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

%% ----------------------------------------------------------------
\textbf{Problem 1.}  Let $f(x) = x^{2x}$.  Numerically approximate $f'(1)$
using difference quotients.  Use at least three successively smaller values of
$h$.

\vspace*{2.0in}

%% ----------------------------------------------------------------
\textbf{Problem 2.}  The period $T$ (in seconds) of a pendulum depends on the
length $\ell$ (in centimeters) of the pendulum, as follows:
$$
	T = 2\pi \sqrt{\frac{\ell}{g}}
$$
where $g$ is the acceleration due to gravity (a positive constant).

\vspace*{0.1in}

\textbf{Part (a).} What are the units of $dT/d\ell$?

\vspace*{0.75in}

\textbf{Part (b).} Find the formula for $dT/d\ell$.

\vspace*{1.2in}

%% ----------------------------------------------------------------
\textbf{Problem 3.}  The horizontal displacement $H$ (in millimeters) of a
weighted spring from its equilibrium position, at time $t$ milliseconds from
the start of an experiment, is
$$
	H(t) = 2.3 + e^{-2 t} \cos(t).
$$

\textbf{Part (a).}  What are the units of $H''(t)$?

\vspace*{0.2in}

\textbf{Part (b).}  Find $H''(t)$.

%% ----------------------------------------------------------------
\newpage
\textbf{Problem 4.}  The reaction time $T$ (in minutes) of a chemical
experiment is a function of the amount $m$ (in grams) of catalyst added to a
solution.  That is, $T = f(m)$.

\vspace*{0.1in}

\textbf{Part (a).}  Describe the significance of $f(200)$, including units
in your answer.

\vspace*{1.5in}

\textbf{Part (b).}  Describe the significance of $f^{-1}(6)$, including units
in your answer.

\vspace*{1.5in}

\textbf{Part (c).}  Suppose you discover the following:  As you repeat the
experiment, with one more gram of catalyst each time, you find that the
reaction time decreases.  However, each additional gram has less and less
effect.  What is sign of $f''(m)$?  Please justify your answer.  (Sketch a
graph if you like.  If you do so, please label your axes and explain verbally
what you are doing.)

\vspace*{1.5in}

%% ----------------------------------------------------------------
\textbf{Problem 5.}  Let $f(x)=\frac{1}{x^2+1}$.

\vspace*{0.1in}

\textbf{Part (a).}  Find $f'(x)$.

\vspace*{1.25in}

\textbf{Part (b).} Find an equation for the tangent line to $f(x)$ at $x=2$.


%% ----------------------------------------------------------------
\newpage
\textbf{Problem 6.}  Some values of $V(t)$ and $H(t)$ and their first
derivatives are given by the following table.
$$
\begin{tabular}{|c||c|c|c|c|}
\hline $t$ & $V(t)$ & $H(t)$ & $V'(t)$ & $H'(t)$ \\
\hline $6$ & $8$    & $-7$   & $0.5$   & $-0.2$  \\
\hline
\end{tabular}
$$

\vspace*{0.1in}

\textbf{Part (a).} Let $E(t) = V(t) + H(t)$.  Find $E'(6)$.

\vspace*{1.0in}

\textbf{Part (b).} Let $E(t) = V(t)H(t)$.  Find $E'(6)$.

\vspace*{1.0in}

\textbf{Part (c).} Let $E(t) = \frac{V(t)}{H(t)}$.  Find $E'(6)$.

\vspace*{1.0in}

%% ----------------------------------------------------------------
\textbf{Problem 7.}
Let $g(x)=x e^{-ax}$.  Assume $a$ is a positive constant.  Over what
interval(s) is $g(x)$ increasing?  (You will need to solve an inequality.)

\vspace*{1.7in}

%% ----------------------------------------------------------------
\textbf{Problem 8.}  Let $C(\theta) = \frac{\cos(\theta) - 1}{\theta}$.

\vspace*{0.1in}

\textbf{Part (a).}  Does $C'(1)$ exist?  If so, find its value exactly.
If not, explain why not.

\vspace*{1.25in}

\textbf{Part (b).}  Does $C'(0)$ exist?  If so, find its value exactly.
If not, explain why not.

%% ================================================================
\end{document}
