\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts,fullpage,pandora,euler}
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\newcommand{\arccot}{\operatorname{arccot}}

\begin{document}
\title{Mathematics 2B, Section G, Winter 1998}
\author{Marcel Oliver}
\date{Final Exam}
\maketitle


Some trigonometric identities:
\begin{align*}
  {} &  \sin^2 \theta + \cos^2 \theta = 1
     && \sin 2x = 2 \, \sin x \, \cos x \\
     &  1 + \tan^2 \theta = \sec^2 \theta
     && \cos 2x = 2 \, \cos^2 x - 1 = 1 - 2 \, \sin^2 x \\
     &  1 + \cot^2 \theta = \csc^2 \theta
     && \tan 2x = \frac{2 \, \tan x}{1 - \tan^2 x} \\
     &  \sin^2 x = \frac{1 - \cos 2 x}{2}
     && \cos^2 x = \frac{1 + \cos 2 x}{2} 
\end{align*}

Useful integrals:
\begin{gather*}
  \int \frac{du}{\sqrt{1 - u^2}} = \arcsin u + C \\
  \int \frac{du}{1 + u^2} = \arctan u + C
                          = - \arccot u + C' \\
  \int \frac{du}{u \sqrt{u^2 - 1}} = \operatorname{arcsec} |u| + C \\
  \int \sec u \, du = \ln \bigl| \sec u + \tan u \bigr| + C  
\end{gather*}


\begin{enumerate}

\item Compute the following integrals.  Be sure to show all your work.
\begin{enumerate}
\item $\displaystyle \int \frac1{1+e^x} \, dx$
\item $\displaystyle \int x^2 \, \arctan x \, dx$
\item $\displaystyle \int \tan^3 z \, dz$
\item $\displaystyle \int \frac{4x-2}{x^3-x} \, dx$
\end{enumerate}
\hfill (10+10+10+10)


\item
\begin{enumerate}
\item Find the volume $V$ of the unbounded solid obtained by revolving
around the $y$-axis the graph of the function $y=1/x$ for $0<x\leq1$.

\item Show that the surface area $A$ of the solid from part (a) is
infinite. 

\end{enumerate}
\hfill (15+10)



\item Find the arclength of the curve
\[
  y = - \ln \cos \theta, \qquad 0 \leq \theta \leq \tfrac\pi3.
\]
Hint:  You may find some of the identities on the front page useful.
To Simplify your answer, note that
\begin{gather*}
  \sin \tfrac\pi6 = \cos \tfrac\pi3=\tfrac12, \\
  \sin \tfrac\pi3 = \cos \tfrac\pi6=\tfrac{\sqrt3}{2}.
\end{gather*}
\hfill (15)



\item The following questions all concern the $\arccot$ function, but
can be worked on independently.
\begin{enumerate}
\item Explain briefly why
\begin{equation*}
  \lim_{x \to \infty} \arccot x = 0, \qquad
  \lim_{x \to -\infty} \arccot x = \pi.
\end{equation*}

\item Find the derivative of $\arccot$.  You may \emph{not} use the
information on the front page for this part.

\item Sketch the graph of $y = \arccot x$. Your drawing must show
clearly where the graph is increasing, where decreasing, where concave
upward, and where concave downward.  Locate, identify, and label all
extrema and inflection points.  Find and label all asymptotes.

\item Compute
\[  \int_{-\infty}^\infty \frac1{1+x^2} \, dx \]
by using the $\arccot$ function. Do \emph{not} use the $\arctan$
function! 
\end{enumerate}
\hfill(5+10+10+5)


\item In Physics, the power $P$ is defined to be the rate of change of
the energy (i.e., the ``energy delivered per unit time''):
\[
  P = \frac{dE}{dt}.
\]
The power of a certain battery under load decays exponentially, i.e.
\[
  P(t) = P_0 \, e^{-kt}.
\]
where $P_0$ is the power delivered by a fresh battery, and $k$ is a
constant.
\begin{enumerate}
\item Show that the total available electrical energy of a fresh
battery is given by the formula
\[
  E_{\text{fresh}} = \frac{P_0}{k}.
\]

\item Derive a formula for the ``half-life'' of the battery.

\item Suppose a fresh battery delivers $4$ Watts, and, after being
used for two hours, delivers only $1$ Watt.  Use this data to find
$E_{\text{fresh}}$.

\end{enumerate}

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

\end{document}
