\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts,fullpage,pandora,euler}
\parindent 0pt

\begin{document}
\title{Mathematics 2B, Section G, Winter 1998}
\author{Marcel Oliver}
\date{Midterm Exam I}
\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 \sec u \, du = \ln \bigl| \sec u + \tan u \bigr| + C \\
  \int \csc u \, du = \ln \bigl| \csc u - \cot u \bigr| + C \\
  \int \frac{du}{1+u^2} = \arctan u + C
  \end{gather*}


\begin{enumerate}

\item Compute the following integrals.  Be sure to show all your work.
\begin{enumerate}
\item $\displaystyle \int \ln (1+x) \, dx$
\item $\displaystyle \int \ln (1+\sqrt x) \, dx$
\item $\displaystyle \int \frac{\cos 2\theta}{\cos \theta} \, d
\theta$
\item $\displaystyle \int \frac{x^4+1}{x^2+1} \, dx$
\item $\displaystyle \int \frac{\tan \theta}{\cos^2 \theta} \, d
\theta$
\end{enumerate}
Hint: You may find some of the identities on the front page useful. \\
\mbox{} \hfill (10+10+10+10+10)

\null
\item Solve the initial value problem
\[
  \frac{dy}{dt} = y \, \ln y, \qquad y(0) = 2.
\]
\hfill(10)


\item
\begin{enumerate}
\item Verify, by using the more common trigonometric identity $\sin^2
\theta + \cos^2 \theta = 1$, that
\[
  \cot^2 \theta = \csc^2 \theta - 1.
\]
\item Use this identity to show that
\[
  \int \frac{1}{x \sqrt{x^2+1}} \, dx
    = - \ln \left| \sqrt{1+\frac{1}{x^2}} + \frac1x \right| + C
\]
Hint: You may find some of the identities on the front page useful.
\end{enumerate}
\hfill(5+10)


\item Let $f(x)$ be a function defined on the interval $[a,b]$.  The
average value of $f$ on this interval is the real number $c$ with the
property that the area under the graph of $f$ equals the area under
the graph of $y=c$.
\begin{enumerate}
\item Draw a picture with illustrates the definition above.  Be sure
to carefully label the graphs and areas involved.  Then explain why
this definition implies that
\[
  c = \frac1{b-a} \int_a^b f(x) \, dx.
\]
\item The electric power $P(t)$ provided by an AC power supply is
given by
\[
  P(t) = P_0 \, \sin^2 (\omega t),
\]
where $P_0$ is the peak electric power, and $\omega$ is the so-called
phase velocity of the current.

Compute the average power.  (How should you choose the interval
$[a,b]$ to get a reasonable average?)
\end{enumerate}
\hfill (5+10)
\end{enumerate}



\end{document}
