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\begin{document}
\title{Mathematics 2B, Section G, Winter 1998}
\author{Marcel Oliver}
\date{Midterm Exam II}
\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}{\sqrt{1 - u^2}} = \arcsin u + C \\
  \int \frac{du}{1 + u^2} = \arctan u + C \\
  \int \frac{du}{u \sqrt{u^2 - 1}} = \operatorname{arcsec} |u| + C
\end{gather*}


\begin{enumerate}

\item Compute the following integrals.  Be sure to show all your work.
\begin{enumerate}
\item $\displaystyle \int \frac{1}{x^2-x+1} \, dx$
\item $\displaystyle \int x^{\tfrac32} \, \ln x \, dx$
\item $\displaystyle \int \frac{e^{2z}}{e^{2z}-1} \, dz$
\end{enumerate}
Hint: You may find some of the identities on the front page useful.
\hfill (10+10+10)



\item Consider the function
\[ f(x) = \ln \left(x + \frac1x \right). \]
Sketch the graph of $f$.  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.  \hfill (25)



\item Find
\begin{enumerate}
\item Volume 
\item Surface Area
\end{enumerate}
of the solid generated by revolving the graph of the equation $y =
\sin x$ from $x=0$ to $x = \pi$ around the $x$-axis.
\hfill (10+10)



\item The population $P$ of a country which permits a fixed number of
immigrants per year grows according to
\[
  \frac{dP}{dt} = k \, P + I, \qquad P(0) = P_0,
\]
where $k$ is a constant, and $I$ is the number of immigrants per year.

\begin{enumerate}
\item Solve the above initial value problem to find an expression for
the number of people $P(t)$.

\item Assume that a country doesn't allow immigration.  Derive a
formula for the time $T$ in which the population of that country
doubles.  (Your answer should give $T$ in terms of the growth constant
$k$.)

\item Explain mathematically why it doesn't make sense to compute this
``doubling time'' $T$ when $I>0$.
\end{enumerate}

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

\end{document}
