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\title{Math 2B, Section A, Winter 1997}
\author{Marcel Oliver}
\date{Final Exam}

\maketitle

\noindent Useful integrals:
\begin{gather*}
  \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*}
\bigskip

\noindent 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*}

\begin{enumerate}

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

\item
\begin{enumerate}
\item Find the volume $V$ of the unbounded solid obtained by revolving
around the $x$-axis the region under the curve $y=x \, e^{-x}$.

\item Find the volume $V$ of the unbounded solid obtained by revolving
around the $y$-axis the region under the curve $y= e^{-x^2}$.

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

\item Use integration to prove that the circumference of the unit
circle is $2\pi$.
\hfill (15)

\item The following questions all concern the $\arctan$ function, but
can be worked on independently.
\begin{enumerate}
\item Explain briefly why
\begin{equation*}
  \lim_{x \to \infty} \arctan x = \frac\pi2, \qquad
  \lim_{x \to -\infty} \arctan x = -\frac\pi2.
\end{equation*}

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

\item Sketch the graph of $y = \arctan x$. 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 $\displaystyle \int_{-\infty}^\infty \frac1{1+x^2} \,
dx$.
\end{enumerate}
\hfill(5+10+10+5)

\item The population of a remote island follows a natural growth law.
\begin{enumerate}

\item Write down an initial value problem which models the number of
births per year, denoted $B(t)$.

\item Solve this initial value problem.  You should find an equation
of the form
\begin{equation*}
  B(t) = B_0 \, e^{kt}.
\end{equation*}
Explain the meaning of $B_0$ and $k$.

\item Find an expression for the time $\tau$ it takes for the birth
rate to double.

\item Birth records show that $55$ people were born in the year $0$,
and $110$ people were born in the year $100$.

How many individuals ever lived on the island until the year $0$?
Assume that the birth rate forever followed the exponential growth
law.  (This may seem strange, but is the best possible assumption in
the absence of further information.)
\end{enumerate}
\hfill (5+5+5+10)

  
\end{enumerate}

\end{document}
