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\begin{document}
\title{Engineering and Science Mathematics I \\ Standard Track}
\author{Midterm II, Version A}
\date{November 11, 2002}
\maketitle

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\begin{enumerate}

\item Consider the function
\[
  f(x) = \frac{\ln x}{x^3} \,.
\]
Find the domain, intercepts, horizontal and vertical asymptotes,
extrema and inflection points of $f$.  Identify all minima and maxima
of $f$, as well regions where the graph is concave upward or concave
downward.  Finally sketch the graph into the coordinate system
provided.
\hfill(20)

\item Compute the following indefinite integrals.
\begin{enumerate}
\item $\displaystyle \int \frac1{(1+x) \, \sqrt x} \, dx$
\item $\displaystyle \int \frac{x^4}{x^2-2} \, dx$
\item $\displaystyle \int \frac{\tan x}{\ln \cos x} \, dx$
\item $\displaystyle \int \frac{x^2}{\sqrt{4-x^2}} \, dx$
\item $\displaystyle \int \sin \sqrt x \, dx$
\end{enumerate}
\hfill(6+6+6+6+6)

\item Compute the following definite integrals.
\begin{enumerate}
\item $\displaystyle \int_{-178}^{178} \,
\sin x \, \ln \Bigl(\frac{2+\cos x}{2-\cos x} \Bigr) \, dx$
\item $\displaystyle \int_1^e x^4 \, (\ln x)^3 \, d x$
\end{enumerate}
\hfill(6+6)

\item Evaluate the following improper integrals.  Careful: some of the
integrals may not converge.
\begin{enumerate}
\item $\displaystyle \int_0^2 \frac1{1-x} \, dx$
\item $\displaystyle \int_1^2 \frac1{\sqrt{x-1}} \, dx$
\end{enumerate}
\hfill(6+6)

\item Find all values for $b$ for which the integral
\[
\int_{-\infty}^\infty
  b \, r \, e^{br^2} \, d r
\]
converges.  What is the value of the integral when it converges?
\hfill(6)

\item Find the volume of the ellipsoid that is obtained by revolving
the ellipse
\[
  \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
about the $y$-axis.
\hfill (10)
\end{enumerate}

\end{document}
