\documentclass[12pt]{article}
\usepackage{ccfonts,euler}
\usepackage{fullpage,amsmath,amsfonts,amssymb,graphicx,color}

\newcommand{\arccot}{\operatorname{arccot}}

\begin{document}

\title{Engineering and Science Mathematics 1A}
\author{Midterm Exam II}
\date{November 11, 2010}
\maketitle

% \vspace*{\fill}
% {\baselineskip=40pt
% \begin{tabbing}
% Student ID Number: \= \kill
% Last Name: \> \rule[-2pt]{10cm}{0.3pt} \\
% First Name:  \> \rule[-2pt]{10cm}{0.3pt} \\
% Signature: \> \rule[-2pt]{10cm}{0.3pt} \\
% %% Code: \> \fbox{\rule{0pt}{1.5em}\rule{1.5ex}{0pt}}
% %% \fbox{\rule{0pt}{1.5em}\rule{1.5ex}{0pt}}
% %% \fbox{\rule{0pt}{1.5em}\rule{1.5ex}{0pt}}
% %% \fbox{\rule{0pt}{1.5em}\rule{1.5ex}{0pt}}
% \end{tabbing}}

% \vspace*{\fill}
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*}

% \vspace*{\fill}

% \newpage

\begin{enumerate}

\item Find all points $(x,y)$ on the graph of $x^2-xy+y^2=1$ where the
tangent line is horizontal.
\hfill (10)
%\newpage
\item Compute the following indefinite integrals
\begin{enumerate}
\item $\displaystyle \int \sin x \, \sec x \, dx$
\item $\displaystyle \int x \, (\ln x)^3 \, dx$
\item $\displaystyle \int \frac{\cos x}{\sqrt{4-\sin^2x}} \, dx$
\item $\displaystyle \int x^3 \, e^{-x^2} \, dx$
\item $\displaystyle \int \frac{4x^2+x+1}{4x^3+x} \, dx$
\end{enumerate}
\hfill (10+10+10+10+10) 
%\newpage\null\newpage
\item Compute the average value of the function $f(x)=\sin 2x$ on the
interval $[0,\pi]$.
\hfill(5)
%\newpage
\item Does the improper integral
\[
  \int_1^\infty \frac1x \, dx
\] 
converge?  If so, compute its value.
\hfill(10)
%\newpage
\item Compute the area between the $x$-axis and the graph of 
\[
  f(x) = \frac1{9+x^2} \,.
\]
\hfill(10)
%\newpage
\item A policeman is positioned at point $P$ alongside a straight
stretch of road to catch speeding cars.  He employs the following
method:  When a car passes the point $S$ closest to him, he starts his
stopwatch.  He then looks through a spotting scope, mounted at an
angle $\theta$ toward the road.  When the car comes into view at point
$F$, he stops the clock.  The setting is shown in this figure:
\begin{center}
\input{road.pdf_t}
\end{center}
Suppose that the distance of the policeman to the road is $h$, that
the angular resolution of the spotting scope is $\Delta \theta$ (in
other words, $\Delta \theta$ is the expected measurement error of the
angle $\theta$), and that all other measurement errors are negligible.
\begin{enumerate}
\item Use linear approximation to derive a formula for the expected
  error $\Delta s$ in the computed traveled distance $s$.
\item Then derive a formula for  the expected error $\Delta v$ in the
  computed velocity $v$ of the car.
\item What angle $\theta$ should the policeman use to minimize
the error in the velocity?
\end{enumerate}
\emph{Hint:} In part (c) you should find that the optimal angle is
independent of the concrete numerical values of $h$, $v$, and $\Delta
\theta$.
\end{enumerate}
\hfill(5+5+5)
%\newpage\null
%\newpage
\end{document}
