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

\def\R{\mathbb{R}}

\begin{document}

\title{Engineering and Science Mathematics I}
\author{Advanced Placement Exam}
\date{September 8, 2010}
\maketitle



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

\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} \\
     & e^{i\theta} = \cos \theta + i \, \sin \theta
\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 Compute 
\begin{enumerate}
\item $\displaystyle \lim_{n \to \infty} 1 + \frac1n + \frac{1}{n^2}$
\item $\displaystyle \lim_{a \to -1} \frac{a^2 + 4a + 3}{a^2 - 2a -
3}$
\item $\displaystyle \lim_{\theta \to 0} \frac{\theta - \sin
\theta}{\theta^3}$
\end{enumerate}
\hfill(5+5+5)

\item Consider the function
\[
  f(x) = \begin{cases}
           0 & \text{for } x \leq 0 \\
           e^{-1/x} & \text{for } x>0 \,.
         \end{cases}
\]
\begin{enumerate}
\item Is $f$ continuous at $x=0$?  Explain.
\item Is $f$ differentiable at $x=0$?  If so, find $f'(0)$.
\end{enumerate}
\hfill(5+5)

\item Consider the graphs of the functions
\[ 
  f(x) = m\, x
\]
and
\[
  g(x) = \arctan x \,.
\]
For which values of $m$ do they intersect in (a) one, (b) two, and (c)
three points? 
\mbox{\qquad} \hfill (10)

\item Show that the square is the rectangle of maximal area for a given
perimeter.
\hfill (10)

\item For each $a>0$, consider the function
\[
  f(p) = p \, a^{1/p} \,.
\]
Find all minima and maxima, points of inflection, and vertical and
horizontal asymptotes on the interval $p \in (0,\infty)$.  
\hfill (10)

\item Find the indefinite integral 
\[
  \int \cos^2 \theta \, d \theta
\]
by
\begin{enumerate}
\item using trigonometric identities, and
\item writing the expression in terms of complex exponentials.  You
should
\item verify that the results from (a) and (b) are compatible.
\end{enumerate}
\hfill(5+5+5)

\item Find the indefinite integrals
\begin{enumerate}
\item $\displaystyle \int \frac1{1+e^x} \, dx$
\item $\displaystyle \int \frac{4x-2}{x^3-x} \, dx$
\end{enumerate}
\hfill(5+5)

\item Compute the Taylor series about the point $x=0$ of
\begin{enumerate}
\item $f(x) = (x+2)(x-2)$,
\item $g(x) = \ln(1+x)$,
\item $h(x) = \ln(x+1)+ \ln (1/(x+1))$.
\end{enumerate}
\hfill(5+5+5)

\item
\begin{enumerate}
\item Does the infinite series
\[
  \sum_{n=0}^\infty \frac{n}{n^2+1}
\]
converge?
\item Show that
\[
  \sum_{n=0}^\infty \biggl( \frac{\sin n}2 \biggr)^n
  \leq 2
\]
\end{enumerate}
\hfill(5+5)

\item \emph{Note:} In parts (a) and (b) of the following problem you
may leave logarithms and exponential functions unevaluated.

The concentration $c(t)$ of a chemical catalyst decays at a rate
$\lambda \, c(t)$, where $\lambda = 1 \, \mathrm{s^{-1}}$.
\begin{enumerate}
\item If the initial concentration of the catalyst is
$c(0)=10\,\mathrm{g}/\mathrm{l}$, what is the concentration
after $2\,\mathrm{s}$?
\item How long does it take until half of the catalyst has decayed?
\item Suppose the catalyst is added to a reactant with concentration
$r(t)$ which undergoes a chemical reaction at rate $\nu \, c(t) \,
r(t)$, where $\nu$ is some given constant.  Show that the fraction of
the reactant which remains after all chemical activity has subsided is
given by the expression $e^{-\nu c(0)/\lambda}$.


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


\end{enumerate}


\end{document}
