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

\begin{document}

\title{Engineering and Science Mathematics 1A}
\author{Midterm Exam I}
\date{October 5, 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}

% \newpage

\def\N{{\mathbb N}}
\def\R{{\mathbb R}}
\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\i{{\mathrm i}}

\begin{enumerate}

\item Compute the following limits provided they exist.
\begin{enumerate}
\item $\displaystyle \lim_{a \to 1}
\frac{\lvert a-1 \rvert}{a-1}$
\item $\displaystyle \lim_{a\to 1} 
\frac{a^3-3a^2+3a-1}{a^2-2a+1}$
\item $\displaystyle \lim_{a\to 0} 
\sin a \, \cos \frac1a$
\end{enumerate}
\hfill (5+5+5) 

\item The function 
\[
  \phi(x) = x \, \ln \lvert x \rvert
\]
is not defined at $x=0$.  

\begin{enumerate}
\item Can you assign a value to $\phi(0)$ in such
a way to make $\phi$ continuous?

\item Can you assign a value to $\phi'(0)$ in such a way to make
$\phi'$ continuous?
\end{enumerate}
\hfill (5+5)

\item Compute the derivative of $g(x)=\sqrt x$ by
\begin{enumerate}
\item using the definition of the derivative as the limit of a
difference quotient,
\item using the power rule of differentiation.
\end{enumerate}
Confirm that the two results coincide.
\hfill(5+10)

\item Consider the function
\[
  f(x) = x \, \e^{-x} \,.
\]
Find horizontal and vertical asymptotes, local minima, local maxima,
and inflection points of $f$.  Identify the regions where the graph of
$f$ is concave upward or concave downward.  Finally, sketch the graph
into the coordinate system provided.
\hfill(10)

\begin{center}\includegraphics{Grid.pdf}
\end{center}

\item A commuter train carries $400$ passengers each day at a ticket
price of \euro{}\,$1.00$ per person.  Market research reveals that for
each $5$ cent increase/decrease of the price $50$ fewer/more people
would take the train.  What fare should be charged to maximize
revenue?  
\hfill (10)
\end{enumerate}

\end{document}
