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\begin{document}
\title{Engineering and Science Mathematics I}
\author{Final Exam -- Standard Track}
\date{December 21, 2002}
\maketitle

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%% \noindent
%% (Choose a code word consisting of four letters or numbers under which
%% your grades will appear on the public grade list.)

%% \vspace*{\fill}
%% \noindent Useful identities:
%% \bigskip
%% \begin{gather*}
%% \cos^2 \theta = \frac{1+\cos 2 \theta}2 \\
%% \sin^2 \theta = \frac{1-\cos 2 \theta}2 \\
%% \sin \theta \, \cos \theta = \frac{\sin2 \theta}2
%% \end{gather*}
%% Taylor's Formula:
%% \begin{equation*}
%%   f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!} \, (x-a)^k
%% \end{equation*}

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\begin{enumerate}

\item Find the following limits. (Some of these limits may not exist!) 
\begin{enumerate}
\item $\displaystyle \lim_{x \to 1^-} \frac{|x-1|}{x-1}$
\item $\displaystyle \lim_{x \to 0} \sqrt x \, \ln x$
\item $\displaystyle \lim_{n \to \infty} 1 + \biggl( \frac32
\biggr)^n$ 
\item $\displaystyle \lim_{n \to \infty} \biggl( \frac{1+n}n
\biggr)^n$ 
\end{enumerate}
\hfill(5+5+5+5)

\item The radius of a circle is measured with a relative error of
$3$\%.  Give an estimate for the relative error in the computed area
of the circle.
\hfill (10)

\item Find the point on the line
\[
  \boldsymbol l = \begin{pmatrix} 2 \\ 0 \end{pmatrix}
  + \lambda \begin{pmatrix} -1 \\ 2 \end{pmatrix}
\]
that is closest to the origin.

Hint: Find the minimum of $f(\lambda) = |\boldsymbol l|^2$.
\hfill(10)

\item Let $f(x) = x^{1/3}$.

\begin{enumerate}
\item Compute the first three non-zero terms of the Taylor series of
$f$ about the point $a=1$. 
\item Find the equation for the line tangent to $f$ at the point
$x=1$.
\end{enumerate}
\hfill(10+10)

\item Consider the function
\[
  f(x) = \arctan x \,.
\]
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. % on the next page.
\hfill(20)

%\mbox{\kern -30pt\includegraphics[width=\textwidth]{Grid.eps}}

\item Compute the following integrals.
\begin{enumerate}
\item $\displaystyle \int \frac{\sec x}{\tan x} \, dx$
\item $\displaystyle \int x^2 \, \cos^2 x \, dx$
\item $\displaystyle \int \frac{x^2}{\sqrt{4-x^2}} \, dx$
\item $\displaystyle \int \frac{\sin x}{\sin 2x} \, dx$
\end{enumerate}
\hfill(10+10+10+10)

\item Compute the area of the surface generated by revolving the curve
\[
  y = \frac13 \, x^{3} \,, \qquad 0 \leq x \leq 1 \,,
\]
about the $x$-axis.
\hfill(10)

\item Determine if the following definite integrals are zero,
positive, or negative.  You do not need to evalute any of them, but a
short explanation of your reasoning is required!
\begin{enumerate}
\item $\displaystyle \int_{-1}^1 \bigl(
x + x^3 + x^5 + x^7 + x^9 + x^{11} + x^{13} \bigr) \, dx$
\item $\displaystyle \int_0^\pi \bigl(
\sin^{99}x - \sin x \bigr) \, dx$
\item $\displaystyle \int_{-5}^5 (e^x-1) \, dx$
\item $\displaystyle \int_0^{2 \pi} \bigl( \sin^{100} x - \cos^{100} x
\bigr) \, dx$
\end{enumerate}
\hfill(5+5+5+5)

\end{enumerate}
\end{document}

\item
\begin{enumerate}
\item Write down the definition of the derivative of a function $f(x)$
as the limit of a difference quotient.
\item Use this definition to compute the derivative of $f(x)=\sqrt x$
at $x=1$.

Note: You may use the power rule of differentiation \emph{only} to
check your answer!
\end{enumerate}



