\documentclass[12pt]{article}

\usepackage{enumerate}

\setlength{\parindent}{0pt}
\setlength{\textheight}{9in}
\setlength{\voffset}{-.75in}

\everymath{\displaystyle}
\newcommand{\answerspace}{\vspace*{2in}}
\newcommand{\problemspace}{\vfil}
\newcommand{\N}{\mathbf{N}}

\begin{document}

{\bf Math 165: Honors Calculus I}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

{\bf Final Exam}\quad {\sl Dec.\ 15, 1998}

\bigskip
There are 12 pages of problems worth a total of 165 points which includes
15 bonus points.

\begin{enumerate}
\item (30 pts) Give complete definitions for the following concepts. Define
any symbols used.
  \begin{enumerate}[a)]
  \item The natural numbers $\N$
  \answerspace

  \item $\sum_{i=1}^n a_i$
  \answerspace

  \item A function $f$ is integrable on $[a,b]$
  \answerspace

\newpage
  \item $\lim_{x\to p} f(x) = A$
  \answerspace

  \item A function $f$ is differentiable at $p$
  \answerspace

  \item A function $f$ is convex on $[a,b]$.
  \answerspace

  \end{enumerate}


\newpage
\item (30 pts) State the following theorems precisely. Define any new
symbols or concepts used.
  \begin{enumerate}[a)]
  \item \textsc{Binomial Theorem}
  \answerspace

  \item \textsc{The Squeezing Principle}
  \answerspace

  \item \textsc{Boundedness Theorem}
  \answerspace

  \item \textsc{Chain Rule}
  \answerspace

  \item \textsc{Mean Value Theorem for Derivatives}
  \answerspace

  \item \textsc{Second Derivative Test for Extrema}
  \answerspace

  \end{enumerate}


\newpage
\renewcommand{\answerspace}{\vspace*{1.5in}}

\item (20 pts) Calculate the following.
  \begin{enumerate}[a)]
  \item $\int_0^{16} [\sqrt{x}]\, dx$ where $[u]$ is the greatest integer
  ${}\le u$.
  \answerspace
 
  \item $\int_0^2 | x^2 - 1 | \,dx$
  \answerspace

  \item The average value of $\sin(x)$ on the interval $[0,\pi]$.
  \answerspace

  \item $\int_{-1}^{1} \frac{\tan(x)}{x^2 + 1}\,dx$
  \answerspace

  \end{enumerate}


\newpage
\item (20 pts) Compute the following limits (do not use L'Hopit\^al's
Rule) or prove they do not exist.

  \begin{enumerate}[a)]
  \item $\lim_{x\to 0} \frac{\sin(x^2)+\sin^2(x)}{x^2}$
  \answerspace

  \item $\lim_{x\to 1^-} \frac{\sqrt{x^3-2x^2+x}}{x-1}$
  \answerspace

  \item $\lim_{x\to 0} (x^2-x)\cos\big(\frac{1}{x}\big)$
  \answerspace

  \item $\lim_{x\to a} \frac{(x^2+1)^{100} - (a^2+1)^{100}}{x-a}$
  \answerspace

  \end{enumerate}

\newpage
\item (15 pts) Compute the derivative, $f'(x)$, of the function $f(x)$.

  \begin{enumerate}[a)]
  \item $f(x) = x^3\cos(x)$
  \answerspace

  \item $f(x) = \frac{x^2+x+1}{x^2-x+1}$
  \answerspace

  \item $f(x) = \sqrt{1+\sqrt{1+\sqrt{x}}}$
  \answerspace

  \end{enumerate}

\problemspace
\item (5 pts) Find the equation of the line tangent to the curve defined by
\[ y^3 + x^2y^2 - x = 1 \]
at the point $(1,1)$.
\answerspace


\newpage
\item (5 pts) Consider the sequence $a_0 = 1$, $a_1 = 1$, $a_2 = 10$,
$a_3 = 28$, \ldots, defined recursively by
\[ a_{n+1} = 2a_n + 8a_{n-1},\quad n\ge1 \]
For example, $a_4 = 2 a_3 + 8 a_2 = 2(28) + 8(10) = 136$.
Use the \textsc{Principle of Mathematical Induction} to prove that 
\[ a_n = \frac{4^n+(-2)^n}{2},\quad n\ge1 \]
\answerspace

\newpage
\item (5 pts)  Use the \emph{definition of a limit} to prove that 
$\lim_{x\to1} \frac{1}{x} = 1$.
\answerspace


\problemspace
\item (5 pts) Prove that if a function $f$ is differentiable at $c$ and
has a relative maximum at $c$ then $f'(c) = 0$.
\answerspace

\newpage
\item (5 pts)
Let $n\ge3$ be a positive integer. Use \textsc{Bolzano's Theorem} and
\textsc{Rolle's Theorem} to prove that the function $f(x) = x^n - nx + 1$
has \emph{exactly one} root in the interval $(0,1)$.
\answerspace

\problemspace
\item (5 pts) Prove that if $f(x)$ is a strictly increasing function, then
it has an inverse, $f^{-1}(x)$ that is also strictly increasing.
\answerspace

\newpage
\item (5 pts) Prove that
$\sup \Big\{\  \frac{x-1}{x+1} \ | \ x \ge 0 \ \Big\} = 1$
\answerspace

\problemspace
\item (5 pts) Find the height~$x$ and circumference~$y$ of a right
circular cylinder of maximum volume that can be legally sent through
the US~mail ($x + y = 108$~in).
\answerspace

\newpage
\item (10 pts) Let $f(x) = \frac{3}{5}x^5 - 5x^3 + 12x - 3$.
  \begin{enumerate}[a)]
  \item Determine the intervals on which $f$ is increasing and decreasing.
  \answerspace

  \item Determine the relative extrema of $f$.
  \answerspace

  \item Determine the intervals on which $f$ is convex and concave.
  \answerspace

  \item Sketch the graph of $f$.
  \answerspace\answerspace

  \end{enumerate}

\end{enumerate}

\end{document}
