\documentclass[12pt]{article}

%\usepackage{times}

\usepackage{enumerate}
\usepackage{calculus}
\setlength{\parindent}{0pt}
\everymath{\displaystyle}
\newcommand{\answerspace}{\vfill}
\newcommand{\problemspace}{\newpage}

\begin{document}

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

{\bf Final Exam}\quad {\sl Dec.\ 19, 1994}

\bigskip
There are 15 problems worth from 5 to 25 points for a total of 170 points.
\bigskip


\begin{enumerate}
\item (25 pts) Give complete definitions for the following concepts.
  \begin{enumerate}[a)]
  \item A function $f$ is integrable on $[a,b]$.
  \answerspace

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

  \item A function $f$ is continuous at $p$.
  \answerspace

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

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

  \end{enumerate}

\problemspace
\item (25 pts) State the following theorems precisely.
  \begin{enumerate}[a)]
  \item \textsc{Principle of Mathematical Induction}.
  \answerspace

 \item \textsc{Binomial Theorem}.
  \answerspace

\item \textsc{Expansion or Contraction of the Interval of Integration}.
  \answerspace

  \item \textsc{Intermediate Value Theorem}.
  \answerspace

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

  \end{enumerate}

\problemspace
\item (20 pts) Calculate the following.
  \begin{enumerate}[a)]
  \item $\int_0^2 [x^2]\, dx$ where $[u]$ is the greatest integer
  ${}\le u$.
\answerspace
  
  \item $\int_1^x t^2 + (t-1)^{1/2}\, dt$, $x \ge 1$.
  \answerspace
  
  \item The average value of the function $f(x) = x^3$ on the interval
  $[-1,2]$.
  \answerspace
  
  \item $\int_{-1}^{1} \frac{x}{x^4 + 4}\,dx$
  \answerspace
  
  \end{enumerate}

\problemspace
\item (20 pts) Compute the following limits or prove they do not exist.
  \begin{enumerate}[a)]
  \item $\lim_{x\to 1}(x^2-x)\bigg(1 - \cos \frac{1}{x-1}\bigg) $.
  \answerspace

  \item $\lim_{x\to 2^-} \frac{\sqrt{x^3 - 4x^2 + 4x}}{x-2}$.
  \answerspace

  \item $\lim_{x\to 0} \frac{(1 + x^2)^n - (1 - x^2)^n}{x^2}$ where $n$
  is a fixed positive integer.
  \answerspace

  \item $\lim_{x\to 0} \frac{\sin(3x)}{\sin(x)}$.
  \answerspace

  \end{enumerate}

\problemspace
\item (20 pts) Compute the following derivatives.

  \begin{enumerate}[a)]
  \item $\frac{d}{dx}\left[x^2(1 - x)^{12}\right]$.
  \answerspace

  \item $\frac{d}{dx}\sqrt{1 + x^2}$
  \answerspace


  \item $\frac{d}{dx}\sin(\sin(x^3))$
  \answerspace

  \item $\frac{dy}{dx}$ where $y$ is defined implicitly by the equation
  $y^3 + x y + 1 = 0$.
  \answerspace

  \end{enumerate}

\problemspace
\item (5 pts) Find the largest interval containing $x = 1/2$ on which
the function $f(x) = (x^2 - 1)^2$ has an inverse. Give a formula for
the corresponding inverse function, $f^{-1}(x)$, as a function of $x$.
\answerspace
 
\problemspace
\item (5 pts) Find an expression for the area between the graphs of the
functions $f(x) = x^2 - 2x$ and $g(x) = 1 - x$ on the interval $[0,3]$.
Write the answer as a sum of integrals without absolute values---do not
evaluate the integrals.
\answerspace
  
\item (5 pts) Find the equation of the line tangent to the curve
defined by $y = x^4 - x^3 + x^2 - x + 1$ at the point $(1,1)$.
\answerspace


\problemspace
\item (5 pts) Consider the Fibonacci sequence, 1, 1, 2, 3, 5, 8, \dots,
where the next term is the sum of the previous two. Let $a_n$ be the
$n$th term of this sequence so that $a_1 = 1$, $a_2 = 1$, and $a_{n+2}
= a_{n+1} + a_n$, $n \ge 1$. Use induction to prove that 
\[
  a_n = \frac{1}{\sqrt{5}}\left[ \left(\frac{1+\sqrt{5}}{2}\right)^n -
                               \left(\frac{1-\sqrt{5}}{2}\right)^n \right]
\]
You may find the following useful:
\[
  \left(\frac{2}{1\pm\sqrt{5}}\right)^2 + \left(\frac{2}{1\pm\sqrt{5}}\right) = 1
\]
\answerspace

\problemspace
\item (5 pts) Prove that the equation $x\cos(x) = 1-x^2$ has a solution
in the interval $[0,\pi]$.
\answerspace


\item (5 pts)  Use the \emph{definition of a limit} to prove that 
$\lim_{x\to1} x^2 = 1$.
\answerspace


\problemspace
\item (5 pts) Let
\[
  f(x) = \cases{6\sqrt{x} - 5, & for $x < 1$\cr x^3, & for $x\ge 1$\cr}
\]
Prove that $f'(1) = 3$. You must use the definition of a derivative for this and
should examine one-sided limits.
\answerspace


\item (5 pts) If $a$ and $b$ are legs of a right triangle whose hypotenuse
is 1, find the largest value of $2a + b$.
\answerspace


\problemspace
\item (10 pts) Let $f(x) = 3x^4 + 4x^3 - 12 x^2$.
  \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}

\problemspace
\item (10 pts) State and prove the \textsc{Mean Value Theorem for
Derivatives}. You may use \textsc{Rolle's Theorem}.


\end{enumerate}

\end{document}

