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

{\bf Final Exam}\quad {\sl December\ 15, 1999}

\bigskip
There are 10 problems on 12 pages worth a total of 165 points.

\begin{enumerate}

\item (40 pts) Give complete definitions for the following concepts. Define
any symbols used.

  \begin{enumerate}

  \item The natural numbers $\N$

  \answerspace

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

  \answerspace

  \item The supremum, $\sup S$, of a subset of real numbers~$S\subset\R$.

  \answerspace

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

  \answerspace
  \answerspace

  \problemspace

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

  \answerspace

  \end{enumerate}


\problemspace

\item (40 pts) State the following theorems precisely. Define any new
symbols or concepts used.

  \begin{enumerate}

  \item \textsc{Binomial Theorem}

  \answerspace

  \item \textsc{The Linearity Property of Integrals}

  \answerspace

  \item \textsc{Sign-Preserving Property of Continuous Functions}

  \answerspace

  \item \textsc{Intermediate Value Theorem}

  \answerspace

  \problemspace

  \item \textsc{Extreme Value Theorem}

  \answerspace

  \item \textsc{Mean Value Theorem for Integrals}

  \answerspace

  \item \textsc{Mean Value Theorem for Derivatives}

  \answerspace

  \item \textsc{First Derivative Test for Maxima and Minima}

  \answerspace

  \end{enumerate}

\problemspace

\renewcommand{\answerspace}{\vfill}

\item (15 pts) Calculate the following.

  \begin{enumerate}

  \item $\int_0^{2} [x^2]\, dx$ where $[u]$ is the greatest integer
  ${}\le u$.

  \answerspace
 
  \item Find the area between the graphs of $\sin(x)$ and $\cos(x)$ on
  $[0,2\pi]$.

  \answerspace

  \item The average value of $f(x)=x(x-1)$ on the interval $[0,2]$.

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) Compute the following limits (do not use L'H\^opital's
Rule!) or prove they do not exist.

  \begin{enumerate}

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

  \answerspace

  \item $\lim_{x\to 0} \sin(x)\cos(1/x)$

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) 

  \begin{enumerate}

  \item Find the maximum and the minimum of $f(x) = \frac{x^2+1}{x^3+2}$
  on $[0,2]$.

  \answerspace

  \item  Find the line tangent to the ellipse $x^2 + 4y^2 = 4$ at the
  point $(\sqrt{3},-1/2)$.

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) Use the definition of the derivative to prove that
  \[ \frac{d}{dx}\sin(x) = \cos(x)\]
You may use trigonometric identities and the fact that
$\lim_{h\to 0}\frac{\sin(h)}{h} = 1$.

\problemspace

\item (10 pts)  Use the definition of the limit to prove that
$\lim_{x\to3} x^2 = 9$.

\answerspace

\problemspace

\item (10 pts) Use the \textsc{Mean Value Theorem} to prove that if
$f'(x) > 0$ on an interval $(a,b)$, then $f(x)$ is strictly increasing
on that interval.

\answerspace

\problemspace

\item (10 pts)

  \begin{enumerate}

  \item Show that if $f(x)$ is a polynomial of degree $n$, then $f'(x)$ is a
  polynomial of degree $n-1$.

  \answerspace

  \item Use Rolle's Theorem, induction, and part a) to prove that if $f(x)$
  is a polynomial of degree $n$ then $f(x)$ has at most $n$ distinct roots.

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) Let $f(x) = \frac{2}{5}x^5 - 3x^3 + 7x$.

  \begin{enumerate}

  \item Determine the intervals on which $f$ is increasing and decreasing.

  \answerspace

  \item Determine the relative maxima and minima 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}
