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{\bf Math 165: Honors Calculus I}
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{\bf Quiz 4}\quad {\sl Sept. 29, 1994}

\begin{enumerate}
\problemspace
\item
  \begin{enumerate}[a)]
  \item State the least upper bound axiom (Axiom \#10).
  \answerspace

  \item Prove that if $A$ is a non-empty subset of $B$ and if $B$ has an
   upper bound, then $\sup A \le \sup B$.
  \answerspace

  \item Let $A = \{ 1/n\, | \, n \in \N\}$. Compute (or prove it
  doesn't exist): $\sup A$ and $\inf A$.
  \answerspace

  \end{enumerate}

\problemspace
\item Let $f$ be a bounded function on $[a,b]$.
  \begin{enumerate}[a)]
  \item Define the lower integral, $\underline{I}(f)$, and the upper integral,
  $\overline{I}(f)$.
  \answerspace

  \item State a condition on $\underline{I}(f)$ and $\overline{I}(f)$ that is
  equivalent to $f$ being integrable.
  \answerspace

  \item Define what it means for $f$ to be monotone on $[a,b]$.
 \answerspace
  \end{enumerate}

\problemspace
\item Let $f$ be increasing on $[a,b]$.
  \begin{enumerate}[a)]
  \item Describe how to approximate $\int_a^b f(x)\,dx$ by dividing $[a,b]$
  into $n$ subintervals of equal length.
  \answerspace

  \item If $f(x) = x + 1/x$ on $[1,2]$, how large must $n$ be in order that
  the approximation in part a) is within 0.001 of the actual value of
  $\int_1^2 f(x)\,dx$.
\end{enumerate}

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