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

\textbf{Quiz 4}\quad \textsl{Sept. 24, 1998}

\begin{enumerate}
\problemspace
\item
  \begin{enumerate}[a)]
  \item Define the supremum, $\sup A$, of a subset of real numbers $A\subset
  \R$.
  \answerspace

  \item Let $A = \bigg\{1-\frac{1}{n}\;\big|\; n \in \N\bigg\}$. Prove
  $\sup A = 1$.
  \answerspace

  \end{enumerate}

\problemspace
\item Let $f(x) = \frac{1}{x}$ for $x\in[1,2]$.
  Find step functions $s_n$ and $t_n$ on $[1,2]$:
  \begin{enumerate}
  \item that are constant on $n$ open subintervals of \emph{equal length},
  \item satisfy $s_n(x) \le f(x) \le t_n(x)$ for all $x\in[1,2]$,
  \item are the best approximation to $f(x)$ by step functions from
  below and above, respectively, subject to constraints (a) and (b).
  \item Calculate $\int_1^2s_n(x)\,dx$ and $\int_1^2 t_n(x)\,dx$, and
  conclude that
    \[ \sum_{k=1}^n \frac{1}{n+k} < \int_1^2 \frac{1}{x}\,dx <
    \sum_{k=0}^{n-1}\frac{1}{n+k}
    \]
  \end{enumerate}
  \answerspace

\newpage
\item Calculate $\int_0^2 |(x-1)^5|\, dx$ using only the basic properties
of integrals covered so far in the course.

\end{enumerate}

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