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\begin{document}

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

{\bf Exam I}\quad {\sl Oct.\ 13, 1994}

\bigskip
There are 9 problems worth 12 points each. Problem 10 is a bonus question
worth an additional 12 points.
\bigskip


\begin{enumerate}
\item \begin{enumerate}[a)]
  \item (4 pts) Define the positive integers, $\N$.
  \answerspace

  \item (4 pts) Define the rational numbers, $\Q$.
  \answerspace

  \item (4 pts) Assume $\sqrt{3}$ is irrational. Prove that  $(\sqrt{3}-1)^3$ is
   irrational.
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace
\item \begin{enumerate}[a)]
  \item (4 pts) Define an inductive set.
  \answerspace

  \item (4 pts) State the {\sc Principle of Mathematical Induction}.
  \answerspace

  \item (4 pts) Prove by induction that for
  $n\in \N$ \[(x-1)\sum_{k=0}^{n-1} x^k = x^n - 1\]
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace
\item \begin{enumerate}[a)]
  \item (4 pts) Define ${n\choose k}$.
  \answerspace

  \item (4 pts) State the {\sc Binomial Theorem}.
  \answerspace

  \item (4 pts) Find the coefficient of $x^{14}$ in $(x^2 + 3)^{20}$.
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace
\item \begin{enumerate}[a)]
  \item (4 pts) Define what it means for a set $S$ of real numbers to be
  bounded from above.
  \answerspace

  \item (4 pts) Define $\sup S$.
  \answerspace

  \item (4 pts) Let $S = \left\{ {n\over n+1}\, |\, n\in \N\right\}$. 
  Prove that $\sup S = 1$.
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace
\item Give a precise definition of the following.
  \begin{enumerate}[a)]
  \item (4 pts) A step function $s$ on $[a,b]$.
  \answerspace

  \item (4 pts) $\int_a^b s(x) \,dx$ where $s$ is a step function on $[a,b]$.
  \answerspace

  \item (4 pts) An integrable function $f$ on $[a,b]$.
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace
\item (12 pts) Let $$f(x) = \cases{0& if $x\in \Q$ and $0 \le x \le 1$\cr
1& if $x \notin \Q$ and $0 \le x \le 1$\cr}$$ 
  \begin{enumerate}[a)]
  \item (4 pts) Show that $\underline{I}(f) = 0$.
  \answerspace

  \item (4 pts) Show that $\overline{I}(f) = 1$.
  \answerspace

  \item (4 pts) Prove that $f$ is {\em not} integrable on $[0,1]$.
  \answerspace
  \end{enumerate}

\problemspace
\item Give precise statements of the following theorems for integrals.
  \begin{enumerate}[a)]
  \item (4 pts) {\sc The Linearity Theorem}.
  \answerspace

  \item (4 pts) {\sc The Comparison Theorem}.
  \answerspace

  \item (4 pts) {\sc The Expansion/Contraction Theorem}.
  \answerspace

  \end{enumerate}

\problemspace
\item Evaluate the following integrals (justify your answers!).
  \begin{enumerate}[a)]
  \item (4 pts) $\int_{-1}^2 (3x^2 - 6x + 1)\,dx$.
  \answerspace

  \item (4 pts) $\int_{1}^3 (x-2)^7\,dx$.
  \answerspace

  \item (4 pts) $\int_{-2}^2 {x^3\over \sqrt{1+x^2}}\,dx$.
  \answerspace

  \end{enumerate}

\problemspace
\item \begin{enumerate}[a)]
  \item (6 pts) Find an expression for the area between the graphs of $f(x) =
  x^2$ and $g(x) = 3x-2$ on the interval $[0,4]$. Write the answer as a sum of
  integrals without absolute values---do {\em not} evaluate the integrals.
  \answerspace

  \item (6 pts) Find the average value of the function $f(x) = \sqrt{x}$
  on the interval $[0,4]$.
  \answerspace

  \end{enumerate}

\problemspace
\item {\sc Bonus Problem}. (12 pts)  Let $$g(x) = \cases{0& if $x
= {1\over n}$ for some $n\in\N$\cr 1 & if $x\ne {1\over n}$ for
some $n\in\N$\cr}$$ Prove that $g$ is integrable on $[0,1]$ and
find the value of $\int_0^1 g(x)\,dx$.


\end{enumerate}

\end{document}
