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\begin{document}

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

{\bf Exam I}\quad {\sl Sept.\ 28, 1995}

\bigskip
There are 9 problems worth of total of 110 points.

\bigskip


\begin{enumerate}
\item \begin{enumerate}[a)]
  \item (5 pts) Define the positive integers, $\N$.
  \answerspace

  \item (5 pts) Define the rational numbers, $\Q$.
  \answerspace

  \item (5 pts) Using only the axioms for the real numbers, prove that
  $a\cdot 0 = 0$ for all $a\in\R$.
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace
\item \begin{enumerate}[a)]
  \item (5 pts) State the {\sc Principle of Mathematical Induction}.
  \answerspace

  \item (5 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 (5 pts) Define completely ${n\choose k}$.
  \answerspace

  \item (5 pts) State the {\sc Binomial Theorem}.
  \answerspace

  \item (5 pts) Find the coefficient of $x^{15}$ in $(x^3 + 2)^{20}$
  (give the prime factorization of this coefficient).
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace
\item \begin{enumerate}[a)]
  \item (2 pts) Define what it means for a set $S$ of real numbers to be
  bounded from below.
  \answerspace

  \item (3 pts) Define $\inf S$.
  \answerspace

  \item (5 pts) Let $S = \left\{ \frac{n}{n^2+1}\, |\, n\in \N\right\}$. 
  Prove that $\inf S = 0$.
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace
\item Give precise definitions of the following.
  \begin{enumerate}[a)]
  \item (5 pts) A step function $s$ on $[a,b]$.
  \answerspace

  \item (5 pts) $\int_a^b s(x) \,dx$ where $s$ is a step function on $[a,b]$.
  \answerspace

  \item (5 pts) The upper integral of $f$ on $[a,b]$, $\overline{I}(f)$.
  \answerspace

  \item (5 pts) An integrable function $f$ on $[a,b]$.
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace
\item (5 pts) Find an approximation for $\int_0^1 \frac{1}{1+x^3}\,dx$
that is accurate to within $\pm0.075$.
\answerspace

\problemspace
\item Give precise statements of the following theorems for integrals.
  \begin{enumerate}[a)]
  \item (5 pts) \textsc{ The Linearity Theorem}.
  \answerspace

  \item (5 pts) \textsc{ The Expansion/Contraction Theorem}.
  \answerspace

  \end{enumerate}

\problemspace
\item Evaluate the following integrals (justify your answers!).
  \begin{enumerate}[a)]
  \item (5 pts) $\int_{-1}^2 |3x^2 - 6x|\,dx$.
  \answerspace

  \item (5 pts) $\int_{1}^3 (x-2)^{17}\,dx$.
  \answerspace

  \item (5 pts) $\int_{-3}^3 {x^5\over \sqrt{2+x^4}}\,dx$.
  \answerspace

  \end{enumerate}

\problemspace
\item \begin{enumerate}[a)]
  \item (5 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 \emph{not} evaluate the integrals.
  \answerspace

  \item (5 pts) Find the average value of the function $f(x) = \sqrt{x}$
  on the interval $[0,a]$.
  \answerspace

  \end{enumerate}

\end{enumerate}

\end{document}
