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

\textbf{Exam I}\quad \emph{October 1, 1998}

\bigskip
There are 8 problems worth of total of 110 points.

\bigskip


\begin{enumerate}

\item (15 pts) Use the axioms for the real numbers\footnote{See the Appendix}
to prove the following statements. Be sure to justify each step.
  \begin{enumerate}[a)]
  \item $0\cdot a = 0$ for all $a\in\R$ \quad[Hint: consider $0 + 0$]
  \answerspace
  \item $(-1)(-1) = 1$ \quad[Hint: consider $1 + (-1)$ and use a)]
  \answerspace
  \item $1>0$ \quad[Hint: consider a proof by contradiction using b)]
  \answerspace
  \end{enumerate}

\problemspace
\item (15 pts)\begin{enumerate}[a)]
  \item Define an inductive set,~$S$.
  \answerspace
  \item Define the positive integers,~$\N$.
  \answerspace
  \item Prove by induction that for $n\in \N$,
    \[\sum_{k=1}^n k(k-1) = \frac{n^3-n}{3} \]
  \answerspace
  \answerspace
  \end{enumerate}

\problemspace
\item (15 pts)
  \begin{enumerate}[a)]
  \item Define completely ${n\choose k}$.
  \answerspace
  \item State the {\sc Binomial Theorem}.
  \answerspace
  \item Prove that for any positive integer~$n$,
  \[ \sum_{k=0}^n (-1)^k{n\choose k}2^{n-k} = 1 \]
  \answerspace
  \answerspace
  \end{enumerate}

\problemspace
\item (10 pts)
  \begin{enumerate}[a)]
  \item For a subset of $S\subset\R$, define $\sup S$ and $\inf S$.
  \answerspace
  \item Let $S = \bigg\{ \frac{n}{n+1}\, \big|\, n\in \N\bigg\}$. 
  Prove that $\sup S = 1$.
  \answerspace
  \answerspace
  \end{enumerate}

\problemspace
\item (15 pts) Give precise definitions of the following.
  \begin{enumerate}[a)]
  \item $\int_a^b s(x)\,dx$ where $s$ is a step function on~$[a,b]$.
  \answerspace

  \item The lower integral, $\underline{I}(f)$, of a bounded
  function $f$ on $[a,b]$.
  \answerspace

  \item An integrable function $f$ on $[a,b]$.
  \answerspace
  \end{enumerate}

\problemspace
\item (15 pts) Give precise statements of the following theorems for integrals.
  \begin{enumerate}[a)]
  \item \textsc{Linearity with Respect to the Integrand}.
  \answerspace

  \item \textsc{Additivity with Respect to the Interval of
  Integration}.
  \answerspace

  \item \textsc{Comparison Theorem}
  \answerspace

  \end{enumerate}

\problemspace
\item (15 pts) Evaluate the following integrals (justify your answers!).
  \begin{enumerate}[a)]
  \item $\int_{2}^3 (x-3)^{16}\,dx$.
  \answerspace

  \item $\int_{-1}^1 {x\over \sqrt{4-x^4}}\,dx$.
  \answerspace

  \item Find the area between the graphs of $f(x) = x^3$ and $g(x)
  = 3x^2-2x$ on the interval $[0,2]$.
  \answerspace
  \answerspace
  \end{enumerate}

\problemspace
\item (10 pts) Use step functions that are constant on subintervals of
equal length to compute a numerical approximation for
$\int_0^1 \frac{1}{1+x^2}\,dx$ that is accurate to within $\pm0.25$.
\answerspace

\end{enumerate}

\newpage
\textbf{Appendix: Axioms for the Real Numbers}

\bigskip
A set $\R$, called the set of real numbers, is assumed to exist
satisfying the ten axioms below.

\bigskip
\emph{The Field Axioms.}
Two operations on $\R$, \emph{addition} and \emph{multiplication}, are
assumed to be defined, so that for each pair, $x$, $y\in \R$, there is
a uniquely determined \emph{sum}, $x+y\in\R$, and a uniqely determined
\emph{product}, $x\cdot y\in\R$, satisfying the following axioms.
  \begin{itemize}
  \item \textsc{Axiom 1.} $x + y = y + x$, $xy = yx$
  \item \textsc{Axiom 2.} $x + (y + z) = (x + y) + z$, $x(yz) = (xy)z$
  \item \textsc{Axiom 3.} $x(y + z) = xy + xz$
  \item \textsc{Axiom 4.}  There exists distinct numbers $0$ and $1$ such
  that for every $x\in\R$, $x+0 = x$ and $1\cdot x = x$.
  \item \textsc{Axiom 5.}  For every $x\in\R$, there is a $y\in\R$ such that
  $x+y = 0$.
  \item \textsc{Axiom 6.}  For every $x\in\R$, $x\ne 0$, there is a
  $y\in\R$ such that $xy = 1$.
  \end{itemize}
\emph{The Order axioms} A subset $\R^+\subset \R$, called the positive
numbers, is assumed to exist satisfying the following axioms.
  \begin{itemize}
  \item \textsc{Axiom 7.} If $x$ and $y$ are in $\R^+$, so are $x+y$ and $xy$.
  \item \textsc{Axiom 8.} For every real $x\ne0$, either $x\in\R^+$ or
  $-x\in\R^+$, but not both.
  \item \textsc{Axiom 9.} $0\not\in\R^+$.
  \end{itemize}
\emph{The Completeness Axiom}
  \begin{itemize}
  \item \textsc{Axiom 10.}   For every non-empty subset $S\subset\R$
  that is bounded above there is a $B\in \R$ that is  the supremum of~$S$,
  $B = \sup S$.
  \end{itemize}

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