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

\textbf{Exam I}\quad \emph{September 30, 1999}

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

\bigskip


\begin{enumerate}

\item (15 pts) Justify your statements using the axioms for the real
numbers (see the Appendix) or previous parts of the problem.
Let $a\in\R$ be an arbitrary real number.

  \begin{enumerate}[a)]

  \item Define $-a$.

  \answerspace

  \item 
  Prove that $0\cdot a = 0$.\qquad [Hint: $0 + 0 = 0$]

  \answerspace
  \answerspace

  \item Prove that $(-1)\cdot a = -a$.\qquad [Hint: $1 + (-1) = 0$]

  \answerspace
  \answerspace

  \end{enumerate}

\problemspace

\item (15 pts)

  \begin{enumerate}[a)]

  \item Define the positive integers,~$\N$.

  \answerspace

  \item Prove that there is no upper bound for~$\N$.

  \answerspace
  \answerspace

  \item Let $a > 0$ and let $T = \Big\{a\Big(1 +
  \frac{1}{n}\Big)\,|\,n\in\N\Big\}$.  Prove that $ a = \inf(T)$.
  
  \answerspace
  \answerspace

  \end{enumerate}

\problemspace

\item (15 pts)

  \begin{enumerate}[a)]

  \item Define $\sum_{k=1}^n a_k$.

  \answerspace

  \item Use induction to prove that for any positive integer~$n$,
  $\sum_{k=1}^n (k-1)^3 < \frac{n^4}{4}$.

  \answerspace
  \answerspace

  \item Use telescoping sums to calculate $\sum_{k=1}^n \frac{2k-1}{2^kk!}$.

  \answerspace
  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) Find the coefficient of $x^{81}$ in $(x^3 - 7)^{30}$
(give its prime factorization).

\problemspace
\item (20 pts) Give precise definitions of the following.

  \begin{enumerate}[a)]

  \item A step function $s$ on $[a,b]$.

  \answerspace

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

  \answerspace

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

  \answerspace

  \item $\int_a^b f(x)\,dx$ where $f$ is a bounded function 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 (10 pts) Evaluate $\int_{-2}^2 3 - |x^2-1| \,dx$.

\problemspace

\item (10 pts) Let $b>0$. Show that $\int_0^b x\,dx = \frac{1}{2}b^2$ by
considering step functions constant on subintervals of equal length.\newline
[Hint: You may find the formula $\sum_{k=1}^n k = \frac{n(n+1)}{2}$ useful.]
\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 uniquely 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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