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\begin{document}





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\begin{Large}

{\bf MATH 335: Real Analysis I -- FALL 2000}\\

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\vspace{5mm}







\bigskip \medskip



\end{center}



\begin{Large}

\centerline{\bf FINAL EXAM }

\end{Large}

\medskip



\medskip







\hspace{2cm}



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\medskip

This  Exam contains 7 problems on 9 sheets of paper, including

this front cover. Do all of your work on the paper provided.

\medskip



\hrule





\bigskip





\hspace{1cm}



\begin{large}

\begin{center}

\begin{tabular}{|c|c|c|}

\hline \hline Problem & Possible points & SCORE \\ \hline \hline

1 & 20 & \\ \hline 2 & 25 & \\ \hline 3 & 15 & \\ \hline 4 & 10 &

\\ \hline 5 & 10 & \\ \hline 6 & 10 & \\ \hline 7 & 10 & \\\hline \hline

TOTAL & 100 & \\

\hline \hline

\end{tabular}

\end{center}

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\newpage



\noindent{\bf Problem 1.}

\medskip

\noi Do four of the following five (5 points each).



\medskip



\noi (a) State the definition of the Cauchy sequence in a metric

space $(M, \rho)$. \vspace{4cm}



\noi (b) Let $f$ be a function defined  on a set $D\subset

\mathbb{R}$ and let $c$ be a point in $D$. Define what it means

for $f$ to be continuous at $c$. \vspace{4cm}



\noi (c) Formulate Rolle's Theorem.

\vspace{4cm}



\noi (d) Give the definition of a Riemann Sum for a bounded

function $f$ defined on $[a,b]$.

\vspace{4cm}



\noi (e) Let $\{f_n\}$ be a sequence of functions defined on the

set $E\subset \mathbb{R}$. Define what it means for $f_n$ to

converge {\bf uniformly} to a function $f$ defined on

$E$.

\vspace{4cm}



\newpage



\noindent{\bf Problem 2.}

\medskip

\noi Do five  of the following seven (5 points each). Explain your

answers. If necessary, provide a counterexample.



\medskip



\noi (a) True or false: if a sequence of uniformly continuous

functions on $[0,1]$ converges {\bf point-wise} to a function $f$,

then $f$ is uniformly continuous.

 \vspace{5cm}



\noi (b) True or false: Riemann integrable function on $[a,b]$ is

continuous on $[a,b]$.

 \vspace{5cm}



\noi (c) Find ${\displaystyle \lim_{x\to 1} \frac{e^{5 (x-1)} -

1}{x^7 -1}}$ \vspace{5cm}



\noi (d) True or false: for $f\in C[a,b]$, $\| f \|_{\infty} \le

1$ implies $|f(x)| \le 1 $ for all $x \in [a,b]$ \vspace{5cm}



\noi (e) True or false: if $f_n\to f$ uniformly on $[0,1]$ and

${\displaystyle \lim_{n\to \infty} \int_0^1 f_n(x) dx = 3}$ , then

$\int_0^1 f(x) dx = 3$. \vspace{5cm}



\noi (f) True or false: every Cauchy sequence in a metric space

converges. \vspace{5cm}



\noi (g) If $f\in C^{1}[1,2]$, $f(1)=2$ and $|f'(x)| \le 0.1$ for

every $x \in [1,2]$, what is the largest possible value for

$f(2)$? \vspace{5cm}



\newpage









\noindent

{\bf Problem 3.} (15 points)

\medskip

\noi State and prove an important theorem from this class.



\newpage





\noindent {\bf Problem 4.} (10 points)

\medskip



\noi Prove that $f(x)=\sqrt{x}$ is uniformly continuous on $[0,

\infty)$.





\newpage





\noindent {\bf Problem 5.} (10 points)



\noi For which $\alpha\ge 0$ an improper integral

$${\displaystyle

\int_0^\infty \frac{1}{(x+1)^\alpha} dx}

$$

exists.

\newpage



\noindent {\bf Problem 6.} (10 points)

\medskip



\noi Let $f$ belong to $ C^1[a,b]$. Assume that $f'(x)< 0$ for

every $x\in [a,b]$. Show that $f(x)$ is strictly decreasing on

$[a,b]$.



\newpage



\noindent {\bf Problem 7.} (10 points)

\medskip



\noi Find the second Taylor polynomial $T^{(2)} (x, x_0)$ for a

function $f(x)=\ln (1+x^2)$ at $x_0=1$.

\end{document}

