\magnification=\magstep1
\overfullrule=0pt
\def\glb{{\rm inf}}
\def\lub{{\rm sup}}
\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Supplementary Problems -- Exam 3}
\centerline{December 1, 2004}
\bigskip
\noindent
I.  
\item{A)} Give the formal $\epsilon$-$\delta$ definition of the 
statement $\lim_{x\to c} f(x) = L$.
\bigskip
\item{B)} Use the definition to show $\lim_{x \to 2} x^2 - 5x = -6$.
\bigskip
\noindent
II.  State and prove the Mean Value Theorem.
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\noindent
III.  Let 
$$f(x) = \cases{1 & if $x < 0$\cr
                1 + x + x^2 & if $x \ge 0$\cr}$$
\item{A)} Using the definition of $f'(0)$, determine whether
$f(x)$ is {\it differentiable} at $x = 0$.
\bigskip
\item{B)} Show using the definition that $f(x)$ is integrable on $[-1,1]$.
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\item{C)} {\it Without using the Fundamental Theorem of 
Calculus}, find the value of $\int_{-1}^1 f(x)\ dx$.
Potentially useful information:  $\sum_{i=1}^n i = {n(n+1)\over 2}$,
and $\sum_{i=1}^n i^2 = {n(n+1)(2n+1)\over 6}$.
\bigskip
\noindent
IV.  True-False.  For {\it any two} of the following, determine
whether the statement is true or false.  If it is true, give a short
proof.  If it is false give a reason or a counterexample.  
\bigskip
\item{A)} The function 
$$f(x) = \cases{{\sin(x)\over x} & if $x \ne 0$\cr
                       0 & if $x = 0$\cr}$$
is continuous at $x = 0$.
\bigskip
\item{B)} If $f(x)$ is a bounded function on $[a,b]$
with the intermediate value property
\footnote{${}^1$}{The ``intermediate value property'' is the statement that
if $m = \glb \{f(x) : x\in [a,b]\}$, $M = \lub \{f(x) : x\in [a,b]\}$,
then for all $y$ with $m \le y \le M$, there exists $x \in [a,b]$ such 
that $f(x) = y$.}, then $f(x)$ is continuous
on $[a,b]$.
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\item{C)}  If $f(x)$ is a function whose {\it second} derivative 
$f''(x)$ exists for all $x \in {\bf R}$ and there are
three distinct points $x_1,x_2,x_3$ such that 
$f(x_1) = f(x_2) = f(x_3)$, then there is some point $c$ where 
$f''(c) = 0$.
\bye