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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Midterm Exam 3}
\centerline{May 3, 2013}
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\noindent
{\it Directions}
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Do all work in the blue exam booklet.  There are 100 possible 
regular points and 10 possible Extra Credit points.
Possibly useful information:
$$\sum_{i=1}^n = {n(n+1)\over 2}, \qquad \sum_{i=1}^n i^2 = {n(n+1)(2n+1)\over 6}.$$
\noindent
I.  
\item{A)}  (25)  Show directly, using upper and lower sums,
that the function $f(x) = x^2$ is integrable on $[0,1]$ and determine
the value of $\int_0^1 x^2\ dx$.  
\item{B)}  (5) Compute $\int_0^1 x^2\ dx$ using the Fundamental Theorem
of Calculus to check your work.

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\noindent
II.   Let $f(x) = {\sin(x)\over x}$ for $x \ne 0$.
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\item{A)}  (5) How should a value $f(0)$ be defined to make $f$ continuous
at $x = 0$?  Explain.  (Note:  If you do not know how to decide, you may 
``buy'' the answer for 5 points so that you can do the next part.)
\item{B)}  (15) Using your value from part A for $f(0)$, use the limit
definition to show that the resulting 
function is differentiable at $0$ and find the value $f'(0)$.  Say what
method you are using to compute this limit.

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\noindent  
III.  
\item{A)}  (10)  State the Mean Value Theorem.   
\item{B)}  (10)  Determine
$${\rm lub} \left\{ {\ln(x)\over x} \mid x \in (e,+\infty)\right\}$$  
and justify your answer completely using facts we have developed in
this course.  

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\noindent
IV.  True-False.  Say whether each of the following statements is 
true or false.  For true statements, give short proofs or reasons; for false ones
give reasons or counterexamples.  
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\noindent
%\item{A)} (10)  Let $f(x) = {\ln(x) \over \ln(x) + 1}$.  
%There exists some $c \in (1,\infty)$ such that $f(c) = {997\over 1000}$.
%(Note: It is not necessary to find such a $c$ explicitly, but if you can, then
%that is one way to answer this!)
%\medskip
\item{A)} (10)  $f(x) = \ln(x)$ is uniformly continuous on $(1,+\infty)$.     
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\item{B)} (10) If $A \subset {\bf R}$ contains
no nonempty interval $(a,b)$, then $A$ is finite or countably infinite.  
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\item{C)} (10) Let $f$ be continuous on $[a,b]$ and assume $f(x) > 1$ for
all $x \in [a,b]$.  It is possible for ${\rm glb}\{f(x)\mid x \in [a,b]\}$ 
to equal $1$. 
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\noindent
{\it Extra Credit} (10)  Show that if $f$ is differentiable on an open 
interval $I$ and $[a,b] \subset I$ where $f'(a) < 0$ and $f'(b) > 0$,
then there must be some $c \in (a,b)$ where $f'(c) = 0$.  
\bye

