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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Midterm Exam 3}
\centerline{May 2, 2014}
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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 (see question IV below).
Possibly useful information:
$$\sum_{i=1}^n = {n(n+1)\over 2}, \qquad \sum_{i=1}^n i^2 = {n(n+1)(2n+1)\over 6}.$$
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\noindent
I. (15)  Let
$$f(x) = \cases{ x^{4/5} \sin(1/x) & if $x \ne 0$\cr
                       0 & if $x = 0$.\cr}$$
Is $f$ continuous at $x = 0$?  Is $f$ differentiable at $x = 0$?  Give complete
reasons for your assertions.
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\noindent
II.  Both parts of this question refer to the function $f : {\bf R} \to {\bf R}$
defined by $f(x) = 1 - x^2$.
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\item{A)}  (20)  Consider the regular partitions ${\cal P}_n$ of the
interval $[1,3]$ and show directly, using the upper and lower sums,
that $f$ is integrable on $[1,3]$.  
\item{B)}  (15)  Explain why the hypothesis of the Mean
Value Theorem is satisfied for $f$ on the interval $[1,3]$
and find the number $c$ mentioned in the conclusion.
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\noindent  
III.  (20)  Prove that if $f$ is continuous on $[a,b]$,
then $f$ is integrable on $[a,b]$.  
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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; for false ones
give reasons or counterexamples.  Do any {\it three} parts.  If you submit
solutions for all four, then I will consider the other one for
Extra Credit.  
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\noindent
\item{A)} (10)  Let $f(x) = e^{2x} - e^{x}$.  There exists some $c \in (0,\ln(2))$
such that $f(c) = 1$.
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\item{B)} (10)  The function $f(x) = \arctan(x)$ is uniformly continuous
on the interval $(-1,1)$.   
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\item{C)} (10) There are continuous functions $f(x)$ on $[a,b]$ for which there exist
no differentiable function $F(x)$ on $[a,b]$ with $F'(x) = f(x)$.  
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\item{D)} (10) Let $f$ be differentiable on an open interval $I$ with $[a,b] \subset I$.
If $f'(a) > 0$ and $f'(b) < 0$, then there must exist some $c \in (a,b)$ where $f'(c) = 0$.     
\bye

