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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Makeup Midterm Exam 3}
\centerline{May 10, 2011}
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{\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.  Both parts of this question refer to the function $f : {\bf R} \to {\bf R}$
defined by $f(x) = x^2 - 4x$.
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\item{A)}  (20)  Consider the regular partitions ${\cal P}_n$ of the
interval $[0,3]$ and show directly, using the upper and lower sums,
that $f$ is integrable on $[0,3]$.  
\item{B)}  (15)  Explain why the hypothesis of the Mean
Value Theorem is satisfied for $f$ on the interval $[1,5]$
and find the number $c$ mentioned in the conclusion.

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\noindent
II. (15) Let
$$f(x) = \cases{ x^a \sin(1/x) & if $x \ne 0$\cr
                       0 & if $x = 0$\cr}$$
 For which $a \ge 0$ does  $\lim_{x\to 0^+} f(x)$ exist?  
Is $f(x)$ differentiable at $x = 0$ when $a = 2/3$?  Explain.

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\noindent  
III.  (20)  Let $f$ be continuous on $[a,b]$.  Show that the
function $F(x) = \int_a^x f(t)\ dt$ is differentiable and $F'(x) = f(x)$.    

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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 $f(x) = \sin(4x) - 3\cos(2x)$.  There exists some $c \in (0,{\pi/4})$
such that $f(c) = -1$.
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\item{B)} (10)  $f(x) = {1\over x}$ is uniformly continuous on $(0,1)$.     
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\item{C)} (10) If $A \subset {\bf R}$ contains
no nonempty interval $(a,b)$, then $A$ is at most countably infinite.  
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\item{D)} (10) Let $f$ be continuous on $[a,b]$ and assume $f(x) < 0$ for
all $x \in [a,b]$.  Then there exists a constant $k < 0$ such that
$f(x) \le k < 0$ for all $x \in [a,b]$.   
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