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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Final Examination}
\centerline{May 9, 2014}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 200 possible 
regular points and 10 possible Extra Credit points.
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I.  Let $A = \{\tan(x) : x \in [0,\pi/4]\}$ and let 
$B = \{x : 1 < |x| < 3\}$.  
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\item{A)} (10) What is the set $A \cup B$?  
\item{B)} (10) What is the least upper bound of the set 
$C = \{|x - 2| : x \in A\}$?
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II.  
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\item{A)}  (10) State the $\varepsilon$, $n_0$ definition for 
convergence of a sequence.
\item{B)}  (20) Identify $L = \lim_{n\to\infty} x_n$ for
the sequence 
$$x_n = 1 + {(-1)^n\over \sqrt{n}}$$
and prove using the definition that $\lim_{n\to\infty} x_n = L$.
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\noindent
III. (10) Let $n \in {\bf N}$ be an integer, and let $x_n = \sin(n)$ 
and $y_n = \cos(n)$.  In this problem you may use without proof any 
facts about sequences, subsequences, etc. that we proved in class.  
Does there exist a strictly increasing index sequence $n_k$ of positive 
integers such that {\it both} subsequences $x_{n_k}$ and $y_{n_k}$ converge?  
Prove your assertion.
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IV.  
\item{A)}  (10) Give the $\varepsilon$, $\delta$ definition for 
the statement $\lim_{x\to c} f(x) = L$ where $c \in {\bf R}$.
How does this change if $c = +\infty$?
\item{B)}  (20)  Identify the limit $L = \lim_{x\to 1} x^2 - 4x + 2$
and prove using the definition that the value of $L$ is correct.  
\item{C)}  (10)  Identify the limit 
$$M = \lim_{x\to \infty} {1 \over 1 + {1\over x^2}}$$
and prove using the definition that $M$ is correct.
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V. 
\item{A)}  (20)  State and prove the Intermediate Value Theorem.
\item{B)}  (10)  For the rest of this question, let $f(x) = {32x \over x^4 + 48}$.
Show that for each $k$ with $0 < k < 1$, the equation $f(x) = k$ 
has {\it at least two\/} solutions $x \in {\bf R}$, with $x > 0$. 
\item{C)}  (10)  Show that there are {\it exactly two\/} solutions of 
the equation $f(x) = k$ from part B for each $k$ with $0 < k < 1$.
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VI.  (20) In this question you may use without proof the summation
formulas:
$$\sum_{i=1}^n 1 = n \quad \sum_{i=1}^n i = {n^2 + n \over 2} \quad
\sum_{i=1}^n i^2 = {2n^3 + 3n^2 + n \over 6}.$$ 
Show that 
$$f(x) = \cases{x + 1 & if $x < 1$\cr
                         -x & if $x \ge 1$\cr}$$
is integrable on $[a,b] = [0,2]$ by 
considering upper and lower sums for $f$ and determine the value of
$\int_0^2 f$.  (Caution:  $f$ is {\it not monotone} and not continuous 
on this interval, so you will need to be careful!)
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\noindent
VII.  True - False.  Do any 4 of the following parts.
For each true statement, give a short proof
or reason.  For each false statement give a reason or a counterexample.
(If you submit solutions for more than 4, they will be considered for
Extra Credit.)
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\item{A)} (10) Let $\sum_{n=1}^\infty a_n$ be an infinite series with
real terms.  If the partial sums $s_N$ are bounded, $|S_N| \le B$ 
for all $N$, then $\sum_{n=1}^\infty a_n$ converges.
\item{B)} (10) Let
$$f(x) = \cases{\cos(2x) & if $x < 0$\cr
                ax^2 + bx + c & if $x \ge 0$\cr}$$
There is exactly one set of constants $a,b,c$ 
for which $f'(0)$ and $f''(0)$ both exist.  
\item{C)} (10) The power series 
$$\sum_{n=1}^\infty {x^n\over \sqrt{n}\cdot 2^n}$$
converges absolutely for all $x \in [-2,2]$.  
\item{D)} (10)  If $f$ is 
differentiable on an interval $I = (a,b)$ and
$f'(x) \ne 0$ for all $x\in I$, then for each $k$ the equation 
$f(x) = k$ has {\it at most one\/} solution with $x \in I$.  
\item{E)} (10) (Note:  This question is suggested by a problem I assigned in my 
MATH 136 class.)  Let $n,k$ be positive integers.
The {\it centroid} of the region $R$ bounded by $y = x^n$ and $y = x^{n+k}$, 
and $0 \le x \le 1$ is the point in ${\bf R}^2$ with coordinates
$$(\overline{x},\overline{y}) = \left({(n+k+1)(n+1)\over(n+k+2)(n+2)}, 
{(n+k+1)(n+1)\over (2n+2k+1)(2n+1)}\right).$$
The centroid
satisfies $\overline{x}^{n+k} \le \overline{y} \le \overline{x}^n$ for each $k$ and $n$.   
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\centerline{\it Have a safe, enjoyable, and productive summer!}
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