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\centerline{Mathematics 242, section 1 -- Principles of Analysis}
\centerline{Final Examination -- May 15, 2013}
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{\it Directions}
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Do all of the following problems I - VII in the blue exam booklet.  
There are 200 possible regular points and 10 Extra Credit points.  
{\it Even if you cannot see how
to answer some question completely, for possible partial credit, 
state a theorem or a result you think applies.}  You have until 5:30pm.
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I. (20) Let $A = \{x^2 - 2 : -1 < x < 2\}$ and $B = \{x : |x - 1| < 4\}$.
Find ${\rm lub} (A \cup B)$ and ${\rm glb}(A \cap B)$.
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II.
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\item{A)} (10) State the $\varepsilon$, $n_0$ definition of convergence for 
a sequence of real numbers.
\item{B)} (10) Identify $\lim_{n\to\infty} {5n^2+1\over n^2 + n + 4}$.
\item{C)} (10) Show that your result in part B is correct using the 
definition.
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III.  
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\item{A)} (15) Show that if $x_n$ is a monotone increasing sequence
of real numbers that is bounded above, then $x_n$ converges
to some real number.
\item{B)}  (10) True/False and prove/give a reason: The sequence
$$x_n = \cases{-1 + {1\over n^2} & if $n$ is a prime integer $> 1$\cr
                1 - {1\over n^2} & if $n$ is not prime\cr}
$$
has convergent subsequences.
\item{C)}  (10)  True/False and prove/give a reason:  
The infinite ``continued radical''
$$\sqrt{3 + \sqrt{3 + \sqrt{3 + \sqrt{3 + \sqrt{ 3 + \cdots }}}}}$$
represents a finite real number.  (Hint: If so, that number would be 
the limit of a sequence defined by $x_1 = \sqrt{3}$ and $x_n = \sqrt{3 + x_{n-1}}$
for all $n\ge 2$.)
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IV.  
\item{A)} (15) Let 
$$f(x) = \cases{ x + 3 & if $x$ is a rational number\cr
                -x^2 + 3 & if $x$ is an irrational number\cr}$$
Is $f$ continuous at $x = 0$?  Why or why not? 
\item{B)} (25) State and prove the Intermediate Value Theorem.  (You may assume
as known the theorem that if $f$ is continuous at $c$ and 
$x_n \to c$ is a sequence contained in the domain of $f$, then $f(x_n) \to f(c)$).  
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V.  
\item{A)} (15)  Using the limit definition of the derivative, compute $f'(c)$ for 
$f(x) = {1\over (x+3)^2}$ at a general $c \ne -3$.  
\item{B)} (10)  What theorem guarantees that 
$$F(x) = \int_1^x {1\over (t+3)^2}\ dt$$
is differentiable at $x = 2$?  Exactly why does it apply here?
What does it say about $F'(2)$?  
\item{C)} (10) Show that if $f(x) = {e^x + e^{-x}\over 2}$, then 
for every real $k$, there exists a solution $c$
of the equation $f'(c) = {e^c - e^{-c}\over 2} = k$.
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VI.  (20) In this question, you may use the summation
formulas:
$$\sum_{i=1}^n 1 = n \quad \sum_{i=1}^n i = {n(n+1) \over 2} \quad
\sum_{i=1}^n i^2 = {n(n+1)(2n+1) \over 6}.$$ 
Show that $f(x) = x^2 + x$ is integrable on $[a,b] = [0,2]$ by 
considering upper and lower sums for $f$ and determine the value of
$\int_0^2 x^2 + x\, dx$ by using a limit of the upper sums.  
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VII.  
\item{A)} (5) State the definition of convergence for an infinite
series $\sum_{n=1}^\infty a_n$. 
\item{B)} (15)  For which $x \in {\bf R}$ does the power series
$$\sum_{n=1}^\infty {3^n x^n\over n 7^n}$$
converge?  (Apply the Ratio Test and then test the series at the endpoints
of your interval separately.) 
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{\it Extra Credit}  (10)  Prove the result mentioned in part B of question 
IV above:  if $f$ is continuous at $c$ and 
$x_n \to c$ is a sequence contained in the domain of $f$, then $f(x_n) \to f(c)$.
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\centerline{\it Have an enjoyable and productive summer!}
\bye              
