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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Information on Final Examination}
\centerline{May 1, 2014}
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\noindent
{\it General Information}
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\item{$\bullet$} The final examination for this class will be given during 
the scheduled period -- 8:00 to 10:30 am on Friday, May 9 
\item{$\bullet$}  The final will be a {\it comprehensive} exam.  It will
cover all the topics from the three midterms, {\it plus the material about
series from the past week.  About $10\%$ of the exam will be devoted to questions
about series.}  See the list of topics
below for more details.
\item{$\bullet$} The exam will be similar in format to 
the midterms (however, read the next bullet point carefully).  It will
be roughly 1.5 to 1.75 times as long as the midterms.  
In other words, it will be written to take about 1.5 hours = 
90 minutes if you work steadily.  But you will have the full 2.5 hour = 150 minute 
period to use if you need that much time.  
\item{$\bullet$}  Since there are many interconnections between topics
we studied earlier in the semester and those we studied more recently, 
a few questions may be posed in different ways and connect sections of
the course in ways you have not seen on the previous exams.  For instance,
I might ask you to evaluate a limit of a sequence by using properties
of functions like the natural logarithm (PS 9) and L'Hopital's Rule (PS 8).  
Similarly, you may need to use properties of integrals to understand whether
an infinite series converges by using the Integral Test. 
\item{$\bullet$}  If there is interest, I would be happy to arrange
an evening review session during exam week -- I think I'm free every evening.  
We can discuss this in class on May 5.
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\noindent
{\it Philosophical Comments and Suggestions on How to Prepare}
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\item{$\bullet$}  The reason we give final exams  
in almost all mathematics classes is to encourage students
to ``put whole courses together'' in their minds. 
Also, preparing for the final should help to 
make the ideas ``stick'' 
so you will have the material at your disposal to use in later courses.  
\item{$\bullet$}  If you approach preparing for a final exam in the right 
way it can be a {\it real learning experience\/} -- especially in a class 
like this one where almost everything we have done ``fits together'' in a 
very tight chain of logical reasoning starting with the Completeness 
Axiom for the 
real number system.  Much of what we did earlier in the semester 
may and should 
make much more sense now than it may have the first time around!
\item{$\bullet$}  {\it Start
reviewing now\/}, and  do some review each day between now and May 15 
(even just 1/2 hour each day will make a big difference).  That way you 
will not be ``crunched'' at the end (and with any luck the ideas we have 
developed in this course will ``stick'' better!)
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\noindent
{\it Topics To Be Included}
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\item{0)}  Sets, functions
\item{1)}  The real number system, rational and irrational numbers, 
the algebraic and order properties, least upper bounds (Axiom of Completeness)
\item{2)}  Mathematical induction
\item{3)}  Sequences, convergence ($\lim_{n\to\infty} x_n$ -- both the
$\varepsilon,n_0$ definition, and computing limits via the 
limit theorems).
\item{4)}  Subsequences, The Nested Interval Theorem, and the 
Bolzano-Weierstrass theorem
\item{5)}  Limits of functions (the $\varepsilon,\delta$ definition), 
the ``big theorem'' for function limits.  The order limit theorems and ``squeeze theorem'' (see Theorems 3.2.8, 3.2.9 in the text for the statements).   
\item{6)}  Continuity, the Extreme and Intermediate Value Theorems, uniform continuity.
\item{7)}  Definition and properties of the derivative, the Mean Value 
Theorem and its consequences
\item{8)}  The definite integral, integrability, the Fundamental Theorem of Calculus
\item{9)}  Infinite series -- convergence and divergence, key examples
such as geometric series, $\sum_{n=1}^\infty {1\over n^p}$-series, etc.  
Absolute vs. conditional convergence. Comparison, alternating 
series, and ratio tests for convergence.
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\noindent
{\it Proofs to Know}
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\noindent
You should be able to give precise statements of {\it all the definitions}
listed on the course homepage and the theorems mentioned in the 
outline above.  Also, be able to give proofs of the following:
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\item{1)} Every monotone increasing sequence of real numbers that is bounded
above converges.
\item{2)}  The Bolzano-Weierstrass Theorem
\item{3)} The Intermediate Value Theorem (the proof of the special case we did in class)
\item{4)} The Mean Value Theorem (including the special case known as ``Rolle's
Theorem;'' the general statement is deduced from that).   
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\noindent
{\it Suggested Review Problems}
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See review sheets for Midterm Exams 1, 2, and 3 for topics 1-8 in the list
above.  (Those review sheets are now reposted on the course homepage
if you need another copy of any of them.)  
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\noindent
{\it Practice Questions}
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\noindent
Comment: This was the final exam for my section of MATH 242 last spring.
That section's final was given at 3:00pm on the final day of the exam period
(i.e. the last possible time).   
Because students can get ``burned out'' in that situation with our finals 
schedule, I decided to take it easy and I would characterize this as a {\it very} 
straight-forward exam.  Even so, as I expected, it did not turn out to be too easy 
for last year's class.  {\it We don't have that schedule constraint this time and I
may decide to include at least one more challenging question.  So be prepared.}  
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\noindent
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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\noindent
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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\noindent
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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\noindent
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 $f(x_n) \to f(c)$).  
\vfill\eject
\noindent
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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\noindent
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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\noindent
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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\noindent
{\it Extra Credit}  (10)  Prove that if $f$ is continuous at $c$ and 
$x_n \to c$ is a sequence contained in the domain of $f$, then $f(x_n) \to f(c)$.
\bye
