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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Information on Final Examination}
\centerline{December 6, 2004}
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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 -- 8:30 to 11:30 am on Wednesday, December 15.  
\item{$\bullet$}  The final will be a {\it comprehensive exam}, covering
all the topics from the three midterms, {\it and the material about
infinite series from the past week.}  See the list of topics
below for more details.
\item{$\bullet$} The exam will be similar in format to 
the midterms but roughly twice as long -- it will written to take about 
2 hours if you work steadily, but you will have the full 3 hour period 
to use if you need that much time.  
\item{$\bullet$}  If there is interest, I would be happy to arrange
an evening review session during exam week (Monday would 
probably be the best day).  We can discuss this in class on December 6.
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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 December 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)}  Logic, sets, functions and relations
\item{1)}  The real number system, rational and irrational numbers, 
the algebraic and order properties, the least upper bound (completeness)
axiom.
\item{2)}  Mathematical induction
\item{3)}  Sequences, convergence ($\lim_{n\to\infty} x_n$ -- both
$\varepsilon$ - $N$ 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 limit theorems (sum, product, quotient,
``squeeze'' rules), one-sided limits
\item{6)}  Continuity, the Extreme and Intermediate Value Theorems
\item{7)}  Definition and properties of the derivative, the Mean Value 
Theorem and its consequences
\item{8)}  The definite integral, integrability, the Fundamental Theorem
\item{9)}  Infinite series -- convergence and divergence, key examples
such as geometric series, $p$-series, etc.  Absolute vs. conditional 
convergence. Comparison, integral, 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 Intermediate Value Theorem (proof we did in class
relying on Bolzano-Weierstrass theorem -- Note: this is {\it different}
from the proof given in the text in Section 22)
\item{3)} The Mean Value Theorem
\item{4)} The Fundamental Theorem of Calculus
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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.)  For topic 9:
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\item{} Section 32/4, 5, 7 
\item{} Section 33/3, 4, 5
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