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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Information on Final Examination}
\centerline{December 5, 2005}
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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 -- 2:30 to 5:30 pm on Wednesday, December 14.  
\item{$\bullet$}  The final will be a {\it comprehensive exam}, covering
all the topics from the three midterms, {\it and the material about
integration 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 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 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
\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
$\varepsilon$ - $N$ definition, and computing limits via the 
limit theorems).
\item{4)}  Subsequences, The Nested Interval Theorem, and the 
Bolzano-Weierstrass theorem
\item{5)}  Infinite series -- convergence and divergence, key examples
such as geometric series, $\sum_{k=1}^\infty {1\over k^\alpha}$-series, etc.  
Absolute vs. conditional convergence. Comparison, alternating 
series, and ratio tests for convergence.
\item{6)}  Limits of functions, the algebraic and order limit theorems,
``squeeze theorem,'' 
\item{7)}  Continuity, the Extreme and Intermediate Value Theorems
\item{8)}  Definition and properties of the derivative, the Mean Value 
Theorem and its consequences
\item{9)}  The definite integral, integrability, the Fundamental Theorem
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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
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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.)  
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\noindent
{\it Practice Questions}
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\noindent
I.  Let $A = \{\cos(x) : x \in [0,3\pi/4]\}$ and let 
$B = \{x : 1 < x^2 < 4\}$.  
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\item{A)}  What is the set $A \cup B$?  
\item{B)}  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)}  State the $\varepsilon$, $N$ definition for 
convergence of a sequence.
\item{B)}  Identify $L = \lim_{n\to\infty} x_n$ for
the sequence 
$$x_n = {3n^2 + n\over n^2 + 1}$$
and prove using the definition that $\lim_{n\to\infty} x_n = L$.
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\noindent
III.  Let $x_n = \sin(2*\pi\cos(n))$.  Show that there exists a convergent
subsequence of $(x_n)$.  (Don't try to find one explicitly!)
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IV.  
\item{A)}  Is $\sum_{k=2}^\infty {{-1}^n \over \ln(k)}$ absolutely
convergent, conditionally convergent, or divergent?
\item{B)}  Same question as in A for $\sum_{k=0}^\infty {(-1)^k k^3 3^k\over k!}$.
\item{C)}  For which $x \in {\bf R}$ does the series
$$\sum_{k=0}^\infty {x^k\over k^2}$$
converge?  
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\noindent
V.  
\item{A)}  Give the $\varepsilon$, $\delta$ definition for 
the statement $\lim_{x\to c} f(x) = L$.
\item{B)}  Identify the limit 
$$L = \lim_{x\to 0} x^{1/2} \sin(1/x)$$
and prove using the definition that $\lim_{x\to 0} x^{1/2} \sin(1/x) = L$.
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\noindent
VI. All parts of this problem refer to the function
$$f(x) = {32 x \over x^4 + 48}$$
\item{A)} What are $f(0)$ and $f(2)$ for this function?
\item{B)}  Show using the Intermediate Value Theorem
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)}   Show, using the Mean Value Theorem, that 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 $x \in I$.  (Hint:
Prove the contrapositive.)
\item{D)}  Show using part C 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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\noindent
VII.  In this question you may use without proof 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 $f(x) = x^2 + x - 1$ is integrable on $[a,b] = [0,3]$ by 
considering upper and lower sums for $f$ and determine the value of
$\int_0^3 x^2 + x - 1\, dx$.
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\noindent
VIII.  True - False.   For each true statement, give a short proof
or reason.  For each false statement give a reason or a counterexample.
\item{A)} Let $\sum_{n=1}^\infty a_n$ be an infinite series with 
positive terms.  If the partial sums $s_N$ are bounded
above by some $B$ for all $N$, then $\sum_{n=1}^\infty a_n$ converges.
\item{B)} If $f$ is differentiable on $[a,b]$
with $f'(a) > 0$, then there is an interval containing $a$ on 
which $f$ is increasing. 
\item{C)} The function 
$$f(x) = \cases{x & if $x$ is rational\cr
                -x^2 & if $x$ is irrational\cr}$$ 
is {\it continuous} at $x = 0$.  
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\noindent
IX.  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.  Find them.
\bye