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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Final Examination}
\centerline{December 15, 2004}
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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 $A = \{\cos(x) : x \in [0,3\pi/4]\}$ and let 
$B = \{x : 1 < x^2 < 4\}$.  
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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 $\varepsilon$, $N$ definition for 
convergence of a sequence.
\item{B)}  (20) 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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III.  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.
\item{A)} (10) Show that
there exists an index sequence of positive integers
$n_k$, $k \in {\bf N}$, such that $x_{n_k} = \sin(n_k)$
converges.   
\item{B)} (10) Does there
exist an index sequence $n_k$ of positive integers such that
{\it both} $x_{n_k}$ and $y_{n_k}$ converge?  Prove your assertion.
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IV.  
\item{A)}  (10) Give the $\varepsilon$, $\delta$ definition for 
the statement $\lim_{x\to c} f(x) = L$.
\item{B)}  (20)  Identify the limit 
$$L = \lim_{x\to 0} x^2 \cos(1/x)$$
and prove using the definition that $\lim_{x\to 0} x^2\cos(1/x) = L$.
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V. All parts of this problem refer to the function
$$f(x) = {32 x \over x^4 + 48}$$
\item{A)}  (5)  What are $f(0)$ and $f(2)$ for this function?
\item{B)}  (10) 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)}  (10)  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)}  (5)  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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VI.
\item{A)}  (20) In this part 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 $f(x) = x^2 + x - 1$ is integrable on $[a,b] = [0,3]$ by 
considering upper and lower sums for $f$ and determine the value of
$\int_0^3 x^2 + x - 1\, dx$.
\item{B)}  (20)  State and prove ``part 1'' of the Fundamental Theorem
of Calculus.
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VII.  True - False.   For each true statement, give a short proof
or reason.  For each false statement give a reason or a counterexample.
\item{A)} (10) 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)} (10)  The infinite series
$$\sum_{n=1}^\infty {(-1)^n \over n^3}$$
converges conditionally.
\item{C)} (10) 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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{\it Extra Credit} (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.  Find them.
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\centerline{\it Have a peaceful and joyous holiday season!}
\bye

