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\begin{document}


\hbox to \hsize{\bf Math 331, Fall 2020\hfil Final Exam}
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\centerline{\it Your work should be submitted before the official final exam time,}
\centerline{\it 1:30 PM, Thursday, December 10.}

\bigskip

{\narrower

\textbf{About the exam:} Taking the final exam is optional.  If you take
it, you will have six grades, counting your homework grade twice.  The
lowest of those six grades will be dropped.  Note that you cannot lower 
your grade for the course by taking the final exam; if your final exam grade
is your lowest grade, it will be dropped.

This is a takehome exam, but with a short Zoom meeting to discuss your responses.
As usual for this course, you can submit your work as an overleaf.com project or
through Canvas.  Your work is due by the officially scheduled final
exam period, 1:30~PM on Thursday, December~10.

You can make an appointment to meet with me during the final exam period through the
Canvas calendar, in the same way that you would for office hours.  The Zoom meeting
will use the same link as the one used for office hours, which can be found on the
Canvas page for this course.  If you cannot meet during the regular exam period for
some reason, you should email me to set up an alternative time.

The work that you submit for this exam should be your own.  You can use
course materials, including the textbook, your notes, videos from online lectures,
and posted solutions to homeworks.  You should not use other books or material from
the Internet.   You can ask me questions about the
exam, but you should not receive help on the exam from
other students, your friends and family, or anyone else.

For the problems and proofs on this exam,  you should present neatly written solutions.
Write out your answers carefully, including explanations to justify your work when appropriate.

For the essay questions on this exam, you should write out clear and well-organized responses
in complete sentences and paragraphs.  The essay questions are meant to give you an opportunity to
display your understanding of some central ideas from the course.
Please type your answers to essay questions in LaTeX or in a word processing application.
You can submit your essays as part of an overleaf.com project or through Canvas as separate 
documents.  If you are submitting your work through Canvas, you must save your essays as a PDF 
file or as a Microsoft Word document.

If you are confused about any of the problems on the test, or if you have other
questions, you can email me.  I will also have some office hour times available in the
Canvas calendar before the exam is due. While I won't give extensive help, I might be able
to give some hints.  

}

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\problem [{\it From textbook problem 1.1.14, 6 points\/}]\ \ Prove that $\sqrt2+\sqrt3$ is
irrational by filling in the details in this outline:  Suppose $\sqrt2+\sqrt3$ is rational.
Then $\sqrt2-\sqrt3$ is also rational.  (Hint: Consider the product of these two numbers.)
But then the sum of $\sqrt2+\sqrt3$ and $\sqrt2-\sqrt3$ is also rational, which leads
to a contradiction.


\problem [{\it 6 points\/}]\ \ Let $f(x) = 3x^2+2x+1$.  Use the definition of derivative
to show directly that for any $a\in\R$, $f'(a) = 6a+2$.


\problem [{\it 6 points\/}]\ \ \textbf{(a)} Let $g(x)=\frac12x|x|$. Show that $g'(x)=|x|$ for all $x$.
(You can use the fact that $g'(0)=0$, which was already shown on a homework problem.
You can also use the facts that for $x>0$, $g(x) = \frac12x^2$, and for $x<0$, $g(x)=-\frac12x^2$. You
do not have to use the definition of derivative here!)

\Item{}\indent \textbf{(b)} Use the result from part \textbf{(a)} and a single application
of the First Fundamental Theorem of Calculus to find the value of $\ds\int_{-5}^{10}|x|\,dx$.


\problem [{\it 6 points\/}]\ \ Let $f$ and $g$ be bounded functions on $[a,b]$. Suppose that 
$f(x)\le g(x)$ for all $x\in [a,b]$.  And let $P=\{x_0,x_1,\dots,x_n\}$ be a partition.
Show that $U(P,f)\le U(P,g)$.


\problem [{\it 6 points\/}]\ \ Find the interval of convergence of the following power series.
(That is, find the radius of convergence, and determine whether the series converges or
diverges at each endpoint.  The ratio test or the root test can be used to find the
radius of convergence.)

\medskip
\centerline{$\ds\sum_{n=1}^\infty\frac{(-1)^{n+1}x^n}{5n}$}


\problem [{\it 8 points\/}]\ \ Suppose that $f$ is a nondecreasing function on $[a,b]$.
That is, for all $x_1,x_2\in[a,b]$, if $x_1<x_2$ then $f(x_1)\le f(x_2)$.  It is then
true that the only kind of discontinuity that $f$ can have on $[a,b]$ is
jump discontinuities  This problem asks you to provide an essential piece of the proof of that
fact.
\ppart Let $c\in(a,b]$.  Show that the set $\{f(x)\,|\,a\le x<c\}$ is non-empty and bounded above
and therefore has a least upper bound.  [Hint:  $f(c)$ is an upper bound for this set.]
\ppart Let $\lambda=lub\{f(x)\,|\,a\le x<c\}$.  Show that $\ds\lim_{x\to c^-}f(x) = \lambda$.
[Hint:  This is similar to the proof of the Monotone Convergence
Theorem.  A picture will help!] 


\problem [{\it 8 points\/}]\ \ Let $f$ and $g$ be functions that are uniformly continuous on $\R$.
We already know that $f\cdot g$ is not necessarily uniformly continuous on $\R$:  Let
$f(x)=g(x)=x$.  Then $f$ and $g$ are uniformly continuous on $\R$, but $(f\cdot g)(x) = f(x)g(x) = x^2$,
which is not uniformly continuous on $\R$ by Example 2.6.1 in the textbook.

\Item{}\indent Now, suppose that $f$ and $g$ are uniformly continuous bounded functions on $\R$.  So, there
are constants $M_f>0$ amd $M_g>0$ such that $f(x)\le M_f$ and $g(x)\le M_g$ for all $x\in\R$.
Prove that $f\cdot g$ is uniformly continuous on $\R$.  [Hint:  You will need to consider
$|f(x)g(x)-f(y)g(y)|$.  Try rewriting that as $|f(x)g(x)-f(x)g(y)+f(x)g(y)-f(y)g(y)|$.]


\problem [{\it 8 points\/}]\ \ One of the following is possible and one is impossible.  Explain why
the impossible one is impossible.  For the one that is possible, give a
specific series that satisfies the stated condition (and show that your example works!). 

\smallskip
\leftline{\qquad\qquad \textbf{(a)} A power series $\ds\sum_{k=0}^\infty a_kx^k$ that converges at $x=3$ but not at $x=1$.}

\smallskip
\leftline{\qquad\qquad \textbf{(b)}  A power series $\ds\sum_{k=0}^\infty a_kx^k$ that converges at $x=1$ but not at $x=3$.}

\problem [{\it 8 points\/}]\ \ Let $\ds\sum_{k=1}^\infty a_k$ be a series.  Define
$\ds\sum_{k=1}^\infty p_k$ and $\ds\sum_{k=1}^\infty n_k$,
the series of positive and of negative terms of this series, as follows:

\medskip
\centerline{\qquad
$p_k=\begin{cases} a_k&\mbox{ if $a_k>0$}\\ 0&\mbox{ if $a_k\le0$} \end{cases}$\hskip 1 in
$n_k=\begin{cases} 0&\mbox{ if $a_k> 0$}\\ -a_k&\mbox{ if $a_k\le0$} \end{cases}$}

\medskip
\Item{}Since $\ds\sum_{k=1}^\infty p_k$ and $\ds\sum_{k=1}^\infty n_k$ are non-negative series,
they must either converge or diverge to $+\infty$.  Prove that if 
$\ds\sum_{k=1}^\infty a_k$ is conditionally convergent, then $\ds\sum_{k=1}^\infty p_k$ and 
$\ds\sum_{k=1}^\infty n_k$ both diverge to $+\infty$.  [Hint: Clearly, $a_k=p_k-n_k$, so we
also get identities like $n_k=p_k-a_k$ and $p_k = a_k + n_k$.  Depending on how you approach
this, you will probably need one at least one of these identities.  Also, consider $|a_k|$.]

\problem [{\it Textbook problem 4.5.7, 8 points\/}]\ \ \textbf{(a)} Let $\{f_n\}_{n=1}^\infty$
be a sequence of functions defined on $[a,b]$. Assume that each $f_n$ is bounded, that is
there are constants $M_n$ such that $|f_n(x)|\le M_n$ for all $x\in[a,b]$. Show: If 
$\{f_n\}_{n=1}^\infty$ converges uniformly to $f$, then $f$ must also be bounded.

\Item{}\indent\textbf{(b)} Show that the hypothesis of uniform convergence is necessary
by finding a sequence of bounded functions that converges pointwise to a function that
is not bounded.  (Drawing a picture might help.)

\problem [{\it 10 points\/}] This is an essay question; see the instructions for essay questions 
on the first page of the exam. To answer this question properly, you will need something
like a half-page essay single-spaced, or a full page double-spaced.

\Item{}\indent The relationship between derivatives and integrals is fundamental to Calculus.
Write an essay discussing the relationship.

\problem [{\it 20 points\/}] This is an essay question; see the instructions for essay questions
 on the first page of the exam.  To answer this question properly, you will need something
close to a full page essay single-spaced, or two pages double-spaced.

\Item{}\indent A central concept in the study of the real numbers is {\it completeness}.
One way of expressing the completeness of the real numbers is the least upper bound property.
Completeness distinguishes the real numbers from similar number systems such as the
rational numbers.  Write an essay discussing the concept of completeness, what it means,
how it was used in different parts of this course, and how it distinguishes the
real numbers from the rational numbers.



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