Exam 2

Exam rules

The exam is modeled on https://mathinthetimeofcorona.wordpress.com/2020/06/11/june-11-day-95-finals-part-ii/.

You can use all available resources to prepare, including but not limited to: each other, books, notes, the internet. During the exam, however, please use only a single 8.5x11 sheet of notes. Your true/false questions for question 1 don’t count toward the sheet of notes (you can have a sheet of notes in addition to your true/false questions).

You will have 20 minutes to present two questions. The first question will be your choice and the second will be mine. You may reject my chosen question for a four point reduction, twice.

At the end of the exam I will ask you for a self-assessment of your performance.

During the exam we will talk in a conversational style. As part of this I will ask you to turn on your microphone and camera.

Please be prepared to present your work, including responding to questions. The following formats are suggested:

Writing on a tablet screen using a stylus
Writing on paper with a document camera
Writing on a chalkboard or whiteboard

Prepared slides may be part of your presentation, but they are not well suited for responding to questions where your response might involve writing. So even if you include slides, please be prepared to also use one of the above formats. If it will not be possible for you to use any of the above, please let me know in advance.

Questions

Of the written homework you have submitted this term, identify the problem that you found most challenging, critique your written submission, and present an improved solution. (Have a copy of the original submission to share and discuss.)
Consider one mathematical idea from the course that you have found beautiful, and explain why it is beautiful to you. Your answer should explain the idea in a way that could be understood by a peer student who is familiar with introductory real analysis but has not yet taken this class.
One of the main subjects in real analysis is the topic of infinite series. How do infinite series, along with the concepts of convergence and divergence, relate to other human experiences outside of mathematics classes? Illustrate with examples.
The Thomae function \(f\) defined by \(f(x)=0\) if \(x\) is irrational, \(f(a/b) = 1/b\) if \(x=a/b\) is rational, in lowest terms, has the property that the set of discontinuity points of \(f\) is exactly \(\mathbb{Q}\), the rational numbers. The function \(g\) defined by \(g(x)=1\) if \(x\) is irrational, \(g(x)=0\) if \(x\) is rational, has the property that the set of zeros of \(g\) is exactly \(\mathbb{Q}\). But the zeros of \(f\) are the irrational numbers, not \(\mathbb{Q}\); and the discontinuity points of \(g\) are all real numbers, not just \(\mathbb{Q}\).

Is it possible to have a function \(h\) with both properties, that the set of discontinuity points of \(h\) is \(\mathbb{Q}\), and the set of zeros of \(h\) is also \(\mathbb{Q}\)? Why or why not?
TBB 13.4.3 and 13.4.4: In these exercises, \(\ell_2\) is the space of sequences \(x = \{ x_1,x_2,x_3,\dotsc \}\) such that \(\sum x_i^2\) converges. It is a metric space with distance function \[ d(x,y) = \sqrt{ \sum (x_i-y_i)^2 } . \]
1. In order for a sequence of points \(x^{(n)} = \{x_1^{(n)},x_2^{(n)},x_3^{(n)},\dotsc\}\) \(n=1,2,3,\dotsc\) in \(\ell_2\) to converge to a point \(y = \{y_1,y_2,y_3,\dotsc\}\) is it necessary that each \(x_k^{(n)}\) converges to \(y_k\) as \(n \to \infty\) for \(k=1,2,3,\dotsc\)? Is it sufficient?
2. The Hilbert cube is the subspace \(H \subset \ell_2\) defined by \[ H = \{ \, \{x_1,x_2,\dotsc\} \in \ell_2 \colon |x_i| \leq 1/i\} . \] Show that a sequence of points \[ x^{(n)} = \{ x_1^{(n)},x_2^{(n)},x_3^{(n)},\dotsc \} \] in \(H\) converges if and only if each \(x_k^{(n)}\) converges for \(k=1,2,3,\dotsc\).
TBB 13.9.8: The space \(C[0,1]\) consists of the continuous functions \([0,1] \to \mathbb{R}\). It is a metric space, with distance function \[ d(f,g) = \max\{|f(t)-g(t)| \colon 0 \leq t \leq 1 \} . \]

Define a function \(A : C[0,1] \to C[0,1]\) by \[ (A(f))(x) = g(x) = \int_0^x f(t) \, dt, \quad 0 \leq x \leq 1. \]
1. Is \(A\) a contraction?
2. Is \(A^2\) a contraction?
3. Does \(A\) have a unique fixed point?

Grading

Questions will be graded out of 50 points (each) using the following rubric (copied from the web page linked above).

A	50	Well-executed, well-communicated, essentially correct. Minor errors quickly corrected. No nontrivial errors.
B	43	Generally well-executed. Some minor errors not recognized or corrected. Nontrivial errors corrected when identified.
C	38	Adequately executed but numerous or repeated errors (minor and nontrivial) without satisfactory resolution.
D	33	Flawed execution with nontrivial errors.
F	25	Unsatisfactory execution with fundamental errors and deep misunderstandings.
0	0	Very little of relevance. No evidence of preparation.

Scheduling

You will schedule an online, one-on-one meeting with me for the exam. Meetings will be scheduled for half-hour blocks either from :00 to :30, or :30 to :00 (ie., aligned with hours). Exams are expected to take 20 minutes, not 30; the extra time is for safety, in case of delay or technical difficulties.

Available meeting times are:

Monday, May 3	11:00am-2:00pm
Tuesday, May 4	2:00pm-4:00pm
Wednesday, May 5	3:00pm-5:00pm
Thursday, May 6	9:30am-11:30am
Thursday, May 6	1:00pm-4:00pm
Friday, May 7	1:00pm-4:00pm

If other days/times are needed, please contact me.

Time limit

Due to the number of enrolled students, it is not possible to extend exam times. It will be very important for you to have your answers prepared and rehearsed; thinking through an answer on the spot is not recommended.

There will be about 10 minutes for each question. As a guideline, plan to spend about 5 minutes giving your prepared answer, then 5 minutes on discussion and follow-up questions. Please don’t prepare answers that will take much longer than 5 minutes. For a discussion/reflection question, this means preparing a short, focused answer, with just a few carefully selected examples. For a proof question, this might mean preparing a summary or overview of the key steps of the proof, not trying to present every single detailed step.

This can be challenging (it’s hard enough trying to figure out a tricky proof, even harder to present it, and then much harder with a strict time limit on top of that). For what it’s worth, focused and to-the-point communication is an important skill. Some tips include:

Work together with other students to plan and practice your answers.
Decide in advance which question you will choose to answer, and which questions you will reject if I choose them. (Don’t use up your time deciding this during the exam!)
Personally, I enjoy clear, well-prepared, well-executed answers, even if they are for “easy” questions. (But if you want to answer a challenging question, that’s fine too!)
Think about what follow-up questions might be asked. Some possibilities include: asking you to expand on a detail of your prepared answer; a “what-if” question asking you for a different version of the main question (e.g., “what if \(f\) has a jump discontinuity at \(x\)”); asking how you would teach the answer you just gave. As part of your preparation, think about how you might handle these or other questions.
Don’t write down or memorize answers to every possible follow-up question. It’s okay to think about the follow-up questions during the exam. In fact, that’s a good thing, since it shows your thought process. The point of thinking about possible follow-up questions ahead of time is just to be prepared and calm during the exam.