MA321 Syllabus

Advanced Calculus I in Fall 2019 as taught by Marko Budišić

  1. Instructor
  2. Learning objectives
  3. Topics and textbook
  4. Prerequisites
  5. Course work
  6. Assessment
  7. Thinking about math
  8. Accommodations
  9. Ethical Behavior
  10. Additional Resources

Instructor

Marko Budišić

  • Office: SC391
  • Walk-in office hours: Mo 12.30-2p, We 9-10.30p, Th 1-2.30p
  • Contact: marko@clarkson.edu (put “MA321” in subject line somewhere) or via Moodle
  • Website: http://people.clarkson.edu/~mbudisic

Learning objectives

This class serves a dual purpose:

  1. It provides students with the rigourous foundation of infinitesimal calculus, including properties of real numbers and their sequences and series, basic function theory and operations of differentiation and integration.
  2. It sharpens the reading and writing of proofs in the context of mathematical analysis.

By the end of this class, successful students will:

  • be able to explain the differences between the set of real numbers and the set of fractions/decimals, including answer questions such as “Is 0.9999… = 1?”
  • be able to discuss various limits of sequences and ordered infinite sums (series) of real numbers,
  • use concepts and theorems, such as Bolzano–Weierstrass and Cauchy sequences, that underlie topology,
  • be able to rigorously analyze continuity properties of functions and their sequences, and formulate proofs based on this property,
  • be able to provide rigorous definition of a single-variable derivative and differentiability of functions, with all its implications on properties of the function
  • be able to provide rigorous definition of integrals, including proving the Fundamental Theorem of Calculus,
  • distinguish between pointwise and uniform convergence of a sequence of functions,
  • integrate all that was learned to prove theorems on existence and uniqueness of solutions of ODEs,
  • sharpen their proof-writing skills to the advanced-undergraduate level.

For those students aiming to enter graduate programs in mathematics, this course is indispensible.

Topics and textbook

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Detailed list of topics can be found in the Schedule section of this site. These topics correspond to topics from n Jiri Lebl’s, Basic Analysis: Introduction to Real Analysis Vol I. The book is completely free to use in PDF form. I strongly encourage you buy the hardcopy: either the lulu.com version (purple) or Amazon version. They are about $15 to defray costs of printing.

https://upload.wikimedia.org/wikipedia/en/0/02/Math_Girls_-_book_cover.jpg Regular and frequent access to textbook is mandatory. Any format of the textbook is fine (ebook, hardcopy, borrowed, owned), but don’t count that “sharing a textbook” and occassionally looking at it will be enough. I specifically chose the book that is inexpensive so that everyone can afford the hardcopy. If you cannot, please come see me and I’ll find a way to get you a printed copy.

Why the hard-copy push? I insist on hard-copy textbooks because writing proofs requires intense concentration, free from distractions. If you need to have a laptop or a tablet open to work on proofs, you expose yourself to distractions that sap your concentration. Be better than that.

In addition to the textbook, I ask you to read the first volume of Hiroshi Yuki’s novel Math Girls. The plot of the novel is thin (love in high school), but the book communicates the passion for and beauty of mathematics as few other accessible works do so. Here’s a review from the AMS Notices (flagship magazine for the American Mathematical Society) and MAA Reviews. Feel free to borrow or buy this book – we won’t refer to it regularly, but it’s good for priming the math pump.

Prerequisites

Introductory proof-based class, like Clarkson’s MA211. If this is the first time you will be writing proofs, then you should first sign up for MA211.

If you are looking for completing a math-minor, this class is likely not the best idea (except for those CS students who are pursuing theoretical CS). Dr. Fulton maintains a list of Fall 2019 courses that are much better suited for math minor.

Course work

Before class: reading and journaling

You are expected to read the textbook ahead of the lecture. The lectures will assume that students have read at least once through the listed chapters.

Each day of the class you will turn in page(s) for a journal on a three-hole-punched paper consisting of:

  • summary of your reading
  • answers to the Journal questions
  • comments, ideas, tangents, etc. prompted by readings

You can omit a journal page only 3 times during the semester. Journal pages are due at the beginning of the class, with no late pages accepted (except at my discretion due to exceptional circumstances).

When I return your journal pages, store them in a three-ring binder and use them to study for the exams.

I suggest that you take some amount of time during the weekend and complete the readings and journal questions for more than one class. That way you won’t risk running out of time during the week, and you’ll need only a quick skim before the class to get up to speed.

Index Cards: While math is rarely about memorizing, unless you know definitions, you cannot start doing anything before you look it up in a book. I suggest you keep a stack of index cards and write out important definitions as you encounter them, and use the stack to keep your memory fresh.

During class: Participation and Note-taking

During class, you should expect to work actively. I will assume you have completed the assigned readings for the class and I will strive not to go over every detail from the textbook. Rather, we’ll work on problems, discuss questions, etc.

Do not be surprised if you have to work in pairs, on the blackboard, etc Despite its reputation, mathematics is a collaborative discipline. Most of your time thinking about this class will likely be spent alone. We’ll use the class time to work together.

You will need a notebook (preferrably physical) in which to take notes during the “lecture” part of the class.

During collaborative work, obviously there is little time to both work and copy the material into your notes. So use your cell phone/tablet to take a photo of the work and share it with both partners.

After Class: Homeworks

There will be 10 homeworks during the semester. Students are expected to work together to solve the problems, but then each of the student has to put together their own individual report that they will submit. There will be several mandatory problems on each homework, and a few optional ones. Students aiming for high-Bs and As should definitely attempt to complete optional problems every time.

It is expected that solutions to the homework will strive for mathematical maturity - the goal is not just to get the solution (solutions are typically known in this class, or can be looked up), but to justify and interpret the steps along the way.

The main purpose of the homework is to expose you to the problems and the level of rigor expected of you in this class, and give you feedback. However, the accuracy of the homeworks will not contribute to your grade.

Assessment

The grade will be based on your score on three non-cumulative in-class midterms and a cumulative final exam. For dates, see Schedule.

You will get feedback on homeworks, but feedback on homeworks will not factor into your final grade. You will have to submit all homeworks on time, however, to get D or higher in this course. (Necessary, but not sufficient condition.)

The exams will have questions labeled according to the level of difficulty (C-level, B-level, A-level); completing higher-level problems will award you the appropriate grade.

I may offer portions of midterm for re-do, in order to boost your grade. Only students who have submitted all homeworks up to that point, and all journal entries (minus the 3 freebies) will be eligible to do so.

  • For a journal to count as “turned-in”, it has to contain a summary of reading (a paragraph), and answers to all Journal questions.
  • For a homework to count as “turned-in”, all problems have to be attempted, except possibly the optional ones. Homeworks are scored for completion, not accuracy, although it’s expected you will do your best on each.
  • Problems will be commented on, and scored as “Exceptional”, “Meets the expectation”, “Partial”, or “Incorrect”. That’s what the letters in the table below stand for,e.g., “50%+ P+” stands for “more than 50% problems on midterm and final partially correct or better”.
  • Students scoring a perfect score on all midterms, turning in all homeworks, and all except 3 journal entries are exempt from the final exam with grade A. Grade A+ cannot be earned without completing the final.
Grade Journals Homeworks C problems B problems A problems Grade Explanation
A All except 3 all 90%+ M+ 90%+ M+ 75% P+, 50%+ M+ Mastery of the material
B All except 3 9+ 90%+ M+ 50%+ M+ 50%+ P+, 25% M+ Competent command of the material
C All except 3 7+ 50%+ M+ 50%+ P+ 1 or more C+ Basic proof-writing skill
D All except 3 5+ 50%+ P+ none none Basic vocabulary

Grade “Exceptional” will be awarded rarely, for particularly innovative, clear, well-rounded answers. They are the route to intermediate grades, e.g., A+, B+, etc. Intermediate grades will also be assigned for positive/negative trends. (Student who aces the first midterm, does OK on the second, and bombs the final, yet still makes the cut for B, may end up with B-. The reverse is true for the opposite trajectories.)

Thinking about math

Instead of Try thinking
I’m not good at this What am I missing?
I give up I’ll use some of the strategies I’ve learned.
This is too hard This may take some time and effort
I can’t make this any better I can always improve so I’ll keep trying
I just can’t do math. I’m going to train my brain.
I made a mistake. Mistakes help me learn.
She’s so smart. I will never be that smart. Let me figure out how she does it so I can try.
It’s good enough. Is this my best work?
Plan A didn’t work. Good thing the alphabet has more letters in it.

(From: Reinholz (2018) https://doi.org/10.1080/10511970.2017.1394944 )

Accommodations

I will do my best to arrange for any necessary accommodations that would enable you to fully participate in this class. If you require assistance during the lectures or during exams, please register with the AccessABILITY Services.

Ethical Behavior

From the Student Manual: The Clarkson student will not present, as his or her own, the work of another, or any work that has not been honestly performed, will not take any examination by improper means, and will not aid and abet another in any dishonesty. Failure to adhere to this code will mean a failing grade and a report to the Dean of Students.

Additional Resources

  1. Thomson, Brian S., Judith B. Bruckner, and Andrew M. Bruckner. Elementary Real Analysis. ClassicalRealAnalysis.com, 2008. – This book was used for MA321 by other instructors.
  2. Rudin, Walter. Principles of Mathematical Analysis. 3rd edition. New York: McGraw-Hill Education, 1976. – A time-tested standard textbook on this topic. AKA “Baby Rudin”. Read this review for the fun of it
  3. Polya, G. How to Solve It. Princeton University Press, Princeton, NJ, 2014. – A classic guide to proof-writing strategies
  4. Velleman, Daniel J. How to Prove It. 2nd ed. Cambridge University Press, Cambridge, 2006. – A structured approach to proof writing
  5. Tao, Terence. Analysis I, 3rd ed. Texts and Readings in Mathematics. Springer Singapore, 2016. – Textbook written by one of the best-known living mathematicians and analysts Terry Tao
  6. Terry Tao’s blog