Instructor: Andrew Obus
Email: Andrew.Obus [at] baruch.cuny.edu
Office: VC 6-223
Office Phone: 646-312-4008
Website: http://blogs.baruch.cuny.edu/aobus
Office Hours: Mondays 2:05 – 3:05, Wednesdays 10:30 – 11:30, in person
Lectures: Mondays and Wednesdays 12:50 – 2:05, VC 9-140
Lectures
Attendance is crucial for learning the material for this course. We will start right at 12:50 and go for the entire 75 minutes — it is imperative that you show up on time so that you don’t get lost! Please ask questions if anything in lecture is unclear.
CELL PHONES MAY NOT BE TAKEN OUT DURING CLASS! If you know that you are expecting an important call (relative in the hospital, etc.), you must let me know ahead of time. Phones in class are *extremely* distracting to you, to me, and the people around you, and they are detrimental to your understanding of the material. I will stop class to ask you to put them away, and repeated cell phone use can count against your grade in borderline situations (see “Final Course Grades”).
Textbook
Understanding Analysis, Second Edition, by Stephen Abbott. Available on Amazon and at the bookstore.
Content
This course is dedicated to the theory behind single-variable calculus. We will dig deeply and rigorously into the concepts of continuity, convergence, differentiation, and series summation, and we will prove many of the fundamental results of calculus. Furthermore, this course will hopefully give you a sense of why it is a good idea to undertake this effort — what is wrong with defining continuity of a function simply as “you can draw the graph without lifting your pen”?
In order to do this, we begin with a study of the real numbers themselves — what are they, really (get it? booooooo), and how should we think about them?
In particular, we will cover most of Chapters 1 through 6 of the book, as well as a brief overview of Chapter 7. As you know by now, math is very cumulative. So if you fall too far behind in this course, it will be difficult to catch up. Please see me promptly if things stop making sense!
This course is quite different in “flavor” than calculus, matrix algebra, probability, statistics, and financial math courses. Namely, it is all proofs, all the time. The homework, exams, and lectures will consist predominantly of written proofs, and reasoning will be much more important than computation (although sometimes it will be necessary to do computations in order to complete a proof). You are already expected to have some degree of familiarity and comfort with writing proofs from a previous class — the point of this course is not to teach you how to write a basic proof, although I will, of course, provide feedback and help.
Prerequisite: A proof-based math course (MTH 4000, 4009, 4030, 4200, 4210, 4215, 4220, 4240, or 4315).
Office Hours
Mondays 2:05 – 3:05 (right after class), Wednesdays 10:30 – 11:30, in person (VC 6-223). I am also happy to make a Zoom appointment with you at some other time if you prefer. I want to help you succeed in this class!
For the second half of the semester (starting the week of March 18), the Wednesday office hour will move to Tuesdays, 11:00 – 12:00 on Zoom. The Monday office hour will remain in person at the same time.
Discussion Forums
We will utilize the Discussion Board feature on Blackboard (in the past I have used third-party software like Ed or Piazza, but to simplify things, I am experimenting this year with doing everything on Blackboard). If you have questions about homework or course material, asking a question on the forum can be of great benefit to your classmates, many of whom probably have a similar question!
Homework
Homework will be assigned almost every week, due Wednesdays in class. I will grade a selection of the problems, and I will also check that the others are completed. The homework is the core of the class! It is where you will solidify concepts that you may have only half-understood in class, and where you will get practice writing. It is not possible to succeed in this course without putting serious mental energy into the homework — if you make a habit of beginning the homework an hour before it’s due or of copying it from the internet, you will almost certainly fall behind and fail the course. See also the section on academic honesty below.
You are encouraged to discuss homework and the extra credit problems with your classmates and with me, but you must write up the solutions on your own. Late homework will not be accepted for credit without a very good prior excuse, but your lowest two homeworks will be dropped. If you know you will not be able to attend class, you can have a friend hand it in for you, you can leave it in my mailbox in the main math department office (VC 6-230), or as a last resort, you can email it to me as a pdf.
As of now, I am not planning on giving quizzes in class. However, if the homework performance is inconsistent or if the work appears to be copied, I reserve the right to start giving quizzes which will count toward your homework grade.
Exams
This class has two midterms, to be given in class on March 13 and April 10 (both Wednesdays). The final exam will take place Monday May 20 from 1:00 – 3:00 in our usual classroom. Exam questions will require written answers (no multiple choice or fill-in-the-blanks). Partial credit will be given for correct work shown. This course will not require a calculator, and calculators will not be permitted on exams.
If you cannot make a midterm (e.g., you are under quarantine), please contact me as early as possible, and we will make alternate arrangements.
Final Course Grades
25% Homework
20% Each Midterm
35% Final Exam
If your final exam is better than your lowest midterm, it will count for 45% and that midterm will count for 10%.
Grading will be based on the Baruch grading scale. It is possible for exceptional class participation to be factored into your numerical grade in borderline cases.
Academic Honesty
Any cheating on exams or homework will result in a grade of 0 on the assignment involved. A second instance of cheating will result in automatic failure of the class. Refer here for more information on Academic Honesty.
In particular, use of artificial intelligence (AI) is strictly prohibited in all coursework and assignments. This includes, but is not limited to, the use of AI-generated text, as well as the use of AI tools or software to complete any portion of an assignment. The only way to master mathematical analysis is to spend time struggling with it, and using AI tools such as ChatGPT to avoid this struggle not only reduces your mastery, but on top of that, often leads to wrong/non-sensical answers. Furthermore, dependency on AI tools will not serve you well on exams, which account for 75% of your grade. Students are expected to use their own knowledge and analysis to complete coursework.
Disabilites
All students with special needs requiring accommodations should present the appropriate paperwork from Student Disability Services. It is the student’s responsibility to present this paperwork in a timely fashion and follow up with the instructor about the accommodations being offered. Accommodations for test-taking (e.g., extended time) should be arranged at least 5 business days before an exam.
Schedule (topics subject to change)
| Class # | Date | Topics | Book Sections (approximate) | Remarks |
| 1 | 1/29 | Rational and irrational numbers | 1.1 | |
| 2 | 1/31 | Review of proofs | 1.2 | |
| 3 | 2/5 | The completeness axiom | 1.3 | |
| 4 | 2/7 | Consequences of the completeness axiom | 1.4 | HW #1 Due |
| 5 | 2/14 | Cardinality, introduction to series | 1.5, 2.1 | |
| 6 | 2/21 | Properties of sequence limits | 2.2 | HW #2 Due |
| 7 | 2/22 | The algebraic and order limit theorems | 2.3 | |
| 8 | 2/26 | Monotone convergence theorem, subsequences | 2.4, 2.5 | |
| 9 | 2/28 | Bolzano-Weierstrass theorem, Cauchy sequences | 2.5, 2.6 | HW #3 Due |
| 10 | 3/4 | Infinite series | 2.7 | |
| 11 | 3/6 | More on infinite series, introduction to topology of the real numbers | 2.7, 3.1 | HW #4 Due |
| 12 | 3/11 | Open and closed sets | 3.2 | |
| 13 | 3/13 | MIDTERM 1 | MIDTERM 1 | MIDTERM 1 |
| 14 | 3/18 | Compact and connected sets | 3.3, 3.4 | |
| 15 | 3/20 | Introduction to continuity | 3.4, 4.1 | HW #5 Due |
| 16 | 3/25 | Limits of functions | 4.2 | |
| 17 | 3/27 | Continuous functions | 4.3 | HW #6 Due |
| 18 | 4/1 | Continuous functions on compact sets, intermediate value theorem | 4.4, 4.5 | |
| 19 | 4/3 | Introduction to derivatives | 4.5, 5.1 | HW #7 Due |
| 20 | 4/8 | Derivatives and the intermediate value property | 5.2 | |
| 21 | 4/10 | MIDTERM 2 | MIDTERM 2 | MIDTERM 2 |
| 22 | 4/15 | The mean value theorem | 5.3 | HW #8 Due (note: Monday!) |
| 23 | 4/17 | Introduction to power series | 6.1, 6.2 | CLASS ON ZOOM! |
| 24 | 5/1 | Uniform convergence | 6.2 | HW #9 Due |
| 25 | 5/6 | Uniform convergence and differentiation, series of functions | 6.3, 6.4 | |
| 26 | 5/8 | Power series | 6.5 | HW #10 Due |
| 27 | 5/13 | Taylor series | 6.6 | |
| 28 | 5/15 | Overview of integration | Ch. 7 | HW #11 Due |
| 29 | 5/20 | FINAL EXAM | FINAL EXAM | FINAL EXAM |