Instructor: Tim Chumley
Office: Clapp 423
Phone: 538-2525
e-mail: tchumley
Office Hours: Mondays 4:00-5:00; Wednesdays 4:30-5:30; Thursdays 1:00-2:00; additional availability by appointment
Textbook: Elementary Analysis: The Theory of Calculus by Kenneth A. Ross, ISBN: 9781461462705;
available as a free e-text when on the campus network.
Announcements
Announcements will be posted on the Course Announcements Moodle forum throughout the semester, but essentially all other materials will be posted on this page.
Syllabus
Check the syllabus for all the important class policies (grades, attendance, etc.).
Homework
There will be weekly homework assignments throughout the semester to be turned in. Please read these guidelines for writing mathematics.
- General information. A selection of problems will be assigned to be written up and turned in each week.
- These problems will be due Fridays at 5 pm.
- You may work with others but the writing should be done on your own unless otherwise specified.
- Gradescope. Homework will be turned in through Gradescope.
- You should be enrolled automatically if you have an account with your MHC email address. Please let me know if you have any issues logging in.
- Gradescope has made a short tutorial on submitting homework.
- Collaboration. I want you to work together on the homework! The process of explaining your ideas to one another really helps in learning math. However, you must write up the assignments on your own and avoid copying others’ work directly. Also, please only write what you understand so that I know where to help, and please avoid using online forums like Math StackExchange, solutions manuals, or similar resources. A huge part of learning in this class comes from figuring out how to get unstuck by spending time thinking on your own or talking to me and classmates; these other resources can end up being counter-productive in the long term.
- Rewrites. Homework is for practice, and you are not expected to write perfect answers from the start!
- You will be allowed to submit revisions of most problems for full credit each week.
- Your revisions will be due on Fridays at 5 pm.
- Please resubmit (only the problems you’re revising) on Gradescope.
| Assignment | Due |
|---|---|
| Homework 0 | Jan 27 |
| Homework 1 | Feb 3 |
| Homework 2 | Feb 10 |
| Homework 3 | Feb 17 |
| Homework 4 | Feb 24 |
| Homework 5 | Mar 3 |
| Homework 6 | Mar 24 |
| Homework 7 | Mar 31 |
| Homework 8 | Apr 7 |
| Homework 9 | Apr 14 |
| Homework 10 | Apr 21 |
| Homework 11 | Apr 28 |
Quizzes
There will be (mostly) weekly quizzes that will be given on Wednesdays. The purpose of these is to check in to see that you’re comfortable with fundamental material like definitions, theorems, and important examples and proofs. Problems will always be related to the previous week’s homework and class topics.
| Quiz | Date | Material |
|---|---|---|
| Quiz 1 | Feb 1 | Week 1 |
| Quiz 2 | Feb 8 | Week 2 |
| Quiz 3 | Feb 15 | Week 3 |
| Quiz 4 | Feb 22 | Week 4 |
| Quiz 5 | Mar 1 | Week 5 |
| Quiz 6 | Mar 29 | Week 9 |
| Quiz 7 | Apr 5 | Week 10 |
| Quiz 8 | Apr 12 | Week 11 |
| Quiz 9 | Apr 19 | Week 12 |
| Quiz 10 | Apr 26 | Week 13 |
Exams
There will be a midterm and a final. The date for the mid-term is subject to change slightly.
| Exam | Due | Format | Material |
|---|---|---|---|
| Exam 1 | Mar 8 | in-class, take-home | Homework 0-5 |
| Exam 2 | May 4-8 | self-scheduled, take-home | Homework 6-11 |
Course plan
Our plan is to cover parts of the textbook chapters 1 – 4. If there’s time, we’ll talk about parts of chapter 5 and 6 too. Below is a rough outline of what is to be covered week by week through the semester. Please check back regularly for precise details on what is covered, as well as postings of class materials like lecture notes.
Introduction
Monday
- Topic: Sections 1, 2: The natural, rational, and real numbers. A hint at why we need real numbers and some review of induction.
- Class materials: Lecture notes
- After class: Read Theorems 3.2 and 3.4 about inequalities. Work on Homework 0.
Wednesday
- Topic: Section 3: Basic inequalities. We discuss the triangle inequality and work on some more basic analysis proof techniques.
- Class materials: Lecture notes, worksheet
- After class: Read Definitions 4.1, 4.2, and 4.3 about min/max, boundedness, and sup/inf. Try Exercise 3.6.
Friday
- Topic: Section 4: Suprema and infima. A look at sup, inf, and the key defining property of the reals.
- Class materials: Lecture notes, worksheet
- After class: Read about the Archimedean Property and read Theorem 4.7.
Sequences
Monday
- Topic: Section 4: The completeness axiom. We discuss the Archmidean property and idea that Q is dense in R.
- Class materials: Lecture notes, worksheet
- After class: Read Examples 1 and 2 and Definition 7.1 in Section 7, and do Exercise 7.3.
Wednesday
- Topic: Sections 7: Introduction to sequences and limits. A look at the formal definition of convergence and how we prove sequences converge.
- Class materials: Lecture notes, worksheet
- After class: Read Examples 3 and 4 in Section 8.
Friday
- Topic: Section 8: Discussion about proofs. We continue our look at introductory proof examples involving limits of sequences.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 2. Read the discussion in Section 7 that follows Example 3 and Theorem 9.2.
Sequences
Monday
- Topic: Section 8: Discussion about proofs. We continue working on introductory proofs involving sequences.
- Class materials: Lecture notes, worksheet
- After class: Read Theorems 9.1 and 9.4 in Section 9.
Wednesday
- Topic: Section 9: Limit theorems. A continued look at how to prove the theorems we used in calculus for computing limits.
- Class materials: Lecture notes
- After class: Read up through Example 2 in Section 10.
Friday
- Topic: Section 10: The monotone convergence theorem. We show that sequences that are bounded and either increasing or decreasing must always converge.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 3 and read Theorems 11.3, 11.4, and 11.5.
Sequences
Monday
- Topic: Section 11: Subsequences. We begin discussing the notion of subsequences, and discuss the deeply important Bolzano-Weierstrass theorem, which says that every bounded sequence has a convergent subsequence.
- Class materials: Lecture notes, worksheet
- After class: Read the definition of Cauchy sequences in Section 10 and read Lemmas 10.9 and 10.10.
Wednesday
- Topic: Section 10: Cauchy sequences. We introduce another useful class of sequences that always converge.
- Class materials: Lecture notes
- After class: Read pages 60-62 and 73-76 on limsup, liminf, and subsequential limits.
Friday
- Topic: Sections 10, 11: Limit supremum and limit infimum. Not all sequences converge, but it’s possible to define notions of limits even when they oscillate.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 4 and read Theorems 11.7, 11.8, 11.9.
Sequences
Monday
- Topic: Sections 10, 11: Subsequential limits. We relate the notions of liminf, limsup and subsequential limits.
- Class materials: Lecture notes
- After class: Read about the Cauchy criterion and the test for divergence in Section 14.
Wednesday
- Topic: Section 14: Series. We begin our discussion of infinite series, looking at some familiar series and proving tests of convergence from calculus. The introduce and prove the Cauchy criterion for series and the Test for Divergence.
- Class materials: Lecture notes
- After class: Read about the comparison test in Section 14.
Friday
- Topic: Section 14: Series. We continue our discussion of infinite series. We prove the comparison test and absolute convergence test and look at some examples.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 5. Read about the ratio test test in Section 14.
Series
Monday
- Topic: Section 14: Series. We discuss and prove the ratio test.
- Class materials: Lecture notes, worksheet
- After class: Refresh yourself on antiderivatives of functions like \(f(x)=1/x^p\).
Wednesday
- Topic: Section 15: Integral test. We discuss the integral test and use it to characterize the convergence of \(p\)-series.
- Class materials: Lecture notes, worksheet
- After class: Read about the alternating series test in Section 15.
Friday
- Topic: Section 15: Alternating series test. We discuss the proof of the alternating series test.
- Class materials: Lecture notes, worksheet
- After class: Begin studying for the exam by memorizing definitions and statements of named theorems we’ve learned up through Section 14.
Exam
Monday
- Topic: Office hours, no class. I’ll be available for office hours during class time but there will be no lecture.
- After class: Study for the in-class portion of the exam.
Wednesday
- Topic: Exam. We’ll spend the entire day on the in-class portion of the exam.
- After class: Work on take-home portion of the exam, which is due Friday.
Friday
- Topic: Office hours, no class. I’ll be available for office hours during class time but there will be no lecture.
- After class: Read Definition 17.1, Theorem 17.2, and Examples 1 and 2 in Section 17.
Spring Break
Monday
- Topic: Spring break, no class.
Wednesday
- Topic: Spring break, no class.
Friday
- Topic: Spring break, no class.
Continuity
Monday
- Topic: Sections 17, 20: Limits of functions. We begin our discussion of limits of functions and get introduced to epsilon-delta proofs.
- Class materials: Lecture notes, worksheet
- After class: Read Examples 1 and 3 in Section 17.
Wednesday
- Topic: Section 17: Continuity. We discuss what it means for a function to be continuous at a point and continuous on its domain, and discuss how this definition is connected to sequences.
- Class materials: Lecture notes, worksheet
- After class: Work on Problems 1 and 2 from the worksheet today.
Friday
- Topic: Section 17: Continuity. We discuss when and when not to use the sequential characterization of continuity.
- Class materials: Lecture notes, worksheet
- After class: Read Theorem 18.1 and its proof. Do Problem 2 from the worksheet today.
Continuity
Monday
- Topic: Section 18: Extreme Value Theorem. We discuss the proof of the Extreme Value Theorem, which connects the concepts of continuity, closed intervals, and the Bolzano-Weierstrass theorem.
- Class materials: Lecture notes, worksheet
- After class: Read Theorem 18.2 and its proof.
Wednesday
- Topic: Section 18: Intermediate Value Theorem. We discuss the proof of the intermediate value theorem.
- Class materials: Lecture notes
- After class: Read through to Definition 19.1 in Section 19.
Friday
- Topic: Section 19: Uniform continuity. We discuss a stronger form of continuity where our choice of \(\delta\) depends only on \(\epsilon\).
- Class materials: Lecture notes, worksheet
- After class: Read Examples 3 and 4 and Theorem 19.2 in Section 19.
Continuity, Differentiability
Monday
- Topic: Section 19: Uniform continuity. We discuss why continuous functions on closed intervals must be uniformly continuous, as well as how to prove a function fails to be uniformly continuous.
- Class materials: Lecture notes, worksheet
- After class: Read Definition 28.1 and Theorems 28.2 and 28.3.
Wednesday
- Topic: Section 28: Differentiability. We introduce the idea of differentiable functions and discuss some theorems that are a consequence of differentiability.
- Class materials: Lecture notes, worksheet
- After class: Try Exercises 28.1ac and 28.3.
Friday
- Topic: Section 28: Differentiability. We discuss how to prove a function is differentiable at a point using the epsilon-delta definition of limits of functions. We also discuss proving a function is not differentiable at a point.
- Class materials: Lecture notes, worksheet
- After class: Read Theorems 29.1, 29.2, and 29.3.
Mean Value Theorem, Fundamental Theorem of Calculus
Monday
- Topic: Section 29: Mean Value Theorem. We prove the Intermediate Extremum Theorem, Rolle’s Theorem, and the Mean Value Theorem.
- Class materials: Lecture notes, worksheet
- After class: Read Corollaries 29.4 and 29.5.
Wednesday
- Topic: Section 29: Mean Value Theorem, continued. We discuss some consequences of the Mean Value Theorem.
- Class materials: Lecture notes, worksheet
- After class: Read Theorems 34.1 and 34.3.
Friday
- Topic: Section 34: Fundamental Theorem of Calculus. We use uniform continuity and the mean value theorem in the context of integration.
- Class materials: Lecture notes, worksheet
- After class: Read Definitions 24.1, 24.2, and Examples 2 and 4 in Section 24.
Sequences of functions
Monday
- Topic: Section 24: Pointwise and uniform convergence. We talk about sequences of functions and two notions of convergence.
- Class materials: Lecture notes, worksheet
- After class: Read Remark 24.4, and Example 7 in Section 24.
Wednesday
- Topic: Section 25: Uniform convergence properties. We discuss some general properties of uniformly convergent sequences of functions.
- Class materials: Lecture notes, worksheet
- After class: Read Theorems 24.3 and 25.2 in Sections 24 and 25.
Friday
- Topic: Section 24: Uniform convergence. We discuss a computational technique for verifying uniform convergence.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 11.
Sequences of functions
Monday
- Topic: Section 25: More on uniform convergence. We discuss uniformly Cauchy sequences and the Weierstrass M-test.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 11 and study for your last quiz.
Wednesday
- Topic: Section 27: Weierstrass Approximation Theorem. We discuss the fact that any continuous function on a closed interval is the uniform limit of a sequence of polynomials. In other words, polynomials are dense in the space of continuous functions on a closed interval equipped with the supremum metric.
- Class materials: Lecture notes
- After class: Study! Enjoy your summer! Keep in touch!
Friday
- Topic: New England Dynamics Seminar, no class. I’ll host an extra office hour 10:00 - 11:15.
- After class: Study! Enjoy your summer! Keep in touch!
Getting help
Here are a few ways to get help:
- Office hours: Mondays 4:00-5:00; Wednesdays 4:30-5:30; Thursdays 1:00-2:00; additional availability by appointment
- TA help: We will have a class TA who will be available for help during drop-in sessions on Wednesdays 7-9 pm in Clapp 420.
- Study groups: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together.
- Piazza: I’ve set up an online forum for asking and answering questions about class material, including homework problems. My plan is to check the forum every day and answer questions as they come up, but I hope everyone in the class will pitch in and answer others’ questions when possible.
Resources
- When away from campus, a college VPN is available to access resources normally only available when on the campus network (eg. free e-books through LITS and Springer Link). However, I understand there might be some accessibility issues even with the VPN, so I will do my best to help everyone access necessary materials through other methods as well.
- I’ve collected some resources to help you when writing homework solutions with LaTeX.
- Here is a LaTeX template file for writing homework solutions and its pdf output.
- A quick reference is available for commonly used symbols.
- Overleaf is a cloud service that lets you edit and compile LaTeX through a web browser so that no local installation is needed. The free version will be sufficient for our class.
- To install LaTeX locally on your personal computer, I recommend installing MacTex if you use a Mac, MikTeX for Windows, or TeXLive for Linux.
- There are many real analysis books out there. Reading our required textbook will be enough to get a strong understanding of the material in class, but if you’re curious, here are some other good books. The first two are available for free as e-books when signed into the campus network or VPN.
- Understanding Analysis by Stephen Abbott. Similar to our textbook, but with some interesting chapter introductions and projects.
- Real Mathematical Analysis by Charles C. Pugh. Nicely written, but with more emphasis on topology and metric spaces than our textbook.
- Principles of Mathematical Analysis by Walter Rudin. A classic textbook that is a bit too terse for a first analysis course but worth reading if you’re really excited about analysis. It’s filled with great, challenging exercises.