** Course: ** MATH 3210, Section 001

** Time/Place: ** M,W,F 8:05 AM - 9:25 AM; JWB 208

** Instructor: ** Thomas Goller

** Course book: ** "Foundations of Analysis", by Joseph Taylor, ISBN: 9780821889848.

** Office Hours: ** M,W: 3-5 PM, H: 11 AM - 12 PM, in JWB 307. Or e-mail me to set up an appointment.

The Syllabus is Up!

My final post: solutions to the Final Learning Celebration.

Grades are posted on CIS!

Please evaluate the course online! You should have received an e-mail prompting you to evaluate your courses. You will need to evaluate the course to view your grade on CIS before the official end of the semester. If you're having trouble, let me know.

If you want to pick up your exam or discuss your grade, I will be in my office Tuesday, 17 December, from 10-12:30 and 3:30-4:30. If those times don't work for you, e-mail me and we can find another time.

Thank you for working hard this semester, for putting up with my eccentricity (I am certainly not a circle), and for rolling out of bed at a much too early hour just to listen to me ramble on about sequences, functions, and quantifiers!

I'd be happy to chat with you about mathematics, courses, chess, life, or anything else. I'm waiting for you in JWB 307.

**The Final Learning Celebration** is on Monday, December 16, from 8-10 AM in the usual room. As usual, we'll start right on time, so come at least 5 minutes early!

Study these solutions to Learning Celebration #5 to prepare for the FLC.

**Learning Celebration #5** is on Wednesday, December 11. It will cover all of Chapter 6, as well as Sections 2.6 and 3.4. The celebration will begin at 8:05, so come a few minutes early! If you are worried, come see me during my Monday office hours (after class or 3-5 PM or by e-mail appointment) and I will ask you some guiding questions and give you tips on how to prepare!

Have a peek at these solutions to Quiz #6. As usual, I graded the quiz out of 10. Consider this quiz as a warning: If you want to do well on the FLC, you need to (1) State all quantifiers clearly, (2) Define all variables you introduce that are not already defined in the problem, and (3) Be careful with inequalities!

**Homework #14**, the last homework assignment, due in class Monday 12/09, is Section 6.4: 4, 5, 7, 8, 13, 14; Section 6.5: 2, 3, 5, 6, 7, 12, 13, 15. The Version 2.5 notes have different exercise numbering for Section 6.4, so check with someone who has the book!

Hint for Homework #13, Section 6.3 Exercise #7: Try to relate the partial sums for the series (Sigma a_k^+) and (Sigma a_k^-) to the partial sums for the series (Sigma a_k) and (Sigma |a_k|). You should get two simple equations. Then use the assumption that (Sigma a_k) converges and (Sigma |a_k|) diverges, together with Theorem 2.4.7 (b) and a bit of trickery (what happens when you add or subtract your equations from each other?). If you're confused by the notation in the problem, try figuring out what it means in an explicit example (try an alternating series!).

On Quiz #6, which will take place on Monday 12/02, you will be given a sequence of functions, asked to guess what function the sequence converges to, and then asked to prove that the convergence is uniform.

**Homework #13**, due in class Monday 12/02, is Section 6.3: 1, 2, 3, 4, 5, 6, 7; Section 3.4: 1, 6, 9, 10. For Section 3.4 Exercise 10, instead of proving uniform convergence on [-r,r], you may do just the special case r=1/2, and for the proof that convergence is not uniform on (-1,1), try using Theorem 3.4.7 with the sequence x_n = 1-1/2^(n+1). The Version 2.5 notes have the correct exercises.

These solutions to Learning Celebration #4 might interest you.

**Homework #12**, due in class Monday 11/25, is Section 2.6: 1, 3, 12; Section 6.2: 1, 2, 3, 4, 5, 6, 7. The Version 2.5 notes have the correct exercises.

**Learning Celebration #4** is on Wednesday, November 20. Here are some (by no means all) of the highlights from the sections that we'll be celebrating:

- Section 5.1: Definition of the Riemann integral. Understand the definition involving the supremum of lower sums and infimum of upper sums over all partitions, and also the two equivalent conditions stated as theorems. Be able to prove simple functions (like constants or low-degree polynomials) are integrable and compute their integrals from the definition of integral (via the 4-step process that was on the quiz). Come up with examples of integrable and non-integrable functions.
- Section 5.2: Existence and properties of the integral. Be able to prove the easier properties (like 5.2.4, 5.2.5, and the theorem of the mean for integrals) and apply these properties to manipulate integrals.
- Section 5.3: The fundamental theorems of calculus. Know what they say and how to use them. FTC1 is incredibly useful for computing integrals when you know an explicit antiderivative of the integrand, while FTC2 guarantees the existence of an antiderivative under certain conditions and tells you how to differentiate certain integrals that are viewed as functions. Know the substitution and integration by parts theorems, how to implement them, and how to prove them.
- Section 5.4: Know the definitions and basic properties of the natural log and exponential functions. Be able to prove basic facts about them. Understand how to define integrals on unbounded intervals and on intervals on which the integrand is unbounded, and how to determine whether such integrals converge.
- Section 6.1: Definition of series, convergence, and divergence. Know when geometric series converge and diverge and how to prove the formula for the partial sums of a geometric series. Understand how the term test can be used to prove a series diverges, and how the comparison test can be used to prove a series converges. Come up with examples of series with various properties (geometric, not geometric, all terms positive, etc.) that converge or diverge.

The celebration will begin at 8:05, so come a few minutes early! If you are worried, come see me during my Monday office hours (after class or 3-5 PM or by e-mail appointment) and I will ask you some guiding questions and give you tips on how to prepare!

**Homework #11**, due in class Monday 11/18, is Section 5.4: 1, 3, 4, 9, 10; Section 6.1: 1, 2, 3, 4, 7, 8, 9, 10, 11, 12. The Version 2.5 notes have the correct exercises.

On Quiz #5, I will ask you to prove the first fundamental theorem of calculus. Remember: the idea is that if f is continuous on [a,b] and differentiable on (a,b), then every partition of [a,b] has a Riemann sum for f' that equals f(b)-f(a) (this Riemann sum is constructed by applying MVT to each subinterval in the partition). Then since you are assuming f' is integrable on [a,b], there is a sequence of partitions P_n such that the limit of any Riemann sums for P_n equals the integral. In particular, taking the Riemann sums that equal f(b)-f(a), we see that the integral is the limit of the constant sequence f(b)-f(a), which equals f(b)-f(a). Amazing!

If you hunger for more, try these solutions to Quiz #4.

**Homework #10**, due in class Monday 11/11, is Section 5.2: 1, 6, 11, 12; Section 5.3: 1, 2, 4, 5, 6, 8, 10. For Exercise 5.3.8, the integral should be from 1 to x, not from 0 to x, and you should assume that x>0. When you use a theorem in the book, check to make sure the assumptions of the theorem (often involving continuity, differentiability, and integrability) are satisfied, and state why they are satisfied in your solution. The Version 2.5 notes have the correct exercises.

Compare your work to these solutions to Learning Celebration #3.

**Homework #9**, due in class Monday 11/4, is Section 5.1: 1, 2, 3, 4, 5, 8, 9, 10, 11. The Version 2.5 notes have the correct exercises. The Quiz on 11/4 will ask you to compute the Riemann integral of a function using the definition of the integral.

**Learning Celebration #3** is on Wednesday, October 30. It will cover sections 3.2, 3.3, 4.1, 4.2, 4.3, and 4.4 of the book. The celebration will begin at 8:05, so come a few minutes early! If you are worried, come see me during my Monday office hours (after class or 3-5 PM or by e-mail appointment) and I will ask you some guiding questions and give you tips on how to prepare!

**Homework #8**, due in class Monday 10/28, is the modest Section 4.3: 1, 2, 4, 6; Section 4.4: 1, 6, 7, 8, 10.

Please check out these solutions to Quiz #3.

The quiz on Monday 10/21 will be on proving functions have limits. You will have to know how to prove all of the 15 definitions of limit described on the worksheet for Section 4.1. I will of course give you fairly easy functions. For instance, I could ask you to prove that the limit of f(x)=-x as x goes to infinity is negative infinity.

Take a look at these solutions to Quiz #2.

**Homework #7**, due in class Monday 10/21, includes a worksheet for Section 4.1 as well as Section 4.1: 1, 2, 3, 4, 5, 6; Section 4.2: 1, 2, 4, 7, 11, 12.

Starting next Monday, we'll have weekly 5-10 minute quizzes on key concepts that we covered during previous weeks. Quizzes will be at the end of class every Monday during weeks where there is no Learning Celebration. Next Monday's quiz will be on proving functions are continuous using the epsilon-delta definition.

For you to peruse: solutions to Learning Celebration #2.

**Connecting Book Exercises to LC Problems** When a book exercise says "prove", you should write a rigorous formal proof and label it "proof". When an exercise says "show", you should write a rigorous argument that can be a little less formal, although if the exercise is making a precise claim, then you might as well write a formal proof. If the exercise only asks you to "find" a value of something, then an informal computation will do, and you should show your work. If an exercise asks you to "find" an example of something, state the example and briefly explain why it works. If the exercise only demands that you "justify your answer", then even intuitive reasoning will suffice. "Prove" and "show" exercises are the kinds of problems I might include in the "Proof" section of an LC. "Find" exercises are similar to the "Computation" section of an LC (please show your work on this section!) or the "Example" section of an LC (although you never need to explain your examples on an LC). Once in a while you might encounter a book exercise asking you to "state" something; these are similar to the "Precision" section of an LC.

**Homework #6**, due in class Monday 10/7, is Section 3.2: 1, 3, 4, 5, 7, 8, 9; Section 3.3: 1, 2, 3, 4, 5, 7, 9, 10. Warning: if you are using the Version 2.5 notes, check with someone who has the book to make sure you are doing the right exercises!

**Learning Celebration #2** is on Wednesday, October 2. It will cover sections 2.3, 2.4, 2.5, and 3.1 of the book. The format will be similar to LC #1. We will start celebrating right at 8:05, so come a few minutes early! If you are worried, come see me during my Monday office hours (after class or 3-5 PM) and I will ask you some guiding questions and give you tips on how to prepare!

**Homework #5**, due in class Monday 9/30, is Section 3.1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (all!). Warning: if you are using the Version 2.5 notes, check with someone who has the book to make sure you are doing the right exercises!

We will begin Chapter 3 on Monday. Chapter 3 is the most important and most difficult chapter in the book because we will be transporting our hard-earned understanding of limits of sequences to the more complicated land of functions. Calculus is all about functions, and Chapter 3 lays a solid foundation, which we will later use to define derivatives (Chapter 4), integration (Chapter 5), and power series (Chapter 6).
Unfortunately, the author of the book assumes we readers will have little difficulty ascending to the land of functions, so Section 3.1 is quite terse. I am planning to devote a whole week to Section 3.1. If you get lost now, you will likely be lost for the rest of the course, so work especially hard this week! Read Section 3.1 carefully multiple times, make sure you understand definitions and theorems using the multi-step process described in the syllabus, get started early on the problems so that you can get help, work in groups, and come to lecture!

Please fill out a suggestion sheet and bring it to class on Monday!

I have vanquished all Section 2.6 exercises in Homework #4!

Here are solutions to Learning Celebration #1.

**Homework #4**, due in class Monday 9/23, is Section 2.4: 8, 10, 11, 12, 13; Section 2.5: 1, 2, 4, 5, 6, 7, 10; and Section 2.6: none!

Make sure you understand all of the nasty comments the grader and I provide on homework and LCs. If you have any questions, ask me!

Homework will be graded as follows: a blue completeness score (0-3), a red score for graded problems (0-3 per problem), and a final green score (0-6) obtained by adding the blue score to the red score divided by the number of graded problems. To get a blue 3, show some honest effort on all problems. To do well on the red, work especially hard on the problems involving proofs, since I will usually select proof problems to grade. You should be aiming to get a green score of at least 5.

Homework #3 has been edited below and is now in final form.

**Homework #3**, due in class Monday 9/16, is Section 2.2: 11; Section 2.3: 1, 2, 3, 4, 5, 7, 8, 9; and Section 2.4: 1, 2.

**Learning Celebration #1** is on Wednesday, 9/11. It will cover sections 1.4, 1.5, 2.1, and 2.2 of the book. We will start celebrating right at 8:05, so come a few minutes early!

I've updated my office hours above!

**Homework #2**, due in class Monday 9/9, is Section 2.1: 2, 3, 4, 5, 6, 7, 8, 9 and Section 2.2: 1, 2, 3, 4, 5, 6, 7, 9, 10, 15.

**Homework #1**, due in class Wednesday 9/4, is Section 1.4: 1, 2, 4, 6, 7, 8 and Section 1.5: 1, 2, 3, 4, 6, 9, 10.

Here are solutions to the "First Week Quiz" . Please go over the problems I circled on your quiz. You can review the relevant material in the logic notes or my discrete math notes, for which you can find links below.

**Summer reading assignment:** Read these notes on logic , as well as sections 1.1, 1.2, and 1.3 of Taylor's book. Do as many exercises in those sections of Taylor as you can. If you'd like more discussion and examples, check out my notes for Math 2200

(Ch. 1 is logic, Ch. 2 is proofs, Ch. 3 is sets and functions, Ch. 5 is induction). In particular, you should be familiar with basic logic (not, and, or, if...then, if and only if, contrapositive, etc.), with basic proof techniques (direct, contrapositive, contradiction, induction), with basic definitions involving sets and functions (set, union, intersection, subset, function, domain, image, composition, etc.), and with the basic properties of the integers and rational numbers.
Foundations of Analysis (FoA) is a fast-paced and difficult course, and there just isn't enough time to cover all of these "background" topics. It is vital that you begin the course on firm footing. If you're having trouble with the material listed above, come see me in my office or send me an e-mail! You may want to take Math 2200, which is designed to prepare you for FoA by covering this material.
You will not be turning in any of your preparatory work, but
**there will be a short quiz on this material during the first week of classes!**

The textbook is good and will be absolutely necessary. The campus bookstore should now have the correct book listed as a requirement for my section.