| This course ended on December 14, 2019 |
Math 331: Foundations of Analysis
Department of Mathematics and Computer Science
Hobart and William Smith Colleges
Fall 2019.
Instructor: David J. Eck (eck@hws.edu)
Syllabus: http://math.hws.edu/eck/courses/math331_f19.html
Monday, Wednesday, Friday, 1:30–2:30 PM
Room Gulick 223.
Fifteenth Week: December 9 and Final Exam
Monday, December 9, is the last day of class. The final exam will be given in our usual classroom on Sunday, December 15, at 8:30 AM.
You might want to review the Test 1 Study Guide and the Test 2 Study Guide.
I will have the following office hours during reading period and exams, and I might well be in my office at other times as well:
Monday, December 9: 10:15 to 10:50 AM, 12:15 to 1:15 PM
Wednesday, December 11: 12:00 to 3:00 PM
Friday, December 13: 12:00 to 3:00 PM
Saturday, December 14: 11:00 AM to 1:20 PM
Homework 11 is due in class on Monday, December 9. If you want to do rewrites on Homework 11, you can pick up the graded homework during my office hours on Wednesday and turn in your rewrites during my office hours on Friday, December 13.
Here are my Sample answers to Homework 10
Here is the web application that I showed briefly in class on Friday, which shows animated sequences of functionss:
http://math.hws.edu/eck/js/graphs/animated-graph.html
Here are my Sample answers to Homework 11
Fourteenth Week: December 2, 4, and 6
The reading for this last full week of classes is Section 4.6, which covers series of functions, including power series and Taylor series. The following homework is due on the last day of class, Monday, December 9:
Homework #11:
Section 4.5: Exercises 2 and 11
Section 4.6: Exercises 3 and 5
Remember that Homework 10 is due in class on Wednesday.
And here are my sample answers to Homework 9.
Thirteenth Week: November 18, 20, and 22
We continue with Chapter 4. We will complete Sections 4.3 and 4.4, and at least start on Section 4.5. Sections 4.3 and 4.4 deal with infinite series of numbers. Section 4.5 looks at infinite sequences of functions, including the important topic of uniform convergence.
The following homework is due in class on Friday, November 22. Note: We have decided to postpone the due date for Homework 10 until after break.
The following homework is due in class, on Wednesday, December 4.
There is no class next week because of Thanksgiving break. Have a great Thanksgiving!
Twelfth Week: November 11, 13, and 15
We will turn to sequences and series for the rest of the course. The topics for this week are monotone and Cauchy sequences (Section 4.2) and complete metric spaces. We should have time by Friday to get a start on infinite series (Section 4.3).
Here is the reading on complete metric spaces:
Handout 5: Complete Metric Spaces
The following homework is due in class on Friday, November 15:
Eleventh Week: November 4, 6, and 8
There is a test on Friday, November 8. A study guide was handed out in class on November 1.
Although there is no assigned homework on Sections 3.5 and 3.6, there is a set of suggested exercises on integration that includes that material.
We will cover Section 3.7 this week. The topic is Taylor polynomials and Taylor's theorem with remainder. This material will not be on the test. It is a natural introduction to the final third of the course, where we will cover sequences, series, and power series.
Here are my sample answers for homework #8.
And here are my sample answers for the suggested exercises. Note the correction to problem number 6!
Tenth Week: October 28 and 30; November 1
Last week, we only just started section 3.5 by proving that continuous functions are Riemann integrable. We will continue with Chapter 3 this week. You should read the remainder of that chapter, Sections 5 through 7, although we will not finish Section 3.7 until next week. The main topics for the week are properties of the Riemann integral (Section 3.5) and the Fundamental Theorems of Calculus (Section 3.6). If time permits, we will start work on Taylor polynomials (Section 3.7).
There is an in-class test coming up next week on Friday, November 8, which will cover material from the text book through Chapter 3.
Homework #8 is due this Friday, November 1.
Here are my sample answers for homework #7.
Ninth Week: October 21, 23, and 25
The take-home midterm exam is due in class on Wednesday.
We will cover L'Hopital's rule on Monday, and we will move on to the Riemann integral by Wednesday. The reading is Sections 3.4 and 3.5, although we probably won't finish 3.5 this week.
My answers for homework 6 are available:
Sample Answers for Homework #6
The following homework is due next Friday, November 1:
Homework #8: Section 3.3, # 10, 12, 14
Section 3.4, # 4, 5, 8
[More might be added!]
Eighth Week: October 16 and 18
There is no class on Monday this week, because of Fall break.
We will continue with Chapter 3 and our discussion of the derivative. You should read Chapter 3, Sections 0 to 3, although we probably won't finish Section 3 this week.
The take-home midterm exam will be handed out in class on Friday and will be due next Wednesday, October 23.
My answers for homework 5 are available:
Sample Answers for Homework #5
The following homework is due this Friday, October 18:
Homework #7: Section 3.1, # 3, 6, 8, 11, and 12
Seventh Week:October 7, 9, and 11
We will be working on Section 2.6 at the beginning of the week, covering the Intermediate Value Theorem, the Extreme Value Theorem [called the Min-Max Theorem in the textbook], and uniform continuity. We will then be moving on to Chapter 3.
The following homework is due on Friday, October 11:
I wrote a handout on connected metric spaces, but looking at the proofs in that handout, I've decided that they are too technical to cover quickly. However, I am making the handout available as an optional reading. You might at least want to take a look at the definitions and the statements of the theorems:
Handout #4: Connected Metric Spaces
Sixth Week: September 30; October 2 and 4
The reading for the week is Section 2.5 and a handout that will be distributed in class on Monday. The topic is continuity. Section 2.5 deals with continuity of real-valued functions on the real numbers, while the handout introduces continuity of functions between metric spaces. Here is the handout:
Handout #3: Sequences and Continuity in Metric Spaces
The following homework from the textbook is due in class on Friday, October 4:
Section 2.2, Exercise 9
Section 2.3, Exercises 5 and 6
Section 2.4, Exercise 10
Section 2.5, Exercises 1, 2, 6, and 7
Fifth Week: September 23, 25, and 27
There is a test on Wednesday, September 25. A study guide for the test was handed out in class on Friday. My answers for Homework #3 and my answers for Homework #4 were also handed out on Friday.
We will spend Monday reviewing and working on limit proofs. The reading for the week is Chapter 2, Sections 3 and 4, but those sections are not on the test, and you can put off reading them until after the test. Section 2.3 covers the theory of limits of functions, some of which we have already done. Section 2.4 covers one-sided limits, infinite limits, and limits at infinity. One-sided limits are important, but we will probably not do very much with infinite limits or limits at infinity.
Fourth Week: September 16, 18, and 20
The new reading for the week is Sections 2.2 and 4.1. The background material in Sections 2.0, 2.1, and 4.0 is interesting but optional. We will finish up our look at compactness. We will then move on to the definitions of limits of sequences and of functions.
There is a test coming up next Wednesday, September 25. A review/information sheet about the test will be handed out on Friday.
Homework #4 is due in class on Friday. I will hand out my answers to Homework 4 at that time — which means that there will be no opportunity for rewrites on this homework.
Third Week: September 9, 11, and 13
We will finish up the handout on metric spaces on Monday by looking at accumulation points and closed sets. We will then cover Section 1.4, which covers the Heine-Borel and Bolzano-Weirstrass Theorems for closed, bounded intervals. There will also be a new handout on compact sets, which we might or might not finish this week:
The reading for the week is to read Section 1.4 and the compactness handout. The following written homework is due on Friday, September 13:
Section 1.4, Exercises 3, 4, and 14
Handout 1 on Metric Spaces, Exercise 1 through 7
Second Week: September 2, 4, and 6
We will continue with Section 1.3. You should be sure to read that section before class on Monday. We will then turn to an extra topic, metric spaces. The reading is a handout, which was given out in class last Friday and is also available here:
The following homework from Section 1.3 is due in class this Friday, September 6:
Section 1.3, Exercises 3, 4, 11, 12, 17
Homework on metric spaces will be due later.
I will need to leave my Thursday office hours early this week, at 1:50. However, I will be available earlier in the day from 10:00 to 12:00.
I have made the LaTeX source code for the Metric Spaces handout available on Overleaf.com. You can use this link to view the source:
https://www.overleaf.com/read/bdcpgxnnwdbb
First Week: August 26, 28, and 30
Welcome to the course!
The reading for the week is Chapter 1, Sections 1.0 through 1.2. Please read them! We might start covering Section 1.3 in class on Friday. From Section 1.1, you should be familiar with irrational numbers and the Fundamental Theorem of Arithmetic, but we will not spend a great deal of time on that. Section 1.2 gives a "construction" of the set of real numbers; that is, it defines a specific mathematical object that has the properties that we expect the real numbers to have.
The following homework exercises are due in class next Monday, September 2. For this first homework assignment, you can write your solutions by hand, or you can type them up using LaTeX if you prefer to do that. We will discuss whether LaTeX should be mandatory for later assignments. As stated in the syllabus, you are allowed and encouraged to discuss homework assignments with other people in the class, but you should write up your own solutions in your own words.
Section 1.1, Exercises 9, 12, 14, 16
Section 1.2, Exercises 4, 5, 13, 17, 18