# Math 331: Foundations of Analysis I

```   Department of Mathematics and Computer Science
Hobart and William Smith Colleges

Spring 2008.

Instructor:  David J. Eck.

Monday, Wednesday, Friday, 12:20 to 1:15.
Room Eaton 105.
```

### About The Course and the Textbook

This course covers the real numbers, limits, continuity, differentiation, integration, and infinite series and sequences -- the same topics as Calculus I and II. It is, however, decidedly not just another calculus course. In your calculus courses, you mostly learned some techniques and used them to solve some problems. For example, you memorized a list of rules for differentiation, and you used them to find the derivatives of a lot of functions (often given by formulas that are unlikely ever to turn up in a real-world application). You might have done some simple proofs, and you probably spent some time developing intuition about why certain things are true, but for the most part, "calculus" meant "calculating" based on some memorized formulas.

In this course, you will do very little calculating. Instead, you will be doing mathematics. That is, we will start from first principles and build up a theory from there, based on rigorous logical reasoning. The first principles are the definition and basic properties of the real numbers. The theory that we will develop is the real Calculus, the theory of the infinite and the infinitesimal that has taken some 2500 years to develop.

The textbook for the course is Foundations of Analysis by Professors David Belding and Kevin Mitchell, which is available from http://www.lulu.com/content/495090. This book was written specifically as a textbook for Foundations of Analysis I and II. We will cover chapters 1 through 4. According the preface, Belding and Mitchell's goals in writing the book were as follows:

• First, we wished to help students develop some understanding and control of the language of mathematics: definitions, theorems, and proofs.
• Secondly, we wanted to present the central concepts and theory of calculus thoroughly and clearly enough to provide a secure foundation for more advanced mathematics courses and enable students to grasp some of the unity and beauty of the subject.
• Finally, we wished to foster an appreciation for the living, human nature of mathematics, and a sense of how mathematics is created and evolves.

### Assignments

Homework will be assigned each week, except when there is a test. You are encouraged to work together on homework problems with other people in the course. However, you should write up your own solutions to the problems in your own words. Most, but not all, of the problems will be taken from the textbook. Most of the problems will ask you to write a proof, and all problems will require you to show your work to get any credit.

Homework will not be accepted late. In general, sample solutions to the homework problems will be handed out at the same time that your solutions are collected.

There will be at least a few student presentations of solutions to homework problems or other material. We will try this at the beginning of the term, see how it goes, and discuss the extent to which we want to continue the practice. Any grades given for presentations will be counted as part of your homework grade.

### Tests

There will be two tests in this course and a final exam at the end. Each of these will consist of two parts: There will be an in-class part that will cover mainly definitions, concepts, statements of theorems, and fairly simple proofs. And there will be a take-home part that will consist of harder proofs and other longer problems. The take-home parts of the tests will be similar to homework assignments, except that you will not be allowed to work together on them.

The first in-class test will be given on Monday, February 25 and will cover Chapters 1 and 2. The second in-class test is on Monday, April 7 and will cover Chapter 3. The take-home part of each test will be handed out during the in-class test and will be due the following Friday.

The in-class part of the final exam will be given during the scheduled final exam period: Sunday, May 11, at 7:00 PM. The schedule for the take-home part of the final exam will be decided later. Note that the final exam will be cumulative, with some emphasis on Chapter 4.

```            Homework:        35%
First Test:      20%
Second Test:     20%
Final Exam:      25%
```

I reserve the right to adjust your grade downwards if you miss more than one or two classes without a good excuse.

In my grading scale, an A corresponds to 90--100%, B to 80--89%, C to 65--79%, D to 50--64%, and F to 0--49%. Grades near the endpoints of a range get a plus or minus.

### Office Hours, E-mail, and Web

My office is room 313 in Lansing Hall. My office phone extension is 3398. I am on campus most days, and you are welcome to come in anytime you can find me there. My regular office hours are Monday, Wednesday, and Friday, 10:30 to 12:00, but I will often be there at other times. Office hours are times when I promise to try my best to be in my office. I do not generally make appointments during my office hours, since they are times when I am available to students on a first-come, first-served basis. When necessary, we can make appointments for meetings outside my scheduled office hours at a mutually agreeable time.

My e-mail address is eck@hws.edu. E-mail is good way to communicate with me, since I usually answer messages within a day of the time I receive them.

There is a short Web page for this course at http://math.hws.edu/eck/math331/. I will post weekly readings and assignments on that page.

### Tentative Schedule

Here is a tentative weekly schedule of topics for the course. We will definitely cover Chapters 1 through 4 of the textbook, but the amount of time spent on each section might end up being rather different from what is indicated here.

Jan. 21 Constructing the Real Numbers. Sections 1.0, 1.1, and 1.2
Jan. 28 Properties of the Real Numbers. Sections 1.3 and 1.4
Feb. 4 Functions and Limits. Sections 2.0, 2.1, and 2.2
Feb. 11 Theory of Limits. Sections 2.3 and 2.4
Feb. 18 Continuity. Sections 2.5 and 2.6
Feb. 25 Test on Monday, February 25.
The Derivative.
Sections 3.0 and 3.1
Mar. 3 Properties of the Derivative. Sections 3.2 and 3.3
Mar. 10 The Riemann Integral.
No class Friday, because of Spring Break.
Section 3.4
Spring Break, March 14 through 23
Mar. 24 Properties of the Integral. Sections 3.5 and 3.6
Mar. 31 Finish Integration; Taylor Polynomials. Sections 3.6 and 3.7
Apr. 7 Test on Monday, April 7.
Infinite Sequences.
Sections 4.0 and 4.1
Apr. 14 More on Sequences; Infinite Series. Sections 4.2 and 4.3.
Apr. 21 Convergence Tests for Infinite Series. Sections 4.3 and 4.4.
Apr. 28 Sequences of Functions. Sections 4.5 and 4.6.
May. 5 Review.
Classes end on Monday, May 5.

Final Exam, Sunday May 11, 7:00 PM