# Math 319 - Foundations of Analysis - Spring 2006

Thise is an introductory courses in analysis. Math majors must take either this course or MAT 320.

Analysis, one of the pillars of modern mathematics, begins with a rigorous development of single-variable calculus. Thus, these courses serve the important purpose of teaching you how to rigorously prove and apply results in calculus, including results related to the notions of limit, continuity, derivative, integral, and infinite series. Inherent in all of these notions is the concept of approximation. As we shall see, a good grasp of this latter concept is essential not only in proving ''pure'' results in analysis, but is also crucial in ''applied'' problems requiring estimations. In any approximation a key question is ``how do you estimate the error''? In the first part of this course, we will look at some types of algebraic manipulations that can be used in error estimation; we will also look at more powerful methods involving the mean value theorem for derivatives.

# Difference between MAT 319 and MAT 320:

MAT 320 is more comprehensive and provides a firm grounding for further study. MAT 319 has more of an emphasis on topics which arise in high-school calculus. Students planning to go on to graduate school in mathematics are advised to take MAT 322 and MAT 324 as well. Students wanting to take MAT 322 or MAT 324 (or the seminars MAT 401 or MAT 402) need to take MAT 320, not MAT 319.

# Prerequisites:

C or higher in MAT 200 or permission of instructor; plus one of the following: MAT 203, 205, 211, AMS 261, or A- or higher in MAT 127, 132, 142 or AMS 161. Anyone lacking these prerequisites risks deregistration.

# Text:

D. Geller, A Bridge to Analysis, available only at the bookstore. on campus. We will cover Chapters 1-3, sections 4.1, 4.2, 4.2.1, 4.3, 4.3.1, Chapter 5 (omitting pages 216-224), and Chapter 6 (omitting pages 288-294 and also section 6.5; but you should know the statement of Theorem 6.21 on pages 311-312).

# Office hours:

Tuesdays and Thursdays from 12:45-2:15, in 4-100B Math or P-143 Math.

The final course grades will be determined as follows: homework 10%, two midterms 25% each, final exam 40%.
The grades of \$A^-\$ and \$A\$ will be reserved for students who demonstrate a substantial ability to apply the concepts of these courses in new and somewhat creative ways.
Please note that, on midterms and exams, ``proofs'' that contain errors will receive only a limited amount of partial credit, if any at all. Also please note that there will be no curve in determining grades. However, the total number of points available on the second midterm and on the final exam will exceed 100; these exams will be taken out of 100. Incompletes will be granted only if documented circumstances beyond your control prevent you from completing the course work.

# Homework:

The only way to learn the material is to work problems for yourself. Each week, you should attempt to do all of the problems from the sections which are covered in class. We will ask you to hand some problems in. Your homework will be graded meticulously and will give you vital feedback on where you are making mistakes.
Homework is a means to an end, the ``end'' being for you to learn the material. We encourage you to work on homework together with friends. In this course, we will never prosecute anyone for academic dishonesty on any issue relating to homework. If you hand in complete, correct solutions, you will get full credit for them, no matter how you obtained them. If someone regularly ``does'' the homework by copying from friends or from solution manuals, they are only cheating themselves, since this is not a way to learn the material. Never be shy to ask us how to do a homework problem, even if you handed in a copied solution that you do not understand. You will not be prosecuted or condemned for this, and we will be only too glad to help you.

# Assignments:

Homework due in recitation on Wed. Feb 1 : problems 12.5, 13, 16, 17, 18 at the end of section 1.4.
Homework due in recitation on Wed. Feb 8 : problems 23, 26, 27, 29 and 32 (in sections 1.5 and 1.6).
Homework due in recitation on Wed. Feb 15 : problems 34, 36 (in section 1.7), problems 2, 3, 5 (in section 2.1), problem 6 (in section 2.2).
Homework due in recitation on Wed. Feb 22 : problems 8, 9, 10, 11 (in section 2.2); problems 17, 19, 20 (in section 2.3); problems 24, 27 (in section 2.4).
Homework due in recitation on Fri. Mar 3 : 33.1, 33.2, 33.5, 33.6, 36, 37 (in chapter 2).
Homework due in class on Tues. Mar 7 : problems 38 and 39 (in section 2.6); problems 1, 2 and 5 (in section 3.1).
First midterm Thursday March 9, in class, on Chapters 1-3.
Homework due in recitation on Fri. Mar 17 : problems 2, 4, 5 (in chapter 4).
Homework due in recitation on Fri. Mar 24: problems 7, 8, 9, 16, 17, 18.
Homework due in recitation on Fri. Mar 31: problems 23, 25, 26, 27, 28 (in chapter 4).
Homework due in recitation on Fri. Apr. 7: problems 2, 3, 7, 8, 9, 10, 12 (in chapter 5).
Homework due in class on Tue. Apr. 25: problems 22, 23, 24, 29, 30 (in chapter 5).
Test on Chapters 4 and 5: in class, Thursday, April 27. Here is the very important three-page study guide. (It's a PDF file, please let me know if you have a problem reading it.) Please learn the material indicated in this study guide!
Homework due in recitation on Fri. May 5: 4, 7, 8, 10(a)(b)(c), and 11 of chapter 6.
Final Exam: Thursday May 11, 11-1:30 p.m., in the regular classroom.
The final exam will be about 50% on Chapters 1-3. The other 50% will put a greater emphasis on Chapter 6, and a lesser emphasis on Chapters 4 and 5. Here again is the study guide for Chapters 4 and 5 which is the same as for the second midterm. Here is the very important two-page study guide for Chapter 6.

I will have a review session for Chapter 6 on Tuesday, May 9, during the regular class time (11:20 a.m. to 12:40 p.m.). Meet at my office 4-100B Math Building (in the hallway leading from the Math to the Physics building), not in our regular classroom.
If, after the course is over, you wish to request a change in your final grade, or would like any further information about how your grade was arrived at, please send me a letter (not an E-mail); you will receive a written reply. Grade change requests will be dealt with in writing only; that way, we have a written record of what the student says, and what we reply.