MATHEMATICAL ANALYSIS I

Mathematical Analysis I
MATH-4200-01, Fall 2012

Instructor: Victor Roytburd (Amos Eaton 405, Ph. 276-6889, roytbv@rpi.edu)

Tentative Office hours: Tu, Fr, 4-5:30 (or by appointment)

Teaching Assistant: Joseph Rosenthal rosenj2 at rpi.edu

Office hours: We 4-5 (Eaton 424)

Class meets: Tu, Fr 10:00-11:50, Room CARNEG 101

Web-page: http://homepages.rpi.edu/~roytbv//ma1/ma1-f12.html

Course Philosophy

In this course we will revisit some basic notions of calculus. The emphasis will be on the conceptual understanding of calculus and on mathematical rigor. One of the goals of the course is helping students to refine their capacities of reading abstract mathematical texts and develop facilities to write brief mathematical proofs.

I don't have a rigid agenda as to how much material to cover. The pace of the course will be mostly determined by our progress, which will be measured by the performance on the homework assignments and tests. I am somewhat skeptical about benefits of doing homework problems in groups; and there are some educational research findings that suggest that students working individually master the material better than those working in groups. What is definitely helpful is telling your solutions to your friends for "quality control."

Textbooks:

(1) (required) The Way of Analysis, by R. S. Strichartz, Jones and Bartlett 2000 ISBN 0-7637-1497-6.

The book by Strichartz provides very good explanations and motivations for many subjects.
(2) (recommended) Introduction to Analysis, by Maxwell Rosenlicht, Dover 1986 ISBN 0-486-65038-3. This very brief (and cheap) book gives a different perspective on the subject;

(3) (recommended) Principles of Mathematical Analysis, 3^rd Edition, by W. Rudin, McGraw-Hill., Cambridge. The book by Rudin (affectionately known as "Baby Rudin") gives a very concise (about 300 pages) presentation of everything you need to know about rigorous mathematical analysis. The definitions and statements of the principal results are very clear, and the problems are excellent. This particular edition of the book has been in print for about 35 years and you can get it on-line at a very low price. You should keep in mind, however, that the proofs are exceedingly brief and the motivations are frequently skipped.

Homework: There will be weekly homework assignments that will be collected and graded.

Tests: There will be three tests and a final.

Grading: Homework 35%, Tests 40%, Final 25%

Course Outline

I expect to cover the material contained in at least six first chapters of the text by Strichartz. Here are the key topics:

- Basic logic. Countable and uncountable sets, cardinality, Cantor's diagonalization.

- Cauchy sequences. Limits, sups and infs. (The real number system.)

- Open sets and closed sets; Cantor sets.

- Compacts.

- Continuity and uniform continuity. Continuity via open sets.

- The intermediate value theorem. Continuous functions on compact domains. Monotone functions.

- Definition of the derivative. Big O and little o.

- Local behavior of a function and the sign of its derivative. The mean value theorem.

- The calculus of derivatives. Taylor's theorem.

- Integrals of continuous functions. Fundamental theorem of calculus.

- The Riemann integral. Improper integrals.

- Basics of complex numbers.

- Numerical series and sequences.

- Uniform convergence of functional sequences.

- Power series.

- Elementary functions

Student Learning Outcomes

This is a list from the Digital Measures course profile. Except for the first two items it basically repeats the course outline.

- The students will refine their capacities of reading abstract mathematical texts and develop facilities to write brief mathematical proofs.

- The students will improve their abilities to select tools from a broad arsenal of analytical techniques for solving theoretical problems of calculus.

- The students will develop understanding of elements of set theory.

- The students will learn some basic facts of topology in one spatial dimension and of properties of continuous functions.

- The students will learn how the differential calculus is applied to the investigation of local properties of functions.

- The students will become familiar with the rigorous introduction of the Riemann integral and with its basic properties.

- The students will learn how to determine convergence of numerical series and sequences.

Tentative course schedule week by week (all text references are to The Way of Analysis):

Week	Topics	Sections
8/28, 8/31	The logic of quantifiers. Countable and uncountable sets.	1.1-1.4
9/4	Cauchy sequences. Limits, sups, and infs.	2.1 (pp. 25-34), 3.1.1
9/7, 9/11	The real number system. Review of different ways of establishing the real number system.	Notes, 2.1.2, 2.4.1-2
9/11, 9/14	Limits, open sets and closed sets.	3.1-3.2
9/18, 9/21	Compact sets.	3.3
9/25, 9/28	Continuity.	4.1-4.2.1
10/2	Concepts of continuity. Exam review	4.2
10/5	EXAM 1
10/9	(Monday schedule). NO CLASS
10/12	Monotone functions	4.2
10/16, 10/19	Derivative and its properties.	5.1-5.2
10/23, 10/26	Properties of the derivative. Calculus rules.	5.2-5.3
10/30	Taylor’s theorem. Integral of continuous functions.	5.3.3-5.4, 6.1
11/2	The Riemann integral. Improper integrals.	6.2-6.3
11/6
11/9	EXAM 2
11/13		6.3
11/16	Primer of complex numbers. Numerical series.	7.1, 7.2
11/20	Numerical series. Functional series	7.2, 7.3
11/22-11/26	Thanksgiving break
11/27	Power series. Elementary Functions	7.4, 8.1
11/30	Elementary functions
12/4	Weierstrass's Approximation Theorem	7.5
12/7	Last day of classes

12/12	Final Exam: 3-6pm in WALKER 5113

Statement of Academic Integrity

The guiding principle is that work that you present for grading as your own should in fact BE your own. The following statement on Academic Integrity is from the course profile on Digital Measures:

Student-teacher relationships are built on trust. For example, students must trust that teachers have made appropriate decisions about the structure and content of the courses they teach, and teachers must trust that the assignments that students turn in are their own. Acts, which violate this trust, undermine the educational process. The Rensselaer Handbook of Student Rights and Responsibilities define various forms of Academic Dishonesty and you should make yourself familiar with these. In this class, all assignments that are turned in for a grade must represent the student's own work. In cases where help was received, or teamwork was allowed, a notation on the assignment should indicate your collaboration. Submission of any assignment that is in violation of this policy may result in a penalty of a grade of F. If you have any question concerning this policy before submitting an assignment, please ask for clarification.

Homework Assignments

Assignment #1, due 4 September 2012. A copy of problems from the textbook for this assignment

Assignment #2, due 11 September

Assignment #3, due 18 September

Solutions to problems from Assignments 1--3

Assignment #4, due 25 September

Assignment #5, due 2 October. Assignment #5 Solutions.

There will be no class on Tuesday October 9 (Monday, Columbus Day schedule)

Assignment #6, due 16 October. The due date is postponed to 10/18

Assignment #7, due 23 October

Assignment #8, due 30 October

There shall be a 5 points penalty per day for late assignments.

Assignment #9, due 6 November

Solutions to homework problems: Set 1 [problems 6.1 (4.2.7), 6.3-4 (5.1.3)]; Set 2 [problem 6.2]; Set 3 [problems 7.5 (5.3.13), 8.1 (5.4.20), 7.3 (5.3.7), 6.5 (5.1.9), 7.2 (5.2.2)]; Set 4 [problems 8.3 (6.1.4), 8.4 (6.1.10)]; Set 5 [Assignment 9]

Assignment #10, due 20 November. Solutions to homework problems

Exams

Test 1 will be given on Friday, 5 October 2012, in class. (Test solutions.) The test will cover the following topics; references are to the Textbook:

- Countable and uncountable sets, cardinality, Cantor's diagonalization. (Sec. 1.2)

- Cauchy sequences. (Know results of Sec. 2.3, see the summary in Sec. 2.5). Limits, sups and infs. (Sec. 3.1)

- Open sets and closed sets; Cantor sets. (3.2)

- Compacts. (3.3)

- Continuity and uniform continuity. Continuity via open sets. (4.1)

- The intermediate value theorem. (4.2.1)

Test questions will be similar to the shorter homework questions from the homework assignments 1-5. You will be allowed the use of one sheet of hand-written notes.

To give you some idea of the possible length and degree of difficulty of the test, here is a copy of the test I gave when I last taught this course several years ago. Test 1 fall 2006. The scope of your test will be different, however.

Test 2 will be given on Friday, 9 November 2012, in class. The test will cover the following topics; references are to the Textbook:

- Definition of the derivative. Big O and little o. (5.1)

- Local behavior of a function and the sign of its derivative. The mean value theorem. (5.2)

- The calculus of derivatives. Inverse function theorem. (5.3)

- Higher derivatives and Taylor's theorem. (5.4)

- Integrals of continuous functions. Fundamental theorem of calculus. (6.1)

- The Riemann integral. Improper integrals. (6.2-6.3)

The test format will be similar to that of Test 1; questions will be similar to the shorter homework questions from the homework assignments 6-9. You will be allowed the use of one sheet of hand-written notes. Test 2 solutions

Final Exam will be given on Wednesday 12/12, 3-6pm in WALKER 5113. The test will cover the material included in the homework assignments; that is, the last two topics of the course outline (power series and elementary functions) are not included.

Here is a copy of the final exam given in 2006. Your exam will be at least one problem shorter than this one. You will be allowed the use of one sheet of hand-written notes.

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Last updated 12/05/2012