Math 203, Fall 2015

Honors Mathematics I

Instructor                 John E. M^cCarthy
Class   MTuThF 11.00-12.00 in 115 Cupples I

Discussion Sections  Th 10-11 in 199 Cupples I

JM Office   105 Cupples I
JM Office Hours M 12.00-1.00, Tu 10.00-11.00, Th 12.00-1.00, and by appointment
Phone  935-6753

Teaching Assistant Tokio Sasaki

TS Office Hours        M 3.00-4.00, W 12.00-1.00, F 2.00-4.00 in Room 6, Cupples I

Exams    There will be three exams in the course:

                        1) Exam 1       In class. Friday September 18.
                        2) Exam 2       Take Home. Due Monday October 26.
                        3) Exam 3       Final exam. Tuesday December 15, 10.30-12.30.

Homework

There will be weekly homework sets during the semester, assigned on Tuesday and due the following Tuesday.

Homework 1 due September 1.
Homework 2 due September 8.
Homework 3 due September 15.
Homework 4 due September 22.
Homework 5 due September 29.
Homework 6 due October 6.
Homework 7 due October 13.
Homework 8 due October 20.
Homework 9 due November 3.
Homework 10 due November 10.
Homework 11 due November 17.
Homework 12 due November 24.
Homework 13 due Thursday December 3.

Prerequisites

AP Calculus BC, Score of 5, or equivalent.

Description

This is the first half of a one-year calculus sequence for first year students with a strong interest in mathematics.
The course will be challenging, with an emphasis on rigor and proofs.
If you complete the year-long sequence, you will not only cover all of Math 233, but most of  Math 318 and Math 310, and some of Math 309.
More importantly, you will have learned what real mathematics is!

In the first semester, we will cover the basics of proofs (including addressing why proofs are important, and not just a formal exercise),
 revisit one variable calculus, and spend some time on vectors and matrices.

Content

Here is a very tentative schedule. We will not stick to it closely.

Week 1:         "And" and "Or". Sets, functions, injections, surjections, bijections. Images and inverses.
Week 2:         Sequences. Russell's paradox. Partial Orderings. Equivalences.
Week 3:         Propositional Logic. Quantifiers. Induction.
Week 4:         Limits. Continuity. Limit laws. Pointwise and uniform convergence.
Week 5:         Integration of step functions, monotonic functions, polynomials. Linearity of the integral.
Week 6:         Integration as area. Integration of trigonometric functions. Applications of integration.
Week 7:         The least upper bound property. The intermediate value theorem. The extreme value theorem.
Week 8:         Differentiation. Tangents. Leibniz's rule. Chain rule. Mean value theorem.
Week 9:         Fundamental theorem of calculus. Integration by substitution. Integration by parts.
Week 10:       Sequences/series of real numbers. Comparison test, ratio test, root test. Integral test. Alternating series
Week 11:       Sequences/series of functions. Power series. Taylor series. Convergence of Taylor series.
Week 12:       Vectors. Dot products. Projections. Cross products.
Week 13:       Vector spaces. Linear independence. Bases.
Week 14:       Linear transformations. Matrices. Matrix algebra.

Basis for Grading

Each midterm and the homework will be 20% of your grade, the final will be 40%. If you do well on the final, this grade can be substituted for one of your midterms.

Homework

Homework is an extremely important part of the course. Whilst talking to other people about it is not dis-allowed, too often this degenerates into one person solving the problem, and other people copying them (often justified to themselves by saying "I provide the ideas, X does the details" - but the details are the key. If you can't translate the idea into a real proof, you don't understand the material well enough). So I shall introduce the following rules:
(a) You can only talk to some-one else about a problem if you have made a genuine effort to solve it yourself.
(b) You must write up the solutions on your own. Suspiciously similar write-ups will receive 0 points.

Class

I do expect you to come to class every day, and to participate in class discussions. I also expect you to stay abreast of the material we are covering, and may call on you at any time to answer a question.

The Discussion Sections are very strongly recommended, but I understand some students will have unavoidable conflicts with them.

Class etiquette: don't be disruptive or discourteous. No beeping, ringing, crunching, rustling, leaving early or arriving late. No texting, sleeping, checkingyour phone.

Texts Transition to Higher Mathematics: Structure and Proof  by Bob Dumas and John McCarthy ( Available free here) 
Calculus, Volume I by Tom Apostol, (Wiley) Second Edition, 1967
Calculus, Volume II  by Tom Apostol, (Wiley) Second Edition, 1969 (just for 204, in the Spring)

Note on the texts: The two books by Apostol are very expensive. They are not required texts for the course, though they will be useful.
I recommend buying used copies if you can.
There will be a copy of both Volume I and II on two-hour reserve at the Library.

Additional Reading

Multivariable Mathematics  by Theodore Shifrin

Vector Calculus by J. Marsden and A. Tromba