**Math 203, Fall 2015 **

**Honors Mathematics I **

**Instructor **John E. M^{c}Carthy

**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.

**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