MATH 633: Advanced Calculus I

Syllabus - Fall 2004

Instructor: Marianne Korten
Office: Cardwell 234, 532-0567
Office Hours: Tuesdays, 11:30-12:20

Textbook:
Advanced Calculus: An Introduction to Analysis - Third Edition, Watson Fulks, John Wiley and Sons, New York, 1978.

Course description:

In this course we will revise, reinforce, and expand the concepts and tehorems covered in Calculus 1 and 2 (we will work on Calculus 3 material in Advanced Calculus II next semester). The focus will now be on the proofs of the results: the goal is to develop skills constructing and writing proofs. You wll be expected to write clean and rigorous arguments with correct logic and grammar.
This is a foundational course fot graduate level courses in mathematics and elsewhere, in particular for Analysis and Topology.

Most graduate schools will give new students a placement test upon arrival, typically the content of Math 633 covers the Analysis part of this exam. Here is the syllabus of the counseling exam of our graduate program; the page contains a link to an archive of our old exams.

Syllabus:

Functions, sequences, limits, continuity. Open and closed sets in R. Bounded sets, supremum, infimum. Cluster points. Accumulation points. Limit superior and limit inferior. Sequences. Functions. Limits, operations with limits of sequences and functions. Monotone sequences. Monotone functions. The Cauchy criterion for convergence. Continuity. Operations with continuous functions. The intermediate value property. Properties of continuous functions on closed intervals. Inverses of strictly monotone, continuous functions. Uniform continuity.

Differentiation of real-valued functions. The derivative. Chain rule. The mean value theorem. Cauchy's mean value theorem. L'Hospital's rule. Taylor's theorem. Estimation of the remainder. Extremes, concavity.

The Riemann Integral. Properties of definite integrals. The Fundamental Theorem of Calculus. Integration by substitution. Integration by parts. Mean value theorems for integrals. Integrals of discontinuous functions.

Numerical series. Absolute and conditional convergence. Alternating series. Series of nonnegative terms: comparison, ratio, and root tests.

Sequences and series of functions. Uniform convergence. Consequences of uniform convergence. Convergence tests. Power series. Interval of convergence. Taylor and Mclaurin series. Arithmetic of power series.

Improper integrals. Conditional and absolute convergence. Improper integrals with nonnegative integrands. Principal value. Convergence tests.

Grading policy:

Attendance and participation are requiered. Homework will be assigned weekely.
Final grades will be based on witten work and participation, as follows:
30% for homework, 30% for the midterm, 10% for your participation in class (this includes working on problems in class), and 30% for the final.

You are encouraged and expected to work on the homework exercises in groups, and to subject your work to revision by classmates before handing it in. You are strongly encouraged to use the advanced help sessions staffed by the mathematics department to clear doubts and questions and fine-tune your proofs.

Journal

Here I will  post your (roughly weekly) assignments.