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Math 414 – Advanced Calculus

Degrees and Courses

4 semester credits

Catalog Description

MATH 414 ADVANCED CALCULUS (4-0-4)(S). Infinite series, sequences and series of functions, uniform convergence, theory of integration (Riemann and Stieltjes), further topics as time permits. PREREQ: MATH 275, MATH 301 and MATH 314. Offered spring of odd-numbered years, subject to sufficient demand.

Prerequisites

This course requires a thorough mastery of the material of MATH 314 (Foundations of Analysis). The prerequisites of MATH 275 (Multivariable and Vector Calculus) and MATH 301 (Linear Algebra) help ensure an additional element of mathematical maturity and breadth of knowledge.

Jurisdiction

This course is not directly controlled by a departmental committee. The instructor has jurisdiction over this course, though some effort is made to coordinate the selection of a text for this course with that for MATH 314.

Objectives

This course is intended for mathematics majors and is also of interest to mathematically strong students in the sciences or engineering; it continues the study of basic principles of real analysis from the point where MATH 314 leaves off. The concept of integration is studied from a much more thorough perspective than that of introductory calculus, beginning with the Riemann integral and later advancing to Riemann-Stieltjes integration. The basic notions of convergence and continuity are extended and combined to study sequences and series of functions. The interaction among convergence, differentiation and integration is a primary theme of the course. Depending on the depth to which these basic topics are covered, there may be time for a glimpse at a more advanced topic such as measure, category, or multivariable differentiation and integration.

At the conclusion of this course, the student should have a solid understanding of these basic principles of analysis, should have become quite adept at writing proofs, and should have at least minimal preparation needed for an introductory real analysis course at the graduate level.

Learning objectives and assessment of learning objectives

Students will be assessed by evaluating their ability to do problems based on the learning objectives. The problems will occur in several contexts:

  • Periodic problem sets for homework serve both as learning and as assessment tools.
  • Problems given on take-home exams are designed to evaluate a student’s ability to solve more complicated and time consuming problems than can be reasonably completed on an in-class exam.
  • Problems given on in-class exams are designed to see if students can use the tools that have been developed to solve straight forward problems in a limited amount of time.
  • Students may also be expected to present solutions in class or to work in small groups to solve problems. Both of these situations permit the instructor to assess the student’s ability to communicate effectively using the language of mathematics.

Text

Most texts suitable for this course begin by covering the material of the prerequisite course MATH 314, so an effort is made to use the same text for both courses. The following two texts are recommended on this basis:
J.A. Friday, Introductory Analysis: The Theory of Calculus, Harcourt Brace Jovanovich, 1987.
Michael J. Schramm, Introduction to Real Analysis, Prentice Hall, 1996.

Topics and Approximate Timeline

The following table is based on a typical semester schedule-60 class meetings of 50 minutes each. The exact order of topics and allocation of time will vary somewhat.

Topic

Number of Lectures

The Riemann Integral

10

            Riemann sums and definition of the integral
            Integrability criteria
            The fundamental theorem of calculus
            Improper integrals
Infinite Series

10

            Convergence tests
            Absolute & conditional convergence
            Rearrangements
Sequences and series of functions

15

            Pointwise & uniform convergence
            Equicontinuous families
            Differentiation and integration of sequences
            Power series
The Riemann-Stieltjes Integral

10

            Functions of bounded variation
            Riemann-Stieltjes sums
            Integration by parts
Further Topics (as time permits)

10

            Introduction to measure theory
            First and second category
            Calculus of several variables
Exams and review

5

Format, Student Activities, and Grades

Class meetings involve a combination of lecture, questions and discussion. Homework is an important part of the course. The instructor chooses the exact grading scheme, with a typical distribution being:

Homework

100

2 Exams

200

Final Exam

200

Total Points

500

Letter grades are usually based on a standard scale in which 90% of the total possible points guarantees an A , 80% a B, and 70% a C, with the instructor having the discretion to raise or lower these cut-offs if warranted.

Updated Fall 2005