Department of Mathematics Generic Syllabus
Boise State University Updated Fall 1998

Math 314
Foundations of Analysis

Catalog Description

MATH 314 FOUNDATIONS OF ANALYSIS (3-0-3)(F). The real number system, completeness and compactness, sequences, continuity, foundations of the calculus. PREREQ: MATH 187and MATH 175 or PERM/INST.

Prerequisites

MATH 314 requires MATH 187 (Discrete and Foundational Mathematics), which provides needed background in elementary set theory, logic, proof techniques and induction. It also requires at least 2 semesters of calculus (MATH 170,171,175) to ensure that the students have at least an intuitive acquaintance with the calculus, up through an introduction to infinite series.

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 M 414.

Learning Objectives

The objectives of Foundations of Analysis coincide with three of the four departmental goals. M 314 does not stress the applications of mathematics or the impact of technology, but it does stress the ideas of abstraction, aesthetics, the development of mathematical tools and the use of the language of mathematics. This course is required of mathematics/secondary education majors, and they usually account for the majority of the enrollment. It is also intended to prepare mathematics majors for further study of analysis in MATH 414, Advanced Calculus. Upon completion of this course, students should:

Be able to write proofs based on logical deduction from the basic principles.
Be able to use the language of mathematics correctly and effectively.
Be able to identify the ideas on which real analysis is based.
Be able to state the fundamental concepts and prove the basic theorems of analysis, principally those involving convergence and continuity.

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.

Topics and Approximate Timeline

The following table is based on a typical semester schedule of 45 class meetings of 50 minutes each. The exact order of topics and allocation of time will vary slightly.


	Number of
Topic	Meetings
Order and completeness properties of the real numbers	6
The topology of the reals	9
Bolzano-Weierstrass Theorem
Compactness
Nested Interval Theorem
Sequences and Series	9
Definition of Convergence
Basic Limit Theorems
Cauchy Sequences
Continuity	9
Definition of Continuity and Uniform Continuity
Extreme and Intermediate Value Theorems
The Derivative	6
The Mean Value Theorem
Taylor Polynomials
Exams and review	6

Text

There are many books suitable for this course, and the final decision is left to the instructor, but the following two are recommended because their later portions may be also be used for MATH 414:
J.A. Friday, Introductory Analysis: The Theory of Calculus, Harcourt Brace Jovanovich, 1987.
Michael J. Schramm, Introduction to Real Analysis, Prentice Hall, 1996.

Format, Student Activities, and Grades

Class meetings involve a combination of lecture, questions and discussion, and sometimes small group activity. Homework is an important part of the course; though students may sometimes work in teams, independent effort is the primary homework mode. The instructor chooses the exact grading scheme, but a typical distribution would be:

Homework (scaled to) 200
2 Exams 200
Final Exam 200
Total 600

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.

File translated from T_EX by T_TH, version 1.56.