MATHS 371:  Intermediate Analysis (3)

Syllabus

 

1.        Prerequisite: MATHS 166, 215, or permission of the department chairperson.

 

Note: Students with strong performance in MATHS 166, 215, and 267 may request permission to substitute MATHS 471 for MATHS 371.

 

2.        Course Description:  Introduction to basic concepts of analysis:  the real numbers, sequences, continuous functions, the derivative, and the Riemann integral.

 

3.        Course Objectives:  Students will see a careful and rigorous treatment of the main ideas of differential and integral calculus of one variable, first encountered in their freshman/sophomore calculus sequence. Students completing this course will be introduced to the theoretical underpinnings of calculus.  In this course, students will gain experience in writing proofs of moderate difficulty and solving problems related to the content of the course.       

 

4.        Course Rationale:  Analysis is a major branch of modern mathematics, with profound connections to other branches of mathematics, including geometry, topology, algebra, and mathematical physics.  Providing an axiomatic framework for the basic ideas seen in calculus, MATHS 371 is an essential course for Mathematics majors and highly recommended for Mathematics Teaching majors.

 

5.        Course Content: 

·          The Real Numbers: sets, functions, algebraic properties and order properties, positive integers, Least Upper Bound Axiom

·          Sequences:  sequences and limits, limit theorems, monotonic sequences, subsequences, Cauchy sequences, infinite limits

·          Continuous Functions:  limit of a function, limit theorems, other limits, continuity, Intermediate Value Theorem, Extreme Value Theorem, uniform continuity

·          The Derivative: definition of the derivative, rules for differentiation, Mean Value Theorem, Inverse Function Theorem, l’Hôpital’s Rule, Taylor’s Theorem

·          The Integral: definition of the Riemann integral, properties of the Riemann integral, existence theory, Fundamental Theorem of Calculus, improper integrals

 

6.        Course Format:  lecture/discussion.

 

7.        Method of Evaluating Student Performance:  Typical components are homework assignments, projects, and take-home exams (possibly involving group work); in-class exams; student presentations and oral exams. The evaluation and weight of these components are at the discretion of the instructor. Students may be evaluated on their understanding of the content material for the course, their ability to solve problems related to the course material, and on their ability to communicate mathematically to others verbally or in writing.

 

8.        Evaluation of the Course:  The instruction of the course is evaluated by departmental student evaluations and peer evaluation.  The course is reviewed and revised periodically by the Departmental Undergraduate Programs Committee.

 

M. Karls, A. Mohammed, B. Hartter, 11/10/04, M. Karls Fall 2006