Summer Session III 2005

MATH 464.02/506.02 (3 credit), MTuWThF, 10:00A-11:30A, Wickersham 226

**Prerequisites:**A grades of C- or better in MATH 310 (formerly numbered 220)

*Introduction to Proof*, MATH 311 (formerly numbered 261)*Calculus III*, and MATH 322 (formerly numbered 242)*Linear Algebra*are the prerequisites for this course.**Instructor:**Dr. Buchanan

Office: Wickersham 218, Phone: 872-3659, FAX: 871-2320

Office Hours: 2:00P-3:00P (M-F), or by appointment

Email:`Robert.Buchanan@millersville.edu`

URL:`http://banach.millersville.edu/~bob`**Textbook:**Charles G. Denlinger,

*Elements of Real Analysis*, July 2005.**Objectives:**Upon successful completion of this course, a student will have

- developed an ability to read and comprehend expository mathematical writing in real analysis at the upper undergraduate level,
- acquired a comprehensive understanding of the basic concepts of real
analysis, including
- its axiomatic/deductive organization and flow of ideas, its foundations in the real number system (a complete ordered field),
- the fundamental notions of general topology and their uses in real analysis,
- the theory of continuous, real-valued functions of a real variable,
- the theory of differentiable, real-valued functions of a real variable,
- the theory of the Riemann integral of real-valued functions of a real variable,
- the fundamental theorem of calculus,

- sharpened his/her ability to critique mathematical ideas with constructive skepticism and in particular, will have developed an intellectual storehouse of instances in which naïve intuition would otherwise lead him/her to believe untrue statements in analysis or disbelieve true statements,
- learned that mathematical proof is the one indispensable tool for separating truth from falsehood in analysis,
- learned the essential role played by definitions in the formulation of methodology and proofs in mathematics,
- developed the specialized techniques of real analysis that allow the above objectives to be realized,
- learned significant applications of the concepts and techniques of real analysis, and
- learned the fundamental role played by real analysis in providing
rigorous justification of the methods taught in elementary calculus
courses.
^{1}

**Course Contents:**- The summer session activities may include exposure to and exploration of
the following topics.
Partial Topic List:

- Real Number System ()
- Definitions and properties of fields and ordered fields
- Natural numbers in ordered fields
- Rational numbers in ordered fields
- Archimedian ordered fields

- Sequences
- Limits and convergence
- Algebra of limits
- Inequalities and limits
- Divergence to infinity
- Monotone sequences
- Subsequences
- Cauchy sequences

- Topology of the Real Number System
- Neighborhoods and open sets
- Interior, exterior, and boundary of a set
- Isolated points
- Closed sets and cluster points

- Limits of Functions
- Definition of the limit of a function
- Algebra of limits of functions
- Limits of polynomial and rational functions
- Inequalities and limits
- One-sided limits

- Continuous Functions
- Definition of continuous function
- Algebra of continuous functions
- Continuity on compact sets
- Continuity on intervals
- Applications of continuity to root finding, fixed points,
*etc.*

- Differentiable Functions
- Definition of derivative and differentiability
- Rules for differentiation
- Relative extrema
- Monotone functions
- Rolle's Theorem and the Mean Value Theorem

- Riemann Integral
- Darboux's definition of the Riemann integral
- Integral as the limit of Riemann sums
- Existence and additivity theorems
- Algebra of integrals
- Fundamental Theorem of Calculus (first and second forms)

- Real Number System ()
**Tests:**There will be a take-home midterm test (given to students on Friday August 5, 2005 and due Monday, August 8, 2005) and a comprehensive final examination (given to students on Friday August 19, 2005 and due Monday, August 22, 2005). Students are not allowed to work together or discuss the midterm and final examination assignments. I will not ``curve'' midterm or exam grades.

**Homework:**Students are expected to do their homework and participate in class. Students should expect to spend a minimum of three hours outside of class on homework and review for every hour spent in class. There will be four graded homework assignments during the summer session. The homework problems will be selected from the exercises in the textbook. Students should submit all homework by the date due. Late homework will not be accepted without valid excuse. It is very important that students work all assigned homework exercises, even those not designated as part of a graded assignment. Because of the pace of the summer session, a significant amount of your learning of the material will take place as you work on assigned problems. Students may work together and/or discuss any homework problems not assigned for a grade. The work you submit for graded homework exercises must represent your individual effort.

**Attendance:**Students are expected to attend all class meetings. If you must be absent from class on the day an assignment is due, you must complete and hand in the assignment prior to the absence. If you know you will be absent on the day that a homework assignment or the midterm test is due, you must submit the work prior to the absence. Students who miss handing in an assignment due to an unforeseen absence should provide a valid excuse, otherwise you will not be allowed to submit the assignment.

**Grades:**Course grade will be calculated as follows.

Midterm test 25% Final exam 25% Homework/Projects 50% I keep a record of students' homework and test scores. Students should also keep an individual record of graded assignments. I will not ``curve'' course grades. The course letter grades will be calculated as follows.

90-92 A 93-100 A 80-82 B 83-86 B 87-89 B 70-72 C 73-76 C 77-79 C 60-62 D 63-66 D 67-69 D 0-59 F **Course Repeat Policy**An undergraduate student may not take an undergraduate course of record more than three times. A course of record is defined as a course in which a student receives a grade of A, B, C, D, (including and ) F, U, Z or W. The academic department offering a course may drop a student from a course if the student attempts to take a course more than three times.

^{2}**Inclement Weather Policy:**If we should miss a class day due to a school closing because of weather, any activities planned for that missed day will take place the next time the class meets. For example, if a test is scheduled for a day that class is canceled on account of snow, the test will be given the next time the class meets.

**Final Word:**Math is not a spectator sport. What you learn from this course and your final grade depend mainly on the amount of work you put forth. Daily contact with the material through homework assignments and review of notes taken during lectures is extremely important.

Robert.Buchanan@millersville.edu

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