FAYETTEVILLE STATE UNIVERSITY
College of Arts and Sciences
Department of Mathematics and Computer Science
COURSE SYLLABUS
I.
Fall 2006
Location Information:
Semester: Spring
Year:
Fall
Every Year
X
X
Both
Every Other Year
Course Number & Name: Math 521 Real Analysis
Semester Hours of Credit:
3
Time Class Meets: TR 6:00-7:15.
Instructor's Name:
Where Class Meet: SBE 213
Dr. Bo Zhang
Office Location: SBE 348
Office Telephone: 672-1786
Office Hours:
MWF: 1:00-3:00pm & TR: 2:00-3:00pm
E-mail Address: bzhang@uncfsu.edu
Instructor’s web page: http://faculty.uncfsu.edu/bzhang
Department Office Location: SBE 338
Department Office Telephone: 672-1294
II.
COURSE DESCRIPTION:
The first course of a three-semester sequence in real analysis,
including such topics as real number system, element of point set
topology and metric spaces, sequences and series of real numbers,
continuity, differentiation, integration, the Reimann-Stieltjes
integral, sequences and series of functions, pointwise and uniform
convergence, functions of several variables, implicit function, and
inverse function theorem. Prerequisite: Math 412 or Math 461 or consent
of department.
III. TEXTBOOK
Manfed Stoll, Introduction to Real Analysis, Addison Wesley, 2001.
Mathematical Software: Maple V9
IV.
Course Objective:
To provide necessary background in real analysis for students in
mathematics computer science, engineering, physics, and teaching of
mathematics. After the completion of this course, students would have a
working knowledge of the basic concepts in Real analysis described in II.
Page #2
V. Course Competencies:
Competencies (DPI):
(2.1) Evaluate limits involving the elementary functions.
(2.2) Demonstrate an awareness that derivative and definite integrals are
defined as limits.
(2.3) Know the relationship between differentiability and continuity.
(2.4) Possess a knowledge of the notations and fundamental concepts used in
studying sequences and series.
(2.5) Possess a knowledge of the basic properties of power series,
especially the properties of geometric series.
(2.6) Possess a knowledge of the relationship between infinite series and
improper integrals.
(2.7) Use the standard tests for convergence to infinite series.
(2.8) Possess a knowledge of the Taylor series representation of the
elementary function.
(8.1) Know the symbolism of mathematical logic.
(8.2) Demonstrate a thorough knowledge of the concepts of equivalence and
implication.
(8.4) Possess a knowledge of the role of proof in the study and development
of mathematics.
(8.6) Create original proofs in the various branches of mathematics
including direct proofs, indirect proof, and proofs using
mathematical induction.
(9.1) Use the set theoretic operations, intersection and complementation.
(9.2) Know the relationship between the logical operations and the set of
theoretic operations.
NCATE Standards
(1.1.1) Use a problem-solving approach to investigate and understand
mathematical concept.
(1.1.2) Formulate and solve problems from both mathematical and everyday
situations.
(1.2.1) Communicate mathematical ideas in writing, using everyday
mathematical language, including symbols.
(1.3.0) Make and evaluate mathematical conjectures/arguments and validate
their own mathematical thinking.
(1.4.1) Show an understanding of interrelationships within mathematics.
(1.4.2) Connect mathematics to other disciplines and real-world situations.
(1.6.1) Use calculators in computational and problem-solving situations.
(1.6.2) Use computer software to explore and solve mathematical problems.
(2.2.2) Use graphing calculators, computers and other technologies as tools
for teaching mathematics.
(2.4.0) Use a variety of resource materials such as software, print
materials, technology, and activity files to enhance the learning
of mathematics.
(2.5.0) Select appropriate mathematical tasks that will stimulate students'
development of mathematical concepts and skills.
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VI. Evaluation Criteria:
Homework 20%,
Tests 60%,
Final Exam 20%
Final grade will be determined on the basis of academic
performance in the following manner:
A = 90-100%, B = 80-89%, C = 70-79%, F = below 70%
VII. Course Outline with assignment schedule:
See attached assignment calendar
VIII. Course requirements:
1. Students are expected to attend all class sessions. Excessive absences
may result in a reduction of your final grade.
2. Students are expected to enter the classroom on time and remain for
the full class period. Students should not make other appointments in conflict
with their class schedule.
3. All tests will be announced prior to their administration. A make-up
exam will be given only if the student has a documented and valid written
justification for unavoidable absence from the exam. There is no more than one
make-up exam for each student during the semester.
4. The Instructor's office hours are times when you may seek
assistance without prior appointment. You are encouraged to seek
help as needed.
5. Students must refrain from smoking, eating and drinking
classroom. The rights of others must be respected at all times.
in the
6. Students are encouraged to ask questions of the instructor in class
and to respond to those posed by the instructor. They should not discourage
others from asking or answering questions. Other students often have the same
questions on their minds, but are hesitant to ask.
7. Students are expected to complete all class assignments, to spend
adequate time on their class work, and to read each topic prior to class
discussion to insure that the course outcomes are met. At least two hours of
home study is expected for each class.
8. Talking in class between students is strictly prohibited.
Discussions should be directed to the instructor. Unacceptable
behavior in the class will result in a reduction of your final
grade.
9.
Dishonesty on graded assignments will not be tolerated.
Students
must neither give nor receive help on any work to be graded. The university
policy on cheating will be applied to any violations. The minimum penalty will
be a grade of zero on the assignment.
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IX.
References:
1. Bartle, R. G. and Sherbert, D. R., Introduction to real analysis, John
Wiley & Sons, Inc., 2000.
2. Fitzpatrick, P., Real Analysis, PWS Publishing Company, 1996.
3. James R. Kirkwood, An Introduction to Analysis, PWS Pub. Company, 1995.
4. Marsden, J. R., Elementary Classical Analysis, Freman and Co.
publishers, 1990.
5. Markarov, B. M, Goluzina M. G., Lodkin, A. A. and Podkorytov, A. N.,
Selected Problems in Real Analysis, American Mathematical Society, 1992.
6. Rudin, W., Principles of Mathematical Analysis, McGraw- Hill, 1976.
7. Gaskill, Herbert S. and Narayanaswami, P. P., Elements of Real
Analysis, Prentice-Hall, 1998.
8. Wade, William R., An Introduction to Real Analysis, Prentice Hall, 2000.
X. Teaching Strategies:
The majority of the material of the course will be given in lecture
format. There is a short review before and after each lecture. There
will be a comprehensive review after the completion of each chapter.
Graphing calculators and Maple (mathematical software) will be used in
the class to help students develop a firm grasp of the underlying
mathematical concepts. Selected student research project will be
introduced during the semester.
*VII Course Outline
Dates
Lecture
THU: 08/24
[1.3] Mathematical Induction
[1.4] The least upper bound property
---------------------------------------------------------------------------TUE: 08/29 [1.7] Countable and uncountable sets
[2.1] Convergent sequences
THU: 08/31 [2.2] Limit theorems
[2.3] Monotone sequences
--------------------------------------------------------------------------TUE: 09/05 [2.4] Subsequences and the Bolzano-Weierstrass theorem
[2.5] Limit superior and inferior of a sequence
THU: 09/07 [2.6] Cauchy sequences
[2.7] Series of real numbers
TUE: 09/12
THU: 09/14
[3.1] Open and closed sets
[3.2] Compact sets
[3.3] The Cantor set
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*VII Course Outline (continued)
Dates
Lecture
TUE: 09/19
THU: 09/21
[4.1] Limit of a function
[4.2] Continuous functions
[4.3] Uniform continuity
TUE: 09/26
THU: 09/28
[4.4] Monotone functions and discontinuity
Exam #1
TUE: 09/03
[5.1]
[5.2]
[5.3]
[5.4]
THU: 09/05
TUE: 10/10
THU: 10/12
The derivative
The mean value theorem
L’Hospital’s rule
Newton’s method
[6.1] The Riemann Integral
Fall Break
TUE: 10/17
[6.2] Properties of the Riemann integral
[6.3] Fundamental Theorem of Calculus
THU: 10/19 [6.4] Improper Riemann integral
---------------------------------------------------------------------------TUE: 10/24
THU: 10/26
[6.5] The Riemann-Stieltjes integral
[7.1] Convergence Tests
[7.2] The Dirichlet Test
TUE: 10/31
[7.3] Absolute and Conditional Convergence
[7.4] Square summable sequence
Exam #2
THU: 11/02
TUE: 11/07 [8.1] Pointwise Convergence and interchange of limits
THU: 11/09 [8.2] Uniform convergence
---------------------------------------------------------------------------TUE: 11/14
THU: 11/16
[8.3] Uniform convergence and continuity
[8.4] Uniform convergence and integration
TUE: 11/21 [8.5] Uniform convergence and differentiation
THU: 11/23 Thanksgiving Holiday
---------------------------------------------------------------------------TUE: 11/28 [8.6] The Weierstrass Approximation Theorem
THU: 11/30 [8.7] Power series expansions
TUE: 12/05
THU: 12/07
[8.8] The gamma Function
Review for final Exam
Tue: 12/12
(6:00-7:50pm) Final Exam