Math 431, Real Analysis
Fall 2007
Instructor:
Dr. Chris Lee
Trexler 270D
375-2347
clee@roanoke.edu
Monday
Tuesday
Wednesday
Thursday
Friday
8:30 - 9:30
Math 121
Trexler 362
8:30 - 10:00
Office Hours
8:30 - 9:30
Math 121
Trexler 362
8:30 - 10:00
Math 121 Lab
Trexler 362
8:30 - 9:30
Math 121
Trexler 362
9:40 - 10:40
Math 431
Life 301
9:40 - 10:40
Math 431
Life 301
9:40 - 10:40
Math 431
Life 301
11:50 - 12:25
CCLS
1:10 - 2:10
Stat 101
Trexler 372
1:10 - 2:10
Stat 101
Trexler 372
1:15 - 2:45
Office Hours
1:10 - 2:10
Stat 101
Trexler 372
2:20 - 3:30
Office Hours
7:00 - 9:00 pm
Office Hours
Course Objectives: Borrowing from the author of the text: The aim of this course will be to challenge and
improve mathematical intuition rather than simply verify it. This will be done by taking a mathematical
rigorous approach to the study of functions of a real variable. While many of the topics will seem familiar
from your calculus courses, the focus will not be on concepts that are easily visualized. Most of our work
will involve complete understanding of definitions, applying them in contexts where simple intuition will fail,
and extensive use of formal proofs as communication.
Text: Understanding Analysis, by Stephen Abbott
Homework: Daily homework assignments will be collected and graded.
Attendance: Attendance is critical to the understanding of the material in the course; it is both required and
expected. Any absence that is not discussed with the instructor prior to the missed class is considered
unexcused. Unexcused absences may result in the lowering of the final grade. I will assume that if you
accumulate 3 unexcused absences you are not interested in completing the course and will drop you from the
class (DF). When absent, excused or unexcused, you are responsible for all material covered in class. You
will not be allowed to make up any work missed due to an unexcused absence.
MCSP Conversations: The Math, Computer Science and Physics department offers a series of discussions
that appeal to a broad range of interests related to these fields of study. These co-curricular sessions will
engage the community to think about ongoing research, novel applications and other issues that face our
discipline. Members of this class are invited be involved with all of these meetings; however participation in
at least three of these sessions is mandatory. After attending, students will submit within one week of the
presentation a one-page paper reflecting on the discussion. This should not simply be a regurgitation of the
content, but rather a personal contemplation of the experience. These writeups will be counted as quizzes.
Quizzes: Quizzes will be given frequently and randomly, serving both as an incentive for the student to keep
up, and as a gauge for the instructor to measure the students understanding of the material.
Tests: Two tests will be given throughout the semester.
Final Exam: The final exam will be cumulative, covering all material presented in the course.
Grading: A students grade will be determined as
Two Tests 25% each , Quizzes & Homework 25%,
Final Exam 25%
No test or quiz scores will be dropped when calculating averages. A tentative guideline for determination of
grade will then be:
A
AB+
> 93
90 – 93
87 – 89.9
B
BC+
83 – 86.9
80 – 82.9
77 – 79.9
C
CD+
73 – 76.9
70 – 72.9
67 – 69.9
D
DF
63 – 66.9
60 – 62.9
< 60
Attendance and class participation will be considered when determining marginal and plus or minus grades.
Academic Integrity: Students are expected to adhere to the Academic Integrity policies of Roanoke College.
All work submitted for a grade is to be your own work! No electronic devices other than the TI-89 can be
taken out during any class or testing period.
Course Outline for Mathematics 431, Fall 2007
The Real Numbers, Sequences and Series
Wed
Fri
Aug 29
Aug 31
Introduction
1.1, 1.2
Preliminaries
Mon
Wed
Fri
Sept 3
Sept 5
Sept 7
1.3
1.4
1.4
The Axiom of Completeness
Consequences of Completeness
Mon
Wed
Fri
Sept 10
Sept 12
Sept 14
2.1, 2.2
2.3
2.4
The Limit of a Sequence
Algebraic and Order Limit Theorems
Monotone Convergence Theorem, Infinite Series Intro
Mon
Wed
Fri
Sept 17
Sept 19
Sept 21
2.5
2.6
2.7
Subsequences and Bolzano-Weierstrass
Cauchy Criterion
Properties of Infinite Series
Mon
Wed
Sept 24
Sept 26
Review
Test 1
Basic Topology of R, Function Limits & Continuity, The Derivative
Fri
Sept 28
3.1, 3.2
Open and Closed Sets
Mon
Wed
Fri
Oct 1
Oct 3
Oct 5
3.2
3.3
3.4
Compact Sets
Perfect Sets and Connected Sets
Mon
Wed
Fri
Oct 8
Oct 10
Oct 12
4.1, 4.2
4.3
4.4
Functional Limits
Combinations of Continuous Functions
Continuous Functions on Compact Sets
Fall Break
Mon
Wed
Fri
Oct 22
Oct 24
Oct 26
4.5
5.1, 5.2
5.2
Intermediate Value Theorem
Are Derivatives Continuous
Derivatives and Intermediate Value Theorem
Mon
Wed
Fri
Oct 29
Oct 31
Nov 2
5.3
Review
Test 2
Mean Value Theorem
Sequences and Series of Functions, The Riemann Integral
Mon
Wed
Fri
Nov 5
Nov 7
Nov 9
6.1, 6.2
6.2
6.3
Convergence of a Sequence of Functions
Uniform Convergence of a Sequence of Functions
Uniform Convergence and Differentiation
Mon
Wed
Fri
Nov 12
Nov 14
Nov 16
6.4
6.5
6.5
Series of Functions
Power Series
Mon
Nov 19
6.5
Thanksgiving Break
Mon
Wed
Fri
Nov 26
Nov 28
Nov 30
7.1
7.2
7.3
How Should Integration Be Defined?
Definition of the Riemann Integral
Integrating Functions with Discontinuities
Mon
Wed
Fri
Dec 3
Dec 5
Dec 7
7.4
7.5
Review
Properties of the Integral
Fundamental Theorem of Calculus
Wed
Dec 12
8:30 – 11:30am
Final Exam