Instructor: Victor Roytburd (Amos Eaton 405, Ph. 276-6889,
roytbv at rpi dot edu)
Class meets: M, Th 2:00 – 3:50PM, RICKETTS 212
Office hours: Tuesday
2-4, Thursday 4-6 (or by appointment)
Web page: http://www.rpi.edu/~roytbv/ma/ma1.html
TEXTS: (1) (required) The Way of Analysis, by R. S. Strichartz, Jones and Bartlett 2000 ISBN 0-7637-1497-6;
(2) (recommended) Introduction to Analysis, by Maxwell
Rosenlicht, Dover 1986 ISBN 0-486-65038-3. This very brief (and cheap) book
gives a different perspective on the subject;
(3) (recommended) How to read and do proofs, by Daniel Sollow,
John Wiley 2002 ISBN 0-471-40647-3. This book is devoted mostly to ”proof
techniques.” Any other book that explains, how to read and do proofs, will work
as well, check, for example,
How to prove it, by Daniel Velleman or
Reading, Writing, and Proving, by Ulrich Daepp and Pamela
Gorkin, Springer 2000 ISBN 0-387-00834.
You may also find helpful the books Professor
Harry McLaughlin wrote for the Fundamentals of Mathematics
class:
Fundamentals of mathematics: Notes
Fundamentals of mathematics: Workbook
In this
course we will revisit some basic notions of calculus. The emphasis will be on
the conceptual understanding of calculus and on mathematical rigor. I don't
have a rigid agenda as to how much material to cover. The pace of the course
will be mostly determined by our progress, which will be measured by the
performance on the homework assignments and tests. I expect to cover the
material of at least six first chapters of the text by Strichartz. Here are the
key topics:
First Exam
·
Basic
logic. Countable and uncountable sets, cardinality, Cantor’s diagonalization.
·
Cauchy
sequences. Limits, sups and infs. (The real number system.)
·
Open
sets and closed sets; Cantor sets.
Second Exam
·
Compacts.
·
Continuity
and uniform continuity. Continuity via open sets.
·
The
intermediate value theorem. Continuous functions on compact domains. Monotone
functions.
·
Definition
of the derivative. Big O and little o.
Third Exam
·
Local
behavior of a function and the sign of its derivative. The mean value theorem.
·
The
calculus of derivatives. Taylor’s theorem.
·
Integrals
of continuous functions. Fundamental theorem of calculus.
·
The
Riemann integral. Improper integrals.
·
Basics of
complex numbers.
·
Numerical
series and sequences.
·
Uniform
convergence of functional sequences.
Next is a
tentative course outline:
|
Week |
Topics |
Sections |
|
8/28, 8/31 |
The logic of quantifiers.
Countable and uncountable sets. Cantor’s diagonalization. |
1.1 – 1.4 |
|
9/4 |
Labor Day |
|
|
9/7 |
Cauchy sequences. Limits, sups, and infs. |
2.1 (pp.
25-34), 3.1.1 |
|
9/11, 9/14 |
The real number system.
Review of different ways of establishing the real number system. |
Notes, 2.1.2,
2.4.1-2 |
|
9/18, 9/21 |
Limits, open sets
and closed sets. |
3.1–3.2 |
|
9/25, 9/28 |
Compact sets. Pre-test review |
3.3 |
|
10/2 |
EXAM 1 |
First 3 items in the list of topics. |
|
10/5 |
Continuity |
4.1 |
|
10/10 |
(Monday schedule). Concepts of continuity |
4.1 – 4.2 |
|
10/12 |
Properties of
continuous functions |
4.2 |
|
10/16, 10/19 |
Properties of continuous functions |
4.2 |
|
10/23, 10/26 |
Derivative and its properties. Pre-test review |
5.1 – 5.2 |
|
10/30 |
EXAM
2 |
|
|
11/2 |
Properties
of the derivative. Calculus rules. |
5.2 – 5.3 |
|
11/6, 11/9 |
Taylor’s
theorem. Integral of continuous functions. |
5.3.3 – 5.4,
6.1 – 6.2 |
|
11/13 |
The
Riemann integral. Improper integrals. Odds and ends. |
6.2 – 6.3 |
|
11/16 |
Primer of complex numbers. |
7.1 |
|
11/20 |
Numerical series. |
7.2 |
|
11/22-11/26 |
Thanksgiving break |
|
|
11/27 |
Uniform convergence. (It will be included
in the final) |
7.3 |
|
11/30 |
Review |
|
|
12/4 |
EXAM 3 |
Blue topics |
|
12/7 |
Last day
of classes |
|
|
Tuesday 12/12 |
Course review: 4:00-7:00 pm, Amos Eaton 216 |
|
|
12/14 |
Final Exam: 6:30 – 9:30 pm. SAGE 4101 Uniform convergence will not be
on the final exam |
|
Homework
problems: There will be weekly homework assignments that will be collected
and graded. As a rule, homework assignments are due on Thursday by 4:00 pm,
every week except exam weeks. Late homework
is not accepted.
Tests: I plan to give two three midterm tests and a final. Only two
best test results will be counted for the course grade. Exam 2 is scheduled for 10/30; Exam 3 is
scheduled for 12/04.
Grading: Homework 30%, Tests 40%, Final 30%
In case I decide to give weekly quizzes on basic
definitions, the weight will change as follows: Homework 25%, Quizzes 10%,
Tests 35%, Final 30%
STATEMENT
OF ACADEMIC INTEGRITY: The
guiding principle is that work that you present for grading as your own should
in fact BE your own. With respect to exams, this means that no
assistance or collaboration of any kind is permitted. With respect to homework,
you are free to seek assistance from any person, or book. But before working together
on the homework, it is in your best interest to think over the problems on your
own. You must write up your solutions on your own. You are not allowed to just
copy from a shared (or someone else's) set of notes.
HOMEWORK ASSIGNMENTS
Note: Extra credit problems
do not have expiration date; you can hand them in separately and after the
homework they first time appeared in.
List of all Extra credit problems
to date.
Assignment #1, due 7 September
Assignment #2, due 14 September
Assignment #3, due 21 September. Solutions to selected homework
problems from first three assignments.
Assignment #4, due 28 September. Solutions
#4
Assignment #5, due 12 October. Solutions #5
Assignment #6, due 19 October. Solutions #6
Assignment #7, due 26 October. Solutions #7
EXAM 2 Solutions
Assignment #8, due 9 November: correction the problem #
should read 5.2.4.2. Solutions
#8
Assignment #9, due November 16, Solutions #9
Assignment #10, due November 30, Solutions #10
EXAM 3 Solutions
Uniform convergence will not be on the final
exam.
|
Tuesday 12/12 |
Course review: 4:00-7:00 pm, Amos Eaton 216 |
TOPICS for BRIEF (extra credit) PRESENTATIONS
Cantor – Bernstein’s theorem.
Continued fractions
Basics of Nonstandard Analysis.
LATEST NEWS
Sept. 1 – An outstanding book written by one of the top
mathematicians of the 20th century: Introductory Real Analysis by
A.N. Kolmogorov and S.V. Fomin, Dover, ISBN 0-486-61226-0, $15.95.
Oct. 20 – Exam 2 is
scheduled for 10/30; Exam
3 is scheduled for 12/04.
Now we are meeting on Wednesdays for problem discussions. Room Amos Eaton 402 is reserved
from 5-7 pm.
OLD EXAMS
Test 1 from 2002
Test 2 from 2002
Final Exam 2002
Final Exam 2005
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