Math 295, Fall 2011.
Honors Mathematics I
My office is 4859 East Hall,
and my office hours are 2-5pm every Wednesday (and also by appointment).
The class meets on Mondays, Tuesdays, Wednesdays and Fridays, 1pm-2pm,
in room 1372 of East Hall.
The course assistant is Nicholas Triantafillou.
His office hours are 5-6pm every Thursday in 4096 East Hall.
His review sessions are 5-6pm every Monday in 4096 East Hall.
Syllabus in PDF
An extended course description in PDF
A list of axioms for the notions that we will encounter in the course
in PDF
If this is your first proof-based course, you may find this text useful
Recommended reading:
- Calculus, Volume I
by Tom M. Apostol
(full title: One-Variable Calculus,
with an Introduction to Linear Algebra)
- Elementary Real and
Complex Analysis by Georgi E. Shilov
This is my favorite book on the subject. The presentation is rather
terse, but if
you do manage to read and understand most of it, the effort will be
worth it.
Another important advantage of this book is that it is cheap
(unlike Apostol's book).
- Principles of
Mathematical Analysis by Walter Rudin
Exams
Final exam problems in PDF
Final exam solutions in PDF
Midterm 1 in PDF
Solutions for Midterm 1 in PDF
Midterm 2 in PDF
Solutions for Midterm 2 in PDF
Homeworks
Homework 0 (due Friday,
September 9, at the beginning of class) in PDF
Solutions for Homework 0 in PDF
Homework 1 (due Friday,
September 16, at the beginning of class) in PDF
Solutions for Homework 1 in PDF
Homework 2 (due Friday,
September 23, at the beginning of class) in PDF
Solutions for Homework 2 in PDF
Homework 3 (due Friday, September 30, at the beginning of class) in PDF
Solutions for Homework 3 in PDF
Homework 4 (due Friday, October 7, at the
beginning of class) in PDF
Solutions for Homework 4 in PDF
Homework 5 (due Friday, October 21, at the beginning of class) in PDF
Solutions for Homework 5 in PDF
Review problems for the first part of the course in PDF
Optional Homework 6 in PDF
Homework 7 (due Friday, October 28, at the beginning of class) in PDF
Solutions for Homework 7 in PDF
Homework 8 (due Friday, November 4, at the
beginning of class) in PDF
Solutions for Homework 8 in PDF
Homework 9 (due Friday, November 11, at the
beginning of class) in PDF
Solutions for Homework 9 in PDF
Review problems for the second part of the course in PDF
Homework 10 (due Friday, November 18, at the
beginning of class) in PDF
Solutions for Homework 10 in PDF
Homework 11 (due Friday, December 2, at the
beginning of class) in PDF
Solutions for Homework 11 in PDF
Homework 12 (due Friday, December 9, at the
beginning of class) in PDF
Solutions for Homework 12 in PDF
Approximate course contents (arranged by
week)
- Introduction to real and natural numbers (week of
September 6)
Suggested reading from Spivak's book: Chapters 1 and 2
- Functions and bijections; finite and countable sets
(week of September 12)
- Sequences of real numbers and their limits (week of
September 19)
- Convergence criteria for sequences (week of September
26)
- Introduction to infinite series (week of October 3)
- Limits and continuity of functions (week of October 10)
- Intermediate and extreme value theorems (week of
October
19 and 21)
- Introduction to differentiation (week of October 24)
- Fundamental theorems of differentiation (week of
October 31)
- Differentiation of sums, products and quotients
- The Chain Rule
- The Mean Value Theorem
- Notes for the 11/02/2011 lecture in PDF
- More on derivatives (week of November 7)
- The Inverse Function Theorem
- Critical points and local extrema
- The Second Derivative Test
- Introduction to complex numbers
- Analysis of trigonometric functions (week of November
14)
- Relation between sin(x), cos(x) and exp(z)
- Derivatives and main properties of sin(x) and cos(x)
- Polar coordinates
- Introduction to Riemann integrals (week of
November 21)
- Integration techniques (week of November 28)
- Integrability of continuous functions
- Fundamental Theorem of Calculus
- Antiderivatives
- Further topics in integration and differentiation (week
of December 5)
- Partial fraction decompositions
- Trigonometric substitutions
- Taylor's formula with remainder
- Lebesgue's criterion for integrability; remarks on the
deficiency of the Riemann integral (week of December 12-13)