Mathematics 104, Spring 2012

Introduction to Analysis

Lectures: Mondays, Wednesdays, and Fridays, 3:10--4pm, Room 2, Evans Hall.

Instructor: Michael VanValkenburgh, 895 Evans Hall.

Office Hours: on Mondays, Wednesdays, and Fridays, 4:15--5:15pm, and by appointment. These times are subject to change, depending on departmental seminars, etc., but I will give notice by email.

A description of the course: Math 104, Spring 2012 (pdf)




Announcements:

The Final Exam will be held on Wednesday, May 9, 7--10pm, in 180 Tan Hall.

There will be no office hours during finals week.

During RRR Week, we will have reviews during the usual class time (MWF, 3:10--4pm), in the usual place (Evans Room 2). I won't present any new material; the purpose is solely to review for the final exam. Please bring questions!

The scores on Problem~6 from Midterm~1 and Problem~2 from Midterm~2 were particularly disappointing--those problems covered central topics in the course: limsup/liminf and uniform convergence, respectively. It is of the utmost importance that you understand those topics in time for the final exam. As an extra incentive: on the final exam there will be a 5-point problem on limsup/liminf (and related topics), your score to be simply added to your Midterm~1 score, and a 10-point problem on uniform convergence, your score to be simply added to your Midterm~2 score. If you end up with over 100% on an exam, which for most of you is an impossibility, it will be taken into consideration when computing your final grade.


Homework:

Only a few selected homework problems will be graded in detail. Your two lowest homework assignments will be dropped from the final grade.
For solutions to selected problems, see the webpage of my good friend Professor Chris Rycroft. If you would like to see the solution to another problem, send me an email, ask before class, or ask in office hours.
Homework 1, Due Friday, January 27.
Homework 1, LaTeX file.

Homework 2, Due Monday, February 6.
Homework 2, LaTeX file.

Homework 3, Due Monday, February 13.
Homework 3, LaTeX file.
Homework 3, Solutions to Selected Problems

Homework 4, Due Friday, February 24.
Homework 4, LaTeX file.
Homework 4, Solutions to Selected Problems

Homework 5, Due Friday, March 2.
Homework 5, LaTeX file.
Homework 5, Solutions to Selected Problems

Homework 6, Due Monday, March 12.
Homework 6, LaTeX file.

Homework 7, Due Monday, March 19.
Homework 7, LaTeX file.

Homework 8, Due Monday, April 9.
Homework 8, LaTeX file.
Homework 8, Solutions to Selected Problems

Homework 9, Due Wednesday, April 18.
Homework 9, LaTeX file.

Homework 10, Due Friday, April 27.
Homework 10, LaTeX file.

Many thanks to those of you who give me LaTeX advice after reading my code!


Exams:

Two Sample Questions for Midterm 1 Review. (Here are the solutions.)
Midterm 1.
Solutions to Midterm 1.
Commentary on Midterm 1.
Raw Scores for Midterm 1.

Midterm 2.
Solutions to Midterm 2.
Commentary on Midterm 2.
Raw Scores for Midterm 2.

Rough Idea of a Practice Final.



Notes:

Lectures 1 and 2: I covered Chapter 0, Sections 1--4, of the book "Advanced Calculus" by Loomis and Sternberg. The book is available free online, on Prof. Sternberg's webpage: www.math.harvard.edu/~shlomo/.
Lecture 3: Counterexample to my purported proof that ``all birds in a flock are the same color.'' (Thanks to one of you for sending this to me!)
Lecture 4: The Triangle Inequality by Squares.
February 15: Our Proof of the Bolzano-Weierstrass Theorem.
February 27: Monomials Are Continuous Functions.






Syllabus

DateTopics BookNotes
1. Wed 1/18 Introduction, Logical Statements. Loomis-Sternberg
2. Fri 1/20 Continuation of Logic, Induction. § 1
3. Mon 1/23 Field Axioms, Rational Numbers, Real Numbers. § 2, 3
4. Wed 1/25 The Completeness Axiom, sup and inf. § 4
5. Fri 1/27 Continuation of Last Time, ``Infinity.'' § 5Homework 1 due
6. Mon 1/30 Limits of Sequences § 7
7. Wed 2/1 Properties of Sequences, I. § 8
8. Fri 2/3 Properties of Sequences, II. § 9
9. Mon 2/6 Monotone Sequences and Cauchy Sequences. § 10Homework 2 due
10. Wed 2/8 Continuation of Last Time. § 10
11. Fri 2/10 Continuation of Last Time, Review.
12. Mon 2/13 MIDTERM 1 Homework 3 due
13. Wed 2/15 Subsequences, Bolzano-Weierstrass § 11
14. Fri 2/17 limsup, liminf. § 12drop deadline
Mon 2/20 NO CLASS (Presidents' Day)
15. Wed 2/22 Infinite Series, I. § 14
16. Fri 2/24 Infinite Series, II. § 14, 15Homework 4 due
17. Mon 2/27 Continuous Functions, I. § 17
18. Wed 2/29 Continuous Functions, II. § 18
19. Fri 3/2 Uniform Continuity, I. § 19Homework 5 due
20. Mon 3/5 Uniform Continuity, II. § 19
21. Wed 3/7 Limits of Functions, I. § 20
22. Fri 3/9 Limits of Functions, II. § 20
23. Mon 3/12 Power Series. § 23Homework 6 due
24. Wed 3/14 Uniform Convergence, I. § 24
25. Fri 3/16 Uniform Convergence, II. § 25
26. Mon 3/19 Uniform Convergence, III. Midterm Review. § 25Homework 7 due
27. Wed 3/21 MIDTERM 2
28. Fri 3/23 Abel's Theorem. § 26
Mon-Fri 3/26-30 NO CLASS (Spring Break)
29. Mon 4/2 Differentiation. § 28
30. Wed 4/4 The Mean Value Theorem. § 29
31. Fri 4/6 Differentiation and Integration of Power Series. § 26
32. Mon 4/9 L'Hospital's Rule. § 30Homework 8 due
33. Wed 4/11 Taylor's Theorem, I. § 31
34. Fri 4/13 Class is cancelled.
35. Mon 4/16 Taylor's Theorem, II. § 31
36. Wed 4/18 The Riemann Integral, I. § 32Homework 9 due
37. Fri 4/20 The Riemann Integral, II. § 32-33
38. Mon 4/23 The Riemann Integral, III. § 33
39. Wed 4/25 The Fundamental Theorem of Calculus, I. § 34
40. Fri 4/27 The Fundamental Theorem of Calculus, II. § 34Homework 10 due
RRR Week Review sessions held in the regular classroom at the regular time.
Wednesday 5/9 FINAL EXAM, 7-10pm, Room 180 Tan Hall.