18.100BC Analysis I -- Fall 2008

IMPORTANT INFORMATION: Solutions for Final Exam. Class Average 78 - well done!
Scores for the final are on stellar; final letter grades will be on websis by Thursday .... happy holidays!

Exams of 100B MWF and 100C can be picked up Wednesday 9:30-11:30 and 3:00-4:00 in 2-246.

 The graph of the Weierstrass function:




Course Information:

Instructor: Gigliola Staffilani (B) Todd Kemp (B)
 Katrin Wehrheim (C)
E-mail: gigliola(at)math.mit.edu tkemp(at)math.mit.edu katrin(at)math.mit.edu
Lectures: MWF 12-1 in 4-237 TR 1-2:30 in 4-163
MWF 11-12 in 2-102, extra C session
Office hours: Tue 2:30-4:30pm,
Thu 3-5pm in 2-246
Thu 10-12am in 2-175
 Thu 12/11 4pm,
Fri 12/12 11am-12:30 in 2-277


Course assistants:
 Tim Nguyen (B and C)
Joel Lewis (C)
Susan Ruff (C)
Email:
 timothyn(at)math.mit.edu
 jblewis(at)math.mit.edu
 ruff(at)mit.edu
Office hours: Tue 1-2pm, Fri 1:15-2:15pm in 2-310 Fri 2-4pm in 2-333


Homework and Practice Problems: (If you are stuck on some problem consider looking at the extensive solution writeups from the C section.)
  • Practice Problems from week 1 in 18.100C
  • Problem Set 1, covering week 1 and 2, due 9/17. Solutions
  • Problem Set 2 and notes on sequential compactness, covering week 3, due 9/24. Solutions
  • Problem Set 3, covering week 4, due 10/1. Solutions
  • Problem Set 4, covering week 5, will not be graded due to Midterm that week.
    (You may however submit solutions as usual to get feedback.) Solutions
  • Practice Problems for Midterm 1. Solutions
  • Problem Set 5, covering week 6, due 10/15. Solutions
  • Problem Set 6, covering week 7, due 10/22. Solutions
  • Problem Set 7, covering week 8, due 10/29. Solutions
  • Problem Set 8, covering week 9, due 11/5. Solutions
  • Problem Set 9, covering week 10, will not be graded due to Midterm that week.
    (You may however submit solutions as usual to get feedback.) Solutions
  • Practice Problems for Midterm 2. Solutions
  • Problem Set 10, covering week 11, due 11/19. Solutions
  • Problem Set 11, covering week 12, due 11/26. Solutions
  • Problem Set 12, covering week 13 and 14, due Friday 12/5. Solutions
  • Practice Exam for the final. Solutions

    • Tentative Syllabus: The following is a rough week-by-week guide to what we plan to cover and when.
  • Week 1 (9/3,5 resp. 9/4): Ordered sets, fields, the real numbers, countability
    Read Rudin sections 1.1-35, 2.1-14
  • Week 2 (9/8,10,12 resp. 9/9,11): euclidean spaces, metric spaces
    Read Rudin sections 1.36-38, 2.15-28.
  • Week 3 (9/15,17,19 resp. 9/16,18): compact sets, connected sets
    Read Rudin sections 2.30-47 and notes on sequential compactness.
    • Optional: additional notes on connected components.
  • Week 4 (9/24,26 resp. 9/23,25): convergence
    Read Rudin sections 3.1-6, 3.15-20.
  • Week 5 (9/29,10/1,10/3 resp. 9/30,10/2 ): completeness, construction of the real numbers
    Read Rudin sections 3.7-14 and notes on construction of the reals.
  • Monday 10/6, 4-6pm in 32-124 : Optional Review Session for Midterm 1 by Tim Nguyen
  • Tuesday 10/7, 7:30-9:00pm in 4-370: Midterm 1, covering week 1-5 : exam and solutions. Scores are also on stellar.
  • Week 6 (10/6,8,10 resp. 10/9,10/11): Sequences and series.
    Read Rudin pgs. 3.20-3.55 and notes on conditional convergence .
  • Week 7 (10/15,17 resp. 10/14,16): l^p spaces, continuity.
    Read Rudin sections 4.1-12; additional material in lectures on l^p spaces, and optional notes on completeness.
  • Week 8 (10/20,22,24 resp. 10/21,25): Continuity and compactness.
    Read Rudin sections 4.13-34.
  • Week 9 (10/27,29,31 resp. 10/28,30): Differentiability, Mean value theorem.
    Read Rudin sections 5.1-15, and optional notes on the chain rule.
  • Week 10 (11/3,5,7 resp. 11/4,6): Taylor series, Riemann-Stieltjes integral.
    Read Rudin sections 8.1-7, 6.1-11.
  • Wednesday 11/12, 4-6pm in 4-163 : Optional Review Session for Midterm 2 by Tim Nguyen
  • Thursday 11/13, 7:30-9:00pm in 1-190: Midterm 2, covering week 5-10 : exam and solutions. Scores are also on stellar.
  • Week 11 (11/12,14 resp. 11/13): Integrability.
    Read Rudin sections 6.12-17. and notes on Riemann integrability and continuity almost everywhere.
  • Week 12 (11/17,19,21 resp. 11/18,20): Fundamental theorem of calculus, Sequences of functions.
    Read Rudin sections 6.18-6.22, 7.1-9.
  • Week 13 (11/24,11/26 resp. 11/25): Uniform convergence.
    Read Rudin sections 7.10-17 and notes on Devil's Staircase function.
  • Week 14 (12/1,3,5 resp. 12/2,4): Uniform convergence, equicontinuity.
    Read Rudin sections 7.18-27 and optional notes on Weierstrass function.
  • Week 15 (12/8,10 resp. 12/9): Final review.
  • December 12, 2-4pm in 2-105 Optional Review Session by Joel Lewis
  • December 15, 1:30 - 4:30PM in Johnson: Final Exam. Solutions.
  • December 16: Scores for the final are on stellar; final letter grades cannot be displayed due to MIT policy .... happy holidays!

    • Literature:
    W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, third edition.

    Grading: Homework and exam grades will be uploaded on stellar.
    Weekly homework will be due each Wednesday at 11 AM in 2-108. Late homework will not be accepted, since solutions will be posted online. You are encouraged to collaborate with other students. However, the solution must be written in your own words, and all collaborators and other sources of information (e.g. online or books other than Rudin) need to be quoted (on the first page or with the specific problem). Each homework is graded on a scale of 0-100. The two lowest scores will be dropped.
    Midterm I is scheduled for Tuesday, October 7, 7:30pm.
    Midterm II is scheduled for Thursday, November 13, 7:30pm.
    Final was scheduled for Monday, December 15, 1:30 - 4:30PM in Johnson.
    The final grade for 18.100B will be based on homework (25%), two midterms (each 20%), and the final (35%). (See here for 18.100C.)