- Lecturer: Amy Cohen
- Course Meetings: Tuesdays 6:10-9:00 pm (with a break) and Thursdays 6:10-7:30pm) in Hardenburgh A4 on CAC
- Office Hours (firm):
- Tuesdays 10:30-11:30am in Hill 530 on Busch (primarily for Math 151)
- Thursdays 7:30-8:00pm, HH-A4 on CAC (primarily for Math 311)
- also by appointment in Hill 530 on Busch -- use e-mail to arrange -- a time

- Textbook:
*Understanding Analysis*by Abbott - Usual Format: lectures, discussion of homework, and workshops

- What are the properties (axioms) we assume for the system of real numbers? Why do we need to make our assumptions explicit?
- What sort of "concrete" mathematical objects do we use to represent real numbers? Why should there be a number represented by 0.999...999... and what is it?
- How and why do we prove (some of) those properties of functions that were asserted without proof in the first two terms of calculus? (a) Why must the image of a continuous function from a closed bounded interval into the reals have a maximum element? (b) Why do the three usual defintions of e given in calculus actually define real numbers? Why are they the same real number?

Syllabus and Homework dates corrected at least for September (9/20/06)

A tentative syllabus is available below. Beware of frequent adjustments.

Read the assigned sections once before each lecture. Read them again after each lecture before starting on the homework. Re-read them as often as necessary! Additional material may be posted on this web page.

Attendance is crucial. I will accept late homework only in special cases and even then only if I have not yet returned the graded set. There may be a stiff penalty (possibly, 50 percent off) for workshop write-ups submitted by students who were absent from the workshop itself.

Make-up exams will be offered only if there is adequate reason to do so. A student's lack of preparation or lack of confidence is not an adequate reason. In most cases, if you must miss an exam you will know in time to discuss the matter with me (in person, by phone, or by email) IN ADVANCE. If we have not discussed the matter in advance, then I will need evidence of an emergency.

EXAMS: There will be two midterm exams and a final exam.

WORKSHOPS: We will usually have a workshop each week in which there is no exam. Workshops are essential to the course. Students will work in groups on specially prepared problem sets. One of these problems will be assigned each week to be written up and submitted the next week. Usually, a write-up will be critiqued and returned for revision before being graded for the record. The term workshop score will be based on the best ten individual write-ups. Workshop grades: 0 - 5 for content and 0 - 5 for exposition. Directions for write-ups

Summary of Directions for workshop writeups

0. Use the guidelines for homework (below) as to paper, margins, etc.

1. For each problem or part of a problem,

- Start with the caption "Task" followed by a self-contained statement of the task. The reader should not have to refer to another piece of paper.
- Next use the caption "Result" and give the result in a sentence or equation or graph as appropriate. The step is omitted if the task asks for a proof.
- Next use the caption "Work" or "Proof" as appropriate and show how you got your result and why it is correct OR give the proof if the task is to prove something. Illustrate your work or proof by any useful special cases, graphs, numerical tables, etc that would help the reader understand what you are doing and why.

HOMEWORK: There will be regular homework assignments from the textbook. These assignments will be made in class. Each graded homework problem will be graded on a 4-point scale. The term homework score will be the percentage of points earned on these assignments after dropping the weakest two homework scores. Directions for homework style are given at the start of the homework listings below.

TERM GRADES: The term grade will be based on a 550 point scale. Each midterm is worth 100 points; the final is worth 200 points; the workshops will be worth 100 points; the homework will be rescaled to be worth 50 points.

Classmeeting | Date | Sections from Text | topics |
---|---|---|---|

Lect 1 | Sept 5 | 1.1-1.3 | Field Axioms |

Wkshp 1 | Sept 5 | - | Decimal representations of reals |

Lect 2 | Sept 7 | 1.3-1.4 | Completeness |

Lect 3 | Sept 12 | 1.3-1.4 | more completeness |

WkShp 2 | Sept 12 | - | why does 5 have a positive square root? |

Lect 4 | Sept 14 | 2.1-2.2 | finish workshop \#2 |

Lect 5 | Sept 19 | 2.3, 2.4 | more on completeness; intro to sequences |

Lect 6 | Sept 21 | 2.3 | properties of limits |

Lect 7 | Sept 26 | 2.4 | more propertiesof limits |

Lect 8 | Sept 28 | 2.2-4 | monotone convergence |

Lect 9 | Oct 3 | 2.4 | Nested Interval Th |

Lect 10 | Oct 5 | 2.5 | basic subsequences |

Lect 11 | Oct 10 | Exam #1 | Ch 1 and Ch 2.1-5 |

Lect 12 | Oct 12 | 2.5-6 | Cauchy sequences |

Lect 13 | Oct 17 | 2.5-6 | Bolzano-Weierstrass |

Lect 14 | Oct 19 | 2.7 | more on series |

Lect 15 | Oct 24 | 3.2 | some topology |

Lect 16 | Oct 26 | 4.2 | limits and continuity |

Lect 17 | Oct 31 | 4.3 | properties of continuity and discontinuity |

Lect 18 | Nov 2 | 4.4 | continuous functions on [a,b] |

Lect 19 | Nov 7 | 4.5 | Intermediate Value Theorem |

Lect 20 | Nov 9 | 5.1-2 | Derivatives |

Lect 21 | Nov 14 | 5.3 | Mean Value Th. |

Lect 22 | Nov 16 | 5.3 | catch-up and review |

Lect 23 | Nov 21 | 2.2-5.3 | Exam 2 probably |

-- | Nov 23 | Thanksgiving | holiday |

Lect 24 | Nov 28 | notes | Exponential series |

Lect 25 | Nov 30 | notes | Exponential series |

Lect 26 | Dec 5 | notes | exponential function |

Lect 27 | Dec 7 | notes | natural logarithm |

Lect 28 | Dec 12 | notes | logs and exp |

Review | tba | all | all |

Final Exam | tba | tba | place tba |

Try to do all problems from sections we are discussing.

I will assign three to five problems to turn in for grading each week. This assignments will be made in class. The grader will grade at least three problems each week, at most five. We will discuss some textbook problems in class.

Directions for writing up homework:

- Write on 8.5-by-11 paper.
- Write on one side only.
- Start each new numbered problem at the top of a fresh sheet.
- Put your name on each sheet.
- Number your sheets carefully, for example as 1 of 3; 2 of 3; 3 of3.
- Staple homework together, being sure that you include all your work in the proper order.
- Write neatly.
- Give reasons for your steps.
- Maintain margins on all four sides of your work.

Remember you will be graded on clarity of mathematical communication as well as on "mathematical content". If the reader cannot understand your content, the reader cannot give credit for it.

VERY TENTATIVE HOMEWORK ASSIGNMENTS by textbook section

Warning. Some of the problems in Abbott have minor technical errors. For example in Problem 1.3.3, there is a missing assumption, namely that A is not empty. If you suspect that the statement of the problem needs fixing -- either fix it yourself, or check with me by e-mail -- but do not oversimplify the problem by adding assumptions.

Due Date | Section | Problems to do | Problems to turn in | |

Sept 14 | 1.2 | 1, 3 ,5, 7, 10 | 1, 5, 10 | |

Sept 21 | 1.3 | 2, 3ab, 4, 6, | 2, 4, | |

Sept 21 | 1.4 | 2, 5 | 2 | |

Sept 28 | 2.2 | 1, 5, 8 | 1b, 5b | |

Sept 28 | 2.3 | 2, 4, 7, 8 | 7, 8 | |

Oct 5 | 2.3 | 10 | 10 | |

Oct 5 | 2.4 | 2, 4 | 4 | |

Oct 19 | 2.5 | 3 | 3 | |

Oct 19 | 2.4 | 3 | 3 | |

Oct 19 | 2.6 | 1, 3 | 1 | |

Oct 26 | 2.7 | 1-5 | 1a, 4, 5 | |

Nov 2 | 3.2 | 2, 3, 7, 8, 12 | 2, 3, 12 | |

Nov 9 | 4.2 | 1, 3a, 6, 8, 9 | 1, 6, 8 | |

Nov 16 | 4.3 | 7, 8, 10 | 7, 8 | |

Nov 21 | 4.4 | 4,, 6, 10 | 6 | |

Nov 30 | 5.2 | 4, 1, 5 | 4 | |

?? | 5.3 | 1, 5 | 1, 5 |

- Exam 1 Solutions p1 Exam 1 Solutions p2 Exam 1 Solutions p3 Exam 1 Solutions p4 Exam 1 Solutions p5 Exam 1 Solutions p6

- Exam 2 Solutions p1 Exam 2 Solutions p2 Exam 2 Solutions p3 Exam 2 Solutions p4 Exam 2 Solutions p5 Exam 2 Solutions p6

The following material will probably not be covered this term.

- R.I. Sec.1 p1 R.I. Sec.1 p2 R.I. Sec.1 p3 R.I. Sec.1 p4 R.I. Sec.1 p5 R.I. Sec.1 p6 R.I. Sec.1 p7 R.I. Sec.1 p8

- R.I. Sec.2 p1 R.I. Sec.2 p2 R.I. Sec.2 p3 R.I. Sec.2 p4 R.I. Sec.2 p5 R.I. Sec.2 p6 R.I. Sec.2 p7 R.I. Sec.2 p8