Math 202: Proving Things: Analysis |
Fall 2018 |
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Faculty: Jerry L. Kazdan

Telephone: (215) 898-5109

email:kazdan AT math.upenn.edu

Office Hours: Wed. 10:30-11:30 (and also by appointment) in DRL 4E15TA: Man Cheung Tsui

Telephone: (215) 746-7326

email:mancheun@sas.upenn.edu

Office Hours: Mon. and Wed., 5:30-6:30 (and also by appointment) in DRL 4C15.

Web page:This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication.

We will study real and complex numbers, sequences, series, continuity, differentiability and integrability for functions of one variable, proving our way as we go, and enjoying a number of challenging problems.

This course is about thinking, attempting to understand. It it for students who enjoy thinking hard, even when completely stumped. Then understanding (if and when it comes) is all the more satisfying.

Intuitionandcomputational skillwill be essential in the discovery and presentation of your ideas.The concept of a

proofwill be vital. There is nothing exotic about a proof. It is simply convincing someone else about your reasoning. You give different proofs to different people, depending on their background — and how well you know them. The first and most important person to convince isyourself.Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening. The evening labs will provide opportunity for all students to present their solutions at the blackboard, and to become comfortable and proficient at doing this.

To be successful in this course, you should be present for all class meetings and plan to take good notes.

Text:John P. D'Angelo and Douglas B. WestMathematical Thinking: Problem-Solving and Proofs, Second edition, Prentice Hall (2000). See also: Typos and corrections. [Note: There is also a newer paperback "Classical Edition", It is the *same* book.]

Prerequisites & Review Material:Some experience with calculus.

To remove rust from your background I suggest doing the problems from recent Math 103 Final Exams. Note that both Math 103 and 104 focus on techniques for solving standard calculus problems. This course will focus on basic ideas and proofs.Problems to think about during August.

On the first day of class some students might be invited to present their solutions at the blackboard.

Notes:

Symmetries of a Square

Big numbers

Axioms for a Field

Sum-of-Squares

Real solution of x^{2}= 2

Logic Notation

Newton's method for square roots.

Some Puzzles

TeX/LaTeX Stuff Creating documents having formulas.

ODE: Existence

MVT: How do you prove that?

The Fundamental Theorem of Algebra

Homework Assignments:

- Set 0 Discussed by
YOUin class on Tues. Aug. 28.(This will not be collected.)- Set 1 Due: Thurs., Sept. 6 in class. [Late papers will be accepted until 1:00 PM Fri.]
- Set 2 Due: Thurs., Sept. 13 in class. [Late papers will be accepted until 1:00 PM Fri.]
- Set 3 Due: Thurs., Sept. 20 in class. [Late papers will be accepted until 1:00 PM Fri.]
- Set 4 Due: Thurs., Sept. 27 in class. [Late papers will be accepted until 1:00 PM Fri.]
- Set 5 Due: Thurs., Oct. 11 in class. [Late papers will be accepted until 1:00 PM Fri.]
- Set 6 Due: Thurs., Oct. 18 in class. [Late papers will be accepted until 1:00 PM Fri.]
- Set 7 Due: Thurs., Oct. 25 in class. [Late papers will be accepted until 1:00 PM Fri.]
- Set 8 Due: Thurs., Nov. 8 in class. [Late papers will be accepted until 1:00 PM Fri.]
- Set 9 Due: Thurs., Nov. 15 in class. [Late papers will be accepted until 1:00 PM Fri.]
- Set 10 Due: Thurs., Nov. 29 in class. [Late papers will be accepted until 1:00 PM Fri.]

Exams:There will be three in-class exams, from 10:30-12:00 onTues., Oct. 2, Thurs. Nov. 1, Thurs. Dec.6 [No Final Exam] You may always use one 3"×5" card withhandwrittennotes on both sides.

Exam 1: Original, condensed, solutions.

Exam 2: Original, condensed, solutions.

Exam 3: Original condensed, solutions.

Old Exams:0

- Exam. 1, Tues. Oct. 1, 2013 Original, condensed, solutions.
- Exam. 2, Tues. Nov. 5, 2013 Original, 0condensed, solutions.
- Exam. 3, Tues. Dec. 10, 2013 Original, condensed, solutions, Make-up Exam 3