Dept. Phone: (631)-632-8290

Dept. FAX: (631)-632-7631

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MAT 319 R01 MW 11:45am-12:40pm Earth-Space Science 181, TA is Jordan Rainone

MAT 319 R02 MW 11:45am-12:40pm Frey Hall 309, TA is Paul Sweeney

MAT 319 R03 MW 11:45am-12:40pm Socbehav Sci N310, Dylan Galt

MAT 320 R01 MW 11:45am-12:40pm Physics P116, TA is Jordan Rainone (this meets after 1st midterm)

Note that all recitations meet at the same time.

Most of the problems are short proofs. In general, a few sentences will be sufficient for most problems. A proof should be a clearly written argument; the graders may penalize organizational or grammatical errors that make the problem hard to read or the reasoning hard to follow. Include a diagram or picture if this helps convey the meaning. For most problems, I would expect that a couple of practice drafts will be necessary. It is hard even for an experienced mathematician to write down a complete, legible proof off the top of their heads (published proofs can take weeks, months or even years of polishing to make the ideas understandable, but I would guess 15-30 minutes would be sufficient for most problems in this course).

1.1 Sets and functions (problems 6, 14, 18, 21)

1.2 Induction (problems 7, 11, 17)

1.3 Finite and infinite sets (problems 4, 12, 13)

A few people did not have the textbook the first week. Here is a scan of problems due Aug 30 from textbook sections 1.1, 1.2, 1.3 ,

Since there is no recitation Monday 9/6, Problem set 2 is due in recitation on Wed., 9/8. Section 2.1 problems only.

2.1 Algebraic and order axioms (problems 4, 8, 17, 19)

Revised problem assignment for Week 3 is given below.

2.2 Absolute value and the real line (problems 17)

2.3 The completeness property (problems 10, 12,13)

2.4 Applications of the Supremum property (problems 5, 7, 11, 19)

2.5 Intervals (no problems assigned)

The others are recommended to try.

Some will be chosen for the midterm, so you should be able to do them.

Problems are still due in recitation on Mondays.

3.1 Sequences and their limits (problems 4, 5*, 11*, 17)

3.2 Limit theorems (problems 2*, 7*, 15, 20)

3.3 Monotone sequences (problems 2, 7*, 9)

3.4 Bolzano-Weierstrass theorem (problems 6, 9, 18*)

3.5 The Cauchy Criterion (problems 4*, 6, 11*)

3.6 Properly divergence sequences (problems 2, 5, 10)

3.7 Introduction to infinite series (problems 5, 11*, 15*, 17)

Tuesday Sept 28, Review of Chapters 1-3

Thursday Sept 30, Midterm 1 in lecture

MAT 320 moves to Math Tower P-131 staring Thursday, October 7.

4.1 Limits of functions

4.1 Limits of functions (problems 6, 13*, 16)

4.2 Limit Theorems (problems 8, 12*, 14)

4.3 Extensions of the limit concept - read on your own (problems 7*, 11)

5.1 Continuous functions (problems 3, 6, 12*, 14*, 15)

5.2 Combinations of continuous functions (problems 3*, 8*, 11, 14)

5.3 Continuous functions on intervals (problems 6, 13*, 18)

5.4 Uniform continuity (problems 2, 6*, 15, 16*)

Short (but not simple) proof of Weierstrass approximation theorem

Sketch of Weierstrass's proof of his approximation theorem

Bernstein's proof of Weierstrass approximation theorem

5.6 Monotone and inverse functions (problems 2, 9, 10, 12*, 13)

6.1 The deriviative (problems 7, 9, 13*, 17)

6.2 The mean value theorem (problems 8, 11, 12, 13*, 15)

A proof that the Weierstrass function is nowhere differentiable is in Section 5.2 of my book Fractals in probability and analysis with Y. Peres.

6.4 Taylor's theorem (problems 8*, 12, 16)

7.1 The Riemann integral (problems 7, 8*)

7.2 Riemann integrable functions (problems 6* (give an example), 8, 9, 15*)

7.3 The fundamental theorem (problems 8, 14, 16*, 21*, 22*, due Mon, Nov 15)

Appendix C, The Riemann and Lebesgue criteria

A proof of Theorem 8.2.5 (omitted from textbook) .

Nov 9 notes

Sample Midterm 2

Midterm 2 (covers chapters 4-7) Histogram of Midterm 2 total scores , Scatter plot of T/F versus Proof scores ,

Nov 16 notes

Nov 18 notes

8.1 Pointwise and uniform convergence (problems 12, 19, 21, 24*)

8.2 Interchange of limits (problems 3*, 14, 16*, 17, 18)

9.1 Absolute convergence (problems 2*, 7, 8*, 9, 15, 16)

9.2 Tests for absolute convergence (problems 15, 17, 19)

Thanksgiving break, no class Thursday.

Tuesday's class in on Zoom. See Blackboard for the link.

Nov 23 notes for online lecture

No homework over Thanksgivng break

9.3 Tests for non-absolute convegence (problems 6, 9, 10, 12)

9.4 Series of functions (problems 2, 7, 11, 12, 15)

Nov 30 notes

online slides for Tuesday

11.1 Open and closed sets (problems 10, 11, 16}

11.2 Compact sets (problems 4, 9, 11)

11.3 Continuous functions (problems 2, 7, 10)

11.4 Metric spaces (problems 7, 9, 12)

program website

Wikipedia page on calculus.

Wikipedia page on Issac Newton.

Wikipedia page on Newton-Leibniz controversy.

Wikipedia page on the discovery of the planet Neptune (using only mathematics).

Wikipedia page on Gauss.

Wikipedia page on Riemann.

Wikipedia page on Lebesgue.

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