Syllabus

Week 1. August 21 - 25

1.1 Properties of the Real Numbers: Introduction

1.2 The Real Number System

1.3 Algebraic Structure

1.4 Order Structure

1.5 Bounds

1.6 Sups and Infs

1.7 The Archimedean Property

1.8 Inductive Property of N

1.9 The Rational Numbers Are Dense

1.10 The Metric Structure of R

2.1 Sequences: Introduction

2.2 Sequences

Skip 2.3

Drop/Add Deadline: August 25

Week 2. August 28 - September 1

2.4 Convergence

2.5 Divergence

2.6 Boundedness Properties of Limits

2.7 Algebra of Limits

2.8 Order Properties of Limits

2.9 Monotone Convergence Criterion

Week 3. September 5 - 8

Labor Day: September 4

2.10 Examples of Limits

2.11 Subsequences

2.12 Cauchy Convergence Criterion

Skip 2.13

4.1 Sets of Real Numbers: Introduction

4.2 Points

4.3 Sets

Week 4. September 11 - 15

4.4 Elementary Topology

4.5 Compactness Arguments (4.5.1 and 4.5.5 only)

4.6 Countable Sets

5.1 Introduction to Limits (skip 5.1.3)

Week 5. September 18 - 22

5.2 Properties of Limits (through 5.2.3)

Review

Exam 1: Thursday, September 21

Week 6. September 25 - 29

5.2 Properties of Limits (from 5.2.4)

Skip 5.3

5.4 Continuity (include 5.4.1, skip 5.4.4)

5.5 Properties of Continuous Functions

5.6 Uniform Continuity

Week 7. October 2 - 6

5.7 Extremal Properties

5.8 Darboux Property

5.9 Points of Discontinuity (skip 5.9.3)

7.1 Differentiation: Introduction

7.2 The Derivative (skip 7.2.3)

7.3 Computations of Derivatives (through 7.3.2)

Week 8. October 9 - 13

7.3 Computations of Derivatives (from 7.3.3)

7.4 Continuity of the Derivative?

7.5 Local Extrema

7.6 Mean Value Theorem (include 7.6.3)

7.7 Monotonicity

Skip 7.8

7.11 L'Hopital's Rule (include 7.11.1, skip 7.11.2 - 7.11.3)

Midterm Grades Due: October 13

Week 9. October 16 - 20

7.9 Inverse Function Theorem

7.10 Convexity

Review

Exam 2: Thursday, October 19

Week 10. October 23 - 27

7.12 Taylor Polynomials

8.1 The Integral: Introduction

8.2 Cauchy's First Method (include 8.2.1)

8.3 Properties of the Integral

8.4 Cauchy's Second Method

Withdrawal Deadline: October 27

Week 11. October 30 - November 3

8.5 Cauchy's Second Method (Continued)

8.6 The Riemann Integral (include 8.6.2 - 8.6.4)

8.7 Properties of the Riemann Integral

8.8 The Improper Riemann Integral

Skip 8.9

3.1 Infinite Sums: Introduction

3.2 Finite Sums

Skip 3.3

Week 12. November 6 - 9

3.4 Ordered Sums: Series

3.5 Criteria for Convergence

3.6 Tests for Convergence (skip 3.6.7 - 3.6.11 and 3.6.13 - 3.6.14)

Skip 3.7 - 3.11

9.1 Sequences and Series of Functions: Introduction

9.2 Pointwise Limits

Veterans' Day: November 10

Week 13. November 13 - 17

9.3 Uniform Limits (skip 9.3.3)

Review

Exam 3: Thursday, November 16

Week 14. November 20 - 22

9.4 Uniform Convergence and Continuity (skip 9.4.1)

9.5 Uniform Convergence and the Integral (skip 9.5.2 - 9.5.3)

9.6 Uniform Convergence and Derivatives (skip 9.6.1)

Skip 9.7 - 9.8

Thanksgiving Break: November 23 - 24

Week 15. November 27 - December 1

epsilon-delta survey

10.1 Power Series: Introduction

10.2 Power Series: Convergence

10.3 Uniform Convergence

10.4 Functions Represented by Power Series

10.5 The Taylor Series

Week 16. December 4 - 5

Review

Final Exam: Friday, December 8, 10:00-11:50 AM