| SES # | TOPICS | READINGS | ASSIGNMENTS |
|---|---|---|---|
| 1 | Monotone sequences; completeness property | Chapter 1; Appendix A, 2.1 (as needed) | 1.4/2 (consider an+1= an); 1.5/2; prob 1-1b, c |
| 2 | Estimations and approximations | Chapter 2 | 2.1/2; 2.4/2; 2.5/3; 2.6/2; prob2-1a |
| 3 | Limit of a sequence | Chapter 3.1-.6 | 3.1/1a; 3.2/4; 3.3/3; 3.4/5; 3.6/1b |
| 4 | Error term; algebraic limit theorems | Chapter 4 (omit 4.3), 5.1 | 4.2/1ab; 4.4/2 (assume A > B); Q5.1/1a,3; 5.1/4 |
| 5 | Limit theorems for sequences | Chapter 5.2-3, .5 | 5.2/3; 5.3/1; 5.3/4 (a: omit hint; b: counterexample?); prob 5-1* |
| 6 | Nested intervals; cluster points | Chapter 5.4, 6.2 | 5.4/1(take k = 2); P5-3; 5.4/3 (below); Q6.2/1(for {cos(n + 1/n)π}; Q6.2/2ab 5.4/3: Do 5.4/2, using p(n)/n; p(n) = highest prime factor of n (e.g., p(12) = 3; p(15) = 5) |
| 7 | Bolzano-Weierstrass theorem; Cauchy sequences | Chapter 6.1-4 | 6.1/1b; 6.3/1; 6.4/2; P6-4 |
| 8 | Completeness property for sets | Chapter 6.5 | 6.4/3(cf.sec. 3.1); 6.5/1ac, 3ag; P6-2ab |
| 9 | Infinite series | Chapter 7.1-4 | 7.2/2,5; 7.1/2, P 7-5 (go together); 7.4/1bdeh |
| 10 | Infinite series (cont.) | Chapter 7.5-7 | 7.4/1acfg; 7.4/3a; P7-6; P7-1 |
| 11 | Power series | Chapter 8.1-.3, (8.4 lightly) | 8.2-3; 8.4 (skip proofs); 8.1/1adh; 8.2/1adh; 8.1/3; 8.3/1, 2; 8.4/1a(i), b |
| 12 | Functions; local and global properties | Chapters 9, 10: 9.2/2; 9.3/3, 5; 9.4/1c; 10.1/5, 6b, 7b; 10.3/2, 4*(= 5 in early ptgs.) | Chapters 9, 10: 9.2/2; 9.3/3, 5; 9.4/1c; 10.1/5, 6b, 7b; 10.3/2, 4*(= 5 in early ptgs.) |
| Exam 1 covering Ses #1-12 | |||
| 13 | Continuity | Chapter 11.1-3 | 11.1/1, 4, 5 (exp.law:ea+b = eaeb); 11.3/1, 3a |
| 14 | Continuity (cont.) | Chapter 11.4-5 | 11.4/1, 2; 11.5/1ab; P11-2 |
| 15 | Intermediate-value theorem | Chapter 12.1-.2 | 12.1/3, 5; 12.2/2, 3, 4; (P12-5 opt'l, for + or gold star) |
| 16 | Continuity theorems | Chapter 13.1-3 | 13.1/Q1 (give ctrexs.); 1, 2; 13.2/1; 13.3/3, 1b* |
| 17 | Uniform continuity | Chapter 13.5 (13.4 lightly) | 13.5/5, 6; P13-2; P13-6 |
| 18 | Differentiation: local properties | Chapter 14 | 14.1/1, 4; 14.3/1, 2a; P14-2; P14-6ab |
| 19 | Differentiation: global properties | Chapter 15 15 | 15.2/1ab; 15.4/1; P15-2; P15-3ab |
| 20 | Convexity; Taylor's theorem (skip proofs) | Chapter 16 to p.225, (17.1-.3 lightly) | 17.4. 16.1/1a; 17.2/4ab (use x2(x - 1)2 ; P16-3, 5 |
| 21 | Integrability | Chapter 18.1-.3, (18.4 lightly) | 18.2/2; Q18.3/3ab; 18.3/1, 3 |
| 22 | Riemann integral | Chapter 19.1-.4; (.5, .6 lightly) | 19.2/1; 19.3/1, 3; 19.4/2 |
| 23 | Fundamental theorems of calculus | Chapter 20.1-4 | 20.1/1; 20.2/4; 20.3/2, 3; P20-2a |
| 24 | Stirling's formula; improper integrals | Chapter 21.1-2 | 21.1/2(set x = 1/u); 21.2/1c, e, h(x = 1/ u), 2, 4 |
| 25 | Gamma function, convergence | Chapter 20.5, 21.3 | 20.5/1a, 2; Q21.3/1, 2; 21.3/1 |
| Exam 2 covering Ses #13-25 | |||
| 26 | Uniform convergence of series | Chapter 22.1-2 | 22.1/1ac, 2; 22.2/2bd, 3 |
| 27 | Integration term-by-term | Chapter 22.3-4 | 22.3/1, 3 (cf. warning in 22.3/2); 22.4/1, 3; P22-3b |
| 28 | Differentiation term-by-term; analyticity | Chapter 22.5-6 | 22.5/1; Q22.5/1; 22.6/2; 22.6/5; P22-2 (just show J0 solves the ODE) |
| 29 | Quantifiers and negation | Appendix B Negation | Not required, no assignment, but recommend trying QB.1, QB.2, QB.3 |
| 30 | Continuous functions on the plane | Chapter 24.1-.5 | 24.1/3; 24.2/2, 3; 24.4/1; 24.5/2, 5 |
| 31 | Continuous functions on the plane (cont.); plane point-set topology | Chapter 24.6-.7 | 25.1-.2 24.7/1, 2; P24-1; 25.1/1; 25.2/2 |
| 32 | Compact sets and open sets | Chapter 25.2-.3 | 25.3/1; P25-1; 25.3/3; 25.2/5; P25-3a |
| 33 | Differentiating finite integrals | Chapter 26.1-.2 | 26.1/1b; 26.2/1ab; (but use: 0∫π cos (xt) dt ; P26-1 (use 20.1 or 20.3A) |
| 34 | Differentiating finite integrals (cont.); Fubini's theorem in rectangular regions | Chapter 26.2-.3 | Q26.2/1; 26.2/5; 26.3/1, 2 |
| 35 | Uniform convergence of improper integrals | Chapter 23.1-.2 | Q27.2/1,2; Q27.3/1,2 (not to hand in) |
| 36 | Differentiation and integration of improper integrals; applications | Chapter 23.3-.4 | Q27.4/1,2 (not to hand in) |
| 37 | Comments; review | Practice final given out (PDF) | |
| Three-hour final exam during finals week |
Table of Contents
Reading, Assignments, and Exams
Reading, Assignments, and Exams
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All readings and daily problem assignments are from the textbook:
Mattuck, Arthur. Introduction to Analysis. Upper Saddle River, NJ: Prentice Hall, 1999. ISBN: 9780130811325.
Table of Contents (PDF)
Key to the Readings
Chapter 1.1-1.3, Appendix A means:
Read: Chapter 1, sections 1, 2, 3, and Appendix A (in the back of the book)
Key to the Assignments
1.4/3,5 Q1.4/2 P1-3,4* means:
Work:
Exercises 3, 5 in Exercise Section 1.4 (at the end of Chapter 1)
Question 1.4/2 (at end of Chapter1, Section 4, answered at very end of Chapter 1)
Problems 3 and 4 at end of Chapter 1
* means Problem 4 is corrected in one or more printings.
Note on the Table below: The daily reading assignments depended on what the class had been able to cover that day, so they differed a little in order and content from the overall plan laid out in the "topics" column.