| For class.. | ..read: | Topic/Notes | Problems | Comments | Due date |
|---|---|---|---|---|---|
| F 9/7 | B 1.1 | What brought on the crisis?
Describe the illustrative problem. | 1.1.2
1.1.3 | Assume calculus theorems. OK to skip #3's psi case. | M 9/10 |
| F 9/7 | B 1.2 | Where do (1.15) and (1.16) come from? | 1.2.2, 1.2.3 | Routine calculation | M 9/10 |
| M 9/10 | B 2.1 B 2.2 | Avoiding infinite series
Newton on Pi | 2.1.5 2.2.11, 2.2.20 | M 9/17 | |
| W 9/12 | B 2.3 | Logarithms, harmonic series | 2.3.1, 2.3.9 | M 9/17 | |
| F 9/14 | B2.4 B 2.5 | Taylor series
Emerging doubts | 2.4.1, 2.4.6, 2.4.11 | M 9/17 | |
| M 9/17 | S 3.9-11 S 7 | Metrics, metric spaces
Cauchy sequences Equivalence classes | J(f-g) is a metric 7-1, 7-2 | The metric space concept abstracts key properties of the real number system. | W 9/26 |
| W 9/19 | S 8 | The real number system | 8-5, 8-7 | W 9/26 | |
| F 9/21 | |||||
| M 9/24 | S 9-10 | Completeness properties of R
Extended real line | 9-4, 9-9 Ex Cr | M 10/1 | |
| W 9/26 | S 17 | "Nowhere dense" etc.: Fri. | 17-1 plus | Neat NASC for A
to be dense in X | M 10/1 |
| F 9/28 | S 18 | Heine-Borel theorem
p. 130: Jon & Ben p. 131: Erik & Eric | 18-14 | Memorable
Insert "relatively" before "closed". | M 10/1 |
| M 10/1 | S 19 | Open sets | 19-1 | F 10/5 | |
| W 10/3 | S 20 S 21 | Perfect sets Fractal space | Extra credit | Curiosity required | |
| F 10/5 | S 22 | Connected sets | |||
| M 10/8 | Review | Bressoud chs. 1-2
Sprecher chs. 3, 5 | In-class test
Take-out test | Concepts & theorems
Problems | M 10/8
W 10/10 |
| W 10/10
F 10/12 | B 3.1
B 3.2 | Newton-Raphson method
Differentiability | 3.1.2, 3.1.12
3.2.6 | Answers | W 10/17 |
| M 10/15 | B 3.4 | Continuity | 3.4: #14, 22, 23, 25, 26 | Most should be easy.
#25 graphs | W 10/24 |
| W 10/17 | B 3.3 | East team: JM, EP, ET
SW team: AG, KP, BS NW team: JH, SH, MS | 3.3.1
3.3.12 3.3.14 | Where does proof fail?
Nbhd = (B,infinity) Use L'Hospital's (3.48). | W 10/17 |
| B 3.3 | Taylor's theorem
Two remainders | 3.3.2, 3.3.17
Prove (3.56-57) | p. 86 | W 10/24 | |
| W 10/24 | B 3.5 | Consequences of continuity | 3.5.5,
3.5.7, 3.5.8 10-24-1 | Part of BIG THEOREM
Cf. Thms 3.6-7 Cf. Ex. 3.1.12 | F 11/2 |
| F 10/26 | S 23-27 | Continuity
esp. uniform continuity | Prove Thm 27.4 | without the Lebesgue
Covering Theorem | F 11/2 |
| M 10/29 | S 28-31 | Differentiability | 10-30-1 | This is extra credit. | M 11/5 |
| W 10/31 | B 4.1 | Convergence/divergence
of infinite series: The basic tests | 4.1.20, 4.1.21, 4.1.25
10-29-1 | Study pp. 125-126. | F 11/2 |
| F 11/2 | B 4.2 | Series of functions | 4.2.3, 4.2.5
11-2-1 Hints | You may assume Stirling's approx. | F 11/9 |
| M 11/5 | B 4.3 | Gauss's test | 11-5-1 | A part of Gauss's proof | F 11/9 |
| W 11/7 | B 4.4 | Finer tests | Extra credit | Suggested 4.4 problems: 3, 5-8, 10-12 | |
| F 11/9 | B 4.3 | Hypergeometric series
Gauss's tests Review | 4.3.1, 4.3.3, 4.3.7 | Each team does all problems.
Teams may divide the work. Make presentations on Fri. | F 11/9 |
| M 11/12 | B 4.5
Review | Abel's lemma
Dirichlet's test | |||
| W 11/14 | Review | Exam 2a
Bressoud chs. 3-4 Sprecher chs. 6-7, 4 | Exam 2b | Problems on
Bressoud chs. 3-4 Sprecher chs. 6-7, 4 | F 11/16 |
| F 11/16 | B 5.1 | Weird series phenomena | 5.1.12
11-16-1 | Class notes | W 11/21 |
| M 11/19 | B 5.2 | Infinite sums of continuous functions | |||
| W 11/21 | B 5.2
B 5.3 | Interchanging limits | 4.5.12, 5.2.8 | F 11/30 | |
| M 11/26 | B 5.3
B 5.4 | Integrating infinite sums
Weierstrass M-test | 5.3.6
Ex. Cr.:5.3.2-5 | 33-11, 33-12 | F 11/30 |
| W 11/28 | B 5.4
S 32 | Verifying uniform convergence | 5.4.7 | Metric version of uniform convergence | F 12/7 |
| F 11/30 | S 35
pp. 227-230 | Weierstrass Approximation Theorem | Proof
in class | Class project
See 11/26 assignment | F 11/30 |
| M 12/3 | B 6.1 | Fourier series:
Dirichlet's theorem | 6.1.5, 6.1.10, 6.1.13 | Comment on the relevance of each problem to Dirichlet's theorem. | M 12/10 |
| W 12/5 | B 6.2 | The Cauchy integral | 6.2.11 | You'll prove the Fundamental Theorem of Calculus. | M 12/10 |
| F 12/7 | B 6.3 | The Riemann integral | |||
| M 12/10 | Various integrals | Extra credit | See class notes | M 12/17 | |
| W 12/12 | Review |