Math 424 Daily Log and Homework

Math 424: Daily Log of Material Covered, Readings and Assigned Homework

Readings are noted on the day when they should be completed. Daily homework problems are noted on the day when they are assigned; they will be discussed on the following day. Weekly (graded) homework assignments are noted on the day when they are assigned, and the due date is provided.

Week Date Reading Daily Homework Weekly Homework

1/16 - 1/20 1/16 No class: MLK Jr day

1/18 No reading (first day of class) 2.3, 2.7

1/20 Chapters 1,2: set theory, real number system, order, least upper bounds 2.8, 2.9 HW 1 (due Wed 1/25):
2.11, 2.12

1/23 - 1/27 1/23 construction of the real numbers (handout) none

1/25 3.1: metric spaces none

1/27 3.1,3.2: metric spaces, open and closed sets 3.1, 3.3 HW 2 (due Fri 2/3):
3.1(b)(c), 3.6, 3.15

1/30 - 2/3 1/30 3.2,3.3: open and closed sets, convergent sequences 3.4, 3.6

2/1 3.3,3.4: convergent sequences, completeness 3.8, 3.14

2/3 3.4: completeness Prove: every closed subset of a
complete metric space is complete HW 3 (due Fri 2/10):
3.24, 3.25, 3.31

2/6 - 2/10 2/6 3.5: compactness Let (X,d) be a discrete metric space.
When is (X,d) sequentially compact?

2/8 3.5: compactness none

2/10 3.5: compactness none
2/13 - 2/17 2/13 review for Midterm Exam 1 (optional)
Midterm Exam 1 from 7:00-8:15pm in 141 Altgeld Hall
Topics: Chapter 2 (all), Chapter 3 (sections 3.1 - 3.5) none

2/15 review midterm exam 1 none

2/17 3.6, 4.1: connectedness, continuity Let X be the set of rational numbers. Show that each connected component of X contains precisely one point.
4.1(a)(c) HW 4 (due Fri 2/24):
4.1(b)(d),4.2,4.8,4.10(b)(c)

2/20 - 2/24 2/20 4.1, 4.2: continuity 4.3

2/22 4.3: continuity: Euclidean range 4.4

2/24 4.4: continuity and compactness, uniform continuity 4.5 HW 5 (due Fri 3/2):
4.14, 4.16, 4.17, extra problem on homeomorphisms

2/27 - 3/3 2/27 4.5: continuity and connectedness none

3/1 homeomorphisms, bi-Lipschitz maps, isometries
(handout) Let (X,d) be a metric space. Show that d'(x,y)=d(x,y)/(1+d(x,y)) defines a new metric on X, that (X,d') is bounded, and that the identity function from (X,d) to (X,d') is a homeomorphism.

3/3 4.6: sequences of functions 4.32 HW 6 (due Fri 3/9):
4.34, 4.35, 4.42, 4.43

3/6 - 3/10 3/6 4.6: sequences of functions, uniform convergence none

3/8 4.6: sequences of functions, function spaces none

3/10 5.1: the derivative 5.1(a)(b)
3/13 - 3/17 3/13 5.2: rules of differentiation none HW 7 (due Wed 3/29):
5.2, 5.6, 5.7

3/14 Midterm Exam 2 from 7:00-8:15pm in 241 Altgeld Hall Topics: Chapter 3 (section 3.6), Chapter 4 (all)

3/15 5.3: Rolle's theorem, Mean Value theorem none

3/17 NO CLASS (Have a good spring break!) none

3/27 - 3/31 3/27 5.4: higher derivatives, Taylor's theorem, L'Hopital's rule 5.8

3/29 6.1: the Riemann integral, definitions and examples 6.2, 6.3 HW 8 (due Mon 4/3):
5.10 (composition of functions, hint: induction), 5.11

3/31 6.2: linearity, order, additivity properties of the integral 6.4

4/3 - 4/7 4/3 6.3: characterization of Riemann integrable functions, integrability of continuous functions none HW 9 (due Fri 4/7):
6.11, 6.16

4/5 6.4: Fundamental Theorem of Calculus 6.15 (Hint: do 6.9 first)

4/7 6.4: FTC (continued) none HW 10 (due Fri 4/14):
6.21, 6.22, 6.26
17.45 from the handout (D'Angelo-West, Chapter 17)

4/10 - 4/14 4/10 6.5: logarithm, exponential, Gamma function derive Gamma(1/2)=sqrt(pi) from the fact that the integral of e^(-u^2) from zero to infinity is sqrt(pi)/2, prove the latter statement

4/12 7.1: integration and differentiation of sequences 7.1, 7.2

4/14 7.2: infinite series 7.7 HW 11 (due Fri 4/21):
7.5, 7.6, 7.11

4/17 - 4/21 4/17 7.2: absolute vs conditional convergence, rearrangements none

4/19 7.3: power series, Taylor series 7.20

4/21 Weierstrass Approximation Theorem Let f be a continuous function on [0,infinity) which tends to zero at infinity. Prove that f can be approximated uniformly by "polynomials in e^(-x)", i.e., functions of the form g(x)=c_1 e^(-x) + c_2 e^(-2x) + ... + c_n e^(-nx). (Hint: change variables) HW 12 (due Wed 5/3):
7.23, 7.35, two additional problems (see handout)

4/24 - 4/28 4/24 Weierstrass Approximation Theorem (continued) none

4/25 Midterm Exam 3 from 7:00-8:15pm in 141 Altgeld Hall Topics: Chapters 5,6 (all)

4/26 Weierstrass nowhere differentiable functions none

4/28 7.5: differentiation under the integral sign none

5/1 - 5/3 5/1 NO CLASS

5/3 Review none

Week	Date	Reading	Daily Homework	Weekly Homework
1/16 - 1/20	1/16	No class: MLK Jr day
	1/18	No reading (first day of class)	2.3, 2.7
	1/20	Chapters 1,2: set theory, real number system, order, least upper bounds	2.8, 2.9	HW 1 (due Wed 1/25): 2.11, 2.12
1/23 - 1/27	1/23	construction of the real numbers (handout)	none
	1/25	3.1: metric spaces	none
	1/27	3.1,3.2: metric spaces, open and closed sets	3.1, 3.3	HW 2 (due Fri 2/3): 3.1(b)(c), 3.6, 3.15
1/30 - 2/3	1/30	3.2,3.3: open and closed sets, convergent sequences	3.4, 3.6
	2/1	3.3,3.4: convergent sequences, completeness	3.8, 3.14
	2/3	3.4: completeness	Prove: every closed subset of a complete metric space is complete	HW 3 (due Fri 2/10): 3.24, 3.25, 3.31
2/6 - 2/10	2/6	3.5: compactness	Let (X,d) be a discrete metric space. When is (X,d) sequentially compact?
	2/8	3.5: compactness	none
	2/10	3.5: compactness	none
2/13 - 2/17	2/13	review for Midterm Exam 1 (optional) Midterm Exam 1 from 7:00-8:15pm in 141 Altgeld Hall Topics: Chapter 2 (all), Chapter 3 (sections 3.1 - 3.5)	none
	2/15	review midterm exam 1	none
	2/17	3.6, 4.1: connectedness, continuity	Let X be the set of rational numbers. Show that each connected component of X contains precisely one point. 4.1(a)(c)	HW 4 (due Fri 2/24): 4.1(b)(d),4.2,4.8,4.10(b)(c)
2/20 - 2/24	2/20	4.1, 4.2: continuity	4.3
	2/22	4.3: continuity: Euclidean range	4.4
	2/24	4.4: continuity and compactness, uniform continuity	4.5	HW 5 (due Fri 3/2): 4.14, 4.16, 4.17, extra problem on homeomorphisms
2/27 - 3/3	2/27	4.5: continuity and connectedness	none
	3/1	homeomorphisms, bi-Lipschitz maps, isometries (handout)	Let (X,d) be a metric space. Show that d'(x,y)=d(x,y)/(1+d(x,y)) defines a new metric on X, that (X,d') is bounded, and that the identity function from (X,d) to (X,d') is a homeomorphism.
	3/3	4.6: sequences of functions	4.32	HW 6 (due Fri 3/9): 4.34, 4.35, 4.42, 4.43
3/6 - 3/10	3/6	4.6: sequences of functions, uniform convergence	none
	3/8	4.6: sequences of functions, function spaces	none
	3/10	5.1: the derivative	5.1(a)(b)
3/13 - 3/17	3/13	5.2: rules of differentiation	none	HW 7 (due Wed 3/29): 5.2, 5.6, 5.7
	3/14	Midterm Exam 2 from 7:00-8:15pm in 241 Altgeld Hall Topics: Chapter 3 (section 3.6), Chapter 4 (all)
	3/15	5.3: Rolle's theorem, Mean Value theorem	none
	3/17	NO CLASS (Have a good spring break!)	none
3/27 - 3/31	3/27	5.4: higher derivatives, Taylor's theorem, L'Hopital's rule	5.8
	3/29	6.1: the Riemann integral, definitions and examples	6.2, 6.3	HW 8 (due Mon 4/3): 5.10 (composition of functions, hint: induction), 5.11
	3/31	6.2: linearity, order, additivity properties of the integral	6.4
4/3 - 4/7	4/3	6.3: characterization of Riemann integrable functions, integrability of continuous functions	none	HW 9 (due Fri 4/7): 6.11, 6.16
	4/5	6.4: Fundamental Theorem of Calculus	6.15 (Hint: do 6.9 first)
	4/7	6.4: FTC (continued)	none	HW 10 (due Fri 4/14): 6.21, 6.22, 6.26 17.45 from the handout (D'Angelo-West, Chapter 17)
4/10 - 4/14	4/10	6.5: logarithm, exponential, Gamma function	derive Gamma(1/2)=sqrt(pi) from the fact that the integral of e^(-u^2) from zero to infinity is sqrt(pi)/2, prove the latter statement
	4/12	7.1: integration and differentiation of sequences	7.1, 7.2
	4/14	7.2: infinite series	7.7	HW 11 (due Fri 4/21): 7.5, 7.6, 7.11

4/17 - 4/21	4/17	7.2: absolute vs conditional convergence, rearrangements	none
	4/19	7.3: power series, Taylor series	7.20
	4/21	Weierstrass Approximation Theorem	Let f be a continuous function on [0,infinity) which tends to zero at infinity. Prove that f can be approximated uniformly by "polynomials in e^(-x)", i.e., functions of the form g(x)=c_1 e^(-x) + c_2 e^(-2x) + ... + c_n e^(-nx). (Hint: change variables)	HW 12 (due Wed 5/3): 7.23, 7.35, two additional problems (see handout)
4/24 - 4/28	4/24	Weierstrass Approximation Theorem (continued)	none

	4/25	Midterm Exam 3 from 7:00-8:15pm in 141 Altgeld Hall Topics: Chapters 5,6 (all)
	4/26	Weierstrass nowhere differentiable functions	none

	4/28	7.5: differentiation under the integral sign	none

5/1 - 5/3	5/1	NO CLASS
	5/3	Review	none