# MTH 421-002: Analysis II

### Lecture location: https://msu.zoom.us/j/4813665416Shared OneNote NotebookYoutube Lecture Videos

Resources for online class
Week 1

Monday 1/6
Make sure you are familiar with the syllabus.
5.1 The Riemann Integral
You should now be able to do Problem 2 in HW1.

Wednesday 1/8
Read Before Class: Def. 5.3, Remark 5.7 and proof, Def. 5.9, Th. 5.10 and proof.
Friday 1/10
Homework Due: HW1

Week 2

Monday 1/13
Read Before Class: Remark 5.14 and proof,  Th. 5.15 and proof.
Wednesday 1/15
5.2
Riemann Sums
Read Before Class: Definition 5.17,  Th. 5.18 and proof.
Friday 1/17
Read Before Class: Th. 5.19, 5:20, 5.22, 5.23, 5.24 and their proofs. You should be able to prove 5.21 on your own.
Homework Due: HW2
Quiz 1
– covers sect. 5.1 (see HW1 and HW2)

Week 3

Monday 1/20 – MLK, no classes

Wednesday 1/22
5.3 The Fundamental Theorem of Calculus

Friday 1/24
Read Before Class: Read Th. 5.34 (Change of Variables) and its proof.
Homework Due: HW3:
Complete the proof of Th. 5.20 by proving the inequality at the bottom of p. 144.
5.2.2 – use the 1st MVT,
5.2.6 – use Comparison Th. + Squeeze Th.,
5.2.11 – What does it mean for the function to be unbounded? You can use the Bolzano-Weierstrass Theorem to show that there exists a convergent sequence s_n, such that f(s_n) goes to infinity.
Quiz 2 – covers sect. 5.2 (see HW3)

Week 4 - Jan. 31 - last day to drop with refund

Monday 1/27
8.1 Algebraic Structure of R^n
Read Before Class: the first 4 pages of Section 8.1 (267-270)

Wednesday 1/29
Read Before Class: pp. 2710-274  of Section 8.1; focus on the proof of Th. 8.5.

Friday 1/31 – Last day to drop with refund (8:00pm)
8.2 Planes and Linear Transformations
Homework Due: HW4: 5.3.0a, 5.3.1b, 5.3.2c, 5.3.5,  5.3.7a-d, (5.3.7e is bonus).
Quiz 3
– covers sect. 5.3

Week 5

Monday 2/3
8.2 – continued
Wednesday 2/5

8.3 Topology of R^n
Read Before Class: the first 4 pages of Section 8.3 (288-291)
Friday 2/7
8.3 – continued
Homework Due: HW5
Quiz 4 – covers sect. 8.1

Week 6
Monday 2/10
8.3 – continued

Wednesday 2/12
Worksheet on open/closed sets in the discrete topology.

Friday 2/14
8.4 Interior, Closure, and Boundary
Read Before Class: the first 3 pages of Section 8.4 (297-299)
Homework Due: HW6: 8.2.2, 8.2.4, 8.2.5a,b, 8.2.10a, 8.2.11-bonus, 8.3.2, 8.3.3b, 8.3.4
Quiz 5 – covers sect. 8.2 and 8.3

Week 7 - Exam 1

Monday 2/17
8.4 – continued

Wednesday 2/19
Review for Exam 1
Old exam for practice – work on this exam for 50 minutes on your own to test yourself,   then use your notes and book to see if you can do even better. Bring your work to class  on Wednesday.

Friday 2/21 – Exam 1
Homework Due: HW7: 8.3.7a,c,d, 8.3.8a, 8.3.9, 8.4.1, 8.4.3, 8.4.5,  8.4.8 – bonus.
Notes on HW: 8.3.7a – provide a rigorous proof; 8.3.7d – picture example is sufficient; 8.4.1 – just give the result, no proof required;

Week 8 - Feb. 26 - last day to drop with no grade reported

Monday 2/24
9.1 – Limits of Sequences
Read Before Class: Section 9.1; focus on the proofs of Th. 9.2 and 9.5.

Wednesday 2/26 – Last day to drop with no grade reported (8:00 pm)
9.1 – continued

Friday 2/28
9.2 – Heine-Borel Theorem
Read Before Class: The Borel Covering Lemma (Lemma 9.9) and its proof (p. 308).
Lecture Notes
NO Homework Due
NO Quiz

Spring Break

Week 9

Monday 3/9
9.2 – continued.
Read Before Class: Read p. 309-311 and focus the Heine-Borel Theorem. Try solving problem 9.2.6.

Wednesday 3/11
Class is postponed. We will have class on Friday on Zoom:
https://msu.zoom.us/j/4813665416

Friday 3/13
9.2 – completed. Video,    Notes for the video,    In-class-notes
No Homework Due
Practice quiz on D2L – please take it by 3/18@4pm
Please take this survey to help me structure our class in a way that helps you in the best possible way.

Week 10

Monday 3/16
9.3 – Limits of Functions
Watch Before Class:
Video (part 1), Video (part 2), Notes for videos 1 and 2,
In-class-notes

Wednesday 3/18
9.3 – continued
Video, Notes for video, In-class-notes

Friday 3/20
9.4 – Continuous Functions
Read Before Class: Read p.321-323 and focus on the proofs of Th. 9.25 and Th. 9.26
Video, Notes for video
Practice quiz (if class was not online, your quiz would have been similar to this…)
Homework Due: HW8: 9.1.2 (you can use your Calc 1, Calc 2 knowledge), 9.1.3a, 9.1.4, 9.1.5b,d, 9.1.7-bonus, 9.2.1, 9.2.4, 9.2.7. Think about 9.2.6, but do not turn in.
Quiz 6 on D2L – covers sect. 9.1 and 9.2. Be sure to know the definitions and statements of main theorems in these sections. Quiz may include problems similar to 9.1.5b.

Week 11

Monday 3/23
9.4 – continued
Read Before Class: Focus on the proofs of Th. 9.26, 9.29, and Th. 9.30
Video (part 1), Video (part 2). All notes can be found in the Shared OneNote Notebook

Wednesday 3/25
9.4 – continued
Before Class: try to solve problems 9.4.4, 9.4.6, 9.4.9. We’ll also discuss them in class.
Blank lecture notes ; Lecture video

Friday 3/27
11.1 – Partial Derivatives and Partial Integrals
Read Before Class: the first 3 pages of Section 11.1 (383-385)
Practice quiz (if class was not online, your quiz would have been similar to this…)
Blank lecture notes ; Lecture video
Homework Due: HW9: 9.3.1c, d, 9.3.2a,b, 9.3.3a,b, 9.3.5, 9.3.8c (the dot product of limits), 9.4.1,  9.4.6 (you can use the fact that lim as t->{0+} e^(-1/t)=0), 9.4.9
Quiz 7 on D2L – covers sect. 9.3 and 9.4.

Week 12

Monday 3/30
11.1 – continued
Read Before Class: Focus on the proofs of Th. 11.2 and 11.4, and Example 11.3.
Blank lecture notes; Lecture video

Wednesday 4/1
11.1 – continued
Blank lecture notes; Lecture video

Friday 4/3
11.2 – The Definition of Differentiability
Read Before Class: the first 3 pages of Section 11.2 (394-396); focus on the proof of Th. 11.13.
Blank lecture notes; Lecture video
Practice quiz
Homework Due: HW10: 11.1.2b, 11.1.3, 11.1.4, 11.1.5
Hint for 11.1.4 – use the fact that g is integrable and therefore bounded and modify the proof of Th. 11.4 to fit the setting of the problem.
In terms of difficulty, I would rank 11.1.5 as easy; 11.1.2 and 11.1.3 – moderate, 11.1.4 – difficult.
Quiz 8
on D2L – covers sect. 11.1.

Week 13 - Exam 2

Monday 4/6
11.2 – continued
Blank lecture notes; Lecture video

Wednesday 4/8
Review for Exam 2;  Lecture video
Old exam for practice

Friday 4/10 – Exam 2 – covers sections 9.1, 9.2, 9.3, 9.4, 11.1.
NO Homework Due

Week 14

Monday 4/13
11.3 – Derivatives, Differentials, and Tangent Planes
Read Before Class: the first 3 pages of Section 11.3 (403-405); focus on the proof of Th. 11.20.
Blank lecture notes; Lecture video

Wednesday 4/15
11.3 – continued
Blank lecture notes; Lecture video

Friday 4/17
11.4 – The Chain Rule
Blank lecture notes; Lecture video
Quiz 9
on D2L – covers sect. 11.2.
Homework Due: HW11

Week 15

Monday 4/20
11.4 – continued
Blank lecture notes; Lecture video

Wednesday 4/22
Review for Chapters 5 and 8. Lecture video
Old exam for practice – for today’s lecture, focus on problems 1, 4a-c, 5, 6, 7

Friday 4/24
Review for Chapter 11.
Homework (not to turn in): 11.3.1b,d, 11.3.2a,b, 11.3.3, 11.4.6, 11.4.7, 11.4.8, 11.4.9

Final Exam

Final Exam: Monday, Apr 27 2020 – ONLINE