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MTH 421-002: Analysis II

Lecture time: MWF 1:50 pm – 2:40 pm

Lecture location: A124 Wells Hall

Instructor:Tsvetanka Sendova
Office:C137 Wells Hall
Email:tsendova@msu.edu
Phone:517-884-1453
Office Hours:Mondays and Wednesdays: 3:00 pm – 4:00 pm
Thursdays: 11:30 am – 12:30 pm
and by appointment
Piazza Forum:https://piazza.com/msu/spring2020/mth421002/home
Syllabus:MTH 421-002 syllabus in pdf format
Book:Textbook: Introduction to Analysis (4th Edition), William Wade, Prentice Hall, ISBN: 9780132296380.
Sections 5.1 and 5.2 are available here.
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
Read Before Class: Read Th. 5.26, 5.27, and 5.28 their proofs. 

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
We will conclude the discussion of Sect. 5.3 and start with
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
  Read Before Class: Sect. 8.2
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: 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
NO Homework Due 
  NO Quiz 

Spring Break

Enjoy your break! 

Week 9

Monday 3/9

Wednesday 3/11

Friday 3/13

Homework Due: 

Week 10

Monday 3/16
Wednesday 3/18
Friday 3/20
Homework Due: 

Week 11

Monday 3/23
Wednesday 3/25
Friday 3/27
Homework Due: 

Week 12

Monday 3/30

Wednesday 4/1

Friday 4/3
Homework Due: 

Week 13 - Exam 2

Monday 4/6

Wednesday 4/8

Friday 4/10 – Exam 2
Homework Due: 

Week 14

Monday 4/13
Wednesday 4/15
Friday 4/17
Homework Due: 

Week 15

Monday 4/20
Wednesday 4/22
Friday 4/24
Homework Due: 

Final Exam

Final Exam: Monday, Apr 27 2020 3:00pm – 5:00pm in A124 Wells Hall