# 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. |

**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**

**Monday 1/13**

**Read Before Class:**Remark 5.14 and proof, Th. 5.15 and proof.

**Wednesday 1/15**

Riemann Sums

**5.2****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)

Quiz 1

**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

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)

**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).

**– covers sect. 5.3**

**Quiz 3**

**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**

**– covers sect. 8.1**

**Quiz 4****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

**– covers sect. 8.2 and 8.3**

**Quiz 5****Monday 2/17** 8.4 – continued

**Review for Exam 1**

Wednesday 2/19

Wednesday 2/19

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;

**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

NO

**Quiz****Enjoy your break! **

**Monday 3/9**

**Wednesday 3/11**

**Friday 3/13**

**Homework Due: **

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

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

**Monday 3/30**

**Wednesday 4/1**

**Friday 4/3****Homework Due: **

**Monday 4/6**

**Wednesday 4/8**

**Friday 4/10 – Exam 2****Homework Due: **

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

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

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