# Assignments Real Analysis II--- Math 441

#### Instructor: Carol S. Schumacher Spring, 2013

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## Homework

Monday, January 14 Differentiation domains and the definition of the derivative * *
Wednesday, January 16 Local linear approximation, differentiation and continuity, differentiation rules Sections 9.1 and 9.2.

Class: Problems 3, 5, 8ac, 11, 12 in Section 9.2
Notebooks: Problems 4, 6, 9, and 10 in Section 9.2

Friday, January 18

Finish section 9.2.
Discussion: Why the Mean Value Theorem?

Section 9.3. *
*
Monday, January 21 Proving the Mean Value Theorem
Corollaries of the Mean Value Theorem
Section 9.4 Class: Problems 1, 2, 3, 4, 5 in Section 9.4
Notebooks: Problems 6, 8, 9 in Section 9.4
Wednesday, January 23 Monotonicity and the Mean Value Theorem Section 9.5 Class: Problems 1 and 3 in Section 9.5
Notebook problems on Sects. 9.2 due.
Friday, January 25 Darboux's Theorem and the nature of derivative functions * Class: Problems 4 and 5 in Section 9.5
Notebooks: Problems 2 and 6 in Section 9.5
*
Monday, January 28 The Intermediate Value Theorem Section 8.1 Class: Problems 1 and 2 in Section 8.1.
Notebook: Problem 3 in Section 8.1
Wednesday, January 30

Introduction to Taylor Polynomials and Taylor's Theorem

Section 9.7
Friday, February 1 Error in Taylor Polynomial Approximations   Notebooks: Problems 1, 2, and 3 in Section 9.7 (Group assignment)
Notebook problems on Sects. 9.4 and 9.5 due
*
Monday, February 4

Experimenting with iteration and cobweb diagrams.

Section 10.1 through the description of "cobweb diagrams"---pg 193. (We will work Exercise 10.1.3 in class.) Notebooks: Problem 2 in Section 10.1
Group notebook assignment on 9.7 due. And individual write-ups from 8.1.
Wednesday, February 6 Iteration and Fixed Points Rest of Section 10.1 Class: Problems 4, 6, and 9 in section 10.1
Notebooks: Problems 5, 8, and 10 in Section 10.1
Friday, February 8 Finish Section 10.1
Contractions
Section 10.2 through the top of page 200. Class: Problems 1, 3(discussion), and 4 in Section 10.2
*
Monday, February 11 The Contraction Mapping Theorem Rest of Section 10.2 Class: Problem 5 in Section 10.2
Notebooks: Problems 2, 6 and 7 in Section 10.2
Wednesday, February 13 More on attracting Fixed Points Section 10.3 Class: Problem 11 in Section 10.1 and Problems 1 and 2 in Section 10.3
Friday, February 15 Iteration and Newton's Method Sections L.1 and L.2 Class: Problems 1 and 2 in Section L.2.
*
Monday, February 18 Defining the integral Sections 11.1 and 11.2 Class: Problems 2, 3, 4 and 6 in Section 11.2
Wednesday, February 20 * Excursion I---food for thought Class: Problems 7 and 8 in Section 11.2
Notebooks:
Problems 1, 5, and 9 in Section 11.2
Notebook problems on Sects. 10.1 and 10.2 due
Friday, February 22     Class problems from 11.2 Continued.
*
Monday, February 25 Arithmetic, order and the integral Section 11.3
Read Section 11.4 through the bottom of pg. 221 (for understanding.)
Class: Problems 1 and 4 in Section 11.3
Notebooks: Problems 3, 5, and 6 in Section 11.3
Wednesday, February 27

Families of Riemann Sums and refinements

Re-Read Section 11.4 through Lemma 11.4.7 Class: Exercises and Problem 1, 2, 4, and 5 in Section 11.4
Notebooks: Problem 3 in Section 11.4
Friday, March 1

### Spring Break

Monday, March 18 Cauchy Criteria for the Existence of the Integral---What are the issues? (Lecture.) Rest of Section 11.4 (don't worry about the proofs; ignore Lemma 11.4.8---read and understand the statements of 11.4.9 and 11.4.10!)

Notebooks: Problems 6 and 7 in Section 11.4

Wednesday, March 20
Existence of the Integral
Section 11.5 Class: Problems 1 and 2 in Section 11.5.
Notebooks: Problems 5, 7, 8 and 10 in Section 11.5.
Notebook problems on Sections 11.2, 11.3 & 11.4 number 3 due
Friday, March 22 The Fundamental Theorem of Calculus Section 11.6 Class: Problems 2 and 3 in Section 11.6.
Notebooks:
Problem 1 in Section 11.6.
*
Monday, March 25 Subsequences and Convergence Excursion G Class: Problems 1 in Section G.1 and 1, 2, 3, 4 and 7 in Section G.2.
Wednesday, March 27 Subsequences and Convergence, cont.   Notebook problems from Sections 11.4 numbers 6 and 7, 11.5 and 11.6 due
Friday, March 29 Relatives of the geometric series---the root and the ratio test. (A quick and dirty discussion.)
Takehome midterm distributed
Carefully read Excursion H.1- H.3 for big picture review. ASK Questions if you are really rusty. **
*
Monday, April 1

## In-class midterm

Wednesday, April 3 Convergence of Series of Functions Switching the order of Limiting Processes---discussion of a delicate business. Section 12. 3
Read Section 12.4 for big picture.

Friday, April 5 Power series---basic definitions
Discussion of Taylor Series.
Excursion J Takehome midterm due by 4 p.m on Thursday, April 4.
Class: Problems 1,3 in Exc. J.1
Notebooks: Problem 4 in Excursion J.1
*
Monday, April 8 Integration and differentiation of power series
Excursion J.2 (again!) Class: Problems 1, 2, 3 and 4 in Section J.2
Notebooks:
Problem 1 in Excursion J.3.
Wednesday, April 10 Finish Power Series
Friday, April 12 Everywhere continuous, nowhere differentiable. Excursion K Class: Theorem K.2.1 steps 1 and 2.
*
Monday, April 15 Everywhere continuous, nowhere differentiable---cont. Excursion K Class: Theorem K.2.1 steps 3, 4 and 5.
Notebook problems due: Excursion J.1 and J.3
Wednesday, April 17 Spaces of Continuous functions Excursion N.1 and N.2

Class: Lemma N.1.1 and Theorem. N.1.3, Theorem N.2.4
Notebooks: Lemma N.1.4, and Thm. N.1.5

Friday, April 19
No Class; Professor Schumacher out of town.
*
Monday, April 22 Compactness in C(K)

Excursion N.2 (again!)

Class: N.2.4, N.2.5, N.2.6, N.2.7 (Assigned in groups.)
Wednesday, April 24 Compactness in C(K)---continued
Friday, April 26 Discussion of Arzela-Ascoli and characterization of compactness in C(K).
*
Monday, April 29 Differential Equations: uniqueness and existence of solutions Excursion O.1 and O.2

Class: Problem 1 in Section O.2
Notebook problems due: Lemma N.1.4 and Thm. N.1.5, Arzela-Ascoli Theorem.

Wednesday, May 1 Picard Iteration Class: Problems 2 and 3 in Section O.2.
Friday, May 3
Takehome final distributed

### Final Examination

In class portion: Monday, May 6 at 1:30 p.m.
Alternate in-class time: 6:30 p.m on Wednesday, May 8
Takehome final due: at 6:30 p.m. on Wednesday, May 8