# Math 402/502: Advanced Calculus II - Spring 2013

## Homework Problems

### MATH 402/502 - HOMEWORK PROBLEMS - Spring 2013

Projects: We will have a 3 day "mini conference" where you will present your projects. Each project will have a 30 minute presentation time. We will use the two lectures on review week and the two hour slot scheduled for the final on Wednesday May 8th.

• Project 1 [Craig (PI), Tim and Andrew (CoPIs)]: Characterization of Riemann Integrable functions (bounded and continuous except on a "set of measure zero"). Part of the project entails understanding what is a set of measure zero.
• Project 2 [Audrey, Cameron and Nuriye (PIs), John (CoPI)] : Cantor Sets and Cantor Function (the Devil's Staircase). Could take a detour on fractals. Need to know what is a set of measure zero (the Cantor set is uncountable with measure zero).
• Project 3 [Patrick (PI), Sulgi (CoPI)]: Construct bumps. These are infinitely differentiable functions defined on R but supported on [a,b] (that is they are zero outside the interval. Once building bumps, build them in R^n supported on a cube or a ball.
• Project 4 [Tairen and Patricio (PIs)]: L^p spaces of sequences and maybe of functions too (show they are complete, understand basic inequalities, etc).
• Project 5 [Karen and Doug (PIs)]: Show that complete and totally bounded implies compact (get started with Exercise 12.5.10).
• Project 6 [Tim and Andrew (PIs), Craig (coPI)]: Axiom of Choice, start reading Chapter 8 and hunt all references in the text to Axiom of Choice and explain why it is necessary to use it.
• Project 7 [Sulgi (PI) and Patrick (coPI)]: Show Weierstrass Approximation Theorem that says that continuous functions on a closed and bounded interval can be approximated uniformly with polynomials (Section 14.8).
• Project 8 [John (PI), and Audrey, Cameron and Nuriye (coPIs)]: About Power Series (Chapter 15): Abel's Theorem (Section 15.3), exponential, log and trigonometric functions (Sections 15.5).

Homework 7 (due on Wednesday April 17):

• Section 17.4: 17.4.1 (if T is linear transformation then T'=T), 17.4.5 (an example using chain rule, be careful with the domain, you want to avoid a zero in the denominator).
• Section 17.5: 17.5.1 (canonical example where function is continuously differentiable, double derivatives exist, but mixed ones are not equal).

Homework 6 (aim for Monday April 8th, but Wed April 10th works too):

• Section 17.2: exercise 17.2.2 (uniqueness of derivative at x_o).
• Section 17.3: exercises 17.3.3 (compares partial derivatives to directional derivatives), 17.3.4 (example of a function that has partial derivatives but is not differentiable at a point).

Midterm on Wed March 27 You turned in what you accomplished in the exam in class, and took the exam home over the weekend and turned in on Monday April 1st, everything you felt you had to redo or not done in class. Here are the review problems

Homework 5 (due Wed 3/6/2013): Exercises from Tao's Book II(hardcover edition) Chapter 13 and 14:

• Section 13.3: exercise 13.3.6 (use the results of 13.3.4 and 13.3.5 as needed)
• Section 13.4: exercise 13.4.1-13.4.2
• Section 14.2: exercise 14.2.2(b)(c)
• Section 14.3: exercises 14.3.4-14.3.5

Homework 4 (due Wed 2/27/2013): Exercises from Tao's Book II(hardcover edition) mostly Chapter 13:

• Section 12.4: exercise 12.4.7 (complete vs closed)
• Section 13.2: exercise 13.2.4 (coordinate functions are continuous), 13.2.10-13.2.11 (jointly continuous implies continuous on each variable but not viceversa).
• Section 13.3: exercise 13.3.1 (continuity preserves compactness), exercise 13.3.2 (continuous functions reach max and min on compact sets).

Homework 3 (due Wed 2/20/2013): Exercises from Tao's Book II(hardcover edition), Chapter 12 on Metric Spaces:

• Section 12.1: exercise 12.1.3 (examples of psudo-metrics, functions failing exactly one of the metric axioms), exercise 12.1.16 (if two sequences converge in a metric space, the distance between nth terms converges to the distance between limit points).
• Section 12.2: exercise 12.2.3 (properties of open sets).
• Section 12.4: exercise 12.4.6 (Cauchy sequences have at most one limit point).

Homework 2 (due Wed 2/5/2013): Exercises from Tao's Book I (hardcover edition), Chapter 11 on Riemann Integration:

• Section 11.6: exercise 11.6.3 (prove integral test)
• Section 11.9: exercise 11.9.2 (two antiderivatives of the same function are equal up to a constant).

Homework 1 (due Wed 1/23/2013): Exercises from Tao's Book I (hardcover edition), Chapter 11 on Riemann Integration:

• Section 11.1: exercise 11.1.3 (about partitions)
• Section 11.2: exercise 11.2.4(g)(h) (piecewise continuous functions are preserved by extensions and restrictions)
• Section 11.3: exercise 11.3.3 only the upper sum (upper Riemann integral coincides witht the Upper Riemann sum)
• Section 11.4: exercise 11.4.1(a)(b)(e) (linear and monotone properties of the Riemann integral), 11.4.2. (if the Riemann integral of a positive continuous function is zero, then the function must be zero)