# Math 402/502: Advanced Calculus II - Spring 2015

## Homework Problems

### MATH 402/502 - HOMEWORK PROBLEMS - Spring 2015

Final Project Reports

• Michael Apers, Trevor Gianini and Angel Poling
Measure Zero in R, the Cantor Set, and the Cantor function, report pdf.

• Abraham Puthuvana Vinod
Lebesgue's criterion for Riemann integrability, report pdf.

• Martin Nemer
Picard-Lindelof Theorem, report pdf .

• Charlie Clauss and Joel Upston
Approximation of compactly supported continuous functions by polynomials,report pdf .

• Jocelyn Gonzales and Steven Kao
The Henstock-Kurzweil Integral, report pdf .

• Michael Brown, Alex Cordova,and Alyssa Sanchez
Metric spaces properties, report pdf .

A non-measurable set in R, report pdf .

• Rebecca Farish and Trace Norris
How to complete a metric space, report pdf .

Homework 9:(due Thursday 4/9):

• Exercise 6.2.2 (uniqueness of derivatives)
• Exercise 6.3.3 (example of a function not differentiable at (0,0) despite being differentiable in every direction).
• Exercise 6.4.1 (linear transformation are continuously differentiable)
• Exercise 6.4.5 (an example)
Reading Assignment: Chapter 6, Sections 6.1-6.4

Homework 8 (due Tuesday 3/31/15):

• Exercise 4.1.2 (examples of power series with same radius of convergence and different behaviours at the endpoints).
• Exercise 4.2.6 (polynomials are real analytic)
• Exercise 4.3.1 (summation by parts formula)
• Exercise 4.5.6 (show that the natural logarithm is real analytic for x>0)

Homework 7 (due on Thursday March 5,2015) I highly recommend you attempt to turn in the homework to me on Thursday March the 5th or the latest on Friday March the 6th to Bishnu, I will be flying out of town on Friday so you want to make sure Bishnu gets the homework so he can grade it over the Spring break and you can get it back on Tuesday before the exam on Thursday March 19th).

• Exercise 3.2.2(c) (exploring convergence and uniform convergence of a geometric power series)
• Exercises 3.3.4-3.3.5 ( convergence of f_n(x_n) given uniform or pointwise convergence of f_n to f and convergence of x_n to x).
• Exercises 3.6.1 and 3.7.3 (interplay between uniform convergence of series, integration and differentiation).
• Bonus: Exercise 3.2.1 (connection between uniform convergence and uniform continuity)

Homework 6 (due on Thursday Feb 26,2015)

• Exercise 2.2.4 (coordinate fcns are continuous),
• Exercises 2.2.10 or 2.2.11 (jointly continuous in (x,y) implies continuous in each variable but not viceversa)
• Exercise 2.3.2 (continuous real-valued functions on a compact set reach max -no need to show they reach min also)
• Exercise 2.3.5 (if f and g are uniformly continuous then their direct sum is uniformly continuous)
Reading Assignment: Chapter 2 Section 2.1, 2.2, 2.3 and 2.4

Homework 5: (due Thursday Feb 19, 2015)

• Exercise 1.4.1 (all subsequences of a convergent sequence converge and to the same limit).
• Exercise 1.4.3 (convergent sequences are Cauchy sequences)
• Exercise 1.5.12 (complete and compact in discrete metric)
• Exercise 2.1.3 (composition preserves continuity)
• (Bonus) Exercise 1.3.1
Reading Assignment: Book II - Sections 1.5 and 2.1

Homework 4: (due Thursday Jan 12, 2015) Problems from Book II 3rd edition Chapter 1 (in earlier editions this is Chapter 12).

• Exercise 1.1.3 (b)(d) (examples of "sets and pseudometrics" that fail to be a metric)
• Exercise 1.1.5 (Cauchy Schwarz in R^n and how to use it to get triangle inequality)
• Exercise 1.1.16 (if two sequence are convergent in a metric space then the distance between the terms in the sequences converges to the distance between their limits).
• Bonus Exercise 1.1.5: in the space of sequences the ell-1 and ell-infinity metrics are NOT equivalent.
Reading Assignment:: Book II - Sections 1.1, 1.2, 1.3 and 1.4.

Homework 3 (Due Thursday Jan 5, 2015)

• Exercise 11.9.2 (two antiderivatives of the same function must be equal up to a constant)
• Exercise 11.9.3 (f monotone, F=int f is differentiable at x iff f is continuous at x)

Homework 2 (Due Tuesday Jan 29, 2015)

• Section 11.3: Exercise 11.3.5 (show only one of the two statements, upper Riemann integral equal infimum of upper Riemann sums)
• Section 11.4: Exercise 11.4.1 (a)(e) (integration laws)
• Exercise 11.4.2 (if f is continuous positive and Riemann integrable, and integral is zero then f must be zero).
• Section 11.6: Exercise 11.6.3 (integral test for series)
Reading Assignment: Sections 11.4, 11.5, 11.6, 11.7

Homework 1 (due on Tue 1/20/2015): Exercises from Tao's Book I (hardcover edition), Chapter 11 on Riemann Integration:

• Exercise 11.1.2 (intersection of two bounded intervals is a bounded interval),
• Exercise 11.1.4 (P1#P2 is a partition and a common refinement for both P1 and P2)
• Exercise 11.2.2 (only show if f,g are piecewise constant (p.c.) on I then max(f,g) is p.c. on I),
• Exercise 11.2.4(g)(h) (laws of p.c. integration).
Reading Assignment Sections 11.1, 11.2 and 11.3