# Math 402/502: Advanced Calculus II - Spring 2016

## Homework Problems

### MATH 402/502 - HOMEWORK PROBLEMS - Spring 2016

Final Projects

• Tuesday May 3, 2016
• 8:00-8:30
Sets of measure zero. The 1/3 Cantor set. The Cantor function.
Michael Sanchez, Jason Baker, Kyle Henke
• 8:40-9:10
Characterization of Riemann Integrable functions: continuous almost everywhere (except on a set of measure zero).
Duc Nguyen
• Thursday May 5, 2016
• 8:00-8:30
Completeness of metric spaces.
Justin Campbell, Matt Robinson
• 8:40-9:10
Weierstrass Approximation Theorem. (Section 14.8).
Peterson Moyo and Cindi Goodman
• Thursday May 12, 2016
• 7:40-8:10
Ultrametric.
Hanna Butler, Zach Stevens, Marshall Branderburg
• 8:20-8:50
Fourier series.
Sarka Blahnik
• 9:00-9:30
Tempered distributions.
Lionel Fiske, Cairn Overturf

Homework 8 (due on Thursday April 28, 2016): Do at least 5 problems (Exercises 6.1.4 and 6.4.1 count like just one problem).

• Exercises 6.1.4 and 6.4.1 (linear transformations are continuous continuous, and continuously differentiable).
• Exercise 6.2.2 (uniqueness of derivatives).
• Exercise 6.3.1 (differentiable implies directional derivatives).
• Exercise 6.3.3 (directional derivatives do not imply differentiable).
• Exercise 6.4.3 (chain rule).
• Exercise 6.5.1 (non example - Clairaut's Theorem).
• Exercise 6.6.7 (Contraction Mapping theorem).
• Exercise 6.6.8 (Nearby contractions have nearby fixed points).
• Exercise 6.7.3 (application of Inverse Function Theorem).

Reading Assignment: Chapter 6.

Homework 7 (due on Thursday April 14, 2016)

• Exercise 4.2.5 and 4.2.6 (polynomials are real analytic. (In 2nd edition there is a typo in the formula in 4.2.5).
• Exercise 4.3.1 (summation by parts formula).
• Exercise 4.5.1 (basic properties of exponential)
• Exercise 4.5.6 (show that the natural logarithm is real analytic for x>0).

Reading Assignment: Chapter 4.

Homework 6 (due on Thursday April 1st, 2016)

• 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).
• Exercise 3.4.1 and 3.4.3 (the space of continuous functions is complete).
• Exercises 3.6.1 and 3.7.3 (interplay between uniform convergence of series, integration and differentiation).

Reading Assignment: Chapter 3.

Homework 5 (due Tuesday March 8, 2016): Do one exercise per section (4 total).

• Section 2.1: Exercises 2.1.5 (inclusion map is continuous).
• Section 2.1: Exercise 2.1.6 (restriction of a continuous map is continuous).
• Section 2.2: Exercise 2.2.2 (prove that the maximum function is continuous only).
• Section 2.2: Exercise 2.2.5 (polynomials of two variables are continuous).
• Section 2.2: Exercise 2.2.11 (continuity for each variable does not ensure continuity on R^2).
• Section 2.3: Exercise 2.3.4 (composition preserves uniform continuity).
• Section 2.4: Exercise 2.4.4 (continuity preserves connectedness)
• Section 2.4: Exercise 2.4.5 (intermediate value theorem)

Reading assignment: Section 2.2, 2.3, and 2.4.

Homework 4 (due Tuesday March 1st, 2016): Problems from Book II 3rd edition Chapter 1 (in earlier editions this is Chapter 12). Do at least 4 of the given exercises.

• 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 1.1.12 (show only that (d) is equivalent to one of the others, because we already showed in class that (a), (b) and (c) are equivalent)
• 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).
• Use the open cover property of a compact set to show that the set must be closed and bounded.

Reading Assignment: Book II - Chapter 1.

Homework 3 (Due Tuesday Feb 16, 2016)

1. Section 11.4: Exercise 11.4.2 (if f is continuous, positive, and Riemann integrable, and it integral is zero then f must be zero).
2. Section 11.6: Exercise 11.6.3 (integral test for series).
3. Section 11.9: Exercise 11.9.2 (two antiderivatives of the same function must be equal up to a constant).
Reading Assignment:Section 11.5, 11.6, 11.7 (an example of a bounded function that is not Riemann integrable), and 11.9 (Fundamental Theorems of calculus).

Homework 2 (Due on Thursday Feb 4th, 2016)

1. Section 11.3: Exercise 11.3.5 (show only that upper Riemann integral is equal to the infimum of the upper Riemann sums).
2. Section 11.4: Exercise 11.4.1 (a) and (d) only (integration laws: linearity and precursor of monotonicity).
Reading Assignment Sections 11.3, 11.4

Homework 1 (due on Tue 1/26/2016): Exercises from Tao's Book I (hardcover edition), Chapter 11 on Riemann Integration:

1. Exercise 11.1.2 (intersection of two bounded intervals is a bounded interval),
2. Exercise 11.1.4 (P1#P2 is a partition and a common refinement for both P1 and P2)
3. 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),
4. Exercise 11.2.4(g)(h) (laws of p.c. integration).
Reading Assignment Sections 11.1, 11.2 and 11.3

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Last updated: January 29, 2016