# Tag Archives: M32BH

teaching

## 32BH Midterm 2 Review

I’ve written up some review questions for the second midterm for Math 32BH, available here. A general hint for the computational problems is that you shouldn’t have to compute any terribly ugly integrals. I just finished writing up solutions for the review questions. As usual, let me know in the comments below if anything is unclear or incorrect.

teaching

## Math 32B/H Midterm 2 Review

I just finished writing up some review materials for the second midterm for Math 32B/H. There are two documents: the problems (here) and solutions (here). If you find any typos or points that need clarification, feel free to let me know in the comments below.

teaching

## Math 32B/H Midterm 1 Review Materials

I have prepared some review materials for the first midterm for Math 32B and Math 32B H. I wrote up some sample problems (here) and solutions (here). As usual, let me know if you find any errors or if an explanation is unclear.

## Limit of Average Values of a Continuous Function

A recent homework assignment asked to prove the following fact: If $$f : B \to \mathbf{R}$$ is continuous where $$B \subset \mathbf{R}^n$$ is open, and $$x_0 \in B$$ then
$\lim_{r \to 0} \frac{1}{V(B_r)} \int_{B_r} f(x)\, dV = f(x_0)$
where $$B_r$$ is the ball of radius $$r$$ centered at $$x_0$$. The limit-and can be interpreted as the average value of $$f$$ on $$B_r$$. Informally, the claim says that the average value of a continuous function tends to the value of the function as we consider smaller and smaller balls containing $$x_0$$.

There were some problems with the solutions I saw on the homework, so I wrote up a careful solution available here. In general, when trying to prove a fact such as this, a good starting point is to write the definitions of terms in the hypotheses and conclusion. In this case, you need to start with the definitions for continuity, limit, and volume in order to have a fighting chance of proving the statement.

## Integrability and Continuity

I just posted a short essay that defines a function with some curious properties. The function is defined by

$f(x) = \begin{cases}\frac{1}{b} & x = \frac{a}{b}\in\mathbf{Q}\text{ in lowest terms}\\ \\0 & \text{if } x \in \mathbf{R}\setminus\mathbf{Q}.\end{cases}$

In the essay I prove that

1. $$f(x)$$ is continuous on $$\mathbf{R}\setminus\mathbf{Q}$$
2. $$f(x)$$ is discontinuous on $$\mathbf{Q}$$
3. $$f(x)$$ is integrable, and in particular $$\int_0^1 f(x)\, dx = 0$$

At first blush, these properties may seem counterintuitive.

The theory of Lebesgue measure completely characterizes Riemann integrable functions: they are functions which are almost everywhere continuous. The above function is almost everywhere continuous because its set of discontinuities ($$\mathbf{Q}$$) is countably infinite, and therefore has Lebesgue measure $$0$$.

teaching

## Welcome to Math 32BH, Winter 2013

This quarter I will be TAing for Math 32BH which covers multivariable integral calculus. Since this is the honors section, the course emphasizes a rigorous development of the theory, as well as computation. Here is a link to the course website. This post and all others relating to my section of Math 32BH will be tagged M32BH.

I am currently planning on holding my office hours on Tuesdays after class (from 10 – 11 AM) in my office, MS 3915B. I am also available by appointment.