# 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.…

# 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.…

# 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.…

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