Showing posts tagged M32BH

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.…

Keep reading

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.…

Keep reading

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.…

Keep reading

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…

Keep reading

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 \(f(x)\) 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…

Keep reading