A Compendium of My Problem Sets

DISCLAIMER: Many of the problems here are my own. However, many of these problems are based on discussions I've had with others, cool things I've read, et cetera. So the organization and structure of each problem set is my own, but I do not make any general claim of responsibility for the problems themselves.

Sequences, Series, and Multivariable Calculus - Emerging Scholars Program Workshop

In Fall 2007, I had the privilege of teaching the Emerging Scholars Program workshop in M408D, the second-semester calculus course for math, science, and engineering majors at UT. One of my responsibilities was to write a problem set for my students to work on each day in class. Over the course of the semester, I developed a collection of calculus problems, some of which are quite challenging. My goal in writing these problems was to help students practice important skills - e.g., determining whether a given series is convergent or divergent - and to give students a sense of the various contexts in which these skills are useful.

Here are the problem sets:

Problem Set 0 - Not technically a problem set; just gives some guidelines for the work we did in class.
Problem Set 1 - Some problems reviewing skills from first-semester calculus.
Problem Set 2 - L'Hopital's Rule and applications.
Problem Set 3 - Improper integrals.
Problem Set 4 - Limits of sequences.
Problem Set 5 - Basic ideas about series convergence, geometric and telescoping series.
Problem Set 6 - Basic Comparison and Limit Comparison Tests for series.
Problem Set 7 - Ratio and Root Tests, Alternating Series Test.
Problem Set 8 - Assorted series convergence problems, and some further explorations on sequences and series.
Problem Set 9 - Representing functions as power series.
Problem Set 10 - Error estimation for power series, introduction to vector geometry.
Problem Set 11 - Applications of the dot product.
Problem Set 12 - Applications of the cross product.
Problem Set 13 - Assorted three-dimensional (and four-dimensional!) geometry problems.
Problem Set 14 - Equations of lines and planes in coordinate space.
Problem Set 15 - Parametric equations.
Problem Set 16 - Applications of parametric equations in physics.
Problem Set 17 - Partial derivatives and linear approximation.
Problem Set 18 - Implicit differentiation and the Chain Rule in several variables.
Problem Set 19 - Gradients, directional derivatives, and local extrema.
Problem Set 20 - Optimization and Lagrange's method.
Problem Set 21 - Double integrals and applications.
Problem Set 22 - Switching the order of integration; integration in polar coordinates.
Problem Set 23 - Triple integrals and moments of inertia.
Problem Set 24 - The All-Ingenuity Problem Set.

Here are some practice exams I gave my class. The mathematician names on the problems indicate which study group submitted each problem. As part of the review process, I had study groups submit problems, which I then tweaked and modified into practice exam problems.

Practice Exam 1 - Sequences, series, Taylor and power series.
Practice Exam 2 - Error estimation, geometry of coordinate space, parametric equations.
Practice Exam 3 - Practice final exam.

Structure of Modern Geometry

I am currently the teaching assistant for M333L, Structure of Modern Geometry, taught by Michael Starbird. Together we are writing a collection of problem sets that help students discover important ideas and problem solving techniques in geometry.

Geometry Problem Sets (In Progress)

Euclidean (Plane) Geometry
Euclidean Constructions

Texas Mathworks Summer Camps

In Summer 2009, I taught two two-week courses during the Texas Mathworks Junior Summer Math Camp. In the mornings, I taught "Level 4," a class in which students between the ages of 10 and 13 learn about discrete mathematics. In the evenings, I taught "Level T," a two-hour workshop in which middle school teachers learned about problem solving techniques that are useful in math contests such as Mathcounts.

Level 4 Problem Sets

Problem Set 1 - Logic puzzles
Problem Set 2 - Applications of modular arithmetic
Problem Set 3 - Arithmetic in bases other than 10
Problem Set 4 - Basic ideas in probability
Problem Set 5 - Counting problems

Level T Problem Sets

Problem Set 1 - Combinatorics
Problem Set 2 - Ratios and algebraic reasoning
Problem Set 3 - Geometry
Problem Set 4 - Number theory

Mock Mathcounts Team Round