In Fall 2010 and Winter 2010 I taught pre-Calculus at the University of Michigan. In Winter 2011 I taught Calculus I and I taught differential equations labs in Fall 2012 and Fall 2013. See the links below to find out more information and some tips on these courses.
This class teaches functions in a conceptual way. It invloves many modeling-type problems which usually include verbal discriptions. "How to Excel in Math at Michigan?" is a presentation that I did in the Preparation Initiative Orientation in Fall 2014. You may find it helpful.
The tips in "How to Excel in Math at Michigan?" could be helpful here as well.3. Differential Equations (Math 216)
This class is an introduction to differential equations. It starts with first order differential equations (DE) and discusses 1st order linear DEs and seprable DEs. If the 1st order equation is linear, integrating factors are used to solve the DE, otherwise it should be checked if the DE is separable. The class then goes over the 1D phase portriat and the concept of stability.
After 1st order DEs it discusses the linear 2nd Order DEs with constant coeffients. The method of charectristic equation is introduced for solving this type of DEs when they are homogeneous. Some numerical methods such as Euler's and Rung-Kutta methods are introduced. Also, applications of the 2nd order DEs, such as mechanical vibrations, is discussed. Then the class moves on to non-homogeneous DEs. The solution to linear non-homogeneous DEs is the sumation of the complementry solution (i.e. solution of the homogeneous part) and a particular solution (i.e. a solution that satisfies the non-homogenous DE). The complementry solution is found using the characteristic equation, and to find a particular solution two methods are discussed: method of undetermined coefficients and variation of parameters.
Next, the system of linear DEs is introduced. The concept of eigen-value and eigen-vector is discussed and it is shown how these concepts can be used to study the stabiity of the critical point of a linear system.
Even though the linear DEs are discussed in detail, most of the equations that we encounter in real life are non-linear! Unfortunately, unlike the linear equations there is no systematic way of finding the solutions of non-linear DEs. For this reason, we study the qualitative behavior of non-linear DEs. This is done through introducing the concepts of phase portrait and linearization. The main idea is that under some conditions the behavior of the non-linear system around its critical point is similar to the behavior of its equivalent linear system. Therefore, instead of studying a given non-linear equation we study a simple equivalent linear system. This analysis won't give us all the information about the non-linear system, but under some conditions it gives us a qualitative measure of how the non-linear system behaves around the critical point. For example, stability of a critical point of the non-linear system could be checked by linearization if some conditions are satisfied.
I taught the afternoon sessions of this class in the Michigan Math and Science Scholars (MMSS) program. MMSS, is a program designed to introduce high school students to current developments and research in the sciences.