University of Washington Applied Mathematics Department
In mathematical modeling, we usually don't have a convenient formula for the functions that we deal with. Instead, we just keep track of the value of the function at a finite number of closely-spaced points. However, we're usually interested in the derivatives of these functions as well, and we have to resort to an approximation. I'll start off exploring the definition of a derivative (the one everyone knows from calculus) as an approximation technique. Then I'll show some simple modifications to this that give us much better results. Afterwards, I'll show how this relates to the work I do modeling brain tumors in Kristin Swanson's lab. I just finished writing code to model tumors on simulated brain slices, and the pictures look really cool.