Discussion 17: Rates of Change

The goal of this discussion is for students to extend their understanding of the rate of change of a function of one variable to functions of two variables and to lead them towards the concept of a directional derivative. This discussion is designed to be used before directional derivatives are formally introduced. The discussion should be preceded by a brief introduction to the question of finding the rate of change of a function of two variables in the context of the exercises.

In the first exercise, the students are asked to review what they know about the average and instantaneous rates of change of a function of one variable. In the second exercise, they are asked to analyze a topographical map in order to determine the average and instantaneous rates of change of the altitude function on a specific path between two points on the map. In particular, they must obtain numerical values for the average rate of change directly from the map. Finally, in the third exercise, they are asked to analyze the instantaneous rate of change of the altitude function at a particular point for different paths through the point. Students will probably need to complete the write-up of this discussion outside of class.