Discussion 16: Analyzing Functions Using Maple

In this discussion students are asked to use the plotting capabilities of Maple to further explore functions of two variables. The discussion assumes that the students have completed Discussion 15 and are familiar with the commands plot3d and contourplot in Maple.

In the first exercise, the students are asked to explore the effect of shifting the independent and dependent variables of a function $f: {\bf R} \rightarrow {\bf R}^2$. The second exercise leads the students to systematically explore the graphs of a family of quadratic functions of the form f(x,y) = ax²+by²+cx+dy+e in order to develop a classification of these functions. In the third exercise they are asked to consider the graphical effect of adding higher degree terms to polynomials of the form ax²+by². The classification that students develop in this discussion and the analysis from Exercise 3 will be used in Chapter 4 when quadratic Taylor polynomials and the second derivative test for extrema of functions of two variables are introduced.

This discussion requires computer exploration and experimentation and can either be introduced in a computer laboratory and completed outside of class or assigned as an extended assignment to be done outside of class.