Home Page	MEI Courses	Registration	Workshops & Consulting	Courseware Archive & Consulting	Guestbook

Experimentation and Play

Many, many problems in Calculus&Mathematica leave room for trying different approaches. They frequently require judgment decisions, and don't have fixed answers. As students experiment with them, and as they talk together and with us about what they are doing, they learn all sorts of interesting things.

Here is a function, f(x) = (x² + x) e^-x2, and its plot on the interval [-2,2].

Try to get a decent approximation to this function on [-2,2] by a polynomial.
Try to get a decent approximation to this function on [-2,2] by Sines and Cosines.

Students are free to try anything they want for this. Most will just write down polynomials and try to adjust the coefficients to get good fits. Some will at least notice the number of ups and downs to get some guess about the appropriate degree for the polynomial. By this time in the course, some will write down a generic quadratic or quintic, several values of f(x_k), set up a system of equations and solve. Whatever they do, they are in charge.