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Setting up and Solving Equations

Well honed paper and pencil skills have been the mainstay of math courses for a long time. New tools allow us to shift the balance a bit, because new tools allow us to perform more complicated calculations with much less effort, and much more accuracy. Examples of this run throughout the Calculus&Mathematica courses in the form of finding what you are after by solving the right equations.

A student is asked to fit a polynomial through a collection of points in the plane as part of a problem. Typically, in this class, the student writes a generic polynomial:

poly[x] = a + b x + c x² +...

uses that to set up a system of equations, solves, and claims victory. (This is a deeper thinking skill than most of us think.)

The idea is used throughout the courses. The advantage is that learners focus on the goal rather than a specialized technique.

Think about integration by parts, for example. There's really only a small number of forms that are presented and practiced in most calculus courses, and students tend to make mistakes, particularly with those darned minus signs, don't they? Here's another way.

Integrate f[u] = u sin(3 u) from 0 to t. Using the integration by parts formula will work fine, of course. However, if F[t] is that integral, you know that F'[t] = t sin(3 t). Make an educated guess, then.

F[t] = a t sin(3t) + b t cos(3 t) + c sin(3 t) + d cos(3 t),

differentiate, match coefficients and solve.