Ma Communicating Mathematics Class 15 Polynomials, pistons and animations Carl Eberhart and Paul Eakin

Today, we continue to investigate polynomials, and learn to use animation in a Maple worksheet. We will also discuss the ‘piston problem’.

plots[animate] can be used to study how the graph of a function f(x) as a parameter in the rule for f changes. plots[animate3d] is like plots[animate], except it works with functions of two variable f(x,y). Much more can be done with animations. If you have any sequence of named plots, then they can be displayed with the option insequence=true to produce a movie.

Sample animations plots[animate](x^2+a*x+1,x=-5..5,a=0..4); As you play this movie, you see the parabola slide down a parabolic? hill. plots[animate3d](x^3+m*x+n,x=- 3..2,n=0..7,m=0..4); This movie shows the effect of changing the two parameters in the cubic.

A piston problem Suppose a wheel with center on the x-axis revolving with constant angular speed is tied at a point on its circumference to a rod which is tied at the other end to a piston constrained to move back and forth along the x-axis. Study the velocity and acceleration of the piston as a function of the length of the rod with an eye to finding the ‘best’ length for the rod.