Let [x(t),y(t)] , t Î [a,b] be a parametric
curve. The rate of change of the position [x(t),y(t)] is given by [x^{¢}(t),
y^{¢}(t)] and is called the velocity
at [x(t), y(t)]. The computer may help us understand the velocity by drawing
the line segments joining [x(t),y(t)] with [x(t)+x^{¢}(t),
y(t)+y^{¢}(t)] as t ranges over
equally spaced points of the interval [a,b]. For example, the velocity
along the circle [cos t, sin t], t Î [0,2p]
is represented as:

The procedure follows these steps in Derive:

x: = cos(t)

y: = sin(t)

m: = [[x,y],[x+dif(x,t),y+dif(y,t)]]

VECTOR(m,t,0,2pi,pi/50)

Then apply approX command to the last expression. Before plotting, make
sure that Plot-Option-State-Mode is selected to Connected. In order
to have all line segments drawn with the same color, be sure that Plot-Color-Auto
is set to NO.

The velocity along the epicycloid [2 cos t-cos 2t, 2 sin t-sin 2t] appears
as:

The velocity along the epicycloid [3 cos t-cos 3t, 3 sin t-sin 3t] appears
as:

The velocity along the cycloid [t-sint,1-cos t] appears as: