Rates of Change Along a Path

A parametric curve comes from two separate functions of time, x(t) and y(t). Neither one alone tells you how steep the path looks on the plane. But their rates compare directly: if y is changing fast while x barely moves, the path climbs steeply; if x is changing fast while y crawls, the path runs nearly flat. The slope between two points on the curve is exactly that comparison, built the same way slope always is: rise over run. Here rise is the change in y over the interval, and run is the change in x over the same interval.

Mechanically, you compute average rate of change of y from t1 to t2, which is (y(t2)-y(t1))/(t2-t1), and average rate of change of x the same way, then divide the first by the second. The t2-t1 pieces sit in both the top and bottom, so really you are dividing total change in y by total change in x, no different in spirit from slope between two points on any curve. As the interval shrinks toward a single instant, this ratio becomes dy/dt divided by dx/dt, the instantaneous slope dy/dx, which is where this topic connects forward to derivatives of parametric curves. For now, stick with average rates over a stated interval of t.

This sets up a contrast worth holding onto as the unit moves toward vectors: speed and steepness are not the same question. Speed asks how fast position changes with time, combining both x and y motion into one magnitude. Steepness asks only about the shape traced on the plane, the ratio of the two rates, with time itself erased from the answer.