Rates of Change in Polar Functions

In polar functions r(theta) gives a distance, not a height, but rate of change still means the same thing it always has: how much the output changes per unit of input. Since r measures distance from the pole, a positive average rate of change over an interval means the curve is, on balance, moving farther from the pole as theta sweeps through that interval. A negative rate means it is pulling in toward the pole. Whether the curve actually spirals outward at a given instant depends on two things together: the sign of r itself and whether r is increasing or decreasing there. If r is positive and increasing, the point is swinging around and away from the origin, tracing an outward spiral.

The formula is the same secant slope you have used for every other function: average rate of change from theta = a to theta = b is (r(b) - r(a))/(b - a). Nothing about polar coordinates changes the arithmetic, only the meaning of the output, since r is a length, not a vertical height. For r = 3cos(theta) from theta = 0 to theta = π/3, you get (3cos(π/3) - 3cos(0))/(π/3 - 0) = (1.5 - 3)/(π/3) = -1.5/(π/3), a negative number showing the curve is pulling toward the pole on average over that sweep.

This connects directly to extrema. Points on a polar curve farthest from the pole occur where r(theta) hits a relative maximum, and points closest to the pole occur at a relative minimum, since r(theta) is literally the distance being tracked. Identifying these extrema is how you locate the outermost loops and innermost pinches of a polar graph without ever plotting a single point.