Rates of Change in Polar Functions

A polar function $r = f(\theta)$ describes a curve by distance and direction. This topic asks how that distance changes as the direction turns. The tools are the same ones you used in Unit 1: average rates of change, secant lines, increasing and decreasing. The new twist is the sign of $r$ .

Average rate of change of $r$

The input is the angle and the output is $r$ , so the average rate of change of $r$ on $[a, b]$ is

$\frac{r(b) - r(a)}{b - a}$

measured in units of $r$ per radian. It is the slope of the secant line through $(a, r(a))$ and $(b, r(b))$ on the graph of $r$ against $\theta$ .

Worked example 1. For $r = 2 + 4\cos\theta$ , find the average rate of change of $r$ on $\left[0, \frac{\pi}{3}\right]$ and say what it means.

$r(0) = 2 + 4 = 6$ and $r\left(\frac{\pi}{3}\right) = 2 + 4\cdot\frac{1}{2} = 4$ .

$\frac{4 - 6}{\pi/3 - 0}$ $= -\frac{6}{\pi}$ $\approx -1.91$ units per radian.

Meaning: as the angle turns from $0$ to $\frac{\pi}{3}$ , $r$ falls by about $1.91$ units per radian on average. Because $r$ stays positive on this interval, the curve moves toward the origin.

As with any function, an average rate gives estimates. Between two known values, the point on the secant line estimates $r$ : halfway in angle, halfway in $r$ . Comparing average rates on equal intervals shows whether the rate of change of $r$ is increasing or decreasing.

Toward or away from the origin

The distance from the origin to the point $(r, \theta)$ is $|r|$ , not $r$ . So "is $r$ increasing" and "is the curve moving away from the origin" are different questions. Put the sign of $r$ and its direction of change together:

$r$ is	and $r$ is	so $\|r\|$ is	the curve is moving
positive	increasing	increasing	away from the origin
positive	decreasing	decreasing	toward the origin
negative	decreasing	increasing	away from the origin
negative	increasing	decreasing	toward the origin

A quick test: if $r$ moves away from $0$ on the graph of $r$ against $\theta$ , the curve moves away from the origin. If $r$ moves toward $0$ , the curve comes back.

Worked example 2. For $r = 1 + 2\sin\theta$ (Figure 1), is the distance from the origin increasing or decreasing on $\left(\frac{7\pi}{6}, \frac{3\pi}{2}\right)$ ?

$r\left(\frac{7\pi}{6}\right)$ $= 1 + 2\left(-\frac{1}{2}\right)$ $= 0$ and $r\left(\frac{3\pi}{2}\right) = 1 - 2 = -1$ . On this interval $r$ is negative and decreasing, moving away from $0$ . So the distance from the origin is increasing: the inner loop grows outward until $\theta = \frac{3\pi}{2}$ .

The polar graph of r = 1 + 2 sin theta: a limacon whose large outer loop reaches highest straight above the origin and crosses the horizontal axis 1 unit on each side of the origin, with a small inner loop above the origin. — **Figure 1.** $r = 1 + 2\sin\theta$ , with an inner loop drawn while $r < 0$ . LevlPrep original diagram.

Farthest and closest points

The curve is farthest from the origin where $|r|$ is largest, and touches the origin where $r = 0$ . Read them from the graph of $r$ against $\theta$ : the highest peak and the lowest valley both count, because a valley at $r = -4$ is $4$ units out.

Worked example 3. For $r = 1 - 3\cos\theta$ , find the farthest point from the origin.

$r$ ranges from $1 - 3 = -2$ (at $\theta = 0$ ) to $1 + 3 = 4$ (at $\theta = \pi$ ). The largest $|r|$ is $4$ , at $\theta = \pi$ : the point $(4\cos\pi, 4\sin\pi)$ $= (-4, 0)$ . The value $r = -2$ at $\theta = 0$ is only $2$ units away.

Saying it in words

Name the function, the interval and both facts: "On $\left(\frac{7\pi}{6}, \frac{3\pi}{2}\right)$ , $r$ is negative and decreasing, so the distance between the point on the curve and the origin is increasing." For rates on equal intervals: "The average rates of change of $r$ decrease, so the rate of change of $r$ is decreasing on $\left[0, \frac{\pi}{3}\right]$ ."

Common slips

Saying the curve moves toward the origin just because $r$ decreases, without checking the sign of $r$ .
Dividing by the wrong thing: the rate is change in $r$ over change in $\theta$ .
Reading only the highest peak of $r$ for the farthest point and missing a deeper valley.