Rates of change in polar functions

For a point sweeping along $r = f(\theta)$, the value of r is its distance from the pole (when r is positive). So on an interval of θ where r is increasing, the point is moving away from the origin; where r is decreasing, it is closing in on the origin. That is the entire dictionary.

What r's change does NOT encode: vertical motion, horizontal motion, or the turning direction. As θ increases the sweep is counterclockwise regardless of what r does, and whether the point happens to be rising or falling in the plane depends on which quadrant the angle is passing through. "r is decreasing, so the graph is going down" is the signature misread of this topic; the correct sentence is "r is decreasing, so the point is approaching the pole."

Rates work exactly as they did in Unit 1, with θ as the input:

1. Average rate of change of r on $[\theta_1, \theta_2]$ is $\dfrac{f(\theta_2) - f(\theta_1)}{\theta_2 - \theta_1}$, in units of distance per radian.
2. A positive AROC means the point ended farther from the pole than it started; negative means nearer.
3. The average over an interval is not the rate at either endpoint, and equal-width θ intervals can carry very different rates.

Worked example: $r = 2 + 2\cos\theta$ from θ = 0 to θ = π/2. Endpoints: r(0) = 4 and r(π/2) = 2. AROC = $(2 - 4)/(\pi/2 - 0) = -4/\pi \approx -1.27$ per radian: over that quarter turn the point averaged about 1.27 units of approach toward the pole per radian of sweep.

Distance from the pole is really $|r|$. While r is positive, r and distance move together and the dictionary above applies verbatim. But once r crosses zero and keeps decreasing into negative values, the distance $|r|$ is now increasing: the point shot through the pole and is receding on the opposite side.

Example: $r = \cos\theta$ on (π/2, π). There r falls from 0 to −1, yet the point moves from the pole out to distance 1 (plotted across the pole, in the fourth-quadrant direction). Two changes at once: r decreasing, distance increasing. Exam items that mix the two are testing whether you know which question was asked, about r, or about distance.

Put the two motions together and you can narrate any polar curve without plotting a point. The angle marches counterclockwise; r's rises and falls script the approach and retreat. A cardioid like $r = 1 + \cos\theta$ reads: start at distance 2 on the polar axis, drift inward all the way to the pole by θ = π, then swing back out to 2 as θ completes the turn. A spiral like $r = 2\theta/\pi$ reads: start at the pole and recede steadily forever while circling.

Comparisons across intervals work the Unit 1 way too: compute each interval's AROC of r and compare magnitudes. Bigger magnitude means faster approach or retreat per radian, no picture required.