For r = 2θ, find the average rate of change of r on [0, π]. [1 point]

College Board AP 2023 (style)-style practice: Section I, Part B (multiple choice, calculator allowed). For a polar function r = f(θ), on an interval where f(θ) is positive and increasing, how does the curve behave as θ increases? (A) It moves toward the pole (B) It moves away from the pole (C) It stays at a constant distance (D) It crosses the pole

The correct answer is (B), it moves away from the pole. The radius r is the distance from the pole. If r = f(θ) is positive and increasing, then as θ increases the distance grows, so the curve spirals outward, moving away from the pole.

College Board AP 2024 (style)-style practice: Section II (free response, calculator allowed). A polar function is r = 3 + 2cosθ. (a) Find the average rate of change of r with respect to θ on [0, (π)/(2)]. (b) State whether the curve is moving toward or away from the pole on that interval and justify.

A 4-point question on the average rate of change of a polar radius. (a) Average rate (2 points): r(0) = 3 + 2cos 0 = 5 and r((π)/(2)) = 3 + 2cos(π)/(2) = 3. The average rate of change is (r(π/2) - r(0))/(π/2 - 0) = (3 - 5)/(π/2) = (-2)/(π/2) = -(4)/(π). (b) Direction (2 points): the average rate of change is negative, so r is decreasing over [0, (π)/(2)]. Since the distance from the pole is shrinking, the curve is moving toward the pole on that interval.

United StatesPrecalculusSyllabus dot point

How does the radius of a polar function change as the angle increases, and what does the average rate of change tell you?

Topic 3.15 Rates of Change in Polar Functions: analyze how r changes as theta increases, using the average rate of change to describe whether the curve moves toward or away from the pole.

A focused answer to AP Precalculus Topic 3.15, covering how the radius of a polar function changes with the angle, the average rate of change of r with respect to theta, and how its sign tells you whether the curve approaches or leaves the pole.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

For a polar function $r = f(\theta)$ , the radius is the distance from the pole, so how $r$ changes as $\theta$ increases tells you whether the curve is spiralling outward or inward. The average rate of change of $r$ on an interval $[\theta_1, \theta_2]$ is $\frac{f(\theta_2) - f(\theta_1)}{\theta_2 - \theta_1}$ , exactly the average-rate-of-change formula from Unit 1 applied to $r$ and $\theta$ . If this is positive, $r$ is increasing on average and the curve moves away from the pole; if negative, $r$ is decreasing and the curve moves toward the pole; if zero (over a full symmetric span), the net distance is unchanged. Where $r$ reaches a maximum the curve is farthest out, and where $r = 0$ it touches the pole. Reading the sign and size of the rate of change describes the curve's motion without plotting it.

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What this topic is asking
The radius as a function of the angle
Average rate of change of r
Maxima, minima and the pole
Try this

What this topic is asking

The College Board (Topic 3.15) wants you to analyze how the radius $r$ of a polar function changes as the angle $\theta$ increases. Using the average rate of change of $r$ with respect to $\theta$ , you describe whether the curve is moving toward the pole (radius shrinking) or away from it (radius growing) over an interval.

The radius as a function of the angle

So all the increasing/decreasing language of Unit 1 transfers directly: $r$ increasing means moving away from the pole, $r$ decreasing means moving toward it.

Average rate of change of r

This is the same difference-quotient used for any function in Topic 1.2; here it measures how fast the distance from the pole changes per unit of angle.

Average rate of change of a polar radius

For $r = 4\sin\theta$ , find the average rate of change of $r$ on $\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$ and describe the curve's motion.

step 1 Evaluate r at the endpoints

$r\left(\frac{\pi}{6}\right) = 4\sin\frac{\pi}{6} = 4 \cdot \frac{1}{2} = 2$ . $r\left(\frac{\pi}{2}\right) = 4\sin\frac{\pi}{2} = 4 \cdot 1 = 4$ .

step 2 Apply the average-rate-of-change formula

\frac{r(\pi/2) - r(\pi/6)}{\pi/2 - \pi/6} = \frac{4 - 2}{\pi/2 - \pi/6} = \frac{2}{\pi/3} = \frac{6}{\pi}.

step 3 Interpret the sign

The rate $\frac{6}{\pi}$ is positive, so $r$ is increasing on the interval: as $\theta$ goes from $\frac{\pi}{6}$ to $\frac{\pi}{2}$ , the radius grows from $2$ to $4$ .

step 4 Describe the motion

Because the distance from the pole is increasing, the curve is moving away from the pole on $\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$ , reaching its farthest point ( $r = 4$ ) at $\theta = \frac{\pi}{2}$ .

Maxima, minima and the pole

The radius reaches a maximum where the curve is farthest from the pole, and the curve touches the pole wherever $r = 0$ . Between a zero and a maximum, $r$ is increasing (moving outward); between a maximum and the next zero, $r$ is decreasing (moving inward). Identifying these turning points lets you describe the whole sweep of a polar curve as alternating outward and inward motion, which is exactly how the petals of a rose or the loops of a limacon are traced.

A point worth stating once is that a negative average rate of change does not mean the radius itself is negative; it means the radius is decreasing. The two are separate questions: the sign of $r$ tells you direction (Topic 3.14), while the sign of the rate of change tells you whether the curve is approaching or leaving the pole. Keeping "is $r$ positive or negative?" apart from "is $r$ increasing or decreasing?" prevents the most common confusion in this topic and mirrors the direction-versus-concavity distinction from Topic 1.1.

Try this

Q1. If the average rate of change of $r$ on an interval is positive, is the curve moving toward or away from the pole? [1 point]

Cue. Away from the pole: a positive rate means $r$ (the distance) is increasing.

Q2. For $r = 2\theta$ , find the average rate of change of $r$ on $[0, \pi]$ . [1 point]

Cue. $r(0) = 0$ , $r(\pi) = 2\pi$ ; rate $= \frac{2\pi - 0}{\pi - 0} = 2$ .

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2023 (style)1 marksSection I, Part B (multiple choice, calculator allowed). For a polar function

r = f(\theta)

, on an interval where

f(\theta)

is positive and increasing, how does the curve behave as

\theta

increases? (A) It moves toward the pole (B) It moves away from the pole (C) It stays at a constant distance (D) It crosses the pole

Show worked answer →

The correct answer is (B), it moves away from the pole.

The radius $r$ is the distance from the pole. If $r = f(\theta)$ is positive and increasing, then as $\theta$ increases the distance grows, so the curve spirals outward, moving away from the pole.

AP 2024 (style)4 marksSection II (free response, calculator allowed). A polar function is

r = 3 + 2\cos\theta

. (a) Find the average rate of change of

r

with respect to

\theta

\left[0, \frac{\pi}{2}\right]

. (b) State whether the curve is moving toward or away from the pole on that interval and justify.

Show worked answer →

A 4-point question on the average rate of change of a polar radius.

(a) Average rate (2 points): $r(0) = 3 + 2\cos 0 = 5$ and $r\left(\frac{\pi}{2}\right) = 3 + 2\cos\frac{\pi}{2} = 3$ . The average rate of change is $\frac{r(\pi/2) - r(0)}{\pi/2 - 0} = \frac{3 - 5}{\pi/2} = \frac{-2}{\pi/2} = -\frac{4}{\pi}$ .
(b) Direction (2 points): the average rate of change is negative, so $r$ is decreasing over $\left[0, \frac{\pi}{2}\right]$ . Since the distance from the pole is shrinking, the curve is moving toward the pole on that interval.

Related dot points

Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)