Mathfolis

Rates of Change in Polar Functions

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

For a polar function r=f(θ)r = f(\theta), the average rate of change over [θ1,θ2][\theta_1, \theta_2] is the familiar formula (r(θ2)−r(θ1))/(θ2−θ1)(r(\theta_2) - r(\theta_1)) / (\theta_2 - \theta_1). The sign tells you whether the curve is moving away from the pole (positive AROC) or toward it (negative AROC). Topic 3.15 is the polar version of Topic 1.2.

Polar AROC

Definition
AROC=r(θ2)−r(θ1)θ2−θ1\text{AROC} = \frac{r(\theta_2) - r(\theta_1)}{\theta_2 - \theta_1}
Sign meaning
+⇒r increasing (moving away from pole);  −⇒decreasing (toward pole)+ \Rightarrow r \text{ increasing (moving away from pole)};\;-\Rightarrow \text{decreasing (toward pole)}
AP Tip: Polar AROC measures how fast rr changes per unit of θ\theta — not the Cartesian path length. It's an analog of slope, but in polar.
Type 1

Average rate of change of r(θ)

Evaluate rr at the endpoints, subtract, divide by the change in θ\theta.

Example 1
For r=1+cos⁡θr = 1 + \cos\theta, find the AROC on [0,π/2][0, \pi/2]. (A) −2π-\tfrac{2}{\pi} (B) 2π\tfrac{2}{\pi} (C) −1-1 (D) −12-\tfrac{1}{2}

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Type 2

Interpret sign of AROC

Positive AROC ⇒ moving away from pole; negative ⇒ moving toward pole.

Example 2
Over [0,π/2][0, \pi/2], is r=1+cos⁡θr = 1 + \cos\theta moving toward or away from the pole? (A) Away from (B) Toward (C) Neither (constant) (D) Alternating both

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Type 3

Real-world AROC in polar

Apply AROC of r(θ)r(\theta) to a contextual problem.

Example 3
A spiral satisfies r=0.5θr = 0.5 \theta for θ∈[0,2π]\theta \in [0, 2\pi]. What is the AROC of rr on this interval? (A) 1/(4π)1/(4\pi) (B) 1/21/2 (C) π\pi (D) 11

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