AP PRECALCULUS — UNIT 3B · TRIGONOMETRIC & POLAR FUNCTIONS
3.15 — Rates of Change in Polar Functions
Notes — How r Changes as θ Changes
💡 Learning Objectives (3.15.A)
By the end of this lesson you will be able to:
- Compute and interpret the average rate of change of r over an interval of θ
- Determine on which intervals r is increasing or decreasing
- Use rate-of-change reasoning to describe how a polar curve evolves
- Compare the rate of change at different θ-values
1. Average Rate of Change
The AVERAGE RATE OF CHANGE of r = f(θ) over an interval [θ₁, θ₂] is:
ARoC = (f(θ₂) − f(θ₁)) / (θ₂ − θ₁)
Because r is the distance from the pole, this rate measures how rapidly the distance is changing per unit increase in angle. Units: distance / radian.
📘 Example — ARoC of r = 4 sin(2θ) on [0, π/4]
- r(0) = 4 sin(0) = 0
- r(π/4) = 4 sin(π/2) = 4
- ARoC = (4 − 0)/(π/4 − 0) = 16/π ≈ 5.09 (units of distance per radian)
- Interpretation: the curve moves AWAY from the pole at an average rate of about 5.09 distance units per radian over this interval.
2. Increasing and Decreasing Intervals
A polar function r = f(θ) is INCREASING on an interval where its values rise as θ rises, and DECREASING where its values fall. To find these intervals, examine the function's behavior just like any function of one variable:
- Find where r reaches a max or min (these are turning points)
- Between consecutive turning points, r is monotonic — either all increasing or all decreasing
- Test a sample point to determine which
📘 Example — Behavior of r = 2 + 2 cos θ on [0, 2π]
- Max at θ = 0 (r = 4); min at θ = π (r = 0)
- Decreasing on (0, π) — curve moves toward the pole
- Increasing on (π, 2π) — curve moves away from the pole
3. Geometric Meaning
In polar context, ‘increasing r’ means the curve is moving FARTHER from the pole as θ grows — it spirals outward. ‘Decreasing r’ means it's drifting INWARD. Understanding this lets you describe a polar curve's shape and motion using the language of rate of change.
4. Comparing Rates
The rate of change is not constant for most polar functions. The curve might move out quickly near θ = 0 and slowly near θ = π/2. To compare, compute ARoC over different small intervals.
📘 Example — Compare rate near θ = 0 vs. near θ = π/2 for r = θ²
- Near θ = 0 (use [0, 0.1]): r(0.1) = 0.01, r(0) = 0. ARoC = 0.01/0.1 = 0.1
- Near θ = π/2 (use [π/2 − 0.1, π/2]): r(π/2) ≈ 2.467, r(π/2 − 0.1) ≈ 2.156. ARoC ≈ 0.311/0.1 = 3.11
- So r changes much faster near θ = π/2 than near θ = 0. Consistent with r = θ² being concave-up — its rate of growth speeds up.
5. Rate of Change and Curve Shape
Connecting rate of change back to shape:
- LARGE positive rate of change: the curve sweeps quickly outward, creating elongated arcs
- SMALL positive rate of change: the curve barely moves outward — it's nearly circular at this θ
- LARGE negative rate of change: the curve plunges inward — often near a pole crossing
- Rate of change = 0: the curve is at a max or min r — a turning point
6. AP-Style Question Structure
AP problems on this topic typically ask:
- ‘Compute the average rate of change of r over [a, b].’ Use the formula directly.
- ‘Is r increasing or decreasing at θ = c?’ Use values just before and just after c, or recognize from the function's structure.
- ‘At which value of θ in [a, b] is r changing most rapidly?’ Compare ARoCs over small subintervals, or check turning points.
7. Worked Example — Cardioid
📘 Example — r = 3 − 3 cos θ
- r(0) = 0; r(π/2) = 3; r(π) = 6 (max); r(3π/2) = 3; r(2π) = 0
- ARoC on [0, π/2]: (3 − 0)/(π/2) = 6/π ≈ 1.91 — moving outward
- ARoC on [π/2, π]: (6 − 3)/(π/2) = 6/π ≈ 1.91 — moving outward, same rate on average
- ARoC on [π, 3π/2]: (3 − 6)/(π/2) = −6/π ≈ −1.91 — moving inward
- Cardioid is symmetric, so the ARoCs in symmetric intervals are equal in magnitude with opposite signs.
8. Summary
- Average rate of change of r = f(θ) over [θ₁, θ₂] is (f(θ₂) − f(θ₁)) / (θ₂ − θ₁)
- Increasing r: curve moves AWAY from pole; decreasing r: curve moves TOWARD pole
- Local max/min of r marks turning points where the curve direction reverses
- Rate of change varies along the curve — compare over small intervals to understand shape
- Pole crossings (r = 0) often correspond to rapidly changing r, since r is dropping from positive values