AP Pre-Calculus: Rates of Change in Polar Functions - Night Before Review 🚀

Hey! Let's get you prepped for the exam. This guide is designed to be your quick, go-to resource. We'll break down polar functions, rates of change, and how to tackle those tricky questions. Let's do this! 💪

3.15 Rates of Change in Polar Functions

🔄 Expanding and Contracting Polar Functions

Key Concept

Remember, in polar coordinates, we're dealing with r = f(θ), where r is the distance from the origin and θ is the angle. This is key to understanding how these functions behave. 💡

Expanding Polar Function 🌖:
- If r is positive and increasing as θ increases, the function moves away from the origin.
Contracting Polar Function 🌘:
- If r is negative and decreasing as θ increases, the function moves towards the origin.

Memory Aid

Think of a snail moving along a spiral. If the spiral is moving outwards, it's expanding, if inwards, it's contracting! 🐌

Polar Graph

Caption: Visualizing polar coordinates. The radius 'r' and angle 'θ' define a point's location.

Exam Tip

Pay attention to whether the function is increasing or decreasing. This tells you if the curve is moving towards or away from the origin. This is often tested in multiple choice questions.

🔀 Relative Extrema

Relative Extrema: These occur when a function changes direction—from increasing to decreasing or vice versa. 🧑‍💻
- Relative Maximum: The function changes from increasing to decreasing, representing a point relatively farthest from the origin.
- Relative Minimum: The function changes from decreasing to increasing, representing a point relatively closest to the origin. 💡

Common Mistake

Remember, relative extrema are local—they only tell you about behavior within a specific interval, not the entire function. 🧐

〽️ Rates of Change

Average Rate of Change: This measures how much r changes with respect to θ over an interval. #️⃣
- Formula: (Δr/Δθ) = (r(θ₂) - r(θ₁)) / (θ₂ - θ₁). 📐
- Graphically, it's the slope of the line connecting two points on the polar graph. ⛰️
  - Positive Slope: r is increasing as θ increases. ⬆️
  - Negative Slope: r is decreasing as θ increases. ⬇️

Average rate of change of r with respect to θ

Caption: The average rate of change is the slope of the line connecting two points on the polar curve.

Quick Fact

The average rate of change helps estimate function values within an interval. If it's positive, r is increasing; if negative, r is decreasing. ➕➖

Memory Aid

Think of a mountain path. A positive rate of change means you're going uphill (r increasing), and a negative rate means you're going downhill (r decreasing). ⛰️

Final Exam Focus 🎯

High-Priority Topics:
- Understanding the relationship between r, θ, and the graph's behavior.
- Identifying relative extrema.
- Calculating and interpreting the average rate of change.
Common Question Types:
- Multiple-choice questions asking about increasing/decreasing behavior.
- Free-response questions involving calculations of average rates of change and interpretations.

Last-Minute Tips ⏰

Time Management: Don't get bogged down on one question. Move on and come back if needed.
Common Pitfalls: Double-check your calculations and make sure you're using the correct formula. Watch out for sign errors!
Challenging Formats: Practice with a variety of questions. Focus on understanding the concepts, not just memorizing formulas.

Practice Question

Multiple Choice Questions:

A polar function r = f(θ) is increasing on the interval [0, π/2] and decreasing on the interval [π/2, π]. Which of the following statements is true? (A) The function has a relative minimum at θ = π/2. (B) The function has a relative maximum at θ = π/2. (C) The function is always positive on the interval [0, π]. (D) The function is always negative on the interval [0, π].
The average rate of change of a polar function r = g(θ) on the interval [π/4, 3π/4] is -2. Which of the following must be true? (A) The radius is increasing as θ increases on the interval. (B) The radius is decreasing as θ increases on the interval. (C) The radius is constant on the interval. (D) The function has a relative maximum at some point in the interval.

Free Response Question:

A polar function is given by r(θ) = 3 + 2cos(θ).

(a) Find the average rate of change of r with respect to θ on the interval [0, π/2].

(b) Determine the interval(s) where the function is increasing and decreasing on [0, 2π].

Answer Key:

Multiple Choice:

Free Response:

(a) r(0) = 3 + 2cos(0) = 5 r(π/2) = 3 + 2cos(π/2) = 3 Average rate of change = (3 - 5) / (π/2 - 0) = -4/π

(b) To find where the function is increasing or decreasing, we need to find the derivative of r(θ) with respect to θ. r'(θ) = -2sin(θ) Set r'(θ) = 0 to find critical points: -2sin(θ) = 0 sin(θ) = 0 θ = 0, π, 2π Test intervals:

(0, π): r'(π/2) = -2sin(π/2) = -2 (decreasing)
(π, 2π): r'(3π/2) = -2sin(3π/2) = 2 (increasing)

Therefore, the function is decreasing on the interval (0, π) and increasing on the interval (π, 2π).

(c) From part (b), we know that the function has a relative minimum at θ = π. To find the value of r at this point, we plug in θ = π into r(θ). r(π) = 3 + 2cos(π) = 3 - 2 = 1

Relative minimum at (1, π)

To check the endpoints: r(0) = 3 + 2cos(0) = 5 r(2π) = 3 + 2cos(2π) = 5

Relative maximum at (5, 0) and (5, 2π)

Scoring Breakdown:

(a) 2 points: - 1 point for correct r(0) and r(π/2) - 1 point for correct calculation of average rate of change

(b) 3 points: - 1 point for finding the derivative r'(θ) - 1 point for finding the critical points - 1 point for correct intervals of increasing and decreasing

You've got this! Go ace that exam! 🌟

Rates of Change in Polar Functions

Tom Green