Short on time? This page is the recap. For the figure, explanations and worked examples, use the full study guide. BC-only material. Prerequisites (parametric dy/dx 9.1, parametric second derivatives 9.2, product and chain rules) are on the Calculus BC roadmap.
Recap
- (r, θ) means: face along the ray at angle θ from the positive x-axis, then go r units. Negative r means go backwards.
- A point has many polar names: (r, θ), (r, θ + 2π) and (−r, θ + π) are all the same point.
- For r = f(θ), write x = f(θ) cos θ, y = f(θ) sin θ and use the parametric methods of Topics 9.1 and 9.2, with θ as the parameter.
- Work in radians.
Key relationships
| Quantity | Formula | Note |
|---|---|---|
| Conversion | x = r cos θ, y = r sin θ, r² = x² + y² | Check the quadrant when finding θ |
| dx/dθ | f′(θ) cos θ − f(θ) sin θ | Product rule |
| dy/dθ | f′(θ) sin θ + f(θ) cos θ | Product rule |
| Slope | dy/dx = (dy/dθ) ÷ (dx/dθ) | Needs dx/dθ ≠ 0 |
| Second derivative | d²y/dx² = [d/dθ (dy/dx)] ÷ (dx/dθ) | Divide by dx/dθ again |
| Horizontal tangent | dy/dθ = 0, dx/dθ ≠ 0 | Both 0: investigate |
| Vertical tangent | dx/dθ = 0, dy/dθ ≠ 0 | |
| Distance from pole | Increasing if r and dr/dθ have the same sign | Decreasing if opposite signs |
Mistakes to avoid
- Treating dr/dθ as the slope. It measures change in r, not in y per unit x.
- Using tan θ as the slope. That is the slope of the ray to the point, not of the tangent.
- Dropping a product-rule term in dx/dθ or dy/dθ.
- Dividing second derivatives: d²y/dx² ≠ (d²y/dθ²) ÷ (d²x/dθ²).
- Reading dr/dθ > 0 as “moving away” when r < 0.
- Degree mode on the calculator.
Quick self-check
- Convert the polar point (6, 7π/6) to rectangular form. ((−3√3, −3))
- For r = 1 + cos θ, find dy/dx at θ = π/2. (r = 1, r′ = −1, dx/dθ = −1, dy/dθ = −1, so dy/dx = 1)
- For r = 5 sin θ, find dy/dx at θ = π/6. (√3)
- At some θ, r = −1 and dr/dθ = 2. Is the point moving towards or away from the pole? (Towards: opposite signs, so |r| is decreasing.)
Next: practice questions.