Parametric Equations, Polar Coordinates and Vector-Valued Functions

Unit 9 of AP Calculus BC, Parametric Equations, Polar Coordinates, and Vector-Valued Functions, takes the derivative and integral you already know and applies them to curves that are not written as y = f(x). Every formula here comes from the chain rule or from one geometric picture.

Parametric curves

A curve given by x(t) and y(t):

dy/dx = (dy/dt) / (dx/dt)

d²y/dx² = [d/dt (dy/dx)] / (dx/dt)

The second derivative is where most points are lost: differentiate dy/dx with respect to t, then divide by dx/dt again.

Arc length from t = a to t = b: ∫ₐᵇ √( (dx/dt)² + (dy/dt)² ) dt

Motion with vector-valued functions

Position r(t) = ⟨x(t), y(t)⟩.

• Velocity: v(t) = ⟨x′(t), y′(t)⟩
• Speed: |v(t)| = √(x′² + y′²), a number, never a vector
• Acceleration: a(t) = ⟨x″(t), y″(t)⟩
• Position later: x(b) = x(a) + ∫ₐᵇ x′(t) dt, and the same for y
• Total distance traveled: ∫ₐᵇ speed dt

Worked example: a particle has x′(t) = 2t, y′(t) = 3t², and is at (1, 0) when t = 0.

1. At t = 2: x = 1 + ∫₀² 2t dt = 1 + 4 = 5 and y = 0 + ∫₀² 3t² dt = 8. Position (5, 8).
2. Speed at t = 2: √(4² + 12²) = √160 ≈ 12.65.

Polar coordinates

A point (r, θ) sits at x = r cos θ, y = r sin θ.

• Slope: write x = r(θ) cos θ and y = r(θ) sin θ, then dy/dx = (dy/dθ)/(dx/dθ). The product rule is needed in both.
• Area: A = ½ ∫ r² dθ over the angles that trace the region once.
• Area between two curves: ½ ∫ (R² − r²) dθ, with R the outer curve.

Worked example: the area of one petal of r = sin 2θ. The petal runs from θ = 0 to θ = π/2, where r returns to 0.

A = ½ ∫₀^(π/2) sin² 2θ dθ = ½ × π/4 = π/8

Finding the limits is the real work: set r = 0, or set two curves equal to find where they meet.

The free response approach

Write the integral with its limits before evaluating it, keep units in motion answers, and separate the displacement (integral of velocity) from the distance (integral of speed). On the calculator section, show the setup; the value alone does not earn the setup point.