AP Calculus BC Unit 9: Parametric & Polar
Study parametric equations, polar coordinates, vector-valued functions, derivatives with exam-format practice and rubric-based scoring.
Start AP Calc BC Practice · Full Study Guide
AP and Advanced Placement are trademarks of College Board. AimFive is not affiliated with or endorsed by College Board.
Inside This Unit: The Full Breakdown
BC extends calculus to curves described parametrically (x(t), y(t)) and in polar coordinates (r(theta)). This unit also introduces vector-valued functions for motion in the plane.
Why it matters
Parametric, polar, and vector questions are exclusive to the BC exam and appear on virtually every free response. These topics test your ability to extend derivative and integral concepts to new coordinate systems.
Key concepts
- Parametric derivatives: dy/dx = (dy/dt)/(dx/dt). Second derivative: d^2y/dx^2 = d/dt[dy/dx] / (dx/dt).
- Arc length of a parametric curve: L = integral of sqrt((dx/dt)^2 + (dy/dt)^2) dt.
- Polar area: A = (1/2) integral of r^2 d(theta).
- Velocity vector is <x'(t), y'(t)>; speed is the magnitude of the velocity vector.
Parametric Curves
A parametric curve is defined by x(t) and y(t) as t varies over an interval. The slope of the tangent line is dy/dx = (dy/dt)/(dx/dt). Horizontal tangents occur where dy/dt = 0 and dx/dt is nonzero; vertical tangents where dx/dt = 0 and dy/dt is nonzero. The second derivative d^2y/dx^2 = [d/dt(dy/dx)] / (dx/dt) describes concavity. Arc length is the integral of speed: L = integral of sqrt((dx/dt)^2 + (dy/dt)^2) dt.
Polar Coordinates
Polar coordinates locate points by distance r from the origin and angle theta from the positive x-axis. Curves are defined by r = f(theta). The area enclosed by a polar curve from theta = a to theta = b is A = (1/2) integral of [f(theta)]^2 d(theta). For the area between two polar curves, integrate (1/2)(R^2 - r^2). To find dy/dx for a polar curve, convert to parametric form: x = r*cos(theta), y = r*sin(theta), then use dy/dx = (dy/d(theta))/(dx/d(theta)).
Vector-Valued Functions
A vector-valued function r(t) = <x(t), y(t)> describes position in the plane. The velocity vector v(t) = <x'(t), y'(t)> gives direction and rate of change; its magnitude is speed. The acceleration vector a(t) = <x''(t), y''(t)> describes how velocity changes. Displacement is the integral of velocity, and total distance is the integral of speed. The AP exam often presents particle motion problems using vector notation and asks for speed, direction, or total distance at specific times.
AP exam tip
For polar area problems, carefully determine the correct bounds for theta by identifying where the curve starts and completes the region. Sketching the curve or making a table of r-values for key angles prevents errors.
Connections to other units
- Unit 2: Parametric derivatives use the chain rule; dy/dx = (dy/dt)/(dx/dt) is a direct application.
- Unit 5: Polar and parametric areas extend the definite integral to new coordinate systems.
- Unit 10: Taylor series can approximate parametric and polar functions locally.
