AP Calc BC Unit 9 Review: Parametric, Polar and Vectors

AP Calculus Unit 9, Parametric Equations, Polar Coordinates, and Vector-Valued Functions, is the first of the two BC-only units. It takes derivatives and integrals off the x-axis and into the plane: a particle moving along a path, a curve given by its angle and distance from the origin, a position written as a vector. The calculus is the same. What changes is how you set it up.

What Unit 9 covers

College Board splits the unit into these topics. Each row links to a lesson where you can practice it with new numbers every time.

The key ideas

A parametric curve is a path traced over time. Both x and y depend on a parameter t. The slope is a ratio of rates:

Vectors bundle motion together. If position is \vec{r}(t) = \langle x(t), y(t)\rangle, then velocity is \vec{v}(t) = \langle x'(t), y'(t)\rangle and acceleration is \vec{a}(t) = \langle x''(t), y''(t)\rangle. Speed is the length of the velocity vector, and total distance is the integral of speed.

Polar curves use angle and distance. A point is (r, \theta) with x = r\cos\theta and y = r\sin\theta. For slopes, differentiate x and y with respect to \theta and divide. For area, add up thin sectors:

Sample questions

1. Parametric slope and concavity. For x = t^2 and y = t^3 - 3t:

At t = 2 the slope is \frac{9}{4}. The curve has horizontal tangents at t = \pm 1. For concavity, \frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{3}{2}\left(1 + \frac{1}{t^2}\right), so at t = 2:

2. Parametric arc length. For x = 2\cos t and y = 2\sin t on [0, \pi], (x')^2 + (y')^2 = 4\sin^2 t + 4\cos^2 t = 4, so the length is \int_0^\pi 2\,dt = 2\pi, half the circumference of a circle of radius 2.

3. Motion in the plane. A particle has position \vec{r}(t) = \langle t^2, 3t\rangle. Then \vec{v}(t) = \langle 2t, 3\rangle and \vec{a}(t) = \langle 2, 0\rangle. At t = 2 the speed is \sqrt{4^2 + 3^2} = 5.

4. Polar area. The area inside the cardioid r = 1 + \cos\theta is

using \int_0^{2\pi}\cos\theta\,d\theta = 0 and \int_0^{2\pi}\cos^2\theta\,d\theta = \pi.

How to study Unit 9

• Divide by \frac{dx}{dt} twice. It appears in the first derivative and again in the second. Leaving it out of \frac{d^2y}{dx^2} is the most common Unit 9 mistake.
• Treat vectors one component at a time. Differentiate, integrate and add initial positions for x and y separately.
• Find polar intersections first. For the area between two polar curves, set the r values equal to get the \theta limits, then use \frac{1}{2}\int \left(r_{\text{outer}}^2 - r_{\text{inner}}^2\right)d\theta.
• Use your calculator where allowed. Many polar and arc length integrals have no simple antiderivative, so they appear on the calculator section.
• Work through problem sets. Try the parametric and polar practice problems, each with full solutions.

For the full course map, see the AP Calculus BC guide, or compare the two courses in the AP Calculus AB guide.

AP and Advanced Placement are registered trademarks of College Board, which is not affiliated with PracticeCalculus and does not endorse this guide. Unit names and weights come from College Board's course framework; check it for any changes.