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

> AP Calculus BC Unit 9 review: parametric derivatives and arc length, vector-valued motion, polar slopes and polar area, with sample questions and practice.

Source: https://practicecalculus.com/ap-calculus/unit-9-parametric-polar-and-vector-valued-functions
Updated: 2026-11-06

## At a glance

- **AB exam weight:** Not on the AB exam
- **BC exam weight:** 10–15%
- **Big ideas:** Parametric derivatives, arc length, motion in the plane, polar area
- **BC only:** The whole unit
- **Comes before:** Unit 10, infinite sequences and series
- **Practice here:** [Calculus 2: Parametric and polar](/learn/calculus-2/power-series-parametric-polar/parametric-curves)

## Key takeaways

- Unit 9 is a BC-only unit worth 10 to 15 percent of the BC exam.
- For a parametric curve, $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, and $\frac{d^2y}{dx^2}$ is the derivative of $\frac{dy}{dx}$ with respect to $t$, divided by $\frac{dx}{dt}$.
- Speed in the plane is $\sqrt{\left(x'(t)\right)^2 + \left(y'(t)\right)^2}$, and integrating speed gives distance traveled.
- The area inside a polar curve is $\frac{1}{2}\int_\alpha^\beta r^2\,d\theta$.

**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.

| | AP Calculus AB | AP Calculus BC |
| --- | --- | --- |
| Exam weight | not on the AB exam | 10–15% (BC only) |
| Extra topics | none | the whole unit is BC only |

## 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.

| Topic | What you learn | Practice |
| --- | --- | --- |
| Parametric equations | Slopes $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ and tangent lines | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) |
| Second derivatives of parametric equations | Concavity of a parametric curve | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) |
| Arc length of parametric curves | $\int_a^b \sqrt{(x')^2 + (y')^2}\,dt$ | [Parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) |
| Vector-valued functions | Differentiating $\langle x(t), y(t)\rangle$ | [Parametric motion](/learn/calculus-2/power-series-parametric-polar/parametric-motion) |
| Integrating vector-valued functions | Position from velocity, component by component | [Parametric motion](/learn/calculus-2/power-series-parametric-polar/parametric-motion) |
| Motion in the plane | Velocity, speed, acceleration and distance traveled | [Parametric motion](/learn/calculus-2/power-series-parametric-polar/parametric-motion) |
| Polar coordinates and differentiation | Converting, and slopes of polar curves | [Polar slopes](/learn/calculus-2/power-series-parametric-polar/polar-slopes) |
| Area of a polar region | $\frac{1}{2}\int r^2\,d\theta$ for one curve | [Polar area](/learn/calculus-2/power-series-parametric-polar/polar-area) |
| Area between two polar curves | Outer squared minus inner squared | [Polar area](/learn/calculus-2/power-series-parametric-polar/polar-area) |

## 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:

$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \qquad \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{dx/dt}$$

```graph
title: 'The parametric circle x = 2cos t, y = 2sin t'
caption: 'As $t$ goes from 0 to $2\pi$ the point traces the circle $x^2 + y^2 = 4$ once, counterclockwise. Its arc length from $t = 0$ to $t = \pi$ is $2\pi$.'
x: [-3, 3]
y: [-3, 3]
functions:
  - expr: sqrt(4 - x^2)
    label: upper half (0 ≤ t ≤ π)
    domain: [-2, 2]
  - expr: -sqrt(4 - x^2)
    label: lower half (π ≤ t ≤ 2π)
    domain: [-2, 2]
    dashed: true
points:
  - [2, 0, 't = 0']
  - [0, 2, 't = π/2']
```

**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:

$$A = \frac{1}{2}\int_\alpha^\beta r^2\,d\theta$$

## Sample questions

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

$$\frac{dy}{dx} = \frac{3t^2 - 3}{2t} = \frac{3}{2}\left(t - \frac{1}{t}\right)$$

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 $t = 2$:

$$\frac{d^2y}{dx^2} = \frac{\frac{3}{2}\cdot\frac{5}{4}}{4} = \frac{15}{32}$$

**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

$$\frac{1}{2}\int_0^{2\pi} (1 + \cos\theta)^2\,d\theta = \frac{1}{2}\left(2\pi + 0 + \pi\right) = \frac{3\pi}{2}$$

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](/practice-problems/parametric-and-polar), each with full solutions.

For the full course map, see the [AP Calculus BC guide](/ap-calculus/bc), or compare the two courses in the [AP Calculus AB guide](/ap-calculus/ab).

> 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.

## Frequently asked questions

### What is AP Calculus Unit 9 about?

Unit 9 extends calculus to curves that are not graphs of $y = f(x)$. You find first and second derivatives and arc length for parametric curves, analyze motion with vector-valued functions, and find slopes and areas for curves in polar coordinates.

### Is Unit 9 on the AP Calculus AB exam?

No. Unit 9 is only in AP Calculus BC, where it is 10 to 15 percent of the multiple-choice score. It also appears on the free-response section, usually as a particle moving in the plane or a polar region.

### What is the hardest part of Unit 9?

The parametric second derivative and polar area. For $\frac{d^2y}{dx^2}$, many students forget to divide by $\frac{dx}{dt}$. For polar area, the common mistakes are forgetting the $\frac{1}{2}$ and using the wrong $\theta$ limits.

### Do I need to convert polar curves to rectangular form?

Sometimes. Use $x = r\cos\theta$ and $y = r\sin\theta$ to find slopes of polar curves. For area, stay in polar form and use $\frac{1}{2}\int r^2\,d\theta$.