AP Calculus BC · Exam guide
AP Calculus BC Parametric, Polar, and Vector FRQ: Slopes, Speed, Arc Length, and Area
One BC free-response question is usually about motion in the plane or a polar curve. The formulas are short, and the points come from writing the right formula with the right limits. This guide covers the formulas and works two examples.
The formulas to know
| Quantity | Formula |
|---|---|
| Slope of the tangent line | \(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}\), when \(dx/dt \ne 0\) |
| Speed | \(\sqrt{(dx/dt)^2 + (dy/dt)^2}\) |
| Arc length (parametric) | \(\int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2}\,dt\) |
| Area in polar coordinates | \(\dfrac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta\) |
| Total distance traveled | The integral of speed |
Example 1: parametric motion
A particle moves in the plane with \(x(t) = t^2\) and \(y(t) = t^3 - 3t\) for \(t \ge 0\). (a) Find \(\dfrac{dy}{dx}\) at \(t = 2\). (b) Find the speed at \(t = 2\). (c) Write and evaluate an integral for the distance traveled from \(t = 0\) to \(t = 2\).
(a) \(\dfrac{dx}{dt} = 2t\) and \(\dfrac{dy}{dt} = 3t^2 - 3\). At \(t = 2\), \(\dfrac{dy}{dx} = \dfrac{9}{4}\).
(b) Speed \(= \sqrt{4^2 + 9^2} = \sqrt{97} \approx 9.849\).
(c) Distance \(= \int_0^2 \sqrt{(2t)^2 + (3t^2 - 3)^2}\,dt \approx 7.605\).
The value is right, but no integral is shown.
It writes the integral, then the value.
Example 2: polar area
Find the area enclosed by the polar curve \(r = 1 + \cos\theta\), for \(0 \le \theta \le 2\pi\).
Area \(= \dfrac{1}{2}\int_0^{2\pi} (1 + \cos\theta)^2\,d\theta = \dfrac{1}{2}\left(2\pi + 0 + \pi\right) = \dfrac{3\pi}{2} \approx 4.712\). The three terms come from \(\int 1\,d\theta = 2\pi\), \(\int 2\cos\theta\,d\theta = 0\), and \(\int \cos^2\theta\,d\theta = \pi\) over a full turn.
Vector-valued functions
A vector-valued function \(\langle x(t), y(t)\rangle\) is the same as a parametric pair. Velocity is \(\langle x'(t), y'(t)\rangle\), speed is its magnitude, and acceleration is the derivative of velocity. The position at time \(t\) is the initial position plus the integral of each component of velocity.
Common mistakes
- Using \(dy/dx = \dfrac{dx/dt}{dy/dt}\), upside down.
- Forgetting the square root in the speed or arc length formula.
- Forgetting the factor \(\dfrac{1}{2}\) in polar area.
- Using the wrong limits, such as a full period when the curve repeats.
Frequently asked questions
Is this on AP Calculus AB?
No. Parametric, polar, and vector-valued functions are BC only.
Do I need to simplify the speed expression?
No. An unsimplified correct expression is usually fine.
Can I use the calculator for arc length?
On Part A, yes, after writing the integral. On Part B, no calculator is allowed, and the integrand is chosen to be workable by hand.
How do I find where a particle is at rest in the plane?
Both components of velocity must be zero at the same time.
Where can I find the official scoring rules?
Exam formats change a little from year to year, so check the current course and exam description on the College Board site before test day.
Keep practicing
Every lesson in the AP Calculus courses ends with practice problems and an Exam focus box that shows how the topic is tested, and every unit has a 10-problem practice set with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Calculus BC course Free lessons and lesson list
More AP Calculus BC guides (1)
Free-response and writing
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