Parametric, polar and vector functions (BC) practice questions (AP Calculus)

Sample questions

All 3 free AP Calculus questions on this unit, from the free AP Calculus BC Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 12 marks · Parametric & vector motion
   A curve is given parametrically by $x(t) = t^2$ and $y(t) = \dfrac{2}{3}t^3$. The length of the curve from $t = 0$ to $t = \sqrt{3}$ is
   • A.
     $\dfrac{7}{3}$
   • B.
     $\dfrac{14}{3}$
   • C.
     $\dfrac{16}{3}$
   • D.
     $\dfrac{28}{3}$
   Show answerFrom the free AP Calculus BC Practice Exam →
2. Question 22 marks · Parametric & vector motion
   A particle moves in the $xy$-plane with velocity vector $\langle 2t,\ 3t^2 \rangle$ for $t \geq 0$. At time $t = 1$ the particle is at the point $(2, 5)$. Its position at time $t = 2$ is
   • A.
     $(3, 7)$
   • B.
     $(4, 8)$
   • C.
     $(5, 12)$
   • D.
     $(6, 13)$
   Show answerFrom the free AP Calculus BC Practice Exam →
3. Question 39 marks · Polar curves
   Polar curves. For $0 \leq \theta \leq \pi$ the polar curve $r = 4\sin\theta$ is the circle of radius $2$ centered at the point $(0, 2)$, and the polar curve $r = 2$ is the circle of radius $2$ centered at the origin. The two circles cross at two points above the $x$-axis. Let $R$ be the region that lies inside the circle $r = 4\sin\theta$ and outside the circle $r = 2$.
   (a)

   Find all values of $\theta$ in $[0, \pi]$ at which the two curves intersect.

   [2]
   (b)

   Find the area of $R$.

   [3]
   (c)

   Find the slope of the line tangent to the curve $r = 4\sin\theta$ at the point where $\theta = \dfrac{\pi}{6}$.

   [2]
   (d)

   A particle moves along the curve $r = 4\sin\theta$ so that $\dfrac{d\theta}{dt} = 3$ for all $t$. Find the rate of change of the particle's distance from the origin with respect to time at the instant when $\theta = \dfrac{\pi}{3}$.

   [2]
   Show answerFrom the free AP Calculus BC Practice Exam →

Key terms

The definitions and formulas AP Calculus expects for parametric, polar and vector functions (bc), typeset as they appear on the paper.

Derivative of a parametric curve

The slope of a curve given by $x(t)$ and $y(t)$ is the ratio of the two derivatives with respect to $t$. The tangent is horizontal where $\frac{dy}{dt} = 0$ (and $\frac{dx}{dt} \ne 0$), vertical where $\frac{dx}{dt} = 0$.

$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$

Second derivative of a parametric curve

Differentiate $\frac{dy}{dx}$ with respect to $t$, then divide by $\frac{dx}{dt}$ again. It is not the ratio of second derivatives.

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

Speed and arc length of a parametric curve

Speed is the magnitude of the velocity vector; the distance travelled (arc length) is its integral over the time interval.

$$\text{speed} = \sqrt{x'(t)^2 + y'(t)^2}, \qquad L = \int_a^b \sqrt{x'(t)^2 + y'(t)^2}\,dt$$

Vector-valued functions and planar motion

Position $\langle x(t), y(t) \rangle$, velocity as its componentwise derivative, acceleration as the derivative of velocity. Position at a later time is the initial position plus the integral of each velocity component.

$$\mathbf{v}(t) = \langle x'(t), y'(t) \rangle, \qquad \mathbf{r}(b) = \mathbf{r}(a) + \int_a^b \mathbf{v}(t)\,dt$$

Polar coordinates

A point given by distance $r$ from the origin and angle $\theta$. Converting to Cartesian coordinates lets you find $\frac{dy}{dx}$ for a polar curve by treating it as parametric in $\theta$.

$$x = r\cos\theta, \qquad y = r\sin\theta, \qquad \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}$$

Area in polar coordinates

The area swept by a polar curve between two angles. For the area between two curves, subtract the squares of the radii inside one integral, over the angles where the curves bound the region.

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

Frequently asked questions

Other AP Calculus topics

• Limits and continuity
• Differentiation: definition and basic rules
• Composite, implicit and inverse functions
• Contextual applications of differentiation
• Analytical applications of differentiation
• Integration and accumulation of change
• Differential equations
• Applications of integration
• Infinite sequences and series (BC)

All AP Calculus topics · AP Calculus exam guide · All free practice exams