Unit 9 Summary (College Board AP® Calculus BC): Study Guide

Parametric equations, vector-valued functions & polar coordinates summary

Key definitions

• Parametric equations define a curve using a third variable

  • $x=f\left(t\right)$

  • $y=g\left(t\right)$

• A vector-valued function is a pair of functions that are grouped together using vector notation

  • $\mathit{r}\left(t\right)=<x\left(t\right),y\left(t\right)>$

• Polar coordinates $\left(r,\theta \right)$describe the position of a point by stating:

  • the distance from the origin to the point

  • the angle from the initial line to the point

Key formulas

• The derivative of a parametric curve is given by $\frac{dy}{dx}=\frac{{\displaystyle \raisebox{1ex}{$dy$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.}}{{\displaystyle \raisebox{1ex}{$dx$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.}}$

• The second derivative of a parametric curve is given by $\frac{d^{2}y}{d{x}^2}=\frac{{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{dy}{dx}}\right)}{{\displaystyle \frac{dx}{dt}}}$

• The arc length of a parametric curve from $t=t_1$ to $t=t_2$ is ${\int }_{t_1}^{t_2}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$

• If $\left(x\left(t\right),y\left(t\right)\right)$ are the coordinates of a particle, then:

  • $\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}$ is the speed

  • ${\int }_{t_1}^{t_2}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$ is the distance traveled

• The relationships between Cartesian coordinates $\left(x,y\right)$ and polar coordinates $\left(r,\theta \right)$ are:

  • $r^2=x^2+y^2$

  • $\tan \theta =\frac{y}{x}$

  • $x=r\cos \theta$

  • $y=r\sin \theta$

• The derivative of a polar curve $r=f\left(\theta \right)$ can be found by:

  • $\frac{dx}{d\theta }=\frac{dr}{d\theta }\cos \theta -r\sin \theta$

  • $\frac{dy}{d\theta }=\frac{dr}{d\theta }\sin \theta +r\cos \theta$

  • $\frac{dy}{dx}=\frac{{\displaystyle \frac{dr}{d\theta }}\sin \theta +r\cos \theta }{\frac{dr}{d\theta }\cos \theta -r\sin \theta }$

• The second derivative of a polar curve $r=f\left(\theta \right)$ can be found by:

  • $\frac{d^{2}y}{d{x}^2}=\frac{{\displaystyle \frac{d}{d\theta }}\left({\displaystyle \frac{dy}{dx}}\right)}{{\displaystyle \frac{dx}{d\theta }}}$

• The area bounded by the polar curve $r=f\left(\theta \right)$ and the straight lines $\theta =\alpha$ and $\theta =\beta$ is:

  • $\frac{1}{2}{\int }_{\alpha }^{\beta }{r}^{2}d\theta$

Key facts

• A parametric curve intersects the $x$-axis when $y=g\left(t\right)=0$

• A parametric curve intersects the $y$-axis when $x=f\left(t\right)=0$

• If $\frac{dy}{dt}=0$ and $\frac{dx}{dt}\ne 0$ at a point, then the tangent line is horizontal

• If $\frac{dx}{dt}=0$ and $\frac{dy}{dt}\ne 0$ at a point, then the tangent line is vertical

• Vector-valued functions can be differentiated and integrated by differentiating or integrating the components separately