AP®MathsCollege BoardCalculus BCRevision NotesUnit 9: Parametric Equations, Vector-Valued Functions & Polar CoordinatesUnit 9 OverviewUnit 9 Summary

Unit 9 Summary (College Board AP® Calculus BC): Revision Note

Written by: Roger B

Reviewed by: Jamie Wood

Updated on 19 May 2026

Parametric equations, vector-valued functions & polar coordinates summary

Key definitions

Parametric equations define a curve using a third variable
- $x = f (t)$
- $y = g (t)$
A vector-valued function is a pair of functions that are grouped together using vector notation
- $r (t) = <x (t), y (t)>$
Polar coordinates $(r, θ)$ 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 $\frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}}$
The second derivative of a parametric curve is given by $\frac{d^{2} y}{d x^{2}} = \frac{\frac{d}{d t} (\frac{d y}{d x})}{\frac{d x}{d t}}$
The arc length of a parametric curve from $t = t_{1}$ to $t = t_{2}$ is $\int_{t_{1}}^{t_{2}} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t$
If $(x (t), y (t))$ are the coordinates of a particle, then:
- $\sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}}$ is the speed
- $\int_{t_{1}}^{t_{2}} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t$ is the distance traveled
The relationships between Cartesian coordinates $(x, y)$ and polar coordinates $(r, θ)$ are:
- $r^{2} = x^{2} + y^{2}$
- $\tan θ = \frac{y}{x}$
- $x = r \cos θ$
- $y = r \sin θ$
The derivative of a polar curve $r = f (θ)$ can be found by:
- $\frac{d x}{d θ} = \frac{d r}{d θ} \cos θ - r \sin θ$
- $\frac{d y}{d θ} = \frac{d r}{d θ} \sin θ + r \cos θ$
- $\frac{d y}{d x} = \frac{\frac{d r}{d θ} \sin θ + r \cos θ}{\frac{d r}{d θ} \cos θ - r \sin θ}$
The second derivative of a polar curve $r = f (θ)$ can be found by:
- $\frac{d^{2} y}{d x^{2}} = \frac{\frac{d}{d θ} (\frac{d y}{d x})}{\frac{d x}{d θ}}$
The area bounded by the polar curve $r = f (θ)$ and the straight lines $θ = α$ and $θ = β$ is:
- $\frac{1}{2} \int_{α}^{β} r^{2} d θ$

Key facts

A parametric curve intersects the $x$ -axis when $y = g (t) = 0$
A parametric curve intersects the $y$ -axis when $x = f (t) = 0$
If $\frac{d y}{d t} = 0$ and $\frac{d x}{d t} \neq 0$ at a point, then the tangent line is horizontal
If $\frac{d x}{d t} = 0$ and $\frac{d y}{d t} \neq 0$ at a point, then the tangent line is vertical
Vector-valued functions can be differentiated and integrated by differentiating or integrating the components separately

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