AP Precalculus / Unit 3: Trigonometric and Polar Functions
3.15 Rates of Change in Polar Functions
Notes
For $r=f(\theta)$, the average rate of change of $r$ on $[\alpha,\beta]$ is $\displaystyle \frac{f(\beta)-f(\alpha)}{\beta-\alpha}$.
- If $r>0$ and increasing, points move away from the origin. If $r>0$ and decreasing, they move toward it.
Practice
$r=\theta$. Average rate of change of $r$ on $[0,\pi]$:
$\displaystyle \frac{\pi-0}{\pi}$.
$r$ is positive and decreasing on an interval. The graph is moving:
Distance from the origin shrinks.
$r=2\cos\theta$ on $\displaystyle \left[0,\frac{\pi}{2}\right]$: as $\theta$ increases, the distance from the origin:
$2\cos\theta$ goes from 2 to 0.
Ready for more? Take the Unit 3 test.