AP Calculus BC / Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.7 Defining Polar Coordinates and Differentiating in Polar Form
Notes
A point $(r,\theta)$ has $x=r\cos\theta$ and $y=r\sin\theta$.
$$\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin\theta+r\cos\theta}{\frac{dr}{d\theta}\cos\theta-r\sin\theta}$$
- $\displaystyle \frac{dr}{d\theta}$ tells whether points move toward or away from the origin.
Practice
The polar point $\displaystyle (2,\frac{\pi}{2})$ in rectangular form is:
$\displaystyle x=2\cos\frac{\pi}{2}=0$, $y=2$.
For $r=3$, the curve is:
Every point is 3 from the origin.
If $\displaystyle \frac{dr}{d\theta}>0$ and $r>0$, the curve is moving:
$r$ is increasing.
Ready for more? Take the Unit 9 test.