Help understanding the velocity of polar curves.

Question

I have been studying for the AP BC Calculus exam (see this previous question) and most of the questions that deal with the first derivative in polar coordinates say that if ${dr\over d\theta}<0$ and $r>0$, then the graph (in polar coordinates) is moving closer to the origin.

What about $r = 4-\theta$, which has $\frac{\mathrm{d}r}{\mathrm{d}\theta} = -1$?

Here is the graph:

Does this disprove the statement?

Answer 1

When the derivative is negative, that means $r$ is decreasing. While $r$ is positive, that does mean that the graph is approaching the origin, but when $r$ is negative, it will continue to decrease, making the distance from the origin increase.

Answer 2

Maybe this will help:

The solid portion (of each) is when $\theta>0$ while the dashed is $\theta<0$.

So, you can see that the example you gave indeed has the property that when $r>0$ and ${dr\over d\theta}<0$, that the polar plot is tending toward the origin.