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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:

wolfram alpha

Does this disprove the statement?

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It does not, since $r$ is not positive in this case. – Gyu Eun Lee Jan 9 '13 at 2:04
@anonymous how do I get wolfram alpha to graph the function for when the absolute value of theta is less than 4? – yiyi Jan 9 '13 at 2:14
Sorry, I'm no expert on working with Wolfram Alpha. – Gyu Eun Lee Jan 9 '13 at 3:46
up vote 1 down vote accepted

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.

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Ok, I am getting the issue, how can I break the graph up in the problem to show when abs(4-theta) >0 – yiyi Jan 9 '13 at 2:16
@MaoYiyi well $|4-\theta|\ge 0$ always... I'm not quite sure what you're asking. In the graph, $r$ is positive from $\theta\in [0,4)$, and is negative when $\theta\in(4,3\pi]$. In the graph above, you can find that when $\theta=4$, it hits the origin. – angryavian Jan 9 '13 at 2:21
I am trying to get wolfram alpha just to show the positive and negative sections in two different graphs, but I just get solutions to the equations. – yiyi Jan 9 '13 at 2:24

Maybe this will help:

Mathematica graphics

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.

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how did you make the dash part? – yiyi Jan 13 '13 at 8:32
@MaoYiyi: In Mathematica. I plotted $r=4-\theta$ for $0<\theta<4$ (solid), then again for $4<\theta<3\pi$ (dashed) and laid the two plots on top of one another. – JohnD Jan 13 '13 at 19:15
@MaoYiyi: Is there anything else needed in order to accept the answer? – JohnD Feb 1 '13 at 3:58

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