Looking at the graphs of polar curves can be quite fascinating. Doing “calculus” kinds of things is different, yet the same as in Cartesian coordinates. A discussion on the AP Calculus Community here got me thinking about extreme values of polar functions.

The terms maximum and minimum here refers to the value of *r*(θ) which may be positive (when on the ray making an angle of θ with the polar axis), zero (at the pole), or negative (on the ray opposite to the one making an angle of θ with the polar axis). The distance from the pole to the point is |*r*(θ)|. As Mark Howell pointed out in the thread linked above, extreme values of *r*(θ) lie on a circle or circles centered at the pole with radius of |*r*(θ)|, and finding the slope of tangent lines at the extremes is relatively easy, requiring no calculus: the slope of the ray is *y/x*, so the slope of the tangent at the extreme is –x/y. As with Cartesian coordinates, at extreme values since *r*(θ) is change from increasing to decreasing at this point (or vice versa).

A quick look at the graph of simple polar functions seems to show obvious maximum values for *r*(θ), but a closer look reveals some complications.

The graph of for , shown below, appears to show 4 *maximums*. However if we trace the graph, we find that these points are (1, π/4), (–1, 3π/4), (1, 5π/4) and (–1, 7π/4). Two of the values are maximums where *r*(θ) = 1 and two are minimums where *r*(θ) = –1.

Thus points may be both maximums *and* minimums.

Polar Curves can be really fun. While working on the idea above, I explored some other curves. Try some yourself using Desmos, GeoGabra, or another graphing app with sliders. Shown below are members of the family of polar curves . The domain is extended to . Notice:

- How very slight changes in the parameters give very different looking graphs
- Other values give far less “organized” curves
- In the third graph, the maximums and minimums on the irregular part of the curve closest to the pole

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Thanks for the information. You make it look like really easy.

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Lin, et al.-

I did a presentation on polar coordinates in calc and precalc at the 2017 NCTM Annual Meeting San Antonio. One of the more widely appreciated features was using TI-84 dynamic graphing to show wether or not places where two polar graphs cross are really intersections, and under what conditions the crossing points are not intersections. Email me (foerster@idworld.net) and I’ll fill you in on the details.

Paul

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