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.
The graph of )$ for is even more fun: the graph starts at an endpoint maximum at (1,0), next goes to a minimum at (–1, π/3), then a maximum at (1, 2π/3), and a minimum at (–1, π). But this is the starting point that was a maximum. Continuing around the graph, (1, 4π/3) a maximum at the same point as the previous minimum (–1, π/3), then a minimum at (–1, 5π/3), previously a minimum, then returning to (1, 2π) = (–1, π) = (1, 0) which was previously both a maximum and a minimum, and is now a maximum again!
Thus points may be both maximums and minimums.
On a tangent…
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
Try some others yourself. Don’t restrict yourself to sines and cosines. for below at different ranges.