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Chapter 6: Problem 81

Use a graphing utility to graph each butterfly curve. Experiment with the range setting, particularly \(\theta\) step, to produce a butterfly of the best possible quality. $$r=\sin ^{5} \theta+8 \sin \theta \cos ^{3} \theta$$

Short Answer

Expert verified
The graph of the function is a butterfly curve. The exact depiction depends on the chosen \(\theta\) step size. Smaller step size will generally yield a higher resolution graph, creating a more detailed butterfly.

Step by step solution

01

- Choose an appropriate range for \(\theta\)

Normally, for such graphs, we take \(\theta\) from 0 to 2\(\pi\). This is because 2\(\pi\) is the equivalent of 360 degrees, covering the full circle about the origin.
02

- Graph the function

Use a graphing utility (like Desmos or a graphing calculator) to plot the function. Input \(r(\theta)= \sin ^{5} \theta+8 \sin \theta \cos ^{3} \theta \) for the range of \(\theta\) as selected.
03

- Tweak \(\theta\) step

Experiment with the step of \(\theta\) to adjust the granularity of your graph. A smaller step (like 0.01) results in a more precise but computation-intensive graph. A larger step (like 0.1) leads to a more general shape with less calculation.
04

- Inspect and refine graph

Upon graphing, you should see a butterfly curve. Angles are taken in the counter-clockwise direction starting from the positive x-axis. If the graph appears to skip or miss points, decrease the step size. If the graph is taking too long to generate, increase the step size. Adjust until a good blend of detail and performance is achieved.

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