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Chapter 6: Problem 80

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. $$r=\sin ^{4} 4 \theta+\cos 3 \theta$$

Short Answer

Expert verified
By following these steps, a proper graphing of the butterfly curve using the given equation should be obtained. Remember that the range of \(\theta\) should usually be set from 0 to 2\(\pi\) to get the full butterfly shape, and adjust the step to improve the quality of the graph.

Step by step solution

01

Understand Polar Coordinates and the Equation

In this equation, \(r=\sin ^{4} 4 \theta+\cos 3 \theta\), \(r\) represents the distance from the origin (center of the graph), and \(\theta\) represents the angle in radians, counter-clockwise from the positive x-axis. This equation will generate points that form the butterfly shape.
02

Launch Graphing Utility

Launch your preferred graphing utility and input the equation in the polar coordinates section.
03

Adjust Range Settings

Range settings often control the extent of the angle \(\theta\). For a complete butterfly, the range should generally be from 0 to 2\(\pi\) because this allows the function to trace a full circle.
04

Adjust the theta Step

The \(\theta\) step is crucial in determining the quality of the graph. The smaller the step, the smoother the curve, but it might take longer to calculate. Experiment with the step until you achieve a quality rendering of the butterfly curve.
05

Observe and Analyze the Butterfly

Noting the displayed graph, understand how changes in the value of \(\theta\) affect the value of \(r\), and how this relation results in the butterfly shape. Also, see the impact of step changes on graph quality.

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