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Chapter 6: Problem 84

Use a graphing utility to graph the curve represented by the parametric equations. Curtate cycloid: \(x=8 \theta-4 \sin \theta, y=8-4 \cos \theta\)

Short Answer

Expert verified
The graph for the parametric equations \(x=8 \theta-4 \sin \theta, y=8-4 \cos \theta\) is a curtate cycloid.

Step by step solution

01

- Understanding the task

The task requires us to use a graphing utility to graph a curtate cycloid, which is represented by the given parametric equations: \(x=8 \theta-4 \sin \theta\) and \(y=8-4 \cos \theta\) . We need to ensure we input these equations correctly to get the accurate graph.
02

- Entering the equations into a graphing utility

A graphing utility, whether it's an online tool or a software application, is a tool to assist in visualizing mathematical equations on a plane. In the graphing utility, input \(x=8 \theta-4 \sin \theta\) as your x-coordinate function and \(y=8-4 \cos \theta\) as your y-coordinate function.
03

- Setting the parameter range

A cycloid (curtate or otherwise) is generated by a rolling wheel. The parameter \(\theta\) generally ranges from 0 to \(2\pi\), as it represents a full revolution of the wheel. Therefore, make sure the parameter range is set to 0 to \(2\pi\) in your graphing tool.
04

- Observing the plot

The graph as per the given parametric equations represents a curtate cycloid. It will be a cycloid plot but 'curtate' because it falls within the diameter of a circle. The points traced out correspond to a point on the wheel, not on the edge, giving us shorter or 'curtate' cycloid.

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