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Chapter 6: Problem 79

Use a graphing utility to graph the curve represented by the parametric equations. Prolate cycloid: \(x=\theta-\frac{3}{2} \sin \theta, y=1-\frac{3}{2} \cos \theta\)

Short Answer

Expert verified
The curve represented by the parametric equations \(x=\theta-\frac{3}{2} \sin \theta\) and \(y=1-\frac{3}{2} \cos \theta\) is a prolate cycloid. It displays a stretched, looping form when graphed with a utility tool.

Step by step solution

01

Understanding the Parametric Equations

The given parametric equations are \(x=\theta-\frac{3}{2} \sin \theta\) and \(y=1-\frac{3}{2} \cos \theta\). These represent the x and y coordinates of each point on the curve in terms of the parameter \(\theta\). We could notice that \(\theta\) is the variable for both equations and sine and cosine functions are used, which means this curve will likely have a periodic nature.
02

Use a Graphing Tool

Input these equations into a graphing tool. Most graphing utilities would allow parametric equations to be entered. Set the range of \(\theta\) that completely shows the curve's pattern. One complete period of sine and cosine is \(2\pi\), but we can extend the range to see how the cycloid is 'prolate' or stretched.
03

Interpreting the Graph

After entering the equations and creating the graph, we can see the shape of the curve. A prolate cycloid has a elongated, looping form due to the modifications we have from the standard cycloid equation.

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