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Chapter 6: Problem 51

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 graph of the given parametric equations will form a Prolate cycloid. The graph represents a curve shaped like the arch of a rounded object, with one end significantly wider than the other. It loops over and over, repeating this pattern. The shape of the curve is primarily determined by the sinusoidal transformations in the equations.

Step by step solution

01

- Identify the Functions

First, you should identify the parametric equations given. In this problem, the parametric equations are given by \(x = \theta - \frac{3}{2} \sin \theta\) and \(y = 1 - \frac{3}{2} \cos \theta\).
02

- Use a Graphing Utility

To plot parametric equations, use a graphing utility. Parametric equation graphing is not straightforward as function plotting. You need to input the equations of both variables 'x' and 'y' depending on the parameter '胃'. Set the graphing utility to parametric mode to do this.
03

- Plotting and Understanding the Curve

Plot the equations. Make sure to have multiple rounds of '胃' for observing the cycloid curve. This will allow the periodic nature of the curve to be seen. The periodicity comes from the use of the sinusoidal functions of (sin 胃 and cos 胃). These kind of curves are produced from a fixed point on a circle rolling along a straight line.

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