Problem 49 Use a graphing utility to graph ... [FREE SOLUTION]

Chapter 6: Problem 49

Use a graphing utility to graph the curve represented by the parametric equations. Cycloid: \(x=4(\theta-\sin \theta), y=4(1-\cos \theta)\)

Short Answer

Expert verified

Once the parametric equations \(x=4(\theta -\sin \theta)\) and \(y=4(1-\cos \theta)\) are entered onto a graphing utility, and the parameter range for \(\theta\) is set from \(0\) to \(2\pi\), the software will produce a graphic representing a cycloid curve.

Step by step solution

Understanding Parametric Equations and Recognize the Type of Curve

A pair of parametric equations assigns different values to the parameter \(\theta\). As \(\theta\) varies, these values specify points in the plane, whose coordinates are given by the equations. The cycloid is a specific curve generated by a point on the rim of a circular wheel as the wheel rolls along without slippage a straight line. Here, the equations \(x=4(\theta -\sin \theta)\) and \(y=4(1-\cos \theta)\) dictate the curve's path. Note the phase shifts in the trigonometric functions. It's these shifts that give us our cycloid pattern.

Enter the Parametric Equations into the Graphing Utility

Start the graphing utility, such as Desmos or GeoGebra. Arrange the screen so both the input space and the graph are visible. For the x-coordinate, enter the equation \(x=4(\theta -\sin \theta)\), replacing \(\theta\) with \(t\) or another variable if your software requires it. Do the same for the y-coordinate, entering the equation \(y=4(1-\cos \theta)\). The graphing software internals handles time increments to draw the curve.

Adjust the Parameter Range and Visual Inspection

Some utilities start the parameter at \(0\) and end at \(1\) by default. For a complete cycle of a cycloid, it is required to adjust this range. For standard cycloids, the parameter \(\theta\) usually runs from \(0\) to \(2\pi\), which corresponds to one complete revolution or cycle of the circle. Make sure you adjust this range in your graphing tool. Once the curve is drawn, visually inspect it to make sure it matches the standard shape of a cycloid (the appearance of arches or loops that open upward).

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91影视

Use a graphing utility to graph the curve represented by the parametric equations. Cycloid: \(x=4(\theta-\sin \theta), y=4(1-\cos \theta)\)

Short Answer

Step by step solution

Understanding Parametric Equations and Recognize the Type of Curve

Enter the Parametric Equations into the Graphing Utility

Adjust the Parameter Range and Visual Inspection

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