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Chapter 6: Problem 55

Use a graphing utility to graph the curve represented by the parametric equations. Witch of Agnesi: \(x=2 \cot \theta, y=2 \sin ^{2} \theta\)

Short Answer

Expert verified
The Witch of Agnesi can be drawn using parametric equations and a graphing utility. It involves inputting the given equations \(x=2 \cot \theta\), \(y=2 \sin ^{2} \theta\) into the tool and adjusting the \( \theta \) range. The drawn curve will mimic a witch's hat shape.

Step by step solution

01

Understand Parameters

Parametric equations use a parameter, in this case \( \theta \), to define a curve. For each value of \( \theta \), there is a corresponding point (x, y) that lies on the curve represented by the equations. Here, \( x = 2 \cot \theta \) and \( y = 2 \sin^2 \theta \).
02

Make use of a Graphing Tool

A graphing utility is the best tool to use for graphing. A suitable one can be found online or a scientific calculator. The key here is that the application must support parametric equations. Input the equations into the graphing utility. For x, put in \( 2 \cot \theta \) and for y, put in \( 2 \sin^2 \theta \) into the graphing tool.
03

Adjust the Range of \( \theta \)

To ensure that the full curve is displayed, adjust the range of \( \theta \) in the graphing tool. As \( \theta \) takes values from \( 0 \) to \( \pi \), the curve will be graphed in the negative and positive x-axis for all values under the x-axis. It is possible to set the range from \( 0 \) to \( 2\pi \), but this will duplicate the graph and possibly distort the results.
04

Display the Curve

After correctly inserting the equations and adjusting the range of \( \theta \), draw the curve by executing the graph function. The Witch of Agnesi curve will be displayed.

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