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Chapter 6: Problem 86

Use a graphing utility to graph the curve represented by the parametric equations. Folium of Descartes: \(x=\frac{3 t}{1+t^{3}}, y=\frac{3 t^{2}}{1+t^{3}}\)

Short Answer

Expert verified
The plotted graph from the parametric equations reveals a curve with a single loop, identifiable as the 'Folium of Descartes'.

Step by step solution

01

Understand the given parametric equations

The given equations are \(x=\frac{3 t}{1+t^{3}}\) and \(y=\frac{3 t^{2}}{1+t^{3}}\). Here, t represents a parameter that controls the movement over the curve.
02

Plug in values for t

In parametric equations, we replace t with a series of values over a specific range to generate different points (x, y). Since the value of t is not specified in the question, we generally start with the range from -10 to 10 to capture most detail of the curve. However, the range can be adjusted based on the specifics of the graph using the graphing utility.
03

Graph the points

Using the graphing utility, these (x,y) pairs are plotted, and the points are connected to form a curve. The order of connection follows the sequence of values allotted to t, which might show any loops in the curve.
04

Identify the curve

From the diagram displayed on the graphing utility, you should be able to identify the curve as a 'Folium of Descartes'. The curve is a folium (so-called because its shape somewhat resembles a leaf) and has a single loop.

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