Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

By inputting the provided parametric equations into a graphing utility and generating the graph, a representation of the folium of Descartes can be visualized.

Step 1: Understanding parametric equations

Parametric equations are a pair of functions, in this case \(x=\frac{3 t}{1+t^{3}}\) and \(y=\frac{3 t^{2}}{1+t^{3}}\), that define the x and y coordinates in terms of a third variable, often time (t). As t varies, these equations generate a curve in the xy-plane.

Step 2: Input parametric equations into the graphing utility

Enter the parametric equations into the graphing utility. Most graphing tools have a specific option for parametric equations, where you can enter the x and y components separately. Make sure to use a wide range of t-values to generate a clear graph of the curve.

Step 3: Generate the graph

After inputting the parametric equations, generate the graph. You should notice a unique figure, which is a characteristic of the folium of Descartes. Analyze and interpret the graph generated.