Problem 56 - Precalculus with Limits

Question

In Exercises 49-56, use a graphing utility to graph the curve represented by the parametric equations. Folium of Descartes: \(\quad x= \dfrac{3t}{1+t^3}, \quad y= \dfrac{3t^2}{1+t^3}\)

Step-by-Step Solution

Answer

The graph of Folium of Descartes can be obtained by substituting various values of parameter \( t \) into the given parametric equations to generate corresponding (x, y) coordinate points, and plotting these points on a graph using a graphing utility. Connect the points with a smooth curve to complete the graph.

Step 1: Understand the Parametric Equations

The given parametric equations are \( x = \dfrac{3t}{1+t^3} \) and \( y = \dfrac{3t^2}{1+t^3} \). These equations represent the coordinate points (x, y) on the curve for various values of the parameter \( t \).

Step 2: Select Values for Parameter \( t \)

Before graphing, choose several values for the parameter \( t \). For this particular curve, it will be most informative to choose both positive and negative values, including \( t = 0 \), and a few values close to 1 and -1 where the curve has interesting features.

Step 3: Calculate Corresponding Coordinate Points

Substitute each selected value of \( t \) into both parametric equations to get the corresponding (x, y) coordinates for each point. Work carefully to avoid calculation errors, especially in the denominator \( 1+t^3 \) where small changes in \( t \) can lead to large changes in the coordinate points.

Step 4: Plot the Points and Draw the Curve

After having prepared the series of points consisting of various (x, y) pairs, plot these points on a graphing utility. Join these points with a smooth curve to fully represent the Folium of Descartes.

Key Concepts

Folium of Descartes

The Folium of Descartes is a classic example of a curve described by parametric equations. Its unique form captures the intersection between algebraic curves and parametric equations in a fascinating way. Named after the French philosopher and mathematician René Descartes, the folium is represented by specific parametric equations, where both x and y depend on a third variable, the parameter \( t \).

For the Folium of Descartes, the equations are given by:

• \( x = \frac{3t}{1+t^3} \)
• \( y = \frac{3t^2}{1+t^3} \)

These equations describe a looped curve in the xy-plane, noted for its distinct looping intersection. By varying \( t \), you can trace out the entire shape of the folium, seeing how the coordinates change as \( t \) shifts from one value to another. This provides a complex visualization that can't be fully captured by static Cartesian equations.

Graphing Utility

Using a graphing utility is an excellent way to visualize the intricate shapes formed by parametric equations like the Folium of Descartes. These tools allow us to go beyond manual plotting, offering an efficient method to see complex curves instantly.

To graph parametric equations:

• Input the equations for \( x \) and \( y \) into the utility.
• Set a range for the parameter \( t \). For the folium, including values from negative to positive is crucial, as this range captures the full extent of the curve's features.
• Adjustments in the range of \( t \) can highlight different portions of the curve, offering deeper insight into its shape and behavior.

Graphing utilities do more than draw curves—they provide an interactive way to understand the relationships between variables. Watching how the graph develops as \( t \) varies is both enlightening and visually engaging, solidifying comprehension.

Coordinate Points

In the study of parametric equations, coordinate points serve as the building blocks for visualizing curves like the Folium of Descartes. Each point on the curve corresponds to a specific set of parametric values, giving us a clear picture of the curve's progression.

To find these points:

• Choose a value for the parameter \( t \).
• Substitute this \( t \) value into the parametric equations for \( x \) and \( y \).
• Calculate the resulting \((x, y)\) coordinate pair.

It is important to carefully select \( t \) values, ensuring both negative and positive inputs to capture all characteristics of the curve. Each calculated point adds to the connectivity of the curve when these points are plotted and joined smoothly on a graph. This detailed method of analyzing coordinate points emphasizes how individual changes in \( t \) reflect geometrically across the curve.