Problem 48 - Essential Calculus

Question

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Folium of Descartes: } x=\frac{3 t}{1+t^{3}}, \quad y=\frac{3 t^{2}}{1+t^{3}} $$

Step-by-Step Solution

Answer

The curve from the parametric equations is the Folium of Descartes. The direction of the curve is indicated by an arrow on the graph, starting from the minimum of \(t\) and moving to the maximum of \(t\). The point \((0,0)\) is the place where the curve is not smooth as the derivative does not exist when \(t = -1\).

Step 1: Understand the Parametric Equations

The given parametric equations are \(x=\frac{3t}{1+t^{3}}\) and \(y=\frac{3t^{2}}{1+t^{3}}\). In these equations, \(t\) is the parameter. By adjusting the value of \(t\), the corresponding \(x\) and \(y\) values and hence the points can be graphed.

Step 2: Graph the Equations

Use a graphing utility to input the equations and set the parameter \(t\) from a reasonable range, for example, \(-10 \leq t \leq 10\). The folium of Descartes should be visibly plotted on the graphing utility.

Step 3: Analyze the Graph

It's important to note the direction of the graph, which is usually indicated by an arrow. The direction shown by the arrow on the graph is determined by increasing \(t\), so it starts where \(t\) is minimum and moves towards where \(t\) is maximum.

Step 4: Identify Non-Smooth Points

The curve is not smooth where there is a 'loop'. In this case, the inflection point at \(0,0\) is where the folium cannot be smoothly driven because the derivative does not exist there when \(t= -1\).

Key Concepts

Graphing Utility

A graphing utility is an invaluable tool for visualizing parametric equations. It allows us to input equations, such as those of the Folium of Descartes, and see their graphical representations on a coordinate plane. When using a graphing utility, you have control over the range of the parameter. For example, with the given parametric equations \(x=\frac{3t}{1+t^{3}}\) and \(y=\frac{3t^{2}}{1+t^{3}}\), setting \(-10 \leq t \leq 10\) is a reasonable choice.
This range helps to plot enough points to see the curve's full structure without missing details. The utility also offers features like zoom and pan to examine specific parts of the graph. By experimenting with different values, students can gain a better understanding of how changing \(t\) influences the curve.

Direction of Curve

The direction of a parametric curve is an essential aspect when analyzing its behavior. Typically, when using a graphing utility, the direction is indicated by arrows that show the path from a starting to an ending point.
In the parametric equations provided, as \(t\) increases from negative to positive values, the curve is traced in a particular direction. This means the graph starts plotting from the smallest value of \(t\), moving smoothly to the largest value.
The direction helps us understand the sequence in which points have been plotted using the parameter, adding to our understanding of the curve's properties.

Non-Smooth Points

Non-smooth points are places on a curve where the graph is not differentiable, leading to changes in direction or undefined slopes. For the Folium of Descartes, these occur at critical points along the graph.
Specifically, at the origin \((0,0)\), the curve is not smooth. The reason is that the curve has a loop at this point due to the parameter's behavior and derivative properties.
Furthermore, for \(t = -1\), the derivative is not properly defined, which confirms the non-smooth nature. Recognizing these points is crucial for fully understanding a curve's complexity.

Folium of Descartes

The Folium of Descartes is a famous algebraic curve defined by specific parametric equations. These equations \(x=\frac{3t}{1+t^{3}}\) and \(y=\frac{3t^{2}}{1+t^{3}}\) describe a leaf-shaped curve with interesting characteristics.
Known for its loop, the Folium of Descartes is often used in examples involving singular points or infinite cusps in calculus and algebra. The curve is not only a challenge to plot but also a rich source of mathematical phenomena, like its infinity cusp at the origin.
The Folium showcases the beauty and intricacies of parametric equations, making them a key topic for students to explore in advanced math courses.