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. $$ \text { Folium of Descartes: } x=\frac{3 t}{1+t^{3}}, \quad y=\frac{3 t^{2}}{1+t^{3}} $$

Short 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-by-step solution

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.

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.

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.

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 \(t= -1\).