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Chapter 10: Problem 51

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. $$\text { Folium of Descartes } x=\frac{3 t}{1+t^{3}}, y=\frac{3 t^{2}}{1+t^{3}}$$

Short Answer

Expert verified
Answer: The Folium of Descartes has a single loop with a cusp at the point (3, 3) and has asymptotes along the positive x and positive y-axes when graphed using the parametric equations and interval given.

Step by step solution

01

Analyze the Parametric Equations

First, take a look at the parametric equations given for the Folium of Descartes. Notice that as t approaches infinity, both x and y approach 0. This means the curve will approach the x and y axes but never actually touch them. The curve has also a point with both coordinates having the same value, which happens when the denominator equals 1 (giving \(x = y = 3\)). This point happens exactly when \(t = 1\).
02

Determine an Appropriate Interval for the Parameter t

Based on our observation in step 1, the curve clearly has some interesting features near the point \(t=1\). Since we want to capture the full Folium of Descartes, it is reasonable to consider a range of t-values around (but not equal to) 1. We need an interval that goes from a negative value to a positive value, covering a sufficient range of t-values. One such interval could be \(t \in [-2, 2]\) excluding 0.
03

Graph the Parametric Equations

Now that we have determined the interval \(t \in [-2, 2]\) excluding 0, use a graphing utility to input the parametric equations and plot the curve: $$x = \frac{3t}{1 + t^3}$$ $$y = \frac{3t^2}{1 + t^3}$$ Make sure to set the t values to be within the interval [-2, 2] excluding 0. You will see that the Folium of Descartes consists of a single loop with a cusp at (3, 3) and has asymptotes along the positive x and positive y-axes.

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Most popular questions from this chapter

Use a graphing utility to graph the hyperbolas \(r=\frac{e}{1+e \cos \theta},\) for \(e=1.1,1.3,1.5,1.7\) and 2 on the same set of axes. Explain how the shapes of the curves vary as \(e\) changes.

Sector of a hyperbola Let \(H\) be the right branch of the hyperbola \(x^{2}-y^{2}=1\) and let \(\ell\) be the line \(y=m(x-2)\) that passes through the point (2,0) with slope \(m,\) where \(-\infty < m < \infty\). Let \(R\) be the region in the first quadrant bounded by \(H\) and \(\ell\) (see figure). Let \(A(m)\) be the area of \(R .\) Note that for some values of \(m\) \(A(m)\) is not defined. a. Find the \(x\) -coordinates of the intersection points between \(H\) and \(\ell\) as functions of \(m ;\) call them \(u(m)\) and \(v(m),\) where \(v(m) > u(m) > 1 .\) For what values of \(m\) are there two intersection points? b. Evaluate \(\lim _{m \rightarrow 1^{+}} u(m)\) and \(\lim _{m \rightarrow 1^{+}} v(m)\) c. Evaluate \(\lim _{m \rightarrow \infty} u(m)\) and \(\lim _{m \rightarrow \infty} v(m)\) d. Evaluate and interpret \(\lim _{m \rightarrow \infty} A(m)\)

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Find an equation of the line tangent to the following curves at the given point. $$y^{2}-\frac{x^{2}}{64}=1 ;\left(6,-\frac{5}{4}\right)$$

Explain and carry out a method for graphing the curve \(x=1+\cos ^{2} y-\sin ^{2} y\) using parametric equations and a graphing utility.
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