The follum of Descartes is the curve with equation x^3+y^3=3...

The follum of Descartes is the curve with equation $x^{3}+y^{3}=3 a x y$, where $a \neq 0$ is a constant (Figure 25 ). (a) Show that the line $y=t x$ intersects the folium at the origin and at one other point $P$ for all $t \neq-1,0 .$ Express the coordinates of $P$ in terms of $t$ to obtain a parametrization of the folium. Indicate the direction of the parametrization on the graph. (b) Describe the interval of $t$ -values parametrizing the parts of the curve in quadrants $\mathbf{I}, \mathbf{I I},$ and $\mathrm{IV}$. Note that $t=-1$ is a point of discontinuity of the parametrization. (c) Calculate $d y / d x$ as a function of $t$ and find the points with horizontal or vertical tangent.

The follum of Descartes is the curve with equation $x^{3}+y^{3}=3 a x y$, where $a \neq 0$ is a constant (Figure 25 ).
(a) Show that the line $y=t x$ intersects the folium at the origin and at one other point $P$ for all $t \neq-1,0 .$ Express the coordinates of $P$ in terms of $t$ to obtain a parametrization of the folium. Indicate the direction of the parametrization on the graph.
(b) Describe the interval of $t$ -values parametrizing the parts of the curve in quadrants $\mathbf{I}, \mathbf{I I},$ and $\mathrm{IV}$. Note that $t=-1$ is a point of discontinuity of the parametrization.
(c) Calculate $d y / d x$ as a function of $t$ and find the points with horizontal or vertical tangent.

Key Concepts

Continuity in Parametric Representations

When parametrizing a curve, it is important to analyze the parameter intervals over which the representation is continuous and valid. Discontinuities may occur at specific parameter values, which can correspond to points of indeterminacy or singularities in the curve. Understanding these aspects ensures a complete and accurate depiction of the curve’s behavior over its entire domain.

Analysis of Tangent Lines

The analysis of tangent lines involves studying the derivative to determine the slope at a given point on the curve. Identifying points where the derivative is zero or undefined helps in detecting horizontal and vertical tangents respectively. This information is crucial for understanding the local geometry of the curve and any potential critical points.

Intersection of Curves and Lines

Determining where a curve intersects a line is a key algebraic method that involves substituting the line’s equation into the curve's equation. This approach transforms the problem into solving an equation in one variable, allowing us to identify common points between the curve and the line. Factoring or other algebraic techniques are often used to isolate the roots corresponding to these intersections.

Implicit Differentiation

Implicit differentiation is used for finding derivatives of functions that are defined by an equation relating x and y, rather than by an explicit formula. By differentiating both sides of the equation with respect to x and solving for dy/dx, this technique enables us to compute the slope of the tangent line at any point on the curve without having to first solve for y explicitly.

Parametrization of Curves

Parametrization involves expressing the coordinates of points on a curve as functions of one or more independent parameters. This is especially useful for converting an implicit equation into a set of parametric equations, thereby simplifying the analysis of the curve's properties and making it easier to compute derivatives or study the curve’s behavior over given intervals.

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