Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
The folium of Descartes is a planar algebraic curve defined by the equation $x^3 + y^3 - 3axy = 0$. It has a characteristic loop and intersects itself at the origin.
5 Must Know Facts For Your Next Test
The standard form of the folium of Descartes is $x^3 + y^3 - 3axy = 0$.
This curve is named after the French philosopher and mathematician René Descartes.
The folium of Descartes contains a singular point at the origin (0,0) where it intersects itself.
It has an asymptote given by the line $x + y + a = 0$.
Implicit differentiation can be used to find its derivatives at various points on the curve.
Review Questions
Related terms
Implicit Differentiation: A technique to differentiate equations where variables cannot be easily separated. It involves taking derivatives with respect to one variable while treating other variables as implicit functions.
Singular Point: A point on a curve where both partial derivatives vanish or are undefined, often leading to intersections or cusps.
Asymptote: A line that a curve approaches arbitrarily closely as it heads towards infinity. Asymptotes can be horizontal, vertical, or oblique.