The folium of Descartes (See Figure 3.27)


b. At what point other than the origin does the folium have a horizontal tangent line?

Verified step by step guidance

1. The folium of Descartes is given by the equation \(x^3 + y^3 - 9xy = 0\). To find where the curve has a horizontal tangent line, we need to find where the derivative \(\frac{dy}{dx} = 0\).
2. Implicitly differentiate the equation \(x^3 + y^3 - 9xy = 0\) with respect to \(x\). This gives \(3x^2 + 3y^2\frac{dy}{dx} - 9(y + x\frac{dy}{dx}) = 0\).
3. Rearrange the differentiated equation to solve for \(\frac{dy}{dx}\): \(3y^2\frac{dy}{dx} - 9x\frac{dy}{dx} = 9y - 3x^2\).
4. Factor out \(\frac{dy}{dx}\) to get \(\frac{dy}{dx}(3y^2 - 9x) = 9y - 3x^2\).
5. Set \(\frac{dy}{dx} = 0\) to find the horizontal tangent. This implies \(9y - 3x^2 = 0\), which simplifies to \(y = \frac{x^2}{3}\). Substitute \(y = \frac{x^2}{3}\) back into the original equation to find the specific point(s) other than the origin.

Key Concepts

Horizontal Tangent Lines

A horizontal tangent line occurs at points on a curve where the derivative of the function is zero. This means that the slope of the tangent line is flat, indicating a local maximum, minimum, or a point of inflection. To find these points, one typically sets the first derivative of the function equal to zero and solves for the variable.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x, rather than explicitly as y = f(x). This method allows us to find the derivative of y with respect to x by differentiating both sides of the equation and applying the chain rule. It is particularly useful for curves defined by equations like the folium of Descartes.

Folium of Descartes

The folium of Descartes is a specific algebraic curve defined by the equation x^3 + y^3 - 9xy = 0. It has a distinctive shape and features, including points where it intersects itself and horizontal tangents. Understanding its geometry and behavior is essential for analyzing its properties, such as finding points with horizontal tangent lines.