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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.4.755
Chapter 4, Problem 4.4.755

{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.


x³ + y³ = 3xy (Folium of Descartes)

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Identify the given equation of the curve: \(x^3 + y^3 = 3xy\). This is known as the Folium of Descartes.
To analyze the curve, we start by using implicit differentiation. Differentiate both sides of the equation with respect to \(x\). Remember that \(y\) is a function of \(x\), so apply the chain rule when differentiating terms involving \(y\).
The differentiation of \(x^3\) with respect to \(x\) is \(3x^2\). For \(y^3\), use the chain rule: \(3y^2 \frac{dy}{dx}\). For the right side, differentiate \(3xy\) to get \(3y + 3x \frac{dy}{dx}\).
Set up the equation from the derivatives: \(3x^2 + 3y^2 \frac{dy}{dx} = 3y + 3x \frac{dy}{dx}\). Solve for \(\frac{dy}{dx}\) to find the slope of the tangent line at any point \((x, y)\) on the curve.
To graph the curve, use a graphing utility. Input the original equation \(x^3 + y^3 = 3xy\) and observe the shape of the curve. Note any points of interest such as intercepts, symmetry, or asymptotic behavior. The Folium of Descartes typically has a loop and an asymptote, which should be visible in the graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not isolated on one side. It allows us to find the derivative of y with respect to x when y is defined implicitly by an equation, such as x³ + y³ = 3xy. This method involves differentiating both sides of the equation with respect to x and applying the chain rule to terms involving y.
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Graphing Utility

A graphing utility is a software tool or calculator that allows users to visualize mathematical functions and equations. In the context of the Folium of Descartes, a graphing utility can help plot the curve defined by the equation x³ + y³ = 3xy, providing insights into its shape, intercepts, and behavior. This visualization aids in understanding the properties of the curve and its intersections with axes.
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Classical Curves

Classical curves refer to well-studied mathematical curves that have significant historical and theoretical importance, such as the Folium of Descartes. These curves often arise from polynomial equations and exhibit unique properties, such as symmetry and specific points of interest. Understanding these curves involves exploring their geometric characteristics and applications in various fields of mathematics.
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