Question

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. \(x^{3}+y^{3}=3 x y\) (Folium of Descartes)

Short answer

Question: Explain the process of graphing the curve of the Folium of Descartes, given by the equation \(x^3 + y^3 = 3xy\). Answer: To graph the Folium of Descartes, we first rewrite the equation in terms of y if possible, but in this case, it is not possible. We then apply implicit differentiation to find the derivative of the curve. Next, we identify critical points such as where the derivative is 0 or undefined, and analyze the behavior of the curve, including any asymptotes. Finally, we create a detailed graph of the curve, including the critical points and any notable features.

Step-by-step solution

Rewrite the equation in terms of y (if possible)

It may not be possible to isolate y in this case, so we consider the given equation as it is: \(x^3 + y^3 = 3xy\).

Apply Implicit Differentiation

Differentiate both sides of the equation with respect to x: \(\frac{d}{dx}(x^3 + y^3) = \frac{d}{dx}(3xy)\) Applying the chain rule, we get: \(3x^2 + 3y^2\frac{dy}{dx} = 3x\frac{dy}{dx} + 3y\) Now, we need to isolate \(\frac{dy}{dx}\): \(3y^2\frac{dy}{dx} - 3x\frac{dy}{dx} = 3y - 3x^2\) Factor out \(\frac{dy}{dx}\): \(\frac{dy}{dx} (3y^2 - 3x) = 3y - 3x^2\) Divide both sides by \(3y^2 - 3x\) to solve for \(\frac{dy}{dx}\): \(\frac{dy}{dx} = \frac{3y - 3x^2}{3y^2 - 3x}\)

Identify Critical Points

Critical points occur when the derivative is either 0 or undefined. Set the derivative equal to 0 and solve for x and y: \(\frac{3y - 3x^2}{3y^2 - 3x} = 0\) Since this is a rational function, it will be 0 when the numerator is 0: \(3y - 3x^2 = 0\) \(y = x^2\) Now, set the denominator equal to zero to find when the derivative is undefined: \(3y^2 - 3x = 0\) \(y^2 - x = 0\) Or \(y^2 = x\) These are two important curves to include when graphing.

Analyze the Behavior of the Curve

Based on the critical points, we know that the curve has a parabolic shape for \(y=x^2\) and a parabolic shape for \(x=y^2\). In addition, as \(x \to \pm\infty\), both sides of the equation \(\frac{dy}{dx} = \frac{3y - 3x^2}{3y^2 - 3x}\) have the same sign, so the curve has no horizontal or vertical asymptotes.

Create a Detailed Graph of the Curve

Now that we have gathered all the necessary information to graph the curve, we can use a graphing utility to graph the Folium of Descartes: 1. Plot the curve \(x^3 + y^3 = 3xy\). 2. Include the critical points \(y = x^2\) and \(x = y^2\) as dashed lines, showing that these are regions where the derivative is either 0 or undefined. 3. Plot the behavior of the curve as it extends to positive and negative infinity, noting that the curve has no horizontal or vertical asymptotes. With all of these components in place, you can create a detailed and accurate graph of the Folium of Descartes, \(x^3 + y^3 = 3xy\).

Key concepts

Implicit Differentiation

The technique of implicit differentiation is essential when dealing with equations where it is difficult or impossible to solve for one variable in terms of the other explicitly. In the case of the Folium of Descartes, represented by the equation \[x^3 + y^3 = 3xy,\]it is challenging to express \(y\) solely in terms of \(x\). Thus, we use implicit differentiation to find the derivative.

Implicit differentiation requires applying the chain rule while treating \(y\) as a function of \(x\). When differentiating the given relation, both \(x\) and \(y\) terms are dealt with using this technique. This results in the derivative: \[\frac{dy}{dx} = \frac{3y - 3x^2}{3y^2 - 3x}.\]This expression allows us to understand how \(y\) changes with respect to \(x\), even though we can't write \(y\) as a function of \(x\) explicitly.

Understanding this process helps in analyzing and sketching the curve comprehensively.

Critical Points

Critical points in a curve are significant as they denote where the slope of the tangent is zero or undefined. For \[\frac{dy}{dx} = \frac{3y - 3x^2}{3y^2 - 3x},\]critical points occur when the derivative equals zero or is undefined.

To find where the slope is zero, set the numerator of the derivative to zero: \[3y - 3x^2 = 0,\]which gives us the equation \[y = x^2.\]This indicates that along the parabola \(y = x^2\), the slope of the tangent is zero.

For points where the derivative is undefined, set the denominator to zero: \[3y^2 - 3x = 0,\]leading to \[y^2 = x.\]Here, the slope becomes infinite. These equations, \(y = x^2\) and \(x = y^2\), represent regions of significant behavioral change in the curve and serve as guides for further analysis.

Graphing Utility

Utilizing a graphing utility can be incredibly helpful when dealing with complex curves like the Folium of Descartes, especially to visualize critical points and overall behavior. When plotting \[x^3 + y^3 = 3xy,\]a graphing tool provides the ability to depict the intricate forms and intersections that might be missed in manual sketching.

Key elements to include when using a graphing utility:

• Plot the critical curves \(y = x^2\) and \(x = y^2\) as dashed lines. These illustrate the locations where the derivative is zero or undefined.
• Observe the behavior of the curve as \(x\) and \(y\) approach infinity, noting any patterns or limits.

Graphing utilities offer a precise representation of the curve, aiding in visualizing complex interactions and critical intersections.

Analytical Methods

Analytical methods involve using calculus and algebra to break down the equation and discover more about the structure of the curve. For the Folium of Descartes:

• Using implicit differentiation, we derived the slope, \(\frac{dy}{dx} = \frac{3y - 3x^2}{3y^2 - 3x},\) which tells us how \(y\) changes as \(x\) changes.
• We identified critical points by setting parts of the derivative to zero, finding the parabolas \(y = x^2\) and \(x = y^2\).
• Sign analysis of \(\frac{dy}{dx}\) helps determine regions of increasing or decreasing behavior on the curve.

Such methods are invaluable in gaining a deeper understanding of the overall shape and behavior of the curve without solely relying on computational tools. They form the foundation for graphical accuracy and analytical precision in mathematical exploration.