Skip to main content
Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.7.46b
Chapter 3, Problem 3.7.46b

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 \(\frac{dy}{dx} = 0\).
Implicitly differentiate the equation \(x^3 + y^3 - 9xy = 0\) with respect to \(x\). This gives \(3x^2 + 3y^2\frac{dy}{dx} - 9(y + x\frac{dy}{dx}) = 0\).
Rearrange the differentiated equation to solve for \(\frac{dy}{dx}\): \(3y^2\frac{dy}{dx} - 9x\frac{dy}{dx} = 9y - 3x^2\).
Factor out \(\frac{dy}{dx}\) to get \(\frac{dy}{dx}(3y^2 - 9x) = 9y - 3x^2\).
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
9m
Was this helpful?

Key Concepts

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

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.
Recommended video:
05:13
Slopes of Tangent Lines

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.
Recommended video:
05:14
Finding The Implicit Derivative

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.
Related Practice
Textbook Question

A sliding ladder


A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec.


b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?


435
views
Textbook Question

Consider the function f graphed here. The domain of f is the interval [−4, 6] and its graph is made of line segments joined end to end.


" style="" width="350">


b. Graph the derivative of f. The graph should show a step function.

227
views
Textbook Question

In Exercises 47 and 48, find an equation for


(b) the horizontal tangent line to the curve at Q.


" style="" width="200">

313
views
Textbook Question

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.


" style="" width="250">


Find the derivatives with respect to x of the following combinations at the given value of x.


b. f(x)g³(x), x = 0

279
views
Textbook Question

Fruit flies (Continuation of Example 4, Section 2.1.) Populations starting out in closed environments grow slowly at first, when there are relatively few members, then more rapidly as the number of reproducing individuals increases and resources are still abundant, then slowly again as the population reaches the carrying capacity of the environment.


b. During what days does the population seem to be increasing fastest? Slowest?


240
views
Textbook Question

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.


b. At what rate is the angle θ changing at this instant (see the figure)?

248
views