Textbook Question
Slopes, Tangent Lines, and Normal Lines
In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.
x²y² = 9, (–1,3)
4. Applications of Derivatives
Implicit Differentiation
9:42 minutesProblem 3.7.46b
Textbook Question
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 guidance1
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.
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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.
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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.
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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.
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