The folium of Descartes (See Figure 3.27)

c. Find the coordinates of the point A in Figure 3.29 where the folium has a vertical tangent line.

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The folium of Descartes is given by the equation \(x^3 + y^3 - 9xy = 0\). To find where the curve has a vertical tangent, we need to find where the derivative \(\frac{dy}{dx}\) is undefined.

Implicitly differentiate the equation \(x^3 + y^3 - 9xy = 0\) with respect to \(x\). This involves using the chain rule for \(y^3\) and the product rule for \(-9xy\).

After differentiating, you will get an expression for \(\frac{dy}{dx}\). Set the denominator of this expression to zero to find where the derivative is undefined, indicating a vertical tangent.

Solve the resulting equation for \(x\) and \(y\) to find the coordinates of the point(s) where the tangent is vertical.

Substitute the values of \(x\) and \(y\) back into the original equation to verify that they satisfy the folium of Descartes equation, confirming the point A where the tangent is vertical.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Vertical Tangent Line

A vertical tangent line occurs at a point on a curve where the slope of the tangent approaches infinity. This typically happens when the derivative of the function is undefined or infinite at that point. In the context of the folium of Descartes, finding the coordinates of point A involves determining where the derivative of the curve is undefined, indicating a vertical tangent.

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). For the folium of Descartes, which is defined by the equation x³ + y³ - 9xy = 0, implicit differentiation allows us to find dy/dx, the slope of the tangent line, at any point on the curve, including where it is vertical.

Folium of Descartes

The folium of Descartes is a specific type of algebraic curve defined by the equation x³ + y³ - 9xy = 0. It has a distinctive shape and features, including points where the tangent lines can be vertical. Understanding the properties of this curve is essential for analyzing its behavior and finding points of interest, such as point A with a vertical tangent.

The folium of Descartes (See Figure 3.27)c. Find the coordinates of the point A in Figure 3.29 where the folium has a vertical tangent line.