60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
a. Find equations of all lines tangent to the curve at the given value of x.
4x³ =y²(4−x); x=2 (cissoid of Diocles)

Verified step by step guidance

1. First, understand the problem: We need to find the equations of tangent lines to the curve defined by the equation 4x³ = y²(4−x) at x = 2. This involves finding the derivative of the curve with respect to x to determine the slope of the tangent line at the given point.
2. Differentiate the equation implicitly with respect to x. Start by differentiating both sides of the equation 4x³ = y²(4−x). Use the product rule and chain rule where necessary. The derivative of the left side with respect to x is straightforward, while the right side requires implicit differentiation.
3. After differentiating, solve for dy/dx, which represents the slope of the tangent line at any point (x, y) on the curve. Substitute x = 2 into this derivative to find the slope of the tangent line at the specific point where x = 2.
4. Next, find the y-coordinate corresponding to x = 2 by substituting x = 2 into the original equation 4x³ = y²(4−x) and solving for y. This will give you the point (2, y) on the curve where the tangent line touches.
5. Finally, use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where m is the slope found in step 3 and (x₁, y₁) is the point found in step 4, to write the equation of the tangent line at x = 2.

Key Concepts

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point. To find the equation of the tangent line, one typically uses the point-slope form of a line, which requires both the slope and the coordinates of the point of tangency.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In cases like the given equation, where y is defined implicitly in terms of x, this method allows us to find dy/dx by differentiating both sides of the equation with respect to x, treating y as a function of x. This is essential for finding the slope of the tangent line.

Cissoid of Diocles

The cissoid of Diocles is a specific type of curve defined by a particular equation, often used in the context of problems involving tangents and areas. In this case, the equation 4x³ = y²(4−x) describes the cissoid, and understanding its geometric properties is crucial for determining the points at which tangent lines can be drawn. Familiarity with the shape and behavior of this curve aids in visualizing the problem.