Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. 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)​​

Accepted Answer

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. 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. 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. 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. 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.

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)

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Key Concepts

Tangent Lines

Implicit Differentiation

Cissoid of Diocles

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)