Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x^(2/3) * y^(2/3) = 4, (-3, 3), 1 (astroid)

Answer 1

By applying implicit differentiation to the given curve x^(2/3)*y^(2/3) = 4 and then solving for dy/dx, we find that the slope m at the point (-3, 3) is 1. Using the point-slope form, the equation of the tangent line at the given point is y = x + 6.

Step-by-step explanation:

To find the equation of the tangent line to the curve x2/3 * y2/3 = 4 at the point (-3, 3), we will use implicit differentiation. First, differentiate both sides of the equation with respect to x:

d/dx(x2/3y2/3) = d/dx(4)

Applying the product rule and chain rule, we get:

(2/3)x-1/3y2/3 + (2/3)x2/3y-1/3(dy/dx) = 0

Then solve for dy/dx to find the slope of the tangent line at (-3, 3):

dy/dx = -(y/x)(dy/dx) = -((3/3)/(3/(-3))) = 1

Now we have the slope of the tangent line, m = 1. To determine the equation of the tangent line, use the point-slope form:

y - y1 = m(x - x1)

y - 3 = 1(x - (-3))

Finally, the equation of our tangent line is y = x + 6.