AP Calculus AB Exam Question:

Consider the implicitly defined function given by the equation:

x^2 + xy + y^3 = 8

a) Find dy/dx in terms of x and y.

b) Use dy/dx to find the slope of the tangent line to the curve at the point (3, 1).

Answer:

a) To find dy/dx, we will use implicit differentiation. We differentiate both sides of the equation with respect to x, treating y as an implicit function of x.

Start by differentiating the left side: d/dx (x^2 + xy + y^3) = d/dx (8)

Using the power rule for differentiation, we get: 2x + y + 3y^2 * (dy/dx) = 0

Next, isolate dy/dx on one side: 2x + y = -3y^2 * (dy/dx)

Finally, divide both sides by -3y^2 to solve for dy/dx: dy/dx = (2x + y) / (-3y^2)

Therefore, dy/dx is given by (2x + y) / (-3y^2).

b) To find the slope of the tangent line at the point (3, 1), substitute x = 3 and y = 1 into the expression for dy/dx:

dy/dx = (2(3) + 1) / (-3(1)^2) = 7 / -3 = -7/3

The slope of the tangent line at the point (3, 1) is -7/3.

Thus, the answers are:

a) dy/dx = (2x + y) / (-3y^2)

b) The slope of the tangent line at the point (3, 1) is -7/3.