2023 AP Calculus AB FRQ #6

Consider the curve given by the equation 6xy = 2 + y^3.
(a) Show that dy/dx = 2y / y^2 − 2x.
(b) Find the coordinates of a point on the curve at which the line tangent to the curve is horizontal, or explain why no such point exists.
(c) Find the coordinates of a point on the curve at which the line tangent to the curve is vertical, or explain why no such point exists.
(d) A particle is moving along the curve. At the instant when the particle is at the point (1/2, -2) its horizontal position is increasing at a rate of dx/dt = 2/3 unit per second. What is the value of dy/dt, the rate of change of the particle’s vertical position, at that instant?
Intro: 00:00
Problem a: 00:12
Problem b: 03:17
Problem c: 05:35
Problem d: 08:29