A curve is defined by the parametric equations x = t^2, y = t^3 - t. Find the values of t where the tangent line is vertical.
Correct: A
For a vertical tangent, dx/dt = 0 and dy/dt ≠ 0. dx/dt = 2t. dy/dt = 3t^2 - 1. dx/dt = 0 when t = 0. When t = 0, dy/dt = -1 ≠ 0. So t=0 is one solution. 3t^2-1 = 0 when t = ±√(1/3) also give dy/dt = 0. Thus we look for vertical tangent lines only where dx/dt = 0 -> 2t = 0 -> t = 0. At t=0, dy/dt = -1 which is nonzero. dx/dt = 0 implies a vertical tangency.