MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
At a point on an implicitly defined curve, both the numerator and the denominator of dxdy evaluate to zero. Which of the following best describes the behavior of the curve at that point?
(a) The curve necessarily has a vertical tangent since the derivative is undefined.
(b) The point may be a cusp or sharp turn where no unique tangent line exists.
(c) Because both parts are zero, one can apply L’Hôpital’s Rule to obtain a finite non-zero slope, so the tangent is smooth and unique.
(d) The tangent line has slope 1 because the zeros in numerator and denominator cancel.