MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Consider the polar curve given by r=3+2cosθ for 0≤θ≤2π. A student claims that the curve is concave upward (with respect to the usual Cartesian x–y plane) whenever θ is in the interval (2π,23π).
(a) Describe in words what it means for a point on a polar curve to be “concave upward” when the curve is viewed in the Cartesian plane.
(b) Explain, without performing any explicit derivative calculations, why the student’s claim must be analyzed rather than accepted at face value.
(c) State at least two specific mathematical quantities that must be considered in order to verify or refute the student’s claim about concavity on the given interval.