MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A polar curve is defined by r=2+3sinθ for 0≤θ≤π.
The graph represents a limaçon whose points move closer to or farther from the pole as θ varies.
(a) Derive an explicit expression for dθdr and determine every value of θ in [0,π] at which r attains either a relative maximum or a relative minimum.
(b) Without computing any second derivatives, explain how the sign of dθdr on the intervals determined in part (a) classifies each critical point found as a location where r is increasing to a maximum or decreasing to a minimum.
(c) Interpret the meaning of each critical point in the context of the limaçon’s shape. Describe, in complete sentences, how the curve’s distance from the pole changes as θ passes each critical value.