Which of the following best explains why solving is a necessary step when determining…
MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Which of the following best explains why solving dθdr=0 is a necessary step when determining the angles at which a polar curve r=f(θ) is farthest from or closest to the pole?
(a) Because dθdr represents the Cartesian slope dxdy of the curve, setting it equal to zero locates horizontal tangents, which correspond to maximum or minimum radii.
(b) Because radial extrema occur where the rate of change of the radius with respect to the angle is zero, indicating that the radius is momentarily neither increasing nor decreasing.
(c) Because when dθdr=0, the radius has already reached its largest value; the derivative never vanishes at a minimum.
(d) Because polar optimization should be done in Cartesian form, and setting dθdr=0 is a required first step in carrying out that conversion.