Consider the polar curve for

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Consider the polar curve

r=1+\sin\theta

for

0<\theta<\pi

. Determine the concavity of the curve at

\theta=\dfrac{\pi}{4}

by evaluating

\dfrac{d^{2}y}{dx^{2}}

. Which of the following is correct? (a)

\dfrac{d^{2}y}{dx^{2}}\approx+10.2

and the curve is concave upward. (b)

\dfrac{d^{2}y}{dx^{2}}\approx-14.5

and the curve is concave downward. (c)

\dfrac{d^{2}y}{dx^{2}}

is undefined because

dx/d\theta=0

at that angle. (d)

\dfrac{d^{2}y}{dx^{2}}\approx+14.5

and the curve is concave upward.