A cycloidal arch is traced by the parametric equations and for

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

Problem

A cycloidal arch is traced by the parametric equations

x=t-\sin t

and

y=1-\cos t

for

0<t<2\pi

. Which of the following best describes the behavior of

y

with respect to

x

t=\pi

? (a)

y

has a relative minimum at

t=\pi

because

\dfrac{dy}{dx}

changes from negative to positive there. (b)

y

has relative maxima at both

t=0

and

t=\pi

because

\dfrac{dy}{dt}=0

at both points. (c)

y

attains a relative maximum only at

t=\pi

because

\dfrac{dy}{dx}

changes from positive to negative there. (d)

y

has no relative extrema on

(0,2\pi)

because

\dfrac{dy}{dx}

never equals

0

on this interval.