A cycloid is traced by the equations and

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

Problem

A cycloid is traced by the equations

x=a\,(t-\sin t)

and

y=a\,(1-\cos t)

, where

a>0

. (i) Derive an expression for

\dfrac{dy}{dx}

in terms of

t

. (ii) Prove that the tangent to the cycloid at

t=0

is vertical. (iii) Find

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

t=\pi

and comment on the concavity of the curve at that point. (iv) Determine the intervals of

t

in which the cycloid is ascending and those in which it is descending.