Consider the parametric equations x=a t

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Consider the parametric equations

x=a\sin t,\qquad y=a\cos t+bt,\qquad 0<t<2\pi,

where

a

and

b

are positive constants. Answer the following: (i) Determine

\dfrac{dy}{dx}

in terms of

t,\;a,\;b.

(ii) Obtain

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

in terms of

t,\;a,\;b.

(iii) Hence evaluate

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

t=\dfrac{\pi}{3}.