A particle moves in the plane so that its position at time is given by x(t)= 2t

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A particle moves in the plane so that its position at time

t

is given by

x(t)=\sin 2t, \qquad y(t)=t+\cos t,\qquad 0\le t\le2\pi.

The curve is traced exactly once on this interval. (1) Calculate

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

. (2) Determine all values of

t

[0,2\pi]

for which the tangent line to the curve is vertical. Show all work leading to your answers. (3) For each value of

t

found in part (b), give the corresponding coordinate point

(x,y)

on the curve.