A particle moves in the plane along the parametric curve x(t)=t

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

A particle moves in the plane along the parametric curve

x(t)=t, \qquad y(t)=\arcsin\!\left(\dfrac{2t}{1+t^{2}}\right), \qquad 0<t<1.

Answer the following. (a) Compute

\dfrac{dy}{dt}

and simplify as much as possible. (b) Compute

\dfrac{dx}{dt}

. (c) Determine

\dfrac{dy}{dx}

t=\dfrac{1}{2}

and provide the exact simplified value. (d) Based on the result from part (c), state whether the curve is rising or falling at

t=\dfrac{1}{2}

and justify your conclusion.