Two variables and depend on time

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

Two variables

x

and

y

depend on time

t

. At a particular instant

t=t_{0}

, measurements give

\dfrac{dx}{dt}=-4\,\mathrm{units/s}

and

\dfrac{dy}{dt}=6\,\mathrm{units/s}

. (i) Find the instantaneous rate of change of

y

with respect to

x

t=t_{0}

. (ii) Interpret the sign of this value in terms of how

y

varies as

x

changes at that instant. (iii) If at

t=t_{0}

the point

(x,y)

lies in the first quadrant, explain whether, immediately after

t_{0}

, the point will move towards or away from the

y

–axis.