A particle moves along the right-opening parabola

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

Problem

y^2 = 8x

, where

x

and

y

are measured in meters and

t

in seconds. At the moment when the particle’s y-coordinate is

3\,\text{m}

, its y-coordinate is increasing at a rate of

1.2\,\text{m/s}

. (a) Develop a formula that links

\frac{dx}{dt}

and

\frac{dy}{dt}

for a particle confined to the parabola

y^2 = 8x

. (b) Calculate

\frac{dx}{dt}

(in m/s) at the described instant and indicate whether the particle is moving to the left or to the right. (c) Determine the instantaneous slope

\frac{dy}{dx}

of the parabola at that point and explain how the slope you found agrees with the directions of

\frac{dx}{dt}

and

\frac{dy}{dt}