Question

A particle moves along the curve $6y=x^3+2$. Find the points on the curve at which y - coordinate is changing 2 times as fast as x - coordinate.

Answer 1

we need to find coordinates of point where $\frac{dy}{dt}=2\frac{dx}{dt}$

$6y=x^3+2$

Differentiating on both sides

$6\frac{dy}{dt}=3x^2\frac{dx}{dt}$

$6\times 2\frac{dx}{dt}=3x^2\frac{dx}{dt}$

$x^2=4$

$x=\pm 2$

If $x=2\Rightarrow y=\frac{5}{3}$

if $x=-2\Rightarrow y=-1$