A particle moves along the curve y = (2/3)x3 + 1. Find the points on the curve at which the y - coordinate is changing twice as fast as the x - coordinate.
Given: a particle moves along the curve .
To find the points on the curve at which the y - coordinate is changing twice as fast as the x - coordinate.
Equation of curve is
Differentiating the above equation with respect to t, we get
When y - coordinate is changing twice as fast as the x - coordinate, i.e.,
Equating equation (i) and equation (ii), we get
⇒ x2 = 1 ⇒ x = ±1
When x = 1,
When x = - 1,
Hence the points on the curve at which the y - coordinate changes twice as fast as the x - coordinate are and