Question

A particle is moving on the curve x^y=1. As it passes through the point (2,1/4), the xcoordinate is increasing at a rate of 2 units per second. At what rate is the y-coordinate changing at that time?

Answer 1

y-coordinate is decreasing at the rate of
`(1)/(2)` unit/sec.

Explanation:

Given that:

The curve of the particle
`x^2y = 1`

Then:

`y = (1)/(x^2)`

Taking the differential of y with respect to t

`(dy)/(dt)= (dx^(-2))/(dx) * (dx)/(dt)`

`= -2x^(-3) (dx)/(dt)`

At (2, 1/4)

`(dx)/(dt) = 2`

This implies that:

`\implies (dy)/(dt) = -(2)/(8)(2)`

`(dy)/(dt) = -(1)/(2) \ \ unit/sec`

Thus, y-coordinate is decreasing at the rate of
`(1)/(2)` unit/sec.