Question

Given a curve y = 7x − x³ and x increases at the rate of 2 units per second. The rate at which the slope of the curve is changing, when x = 5 is ______.

Options

• −60 units/sec

• 60 units/sec

• −70 units/sec

• −140 units/sec

Answer 1

Given a curve y = 7x − x³ and x increases at the rate of 2 units per second. The rate at which the slope of the curve is changing, when x = 5 is −60 units/sec.

Explanation:

Given, y = 7x − x³

Differentiating both sides w.r.t. x, we get

`dy/dx = 7 - 3x^2`

Thus, slope, s = 7 - 3x²

Now, differentiating w.r.t. 't', we get

`(ds)/dt = -6xdx/dt`

∴ `(ds)/dt|_(at  x = 5)` = −6 × 5 × 2    ...[∵ `(dx)/dt` = 2 unit/s (given)]

= −60 unit/s