Question

For the curve \( y = 5x - 2x^3 \), if \( x increases at the rate of 2 units/s, then how fast is the slope of the curve changing when x = 2?

Hint:

The rate of change of the slope (or curvature) can be found by differentiating the slope of the curve. When the rate of change of \( x \) is known, use the chain rule to connect the second derivative to time.

Answer 1

The first derivative of \( y \) yields the slope of the curve: \[ \frac{dy}{dx} = 5 - 6x^2 \] The second derivative, \( \frac{d^2y}{dx^2} \), represents the rate of change of the slope: \[ \frac{d^2y}{dx^2} = -12x \] Applying the chain rule: \[ \frac{d^2y}{dt^2} = \frac{d^2y}{dx^2} \cdot \frac{dx}{dt} \] With \( x = 2 \) and \( \frac{dx}{dt} = 2 \): \[ \frac{d^2y}{dt^2} = -12(2) \cdot 2 = -48 \] Therefore, the slope of the curve is changing at a rate of \( -48 \, \text{units/s} \) at \( x = 2 \).