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 slope of the curve is given by the first derivative of \( y \): \[ \frac{dy}{dx} = 5 - 6x^2 \] The rate of change of the slope is the second derivative \( \frac{d^2y}{dx^2} \): \[ \frac{d^2y}{dx^2} = -12x \] Using the chain rule: \[ \frac{d^2y}{dt^2} = \frac{d^2y}{dx^2} \cdot \frac{dx}{dt} \] Substituting \( x = 2 \) and \( \frac{dx}{dt} = 2 \), we get: \[ \frac{d^2y}{dt^2} = -12(2) \cdot 2 = -48 \] Thus, the slope of the curve is changing at a rate of \( -48 \, \text{units/s} \) when \( x = 2 \).