The teacher hasn't uploaded a solution for this question yet.
The slope of the curve is given by the derivative of $y$ with respect to $x$, i.e., $\frac{dy}{dx}$. $$y = 5x - 2x^3$$ $$\frac{dy}{dx} = 5 - 6x^2$$ Let $m$ be the slope, so $m = 5 - 6x^2$.
We need to find how fast the slope is changing, which is $\frac{dm}{dt}$. We are given that $\frac{dx}{dt} = 2$. We can use the chain rule to find $\frac{dm}{dt}$: $$\frac{dm}{dt} = \frac{dm}{dx} \cdot \frac{dx}{dt}$$ First, find $\frac{dm}{dx}$: $$m = 5 - 6x^2$$ $$\frac{dm}{dx} = -12x$$
Now, we can find $\frac{dm}{dt}$ when $x=2$ and $\frac{dx}{dt} = 2$: $$\frac{dm}{dt} = \frac{dm}{dx} \cdot \frac{dx}{dt} = (-12x) \cdot (2) = -24x$$ When $x=2$: $$\frac{dm}{dt} = -24(2) = -48$$
The slope of the curve is changing at a rate of $-48$ units/s when $x=2$.
Final Answer: -48 units/s
AI generated content. Review strictly for academic accuracy.