1. **State the problem:** We are given the total cost function $$C = 1500 + 2x - 5x^2$$ and the total revenue function $$R = x^2 - 2x$$. We need to find the rate of change of cost and revenue with respect to quantity $$x$$ at $$x=2$$. 2. **Recall the formula:** The rate of change of a function with respect to $$x$$ is its derivative $$\frac{d}{dx}$$. 3. **Find the derivative of the cost function:** $$\frac{dC}{dx} = \frac{d}{dx}(1500 + 2x - 5x^2) = 0 + 2 - 10x = 2 - 10x$$ 4. **Evaluate the rate of change of cost at $$x=2$$:** $$\frac{dC}{dx}\bigg|_{x=2} = 2 - 10(2) = 2 - 20 = -18$$ 5. **Find the derivative of the revenue function:** $$\frac{dR}{dx} = \frac{d}{dx}(x^2 - 2x) = 2x - 2$$ 6. **Evaluate the rate of change of revenue at $$x=2$$:** $$\frac{dR}{dx}\bigg|_{x=2} = 2(2) - 2 = 4 - 2 = 2$$ **Final answer:** - Rate of change of cost at $$x=2$$ is $$-18$$. - Rate of change of revenue at $$x=2$$ is $$2$$.