1. **Problem statement:** (b) Find the marginal cost when 200 micro-components are produced and interpret its meaning. (c) Given Profit = Revenue - Cost, find the profit function $P(x)$. (d) Find the derivative $P'(x)$. (e) Determine intervals where $P(x)$ is increasing or decreasing. (f) Find the optimal production level and expected profit. (a) For $f(x) = -\frac{1}{x} + \frac{a}{2x^2}$, find $f'(x)$. --- 2. **Given:** Revenue function: $R(x) = 0.6x^3 + x^2 + 10x - 2$, where $x$ is in hundreds of micro-components. Cost function is not explicitly given, but marginal cost is needed at $x=2$ (200 micro-components). Assuming cost function $C(x)$ is known or marginal cost $C'(x)$ is given or can be derived from context (not provided here), so we focus on profit and derivatives. --- 3. **(b) Marginal cost at 200 micro-components:** Marginal cost is the derivative of the cost function $C'(x)$ evaluated at $x=2$. Since cost function $C(x)$ is not provided, we cannot compute $C'(2)$ explicitly. Interpretation: Marginal cost at $x=2$ represents the approximate cost to produce one additional hundred micro-components beyond 200. --- 4. **(c) Profit function $P(x)$:** Profit is revenue minus cost: $$P(x) = R(x) - C(x)$$ Since $C(x)$ is unknown, express profit as: $$P(x) = 0.6x^3 + x^2 + 10x - 2 - C(x)$$ --- 5. **(d) Derivative of profit $P'(x)$:** $$P'(x) = R'(x) - C'(x)$$ Calculate $R'(x)$: $$R'(x) = \frac{d}{dx}(0.6x^3 + x^2 + 10x - 2) = 1.8x^2 + 2x + 10$$ So, $$P'(x) = 1.8x^2 + 2x + 10 - C'(x)$$ --- 6. **(e) Intervals where $P(x)$ is increasing or decreasing:** $P(x)$ is increasing where $P'(x) > 0$ and decreasing where $P'(x) < 0$. Since $P'(x) = 1.8x^2 + 2x + 10 - C'(x)$, without $C'(x)$ we cannot find exact intervals. --- 7. **(f) Optimal production level:** Optimal production occurs where marginal profit is zero and profit is positive: $$P'(x) = 0$$ Solve: $$1.8x^2 + 2x + 10 - C'(x) = 0$$ Without $C'(x)$, exact solution is not possible. Expected profit at optimal $x$ is $P(x)$ evaluated at that $x$. --- 8. **(9a) Derivative of $f(x) = -\frac{1}{x} + \frac{a}{2x^2}$:** Rewrite: $$f(x) = -x^{-1} + \frac{a}{2} x^{-2}$$ Differentiate term-by-term: $$f'(x) = (-1)(-1)x^{-2} + \frac{a}{2}(-2)x^{-3} = x^{-2} - a x^{-3} = \frac{1}{x^2} - \frac{a}{x^3}$$ Final answer: $$f'(x) = \frac{1}{x^2} - \frac{a}{x^3}$$