Find the marginal profit function if cost and revenue are given by \( C(x)=204+0.7 x \) and \( R(x)=8 x-0.03 x^{2} \)

To find the marginal profit function, we first need to determine the profit function \( P(x) \). The profit function is defined as the difference between revenue and cost: \[ P(x) = R(x) - C(x) \] Given the cost function \( C(x) = 204 + 0.7x \) and the revenue function \( R(x) = 8x - 0.03x^2 \), we can substitute these into the profit function: \[ P(x) = (8x - 0.03x^2) - (204 + 0.7x) \] Now, simplify the expression: \[ P(x) = 8x - 0.03x^2 - 204 - 0.7x \] Combine like terms: \[ P(x) = (8x - 0.7x) - 0.03x^2 - 204 \] \[ P(x) = 7.3x - 0.03x^2 - 204 \] Next, we find the marginal profit function \( P'(x) \) by taking the derivative of the profit function \( P(x) \): \[ P'(x) = \frac{d}{dx}(7.3x - 0.03x^2 - 204) \] Calculating the derivative: \[ P'(x) = 7.3 - 0.06x \] Thus, the marginal profit function is: \[ \boxed{7.3 - 0.06x} \]