91Ó°ÊÓ

Find the profit function for the given marginal profit and initial condition. $$ \begin{array}{ll}{\text {Marginal Profit}} & {\text { Initial Condition }} \\\ {\frac{d P}{d x}=-24 x+805} & {P(12)=\$ 8000}\end{array} $$

A company produces a product for which the marginal cost of producing \(x\) units is modeled by \(d C / d x=2 x-12,\) and the fixed costs are dollar 125 . (a) Find the total cost function and the average cost function. (b) Find the total cost of producing 50 units. (c) In part (b), how much of the total cost is fixed? How much is variable? Give examples of fixed costs associated with the manufacturing of a product. Give examples of variable costs.

Use a symbolic integration utility to evaluate the definite integral. \(\int_{1 / 2}^{1}(x+1) \sqrt{1-x} d x\)