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Calculus for Scientists and Engineers: Early Transcendental

Average and marginal profit Let $C(x)$ represent the cost of producing $x$ items and $p(x)$ be the sale price per item if $x$ items are sold. The profit $P(x)$ of selling $x$ items is $P(x)=x p(x)-C(x)$ (revenue minus costs). The average profit per item when $x$ items are sold is $P(x) / x$ and the marginal profit is dP/dx. The marginal profit

approximates the profit obtained by selling one more item given that $x$

items have already been sold. Consider the following cost functions $C$ and price functions $p$

a. Find the profit fiunction $P$.

b. Find the average profit function and marginal profit function.

c. Find the average profit and marginal profit if $x=$ a units have been sold.

d. Interpret the meaning of the values obtained in part (c).

$$\begin{aligned}

&C(x)=-0.04 x^{2}+100 x+800, p(x)=200-0.1 x\\

&a=1000

\end{aligned}$$