A company finds that the rate at which the quantity of a product that consumers demand changes with respect to price is given by the marginal-demand function
$D^{'} (x) = - \frac{2000}{x^{2}}$
where $x$ is the price per unit, in dollars. Find the demand function if it is known that 1004 units of the product are demanded by consumers when the price is $$ 2$ per unit.
$D (x) =$

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$D (x) = \frac{2000}{x} + 4$ is the final answer.

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Step 1 ：Given the marginal-demand function $D^{'} (x) = - \frac{2000}{x^{2}}$ , where $x$ is the price per unit in dollars.

Step 2 ：To find the demand function, we need to integrate the derivative function.

Step 3 ：The integral of $- \frac{2000}{x^{2}}$ is $\frac{2000}{x}$ .

Step 4 ：We also know that 1004 units of the product are demanded when the price is $2 per unit. This gives us the equation $C + \frac{2000}{2} = 1004$ .

Step 5 ：Solving this equation, we find that the constant of integration, $C$ , is 4.

Step 6 ：Therefore, the demand function is $D (x) = \frac{2000}{x} + 4$ .

Step 7 ： $D (x) = \frac{2000}{x} + 4$ is the final answer.

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