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 \[ D^{\prime}(x)=-\frac{2000}{x^{2}} \] where $\mathrm{x}$ 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. \[ \mathrm{D}(\mathrm{x})= \]

Step 1 ：Given the marginal-demand function \(D^{\prime}(x)=-\frac{2000}{x^{2}}\), 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 ：\(\boxed{D(x) = \frac{2000}{x} + 4}\) is the final answer.