Question

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 1006 units of the product are demanded by consumers when the price is ( $ 2 ) per unit

d(x)=square

Explanation:

Step1: Integrate the marginal - demand function

We know that if \(D^{\prime}(x)=-\frac{2000}{x^{2}}=-2000x^{-2}\), then by the power rule of integration \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)).

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Step2: Use the initial condition to find \(C\)

We are given that when \(x = 2\), \(D(2)=1006\). Substitute \(x = 2\) and \(D(2)=1006\) into \(D(x)=\frac{2000}{x}+C\).

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Answer:

\(D(x)=\frac{2000}{x}+6\)