The consumer expenditure for a product is \( E(p)=p x \). Suppose that the demand for \( x \) units of a product is \( x=9,000-90 p \) where \( p \) dollars is the price per unit. What is the instantaneous rate of change of consumer expenditure with respect to price at the following values? (a) any price \( p \) (b) \( p=10 \) (c) \( p=35 \)

To find the instantaneous rate of change of consumer expenditure \( E(p) = p \cdot x \) with respect to price \( p \), we first need to express \( E(p) \) in terms of \( p \) using the demand function \( x = 9,000 - 90p \). Substituting for \( x \), we get: \[ E(p) = p(9,000 - 90p) = 9,000p - 90p^2. \] Now, we can differentiate \( E(p) \) with respect to \( p \): \[ E'(p) = 9,000 - 180p. \] Now, we can calculate the instantaneous rate of change for the specified values: (a) For any price \( p \): \[ E'(p) = 9,000 - 180p. \] (b) For \( p = 10 \): \[ E'(10) = 9,000 - 180 \cdot 10 = 9,000 - 1,800 = 7,200. \] (c) For \( p = 35 \): \[ E'(35) = 9,000 - 180 \cdot 35 = 9,000 - 6,300 = 2,700. \] In summary: - The instantaneous rate of change of consumer expenditure is \( 9,000 - 180p \). - At \( p = 10 \), it is \( 7,200 \). - At \( p = 35 \), it is \( 2,700 \).