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 determine the instantaneous rate of change of consumer expenditure \( E(p) \) with respect to price \( p \), we'll follow these steps: 1. **Express Consumer Expenditure:** \[ E(p) = p \times x \] Given the demand function: \[ x = 9,000 - 90p \] Substitute \( x \) into the expenditure equation: \[ E(p) = p \times (9,000 - 90p) = 9,000p - 90p^2 \] 2. **Find the Derivative \( \frac{dE}{dp} \):** \[ \frac{dE}{dp} = \frac{d}{dp}(9,000p - 90p^2) = 9,000 - 180p \] Now, let's address each part of the question: ### (a) Instantaneous Rate of Change for Any Price \( p \) The derivative \( \frac{dE}{dp} \) gives the instantaneous rate of change of consumer expenditure with respect to price: \[ \frac{dE}{dp} = 9,000 - 180p \] ### (b) Instantaneous Rate of Change at \( p = \$10 \) Substitute \( p = 10 \) into the derivative: \[ \frac{dE}{dp}\bigg|_{p=10} = 9,000 - 180 \times 10 = 9,000 - 1,800 = 7,200 \] ### (c) Instantaneous Rate of Change at \( p = \$35 \) Substitute \( p = 35 \) into the derivative: \[ \frac{dE}{dp}\bigg|_{p=35} = 9,000 - 180 \times 35 = 9,000 - 6,300 = 2,700 \] ### **Summary of Results:** - **(a)** For any price \( p \), the instantaneous rate of change is: \[ \frac{dE}{dp} = 9,000 - 180p \] - **(b)** At \( p = \$10 \): \[ \frac{dE}{dp} = 7,200 \] - **(c)** At \( p = \$35 \): \[ \frac{dE}{dp} = 2,700 \]