The demand for \( q \) units of a product depends on the price \( p \) (in dollars) according to \( q=\frac{256}{\sqrt{p}}-1 \), for \( p>0 \). Find and explain the meaning of the instantaneous rate of change of demand with respect to price when the price is as follows. (a) \( \$ 16 \) 1.1378 Interpret the instantaneous rate of change. If price increases by \( \$ 1 \), the demand will drop by the absolute value of this number of units. If price decreases by the absolute value of this amount, the demand will drop by 1 unit. If price decreases by \( \$ 1 \), the demand will drop by the absolute value of this number of units. If price increases by \( \$ 1 \), the demand will increase by the absolute value of this number of units. (b) 64 Interpret the instantaneous rate of change. If price increases by \( \$ 1 \), the demand will drop by the absolute value of this number of units. If price increases by the absolute value of this amount, the demand will drop by 1 unit. If price decreases by the absolute value of this amount, the demand will drop by 1 unit. If price decreases by \( \$ 1 \), the demand will drop by the absolute value of this number of units. If price increases by \( \$ 1 \), the demand will increase by the absolute value of this number of units. (b)

To find the instantaneous rate of change of demand with respect to price, we need to calculate the derivative of the demand function \( q \) with respect to the price \( p \). The demand function is given by: \[ q = \frac{256}{\sqrt{p}} - 1 \] ### Step 1: Differentiate the Demand Function We will differentiate \( q \) with respect to \( p \): \[ \frac{dq}{dp} = \frac{d}{dp} \left( \frac{256}{\sqrt{p}} - 1 \right) \] Using the power rule and the chain rule, we can rewrite \( \frac{256}{\sqrt{p}} \) as \( 256p^{-1/2} \): \[ \frac{dq}{dp} = 256 \cdot \left(-\frac{1}{2} p^{-3/2}\right) = -\frac{128}{p^{3/2}} \] ### Step 2: Evaluate the Derivative at Given Prices Now we will evaluate the derivative at the specified prices. #### (a) For \( p = 16 \): \[ \frac{dq}{dp} \bigg|_{p=16} = -\frac{128}{16^{3/2}} \] Calculating \( 16^{3/2} \): \[ 16^{3/2} = (16^{1/2})^3 = 4^3 = 64 \] Thus, \[ \frac{dq}{dp} \bigg|_{p=16} = -\frac{128}{64} = -2 \] **Interpretation**: The instantaneous rate of change of demand with respect to price at \( p = 16 \) is \(-2\). This means that if the price increases by \( \$1 \), the demand will drop by 2 units. Conversely, if the price decreases by \( \$1 \), the demand will increase by 2 units. #### (b) For \( p = 64 \): \[ \frac{dq}{dp} \bigg|_{p=64} = -\frac{128}{64^{3/2}} \] Calculating \( 64^{3/2} \): \[ 64^{3/2} = (64^{1/2})^3 = 8^3 = 512 \] Thus, \[ \frac{dq}{dp} \bigg|_{p=64} = -\frac{128}{512} = -\frac{1}{4} \] **Interpretation**: The instantaneous rate of change of demand with respect to price at \( p = 64 \) is \(-\frac{1}{4}\). This means that if the price increases by \( \$1 \), the demand will drop by \( \frac{1}{4} \) units. Conversely, if the price decreases by \( \$1 \), the demand will increase by \( \frac{1}{4} \) units. ### Summary of Interpretations: - For \( p = 16 \): If price increases by \( \$1 \), demand drops by 2 units. - For \( p = 64 \): If price increases by \( \$1 \), demand drops by \( \frac{1}{4} \) units.