Find the average rate of change for the given function. \( f(x)=0.5703 e^{1.298 \sqrt{x}} \) between \( x=2 \) and \( x=2.5 \)

To find the average rate of change for the function \( f(x) = 0.5703 e^{1.298 \sqrt{x}} \) between \( x = 2 \) and \( x = 2.5 \), we use the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] where \( a = 2 \) and \( b = 2.5 \). First, we calculate \( f(2) \) and \( f(2.5) \): 1. \( f(2) = 0.5703 e^{1.298 \sqrt{2}} \) 2. \( f(2.5) = 0.5703 e^{1.298 \sqrt{2.5}} \) Now, we compute these values: - \( \sqrt{2} \approx 1.414 \), so \( f(2) \approx 0.5703 e^{1.298 \times 1.414} \). - \( \sqrt{2.5} \approx 1.581 \), so \( f(2.5) \approx 0.5703 e^{1.298 \times 1.581} \). Calculating these gives us: 1. For \( f(2) \), \( f(2) \approx 0.5703 e^{1.831} \approx 0.5703 \times 6.251 \approx 3.57 \). 2. For \( f(2.5) \), \( f(2.5) \approx 0.5703 e^{2.053} \approx 0.5703 \times 7.66 \approx 4.37 \). Now, substitute these values into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{4.37 - 3.57}{2.5 - 2} = \frac{0.80}{0.5} = 1.6 \] Thus, the average rate of change of the function between \( x = 2 \) and \( x = 2.5 \) is approximately **1.6**.