Find the average rate of change for the given function. \[ f(x)=\sqrt{5 x+1} ; \quad \text { between } x=3 \text { and } x=16 \]

To find the average rate of change for the given function \( f(x) = \sqrt{5x+1} \) between \( x=3 \) and \( x=16 \), we can use the formula for the average rate of change: \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \] Given: - \( f(x) = \sqrt{5x+1} \) - \( x_1 = 3 \) - \( x_2 = 16 \) Let's calculate the average rate of change using the formula. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\sqrt{5\times 16+1}-\sqrt{5\times 3+1}\right)}{\left(16-3\right)}\) - step1: Remove the parentheses: \(\frac{\sqrt{5\times 16+1}-\sqrt{5\times 3+1}}{16-3}\) - step2: Multiply the numbers: \(\frac{\sqrt{80+1}-\sqrt{5\times 3+1}}{16-3}\) - step3: Multiply the numbers: \(\frac{\sqrt{80+1}-\sqrt{15+1}}{16-3}\) - step4: Add the numbers: \(\frac{\sqrt{81}-\sqrt{15+1}}{16-3}\) - step5: Add the numbers: \(\frac{\sqrt{81}-\sqrt{16}}{16-3}\) - step6: Simplify the root: \(\frac{9-\sqrt{16}}{16-3}\) - step7: Simplify the root: \(\frac{9-4}{16-3}\) - step8: Subtract the numbers: \(\frac{5}{16-3}\) - step9: Subtract the numbers: \(\frac{5}{13}\) The average rate of change for the function \( f(x) = \sqrt{5x+1} \) between \( x=3 \) and \( x=16 \) is \( \frac{5}{13} \) or approximately \( 0.384615 \).