Find the average rate of change for the following function \( f(x)=2 \sqrt{x} \) between \( x=9 \) and \( x=25 \)

To find the average rate of change of the function \( f(x) = 2 \sqrt{x} \) between \( x = 9 \) and \( x = 25 \), we use the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] Here, \( a = 9 \) and \( b = 25 \). First, calculate \( f(9) \) and \( f(25) \): \[ f(9) = 2 \sqrt{9} = 2 \times 3 = 6 \] \[ f(25) = 2 \sqrt{25} = 2 \times 5 = 10 \] Now, plug these values into the formula: \[ \text{Average Rate of Change} = \frac{10 - 6}{25 - 9} = \frac{4}{16} = \frac{1}{4} \] So, the average rate of change of the function \( f(x) \) between \( x = 9 \) and \( x = 25 \) is \( \frac{1}{4} \). Here's a nugget of historical context: The concept of average rate of change is foundational in calculus but has its roots in ancient Greek mathematics, where scholars like Archimedes explored the idea of infinitesimals in trying to understand curves and areas. Fast forward to the 17th century, and this laid the groundwork for Newton and Leibniz's development of calculus. Now, let’s dive into a real-world application! Understanding rates of change is super helpful in various fields like economics, physics, and biology. For instance, in economics, calculating the average rate of change can help businesses understand how quantities fluctuate over time, such as sales growth or population change. So the next time you're pondering life’s averages, remember – it’s not just numbers; it’s all about trends in everyday life!