Find the average rate of change of f(x)=√x from x1=4 to x2=9.

Verified step by step guidance

1. Identify the formula for the average rate of change of a function, which is \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).
2. Substitute the given values into the formula: \( x_1 = 4 \) and \( x_2 = 9 \).
3. Calculate \( f(x_1) = f(4) = \sqrt{4} \).
4. Calculate \( f(x_2) = f(9) = \sqrt{9} \).
5. Substitute \( f(x_1) \) and \( f(x_2) \) into the average rate of change formula and simplify.

Key Concepts

Average Rate of Change

The average rate of change of a function over an interval [x1, x2] is calculated as the change in the function's value divided by the change in the input values. Mathematically, it is expressed as (f(x2) - f(x1)) / (x2 - x1). This concept is essential for understanding how a function behaves over a specific interval.

Square Root Function

The square root function, denoted as f(x) = √x, is a fundamental mathematical function that returns the non-negative square root of x. It is defined for x ≥ 0 and has a characteristic shape that increases at a decreasing rate. Understanding this function is crucial for evaluating its behavior and calculating changes over specified intervals.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For the function f(x) = √x, evaluating it at x1 = 4 and x2 = 9 means calculating f(4) and f(9). This step is necessary to find the values needed to compute the average rate of change over the given interval.