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

Verified step by step guidance

1. Identify the function and the interval: The function is $f\left(x\right)=\sqrt{x}$, and the interval is from $x$₁ = 4 to $x$₂ = 9.
2. Recall the formula for the average rate of change of a function $f$ over an interval $[x\_1,\; x\_2]$: $\backslash frac\{f(x\_2)\; -\; f(x\_1)\}\{x\_2\; -\; x\_1\}$.
3. Calculate the function values at the endpoints: find $f(4)\; =\; \backslash sqrt\{4\}$ and $f(9)\; =\; \backslash sqrt\{9\}$.
4. Substitute the values into the average rate of change formula: $\backslash frac\{f(9)\; -\; f(4)\}\{9\; -\; 4\}$.
5. Simplify the expression by evaluating the square roots and performing the subtraction and division to find the average rate of change.

Key Concepts

Average Rate of Change

The average rate of change of a function between two points measures how much the function's output changes per unit change in input. It is calculated as the difference in function values divided by the difference in input values, similar to the slope of a secant line connecting the two points on the graph.

Square Root Function

The square root function, denoted as f(x) = √x, outputs the non-negative number whose square is x. It is defined for x ≥ 0 and is a nonlinear function that increases at a decreasing rate, meaning its graph is a curve that flattens as x increases.

Evaluating Function Values

To find the average rate of change, you must evaluate the function at the given points x1 and x2. This involves substituting the x-values into the function to find f(x1) and f(x2), which are then used to compute the difference quotient.