Question

find the average rate of change of the function f(x) = √x from x₁ = 4 to x₂ = 36.
the average rate of change is (simplify your answer.)

Explanation:

Step1: Recall the average - rate - of - change formula

The formula for the average rate of change of a function $y = f(x)$ from $x_1$ to $x_2$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$.

Step2: Calculate $f(x_1)$ and $f(x_2)$

Given $f(x)=\sqrt{x}$, when $x_1 = 4$, $f(x_1)=\sqrt{4}=2$. When $x_2 = 36$, $f(x_2)=\sqrt{36}=6$.

Step3: Substitute into the formula

Substitute $f(x_1) = 2$, $f(x_2)=6$, $x_1 = 4$ and $x_2 = 36$ into $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$, we get $\frac{6 - 2}{36-4}=\frac{4}{32}$.

Step4: Simplify the result

$\frac{4}{32}=\frac{1}{8}$.

Answer:

$\frac{1}{8}$