The average rate of change of a function $f(x)$ over an interval $[a, b]$ is calculated using the formula:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

For the function $f(x) = x^2 + x$, the interval is $[1, 5]$.

Step 1: Evaluate $f(1)$ and $f(5)$

• $f(1) = 1^2 + 1 = 1 + 1 = 2$
• $f(5) = 5^2 + 5 = 25 + 5 = 30$

Step 2: Apply the formula

$\text{Average Rate of Change} = \frac{f(5) - f(1)}{5 - 1} = \frac{30 - 2}{5 - 1} = \frac{28}{4} = 7$

Thus, the average rate of change of $f(x)$ over the interval $[1, 5]$ is 7.

Would you like more details or have any other questions? Here are some related questions:

1. How would the average rate of change change if the interval was $[2, 6]$?
2. How can you determine the instantaneous rate of change at a specific point on the function?
3. What would be the average rate of change for a different function like $f(x) = x^3 + 2x$ over the interval $[1, 5]$?
4. What happens to the average rate of change if the function is linear?
5. How does the average rate of change relate to the slope of the secant line between two points?

Tip: The average rate of change is essentially the slope of the line connecting two points on the graph of the function.