The problem asks to find the average rate of change of the function $f(x) = x^2 + 8x$ from $x_1 = 1$ to $x_2 = 4$.

The formula for the average rate of change of a function between two points $x_1$ and $x_2$ is given by:

$\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Let's break it down step by step:

1. First, find $f(1)$ and $f(4)$.

• $f(1) = (1)^2 + 8(1) = 1 + 8 = 9$
• $f(4) = (4)^2 + 8(4) = 16 + 32 = 48$

2. Now, use the formula for the average rate of change:

$\text{Average rate of change} = \frac{f(4) - f(1)}{4 - 1} = \frac{48 - 9}{4 - 1} = \frac{39}{3} = 13$

Thus, the average rate of change is $13$.

Do you have any questions or would you like further details? Here are 5 related questions:

1. How do you find the rate of change for more complex functions like $f(x) = \sin(x)$?
2. What is the geometric interpretation of the average rate of change?
3. How does the average rate of change relate to the derivative of a function?
4. Can the average rate of change be negative, and what would that signify?
5. How would you compute the average rate of change for a piecewise function?

Tip: The average rate of change is similar to finding the slope of the secant line between two points on the graph of the function.