To find the average rate of change of the function $f(x) = x^2 + 7x$ between $x = 0$ and $x = 6$, we use the formula:

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

where:

• $f(x) = x^2 + 7x$
• $a = 0$
• $b = 6$

First, calculate $f(0)$:

$f(0) = 0^2 + 7(0) = 0$

Next, calculate $f(6)$:

$f(6) = 6^2 + 7(6) = 36 + 42 = 78$

Now, use the average rate of change formula:

$\text{Average Rate of Change} = \frac{f(6) - f(0)}{6 - 0} = \frac{78 - 0}{6 - 0} = \frac{78}{6} = 13$

So, the average rate of change of the function $f(x) = x^2 + 7x$ between $x = 0$ and $x = 6$ is 13.

Would you like further details or have any questions about this? Here are some related questions you might find helpful:

1. What is the average rate of change of $f(x) = x^2 + 7x$ between $x = 2$ and $x = 5$?
2. How does the average rate of change compare to the instantaneous rate of change?
3. How do you interpret the average rate of change in a real-world context?
4. What does the derivative of $f(x) = x^2 + 7x$ tell us about the rate of change?
5. Can you find the average rate of change for a function over a non-linear interval?

Tip: The average rate of change of a function over an interval gives you the slope of the secant line that passes through the points on the function corresponding to the endpoints of the interval.