The average rate of change of a function $f(x)$ over a closed interval $[a, b]$ is given by the formula:

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

For the function $f(x) = x^4 - 5x$ on the interval $[0, 3]$, we need to calculate:

1. $f(0)$
2. $f(3)$
3. Plug the values into the formula.

Step 1: Calculate $f(0)$

$f(0) = 0^4 - 5(0) = 0$

Step 2: Calculate $f(3)$

$f(3) = 3^4 - 5(3) = 81 - 15 = 66$

Step 3: Use the average rate of change formula

$\text{Average Rate of Change} = \frac{f(3) - f(0)}{3 - 0} = \frac{66 - 0}{3 - 0} = \frac{66}{3} = 22$

Thus, the average rate of change of the function $f(x) = x^4 - 5x$ on the interval $[0, 3]$ is 22.

Do you want more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How do you find the instantaneous rate of change of $f(x) = x^4 - 5x$ at $x = 3$?
2. What is the average rate of change of $f(x) = x^4 - 5x$ on the interval $[1, 3]$?
3. How does the average rate of change differ from the derivative at a point?
4. How would the average rate of change change if the interval was $[-1, 3]$?
5. What is the significance of a positive average rate of change in terms of the behavior of the function?

Tip: The average rate of change over an interval is the slope of the secant line connecting the two points on the function graph over that interval.