To find the average rate of change of the function $f(x) = 2^x - x$ over the interval $[0, 6]$, we can use the formula for the average rate of change:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

where $a = 0$ and $b = 6$ in this case.

1. Calculate $f(0)$: $f(0) = 2^0 - 0 = 1$

2. Calculate $f(6)$: $f(6) = 2^6 - 6 = 64 - 6 = 58$

3. Apply the values in the formula: $\text{Average rate of change} = \frac{f(6) - f(0)}{6 - 0} = \frac{58 - 1}{6} = \frac{57}{6} = 9.5$

So, the average rate of change for the interval $[0, 6]$ is 9.5.

Would you like further details or explanations?

Here are five related questions to deepen your understanding:

1. What does the average rate of change represent in the context of a function?
2. How would the average rate of change differ if we calculated it over a different interval, like $[1, 5]$?
3. How can you determine whether a function is increasing or decreasing over an interval?
4. How does the concept of average rate of change relate to the derivative of a function?
5. How does exponential growth affect the average rate of change compared to linear functions?

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