To find the average rate of change of the function $f(x) = e^x - 1 - 7$ between $x = 1$ and $x = 7$, we use the formula for the average rate of change:

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

Here, $a = 1$ and $b = 7$. Let's calculate:

1. Compute $f(1)$: $f(1) = e^1 - 1 - 7 = e - 8$

2. Compute $f(7)$: $f(7) = e^7 - 1 - 7 = e^7 - 8$

3. Calculate the average rate of change: $\text{Average rate of change} = \frac{f(7) - f(1)}{7 - 1}$ $= \frac{(e^7 - 8) - (e - 8)}{6}$ $= \frac{e^7 - e}{6}$

Thus, the average rate of change is: $\boxed{\frac{e^7 - e}{6}}$

Would you like a numerical approximation of this result or further details?

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Relative Questions:

1. What does the average rate of change represent geometrically in this context?
2. How does this result compare to the instantaneous rate of change at a specific $x$-value?
3. What are the implications of the function $e^x - 1 - 7$ being exponential on its rate of change?
4. Can we use this formula to calculate the average rate of change for other intervals?
5. How would the answer change if the interval were shifted, e.g., $x = 2$ to $x = 5$?

Tip:

When working with exponential functions, consider expressing results in exact forms involving $e$ unless specifically asked for numerical approximations.