To find the average rate of change of the function $f(x) = e^{x - 1} - 9$ between $x = 1$ and $x = 9$, we use the formula for the average rate of change over an interval $[a, b]$:

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

For this problem:

1. Set $a = 1$ and $b = 9$.

2. Evaluate $f(a)$ and $f(b)$:

   • $f(1) = e^{1 - 1} - 9 = e^0 - 9 = 1 - 9 = -8$
   • $f(9) = e^{9 - 1} - 9 = e^8 - 9$

3. Calculate the average rate of change:

$\text{Average Rate of Change} = \frac{f(9) - f(1)}{9 - 1} = \frac{(e^8 - 9) - (-8)}{8} = \frac{e^8 - 1}{8}$

4. Approximate the value:

Using the approximate value $e^8 \approx 2980.96$:

$\text{Average Rate of Change} \approx \frac{2980.96 - 1}{8} = \frac{2979.96}{8} \approx 372.5$

Therefore, the average rate of change is approximately 372.5.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How is the average rate of change different from the instantaneous rate of change?
2. How would the rate of change differ if the interval was $[1, 5]$ instead of $[1, 9]$?
3. What does the average rate of change represent in the context of exponential functions?
4. Can the average rate of change ever be negative for exponential functions like $f(x) = e^{x - 1} - 9$?
5. How would we find the instantaneous rate of change at a specific point within the interval?

Tip: When working with exponential functions, knowing approximate values of constants like $e^8$ can simplify calculations when exact values are not necessary.