To find the average rate of change of the exponential function $f(x) = e^{x-1} - 5$ between $x = 1$ and $x = 5$, we can use the formula for the average rate of change:

$\text{Average rate of change} = \frac{f(5) - f(1)}{5 - 1}$

Step 1: Calculate $f(5)$

$f(5) = e^{5-1} - 5 = e^4 - 5$ Using $e^4 \approx 54.598$, $f(5) \approx 54.598 - 5 = 49.598$

Step 2: Calculate $f(1)$

$f(1) = e^{1-1} - 5 = e^0 - 5 = 1 - 5 = -4$

Step 3: Find the average rate of change

$\text{Average rate of change} = \frac{49.598 - (-4)}{5 - 1} = \frac{49.598 + 4}{4} = \frac{53.598}{4} \approx 13.40$

Thus, the average rate of change is approximately 13.40.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the general formula for the average rate of change?
2. How do you differentiate an exponential function?
3. What happens to the average rate of change as $x$ increases for this function?
4. Can you calculate the instantaneous rate of change at $x = 3$?
5. How does the average rate of change relate to the slope of the secant line?

Tip: Always remember to round to the nearest hundredth when required, especially in real-world problems for clarity and precision.