To find the average rate of change of the function $f(x) = e^{x - 1} - 7$ from $x = 1$ to $x = 7$, we use the formula:

$\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

where $x_1 = 1$ and $x_2 = 7$.

Step 1: Calculate $f(1)$

Substitute $x = 1$ into the function: $f(1) = e^{1 - 1} - 7 = e^0 - 7 = 1 - 7 = -6$

Step 2: Calculate $f(7)$

Substitute $x = 7$ into the function: $f(7) = e^{7 - 1} - 7 = e^6 - 7$ Since $e^6$ is approximately 403.43: $f(7) \approx 403.43 - 7 = 396.43$

Step 3: Apply the Average Rate of Change Formula

$\text{Average rate of change} = \frac{f(7) - f(1)}{7 - 1} = \frac{396.43 - (-6)}{6} = \frac{396.43 + 6}{6} = \frac{402.43}{6} \approx 67.1$

Final Answer

The average rate of change is approximately 67.1.

Would you like more details on the steps or have any questions?

Here are five related questions you might consider:

1. How is the average rate of change in an exponential function different from a linear function?
2. What happens to the average rate of change if the interval changes?
3. Can the average rate of change for an exponential function be negative?
4. How does the value of $e$ affect the rate of change in exponential functions?
5. What are other applications of the average rate of change in real-world problems?

Tip: For exponential functions, the average rate of change over large intervals often grows rapidly due to the nature of the exponential growth.