The average rate of change of a function $f(x)$ over an interval $[a, b]$ is calculated using the formula:

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

Here, the function is $f(x) = e^{x-1} - 7$, and the interval is $[1, 7]$.

Step 1: Compute $f(1)$

Substitute $x = 1$ into $f(x)$:

$f(1) = e^{1-1} - 7 = e^0 - 7 = 1 - 7 = -6.$

Step 2: Compute $f(7)$

Substitute $x = 7$ into $f(x)$:

$f(7) = e^{7-1} - 7 = e^6 - 7.$

Step 3: Find the Average Rate of Change

Substitute $f(1) = -6$ and $f(7) = e^6 - 7$ into the formula:

$\text{Average Rate of Change} = \frac{f(7) - f(1)}{7 - 1}.$

Simplify the numerator:

$f(7) - f(1) = (e^6 - 7) - (-6) = e^6 - 7 + 6 = e^6 - 1.$

Simplify the denominator:

$7 - 1 = 6.$

So the average rate of change is:

$\text{Average Rate of Change} = \frac{e^6 - 1}{6}.$

Final Answer:

The average rate of change of $f(x)$ over $[1, 7]$ is:

$\boxed{\frac{e^6 - 1}{6}}.$

Would you like a step-by-step numerical evaluation for $e^6$, or do you have further questions?

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1. How is the average rate of change related to the derivative of the function?
2. Can the average rate of change ever equal zero? If so, under what conditions?
3. How does the exponential function $e^x$ differ from polynomial functions in terms of growth rate?
4. What is the geometric interpretation of the average rate of change on a graph of $f(x)$?
5. How would this calculation change if the interval were modified, e.g., $[2, 5]$?

Tip: The average rate of change represents the slope of the secant line between two points on the graph of the function!