To find the average rate of change of the function $f(x) = e^{x-2} - 6$ from $x = 2$ to $x = 6$, we use the formula:

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

where $x_1 = 2$ and $x_2 = 6$.

Step 1: Calculate $f(2)$ and $f(6)$

1. Finding $f(2)$: $f(2) = e^{2-2} - 6 = e^0 - 6 = 1 - 6 = -5$

2. Finding $f(6)$: $f(6) = e^{6-2} - 6 = e^4 - 6$

To approximate $e^4$, we know that $e^4 \approx 54.6$.

$f(6) \approx 54.6 - 6 = 48.6$

Step 2: Calculate the Average Rate of Change

$\text{Average rate of change} = \frac{f(6) - f(2)}{6 - 2} = \frac{48.6 - (-5)}{4} = \frac{48.6 + 5}{4} = \frac{53.6}{4} \approx 13.4$

Final Answer

The average rate of change is approximately 13.4.

Would you like further clarification on this process or have any questions?

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Here are related questions for deeper understanding:

1. How do we interpret the average rate of change for different types of functions?
2. What is the significance of the constant $e$ in exponential functions?
3. How can we estimate values like $e^4$ if not using a calculator?
4. How does the average rate of change compare to the instantaneous rate of change at a single point?
5. Can we apply the same approach to logarithmic functions?

Tip: The average rate of change gives a sense of how much the function increases or decreases over an interval.