To find the average rate of change of the function $f(x) = e^{x-2} - 4$ between $x = 2$ and $x = 4$, we use the formula for the average rate of change over an interval $[a, b]$:

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

Step 1: Calculate $f(2)$ and $f(4)$

1. Calculate $f(2)$: $f(2) = e^{2-2} - 4 = e^0 - 4 = 1 - 4 = -3$

2. Calculate $f(4)$: $f(4) = e^{4-2} - 4 = e^2 - 4$ Approximating $e^2 \approx 7.39$: $f(4) \approx 7.39 - 4 = 3.39$

Step 2: Apply the Formula

$\text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{3.39 - (-3)}{2} = \frac{3.39 + 3}{2} = \frac{6.39}{2} = 3.195$

Rounded to the nearest hundredth, the average rate of change is approximately:

$\text{Average Rate of Change} \approx 3.20$

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Would you like a breakdown of these steps or further assistance?

Here are five related questions:

1. How do we interpret the average rate of change in the context of exponential functions?
2. What if we calculated the instantaneous rate of change at $x = 3$?
3. How does the behavior of exponential functions affect their rate of change?
4. How would this calculation differ for a logarithmic function?
5. What are the implications of this average rate of change for values of $x$ beyond 4?

Tip: When working with exponential functions, approximating constants like $e$ can simplify calculations, but always remember to check if a more precise value is needed.