To find the average rate of change of the function $f(x) = e^{x-2} - 4$ between $x = 2$ and $x = 4$, we’ll use the formula for the average rate of change between two points $x = a$ and $x = b$:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

Step-by-Step Solution

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

2. Find $f(4)$: $f(4) = e^{4-2} - 4 = e^2 - 4$ Approximating $e^2 \approx 7.389$, $f(4) \approx 7.389 - 4 = 3.389$

3. Calculate the Average Rate of Change: Using $a = 2$, $b = 4$, $f(2) = -3$, and $f(4) \approx 3.389$, $\text{Average rate of change} = \frac{f(4) - f(2)}{4 - 2} = \frac{3.389 - (-3)}{2} = \frac{3.389 + 3}{2} = \frac{6.389}{2} \approx 3.2$

Answer

The average rate of change is approximately 3.2.

Would you like additional details or explanations?

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Here are five related questions to expand on this topic:

1. What is the average rate of change for $f(x) = e^{x} - 4$ from $x = 0$ to $x = 3$?
2. How does the average rate of change of $f(x) = e^{x-2} - 4$ vary as the interval widens?
3. What is the interpretation of the average rate of change for exponential functions?
4. How would the rate of change change if the function were $f(x) = e^{x-2} + 4$ instead?
5. Can the average rate of change be used to estimate the slope of the tangent line at a midpoint?

Tip: The average rate of change between two points in an exponential function can approximate the function's behavior, but exponential functions typically increase more steeply with larger intervals.