To find the average rate of change of the function $f(x) = e^{x - 1} - 3$ over the interval $x = 1$ to $x = 3$, we use the formula for the average rate of change between two points:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

where $x_1 = 1$ and $x_2 = 3$.

Step 1: Calculate $f(x_1)$

$f(1) = e^{1 - 1} - 3 = e^0 - 3 = 1 - 3 = -2$

Step 2: Calculate $f(x_2)$

$f(3) = e^{3 - 1} - 3 = e^2 - 3$

Using the approximate value of $e^2 \approx 7.389$, $f(3) \approx 7.389 - 3 = 4.389$

Step 3: Substitute into the formula

$\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{4.389 - (-2)}{2} = \frac{4.389 + 2}{2} = \frac{6.389}{2} \approx 3.2$

Answer

The average rate of change is approximately 3.2.

Would you like further details or have any questions?

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Here are some related questions for practice:

1. What is the average rate of change of $f(x) = e^{x} - 4$ from $x = 0$ to $x = 2$?
2. How do you interpret the average rate of change of an exponential function in real-world contexts?
3. What is the significance of the average rate of change for non-linear functions like exponential functions?
4. How would the rate of change differ if we extended the interval to $x = 1$ and $x = 4$?
5. Can the average rate of change be negative for exponential functions?

Tip: For exponential functions, the rate of change increases as $x$ increases, making these functions grow rapidly.