To find the average rate of change of the function $f(x) = e^{x-1} - 5$ between $x = 1$ and $x = 5$, we use the formula:

$\text{Average rate of change} = \frac{f(5) - f(1)}{5 - 1}$

Step 1: Compute $f(1)$ and $f(5)$

The function is $f(x) = e^{x-1} - 5$.

• At $x = 1$: $f(1) = e^{1-1} - 5 = e^0 - 5 = 1 - 5 = -4$

• At $x = 5$: $f(5) = e^{5-1} - 5 = e^4 - 5$

The value of $e^4$ is approximately 54.598.

$f(5) \approx 54.598 - 5 = 49.598$

Step 2: Calculate the average rate of change

$\text{Average rate of change} = \frac{f(5) - f(1)}{5 - 1} = \frac{49.598 - (-4)}{4} = \frac{49.598 + 4}{4} = \frac{53.598}{4} \approx 13.40$

Final Answer:

The average rate of change is approximately 13.40.

---

Would you like more details or have any questions?

Expanding Questions:

1. What does the average rate of change represent in this context?
2. How does the exponential growth of $e^x$ influence the rate of change?
3. Can you find the instantaneous rate of change at $x = 3$?
4. What would the average rate of change be over a different interval, say $x = 0$ to $x = 2$?
5. How does the subtraction of 5 impact the graph of $f(x)$?

Tip:

When computing exponential functions, it's helpful to keep a calculator or approximation table for $e^x$ values handy for faster calculations!