To find the average rate of change of the exponential function $f(x) = e^{x-1} - 7$ between $x = 1$ and $x = 7$, we can use the formula for the average rate of change:

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

Here:

• $a = 1$
• $b = 7$

Step 1: Compute $f(1)$

$f(1) = e^{1-1} - 7 = e^0 - 7 = 1 - 7 = -6$

Step 2: Compute $f(7)$

$f(7) = e^{7-1} - 7 = e^6 - 7$ Using an approximate value of $e^6 \approx 403.43$: $f(7) \approx 403.43 - 7 = 396.43$

Step 3: Compute the average rate of change

$\text{Average Rate of Change} = \frac{f(7) - f(1)}{7 - 1} = \frac{396.43 - (-6)}{6} = \frac{396.43 + 6}{6} = \frac{402.43}{6} \approx 67.07$

Final Answer:

The average rate of change is approximately 67.07.

Would you like additional details or clarification?

Related Questions:

1. How does the formula for average rate of change relate to the slope of a line?
2. Why do exponential functions typically exhibit rapid growth or decay?
3. What is the significance of the base $e$ in this function?
4. How would the average rate of change differ if calculated over smaller intervals (e.g., $x = 1$ to $x = 3$)?
5. Can you generalize this process to calculate average rate of change for other functions?

Tip:

To get more accurate results for exponential functions, use a calculator or software for precise values of $e^x$.