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

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

Step 1: Identify $a$ and $b$

• $a = 1$
• $b = 7$

Step 2: Calculate $f(1)$ and $f(7)$

The function is $f(x) = e^{x-1} - 7$.

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

2. For $f(7)$: $f(7) = e^{7-1} - 7 = e^6 - 7$

Approximating $e^6$ using a calculator: $e^6 \approx 403.4288$ $f(7) = 403.4288 - 7 = 396.4288$

Step 3: Compute the average rate of change

$\text{Average rate of change} = \frac{f(7) - f(1)}{7 - 1}$ Substitute the values: $\text{Average rate of change} = \frac{396.4288 - (-6)}{7 - 1}$ $\text{Average rate of change} = \frac{396.4288 + 6}{6}$ $\text{Average rate of change} = \frac{402.4288}{6} \approx 67.07$

Final Answer:

The average rate of change is approximately 67.07.

Do you want a more detailed breakdown or assistance with similar problems?

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1. What does the average rate of change signify in exponential functions?
2. How does $e^x$ influence the steepness of the curve?
3. What is the role of the constant $-7$ in the function $f(x)$?
4. How can we generalize the average rate of change formula for other intervals?
5. How does the base $e$ differ from other exponential bases?

Tip: Always calculate $e^x$ using at least four decimal places to ensure precision in intermediate steps.