For the given exponential​ function, find the average rate for x equals 1 and x equals 7.
f left parenthesis x right parenthesis equals e Superscript x minus 1 Baseline minus 7
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Part 1
The average rate of change is approximately 
  
enter your response here.
​(Type an integer or decimal rounded to the nearest tenth as ​needed.)

To find the average rate of change of the function \( f(x) = e^{x - 1} - 7 \) from \( x = 1 \) to \( x = 7 \), we use the formula:

\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

where \( x_1 = 1 \) and \( x_2 = 7 \).

### Step 1: Calculate \( f(1) \)
Substitute \( x = 1 \) into the function:
\[
f(1) = e^{1 - 1} - 7 = e^0 - 7 = 1 - 7 = -6
\]

### Step 2: Calculate \( f(7) \)
Substitute \( x = 7 \) into the function:
\[
f(7) = e^{7 - 1} - 7 = e^6 - 7
\]
Since \( e^6 \) is approximately 403.43:
\[
f(7) \approx 403.43 - 7 = 396.43
\]

### Step 3: Apply the Average Rate of Change Formula
\[
\text{Average rate of change} = \frac{f(7) - f(1)}{7 - 1} = \frac{396.43 - (-6)}{6} = \frac{396.43 + 6}{6} = \frac{402.43}{6} \approx 67.1
\]

### Final Answer
The average rate of change is approximately **67.1**.

Would you like more details on the steps or have any questions?

Here are five related questions you might consider:
1. How is the average rate of change in an exponential function different from a linear function?
2. What happens to the average rate of change if the interval changes?
3. Can the average rate of change for an exponential function be negative?
4. How does the value of \( e \) affect the rate of change in exponential functions?
5. What are other applications of the average rate of change in real-world problems?

**Tip:** For exponential functions, the average rate of change over large intervals often grows rapidly due to the nature of the exponential growth.

For the given exponential function, find the average rate for x equals 1 and x equals 7.
f(x) = e^(x - 1) - 7
The average rate of change is approximately ___ (Type an integer or decimal rounded to the nearest tenth as needed.)

Find the average rate of change for the exponential function f(x) = e^(x - 1) - 7 from x = 1 to x = 7. This example guides you through calculating function values, applying the rate of change formula, and approximating the answer to 67.1, rounded to the nearest tenth. Suitable for students in Grades 10-12.

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