For the given exponential​ function, find the average rate for x equals 1 and x equals 9.
f left parenthesis x right parenthesis equals e Superscript x minus 1 Baseline minus 9
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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 hundredth as ​needed.)

To find the average rate of change of the function \( f(x) = e^{x - 1} - 9 \) between \( x = 1 \) and \( x = 9 \), we use the formula for the average rate of change over an interval \([a, b]\):

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

For this problem:

1. **Set** \( a = 1 \) and \( b = 9 \).
2. **Evaluate \( f(a) \) and \( f(b) \):**

   - \( f(1) = e^{1 - 1} - 9 = e^0 - 9 = 1 - 9 = -8 \)
   - \( f(9) = e^{9 - 1} - 9 = e^8 - 9 \)

3. **Calculate the average rate of change:**

\[
\text{Average Rate of Change} = \frac{f(9) - f(1)}{9 - 1} = \frac{(e^8 - 9) - (-8)}{8} = \frac{e^8 - 1}{8}
\]

4. **Approximate the value:**

Using the approximate value \( e^8 \approx 2980.96 \):

\[
\text{Average Rate of Change} \approx \frac{2980.96 - 1}{8} = \frac{2979.96}{8} \approx 372.5
\]

Therefore, the average rate of change is approximately **372.5**.

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. How is the average rate of change different from the instantaneous rate of change?
2. How would the rate of change differ if the interval was \([1, 5]\) instead of \([1, 9]\)?
3. What does the average rate of change represent in the context of exponential functions?
4. Can the average rate of change ever be negative for exponential functions like \( f(x) = e^{x - 1} - 9 \)?
5. How would we find the instantaneous rate of change at a specific point within the interval?

**Tip:** When working with exponential functions, knowing approximate values of constants like \( e^8 \) can simplify calculations when exact values are not necessary.

For the given exponential function, find the average rate for x equals 1 and x equals 9. f(x) = e^(x - 1) - 9

Learn how to calculate the average rate of change for the exponential function f(x) = e^(x - 1) - 9 between x = 1 and x = 9. This solution demonstrates the application of the average rate of change formula with detailed steps and approximations. Suitable for high school students studying exponential functions.

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