For the given exponential​ function, find the average rate for x equals 1 and x equals 3.
f left parenthesis x right parenthesis equals e Superscript x minus 1 Baseline minus 3
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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 \( f(x) = e^{x - 1} - 3 \) between \( x = 1 \) and \( x = 3 \), 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 \( x_2 = 3 \).

1. **Calculate \( f(1) \):**

   \[
   f(1) = e^{1 - 1} - 3 = e^0 - 3 = 1 - 3 = -2
   \]

2. **Calculate \( f(3) \):**

   \[
   f(3) = e^{3 - 1} - 3 = e^2 - 3
   \]

   The value of \( e^2 \) is approximately \( 7.389 \), so:

   \[
   f(3) \approx 7.389 - 3 = 4.389
   \]

3. **Calculate the Average Rate of Change:**

   \[
   \text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{4.389 - (-2)}{2} = \frac{4.389 + 2}{2} = \frac{6.389}{2} = 3.2
   \]

So, the average rate of change is approximately **3.2**. 

Would you like further explanation or details?

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Here are some related questions:
1. How does the average rate of change differ from the instantaneous rate of change?
2. How can we use derivatives to find the rate of change at any given point?
3. What would the average rate of change be over a different interval, say from \( x = 1 \) to \( x = 4 \)?
4. What are the applications of the average rate of change in real-world problems?
5. How does the exponential base \( e \) affect the growth rate of exponential functions?

**Tip:** When finding average rates of change, make sure to evaluate the function accurately at each endpoint before calculating the difference.

For the given exponential function, find the average rate for x = 1 and x = 3. f(x) = e^(x - 1) - 3. The average rate of change is approximately (Type an integer or decimal rounded to the nearest tenth as needed.)

Discover how to find the average rate of change for the exponential function f(x) = e^(x - 1) - 3 between x = 1 and x = 3. This problem explores essential calculus concepts using the formula for average rate of change. Suitable for students in Grades 10-12.

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