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 \( f(x) = e^{x - 1} - 3 \) over the interval \( x = 1 \) to \( x = 3 \), we use the formula for the average rate of change between two points:

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

where \( x_1 = 1 \) and \( x_2 = 3 \).

### Step 1: Calculate \( f(x_1) \)
\[
f(1) = e^{1 - 1} - 3 = e^0 - 3 = 1 - 3 = -2
\]

### Step 2: Calculate \( f(x_2) \)
\[
f(3) = e^{3 - 1} - 3 = e^2 - 3
\]

Using the approximate value of \( e^2 \approx 7.389 \),
\[
f(3) \approx 7.389 - 3 = 4.389
\]

### Step 3: Substitute into the formula
\[
\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{4.389 - (-2)}{2} = \frac{4.389 + 2}{2} = \frac{6.389}{2} \approx 3.2
\]

### Answer
The average rate of change is approximately **3.2**.

Would you like further details or have any questions?

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Here are some related questions for practice:
1. What is the average rate of change of \( f(x) = e^{x} - 4 \) from \( x = 0 \) to \( x = 2 \)?
2. How do you interpret the average rate of change of an exponential function in real-world contexts?
3. What is the significance of the average rate of change for non-linear functions like exponential functions?
4. How would the rate of change differ if we extended the interval to \( x = 1 \) and \( x = 4 \)?
5. Can the average rate of change be negative for exponential functions?

**Tip:** For exponential functions, the rate of change increases as \( x \) increases, making these functions grow rapidly.

For the given exponential function, find the average rate for x equals 1 and x equals 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.)

Calculate the average rate of change for the exponential function f(x) = e^(x - 1) - 3 between x = 1 and x = 3. This problem covers key calculus concepts like exponential functions and average rate of change, ideal for students in Grades 11-12.

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