For the given exponential​ function, find the average rate for x equals 1 and x equals 5.
f left parenthesis x right parenthesis equals e Superscript x minus 1 Baseline minus 5
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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 exponential function \( f(x) = e^{x-1} - 5 \) between \( x = 1 \) and \( x = 5 \), we can use the formula for the average rate of change:

\[
\text{Average rate of change} = \frac{f(5) - f(1)}{5 - 1}
\]

### Step 1: Calculate \( f(5) \)
\[
f(5) = e^{5-1} - 5 = e^4 - 5
\]
Using \( e^4 \approx 54.598 \),
\[
f(5) \approx 54.598 - 5 = 49.598
\]

### Step 2: Calculate \( f(1) \)
\[
f(1) = e^{1-1} - 5 = e^0 - 5 = 1 - 5 = -4
\]

### Step 3: Find the average rate of change
\[
\text{Average rate of change} = \frac{49.598 - (-4)}{5 - 1} = \frac{49.598 + 4}{4} = \frac{53.598}{4} \approx 13.40
\]

Thus, the average rate of change is approximately **13.40**.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the general formula for the average rate of change?
2. How do you differentiate an exponential function?
3. What happens to the average rate of change as \( x \) increases for this function?
4. Can you calculate the instantaneous rate of change at \( x = 3 \)?
5. How does the average rate of change relate to the slope of the secant line?

**Tip**: Always remember to round to the nearest hundredth when required, especially in real-world problems for clarity and precision.

For the given exponential function, find the average rate for x equals 1 and x equals 5. f(x) = e^(x-1) - 5.

Learn how to calculate the average rate of change for the exponential function f(x) = e^(x-1) - 5 between x = 1 and x = 5. This step-by-step solution covers key concepts such as exponential functions and rate of change, suitable for high school students in Grades 10-12.

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