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Part 1
For the given exponential​ function, find the average rate for x equals 2 and x equals 6.
f left parenthesis x right parenthesis equals e Superscript x minus 2 Baseline minus 6
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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 \( f(x) = e^{x-2} - 6 \) from \( x = 2 \) to \( x = 6 \), we use the formula:

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

where \( x_1 = 2 \) and \( x_2 = 6 \).

### Step 1: Calculate \( f(2) \) and \( f(6) \)

1. **Finding \( f(2) \):**
   \[
   f(2) = e^{2-2} - 6 = e^0 - 6 = 1 - 6 = -5
   \]

2. **Finding \( f(6) \):**
   \[
   f(6) = e^{6-2} - 6 = e^4 - 6
   \]

To approximate \( e^4 \), we know that \( e^4 \approx 54.6 \).

\[
f(6) \approx 54.6 - 6 = 48.6
\]

### Step 2: Calculate the Average Rate of Change

\[
\text{Average rate of change} = \frac{f(6) - f(2)}{6 - 2} = \frac{48.6 - (-5)}{4} = \frac{48.6 + 5}{4} = \frac{53.6}{4} \approx 13.4
\]

### Final Answer

The average rate of change is approximately **13.4**.

Would you like further clarification on this process or have any questions?

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Here are related questions for deeper understanding:
1. How do we interpret the average rate of change for different types of functions?
2. What is the significance of the constant \( e \) in exponential functions?
3. How can we estimate values like \( e^4 \) if not using a calculator?
4. How does the average rate of change compare to the instantaneous rate of change at a single point?
5. Can we apply the same approach to logarithmic functions?

**Tip:** The average rate of change gives a sense of how much the function increases or decreases over an interval.

For the given exponential function, find the average rate for x = 2 and x = 6. f(x) = e^(x-2) - 6.

Find the average rate of change for the exponential function f(x) = e^(x-2) - 6 over the interval from x = 2 to x = 6. This solution involves calculus concepts, rate of change, and exponential functions, suitable for high school calculus students.

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