For the given exponential​ function, find the average rate for

x equals 2x=2

and

x equals 4x=4.

f left parenthesis x right parenthesis equals e Superscript x minus 2 Baseline minus 4f(x)=ex−2−4

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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 \( f(x) = e^{x-2} - 4 \) between \( x = 2 \) and \( x = 4 \), we use the formula for the average rate of change over an interval \([a, b]\):

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

### Step 1: Calculate \( f(2) \) and \( f(4) \)

1. **Calculate \( f(2) \):**
   \[
   f(2) = e^{2-2} - 4 = e^0 - 4 = 1 - 4 = -3
   \]

2. **Calculate \( f(4) \):**
   \[
   f(4) = e^{4-2} - 4 = e^2 - 4
   \]
   Approximating \( e^2 \approx 7.39 \):
   \[
   f(4) \approx 7.39 - 4 = 3.39
   \]

### Step 2: Apply the Formula

\[
\text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{3.39 - (-3)}{2} = \frac{3.39 + 3}{2} = \frac{6.39}{2} = 3.195
\]

Rounded to the nearest hundredth, the average rate of change is approximately:

\[
\text{Average Rate of Change} \approx 3.20
\]

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Would you like a breakdown of these steps or further assistance?

Here are five related questions:
1. How do we interpret the average rate of change in the context of exponential functions?
2. What if we calculated the instantaneous rate of change at \( x = 3 \)?
3. How does the behavior of exponential functions affect their rate of change?
4. How would this calculation differ for a logarithmic function?
5. What are the implications of this average rate of change for values of \( x \) beyond 4?

**Tip:** When working with exponential functions, approximating constants like \( e \) can simplify calculations, but always remember to check if a more precise value is needed.

For the given exponential function, find the average rate for x equals 2 and x equals 4. f(x) = e^(x-2) - 4

Learn how to calculate the average rate of change for the exponential function f(x) = e^(x-2) - 4 between x = 2 and x = 4. This step-by-step solution demonstrates key concepts such as exponential functions and the formula for average rate of change, with a final answer of approximately 3.20.

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