Determine the average value of the function $$ f(x) = e^{x} $$ from $$ x = 0 $$ to $$ x = 2 $$.

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To determine the **average value** of the function $$ f(x) = e^{x} $$ over the interval $$[0, 2]$$, we'll use the formula for the average value of a function on a closed interval $$[a, b]$$:

$$
\text{Average Value} = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx
$$

Let's apply this step-by-step.

### Step 1: Set Up the Average Value Formula
For the given function $$ f(x) = e^{x} $$ on the interval $$[0, 2]$$, the values are:
- $$ a = 0 $$
- $$ b = 2 $$

Plugging these into the formula:

$$
\text{Average Value} = \frac{1}{2 - 0} \int_{0}^{2} e^{x} \, dx = \frac{1}{2} \int_{0}^{2} e^{x} \, dx
$$

### Step 2: Compute the Definite Integral
Evaluate the integral $$ \int_{0}^{2} e^{x} \, dx $$:

$$
\int e^{x} \, dx = e^{x} + C
$$

Applying the limits from 0 to 2:

$$
\int_{0}^{2} e^{x} \, dx = e^{2} - e^{0} = e^{2} - 1
$$

### Step 3: Calculate the Average Value
Now, substitute the evaluated integral back into the average value formula:

$$
\text{Average Value} = \frac{1}{2} (e^{2} - 1)
$$

### Step 4: Numerical Approximation (Optional)
If you require a numerical approximation:

- The value of $$ e $$ (Euler's number) is approximately 2.71828.
- Therefore, $$ e^{2} \approx 7.3891 $$.

Plugging this into the equation:

$$
\text{Average Value} \approx \frac{1}{2} (7.3891 - 1) = \frac{1}{2} \times 6.3891 \approx 3.1946
$$

### Final Answer

- **Exact Average Value:** $$ \frac{e^{2} - 1}{2} $$
- **Approximate Average Value:** $$ 3.1946 $$

So, the average value of the function $$ f(x) = e^{x} $$ from $$ x = 0 $$ to $$ x = 2 $$ is $$ \frac{e^{2} - 1}{2} $$, which is approximately **3.1946**.
The average value of $$ f(x) = e^{x} $$ from $$ x = 0 $$ to $$ x = 2 $$ is $$ \frac{e^{2} - 1}{2} $$, which is approximately **3.1946**.

Determine the average value of the function from to .

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