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

To find the average value of the function \( f(x) = e^{x} \) from \( x = 0 \) to \( x = 2 \), we can use the formula for the average value of a continuous function over an interval \([a, b]\): \[ \text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \] In this case, \( a = 0 \) and \( b = 2 \). Thus, we need to evaluate: \[ \text{Average value} = \frac{1}{2-0} \int_{0}^{2} e^{x} \, dx \] Now, we calculate the integral \( \int_{0}^{2} e^{x} \, dx \): \[ \int e^{x} \, dx = e^{x} + C \] Evaluating from \( 0 \) to \( 2 \): \[ \int_{0}^{2} e^{x} \, dx = e^{2} - e^{0} = e^{2} - 1 \] Now, substituting this back into the average value formula: \[ \text{Average value} = \frac{1}{2} (e^{2} - 1) \] Therefore, the average value of the function \( f(x) = e^{x} \) from \( x = 0 \) to \( x = 2 \) is: \[ \text{Average value} = \frac{e^{2} - 1}{2} \]