To find the average value of the function $f(x) = \frac{e^x}{2}$ over the interval $[0, 25]$, we use the formula for the average value of a continuous function over a given interval:

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

For this problem, $f(x) = \frac{e^x}{2}$, $a = 0$, and $b = 25$. Therefore, the average value is:

$\text{Average value} = \frac{1}{25-0} \int_0^{25} \frac{e^x}{2} \, dx$

First, we compute the integral:

$\int_0^{25} \frac{e^x}{2} \, dx$

To integrate $\frac{e^x}{2}$, we can factor out the constant $\frac{1}{2}$:

$\frac{1}{2} \int_0^{25} e^x \, dx$

The integral of $e^x$ is simply $e^x$, so we have:

$\frac{1}{2} \left[ e^x \right]_0^{25}$

Evaluating this at the bounds $x = 0$ and $x = 25$:

$\frac{1}{2} \left( e^{25} - e^0 \right)$

Since $e^0 = 1$:

$\frac{1}{2} \left( e^{25} - 1 \right)$

Next, we divide this result by the length of the interval $25 - 0 = 25$:

$\text{Average value} = \frac{1}{25} \cdot \frac{1}{2} \left( e^{25} - 1 \right)$

$= \frac{1}{50} \left( e^{25} - 1 \right)$

To find a numerical value, we need to compute $e^{25}$. Using a calculator, we find:

$e^{25} \approx 7.20048993 \times 10^{10}$

Substituting this value back in:

$\text{Average value} = \frac{1}{50} \left( 7.20048993 \times 10^{10} - 1 \right)$

$\approx \frac{1}{50} \left( 7.20048993 \times 10^{10} \right)$

$\approx 1.44009799 \times 10^9$

Rounding to two decimal places:

$\text{Average value} \approx 1440097990.00$

So, the average value of the function $f(x) = \frac{e^x}{2}$ on the interval $[0, 25]$ is approximately 1,440,097,990.00.

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How does the exponential function $e^x$ affect the average value compared to polynomial functions?
2. What are the applications of finding the average value of an exponential function in real-world problems?
3. How would the result change if the interval was different, such as $[0, 10]$?
4. What is the significance of the constant $\frac{1}{2}$ in the function $f(x) = \frac{e^x}{2}$?
5. How do you interpret the average value of an exponentially growing function over a large interval?
6. What are some common methods for approximating integrals if an exact solution is difficult to compute?
7. How does the value of $e^{25}$ compare to other large exponential values?
8. What other types of functions might exhibit similar rapid growth over large intervals?

Tip: When dealing with functions involving $e^x$, be aware of the rapid growth rate of the exponential function, especially over large intervals, as it can lead to very large values.