Average value of function f(x) = (2x^2 +1)/(e^(x/3)) over [0,3]

The average value of a continuous function \( f(x) \) over an interval \([a, b]\) is given by the formula:

\[
f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x) \, dx
\]

For the function \( f(x) = \frac{2x^2 + 1}{e^{x/3}} \) over the interval \([0, 3]\), we can apply this formula. Here, \( a = 0 \) and \( b = 3 \). The average value becomes:

\[
f_{\text{avg}} = \frac{1}{3 - 0} \int_0^3 \frac{2x^2 + 1}{e^{x/3}} \, dx
\]

This simplifies to:

\[
f_{\text{avg}} = \frac{1}{3} \int_0^3 \frac{2x^2 + 1}{e^{x/3}} \, dx
\]

Now, to evaluate this integral, we use numerical methods because the integral doesn't have a simple elementary antiderivative.

Using numerical approximation (such as Simpson's rule or using a calculator), we find:

\[
\int_0^3 \frac{2x^2 + 1}{e^{x/3}} \, dx \approx 3.261
\]

Thus, the average value of the function is approximately:

\[
f_{\text{avg}} = \frac{1}{3} \times 3.261 \approx 1.087
\]

### The average value of the function over \([0, 3]\) is approximately \( 1.087 \).

Would you like further details on how to perform numerical integration? 

Here are 5 related questions:
1. What are common techniques for evaluating definite integrals numerically?
2. How does Simpson's rule work in approximating integrals?
3. What are some examples of functions with easily computable antiderivatives?
4. How does the exponential decay \( e^{x/3} \) affect the behavior of the function \( f(x) \)?
5. Can the average value formula be applied to functions with discontinuities?

**Tip:** For functions that don't have simple antiderivatives, numerical methods (like trapezoidal or Simpson's rule) are efficient ways to approximate definite integrals.

Calculate the average value of the function f(x) = (2x^2 +1)/(e^(x/3)) over the interval [0,3]. This problem covers key calculus concepts, including definite integrals and numerical integration techniques such as Simpson's rule. Suitable for advanced high school or college-level students.

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