Math Problem Statement

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

Solution

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

favg=1baabf(x)dxf_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x) \, dx

For the function f(x)=2x2+1ex/3f(x) = \frac{2x^2 + 1}{e^{x/3}} over the interval [0,3][0, 3], we can apply this formula. Here, a=0a = 0 and b=3b = 3. The average value becomes:

favg=130032x2+1ex/3dxf_{\text{avg}} = \frac{1}{3 - 0} \int_0^3 \frac{2x^2 + 1}{e^{x/3}} \, dx

This simplifies to:

favg=13032x2+1ex/3dxf_{\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:

032x2+1ex/3dx3.261\int_0^3 \frac{2x^2 + 1}{e^{x/3}} \, dx \approx 3.261

Thus, the average value of the function is approximately:

favg=13×3.2611.087f_{\text{avg}} = \frac{1}{3} \times 3.261 \approx 1.087

The average value of the function over [0,3][0, 3] is approximately 1.0871.087.

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 ex/3e^{x/3} affect the behavior of the function f(x)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.

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Math Problem Analysis

Mathematical Concepts

Calculus
Definite Integrals
Average Value of a Function
Numerical Integration

Formulas

f_avg = (1 / (b - a)) * ∫[a to b] f(x) dx

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate Calculus (Grades 11-12 and beyond)