To derive the trapezoidal method using the method of undetermined coefficients and obtain its truncation error, we’ll follow these steps:

1. Overview of the Trapezoidal Method

The trapezoidal rule is used to approximate the integral of a function $f(x)$ over an interval $[a, b]$. The formula for the trapezoidal rule is:

$\int_a^b f(x) \, dx \approx \frac{h}{2} \left( f(a) + f(b) \right)$

where $h = b - a$ is the width of the interval.

2. Using the Method of Undetermined Coefficients

To derive this formula using undetermined coefficients, we approximate the function $f(x)$ over the interval $[a, b]$ by assuming a linear combination of function values at the endpoints $a$ and $b$.

1. Assume that the integral $\int_a^b f(x) \, dx$ can be approximated as a weighted sum: $\int_a^b f(x) \, dx \approx A f(a) + B f(b)$ where $A$ and $B$ are coefficients that we need to determine.

2. To find $A$ and $B$, we use the Taylor expansions of $f(a)$ and $f(b)$ around the midpoint $\frac{a+b}{2}$:

   • $f(a) = f\left( \frac{a+b}{2} \right) - \frac{h}{2} f'\left( \frac{a+b}{2} \right) + \frac{h^2}{8} f''\left( \frac{a+b}{2} \right) + O(h^3)$
   • $f(b) = f\left( \frac{a+b}{2} \right) + \frac{h}{2} f'\left( \frac{a+b}{2} \right) + \frac{h^2}{8} f''\left( \frac{a+b}{2} \right) + O(h^3)$

3. Adding these expressions gives: $f(a) + f(b) = 2f\left( \frac{a+b}{2} \right) + \frac{h^2}{4} f''\left( \frac{a+b}{2} \right) + O(h^3)$

4. Now, we want this approximation to match the exact integral, up to an error term of $O(h^3)$, for the case where $f(x)$ is a polynomial of degree at most 1. By comparing terms, we find that the correct choice for $A$ and $B$ is $A = B = \frac{h}{2}$.

Therefore, the approximation becomes: $\int_a^b f(x) \, dx \approx \frac{h}{2} \left( f(a) + f(b) \right)$

This is the trapezoidal rule.

3. Truncation Error

To find the truncation error, we analyze the error term in the Taylor expansions:

1. From the previous expansions, we have: $f(a) + f(b) = 2f\left( \frac{a+b}{2} \right) + \frac{h^2}{4} f''\left( \frac{a+b}{2} \right) + O(h^3)$

2. The exact integral over $[a, b]$ using the midpoint would be: $\int_a^b f(x) \, dx = h f\left( \frac{a+b}{2} \right) + \frac{h^3}{24} f''\left( \frac{a+b}{2} \right) + O(h^5)$

3. Comparing the approximation to the exact integral, the error term is: $\text{Error} = \int_a^b f(x) \, dx - \frac{h}{2} \left( f(a) + f(b) \right) = -\frac{h^3}{12} f''\left( \frac{a+b}{2} \right) + O(h^5)$

Thus, the truncation error for the trapezoidal method is: $\text{Truncation Error} = -\frac{h^3}{12} f''(c)$ where $c$ is some point in the interval $[a, b]$. This indicates that the trapezoidal rule is accurate to order $O(h^3)$ for smooth functions.

Would you like more detailed explanations on any part, or have any other questions?

Here are some related questions that may be helpful:

1. How does the trapezoidal rule compare to the midpoint rule in terms of accuracy?
2. What is the derivation of the Simpson’s rule using undetermined coefficients?
3. How does the error term affect the accuracy of numerical integration for functions with high curvature?
4. How can adaptive integration methods improve the trapezoidal rule?
5. How is the trapezoidal rule applied to composite intervals?

Tip: When dealing with numerical integration, always check the behavior of the function. Sharp changes or high curvature may require a smaller $h$ for accurate results.