The basic trapezoidal rule for approximating $I_f = \int_{a}^{b}f(x)dx$ is based on linear interpolation of $f$ at $x_0=a$ and $x_1 = b$. The Simpson rule is likewise based on quadratic polynomial interpolation. Consider now a a cubic Hermite polynomial, interpolating both $f$ and its derivative $f'$ at $a$ and $b$. The osculating interpolation formula gives
$$p_3(x) = f(a) +f'(a)(x-a)+f[a,a,b](x-a)^2+f[a,a,b,b](x-a)^2(x-b)$$ and integrating this yields(after some algebra)
$$I \approx \int_{a}^{b}p_3(x)dx = \dfrac{b-a}{2}[f(a)+f(b)]+\dfrac{(b-a)^2}{12}[f'(a)-f'(b)]$$
This formula is called the corrected trapezoidal rule.
Show that the error in the basic corrected trapezoidal rule can be estimated by
$$E(f) = \dfrac{f''''(\eta)}{720}(b-a)^5$$
I have no idea how to start on this question, any help would be appreciated.
Best Answer
I think you're in my class, so here's what I've scraped together
The error in polynomial interpolation is known to be
$$ f(x) - p_{n}(x) = f[x_{0},...,f_{n},f_{x}]\prod_{i=0}^{n}(x-x_{i}) $$
For the corrected trapezoidal rule, we have
$$ \int_{a}^{b} f[a,a,b,b,x](x-a)^{2}(x-b)^{2}dx $$
With the mean value theorem with divided differences, we can see that there must exist an $\eta$ in $[a,b]$ such that
$$ \int_{a}^{b} f[a,a,b,b,x](x-a)^{2}(x-b)^{2}dx = f[a,a,b,b,\eta]\int_{a}^{b}(x-a)^{2}(x-b)^{2}dx $$
The only thing remaining to do is the apply the theorem and simplify the integral, and the product should result in the error given.