91Ó°ÊÓ

Find the degree of precision of the degree four Newton-Cotes Rule (often called Boole's Rule) $$ \int_{x_{0}}^{x_{4}} f(x) d x \approx \frac{2 h}{45}\left(7 y+32 y_{1}+12 y_{2}+32 y_{3}+7 y_{4}\right) $$

Develop a second-order method for approximating \(f^{\prime}(x)\) that uses the data \(f(x-2 h), f(x)\), and \(f(x+3 h)\) only.

Prove the second-order formula for the third derivative $$ f^{\prime \prime \prime}(x)=\frac{f(x-3 h)-6 f(x-2 h)+12 f(x-h)-10 f(x)+3 f(x+h)}{2 h^{3}}+O\left(h^{2}\right) . $$

The error term in the two-point forward-difference formula for \(f^{\prime}(x)\) can be written in other ways. Prove the alternative result $$ f^{\prime}(x)=\frac{f(x+h)-f(x)}{h}-\frac{h}{2} f^{\prime \prime}(x)-\frac{h^{2}}{6} f^{*}(c) $$ where \(c\) is between \(x\) and \(x+h\). We will use this error form in the derivation of the Crank-Nicolson Method in Chapter \(8 .\)