基本求积公式的简单推导
Simple derivation of basic quadrature formulas
AI总结：
本文通过分部积分法，简洁地证明了中点法则、梯形法则和辛普森法则的误差估计，并推广到包含端点一阶导数的修正梯形法则。
摘要：
对于紧区间上的数值积分，证明了中点法则、梯形法则和辛普森法则的简单证明。中点法则和梯形法则假设被积函数二次连续可微，辛普森法则假设被积函数四次连续可微。误差用被积函数的二阶或四阶导数的一致范数估计。证明仅使用分部积分法，应用于被积函数的二阶或四阶导数乘以适当的多项式或分段多项式函数。还以这种方式证明了包含积分区间端点处被积函数一阶导数的修正梯形法则，其误差估计中的系数小于中点法则和梯形法则。这些证明适合在微积分或初等数值分析课程中展示。还提出了几个学生项目建议。
英文摘要：
Simple proofs of the midpoint, trapezoidal and Simpson's rules are proved for numerical integration on a compact interval. The integrand is assumed to be twice continuously differentiable for the midpoint and trapezoidal rules, and to be four times continuously differentiable for Simpson's rule. Errors are estimated in terms of the uniform norm of second or fourth derivatives of the integrand. The proof uses only integration by parts, applied to the second or fourth derivative of the integrand, multiplied by an appropriate polynomial or piecewise polynomial function. A corrected trapezoidal rule that includes the first derivative of the integrand at the endpoints of the integration interval is also proved in this manner, the coefficient in the error estimate being smaller than for the midpoint and trapezoidal rules. The proofs are suitable for presentation in a calculus or elementary numerical analysis class. Several student projects are suggested.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
