Numerical Simulations to Illuminate the Sufficient Conditions of Integral Approximation Theorems

Abstract

One of the crucial issue in the mathematical reasoning is that many students do not really understand the mathematical rules underlying the conditional statement p → q, whereas most of mathematical theorems are of this pattern. They believe that the falsity of p implies the falsity of q. This is tantamount to presuming that implication is equivalent to the inverse, but that’s not the case. This fact suggests that students have not been able in distinguishing the sufficient and the necessary conditions of a conditional statement. This article is concerned with this problem by investigating the sufficient conditions on the basic quadrature formulas of integral approximation and their implication to the convergence order attainable. In order to instill students’ critical thinking skills, theorems relating to the convergence orders of three basic methods are proven. Furthermore, the cases where the sufficient conditions are fulfilled as well as cases where they are not satisfied are examined by a series of numerical simulation. Mathematics Subject Classification: 03F03 , 65D30 , 65D32 , 65G50 , 65Y20