Application of Newton Polynomial Interpolation Method in Determining the Continuity of Functions Represented by Tabulated Discrete Points

Abstract

Most people are only familiar with functions that have been formulated explicitly y=f(x)) or implicitly (f(x,y)=0) However, functions obtained by researchers and engineers based on experimental data or field observations often do not have a known formula, and are therefore only represented in the form of tabulated discrete points. To determine the continuity of a tabulated function obtained from observational data, a function formula from the data is required. Consequently, the condition for the continuity of a function, where 〖lim〗┬(x→c)⁡〖f(x)=f(c)〗cannot be met. The problem in this research is to utilize data on the number of poor people from 2015-2021 and predict the monthly number of poor people using the Newton polynomial interpolation method with Maple. It also aims to prove the continuity of functions represented by tabulated discrete points by showing whether 〖lim〗┬(x→c)⁡〖p(x)=p(c)〗 or 〖lim〗┬(x→c)⁡〖p(x)≠p(c)〗. Based on the research results, it was found that Newton polynomial interpolation can be used to estimate the monthly number of poor people based on annual data, provided the conditions are met: the data represents a function, and the data table interval is changed to (1 )/12. The estimated function (Newton polynomial) p(x) obtained, in the form: (0.0121666) x^5-(122.7059942) x^4+ ( 4.950193546*〖10〗^5 ) x^3-( 0.985008654*〖10〗^8 ) x^2+(1.0070 35027*〖10〗^12)x-(4.062567218*〖10〗^14 ) has been proven to demonstrate the continuity of a function represented by tabulated discrete points by showing that 〖lim〗┬(x→c)⁡〖p(x)=p(c)〗 for each point.