Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines
Issue:
Volume 7, Issue 1, March 2021
Pages:
1-6
Received:
21 November 2020
Accepted:
13 January 2021
Published:
10 March 2021
Abstract: We reveal one relationship between each degree algebraic function and its tangent line, via its derivative. In particular, it is easy to see and well known that asymmetry (resp. symmetry) of tangent lines of a quadratic (resp. cubic) function at its minimum and maximum zero points, but it is not easy to investigate symmetry and asymmetry of them of nth-degree functions if n is 4 or more. We thus investigate the relationship between the slopes of the tangent lines at minimum and maximum zero points of the nth-degree function. We will in this note be able to know some sufficient conditions for the ratio of their slopes to be 1 or -1. By these, we can understand that tangent lines at minimum and maximum zero points have a symmetrical (resp. asymmetrical) relationship if the ratio of their slopes is -1 (resp. 1). In other words, these properties give us symmetry and asymmetry of the functions. Furthermore, we also mention the property of the discriminant of a quadratic function.
Abstract: We reveal one relationship between each degree algebraic function and its tangent line, via its derivative. In particular, it is easy to see and well known that asymmetry (resp. symmetry) of tangent lines of a quadratic (resp. cubic) function at its minimum and maximum zero points, but it is not easy to investigate symmetry and asymmetry of them of...
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Cubic Transmuted Dagum Distribution: Properties and Applications
Issue:
Volume 7, Issue 1, March 2021
Pages:
7-18
Received:
26 November 2020
Accepted:
10 December 2020
Published:
30 April 2021
Abstract: In this article, a generalization of the new cubic transmuted Dagum (CTD) distribution is derived and have developed a cubic transmuted family of distribution. The proposed distribution includes a special cases of the new Transmuted Dagum distribution (TDD). The objectives are to model a new distribution called cubic transmuted Dagum distribution to take care of the flexibility and multimodal (complex) effect which the transmuted form of the distribution cannot handle. The work comprises of the probability density function of cubic transmuted Dagum distribution and its Cumulative distribution function and attempt was made to compare the new distribution with the transmuted form of the distribution. Various structural properties of the new distribution, including the moments, characteristic function, quantile, moment generating function, mean, variance, reliability analysis, order statistics are derived. The maximum likelihood estimation method has been proposed for the estimation of the parameters of the Cubic transmuted Dagum distribution. The usefulness of the derived model is illustrated using two data sets to compare the performance of the new distribution with the transmuted form of the distribution and also with the parent (Dagum) distribution, and it is proved that CTD distribution is a better distribution than the transmuted Dagum distribution and the Dagum distributions based on some goodness of fit measures. Therefore, we conclude that the new model fits real life data better than the transmuted form and the base distribution. Also cubic transmuted Dagum distribution attracts more applications in several areas such as engineering, survival data, economics and others.
Abstract: In this article, a generalization of the new cubic transmuted Dagum (CTD) distribution is derived and have developed a cubic transmuted family of distribution. The proposed distribution includes a special cases of the new Transmuted Dagum distribution (TDD). The objectives are to model a new distribution called cubic transmuted Dagum distribution t...
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