Tangencies for Power Functions with Integer Exponent
DOI:
https://doi.org/10.17268/sel.mat.2025.01.13Keywords:
Power Function, Tangency, Binomial TheoremAbstract
Considering a power function f(x) = x^n with exponent n as a positive integer, we show that, at each of its points, there exists a unique polynomial function of degree n − 1 that is tangent to it at that point. Similarly, we verify that every power function h(x) = x^k with exponent k as a negative integer is tangent, at each of its points, to a function of the form l(x) =Sa^t.x^t, where the exponents t are integers between k + 1 and −1.
References
Stewart J. Cálculo, sétima edicao, Cengage Learning, 2013.
Valdes JEN. La Resolución de Problemas de Cálculo Diferencial y La Construcción de la Recta Tangente a Una Curva. Vidya. 2024; 44(2):247-–264.
Ackerman E, Keszegh B. On the Number of Tangencies Among 1-Intersecting x-Monotone Curves European Journal of Combinatorics. 2024; 118.
Hino M, Namba R. Fractional Binomial Distributions Induced by the Generalized Binomial Theorem and Their Applications. Arxiv: 2408.12011v1[math.PR] (2024). Available from https://arxiv.org/html/2408.12011v1.
Beauduin K. Old and New Powerful Tools for the Normal Ordering Problem and Noncommutative Binomials. Arxiv: 2405.03001v3[math.CO] (2024). Available from https://arxiv.org/abs/2405.03001.
Spiegel MR, Lipschutz S, Liu J. Mathematical Handbook of Formulas and Tables. third edition, Schaum’s Outline Series, New York, 2013.
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