CÁLCULO DISCRETO - Teorema de Newton

This video concludes a series of 10 lessons dedicated to the study of finite differences within the framework of discrete calculus. This session presents and conceptually proves Newton's Theorem, which is the discrete equivalent of Taylor series (specifically, the Maclaurin series when evaluated at zero). Through a practical example with the function $f(x) = x^3$, the video explains in detail how to construct a finite difference table and how to use descending factorials ($x^{\underline{k}}$) alongside standard factorials to expand and reconstruct any polynomial function algebraically. Emphasis is also placed on the application of the Fundamental Theorem of Discrete Calculus to determine the limits of the summation and how successive differences progressively reduce the degree of the polynomial until it reaches zero. Table of Contents 0:00 - Introduction to Newton's Theorem in discrete calculus and its analogy with Taylor/Maclaurin series. 0:35 - Formal definition of the summation and components of the polynomial expansion. 1:47 - Explanation of the weights: the k-th finite difference, the standard factorial, and the descending factorial. 3:22 - Presentation of the practical example with the dominant function f(x) = x³. 3:48 - Relationship with the Fundamental Theorem of Discrete Calculus and the elimination of higher-degree terms (k+1). 5:22 - Determining the number of terms in the summation and calculating the necessary descending factorials. 6:05 - Construction of the finite difference table by evaluating the function from x = 0 to x = 3. 9:06 - Substituting the values ​​obtained from the table into Newton's Theorem. 11:50 - Algebraic expansion and simplification of polynomial terms. 14:50 - Grouping like terms, canceling linear and quadratic factors, and reconstructing the original function. #DiscreteCalculus #NewtonsTheorem #FiniteDifferences #Mathematics #FactorialDeclining #Algebra #TaylorSeries