Fórmula de Taylor e Aplicações de Derivadas

Abstract

The theme of this work is the Taylor formula and applications of derivatives. From this theme, the objective was, therefore, to carry out a bibliographical study on the Taylor formula, presenting some applications of this formula in mathematics itself. It begins with a brief historical background on the origin of the Taylor Formula. Consequently, a brief review of the main de nitions and important results for the study of the theory involved in the theme is presented. Then, Taylor's formula is approached through the local approximation of a function derivable by an a ne function, as initial motivation. After de ning the Taylor polynomials of Orders 1 and 2, the Taylor Formula is generalized by de ning the Taylor Polynomial of Order n. Therefore, it is veri ed the existence of an error made between the value obtained by the formula and the real value of the function, which can be estimated in terms of the derivative of order n + 1 of the function utilizing Taylor's Formula Theorem with the remainder of Lagrangian. When estimating this error, it can be seen that it gets smaller enough when we use polynomials with higher and higher orders. Finally, we show some applications of the Taylor formula, such as the local approximation of functions, representation of functions through the Maclaurin Polynomial of Order n, proving that the Euler number is irrational and demonstrates a criterion for determining maxima and minima of functions.