tailieunhanh - Some approximation problems by first degree derivative

In this paper, I mention the approximate calculation of the function value at a point, based on the first derivative at its neighborhood points. The article shows which function classes can be applied to approximate. The paper also points out the importance of approximation as an estimate of the error of that calculation. | SOME APPROXIMATION PROBLEMS BY FIRST-DEGREE DERIVATIVE Nguyen Van Dung and Giap Kim Anh Falculty of Basic Science - University of Transport and Communications Abstract. In this paper I mention the approximate calculation of the function value at a point based on the first derivative at its neighborhood points. The article shows which function classes can be applied to approximate. The paper also points out the importance of approximation as an estimate of the error of that calculation. Keywords. Function Aapproximation First derivative Error. 1. Introduction In Mathematics the approximation is very important because sometimes it is impossible to accurately calculate a specific value. The simplest problem is to approximate the function value at a point through the first derivative. This is a primary issue that the basic textbooks on calculus have all mentioned but these textbooks often do not specify the details of this formula. In this article we will go into solving those problems Firstly given the point x0 we approximate the function value at the points quot very close quot to that point x0 The second given the point x0 we approximate the function value at the point x0 through the points quot very close quot x0 The third given the point x0 does choosing the point closer to x0 give us a better approx- imation 2. Basic Knowledge Assuming f x is defined in the neighborhood of x0 f x has a derivative at x0 we have f x0 x f x0 f 0 x0 lim x 0 x f x0 x f x0 f 0 x0 x lim 0 x 0 x gt 0 δ gt 0 such that f x0 x f x0 f 0 x0 x x lt δ we have 2 Nguyen Van Dung and Giap Kim Anh So when x 0 f x0 x f x0 f 0 x0 x 0 or when x is small enough f x0 x f x0 f 0 x0 x 1 The Formula 1 is the formula for approximation by first-degree derivative It allows us to approximate the value of the function at points quot very close quot to the given point x0 . Example . Approximation value A 4 01 1 Let f x x we have f 0 x 2 x Apply the Formula 1 with x0 4 x we obtain 1 A 4 . 2 0025 2 4