Taylor's approximation theorem allows functions to be approximated by polynomials in a specific point where the function is differentiable. The theorem provides a finite series with a residual term to account for higher-order terms. Taylor series are useful for obtaining close approximations of functions when a finite number of derivative terms are included in the series. Increasing the number of derivative terms causes the approximation to more closely match the real value of the function.

1. RUBEN DARIO ARISMENDI RUEDA

2. CHAPTER 3: ‘TAYLOR’S APPROXIMATION’

3. Taylor's Series. Is a theorem that let us to obtain polynomics approximations of a function in an specific point where the function is diferenciable. As well, with this theorem we can delimit the range of error in the estimation. This is a finitive serie, and the Residual term is include to considerate all the terms from (n+1) to infinitive. Taylor's Serie Residual term

4. McLaurin Serie.

5. How is Taylor’s Serie Used and Why is it important? Taylor’s serie is used with a finitive number of terms that will provide us an approximation really close to the real solution of the function.

6. 1 2 3 4 When the number of derivates (number of terms) in the Serie increase, the result is goning to be closer to the real value of the function.

7. NUMERICAL DIFFERENTIATION . From the Taylor’s serie of first order. We reflect the First derivate: ; = h PROGRESSIVE DIFFERENTIATION

8. From the Taylor’s serie of first order. We reflect the First derivate: ; =h REGRESSIVE DIFFERENTIATION

9. From the Taylor’s serie of first order (Progressive and Regressive) - We reflect the First derivate: CENTRATE DIFFERENTIATION

10. EXAMPLE. Determine the Taylor’s Polynom n = 4 , c = 1 = x i DEVELOPMENT. 1.Find all the derivates that is needed.

11. 2. Replace the values of the derivates in the Taylor’s Serie To find the Polynom. At the end, We will have the polynom to get the approximate value of the function