Taylor's Series is based on the fact that, if a function is continuous and differentiable, The value of that function a small distance, h, from point x will be equal to the value of the function at x, plus a "fudge factor," or really a series of fudge factors. This is stuff you should know, because it is used extensively in math, physics, and geophysics.<br /> Taylor's series can be written in several forms. The first is:<br />TAYLOR SERIES<br />
It includes the residual term to consider all terms from n +1 to infinity<br />The Taylor Series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. If the series is centered at zero, the series is also called a Maclaurin Series.<br />TAYLOR SERIES<br />
The greater amount of n-terms are included the better the approximation of the true solution.<br />It is common practice to use a finite number of terms of the series to approximate a function.<br />
Exponential Function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red).<br />
The practical value of the Taylor series is the use of a finite number of terms that give a close enough approximation to the true solution.<br />The decision on how many terms are required to obtain a reasonable approximation is based on the residual term of the expansion<br />Taylor series and truncation errors<br />
The terms of the series that are neglected as waste, which depends on two factors:<br />1. The number of terms in the series (n), since the higher the value of n, the less waste and better the approximation to the value of the function.<br />2. The step size or distance between the value of the variable (h) because the lower the value of h, the greater the proximity between xi and xi +1 and thus, the better the approximation to the value of the function.<br />Taylor series and truncation errors<br />It represents the next step for the approximation series<br />
Example Numerical Simulation applied to the reservoir.<br />The use of Taylor series are the starting point for numerical differentiation of a simulation model. <br />
Example<br />To view the usefulness of Taylor series, Figures 1, 2, and 3 show the 0th-, 1st-, and 2nd-order Taylor series approximations of the exponential function f(x) = ex at x = 0. While the approximation in Figure 1 becomes poor very quickly, it is quite apparent that the linear, or 1st-order, approximation in Figure 2 is already quite reasonable in a small interval around x = 0. The quadratic, or 2nd-order, approximation in Figure 3 is even better.<br />We will use Taylor series for two purposes: To linearize a system, using the 1st-order Taylor-series approximation, and to perform error analysis on numerical method.<br />
Figure 1. The zeroth-order Taylor series approximation of ex around x = 0. <br />
Figure 2. The first-order Taylor series approximation of ex around x = 0.<br />
Figure 3. The second-order Taylor series approximation of ex around x = 0.<br />
BIBLIOGRAPHY<br />CHAPRA, Steven C. y CANALE, Raymond P.: Métodos Numéricos para Ingenieros. McGraw Hill 2002.<br />Schlumberguer Floviz-Eclipse 2008.1<br />http://demonstrations.wolfram.com/TaylorApproximationsInTwoVariables/<br />
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