Physics
Taylor Expansions
Taylor expansions are a method used to approximate a function by representing it as an infinite sum of terms. This allows for the estimation of a function's value at a particular point based on its derivatives at that point. In physics, Taylor expansions are commonly used to simplify complex functions and make them easier to work with in calculations.
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eBook - PDFTime Series Data Analysis in Oceanography
Applications using MATLAB
- Chunyan Li(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
6 Taylor Series Expansion and Application in Error Estimate About Chapter 6 The objective of this chapter is to review the Taylor series expansion and discuss its usage in error estimation. The unique value of Taylor series expansion is often neglected. The major assumption is that a function must be infinitely differentiable to use the Taylor series expansion. In real applications in oceanography, however, hardly there is a need to worry about a derivative higher than the 3rd order, although one may think of some exceptions. The point is, there is rarely a need in oceanography and other environmental sciences to actually consider calculating a very high order derivative, unless for theoretical investigations or under special situations. So the application of Taylor series expansion usually only involves the first two derivatives. In this chapter, some simple examples are included for a better understanding of the applications. 6.1 Taylor Series Expansion Taylor series expansion is a formula which we use to approximate a differentiable function with a polynomial function in a neighborhood region of a point where the function value and derivatives are somehow known. This can be useful in error estimations in laboratory experiments, data analysis, and stability analysis for computational schemes in numerical models. For instance, it can be used for error estimates for interpolation or extrapolation. Often we do not know the form of a function, but we may know the function value, its slope, and its higher-order derivatives at a given point based on measurements or computer model output. The question is: can we approximate an unknown function within a certain range of the independent variable with a simple function based on knowledge of the values and derivatives of the function at a given point? 110 The answer is yes, with Taylor series expansion, which provides the solution. It is an approximation valid within a certain range of the independent variable.
eBook - PDFFundamentals of Physics
Mechanics, Relativity, and Thermodynamics
- R. Shankar(Author)
- 2014(Publication Date)
- Yale University Press(Publisher)
chapter 16 Mathematical Methods 16.1 Taylor series of a function I am going to introduce you to some mathematical tricks. As you’ve prob-ably noticed by now, a lot of physics has to do with mathematics, and if you’re not good in math, you’re not going to be good in physics. The first important trick is called the Taylor series. The philosophy of the Taylor series is the following: There is some function f ( x ) depicted in Figure 16.1. But I’m going to imagine that you can only zero in on a tiny region near x = 0. And the question is, how will you write an approx-imation for this whole function, valid away from x = 0? Suppose I don’t show you anything except what’s happening at x = 0; I show you only f (0). The value of the function is 92. What should you do away from x = 0? You have no additional information about this function; you don’t know if it’s going up or going down. The best approximation you can make is the flat line in Figure 16.1. There’s no reason to tilt it one way or the other, given the information you have. If you do not pick the constant to be 92 you will even miss the answer at the one place ( x = 0) where you were given the value. 255 256 Mathematical Methods Figure 16.1 A sample function f ( x ) and two approximations to it near x = 0 based on the Taylor series. The flat line f ( x ) = f (0) assumes f ( x ) does not change as we move away from x = 0. It is based on the first term in what is called the Taylor series. The second line f ( x ) = f (0) + f (0) x is the linear approximation based on two terms, and it matches the function and its derivative at x = 0. It assumes the slope does not change. So, the first approximation of the function you will say is f ( x ) = f (0) + ... (16.1) where the ellipsis means there may be corrections away from x = 0 but you do not know them. The left-hand side stands for the actual function and the right to an approximation that matches the real thing at x = 0 with possible unknown corrections as we move away.
eBook - PDFCalculus
Single Variable
- Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum, Daniel E. Flath, Patti Frazer Lock, David O. Lomen, David Lovelock, Brad G. Osgood, Douglas Quinney, Karen R. Rhea, Jeff Tecosky-Feldman, Thomas W. Tucker, Otto K. Bretscher, Sheldon P. Gordon, Andrew Pasquale, Joseph Thrash(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
Taylor series can often be used to answer this question over a small interval. If the constant terms of the series for two functions are the same, compare the linear terms; if the linear terms are the same, compare the quadratic terms, and so on. Example 7 By looking at their Taylor series, decide which of the following functions is largest for t near 0. (a) e t (b) 1 1 − t 574 Chapter Ten APPROXIMATING FUNCTIONS USING SERIES Solution The Taylor expansion about t = 0 for e t is e t = 1 + t + t 2 2! + t 3 3! + · · · . Viewing 1/(1 − t) as the sum of a geometric series with initial term 1 and common ratio t, we have 1 1 − t = 1 + t + t 2 + t 3 + · · · for − 1 < t < 1. Notice that these two series have the same constant term and the same linear term. However, their remaining terms are different. For values of t near zero, the quadratic terms dominate all of the subsequent terms, 4 so we can use the approximations e t ≈ 1 + t + t 2 2 1 1 − t ≈ 1 + t + t 2 . Since 1 + t + 1 2 t 2 < 1 + t + t 2 , and since the approximations are valid for t near 0, we conclude that, for t near 0, e t < 1 1 − t . See Figure 10.14. −1 1 3 1 1−t e t t y Figure 10.14: Comparing two functions near t = 0 Example 8 Two electrical charges of equal magnitude and opposite signs located near one another are called an electrical dipole. The charges Q and −Q are a distance r apart. (See Figure 10.15.) The electric field, E, at the point P , at a distance R from the dipole is given by E = Q R 2 − Q (R + r) 2 . Use series to investigate the behavior of the electric field far to the left along the line through the dipole. Show that when R is large in comparison to r, the electric field is approximately proportional to 1/R 3 . ✲ ✛ R ✲ ✛ r P Q −Q Figure 10.15: Approximating the electric field at P due to a dipole consisting of charges Q and −Q a distance r apart Solution In order to use a series approximation, we need a variable whose value is small.
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