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Chapter 9: Problem 2

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

Short Answer

Expert verified
Answer: The accuracy of an approximation given by a Taylor polynomial generally increases with the order of the approximation. This is because higher-order polynomials capture more information about the function's behavior, and the approximation error decreases as the polynomial order increases according to Taylor's Remainder Theorem.

Step by step solution

01

Understanding Taylor Polynomials

Taylor polynomials are used to approximate a function using a finite number of the function's derivatives at a specific point. The nth-degree Taylor polynomial of a function f(x) centered at a point a is given by: P_n(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^n(a)(x-a)^n/n! Where n is the order of the polynomial and f^i(a) denotes the ith derivative of f(x) evaluated at x=a.
02

Effect of Polynomial Order on Approximation Accuracy

As we increase the order of a Taylor polynomial, we are effectively adding more terms to the polynomial. These terms are based on higher-order derivatives of the function being approximated. In general, the more terms we include in the Taylor polynomial (i.e., the higher the order of the polynomial), the better the approximation of the function will be. This is because, as we increase the order of the polynomial, we are able to capture more information about the behavior of the function. Higher-order derivatives give us information about the curvature and other features of the function that cannot be captured with a lower-order polynomial.
03

Taylor's Remainder Theorem

To formally analyze the accuracy of a Taylor polynomial, we use Taylor's Remainder Theorem. This theorem tells us that the error in approximating a function with a Taylor polynomial of order n is given by: R_n(x) = f(x) - P_n(x) = (f^{(n+1)}(c)(x-a)^{(n+1)})/((n+1)!) for some value c between a and x. The error R_n(x) goes to 0 as n goes to infinity (provided that the function's derivatives are continuous and bounded). In other words, the accuracy of the approximation generally increases as we increase the order of the polynomial.
04

Conclusion

In general, the accuracy of an approximation given by a Taylor polynomial increases with the order of the approximation. This is because higher-order polynomials capture more information about the function's behavior, and Taylor's Remainder Theorem shows that the approximation error decreases as the polynomial order increases.

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Most popular questions from this chapter

Inverse hyperbolic sine The inverse hyperbolic sine is defined in several ways; among them are $$ \sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1})=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}} $$ Find the first four terms of the Taylor series for \(\sinh ^{-1} x\) using these two definitions (and be sure they agree).

Find the remainder in the Taylor series centered at the point a for the following functions. Then show that \(\lim _{n \rightarrow \infty} R_{n}(x)=0\) for all \(x\) in the interval of convergence. $$f(x)=\cos 2 x, a=0$$

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