Question

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

Short answer

Answer: The accuracy of an approximation given by a Taylor polynomial generally increases with the order of the approximation. This is because a higher-order polynomial captures more detailed information about the function's behavior near the centerpoint x=a and the error term (remainder) decreases as the order increases, providing a better local approximation of the function. However, this holds true for functions that are sufficiently smooth, meaning they have continuous derivatives up to the required order.

Step-by-step solution

Understand Taylor Polynomials

The Taylor polynomial is a mathematical method to approximate a function using a series of polynomial terms, where each term comes from higher-degree derivatives of the function. The Taylor polynomial of a function f(x) centered at x = a is given by: P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n Where P_n(x) is the nth-degree Taylor polynomial, f^{(n)}(a) represents the nth derivative of the function f(x) evaluated at x=a, and n! is the factorial of n.

Analyze the influence of the order

As we increase the order of the Taylor polynomial, more terms are added to the approximation. These terms take into account higher-order derivatives, resulting in a more accurate representation of the function's behavior near the centerpoint x=a. In general, higher-order derivatives capture the changes in curvature of the function, which are not considered in lower-order approximations. Therefore, a higher-order Taylor polynomial will provide a better local approximation of the function as it includes more information about the function's behavior.

Error in Taylor Polynomial Approximation

The error in Taylor polynomial approximation, called the remainder, is given by: R_n(x) = f(x) - P_n(x) Where f(x) is the actual function, P_n(x) is the nth-degree Taylor polynomial, and R_n(x) is the remainder. According to Taylor's Remainder Theorem, there exists a number ξ between x and a such that: R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1} The term f^{(n+1)}(\xi) is the (n+1)th derivative of the function f(x) evaluated at x=ξ. As we increase the order of the Taylor polynomial, the term (x-a)^{n+1} will generally decrease given that |x-a|<1, reducing the error in the approximation.

Conclusion

In general, the accuracy of an approximation given by a Taylor polynomial increases with the order of the approximation because a higher-order polynomial captures more detailed information about the function's behavior near the centerpoint x=a and the error term (remainder) decreases as the order increases. However, this holds true for functions that are sufficiently smooth, meaning they have continuous derivatives up to the required order.