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

Verified step by step guidance

1. Recall that a Taylor polynomial of order \(n\) approximates a function by matching the function's value and its first \(n\) derivatives at a specific point.
2. Understand that increasing the order \(n\) means including higher-degree terms in the polynomial, which capture more details about the function's behavior near the point of expansion.
3. Recognize that, generally, as the order of the Taylor polynomial increases, the approximation becomes more accurate near the point of expansion because the polynomial better matches the function's shape.
4. Note that the accuracy improvement depends on the function being sufficiently smooth (infinitely differentiable) and the approximation being considered close to the expansion point.
5. Keep in mind that while higher-order polynomials improve accuracy locally, the approximation might not improve or could even worsen far from the expansion point due to issues like Runge's phenomenon.

Key Concepts

Taylor Polynomial

A Taylor polynomial approximates a function near a specific point using a finite sum of derivatives at that point. The polynomial's degree determines how many terms are included, capturing more details of the function's behavior as the degree increases.

Order of Approximation

The order of a Taylor polynomial refers to its degree, indicating how many derivative terms are used. Higher-order polynomials generally provide better approximations because they incorporate more information about the function's curvature and higher derivatives.

Error and Accuracy in Taylor Approximations

The accuracy of a Taylor polynomial depends on the remainder term, which measures the difference between the function and its polynomial approximation. Typically, increasing the order reduces this error near the expansion point, improving accuracy within a certain interval.