Problem 55 - Essential Calculus

Question

In general, how does the accuracy of a Taylor polynomial change as the degree of the polynomial is increased? Explain your reasoning.

Step-by-Step Solution

Answer

Generally, the accuracy of a Taylor polynomial increases when the degree of the polynomial is increased, providing a closer approximation of the function near the point it's centered on. However, it's important to note that moving away from this point can result in a decrease in accuracy, regardless of the degree of the polynomial, due to the 'Runge's Phenomenon'.

Step 1: Understanding Taylor Polynomial

A Taylor polynomial attempts to approximate a function with a polynomial near some point. Taylor serier/polynomial consists of the terms of the infinite series of the function's n-th derivative. So, the higher the degree of the polynomial, the more terms of the series are included.

Step 2: Relating Degree of Polynomial and Accuracy

The more terms included (i.e., the higher the degree of the polynomial), the more closely the polynomial approximates the function. Therefore, generally, the accuracy of a Taylor polynomial increases when the degree of the polynomial is increased.

Step 3: Understanding the Limitation

However, there is an important caveat: this increased accuracy is best around the point that the Taylor series is centered on. As one moves away from this point, the accuracy can decrease significantly, regardless of the degree of the polynomial. This phenomenon is called the 'Runge's Phenomenon'. So, while the degree of the polynomial contributes to the accuracy of approximation, it's not the only factor to consider.