Finding Accuracy Range for Taylor Polynomial of Cosine

Look at the Taylor polynomial for $\cos(x)$ and we're cutting it off at degree 4, $T_4(x)$. We want to figure out what values of $x$ you can plug in there for it to be accurate to two decimal places.

When dealing with Taylor polynomials, particularly for functions like the cosine, a primary focus is understanding the behavior of the function in terms of both approximation and error. For this specific problem, the task is to determine the values of $x$ for which the Taylor polynomial for cosine, truncated at the fourth degree, remains accurate to two decimal places. The Taylor polynomial provides a way to approximate functions using polynomials, and the accuracy of this approximation depends on the degree of the polynomial used and the range of $x$ values considered. In this context, we are using a fourth-degree polynomial, which means it's a polynomial built using terms up to $x^4$. Because cosine is an even function, this approximation will only include even powers of $x$, specifically leaving out the linear and cubic terms.

Understanding the Taylor series expansions helps in visualizing how well this polynomial can represent the cosine function over specific intervals. This also involves recognizing the role of the remainder or error term in Taylor's theorem which essentially tells us how close our approximation (polynomial) is to the function (cosine) itself. To solve this problem, one must utilize the error bounds of Taylor polynomials, possibly by considering the next term that would appear in the series if it were not truncated, and ensuring that this term is small enough to meet the given precision constraint of two decimal places. Conceptually, this involves calculating derivatives of the cosine function up to the desired degree and evaluating these derivatives at a specified point, typically $x = 0$ for Taylor series centered at the origin. Such problems challenge one’s understanding of both the behavior of trigonometric functions and polynomial approximations, contextualizing learning within deeper studies involving calculus and analysis.