91Ó°ÊÓ

The Taylor series is a powerful tool in calculus used for approximating complex functions with polynomials. A Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point. For the function \(\sin x\), its Taylor series expansion at the point \(x=0\) is:

• \(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\)

In practical applications, we often truncate the series to include only a finite number of terms, which results in a polynomial approximation of the original function. The approximation of \(\sin x\) using the expansion \(\sin x \approx x - \frac{x^3}{3!}\)effectively simplifies calculations while maintaining a level of precision within a given interval around the center point. This simplicity is particularly useful for quick estimations or when the exact solution is not readily required.

Approximation accuracy is essential when using polynomials to represent more complex functions. It refers to how closely the polynomial approximation approaches the true value of the function within a chosen interval. The Remainder Estimation Theorem helps determine the error bound, or the difference between the actual function \(f(x)\)and its approximation \(p(x)\). For instance, in using the Taylor polynomial \(p(x) = x - \frac{x^3}{3!}\) to approximate \(\sin x\), the remainder term \(|R_n(x)|\) is given by:

• \(\left|\frac{f^{(n+1)}(c)\cdot x^{n+1}}{(n+1)!}\right|\)

where \(f^{(n+1)}(c)\)is the \\(n+1\)th derivative evaluated at some point \(c\) in the interval. Ensuring that \(|R_n(x)|\) is below a desired threshold, such as 0.0005 for three-decimal-place accuracy, allows us to confidently use the polynomial for calculations within the specified range.

Interval estimation is pivotal when applying Taylor series for approximation. It identifies the range around a point, often \(x=0\), within which the approximation remains accurate to a specific degree. For our problem, to ensure three-decimal-place accuracy for \(\sin x\) using \(p(x)\), we derived the condition\(\frac{x^4}{24} < 0.0005\). Solving this inequality involves evaluating the maximum error where it occurs and simplifying it to find \(x^4 < 0.012\). By extracting the fourth root of 0.012, we determined the interval

• \([-0.182, 0.182]\)

Understanding this interval ensures that whenever \(x\) lies within these bounds, the polynomial approximation retains its intended accuracy. For a visual confirmation, plotting the absolute difference \(|f(x)-p(x)|\)over the interval confirms that the discrepancy remains minimal, validating the accuracy of the estimation.