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Polynomial approximation is a technique frequently used in mathematics to estimate the value of complex functions by simpler polynomials. This approach simplifies calculations, especially for functions that are otherwise difficult to compute directly.

A polynomial approximation usually involves finding a polynomial, which is a sum of power functions, that closely follows a desired function across a specified range.

The goal is to have the polynomial as close as possible to the target function. Here, this idea is applied by using the given polynomial, \( P_{3}(x) = x - \frac{x^3}{6} \), to approximate \( \sin x \).
The point of interest is \( x = 0.1 \), which is close to zero, ensuring the approximation remains accurate.

The Taylor series expansion is an essential concept in calculus and it represents functions as infinite sums of their derivatives at a single point. In simple terms, it’s a method of creating a polynomial representation of a function.

This not only makes calculations easier but also provides insights into the function's behavior.

For \( \sin x \), the expansion around \( x = 0 \) (also known as the Maclaurin series) involves its derivatives:

• First derivative: \( \cos x \)
• Second derivative: \( -\sin x \)
• Third derivative: \( -\cos x \)
• Fourth derivative: \( \sin x \)

The series ends at the nearest integral power where further terms contribute to a negligible impact on the approximation.

The given polynomial, \( P_{3}(x) = x - \frac{x^3}{6} \), utilizes terms from the Maclaurin series up to \( n = 3 \). This makes it a truncated version of the Taylor series for \( \sin x \).

Calculating the remainder term is crucial for understanding the "error" or "accuracy" of a polynomial approximation. Known also as the Lagrange remainder, it provides a bound for the error.

The formula used is: \[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1} \] This involves evaluating the \((n+1)\)th derivative of the function at an unknown point \(c\) within the interval from the expansion point to the point of approximation.

For our function \( \sin x \), and at the point \( x = 0.1 \), we find:

• \( f^{(4)}(c) = \pm \sin c \)
• The realistic 'worst-case' estimate is: \( \sin c \approx c \)

This leads to:\[ R_3(0.1) = \frac{0.1}{24} \times 0.0001 \approx 0.0000004167 \]By calculating \( R_3(0.1) \), we obtain an error estimate that indicates how close the approximation \( P_{3}(x) \) is to \( \sin x \) at \( x = 0.1 \).

Essentially, this term makes sure that users of the approximation understand the degree of accuracy they can expect.