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The sine function, often denoted as \(\sin x\), is a fundamental mathematical function with a periodic wave-like pattern. Approximating \(\sin x\) using Taylor polynomials can provide a useful estimate of this function鈥檚 value near a given point. This technique is especially powerful when computing the sine function with limited computational resources. For example, in this exercise, we're using Taylor polynomials \(p_3(x)\) and \(p_5(x)\) to approximate \(\sin x\) at the points \(x = -0.2, -0.1, 0.0, 0.1,\) and \(0.2\).

Each Taylor polynomial uses a series expansion derived from the function's derivatives at a specific point, usually zero. This means that for small values of \(x\), the approximations \(p_3(x) = x - \frac{x^3}{6}\) and \(p_5(x) = x - \frac{x^3}{6} + \frac{x^5}{120}\) become particularly effective. Knowing how these expansions work can greatly help to understand their applications.

When we approximate a function, it鈥檚 important to measure how close our approximation is to the actual value. This is where absolute error comes into play. Absolute error is the difference between the exact value and the approximated value, expressed as an absolute value to ensure it鈥檚 always positive.

In this exercise, the absolute errors are calculated using the formula \(|\sin x - p_n(x)|\), where \(p_n(x)\) is our Taylor polynomial approximation. Performing this calculation for each value of \(x\) gives us a sense of the accuracy of the approximation.

For instance, using a calculator to find \(\sin(-0.1)\) and subtracting it from its approximation \(p_3(-0.1)\) results in a specific error value, which must then be rounded to two significant digits for precision consistency.

Taylor polynomials, named after mathematician Brook Taylor, provide a method to approximate functions using polynomials. They employ derivatives of a function at a single point to create polynomials that closely estimate the function around that point.

• Third-Degree Polynomial \(p_3(x)\): This uses the first two non-zero terms of the series, \(x\) and \(-\frac{x^3}{6}\).
• Fifth-Degree Polynomial \(p_5(x)\): This extends the approximation with an additional term \(+\frac{x^5}{120}\).

Since these polynomials include higher degrees of \(x\), they often provide a closer match to \(\sin x\), especially as more terms are added. The degree of the polynomial directly influences the range and accuracy of the approximation. With the degree increasing, like from 3 to 5, we generally get smaller errors, meaning closer approximations for larger \(|x|\) values.

Approximating functions often comes with errors, and understanding these errors helps us improve the approximations. Two main factors affect the errors: the degree of the Taylor polynomial and the point of approximation.

Generally, the error becomes smaller when a higher-degree polynomial is used. This is evident when comparing \(p_3(x)\) to \(p_5(x)\) across our selected \(x\) values.

• Errors are naturally minimized around \(x = 0\) due to the nature of Taylor series expansions centered at this point.
• Largest errors tend to occur as \(x\) moves further away from the center (0 in our case), as the series represents the function less accurately.

This exercise shows how the choice of polynomial impacts the size of approximation error, prompting considerations on balance between polynomial complexity and desired accuracy for different scenarios.