Question

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Find the terms through degree five of the Maclaurin series for f(x)=8 sin xcosx T_5x+

Answer 1

The terms through degree five of the Maclaurin series for f(x)=8sin⁡(x)cos⁡(x)f(x)=8sin(x)cos(x) are T5(x)=8x−1283x3+102445x5T5​(x)=8x−3128​x3+451024​x5.

Explanation:

The Maclaurin series is a representation of a function as an infinite sum of terms calculated from its derivatives at a specific point (usually x=0x=0). In this case, f(x)=8sin⁡(x)cos⁡(x)f(x)=8sin(x)cos(x), and the Maclaurin series expansion, T5(x)T5​(x), involves the function's derivatives up to the fifth degree.

The terms through degree five of the Maclaurin series are obtained by calculating the function's derivatives and evaluating them at x=0x=0, ensuring that each term represents the contribution of the corresponding derivative to the series.

Understanding Maclaurin series expansions is crucial in approximating functions, especially in calculus and mathematical analysis.