Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = x³ − 2x + 4,a = 2

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Identify the function and the point about which the Taylor series is generated: here, the function is \(f(x) = x^{3} - 2x + 4\) and the center is \(a = 2\).

Recall the formula for the Taylor series of a function \(f\) about \(x = a\): \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^{n}\] where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).

Compute the derivatives of \(f(x)\) up to the order needed (usually until the derivatives become zero or a pattern emerges): - \(f(x) = x^{3} - 2x + 4\) - \(f'(x) = 3x^{2} - 2\) - \(f''(x) = 6x\) - \(f^{(3)}(x) = 6\) - \(f^{(4)}(x) = 0\) (and all higher derivatives are zero).

Evaluate each derivative at \(x = 2\): - \(f(2) = 2^{3} - 2(2) + 4\) - \(f'(2) = 3(2)^{2} - 2\) - \(f''(2) = 6(2)\) - \(f^{(3)}(2) = 6\) - \(f^{(4)}(2) = 0\).

Write the Taylor series expansion using the formula and the evaluated derivatives: \[f(x) = f(2) + \frac{f'(2)}{1!}(x - 2) + \frac{f''(2)}{2!}(x - 2)^{2} + \frac{f^{(3)}(2)}{3!}(x - 2)^{3} + \cdots\] Since higher derivatives are zero, the series will terminate here.

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주요 개념

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Taylor Series

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point a. It approximates the function near that point using polynomial terms, where each term involves the nth derivative evaluated at a, multiplied by (x - a)^n and divided by n!.

Maclaurin Series

The Maclaurin series is a special case of the Taylor series centered at a = 0. It expresses a function as an infinite sum of derivatives evaluated at zero, useful for approximating functions near zero. Understanding this helps distinguish between general Taylor expansions and those specifically at zero.

Derivatives and Their Role in Series Expansion

Derivatives of a function at the point a determine the coefficients of the Taylor series. Calculating successive derivatives and evaluating them at a provides the necessary values to build each term of the series, reflecting the function's behavior near a.

Finding Taylor and Maclaurin SeriesIn Exercises 25–34, find the Taylor series generated by f at x = a.f(x) = x³ − 2x + 4,a = 2