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The Taylor series is a powerful tool in mathematical analysis used to express functions as infinite sums of terms. These terms are derived from the function's derivatives at a single point, called the center of the series. The general formula for a Taylor series of a function \( f(x) \) centered at a point \( a \) is given by:\[ f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + R_n(x) \]Here, \( n \) represents the number of terms, and \( R_n(x) \) is the remainder term, accounting for the error between the original function and the approximation made by the series.The Taylor series is particularly useful when you want to approximate a function that is complicated to evaluate directly, utilizing only a few simple arithmetic operations involving derivatives.

Derivatives are a core concept in calculus, representing the rate at which a function changes at a certain point. They are crucial in forming Taylor and Maclaurin series, as each term in the series is constructed using a derivative of the original function.To find the Maclaurin series, which is a Taylor series centered at \( x=0 \), the derivatives of the function are evaluated at zero. For a given function \( f(x) = 2x^4 - 5x^3 \), we need to compute its derivatives:

• First derivative: \( f'(x) = 8x^3 - 15x^2 \)
• Second derivative: \( f''(x) = 24x^2 - 30x \)
• Third derivative: \( f'''(x) = 48x - 30 \)
• Fourth derivative: \( f^{(4)}(x) = 48 \),
• Fifth derivative: \( f^{(5)}(x) = 0 \)

The derivatives allow us to explore how the function behaves locally and enriches the approximation accuracy when constructing a Taylor or Maclaurin series.

The remainder term in a Taylor series, denoted as \( R_n(x) \), provides the difference between the actual function value and the approximated value obtained from truncating the series after \( n \) terms.For the Maclaurin series, different forms of the remainder term can be utilized, but one common form as described in Taylor's Theorem is:\[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} \]where \( c \) is some value in the interval between the center \( a \) and the point \( x \).In our example, when \( n=4 \), the fifth derivative \( f^{(5)}(x) \) is zero, resulting in a remainder term of:\[R_4(x) = 0 \]This tells us that the fourth-order approximation is exact, as all higher derivatives are zero. The simplicity of our given function leads to a perfect match between the polynomial approximation and the actual function.

Function evaluation is the process of finding the value of a function at a specific point. In the context of Maclaurin and Taylor series, evaluating derivatives at a point is necessary to determine the coefficients for the series.For the Maclaurin series, we evaluate the function \( f(x) \) and its derivatives at \( x=0 \). This transforms the series terms into simpler forms:

• \( f(0) = 0 \)
• \( f'(0) = 0 \)
• \( f''(0) = 0 \)
• \( f'''(0) = -30 \)
• \( f^{(4)}(0) = 48 \)
• \( f^{(5)}(0) = 0 \)

The zero evaluations simplify our task by indicating direct calculations, reducing the series to:\[ f(x) = -5x^3 + 2x^4 \]Evaluating derivatives and function values at specific points makes the series calculation feasible and manageable for manual computations.