91Ó°ÊÓ

When working with Taylor series, understanding derivatives is key. Derivatives are essentially a measure of how a function changes as its input changes. To start, the 0-th derivative is simply the function evaluated at a particular point, say where you expand the Taylor series, like point \( a \).

We begin by finding the function’s value at \( a = -3 \). As shown, this is calculated as \( f(-3) = 43 \). Next, we calculate the first derivative \( f'(x) \), which is the rate of change of the original function. Here, \( f'(x) = 8x - 2 \). Evaluating this at \(-3\) gives \( f'(-3) = -26 \).

Continuing, the second derivative is the derivative of the first derivative, and so on. For our function, \( f''(x) = 8 \) shows us how quickly \( f'(x) \) changes, hinting that \( f(x) \) is a quadratic polynomial. Each step helps reveal how \( f \'s \) behavior can be mapped to the Taylor series terms.

Polynomial functions form a core part of many mathematical concepts, including Taylor series. A polynomial is a mathematical expression consisting of variables and coefficients, structured in terms of powers. For example, in the exercise, \( f(x) = 4x^2 - 2x + 1 \) is a polynomial function of degree 2.

When dealing with polynomials in calculus, remember that derivatives simplify as you move from one derivative to the next. As demonstrated, \( f''(x) = 8 \) occurs due to the declining power of \( x \). Higher derivatives, such as the third derivative, become zero for a quadratic polynomial function. After a certain point, any additional derivative of a polynomial function will naturally result in zero.

This means the Taylor series expansion uses non-zero derivatives only up to the order of the polynomial’s degree. These properties simplify calculations and are vital for finding polynomial expansions.

Series expansion is a powerful tool in mathematics to represent a complex function as a sum of simple terms. The Taylor series is a type of series expansion that helps approximate functions using their derivatives at a single point. For this series expansion, the function is expressed as a sum of its derivatives' values at point \( a \).

The expansion for \( f(x) \) about \( x = -3 \) is expressed as follows:

• Zero derivative term: \( 43 \)
• First derivative term: \( -26(x + 3) \)
• Second derivative term: \( \frac{8}{2}(x + 3)^2 \)

Higher terms are not included as derivatives beyond the second become zero.
The final Taylor series becomes a simplified form: \( 43 - 26(x + 3) + 4(x + 3)^2 \), capturing the essence of \( f \) near \( x = -3 \). Utilizing series expansion, especially the Taylor series, makes it easier to handle complex calculations and understand the behavior of functions.