91Ó°ÊÓ

Cubic approximation is a tool used to estimate the behavior of a function near a specific point using a polynomial of degree three. This is particularly useful when we want to simplify complex functions into more manageable forms for calculations or predictions.

When creating a cubic approximation for a function like \( f(x) = \frac{1}{1-x} \), we use Taylor's formula. Taylor's formula generates a polynomial by taking the sum of derivatives of the function at a specific point. Here, our point \( a \) is 0, and the order \( n \) is 3, indicating we're creating a cubic polynomial.

Thus, the cubic approximation is simplified into a polynomial which will look something like: \( P_3(x) = 1 + x + x^2 + x^3 \). This representation helps to estimate \( f(x) \) values when close to the point \( x = 0 \), giving us a polynomial function that is easier to handle and compute with.

Error estimation is a critical aspect when dealing with approximations like those found in Taylor polynomials. It provides us with an understanding of how close our approximation is to the actual function.

In this context, the error term \( R_n(x) \) gives the leftover part of the function that isn't accounted for by the polynomial. To estimate it, we consider the next derivative (beyond those used to create our polynomial). Here the fourth derivative was used since we're approximating up to the third degree.

To estimate the maximum error when \(|x| \leq 0.1\), we consider the worst-case scenario by evaluating the fourth derivative at \(x = 0.1\). The calculated error \( |R_3(x)| \) shows how far the cubic approximation \( P_3(x) \) might deviate from \( f(x) \).

An approximation like \( \approx 0.00012 \) indicates that the cubic polynomial gives a close estimate of the function for values of \( x \) close to zero, ensuring our predictions under this model are reliable.

A Taylor polynomial is a series expansion used to approximate functions. It uses derivatives calculated at a specific point to build a polynomial that represents the function’s behavior near that point. In this problem, Taylor's formula is applied to derive the third-degree Taylor polynomial of \( f(x) = \frac{1}{1-x} \) around \( x = 0 \).

This involves calculating several derivatives and evaluating them at zero. The Taylor polynomial then becomes a sum: \[ P_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3. \]

For our function, this simplifies to \( 1 + x + x^2 + x^3 \). This polynomial serves as an approximation of \( f(x) \) close to zero, offering a simpler form that can be used in various calculations where exact expressions are cumbersome.

Derivative evaluation involves calculating the derivatives of a given function up to a certain order, which are essential for constructing Taylor polynomials. In this specific exercise, derivatives of the function \( f(x) = \frac{1}{1-x} \) have been computed up to the third order:

• The first derivative is \( f'(x) = \frac{1}{(1-x)^2} \).
• The second derivative is \( f''(x) = \frac{2}{(1-x)^3} \).
• The third derivative is \( f'''(x) = \frac{6}{(1-x)^4} \).
• The fourth derivative, important for error estimation, is \( f''''(x) = \frac{24}{(1-x)^5} \).

Evaluating these derivatives at \( x = 0 \), we find \( f(0) = 1 \), \( f'(0) = 1 \), \( f''(0) = 2 \), and \( f'''(0) = 6 \). These values are then substituted into the Taylor polynomial, forming an approximate expression of \( f(x) \) around the desired point.