Dependence of errors on x Consider f(x)=ln(1-x) and its...

Dependence of errors on $x$ Consider $f(x)=\ln (1-x)$ and its Taylor polynomials given in Example $8 .$ a. Graph $y=\left|f(x)-p_{2}(x)\right|$ and $y=\left|f(x)-p_{3}(x)\right|$ on the interval $[-1 / 2,1 / 2]$ (two curves). b. At what points of $[-1 / 2,1 / 2]$ is the error largest? Smallest? c. Are these results consistent with the theoretical error bounds obtained in Example $8 ?$

Dependence of errors on $x$ Consider $f(x)=\ln (1-x)$ and its Taylor polynomials given in Example $8 .$
a. Graph $y=\left|f(x)-p_{2}(x)\right|$ and $y=\left|f(x)-p_{3}(x)\right|$ on the interval $[-1 / 2,1 / 2]$ (two curves).
b. At what points of $[-1 / 2,1 / 2]$ is the error largest? Smallest?
c. Are these results consistent with the theoretical error bounds obtained in Example $8 ?$

Key Concepts

Taylor Series

A Taylor series is an infinite series expansion of a function about a specific point (usually 0 for a Maclaurin series). It represents a function as a sum of its derivatives evaluated at that point, multiplied by corresponding powers of the deviation from that point and divided by factorial terms. This series allows for an approximation of functions by polynomials, with the accuracy improving as more terms are included.

Taylor Polynomial Approximation

A Taylor polynomial is a finite truncation of the Taylor series used to approximate a function. The degree of the polynomial determines how many terms of the series are used, which in turn affects the accuracy of the approximation. Lower-degree polynomials provide rough approximations, while higher-degree ones tend to be more accurate within the interval of convergence.

Error Estimation in Taylor Polynomials

Error estimation in the context of Taylor polynomials involves quantifying the difference between the actual function value and the polynomial approximation. This is often done using a remainder term, such as the Lagrange error bound, which provides a theoretical upper bound for the error over a given interval. Understanding these bounds is crucial for determining where the approximation is most and least accurate.

Graphical Analysis of Approximation Errors

Graphing the absolute difference between a function and its Taylor approximation over a specified interval helps visualize the error behavior. By comparing error curves for different degrees of Taylor polynomials, one can identify regions where the approximation has maximum error and where it is most accurate. This visual analysis underscores the theoretical error bounds and their dependence on the choice of approximation degree and the interval considered.

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