Fourth Degree Taylor Polynomial for Natural Logarithm

Find the fourth degree Taylor polynomial for the function $f(x) = \ln x$ centered at $c = 1$ and use it to approximate $\ln(1.1)$.

The Taylor series is a powerful tool in calculus, used to approximate functions with polynomials that match the original function at a specific point. In this problem, we focus on constructing the fourth degree Taylor polynomial for the natural logarithm function centered at $x = 1$. The natural logarithm, log base $e$, is a fundamental mathematical function, frequently encountered in calculus and related fields. By expanding this function in a Taylor series, you can approximate its values for points close to the center of expansion, $c = 1$.

To construct the Taylor polynomial, you will be utilizing derivatives of the natural logarithm function at the center point. The series involves calculating the zero through fourth derivatives of $\ln(x)$ and evaluating these derivatives at $x = 1$. These values help form the polynomial coefficients, which are used to create the polynomial approximation for $\ln(x)$ about $x = 1$. The Taylor polynomial you derive in this process provides an approximation for these values, and you can use this to find an estimate of $\ln(1.1)$.

The exercise of forming a Taylor polynomial is beneficial in understanding how functions can be approximated by polynomials, which are often simpler to analyze and use for estimations. This specific problem not only helps illustrate the series construction but also shows an application of using the polynomial to approximate nearby function values, demonstrating the practical usage of Taylor polynomials in approximating complex functions with simpler expressions. This is particularly useful in situations where direct computation of the logarithm is difficult, and a suitable polynomial approximation can yield accurate results efficiently.