(a) Repeat the derivation of the Taylor expansion for cosx...

(a) Repeat the derivation of the Taylor expansion for $\cos x$ given in the chapter, but now expand around the point $x=\pi / 2$ (b) Do a Taylor expansion of the function $e^{x}$ around zero. Generate a plot of the fractional error as a function of $x$ for different orders of the approximation and a comparison of the function and the different-order expansions such as was shown in Figure 5.22 (c) A one-dimensional potential-energy landscape is given by the equation $f(x)=x^{4}-2 x^{2}-1 .$ Find the two minima and do a Taylor expansion around one of them to second order. Show the original function and the approximation on the same plot in the style of Figure 5.21

(a) Repeat the derivation of the Taylor expansion for $\cos x$ given in the chapter, but now expand around the point $x=\pi / 2$
(b) Do a Taylor expansion of the function $e^{x}$ around zero. Generate a plot of the fractional error as a function of $x$ for different orders of the approximation and a comparison of the function and the different-order expansions such as was shown in Figure 5.22
(c) A one-dimensional potential-energy landscape is given by the equation $f(x)=x^{4}-2 x^{2}-1 .$ Find the two minima and do a Taylor expansion around one of them to second order. Show the original function and the approximation on the same plot in the style of Figure 5.21

Key Concepts

Taylor Series Expansion

The Taylor series provides a method to represent a smooth function as an infinite sum of terms calculated from its derivatives at a single point. This concept is foundational for understanding how functions can be approximated locally by polynomials, with each term in the series contributing to the accuracy of the approximation based on the function's behavior at the expansion point.

Series Expansion About Different Centers

Expanding a function around different points, whether it is zero (known as the Maclaurin series) or another nonzero point, is an important concept. Changing the center of expansion adjusts the series to be most accurate near that specific point, which is beneficial when the function’s behavior is best captured or of interest away from the origin.

Series Truncation and Error Analysis

In practical applications, only a finite number of terms in a Taylor series are used, leading to truncation error. Analyzing the fractional error—that is, the ratio between the error and the true function value—helps determine the necessary number of terms for a desired accuracy and provides insight into the convergence behavior of the series approximation over the interval of interest.

Local Quadratic (Harmonic) Approximation

Approximating a function by its second-order Taylor expansion around a minimum point simplifies the analysis of systems, especially in studying potential-energy landscapes. This quadratic or harmonic approximation captures the essential curvature of the function near the minimum, making it a powerful tool for modeling small oscillations and understanding local stability in various physical and mathematical contexts.

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