A Taylor expansion is a way to approximate a function by representing it as an infinite sum of terms calculated from the values of the function's derivatives at a single point. It allows for the representation of a function as a polynomial, which can be used to estimate the function's value near that point.
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The $n$th-degree Taylor polynomial approximation of a function $f(x)$ around a point $x = a$ is given by $$T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$$
The Taylor expansion converges to the original function $f(x)$ as the degree $n$ approaches infinity, provided the function is infinitely differentiable at the point $x = a$.
Taylor expansions are useful for approximating functions near a point of interest, as they provide a simple polynomial representation that can be easily evaluated and manipulated.
Maclaurin series are a special case of Taylor series where the expansion is centered at the origin, $x = 0$, simplifying the formula to $$T_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n$$
The remainder term in a Taylor expansion can be used to estimate the error in the approximation, allowing for the determination of the accuracy of the polynomial representation.
Review Questions
Explain the purpose and construction of a Taylor expansion.
The purpose of a Taylor expansion is to approximate a function by representing it as an infinite sum of terms calculated from the values of the function's derivatives at a single point. The $n$th-degree Taylor polynomial approximation is constructed by taking the function value at the point of interest, $f(a)$, and adding terms that incorporate the successive derivatives of the function, $f'(a), f''(a), \dots, f^{(n)}(a)$, each multiplied by a power of the difference between the variable $x$ and the point of expansion $a$. This allows for the representation of the function as a simple polynomial, which can be used to estimate the function's value near the point of expansion.
Describe the relationship between Taylor expansions and Maclaurin series, and explain how they differ.
A Maclaurin series is a special case of a Taylor series where the expansion is centered at the origin, $x = 0$. The Maclaurin series formula simplifies the Taylor expansion by setting $a = 0$, resulting in $$T_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n$$. The key difference is that Maclaurin series are centered at the origin, whereas Taylor expansions can be centered at any point $x = a$. This allows Maclaurin series to be used to approximate functions that are well-behaved near the origin, while Taylor expansions provide more flexibility in approximating functions around a specific point of interest.
Discuss the properties of the remainder term in a Taylor expansion and explain how it can be used to determine the accuracy of the approximation.
The remainder term in a Taylor expansion represents the difference between the true function value and the approximation provided by the Taylor polynomial. This remainder term can be used to estimate the error in the approximation, allowing for the determination of the accuracy of the polynomial representation. The remainder term is typically expressed using Lagrange's form of the remainder, which provides an upper bound on the error. By analyzing the behavior of the remainder term, one can understand the conditions under which the Taylor expansion will converge to the original function and determine the range of $x$ values for which the approximation is sufficiently accurate for a given application. This information is crucial in assessing the validity and limitations of the Taylor expansion as an approximation tool.
The remainder term in a Taylor expansion represents the difference between the true function value and the approximation provided by the Taylor polynomial.