Applications of Taylor Polynomials

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Applications of Taylor Polynomials

What are Taylor Polynomials?
How Accurate are Taylor Polynomials?
What can go Wrong?
Other Uses of Taylor Polynomials

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What are Taylor Polynomials?

An

$n$ -th order Taylor polynomial is the first

$n$ terms of a Taylor series:

$T_n(x) = c_0 + c_1(x-a) + \cdots + c_n (x-a)^n.$

Using first-order Taylor polynomials is nothing new. Back in first semester calculus, we called this approximation the linearization:

$f(x) \approx L(x) = f(a) + f'(a) (x-a)=T_1(x).$

Note that the $n$ -th derivative of $T_n$ is a constant. Using $T_n$ to approximate $f$ ignores the fact that $f^{(n)}$ can change. In particular, $T_0$ pretends that the function doesn't change; $T_1$ pretends that the derivative doesn't change; etc.

To approximate $f(x)$ , we first need to find a point $a$ near $x$ where we understand the function well, then compute the first few terms of the Taylor series for $f$ around $a$ , and finally plug $x$ into that polynomial.