Applications of Taylor Polynomials

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Applications of Taylor Polynomials

What are Taylor Polynomials?
How Accurate are Taylor Polynomials?
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Other Uses of Taylor Polynomials

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How Accurate are Taylor Polynomials?

So far we have assumed that our functions were analytic. But how can we tell which are?
Taylor's theorem with remainder helps us determine how close a Taylor polynomial comes to the original function.

Theorem: If $f(x)$ is $(k+1)$ -times differentiable on an interval around $a$ , and if $x$ is in that interval, then

$f(x) = \sum_{n=0}^k \frac{f^{(n)}(a)}{n!} (x-a)^n + R_k(x),$ where the remainder is given by

$R_k(x) = \frac{f^{(k+1)}(z)}{(k+1)!} (x-a)^{k+1}$ for some point

$z$ between

$a$ and

$x$ .

If a function $f$ can be differentiated infinitely many times, and

$\lim_{k\to\infty}R_k(x)=0,$ then

$f$ is analytic.

Corollary: If the $(k+1)$ st derivative of $f$ is bounded by $M$ on an interval of radius $d$ around $x=a$ , then the remainder $R_k(x)$ is bounded by

$\frac{M}{(k+1)!}(x-a)^{k+1}.$

From this corollary, we can see that the remainders for $e^x$ and $\sin(x)$ , for example, go to zero as $k \to \infty$ , so these functions are analytic.

Taylor's theorem is similar to Rolle's theorem and the Mean Value Theorem, both of which involve a mystery point between $a$ and $b$ . The proof of Taylor's theorem involves repeated application of Rolle's theorem.