Calculus Small Changes, Big Ideas

Section 10.1 Taylor approximations and series

• We saw when we were doing curvature that there may be better approximations of a function than just lines; the osculating circle was one such example. So what other kinds of approximations can we have? (Also, we can note that the osculating circle depends only on the first and second derivatives, so it’s a second-order approximation.)

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• Given a function \(f\text{,}\) we can find a polynomial function \(p\) of degree \(n\) such that

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  \begin{align*} p(a) \amp = f(a)\\ p'(a) \amp = f'(a)\\ p''(a) \amp = f''(a)\\ \vdots\\ p^{(n)}(a) \amp = f^{(n)}(a) \end{align*}
  The resulting polynomial is called a Taylor polynomial.

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• Formula:

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  \begin{align*} p(x) \amp = f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^{n} \\ \amp = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^{k} \end{align*}
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• If we add more and more terms, we get a better and better approximation of the original function.

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• So if we had infinitely many terms, we should match the function perfectly! This is called a Taylor series.

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• If \(a=0\text{,}\) replace “Taylor” with “Maclaurin”.

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• Useful Maclaurin series:

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  \begin{align*} \ee^{x} \amp = \sum_{k=0}^{\infty} \frac{x^{k}}{k!}\\ \sin x \amp = \sum_{k=0}^{\infty} \frac{(-1)^{k+1}x^{2k+1}}{(2k+1)!}\\ \cos x \amp = \sum_{k=0}^{\infty} \frac{(-1)^{k+1}x^{2k}}{(2k)!} \end{align*}
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