A Taylor series approximation uses a Taylor series to represent a number as a polynomial that has a very similar value to the number in a neighborhood around a specified value:
Taylor series are extremely powerful tools for approximating functions that can be difficult to compute otherwise, as well as evaluating infinite sums and integrals by recognizing Taylor series.
Suggested steps for approximating values:
- Identify a function to resemble the operation on the number in question.
- Choose to be a number that makes easy to compute.
- Select to make the number being approximated.
Using the first three terms of the Taylor series expansion of centered at , approximate
The first three terms shown will be sufficient to provide a good approximation for . Evaluating this sum at gives an approximation for
With just three terms, the formula above was able to approximate to six decimal places of accuracy.
Using the quadratic Taylor polynomial for approximate the value of
The quadratic Taylor polynomial is
First, write down the derivatives needed for the Taylor expansion:
But what about and Choose so that the values of the derivatives are easy to calculate. Rewriting the approximated value as
The actual value is
so the approximation is only off by about 0.05%.
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