Calculus II For Dummies
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The Taylor series provides a template for representing a wide variety of functions as power series. It is relatively simple to work with, and you can tailor it to obtain a good approximation of many functions.

Here’s the Taylor series in all its glory:

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The Taylor series uses the notation f(n) to indicate the nth derivative. Here’s the expanded version of the Taylor series:

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The presence of the variable a provides the Taylor series with a lot of flexibility, as the next example illustrates.

Suppose you want to approximate the value of sin 10. You can use only four terms of the Taylor series to make a good approximation. The key to this approximation is a shrewd choice for the variable a:

Let a = 3

This choice has two advantages: First, this value of a is close to 10 (the value of x), which makes for a good approximation. Second, it’s an easy value for calculating sines and cosines, so the computation shouldn’t be too difficult.

To start off, substitute 10 for x and 3 for a in the first four terms of the Taylor series:

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Next, substitute in the first, second, and third derivatives of the sine function and simplify:

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The good news is that sin 3 = 0, so the first and third terms fall out:

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At this point, you probably want to grab your calculator:

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This approximation is correct to two decimal places.

About This Article

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About the book author:

Mark Zegarelli, a math tutor and writer with 25 years of professional experience, delights in making technical information crystal clear — and fun — for average readers. He is the author of Logic For Dummies and Basic Math & Pre-Algebra For Dummies.

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