In this video lesson we will learn about the Taylor and Macluarin Series.
The Taylor Series, or Taylor Polynomial, is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. A Maclaurin Polynomial, is a special case of the Taylor Polynomial, that uses zero as our single point.
The basic idea behind this lesson is that we like polynomials because they’re “easy” and we know how to differentiate and integrate them quickly. What a Taylor Polynomial does for us is to take something that is “hard” and turn it into something easier.
As nicely stated by Wikipedia, Brook Taylor and Colin Maclaurin gave us such a way. The general premise is that we start at a single point, and calculate an infinite number of derivatives knowing that the sum of these terms approximates the function itself.
We will start with the general formula for a Taylor Series, and recognize the specific pattern that is used. We will carefully walk through several examples of how to generate the Taylor Polynomial and see how to use it to approximate a function.
Then we will look at the Maclaurin series, and learn some very important Expansion Formulas that we will want to memorize in order to use quickly rather than having to generate the Taylor Polynomial by hand.
Then we will practice using the Taylor and Maclaurin Expansions for several questions, and see how this enables us to take derivatives and integrals of functions that previously would be too challenging.
Maclaurin and Taylor Series Video
Maclaurin and Taylor Series
Maclaurin and Taylor Series Video
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