Taylor Polynomials & Approximations - Calculus 2

In a previous lesson we established two special types of power series known as Taylor series and Maclaurin series. One of the applications of these series is the ability to approximate the value of a function at a particular value of x. This is done by using what are known as Taylor polynomials, which are just part of a Taylor series.

Taylor polynomials are what result from cutting off the terms of a Taylor series at a particular term for a value of n. Doing so produces a what is called a "Taylor polynomial of degree n". Since an entire Taylor series with its infinite number of terms represents a function f(x), if we cut off some of those terms to create a Taylor polynomial, the remaining terms no longer exactly represent the function, but rather approximate it.

If we could plug in a particular value of x into the Taylor series for a function, we would get the exact value of that function at that value of x. However, from a realistic standpoint, this would be impossible for us to accomplish as there are an infinite number of terms to a Taylor series. This is why we have a need for Taylor polynomials. Shortening the Taylor series to just a few terms will be enough to get us an approximation of the value of the function at a particular value of x that is fairly accurate to the actual value. But just how accurate will it be?

To answer the question of accuracy, we have to take a look at the error/remainder associated with the approximation from a Taylor polynomial. Now, it is always possible to simply take the absolute value of the difference between the actual value and approximation to find the exact error, but what if we don't know, or can't find the actual value in question? This is where the idea of approximating the error/remainder comes into play.

We can use a particular formula to find an error bound, or the maximum error that represents the highest possible difference between the approximation and the actual value for a certain Taylor polynomial. Being able to determine this error bound, as well as find Taylor polynomials and use them to approximate values is the focus of this lesson.

As a side note, this lesson is the last lesson in the second major group of concepts in Calculus 2 of sequences and series. It all began with sequences in lesson 18, and we have slowly built upon that topic with each additional lesson, leading to this final lesson about Taylor polynomials and approximations. I image by this point you are probably sick and tired of series, so the good news is that next lesson will begin the third and final major group of concepts in Calculus 2, which is parametric equations and polar coordinates. So look forward to that next!

But before we get to those topics, we need to talk about Taylor polynomials! So let's get to it!

In this lesson, you will learn:

• How to find an nth degree Taylor polynomial for a function centered at a certain value of c
• How to use Taylor polynomials to approximate values for functions
• How to calculate the error bound/maximum error associated with Taylor approximations

I hope you find this video to be helpful!

-Josh