{"id":10714,"date":"2017-07-28T18:49:30","date_gmt":"2017-07-29T01:49:30","guid":{"rendered":"https:\/\/magoosh.com\/hs\/?p=10714"},"modified":"2017-07-28T18:49:30","modified_gmt":"2017-07-29T01:49:30","slug":"ap-calculus-bc-review-taylor-polynomials","status":"publish","type":"post","link":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-bc-review-taylor-polynomials\/","title":{"rendered":"AP Calculus BC Review: Taylor Polynomials"},"content":{"rendered":"

Taylor polynomials are certain polynomials that can approximate functions. The more terms in the Taylor polynomial, the greater the accuracy of the approximation.<\/p>\n

\"Brook
Brook Taylor, one of the originators of Taylor polynomials<\/figcaption><\/figure>\n

In this article we will define Taylor polynomials and work out a number of examples similar to those you might see on the AP Calculus BC exam.<\/p>\n

Taylor Polynomials<\/h2>\n

Before we define the Taylor polynomial, let’s remember what a normal, everyday kind of polynomial is.<\/p>\n

Polynomials<\/h3>\n

A polynomial<\/strong> is any expression of the following form.<\/p>\n

\"General<\/p>\n

Here, n<\/em> is a whole number. Each coefficient a<\/em>0<\/sub>, a<\/em>1<\/sub>, a<\/em>2<\/sub>, …, an<\/sub><\/em>, must be constant. Furthermore, we always insist that an<\/sub><\/em> is nonzero. (Otherwise, we really wouldn’t have an n<\/em>th term at all, right?) Then we say that the degree<\/strong> of the polynomial is n<\/em>.<\/p>\n

Often, you’ll see the terms written in the opposite order, from high degree down to low, which is called standard form<\/strong>. But a Taylor polynomial is usually written starting from low degree and then progressing to higher degrees with each term. However you slice it, a polynomial is simply the sum of a terms, each of which involving a whole number power of the variable.<\/p>\n

Here are a few example polynomials:<\/p>\n

\"polynomial,<\/p>\n

\"polynomial,<\/p>\n

\"Polynomial<\/p>\n

Here, f<\/em> has degree 6, g<\/em> has degree 8, and h<\/em> has degree 11.<\/p>\n

Polynomials and Derivatives<\/h3>\n

The nice thing about a polynomial is that it’s very easy to take its derivative. All you need are three simple rules.<\/p>\n

    \n
  1. The Sum\/Difference Rule\n

    \"Sum<\/li>\n

  2. the Constant Multiple Rule\n

    \"Constant<\/p>\n<\/li>\n

  3. and the Power Rule\n

    \"Power<\/p>\n<\/li>\n<\/ol>\n

    Now, even if you replace x<\/em> by (x<\/em> – c<\/em>) in a polynomial, you still get a polynomial. Let’s explore the derivatives of such a function.<\/p>\n

    \"Derivatives<\/p>\n

    As you keep taking higher-order derivatives, notice how the degree of the polynomial steps down. <\/p>\n

    Another curiousity is how the constant terms behave. By plugging x<\/em> = c<\/em> into the function and each successive derivative, you can recover each constant term. Take a look at the pattern that emerges.<\/p>\n