{"id":10714,"date":"2017-07-28t18:49:30","date_gmt":"2017-07-29t01:49:30","guid":{"rendered":"\/\/www.catharsisit.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":"\/\/www.catharsisit.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