Motivating Taylor Polynomials Via the Binomial Theorem

By Dobbs, David E. | Mathematics and Computer Education, Fall 2010 | Go to article overview

Motivating Taylor Polynomials Via the Binomial Theorem


Dobbs, David E., Mathematics and Computer Education


(ProQuest: ... denotes formulae omitted.)

1. INTRODUCTION

The material of this note can find classroom use when students are studying Taylor polynomials in a Calculus or Elementary Analysis course. Taylor polynomials play an important role in calculus and its applications. For instance, it was shown in [3, Theorem 2] that Taylor polynomials can arise naturally in a differential equations course when one seeks polynomial approximations to power series solutions of certain initial value problems by using the method of undetermined coefficients. That paper was motivated by an example in a calculus reform textbook [1, Example 14, pp. 587-588]. One calculus textbook that is widely used today motivates the introduction of the nth Taylor polynomial as the polynomial of degree at most ? that gives the "best approximation" (in a certain sense) to a given infinitely differentiable function [4, p. 254] and, later, as a partial sum of the Taylor series of the given function [4, p. 607] . The main purpose of the present note is to motivate the supplement of the study of Taylor polynomials by means of the Binomial Theorem. We next explain how doing so in the typical calculus course would have at least three advantages over the other strategies that were described above.

First, our proposed method uses a tool that is already familiar, as the Binomial Theorem is typically used early in a calculus course to prove the formula for the derivative of x^sup n^, as in [4, p. 184]. Second, our method reinforces the standard rules for differentiating sums or constant multiples of differentiable functions, which are generally covered shortly after the derivative of x^sup n^, as in [4, pp. 186-187]. Third, our method does not need time-consuming calculations of n01 derivatives. Thus, in regard to the third point, this note contributes to the view expressed in [2] that the Taylor polynomials/series of the most useful functions can often be obtained without the explicit calculations of higher derivatives of the kind that are found in [4, pp. 610-612].

2. THE BINOMIAL THEOREM IMPUES TAYLOR'S THEOREM FOR POLYNOMIALS

We will show how the Binomial Theorem, ... leads to a proof that any nth degree polynomial function, f(x)-a^sub n^x^sup n^+a^sub n-1^x^sup n-1^+ ... + a^sub 0^, can be expressed as ..., which happens to be (the definition of) the ? * Taylor polynomial of f (at x^sub 0^). The proof given below is thus, in effect, a discovery activity that can lead to the definition of the Taylor polynomials/series for any sufficiently/infinitely differentiable function.

Let us begin with the special case f(x) = x^sup n^, for some positive integer n. As in [2, Example 1 (b)], the key is to rewrite x as the sum x^sub 0^ +(x-x^sub 0^) . Then, raising to the n* power and applying the Binomial Theorem, we have

Next, observe that the factor n(n - 1)(n - 2) ... (n - i + 1)(x^sub 0^)^sup n-i^ is the same as f^sup (i)^(x^sub 0^) for me particular function f(x) = x^sup n^. …

The rest of this article is only available to active members of Questia

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this article

This article has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this article

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited article

Motivating Taylor Polynomials Via the Binomial Theorem
Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Help
Full screen

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Style
    Citations are available only to our active members.
    Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

    1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

    Cited passage

    Thanks for trying Questia!

    Please continue trying out our research tools, but please note, full functionality is available only to our active members.

    Your work will be lost once you leave this Web page.

    Buy instant access to save your work.

    Already a member? Log in now.

    Oops!

    An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.