Lagrange Error Bound

Benjamin Wright

6 min read

Next Topic - Radius and Interval of Convergence of Power Series

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Study Guide Overview

This guide covers the Lagrange Error Bound for Taylor and Maclaurin Polynomials. It reviews constructing Taylor polynomials and explains how to use the Lagrange Error Bound to determine the maximum error of a Taylor polynomial approximation. The guide includes examples and a practice AP FRQ problem involving the Lagrange Error Bound.

#10.12 Lagrange Error Bound

At this point in your Calculus journey, you should have learned what exactly a Maclaurin and Taylor polynomial is, how to write one, and how to use them to approximate non-algebraic functions. If you’ve ever wondered just how good these approximations are, and by what value this approximation is wrong by, this lesson is for you.

This guide will discuss the Lagrange Error Bound and how this can be used to determine the largest, or maximum, number of error your Taylor polynomial is.

#🟥Review: Taylor Polynomials

In case you’re not familiar with them, or would like to gain further proficiency in writing them, we will touch up on our knowledge of Taylor Polynomials. To generalize what they are, Taylor polynomials are approximations of functions using polynomial expressions by finding derivatives. All Taylor series are centered at a certain point, which is what’s used to estimate the functions behavior. Here’s the general formula for the Taylo...

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Previous Topic - Finding Taylor Polynomial Approximations of Functions Next Topic - Radius and Interval of Convergence of Power Series

Question 1 of 9

What is the general formula for a Taylor polynomial of a function f(x) centered at x=a?

$f(x)=f(a)+f'(a)(x-a)+f''(a)(x-a)^2+...$

$f(x)=f(a)+f'(a)x+\frac{f''(a)x^2}{2!}+\frac{f'''(a)x^3}{3!}+...$

$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)(x-a)^2}{2!}+\frac{f'''(a)(x-a)^3}{3!}+...+ \frac{f^{(n)}(a)(x-a)^n}{n!}$

$f(x)=f(a)+f'(a)(x-a)+f''(a)(x-a)^2+f'''(a)(x-a)^3+...$