example of use of Taylor’s theorem

In this entry we use Taylor’s Theorem in the following form:

Theorem 1 (Taylor’s Theorem: Bounding the Error).

Suppose f and all its derivatives are continuousMathworldPlanetmath. If Tn(x) is the n-th Taylor polynomialMathworldPlanetmath of f(x) around x=a, then the error, or the difference between the real value of f and the values of Tn(x) is given by:


where M is the maximum value of f(n+1) (the n+1-th derivative of f) in the interval between a and x.

Example 2.

Suppose we want to approximate e using the Taylor polynomial of degree 4, T4(x), around x=0 for the function ex. We know that


so we are asking how close are e and T4(1)=1+1+1/2+1/6+1/24. In order to use the formula in the theorem, we just need to find M, the maximum value of the 4th derivative of ex between a=0 and x=1. Since f(4)=ex and ex is strictly increasing, the maximum in (0,1) happens at x=1. Thus M=e which is a number, say, less than 3. Therefore:


Thus the approximation has an error of less than 0.025. Actually, if we use a calculator we obtain that the error is exactly 0.0099. But, of course, the whole point of the theorem is not to use a calculator.

Example 3.

What Taylor polynomial Tn(x) (what n) should we use to approximate e within 0.0001? As above, we will be using the Taylor polynomial Tn(x) for ex, evaluated at x=1. Thus, we want the error |En(1)|<0.0001. Notice all derivatives are ex and the maximum happens at x=1, where e1=e, so for all derivatives M<3. Hence by the theorem:


So we need 3/(n+1)!<0.0001. Try several values of n until that is satisfied:

3/2=1.5, 3/3!=0.5, 3/4!=0.125, 3/5!=0.025, 3/6!=0.00416

Thus n=7 should work. So we just need T7(x), or add 1+1+1/2++1/7!.

Title example of use of Taylor’s theorem
Canonical name ExampleOfUseOfTaylorsTheorem
Date of creation 2013-03-22 15:05:51
Last modified on 2013-03-22 15:05:51
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Example
Classification msc 41A58