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 we want to approximate using the Taylor polynomial of degree 4, , around for the function . We know that
so we are asking how close are and . In order to use the formula in the theorem, we just need to find , the maximum value of the th derivative of between and . Since and is strictly increasing, the maximum in happens at . Thus which is a number, say, less than . Therefore:
What Taylor polynomial (what ) should we use to approximate within ? As above, we will be using the Taylor polynomial for , evaluated at . Thus, we want the error . Notice all derivatives are and the maximum happens at , where , so for all derivatives . Hence by the theorem:
So we need . Try several values of until that is satisfied:
Thus should work. So we just need , or add .
|Title||example of use of Taylor’s theorem|
|Date of creation||2013-03-22 15:05:51|
|Last modified on||2013-03-22 15:05:51|
|Last modified by||alozano (2414)|