1. The LaGrange Error Estimate

2. THEOREM 9.19 TAYLOR’S THEOREM
If a function f is differentiable through order n + 1 in an interval I containing c, then, for each x in I, there exists z between x and c such that
Where .

9. quod erat demonstrandum- which was to be demonstrated

10. Consider the function
The Taylor series for f centered at x = 3 is
We could use the fourth degree polynomial to estimate f(2).
Error =

11. We had the advantage of having a very basic function to evaluate at x = 2. What if the function were not so easy to evaluate? We may not know the actual value of the function at x = 2. Could we have any confidence in our estimate using this polynomial?
The LaGrange Error Estimate

12. We will need the maximum of the absolute value of the fifth derivative of our function.
We will be looking for this maximum in the interval between the center of the polynomial and the value of x we are using for our approximation.
What is the maximum value of
on the interval
We could use a Graphing Calculator, look for critical numbers by setting the 6th derivative equal to zero, or recognize that the 6th derivative is always positive therefore the max of the 5th derivative occurs at the right endpoint.
The maximum will occur at x = 3
The maximum is 22.5

13. The series we used to estimate f(2) was
Did you notice the series was alternating?
The error of an alternating series can be estimated by evaluating the first neglected term.
Remember this is not the actual error (or remainder ). Instead it bounds the error. In other words we know the remainder (error) must be less than or possibly equal to

14. Example 7: Determining the Accuracy of the Approximation
The third Maclaurin polynomial for is given by
Use Taylor’s Theorem to approximate by and determine the accuracy of the approximation.

15. Example 8: Approximating a Value to a Desired Accuracy
Determine the degree of the Taylor polynomial Pn(x) expanded about c = 1 that should be used to approximate
so that the error is less than