The La Grange Error Estimate THEOREM 9 19

THEOREM 9. 19 TAYLOR’S THEOREM If a function f is differentiable through order n

quod erat demonstrandum- which was to be demonstrated

Consider the function The Taylor series for f centered at x = 3 is

We had the advantage of having a very basic function to evaluate at x

We will need the maximum of the absolute value of the fifth derivative of

The series we used to estimate f(2) was Did you notice the series was

Example 7: Determining the Accuracy of the Approximation The third Maclaurin polynomial for Use

Example 8: Approximating a Value to a Desired Accuracy Determine the degree of the

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The La. Grange Error Estimate

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.

quod erat demonstrandum- which was to be demonstrated

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 =

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 La. Grange Error Estimate

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 6 th derivative equal to zero, or recognize that the 6 th derivative is always positive therefore the max of the 5 th derivative occurs at the right endpoint. The maximum will occur at x = 3 The maximum is 22. 5

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 0. 1875.

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

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 0. 001.