1 10.9: Error Analysis in Taylor PolynomialsRecall: We can find the Taylor series of any differentiable function f(x) =∞Xn=0cn(x − a)nwhere cn=However, this is not very practical. It is true, however, that we can approximate the function with afinite polynomial by looking at partial sums.Recall: The N th degree Taylor polynomial of f at x = a:The Remainder of the N th degree Taylor polynomial is given byRN(x) =The question, then, is how large a polynomial is necessary to acheive a desired accuracy for the functionon a given interval? That is, how far off are we at most at any point on a given interval when we stopthe series at a given value of N ?Recall that if the series is an Alternating Series, then |RN(x)| ≤Also recall Taylor’s Inequality:Graphical analysis of the error is another method which will be done in Matlab.1Examples:Use a 3rd degree Taylor polynomial at a = 0 to approximate exon the interval [−1, 1] (which can thenbe used to approximate e) and determine the accuracy of your results using the remainder theorem.Approximate f(x) =√x by a Taylor polynomial of degree 2 at x = 4. Determine the accuracy of theapproximation on the interval [2, 6].2On Beyond Average:Determine the degree of the Taylor Polynomial needed to approximateˆ1/20ln(1 + x2) dx to within0.001 accuracy. (Calc required)In Einstein’s special theory of relativity, the relativistic generalization of the kinetic energy of an objectis given byK = mc2 1 −v2c2−1/2− 1!Here m is the object’s mass, c is the speed of light, and v is the speed of the object. Show that, foreveryday speeds (i.e., whenever v is VERY MUCH LESS than c), the above expression reduces to theclassical kinetic energy of Newtonian theory, K =12mv2:(a) Compute the first 3 terms of the Maclaurin series for f (x) = (1 + x)−1/2(b) Substitute x = −v2c2into (a) to get an approximate series for1 −v2c2−1/2(c) Substitute (b) into the original
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