1 10 9 Error Analysis in Taylor Polynomials Recall We can find the Taylor series of any differentiable function f x X cn x a n where cn n 0 However this is not very practical It is true however that we can approximate the function with a finite 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 by RN x The question then is how large a polynomial is necessary to acheive a desired accuracy for the function on a given interval That is how far off are we at most at any point on a given interval when we stop the 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 1 Examples Use a 3rd degree Taylor polynomial at a 0 to approximate ex on the interval 1 1 which can then be 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 the approximation on the interval 2 6 2 On Beyond Average 1 2 ln 1 x2 dx to within Determine the degree of the Taylor Polynomial needed to approximate 0 0 001 accuracy Calc required In Einstein s special theory of relativity the relativistic generalization of the kinetic energy of an object is given by 2 1 2 v K mc2 1 2 1 c Here m is the object s mass c is the speed of light and v is the speed of the object Show that for everyday speeds i e whenever v is VERY MUCH LESS than c the above expression reduces to the 1 classical kinetic energy of Newtonian theory K mv 2 2 a Compute the first 3 terms of the Maclaurin series for f x 1 x 1 2 v2 b Substitute x 2 into a to get an approximate series for c c Substitute b into the original expression 3 v2 1 2 c 1 2
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