Estimate the maximum error made in approximating [tex]e^x[/tex] by the polynomial [tex]1 + x + \frac{1}{2}x^2[/tex] over the interval [tex]x \in [-0.4, 0.4][/tex].

e^x = 1 + x + x² / 2 + x³/ 3! + x^4 / 4! + ..... = (1 + x + x²/2 ) + x³ [ 1/6 + x /4! + x² / 5! + .... ] Error = e^x - (1+ x + x² ) = x³ [ 1/6 + x /4! + x² / 5! + .... ] x / 4! < x / 6 x² / 5! < x² / 6 and so on So if we replace all factorials by 1/6 .. error < x² [ 1/6 + x/6 + x²/6 + ... ] < x² / 6 [ 1 + x + x² ..... ] < x² / 6 * 1 / (1 -x) = x² / 6 (1-x) if x < 1 maximum error = x² /6(1-x) occurs at 0.4 or -0.4 in the given interval. = 0.0444444

Answered by Everest2017 | 2024-06-10

The maximum error in approximating e^x with a second-degree polynomial over the interval [-0.4,0.4] can be found using Taylor's theorem, looking at the next term in the Taylor series, which is (1/6)x^3 for a third-degree term. Evaluating e^0.4/6 * (0.4)^3 gives an estimate of the error bound. ;

Answered by adipratapsingh12 | 2024-06-18

The maximum error in approximating e x with the polynomial 1 + x + 2 1 ​ x 2 over x ∈ [ − 0.4 , 0.4 ] can be estimated as approximately 0.01067 . This error arises from the first omitted term in the Taylor series expansion. Thus, we conclude that the maximum error is about 0.01067 .
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Answered by Everest2017 | 2024-12-24