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11 August, 14:19

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

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  1. 11 August, 14:26
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    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
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