Question
Write the error in linear interpolation. Hence, show that
where
Answer :
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Consider two points ((x_0, f(x_0))) and ((x_1, f(x_1))). The linear interpolation (L(x)) between them is [ L(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0}(x - x_0). ] The error in linear interpolation is [ E(x) = f(x) - L(x). ] By Taylor's theorem, for some (\xi \in (x_0, x_1)), we have [ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{f''(\xi)}{2}(x - x_0)(x - x_1), ] because ((x - x_0)(x - x_1) =
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विश्रामावस्था से आरंभ कर रही एक गाड़ी का वेग पहले घंटे के लिए निम्नलिखित तालिका में दिया गया है। सिम्पसन
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