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workshop 1 problem 2

workshop 1 problem 2 - -1 Problem Derivatives re-visited 2...

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1 Problem. Derivatives re-visited 2 In Problem 1 we have seen that we can use higher order derivatives (or their approximations in the form of finite differences) to detect an isolated error in tabulated or experimental data. We will now try to understand why that happens. This will also give us a way to estimate the order of magnitude of the initial error. a) In order to look just at the error and how it changes, we first assume we measure a constant function of value 0 , f ( x ) = 0 . We sample it on 11 equidistant points from x 0 = 0 to x 10 = 1 in intervals of Δ = 0 . 1 . Assume that all measurements are exact except for the middle one at x 6 = 0 . 5 which has an error ε . This is represeneted on the next page in the second column of Table 1. Fill out the remaining coloumns of the table using the same “finite difference” approximation for the derivatives as in Problem 1 (i.e., use f x i + x i +1 2 f ( x i +1 ) - f ( x i ) x i +1 - x i and similarly for the higher order derivatives).
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