1) Calculate the error, relative error in the following approximation x_T ? x_A: (a) x_T = 2.71828, x_A = 2.7182 (b) x_T = 98,350, x_A = 98,000 (c) x_T = ln(4), x_A = 1.386002 (d) x_T = ?2, x_A = 1.2598 (e) x_T = ?, x_A = 3.142 2) Sometimes the loss of significance error can be avoided by rearranging terms in the function using a known identity from trigonometry or algebra. Find an equivalent formula for the following functions that avoids a loss of significance. (a) sqrt(x^2 + 1) - x for large x (b) sqrt((1 + cos(x)) / 2) (c) cos^2(x) - sin^2(x) for x ? ?/4 (d) (1 - cos(x)) / x^2 for x ? 0 3) Use Taylor approximations to avoid the loss-of-significance errors in the following formulas when x is near 0. In each case your final answer should have at least four terms. (a) (e^x - 1 - x) / x^2 (b) (e^x - e^{-x}) / x^2 (c) (1 - cos(x^2)) / x (d) (1 - e^x) / x^2
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In this case, we don't have the true value, so we can't calculate the error. The relative error is calculated as the error divided by the true value, so again, we can't calculate it without the true value. Show more…
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