During a motor's start-up

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Integration: error expansions, Gauss rules and adaptivity

Problem

During a motor's start-up, the instantaneous power drawn is recorded at three unequally spaced times t0=0 st_0=0\,\mathrm{s}, t1=1 st_1=1\,\mathrm{s}, t2=4 st_2=4\,\mathrm{s}. To estimate the energy E=∫04P(t) dtE=\int_0^4 P(t)\,\mathrm{d}t (in kJ, with PP in kW) an interpolatory quadrature rule E≈w0P(t0)+w1P(t1)+w2P(t2)E\approx w_0P(t_0)+w_1P(t_1)+w_2P(t_2) is built by integrating the quadratic Lagrange interpolant through these three points. ((a)) Find w0w_0, w1w_1, w2w_2 (in seconds). ((b)) Compute ∑j∣wj∣/(b−a)\sum_j|w_j|/(b-a) with b−a=4 sb-a=4\,\mathrm{s} and comment on the stability of this rule, given the sign of one of the weights.

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