A radioactive source has count rate with and

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

Error, Chance and the Machine, Ch Approximating a function: interpolation and differentiation

Problem

A radioactive source has count rate R(t)=R0e−t/τR(t) = R_0 e^{-t/\tau} with R0=1000 s−1R_0 = 1000\,\mathrm{s^{-1}} and τ=5.0 s\tau = 5.0\,\mathrm{s}. A degree-4 polynomial model of RR on t∈[0,10] st \in [0,10]\,\mathrm{s} is to be built from samples at the five Chebyshev points of the second kind, mapped from [−1,1][-1,1] to [0,10][0,10] by tj=5+5cos⁡(jπ/4)t_j = 5 + 5\cos(j\pi/4), j=0,1,2,3,4j = 0,1,2,3,4. ((a)) Compute the five nodes tjt_j, to two decimal places. ((b)) Compute R(tj)R(t_j) at each node, to three significant figures.

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