A drug response curve describes the level of medication in the bloodstream after

Question

A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S(t) = Atpe?kt (where t > 0) is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A = 0.03, p = 4, k = 0.07, and t is measured in minutes, estimate the times t corresponding to the inflection points. (Round your answers to two decimal places.) t=_____min (smaller value), t=______min (larger value)

Explanation / Answer

d2S/dt2 = d/dt (dS/dt) = d/dt [e-kt (Aptp-1 - kAtp)] = -ke-kt (Aptp-1 - kAtp) + e-kt (Ap(p-1)tp-2 - kpAtp-1)] = = e-kt [k2Atp -2 kpAtp-1 + Ap(p-1)tp-2] For A = 0.02, p = 3, k = 3: d2S/dt2 = e-3t [0.18t3 -0.36At2 + 0.12t] . The zeros of this second derivative are the inflection points: e-3t [0.18t3 -0.36At2 + 0.12t] = 0. e-3t t(3t2 - 6t + 2) = 0. t1 = 0 3t2 - 6t + 2 = 0; D = 9-6=3 t2 = (3 - v3)/3 = 1 - 1/v3 = 0.423; t3 = (3 + v3)/3 = 1 + 1/v3 = 1.577; Thus the points of inflection are 0, 0.423, 1.577

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