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) = A*t^p*e^-kt , t>= 0 is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. The parameters A, p, k are all positive and depend on the particular drug. (a) Find the intervals where S is increasing, and where it is decreasing. (b) Use your result in (a) to specify the time and the value of maximal response.

Explanation / Answer

S(t) = 0.02(t^6 e^-9t) ? S'(t) = 0.02(6t^5 e^-9t + t^6 (-9e^-9t)) = 0.02 (-9t^6 + 6t^5) e^-9t ? S''(t) = 0.02[(-54t^5 + 30t^4) e^-9t + (-9t^6 + 6t^5) (-9e^-9t)] = 0.02 (81t^6 - 108t^5 + 30t^4) e^-9t = 0.06 t^4 (27t^2 - 36t + 10) e^-9t = 0 ? the inflection points are zeros of 27t^2 - 36t + 10 = 0 ? = 36^2 - 4(27)(10) = 216 = 6^3 ? t = (36 + 6v6)/54 and t = (36 - 6v6)/54 ? t = (6 + v6)/9 and t = (6 - v6)/9

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