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) = Atpekt

(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 = 0.09, and t is measured in minutes, estimate the times t corresponding to the inflection points. (Round your answers to two decimal places.)

(smaller value)

(smaller value)

Explanation / Answer

The inflection points of the function.

The the second derivative :
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 = 0.03, p = 4, k = 0.09:
d2S/dt2 = e-0.09t [0.000243t3 -0.0216t2 + 0.36t2 ]

The zeros of this second derivative are the inflection points:
e-0.09t [0.000243t3 -0.0216t2 + 0.36t2 ] = 0

e-0.09t [0.000243t3 -0.3384 t2 ] = 0

e-0.09t [0.000243t3 -0.3384 t2 ] =0

t2 = (3 - √3)/3 = 1 - 1/√3 = 0.423;
t3 = (3 + √3)/3 = 1 + 1/√3 = 1.577;
Thus the points of inflection are 0, 0.423, 1.577

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