Answer to Question #1858 in Calculus for Abhilash

Question

A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function, given below, 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 = 0.02, p = 3, k = 3, and t is measured in minutes, estimate the times, t corresponding to the inflection points. (Round the answers to three decimal places.)

S(t) = At^p e^-kt

t > 0

Answer 1

Let's find the inflection points of the function.
The the second derivative :
d2S/dt2 = d/dt (dS/dt) = d/dt [e^-kt (Apt^p-1 - kAt^p)] = -ke^-kt (Apt^p-1 - kAt^p) + e^-kt (Ap(p-1)t^p-2 - kpAt^p-1)] =

= e^-kt [k²At^p -2 kpAt^p-1 + Ap(p-1)t^p-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 - √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