Question #1858

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

Expert's answer

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} [k^{2}At^{p} -2 kpAt^{p-1} + Ap(p-1)t^{p-2}]

For A = 0.02, p = 3, k = 3:

d^{2}S/dt^{2} = e^{-3t} [0.18t^{3} -0.36At^{2} + 0.12t] . The zeros of this second derivative are the inflection points:

e^{-3t} [0.18t^{3} -0.36At^{2} + 0.12t] = 0.

e^{-3t} t(3t^{2} - 6t + 2) = 0.

t_{1} = 0

3t^{2} - 6t + 2 = 0;

D = 9-6=3

t_{2} = (3 - √3)/3 = 1 - 1/√3 = 0.423;

t_{3} = (3 + √3)/3 = 1 + 1/√3 = 1.577;

Thus the points of inflection are**0, 0.423, 1.577**

The the second derivative :

d2S/dt2 = d/dt (dS/dt) = d/dt [e

= e

For A = 0.02, p = 3, k = 3:

d

e

e

t

3t

D = 9-6=3

t

t

Thus the points of inflection are

## Comments

Assignment Expert14.10.14, 18:38Dear Cara. The first derivative with the given particular parameters A=0.02, p=k=3 is

3*0.02t

^{2}*e^{-3t}-0.06t^{3}e^{-3t}=(0.06t^{2}-0.06t^{3})e^{-3t}Cara14.10.14, 07:46How did you take the first derivative of the surge function? I'm not quite sure how to take the derivative of the function because it has three variables and my derivative isn't coming out the same as yours for the first or second derivative.

Assignment Expert10.10.14, 19:33Dear Emma.

An inflection point on the graph of a function y=f(x) is defined as a point at which the graph passes from one side of its tangent line to another. At an inflection point, the graph changes the direction of its convexity. If the second derivative at some x0 is equal to zero and second derivative changes sign as x passes through x0, then (x0, f(x0)) is an inflection point.

Emma08.10.14, 16:56What is the significance of the points of inflection?

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