Answer to Question #316861 in Calculus for jash

Question

A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S(t) = At^pe^−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 = 0.05, 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)

Answer 1

$S(t)=At^pe^{-kt}=0.03\cdot t^4\cdot e^{-0.05t}$; $t>0$

Find inflection points.

Solution:

Inflection points can only occur when the second derivative is zero or undefined.

Here we have:

$S'(t)=0.03\cdot 4t^3e^{-0.05t}-0.03\cdot 0.05t^4e^{-0.05t}=$$0.03e^{-0.05t}t^3( 4-0.05t)$;

$S''(t)=-0.03\cdot 0.05e^{-0.05t}t^3( 4-0.05t)+$ $0.03\cdot 3e^{-0.05t}t^2( 4-0.05t)-$$0.03\cdot 0.05e^{-0.05t}t^3=$ $t^2\cdot e^{-0.05t}( 0.000075t^2-0.012t+0.36)$;

$t^2\cdot e^{-0.05t}( 0.000075t^2-0.012t+0.36)=0$

Therefore possible inflection points occur at

$t=40$ and $t=120$; ($t=0$ not interested because we are looking for inflection points only for $t>0$ ).

However, to have an inflection point we must check that the sign of the second derivative is different on each side of the point. Here we have

$S''(10)\approx 15.01$;

$S''(100)\approx-6.06$;

$S''(150)\approx3.08$.

Hence, both are inflection points.

Answer: there are two inflection points $t=40$.00 and $t=120$.00 .