Problem

Drug pharmacokinetics The level of medication in …

18:46

Problem 53 Hard Difficulty

Drug pharmacokinetics So-called two-compartment models are often used to describe drug pharmacokinetics,with the blood being one compartment and the internal organs being the other (see Section $10.3 ) .$ If the rate of flow from the blood to the organs is $\alpha$ and the rate of metabolism from the organs is $\beta,$ then under certain conditions the concentration of drug in the organ at time $t$ is given by
$$\frac{e^{-\alpha t}-e^{-\beta t}}{\beta-\alpha}$$
What is the predicted concentration at time $t$ if the values of $\alpha$ and $\beta$ are very close to one another?

Answer

$t e^{-\beta t}$

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Calculus 1 / AB

Biocalculus Calculus for the Life Sciences

Chapter 4

Applications of Derivatives

Section 3

L'Hospital's Rule: Comparing Rates of Growth

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Video Transcript

Okay, so here we are, given the equation e to the negative. Alfa T minus e to the negative beta t over beta minus alfa and were asked what's happening when it alpha and beta are very close to one another's. We're gonna take the limit as Alfa approaches beta. Okay, so we're treating Alfa like our way we would normally treat, say X and then t is gonna be treated more like a constant. Okay, so when we put beta in when we evaluate this limit, we, of course, get 0/0. So we compile Opie atolls rule case will then evaluate the limit as Alfa approaches beta. So we'll take the derivative of E the power of negative offer times T. And so again, this time we're treating tea later, Constance is gonna be negative t e to the negative, Al Fatih. And this time for violating negative, uh, e to native Bt both being tear being treat Lake Constance. And so this whole expression is a constant. That's gonna be a zero data we are treating as a constant will fall off. And then this native Alfa will take the driven in love. It's a negative one. Okay, so the zero we can just kind of say goes away in both. These negatives will cancel each other out. And so, of course, the one on the bottom doesn't really mean anything. Get that. And so we're left with Is the limit as awful? Purchase beta of tea times e to the negative offer tea. So now we evaluate it. Okay? And this is gonna be tee times e to the negative. Beta t, and that is thesame. Ooh, Shin.

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