Problem

Prove that $$\quad \lim _{x \rightarrow \infty} …

03:12

Problem 54 Hard Difficulty

Drug pharmacokinetics The level of medication in the bloodstream after a drug is administered is often modeled by a function of the form
$$S(t)=A t^{p} e^{-k t}$$
where $A, p,$ and $k$ are positive constants. This is called a surge function because its graph shows an initial surge in the drug level followed by a more gradual decline. (Particular cases have been investigated in Example 9 and Exercise 4.2.51.)
(a) Use l'Hospital's Rule to show that lim $_{t \rightarrow \infty} S(t)=0$ for all positive values of $A, p,$ and $k .$
(b) Investigate the family of surge functions for $A=1$ and positive values of $p$ and $k .$ What features do these curves have in common? How do they differ from one another? In particular, what happens to the maximum and minimum points and inflection points as $p$ and $k$ change? Illustrate by graphing several members of the family.

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Calculus 1 / AB

Biocalculus Calculus for the Life Sciences

Chapter 4

Applications of Derivatives

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Video Transcript

um, we have Ah, dysfunction. Mrs D. It's, uh it's morning. The concentration of blood, the specter time, you know, love streams. You have a steam power. Bi exponential, minus key d We're a be monte on. Uh, okay. Are positive constants, but so you want to look good for a start, or is the limit studios changed off us? Uh, so what he's mean is or what happened with what happens with blood Constant? Because attritional drugs bloodstream lis, the time good divinities of you in that? No. Huh? Well, Jenny, the city to some Boston power is gonna be good. You could remind us he's gonna go to zero. Um, should have immunity them zero. So you don't know. What is that? So after many related to be able to have that. So he's leaning from the Motor city. The negative power community in us. Mmm. Today. Okay, g. So with these former, uh, over the limit us t goes to infinity. You have ah, over these expression we you know, we'll be here on data. Also boasted Katie. Okay. It's positive. 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Whereas here it was a B minus one. We said it would just be so We're going decreasing in Power's off. He added. Step here. Where us? Here for day. For days. The Terminator, the borough, The experiential remains the same. So we'll, uh, at some point you're gonna get that B minus some into your number. It's gotta be mortar than zero. So that that would bring the power off t down. And then you get something off the farm ghostlike a first t e t the guns. So what's he goes to infinity here? This expression is gonna go to some constant or anything. So I'm gonna go to Syria. Um, so, uh oh, you don't know obstructed strictly of which point we will stop. But definitely there's gonna be some point. We're get something over form from Causton, but by an exponential. So Oh, but let's you conclude that he's I mean it. These people, it's you. So uh, was expected. Well, they get a time go to Haiti. The drug concentration in the blood I should go, is you and then we'll you differentiate dysfunction. 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So if prime or you make ready from determined by the side of these so well, if kmp are positive if prime will affinity for Thio be here Well okay. So is that determines their Where is that mark? Smart like should look something like grazes which is a much mind grease is with this point being okay on their four day inflection points you want to compute? What he's is no crime to look at were were they uncovered changes. So for his little crime, it's easier to differentiate his expression. Some function expresses this. I'm off those functions. Uh, they're here on your friendship. These? Oh, you differentiating friend shaking First day power functions will be and being like this one Then you grease the power off t buy one Does get you know d to the boat off being witness to times your exponential. I know when we differentiating way differentiate Hobart dysfunction explanation. She'll be here minus K A minus K comes from the inner reality here to mine escaped and speed James. He did a team. I was one. That's the exponential. So now we want the differential. This part. So you differentiating this part well, First friendship the power functions where you get out. Oh Waas Also there is someone there Some witness key then spew from the power on the guns. He might want the exponential. You might just kill you d You didn't might miss Katie. I know we want to differentiate the exponential here. So you're gonna get oh, my nose. Okay. Damn something or minus case of the That gives you last case, squirt. Damn Stevie for you. Yeah, my nose. Okay. Do you? So to write these you Perhaps I'm easier for we can fuck through the exponential Sze, right? Is that? Come on, doctor. There. So we have Yeah, to the minus key. And as you can see there, the lowest power off is Peter my store. All right, so So you can write Jesus. He's square times. He did it. P minus stool. This board that I'm this polar like tea. I'm Steve today. He might still from the same for the sport, so that well, Dr Dennis Dee, put a B minus two. I'm being that multiplying the Polina meal. Ah, well, see, comes know this part. Good. That speech was one No, no, for this part also, that accounts for it team. So we'll let you get you here. There is minus K p minus cape you so minus two times keeping that to the boy off team because he's one of us. That fucker on in here, So plus k square square, plus Okay. Squared, squared. So these are secondly relative. So was you can see here it's gonna be gonna give up 40 going to infinity. So look, some from base give up. But it's gonna be some point. We've seen what's going down Cooking up, determined by over the zeros for this building on Mel Uh, so looking dead your friend grasps. Well, you can see here, here for the green Want we have X Be ableto for the green one, people toe 0.15. I'm, uh, Katie's minus one case one. So for this green one, you see that there's inflection points like there. I'm there with these green one will be friction. Points are for part of what a black one in kitchens get your points are pushed further, uh, on, uh, all day one for the red one. Is this Well, no collection points like, Oh, there, there for that red one. Piece. 15 Okay, she got one. So going back to what is little prime? Just, uh Well, we're the serious for this bull, you know? You know these? He took the d a. Go to one from these, uh, expression

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