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Problem 58 Hard Difficulty

A model for the concentration at time $ t $ of a drug injected into the bloodstream is
$$ C(t) = K(e^{-at} - e^{-bt}) $$
where $ a $, $ b $, and $ K $ are positive constants and $ b > a $. Sketch the graph of the concentration function. What does the graph tell us about how the concentration varies as time passes?

Answer

As time passes, the concentration first increases and reaches a maximum value, after that concentration keeps on decreasing forever.

$ $

SEE GRAPH

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Video Transcript

So we're given this function to model concentration over time and were given these constraints A B and K or positive. Constance and B is greater than K. So I just picked these random numbers that do work for these constraints. And I plugged him into the function and they got this function. So if you want to grab that, the first thing I'm gonna look at is the first derivative. So see of tea, See, prime of tea. So the first derivative is gonna be negative e to the power of negative T plus to e to the negative to t. Okay. And I'm gonna set this equal to zero to find my critical points. And I see that c is equal to Ellen of two. All right, so Ellen of two is approximately 20.693 So if we test values, this is Ellen, too. If we test values, um, into our first derivative for the first derivative test for less than Ellen to, we're going to see that it's increasing and anything greater than is gonna be decreasing. So it's a local Max. All right, So this could be useful for later and then see double prime toe. Look a ah kong cavity. See, double prime of tea. We're going to get e to the negative. T minus four e to the negative to t If we said that equal to zero, we're going to see, um, t equals to Ellen too, which is approximately, um, 1.39. And if we do the second derivative test of double prime and we test values around 1.39 we see that, um, it's gonna be con cave down. And then Khan gave up. So this is an inflection point. All right, so we're going to use this while we graph in just a moment, but I'm gonna look at ah, one more thing, because our graph actually does have an accent. Oh, it has. Ah. Horizontal ass in tow. So let's test, um, the limit, as are variable t approaches infinity over function, which is one times e to the negative one t minus e to the negative to t. All right. And we're gonna So let's just say that we do, um, the limit as t approaches infinity, I'm gonna rewrite this in a way that looks easier to understand. So this is the same thing as one over e to the t because it's, ah, negative exponents bring into the denominator. So one over E to the T minus one over each of the two tea. So this is the same thing, all right. And if you take a look at this, it's every substitute. Wherever we see tease, we're gonna substitute infinity. So as t approaches infinity, we see that one over. A very large number is close to zero. So this is gonna approach zero. So as T approaches infinity, our why values, which is concentration over time, is gonna approach zero. So we have a horizontal ask himto at zero. Okay, And now we can graph. And since we're looking at time, I'm just gonna show the positive part of this cause negative time doesn't really make sense to me. So this is concentration overtime for this access. And this is just time over here, and we know that we do have to say this is approximately one. We know we have a maximum of Ellen, too. And if you plug in Ellen to into the original function, you'll see that it's equal to 1/4. So we know that about Ellen two and 1/4 is our maximum and the sister of sketch. So I'm gonna say this is Ellen too. 1/4 at this point. And this is our maximum. Okay? And we see that, um, according to the second derivative test that it's gonna be conquered down. And then Khan gave up in our inflection point is 1.39. So that's what we're gonna seethe. Sign change at about 1.39. So let's say it's right there. Okay? So it's gonna be con cave down. Oh, and we also haven't Don't forget about the ass in tow. The horizontal asking to is right here. Okay, so we have, um, con cave down. It's gonna first of all, increase until it hits are maximum local, maximum, and then it's gonna starts a decrease after hits the max, but it's still gonna be con cave down until it hits our inflection point where it's gonna be Khan gave up an approach, but never touch are asking too. And this is our graph. All right, so this is our graph for concentration over time, and the question asked asks us, um, what's happening in this graph. So we see as timing is increasing. So as time goes on, um, concentration increases until we hit the. So as time goes on, concentration increases until we hit the maximum concentration, and then after that, the concentration continues to decrease forever.