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Solve the given problems.The concentration $C$ (in $m g / c m^{3}$ ) of a certain drug in a patient's bloodstream is $C=\frac{d .15 t}{t^{2}+1},$ where $t$ is the time (in $h$ ) after the drug is administered. Sketch the graph.

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 6

More on Curve Sketching

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

03:09

In mathematics, precalculu…

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Drug Concentration A drug …

05:54

A model for the concentrat…

01:23

Drug Concentration After a…

07:28

A drug concentration curve…

05:42

03:14

Find the derivatives of th…

04:02

Drug Testing The concentra…

02:24

The concentration $C(t)$ (…

02:21

Drug concentration The con…

03:34

Solve each problem. See Ex…

for problem 89. We're looking at a rational functions. See, if t is equal to five t over T squared plus one or tea is greater than equal zero. And what it represents is the concentration of a drug over time after administration. Okay, so we can definitely go through. Our typical analysis of this function on the X intercept is when the actual function of zero. So it's when the the numerator is zero. And that happens when t A zero. Okay, so you see, I already have the graph up here. The text. What best say to use graphic device. Um, so here it is. The orange reddish graph Here is the actual function I explained that blew ours outline here in a second BC. It does go through the origin of the Y, intercept his labs when you plug in zero. And so that would equal zero as well. So we have that on the craft. Vertical assam tote. Um, that is depends on the degree of the numerator and denominator, since a decree of the numerous less than the Curia Dominator that would imply that the horizontal acid is the ex axes like a zero Okay, so we have a horizontal ask himto, um, of the x axis, and you see that the rational function does approach that had an affinity. Um Well, sorry. It was like that is the horrors I'll ask himto vertical assam tote. There is none. Okay, sack, for messing that up. That was the orders on last until the bird a glass and toast. There is none. Since you cannot make the denominator zero. Anytime you square tea, it'll be positive. And one that's positive. So there is no vertical. Ask himto here and you see it in the crap. Oh, Max can take on any value here. There is no vertical. Ask himto. Okay, so we're asked a few things here. Definitely. Using the graphing utility makes this, um quite a bit easier. Um, in part, a were asked what is the maximum amount of concentration in the bloodstream so it occur at a maximum? Uh, and using a graphing utility like best most. You can just click on this point, or some graphing utility is going to find a function maximum and and use that to find it. But that would occur. Um, at the 0.1 comma 2.5, Which means after one year of time in this case, hours, uh, the maximum concentration that in the bloodstream be 2.5 and the units are milligrams per liter. Okay, Bart, be asked what happens to the amount of drug in the in the bloods Assist bloodstream over time. And so we're looking at the red graph. What happens after many, many, many hours? We see that it approaches zero. Now, according to this model, we could never way. We would say it would never totally disappear because it does not approach. It doesn't equal zero. It approaches, it gets infinitely small. Okay. And lastly, part see, um asked when would the amount of blood stream So when would see a t be less than 0.3 milligrams pre leader? Okay, so now this blue line comes And what I what I did here was I actually graft. Um, just half attacks equals 0.3. That's this blue line here. Okay. And so what I What I was able to do then, is to just click somewhere on this craft and this dock came up right here, which is where the rational function crosses the blue eye, and that happens at 16.6 about grounded 0.3. And what that tells me is that after 16.6 hours approximately, it will be equal 2.3. So answer the question that will occur when the time is greater than 16.6 hours. Okay, after that, the red line will be below, and you start to see here in the graph a little bit below 0.3 milligrams per liter.

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