Drug Concentration After a certain drug is injected into a...

Drug Concentration After a certain drug is injected into a patient, the concentration $c$ of the drug in the bloodstream is monitored. At time $t \geq 0$ (in minutes since the injection) the concentration (in $\mathrm{mg} / \mathrm{L}$ ) is given by $$c(t)=\frac{30 t}{t^{2}+2}$$ (a) Draw a graph of the drug concentration. (b) What eventually happens to the concentration of drug in the bloodstream?

Drug Concentration After a certain drug is injected into a patient, the concentration $c$ of the drug in the bloodstream is monitored. At time $t \geq 0$ (in minutes since the injection) the concentration (in $\mathrm{mg} / \mathrm{L}$ ) is given by
$$c(t)=\frac{30 t}{t^{2}+2}$$
(a) Draw a graph of the drug concentration.
(b) What eventually happens to the concentration of drug in the bloodstream?

Key Concepts

Rational Functions

Rational functions, which are ratios of two polynomials, are essential in understanding how the function behaves over different intervals. They often exhibit characteristic behavior such as intercepts, asymptotes, and turning points, which are critical when analyzing curves and their long-term trends.

Graphing Functions

Graphing functions involves plotting key features such as intercepts, local extrema, and asymptotic behavior to visualize the overall trend of a function. This process is fundamental in understanding how a function behaves in different domains and how it changes over time.

Limits and Asymptotic Behavior

Evaluating limits, particularly as the independent variable approaches infinity, helps to determine the eventual behavior of a function. This concept explains the steady state or long-term trend of a system, often identifying horizontal asymptotes that indicate the function’s value in the long run.

Transcript

00:02 So c of t is a formula for the concentration of a drug in your system, t minutes after it has been injected.

00:11 And our job is to graph it and then see what the end behavior is or what happens as time goes on.

00:16 So this is a rational function.

00:18 The first thing we would do is reduce it.

00:20 It can't be reduced.

00:21 There are no factors we can cancel out.

00:23 So next we're going to determine the asymptotes.

00:26 Vertical asymptotes happen when the denominator is equal to zero.

00:30 I think we can see that there are no real t's, but we'll solve it just for proving.

00:37 So subtract two from both sides, take the square root of both sides, and we get that t equals plus or minus the square root of negative two, which is imaginary.

00:48 So there are no vertical asymptotes.

00:54 Next thing is horizontal asymptotes.

00:56 So horizontal asymptotes tell us what happens as t gets increasingly large.

01:01 And we can see as t approaches infinity, this function is essentially going to look like one over t.

01:11 All right.

01:11 The coefficients are not going to be interesting, and that little constant two is not going to be very interesting as t is super large.

01:18 And one over t approaches zero.

01:21 So we have a horizontal asymptote at y equals zero.

01:29 So we can go ahead and put that on there.

01:35 We go.

01:37 That's our horizontal asymptote.

01:41 And what next? we can, we normally we would look for the zeros and we would look for the y intercepts, but it's not meaningful for time to be negative.

01:53 So the y intercept is kind of boring.

01:55 It's just zero, right? so we know t equals zero, y equals zero.

02:01 So we have a point right there.

02:02 Let's do those in blue maybe.

02:05 And it's not going to cross the x.

02:08 Axis, again, it wouldn't be meaningful for a concentration to go negative.

02:12 So all that's left for us to do is to make a little table of values so that we can graph our points and see what the shape is as it approaches the asymptote.

02:25 Now, we know at the far reach is what it's going to be like.

02:27 It's going to hug the asymptote from the top.

02:32 But typically, functions like this will do some interesting things at the beginning.

02:36 So we're going to figure out a couple values.

02:40 Of t and c of t, basically some coordinate pairs.

02:43 And then we'll try to take a stab at the shape.

02:45 So when t is zero, there is nothing in there, right? or maybe right before the injection happens.

02:53 Then when t is one, we get 30 over three.

02:58 C of t is 10, so we can plot that.

03:01 One, 10.

03:05 And how about when t is two? we're going to get, let's see, 60 over, two squared is four plus two is six...