After a certain drug is injected into a patient, the...

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?

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?

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Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. Understanding rational functions is important because they often have features such as vertical and horizontal asymptotes, and their behavior at extremes (such as large values of the independent variable) helps predict long-term trends.

Graphing Functions

Graphing functions involves plotting points based on the input-output pairs of a mathematical function. It helps in visualizing the behavior of a function over time, identifying key features such as maximum or minimum values, intercepts, and overall trends, which is crucial in understanding how the drug concentration changes after injection.

Asymptotic Behavior and Limits

Asymptotic behavior refers to how a function behaves as the independent variable approaches a particular value or infinity. In the context of the drug concentration function, analyzing limits and asymptotes allows us to determine what happens to the concentration as time progresses indefinitely, which is essential for understanding the eventual decrease in drug concentration.

Transcript

00:01 All righty.

00:02 So this problem is number 76, and it's talking about how a drug is getting injected into a patient.

00:12 It follows this specific concentration in the bloodstream over time and minutes.

00:21 And so it's using the model of this following concentration of time is equal to 30.

00:32 Times the time and minutes divided by time and minutes squared plus two.

00:39 And the first thing we need to do is draw a graph when time is greater than or equal to zero because you can't have negative time.

00:50 So at the initial time to and like kind of see what's happening.

00:55 So it wants us to draw a sketch a graph of that situation.

00:59 So in order to do that, we're going to have t is our x and c is our y and make a table.

01:07 So when we plug in a zero into the equation here and here, 30 times zero is on the top, so that's going to lead us to zero.

01:16 And then we have one, one and a half, two, and we'll see what happens from there.

01:25 So when i plug in a one, i have 30 times one on top over.

01:31 1 squared plus 2 on the bottom.

01:34 So basically it ends up being 30 divided by 3, which is 10.

01:40 When i plug in 1 .5, i do 30 times 1 .5 on the top over 1 .5 squared plus 2 on the bottom, which gets me 45 on top, 4 .25 on the bottom...

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