![]() In applications of differential equations, the functions represent physical quantities, and the derivatives, as we know, represent the rates of change of these qualities. The way that a drug’s concentration over time is calculated is using calculus ! In fact, a drugs course over time can be calculated using a differential equation. In order for doctors to prescribe the correct dosage of a drug and provide a regimen for treatment (ie., “take 2 capsules twice a day”), the drug’s concentration over time must be tracked. The drug no longer provides pain relief and is excreted by the body.During this point, the pain relieving effects begin to wear off slightly.Metabolism – the drug is processed and broken down by the body. ![]() During this point, the pain-relieving gel provides pain relief as it is being spread through the blood stream by the body.Distribution – the drug is dispersed through the body.This would occur when the medicinal components of the gel enter the bloodstream.Absorption – the drug enters the body through blood circulation.For example, this would be the point in which the outer core of a pain-relieving gel capsule (ie., Advil) disintegrates to release the medicinal components inside.Liberation – the drug is released from its pharmaceutical formulation.Pharmacokinetics can be broken down into five general steps in which a drug takes its course: There is no cost to you for having an account, other than our gentle request that you contribute what you can, if possible, to help us maintain and grow this site.Pharmacokinetics is the study of how drugs (or any other substances that can be consumed) are processed within the body. We believe that free, high-quality educational materials should be available to everyone working to learn well. You will also be able to post any Calculus questions that you have on our Forum, and we'll do our best to answer them! We do use aggregated data to help us see, for instance, where many students are having difficulty, so we know where to focus our efforts. Your selections are for your use only, and we do not share your specific data with anyone else. Your progress, and specifically which topics you have marked as complete for yourself.Your self-chosen confidence rating for each problem, so you know which to return to before an exam (super useful!).Your answers to multiple choice questions.Once you log in with your free account, the site will record and then be able to recall for you: But otherwise, the conclusion you reach with the Second Derivative test is indeed conclusive. The one exception is if the second derivative is zero at the point of interest (f”(c)=0), in which case the Second Derivative Test is inconclusive and you have to revert to the First Derivative Test. That test is just as conclusive as the First Derivative Test, and is often easier to use. The only thing that you wrote that isn’t quite right are the very last words, “in the first derivative test” instead, you’re using the Second Derivative Test. And the fact that there’s no point of inflection anywhere doesn’t affect those conclusions. (See the figure below.) Similarly, if the second derivative is a positive constant, then the function is concave up everywhere, and so the point x=c where f'(c) = 0 is guaranteed to be a minimum. The answer to all of your questions is: yes! If the second derivative is a negative constant, then the function is concave down everywhere, and so you’re guaranteed that the point x=c you found where f'(c) = 0 is a maximum. And agreed about getting the problem set-up right as the vast majority of the work here. We’re glad to know you liked our explanation and approach. Here’s a key thing to know about how to solve Optimization problems: you’ll almost always have to use detailed information given in the problem to rewrite the equation you developed in Step 2 to be in terms of one single variable.Ībove, for instance, our equation for $A_\text \quad \cmark Optimization Problems & Complete Solutions.
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