One important area of application is to deciding drug dosages.
Suppose the dose of a drug is Q milligrams (per pill) and that the patient is supposed to take the drug every h hours. Futhermore, suppose the half-life of the drug (the amount of time for the amount of the drug to decay to 50% of the starting amount) in a person's bloodstream is T hours (for simplicity, assume the drug enters the person's bloodstream instantaneously).
Let Qn be the amount of the drug in the body right after the nth dose and let Pn be the amount of the drug in the body right before the nth dose so that Qn=Pn+Q.
Let's seek a pattern: P1=0, Q1=0+Q=Q, P2=Q⋅2−hT, Q2=Q⋅2−hT+Q, P3=(Q⋅2−hT+Q)⋅2−hT=Q(2−2hT+2−hT), Q3=Q(2−2hT+2−hT)+Q, P4=(Q(2−2hT+2−hT)+Q)⋅2−hT=Q(2−3hT+2−2hT+2−hT), Q4=Q(2−3hT+2−2hT+2−hT)+Q, etc...
The patterns indicate that Pn=Qn−1∑k=12−khT and Qn=Qn−1∑k=02−khT.
Here's the calculus-related part. As n→∞, it can be shown that Pn→Q2hT−1 and Qn→Q2hT2hT−1.
What's the application to medicine? You want to choose h and Q so that Q2hT−1 is large enough to be effective in the patient's body and so that Q2hT2hT−1 is small enough that it is not dangerous to the patient.
