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 Q_{n} be the amount of the drug in the body right after the n^{\mbox{th}} dose and let P_{n} be the amount of the drug in the body right before the n^{th} dose so that Q_{n}=P_{n}+Q.
Let's seek a pattern: P_{1}=0, Q_{1}=0+Q=Q, P_{2}=Q\cdot 2^{-h/T}, Q_{2}=Q\cdot 2^{-h/T}+Q, P_{3}=(Q\cdot 2^{-h/T}+Q)\cdot 2^{-h/T}=Q(2^{-2h/T}+2^{-h/T}), Q_{3}=Q(2^{-2h/T}+2^{-h/T})+Q, P_{4}=(Q(2^{-2h/T}+2^{-h/T})+Q)\cdot 2^{-h/T}=Q(2^{-3h/T}+2^{-2h/T}+2^{-h/T}), Q_{4}=Q(2^{-3h/T}+2^{-2h/T}+2^{-h/T})+Q, etc...
The patterns indicate that P_{n}=Q\sum_{k=1}^{n-1}2^(-\frac{kh}{T}) and Q_{n}=Q\sum_{k=0}^{n-1}2^(-\frac{kh}{T}).
Here's the calculus-related part. As n->\infty, it can be shown that P_{n}->\frac{Q}{2^{h/T}-1} and Q_{n}->\frac{Q2^{h/T}}{2^{h/T}-1}.
What's the application to medicine? You want to choose h and Q so that \frac{Q}{2^{h/T}-1} is large enough to be effective in the patient's body and so that \frac{Q2^{h/T}}{2^{h/T}-1} is small enough that it is not dangerous to the patient.
