One important area of application is to deciding drug dosages.
Suppose the dose of a drug is QQ milligrams (per pill) and that the patient is supposed to take the drug every hh 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 TT hours (for simplicity, assume the drug enters the person's bloodstream instantaneously).
Let Q_{n}Qn be the amount of the drug in the body right after the n^{\mbox{th}}nth dose and let P_{n}Pn be the amount of the drug in the body right before the n^{th}nth dose so that Q_{n}=P_{n}+QQn=Pn+Q.
Let's seek a pattern: P_{1}=0P1=0, Q_{1}=0+Q=QQ1=0+Q=Q, P_{2}=Q\cdot 2^{-h/T}P2=Q⋅2−hT, Q_{2}=Q\cdot 2^{-h/T}+QQ2=Q⋅2−hT+Q, P_{3}=(Q\cdot 2^{-h/T}+Q)\cdot 2^{-h/T}=Q(2^{-2h/T}+2^{-h/T})P3=(Q⋅2−hT+Q)⋅2−hT=Q(2−2hT+2−hT), Q_{3}=Q(2^{-2h/T}+2^{-h/T})+QQ3=Q(2−2hT+2−hT)+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})P4=(Q(2−2hT+2−hT)+Q)⋅2−hT=Q(2−3hT+2−2hT+2−hT), Q_{4}=Q(2^{-3h/T}+2^{-2h/T}+2^{-h/T})+QQ4=Q(2−3hT+2−2hT+2−hT)+Q, etc...
The patterns indicate that P_{n}=Q\sum_{k=1}^{n-1}2^(-\frac{kh}{T})Pn=Qn−1∑k=12−khT and Q_{n}=Q\sum_{k=0}^{n-1}2^(-\frac{kh}{T})Qn=Qn−1∑k=02−khT.
Here's the calculus-related part. As n->\inftyn→∞, it can be shown that P_{n}->\frac{Q}{2^{h/T}-1}Pn→Q2hT−1 and Q_{n}->\frac{Q2^{h/T}}{2^{h/T}-1}Qn→Q2hT2hT−1.
What's the application to medicine? You want to choose hh and QQ so that \frac{Q}{2^{h/T}-1}Q2hT−1 is large enough to be effective in the patient's body and so that \frac{Q2^{h/T}}{2^{h/T}-1}Q2hT2hT−1 is small enough that it is not dangerous to the patient.
