Compound Interest

Suppose an amount is deposited in a bank account at an annual interest rate $r$, and — instead of interest being credited once a year or once a quarter — it is compounded continuously, i.e., added to the balance at every instant. Over a short time interval, the interest earned is approximately proportional to the amount already in the account, the interest rate, and the length of the interval, which in the limit gives the differential equation

where $A$ is the amount in the account at time $t$ (in years) and $r$ is the annual interest rate. This is once again the growth model from before, and its solution is