Compound Interest as Continuous Growth

Calculus 2 · Axiom Academy

EXAMPLE Compound Interest as Continuous Growth Model a continuously compounded account as the differential equation A' = rA , solve it, and evaluate the balance. You deposit 10,000 into an account paying an annual rate of r = 0.06 (6%), compounded continuously . How much is the account worth after 10 years — and how does that compare with compounding just once a year? Nicely done. You turned "compounded continuously" into a differential equation, solved it, and read off a real dollar figure. The model: Continuous growth means the rate of change is proportional to the current balance: A' = rA . The solution: Separating variables and integrating gives A = Ce^ rt ; the condition A(0) = P forces C = P , so A = Pe^ rt . The answer: 18,221.19 after 10 years — about 8,221 of interest. Why "continuous" wins: Compounding once a year gives only 17,908.48 , so going continuous earns about 313 more on the same deposit. The same formula A = Pe^ rt describes every process where growth is proportional to size — populations, radioactive decay, and cooling all reuse this exact differential equation.

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