Suppose that money is deposited daily in a savings account...

Suppose that money is deposited daily in a savings account at an annual rate of $\$ 2000 .$ If the account pays $6 \%$ interest compounded continuously, approximately how much will be in the account at the end of 2 years?

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Key Concepts

Continuous Compound Interest

Continuous compounding is a process in financial mathematics where interest is calculated and added to the principal at every possible instant. Instead of discrete intervals like monthly or annually, the interest grows exponentially according to the formula A = Pe^(rt), where P is the principal, r is the annual interest rate, and t is the time in years.

Modeling Regular Deposits with Differential Equations

When funds are added continuously or very frequently to an account, the growth of the account can be modeled using differential equations. This approach incorporates both the interest earned on the current balance and the constant rate of deposits, providing a dynamic model that accurately reflects the cumulative effect of continuous contributions.

Integration in Accumulated Value Calculations

Integration is employed to calculate the total accumulated value when deposits are made continuously. It allows for summing the contributions of infinitesimal deposits over a period of time, each of which earns interest at a continuous rate. This technique is essential in determining how regularly added amounts grow over time when compounded continuously.

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