When interest is compounded continuously, the amount of...

When interest is compounded continuously, the amount of money increases at a rate proportional to the amount $S$ present at time $t$ that is, $d S / d t=r S,$ where $r$ is the annual rate of interest. (a) Find the amount of money accrued at the end of 5 years when $\$ 5000$ is deposited in a savings account drawing $5 \frac{3}{4} \%$ annual interest compounded continuously. (b) In how many years will the initial sum deposited have doubled? (c) Use a calculator to compare the amount obtained in part (a) with the amount $S=5000\left(1+\frac{1}{4}(0.0575)\right)^{5(4)}$ that is accrued when interest is compounded quarterly.

When interest is compounded continuously, the amount of money increases at a rate proportional to the amount $S$ present at time $t$ that is, $d S / d t=r S,$ where $r$ is the annual rate of interest.
(a) Find the amount of money accrued at the end of 5 years when $\$ 5000$ is deposited in a savings account drawing $5 \frac{3}{4} \%$ annual interest compounded continuously.
(b) In how many years will the initial sum deposited have doubled?
(c) Use a calculator to compare the amount obtained in part (a) with the amount $S=5000\left(1+\frac{1}{4}(0.0575)\right)^{5(4)}$ that is accrued when interest is compounded quarterly.

Key Concepts

Doubling Time

Doubling time is the period required for a quantity to double in size. In the context of exponential growth, it can be determined by setting the growth function equal to twice the original amount and solving for time. This involves the natural logarithm and illustrates a practical measure to assess the rapidity of growth, an essential concept in both finance and broader applications of exponential models.

Differential Equations

The equation dS/dt = rS is a first-order linear differential equation that models continuous growth phenomena. Its solution, obtained through techniques in calculus, demonstrates that the rate of change of the amount is directly proportional to the amount itself. This concept not only applies to interest accumulation but also to various natural and economic systems that exhibit growth proportional to their size.

Compounding Frequencies

Compounding frequencies refer to the number of times interest is calculated and added to the principal within a given period, typically a year. While continuous compounding is an idealized limit, discrete compounding (such as quarterly, monthly, or annual) is more commonly used in practice. Comparing these methods shows how the frequency of compounding affects the final accrued amount and helps in understanding how theoretical models translate into real-world financial scenarios.

Continuous Compounding

Continuous compounding refers to the mathematical model of interest accumulation where interest is calculated and added to the principal continuously rather than at discrete intervals. This is modeled by the formula S(t) = S0 * e^(rt), where S0 is the initial amount, r is the annual interest rate, and t is time in years. This concept is fundamental in finance and calculus as it provides a natural model for growth processes that are continuously updated.

Exponential Growth

Exponential growth describes the process by which a quantity increases at a rate proportional to its current value. In the context of continuously compounded interest, this results in an exponential function, where the amount grows according to e raised to the product of the interest rate and time. This explains why investments under continuous compounding grow faster than those compounded at discrete intervals given the same nominal rate.

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