Continuously compounded interest and exponential growth are closely related concepts in mathematics and finance, both involving exponential functions. Here’s how they are connected:
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Mathematical Representation: In continuously compounded interest, the future value \( A \) of an investment is given by the formula \( A = Pe^{rt} \), where \( P \) is the principal amount, \( r \) is the annual interest rate, \( t \) is the time in years, and \( e \) is the base of the natural logarithm (approximately equal to 2.71828). This formula is a specific application of the general formula for exponential growth, which is \( y = ae^{kt} \), where \( a \) is the initial amount, \( k \) is the growth rate, and \( t \) is time.
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Conceptual Relationship: Continuously compounded interest can be viewed as a particular case of exponential growth, where the growth occurs continuously over time rather than at discrete intervals. This means that, just like populations or investments that grow exponentially, the amount in an account earning continuously compounded interest increases at a rate proportional to its current value, leading to a rapid increase over time.
In summary, continuously compounded interest is an application of the principles of exponential growth, characterized by continuous compounding at every moment rather than at set intervals.