The rate of change of an exponential function, especially in the context of compounding interest, can be determined by examining the function itself. In this case, we can consider a general form of a compound interest function, which can be represented as:
A(t) = A_0 e^{rt}
Where:
- A(t): The amount of money accumulated after time t, including interest.
- A_0: The principal amount (the initial amount of money).
- r: The annual interest rate (decimal).
- t: The time (in years).
- e: The base of the natural logarithm, approximately equal to 2.71828.
To determine the rate of change, we focus on how the accumulated amount A(t) changes with respect to time t. We can find the derivative of A(t) with respect to t, which gives us the rate of change:
dA/dt = A_0 r e^{rt}
This derivative indicates that the rate of change of the accumulated amount over time is directly proportional to both the current amount and the interest rate. Therefore:
- As time t increases, the accumulated amount A(t) grows exponentially.
- The rate of change dA/dt will increase over time due to the exponential factor e^{rt}, illustrating how interest compounds.
In summary, the rate of change for an exponential function representing compounding interest is defined by the derivative dA/dt, which reflects the influence of the interest rate and the nature of exponential growth. This means that the faster the rate, the more significant the change in the accumulated amount over time.