Final answer:
The rate-of-change function for the future value of an investment compounded continuously is the derivative of the original function. For the given function of $1500 invested at 2% compounded continuously, the rate-of-change function is f'(t) = 30e^{0.02t} dollars per year.
Step-by-step explanation:
The rate-of-change function for the value of an investment that is compounded continuously is given by the derivative of the future value function with respect to time, t. Since the given future value function is f(t)=1500e^{0.02t}, the rate-of-change function will be the derivative of this function with respect to t.
To find the rate-of-change function, we use the fact that the derivative of e^{kt} with respect to t is ke^{kt}, where k is a constant. Thus, the derivative of 1500e^{0.02t} is 1500 × 0.02 × e^{0.02t}, which simplifies to 30e^{0.02t}. Therefore, the rate-of-change function for the investment's value is f'(t) = 30e^{0.02t} dollars per year.