91Ó°ÊÓ

Calculus is a branch of mathematics that studies continuous change. It provides the tools necessary to understand how things evolve over time, especially when dealing with functions that describe changing quantities. In the context of exponential decay, calculus helps us analyze rates of change and accumulated quantities over a period.

In our exercise, the function given by \( s(t) = 100 e^{-t} \) represents sales decreasing exponentially. The exponential nature signifies that sales decrease at a rate proportional to their current value. The use of calculus allows us to calculate the total sales for a certain period, which involves finding the integral of this function. By integrating, we accumulate the continuous sales over 2 years, giving us a total revenue value.

Calculus enables precise and efficient calculations in scenarios where change happens continuously, making it indispensable in financial and business applications.

A definite integral finds the accumulation of quantities over an interval, in our case, sales revenue over a two-year period. It is represented as \( \int_{a}^{b} f(x) \, dx \), where \([a, b]\) signifies the range of integration.

The exercise involves calculating the definite integral \( \int_{0}^{2} 100 e^{-t} \, dt \). This integral represents the total sales from time \( t = 0 \) to \( t = 2 \) years.

• The integral provides a sum of infinite infinitesimally small quantities, symbolizing continuous sales over time.
• In the solution, \( 100 \) is a constant factor, and the integral of \( e^{-t} \) with respect to \( t \) results in \( -100 e^{-t} \).

Evaluated from 0 to 2, the definite integral gives a clear net value for sales over the period, which approximates to 86.47 thousand dollars.

This value is essential for subsequent financial calculations, such as investment returns.

Continuous compounding refers to the process of computing interest on an investment continuously rather than at discrete intervals like monthly or annually.

The formula for continuous compounding is \( A = Pe^{rt} \), where:

• \( P \) is the initial principal.
• \( r \) is the annual interest rate.
• \( t \) is the time in years.
• \( e \) is the base of natural logarithms.

In the exercise, after calculating total sales of 86.47 thousand dollars over two years, this amount is compounded continuously at a 4% interest rate.

By applying the continuous compounding formula, the future value of this investment is calculated as \( 86.47 \times e^{0.08} \), approximating to 93.72 thousand dollars.

Continuous compounding is advantageous in maximizing returns due to constant application of interest, especially in scenarios involving exponential growth or decay.