Problem 11 - Applied Calculus

Question

Sales of Version \(6.0\) of a computer software package start out high and decrease exponentially. At time \(t\), in years, the sales are \(s(t)=50 e^{-t}\) thousands of dollars per year. After two years, Version \(7.0\) of the software is released and replaces Version \(6.0\). You can invest earnings at an interest rate of \(6 \%\), compounded continuously. Calculate the total present value of sales of Version 6,0 over the two-year period.

Step-by-Step Solution

Answer

The total present value of sales over two years is approximately $41,470.

Step 1: Define Present Value Formula

To calculate the present value, we will use the formula: \( PV = \int_{0}^{T} s(t) e^{-rt} \, dt \), where \( PV \) is the present value, \( s(t) \) is the sales function, \( r \) is the interest rate, and \( T \) is the time period.

Step 2: Setup the Integration

Plug the sales function \( s(t) = 50 e^{-t} \) and the interest rate \( r = 0.06 \) (6\%) into the present value formula, resulting in: \( PV = \int_{0}^{2} 50 e^{-t} e^{-0.06t} \, dt \). Simplify the integrand to \( 50 e^{-(1.06)t} \).

Step 3: Integrate the Function

Integrate \( 50 e^{-(1.06)t} \) from \( t = 0 \) to \( t = 2 \):\[PV = \int_{0}^{2} 50 e^{-(1.06)t} \, dt\]This integration is of the form \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} \).

Step 4: Compute the Definite Integral

Compute the integral:\[PV = \left[ -\frac{50}{1.06} e^{-(1.06)t} \right]_{0}^{2}\]Evaluate the limits:\[PV = -\frac{50}{1.06} \left( e^{-2.12} - e^{0} \right)\]

Step 5: Calculate the Present Value

Substitute the values to find the present value:\[PV = -\frac{50}{1.06} (0.12029 - 1)\]\[PV = 47.17 (0.87971)\]\[PV \approx 41.47\] thousand dollars.

Key Concepts

Exponential Decay

Exponential decay represents a mathematical model where quantities reduce at a consistent percentage rate over time. In this exercise, sales of a software package diminish yearly. This decrease is defined by the sales function \( s(t) = 50 e^{-t} \), reflecting how sales start high and decline exponentially. Exponential decay is commonly used to describe phenomena such as radioactive decay, depreciation of assets, and biological processes.

• The base of the natural logarithm \( e \), approximately 2.71828, is pivotal in these calculations.
• A negative exponent results in a decay, meaning as time \( t \) increases, the sales decrement progressively.
• Exponential functions like this help predict future sales. For instance, a new software version often leads older versions' sales to shrink, as new products capture the market.

Understanding exponential decay allows students to model real-world processes mathematically, making predictions based on changing conditions over time.

Continuous Compounding

Continuous compounding is the process of calculating interest income assuming it is constantly being added to the principal. Unlike annual or monthly compounding, continuous compounding uses the formula \( A = Pe^{rt} \), where:

• \( A \) is the amount of money accumulated after \( n \) years, including interest.
• \( P \) is the principal amount (initial investment).
• \( r \) is the annual interest rate (in decimal form).
• \( t \) is the time in years.

In the exercise, the investment from the sales earnings has a continuous interest rate of 6%. This approach maximizes the return over the investment period. Continuous compounding is ideal for theoretical calculations, providing the best-case scenario for investment growth.

By assuming interest is compounded continuously, the returns are slightly higher compared to other compounding methods when the same initial investment, rate, and time period are used.

Definite Integral

The definite integral is a vital concept in calculus that allows the calculation of the total accumulation of quantities. When dealing with the present value of sales over time, the integral sums up all instantaneous sales values adjusted for interest rate over the period from 0 to 2 years. The definite integral \( \int \) produces a single number representing the accumulated quantity.

To calculate the present value (PV) of sales in the exercise, integrate the function \( 50 e^{-(1.06)t} \) with respects to time from \( t = 0 \) to \( t = 2 \). This operation is pivotal to determine how much the future sales earnings are worth today, considering the decaying sales and continuous interest compounding.

• The definite integral considers both the continuous decrease in the sales and the effect of discounting due to the 6% interest rate.
• Performing the integration provides the total sales value adjusted to present terms over the indicated period.

Understanding definite integrals prepares students to solve complex real-life mathematical models by encapsulating variable factors into a single, quantifiable value.

Sales Function

The sales function in this problem, \( s(t) = 50 e^{-t} \), models the sales behavior over time for a specific software version. This function is an example of an exponential function, where the initial sales value starts at \( 50 \) (in thousands of dollars) and decreases as time goes on.

• The coefficient "50" indicates the sales level at the starting point (\( t = 0 \)). It sets the initial condition for the sales trajectory.
• The exponential factor \( e^{-t} \) dictates the speed of sales decline. As time increases, the exponential term decreases, reflecting reduced sales.

The purpose of the sales function is to mathematically describe how sales revenue for the software diminishes over time. This is crucial in financial planning and forecasting. By understanding and predicting sales behavior, companies can make informed decisions about product development cycles, marketing strategies, and financial investment.

Overall, comprehending such functions equips students with the tools to analyze trends and make forecasts based on mathematical models.