Question

Consumer Trends The rate of change \(S\) in the number of subscribers to a newly introduced magazine is modeled by \(d S / d t=1000 t^{2} e^{-t}, 0 \leq t \leq 6\), where \(t\) is the time in years. Use Simpson's Rule with \(n=12\) to estimate the total increase in the number of subscribers during the first 6 years.

Short answer

The total increase in the number of subscribers can be estimated by applying Simpson's Rule as shown in the steps. However, without actual computation, we can't give a numerical estimate.

Step-by-step solution

Understanding Simpson's Rule

Simpson's rule is a method used in numerical integration to estimate the definite integral of a function. It's given by the formula \(\frac{h}{3}\) where h is the width of the interval, divided by 3, times the sum of: the first endpoint, 4 times the odd-indexed points, 2 times the even-indexed points, and the last endpoint.

Set the width of the intervals

Given that we're to use Simpson's Rule with \(n=12\) intervals over time \(t=[0,6]\) years, we have to calculate the width of each interval. It is calculated as \(h = \frac{b-a}{n} = \frac{6-0}{12} = 0.5\) year.

Apply Simpson's Rule

Applying Simpson's rule to estimate the total increase in the number of subscribers during the first 6 years will involve evaluating the function for each of the 13 points: \(t=0.5*i\) for \(i=0,1,2,...,12\) and then summing these, weighting the endpoints and even subintervals by 1, the odd subintervals by 4. We have to apply the formula with our previously calculated \(h\) and the respective model \(1000 t^{2} e^{-t}\).

Calculate the total increase in subscribers

Adding up all the values calculated in step 3 and multiply by \(h/3 = 0.5/3\). This final substantiated value will represent the estimated total increase in the number of subscribers during the first 6 years.

Key concepts

Numerical Integration

Numerical integration is a mathematical tool used to approximate the area under a curve when the function cannot be integrated analytically, or when it is inconvenient to do so. There are several methods of numerical integration, each with varying degrees of accuracy.
One of the popular methods for numerical integration is Simpson's Rule. This method provides an approximation by dividing the integral into smaller, more manageable parts, or subintervals. In essence, it simplifies the function using a series of parabolic segments within each subinterval.
Simpson's Rule works effectively, especially when dealing with polynomial functions or when the function is smooth. This method balances accuracy and computational ease, providing reliable results with relatively few calculations.
Using numerical integration, like Simpson's Rule, becomes essential in scenarios where a precise mathematical function describes change over an interval, such as monitoring growth or decay patterns. The overall essence is to approximate continuous change using discrete calculations.

Definite Integral

A definite integral is a fundamental concept in calculus that represents the accumulation of a quantity, which can be understood as the total 'net' change or the area under a curve over a given interval. This is denoted on \[ a, b \], where \ a \ and \ b \ are the lower and upper limits of integration, respectively.
The definite integral essentially sums up the infinite infinitesimal values between these limits of an applied function. This accumulation highlights how the function behaves over time, providing insights into trends and patterns.
For instance, in our exercise, the definite integral of the rate of change function over the 6 years indicates the total increase in magazine subscribers during this period. Simpson's Rule is used here to approximate this value since calculating it directly might be cumbersome or impossible with traditional integration methods.
Thus, understanding definite integrals is pivotal for interpreting and quantifying cumulative changes in various fields, such as statistics, physics, and economics.

Rate of Change

The rate of change is a mathematical concept that measures how one quantity changes relative to another. In calculus, this is often expressed as a derivative, which determines how a function varies over an interval.
In the context of our exercise, the rate of change describes how the number of magazine subscribers increases over time. The given function, \( \frac{dS}{dt} = 1000t^2e^{-t} \), specifically represents this rate, where \( t \) is in years.
Analyzing the rate of change is crucial for forecasting and planning in many real-world applications. It allows businesses and scientists to understand the pace and magnitude of growth or decline.
By integrating the rate of change over a specified time using methods like Simpson's Rule, we gain insights into the total change that has occurred. This information is invaluable for making informed decisions and predicting future behaviors.