Problem 49 - Calculus

Question

Let \(f(t)\) be the number of US billionaires in the US in year \(t\) (a) Express the following statements \(^{4}\) in terms of \(f\) (i) In 1985 there were 13 US billionaires. (ii) In 1990 there were 99 US billionaires. (b) Find the average yearly increase in the number of US billionaires between 1985 and \(1990 .\) Express this using \(f\) (c) Assuming the yearly increase remains constant, find a formula predicting the number of US billionaires in year \(t\)

Step-by-Step Solution

Answer

(i) \( f(1985) = 13 \); (ii) \( f(1990) = 99 \). Average increase: 17.2/year. Formula: \( f(t) = 13 + 17.2(t - 1985) \).

Step 1: Express Year 1985 Statement using f

Given that in 1985, there were 13 US billionaires, we express this as \( f(1985) = 13 \).

Step 2: Express Year 1990 Statement using f

In 1990, there were 99 US billionaires, so it is written as \( f(1990) = 99 \).

Step 3: Find the Time Interval

The time period from 1985 to 1990 is \(1990 - 1985 = 5\) years.

Step 4: Calculate Average Yearly Increase

The average yearly increase is calculated by taking the difference in the number of billionaires between 1990 and 1985, and dividing by the number of years: \[ \frac{f(1990) - f(1985)}{1990 - 1985} = \frac{99 - 13}{5} = 17.2 \text{ billionaires per year} \]

Step 5: Create Formula for Constant Increase

Assuming a constant yearly increase, the formula for predicting the number of billionaires in year \( t \) is:\[ f(t) = f(1985) + \frac{f(1990) - f(1985)}{1990 - 1985} \times (t - 1985) \] Substitute the known values:\[ f(t) = 13 + 17.2 \times (t - 1985) \]

Key Concepts

Linear Growth Model

In real-world scenarios, linear growth models help us understand situations where growth occurs at a consistent rate. This model simplifies the prediction process by assuming constant growth over time. For example, if the number of billionaires in the US increases steadily every year, we can use a linear growth model to predict future numbers. The mathematical representation takes the form:

• Initial Value: the starting point, like 13 billionaires in 1985.
• Growth Rate: the constant increase per time unit, such as 17.2 billionaires per year.

To create a linear model, determine the initial value and the rate of change. Then, formulate a function that reflects these parameters. In this exercise, the formula is:

\[f(t) = 13 + 17.2 imes (t - 1985)\]
This equation takes a given year, computes the time elapsed since 1985, and applies the steady growth rate to predict billionaire numbers.

Function Notation in Calculus

Function notation is a compact way of representing mathematical operations. It clearly defines relationships between variables within equations. In this exercise, we label the number of billionaires with a dependent variable, \(f(t)\), where:

• \(f\) is the function name, representing a specific calculation or operation.
• \(t\) serves as the independent variable, usually time in this context.
• Values like \(f(1985)\) and \(f(1990)\) specify instances where the function applies.

Utilizing function notation is valuable because it aids in visualizing and analyzing how changes in the independent variable affect the dependent one. In this scenario, knowing \(f(t)\) provides insights into the fluctuation of billionaire counts relative to time.

Predictive Modeling

Predictive modeling is a statistical technique used to forecast future events based on historical data. In the context of this exercise, we employ predictive modeling to estimate the number of billionaires in future years. This involves several key steps:

• Gathering historical data, which in this case includes billionaire counts in 1985 and 1990.
• Identifying a trend, such as a consistent yearly increase, as seen here.
• Developing a prediction formula, like the linear model \[f(t) = 13 + 17.2 imes (t - 1985)\].

The model's reliability depends on whether the assumption of constant growth holds true. Predictive modeling allows us to make educated guesses about the future. However, it is important to note that any deviation from assumed trends can affect prediction accuracy.