Problem 71
Question
ECONOMICS: National Debt The national debt (the amount of money that the federal government has borrowed from and therefore owes to the public) is approximately \(D(x)=73 x^{2}+1500 x+13,300\) billion dollars, where \(x\) is the number of years since \(2010 .\) The population of the United States is approximately \(P(x)=2.73 x+309\) million. a. Enter these functions into your calculator as \(y_{1}\) and \(y_{2}\), respectively, and define \(y_{3}\) to be \(y_{1} \div y_{2}\), the national debt divided by the population, so that \(y_{3}\) is the per capita national debt, in thousands of dollars (since it is billions divided by millions). Evaluate \(y_{3}\) at 5 and at 10 to find the per capita national debt in the years 2015 and \(2020 .\) This is the amount that the government would owe each of its citizens if the debt were divided equally among them. b. Use the numerical derivative operation NDERIV to find the derivative of \(y_{3}\) at 5 and at 10 and interpret your answers.
Step-by-Step Solution
Verified Answer
Evaluate \(y_3\) at 5 and 10, and use NDERIV for derivatives at 5 and 10 for rates of change.
1Step 1: Understand the Functions
The task involves two functions. The national debt function, \(D(x) = 73x^2 + 1500x + 13,300\), is given in billions of dollars. The population function, \(P(x) = 2.73x + 309\), is given in millions of people. \(x\) is the number of years since 2010.
2Step 2: Define Per Capita Debt Function
The per capita debt is defined as the national debt divided by the population: \(y_3 = \frac{y_1}{y_2} = \frac{73x^2 + 1500x + 13,300}{2.73x + 309}\). This represents the national debt per person (in thousands of dollars) because it is billion dollars divided by million people.
3Step 3: Calculate Per Capita Debt for 2015 and 2020
To find the per capita debt for 2015, substitute \(x = 5\) into \(y_3\), and for 2020, substitute \(x = 10\). This gives:\[y_3(5) = \frac{73(5)^2 + 1500(5) + 13,300}{2.73(5) + 309}\]\[y_3(10) = \frac{73(10)^2 + 1500(10) + 13,300}{2.73(10) + 309}\]
4Step 4: Use a Calculator for Evaluations
Enter the expressions from Step 3 into your calculator to obtain the numerical value for each. The calculations help find how much debt each citizen would theoretically owe.
5Step 5: Compute the Derivative of Per Capita Debt
Use the numerical derivative operation NDERIV on a calculator to find \(\left.\frac{dy_3}{dx}\right|_{x=5}\) and \(\left.\frac{dy_3}{dx}\right|_{x=10}\). These derivatives reflect the rate of change in per capita debt over time at the specified years.
6Step 6: Interpret the Derivative
The derivative values tell you how quickly the per capita debt is increasing at the respective years (2015 and 2020). A positive value indicates an increase in the per capita debt over time.
Key Concepts
Per Capita CalculationNumerical DerivativeFinancial Functions in Calculus
Per Capita Calculation
Per capita calculations help to understand the share of a total amount each individual would be responsible for, in a simplified sense. It makes complex totals more relatable by breaking them down per person. This is highly useful in economics when dealing with large figures like national debt. In this case, the per capita national debt is essentially the total national debt divided by the population at any given time, providing a debt obligation figure per person.
Given the national debt function, \(D(x) = 73x^2 + 1500x + 13,300\) billion dollars, and the population function, \(P(x) = 2.73x + 309\) million people, the per capita debt can be determined by dividing these two functions:
\[ y_3(x) = \frac{D(x)}{P(x)} = \frac{73x^2 + 1500x + 13,300}{2.73x + 309} \]
This formula gives us the national debt per person in thousands of dollars. Essentially, if we evaluate this at specific years like 2015 or 2020, it can help us gain insight into economic conditions and how much debt an average citizen theoretically shares.
Given the national debt function, \(D(x) = 73x^2 + 1500x + 13,300\) billion dollars, and the population function, \(P(x) = 2.73x + 309\) million people, the per capita debt can be determined by dividing these two functions:
\[ y_3(x) = \frac{D(x)}{P(x)} = \frac{73x^2 + 1500x + 13,300}{2.73x + 309} \]
This formula gives us the national debt per person in thousands of dollars. Essentially, if we evaluate this at specific years like 2015 or 2020, it can help us gain insight into economic conditions and how much debt an average citizen theoretically shares.
Numerical Derivative
A numerical derivative essentially provides the rate at which a function is changing at a particular point. Unlike conceptual derivatives calculated via algebraic manipulation, numerical derivatives are obtained by computing values using numerical algorithms, often with a calculator.
In the case of the per capita national debt, understanding how this debt changes over time can be assessed by determining its derivative. When calculating the derivative of the per capita function \(y_3(x)\), we find:
\[ \left.\frac{dy_3}{dx}\right|_{x=5} \] and \[ \left.\frac{dy_3}{dx}\right|_{x=10} \]
These values indicate how fast or slow the per capita debt is increasing in 2015 and 2020. The higher the derivative value, the faster the increase in debt per person. Positive values mean the debt is growing, while negative values (hypothetically) would tell us it's decreasing.
In the case of the per capita national debt, understanding how this debt changes over time can be assessed by determining its derivative. When calculating the derivative of the per capita function \(y_3(x)\), we find:
\[ \left.\frac{dy_3}{dx}\right|_{x=5} \] and \[ \left.\frac{dy_3}{dx}\right|_{x=10} \]
These values indicate how fast or slow the per capita debt is increasing in 2015 and 2020. The higher the derivative value, the faster the increase in debt per person. Positive values mean the debt is growing, while negative values (hypothetically) would tell us it's decreasing.
Financial Functions in Calculus
Financial functions in calculus are powerful tools used in economics to model and predict changes over time. They allow us to study the behavior of financial indicators, like national debt, within a defined mathematical framework.
The key advantage is the ability to handle non-linear and often complex relationships, offering insights that are not obvious through simpler calculation methods. The national debt function \(D(x)\) provides a clear example of this, as it models debt as a second-degree polynomial, accounting for various contributing factors more accurately than linear models might.
Calculus comes handy to:
The key advantage is the ability to handle non-linear and often complex relationships, offering insights that are not obvious through simpler calculation methods. The national debt function \(D(x)\) provides a clear example of this, as it models debt as a second-degree polynomial, accounting for various contributing factors more accurately than linear models might.
Calculus comes handy to:
- Understand trends and predict future financial outcomes.
- Determine how small changes in input variables affect the overall outcome using derivatives.
- Assess rates of change, like the increase in debt over time.
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