Problem 42 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

The gross domestic product of a certain country was \(N(t)=t^{2}+6 t+300\) billion dollars \(t\) years after 2004\. Use calculus to predict the percentage change in the GDP during the second quarter of \(2012 .\)

Step-by-Step Solution

Answer

The percentage change in GDP during the second quarter of 2012 is approximately 1.36%.

Step 1: Determine the interval for the second quarter of 2012

The year 2012 is 8 years after 2004, so we have t = 8. The second quarter of 2012 runs from April to June. This corresponds to t in the interval \[8.25, 8.5\].

Step 2: Compute the derivative of the GDP function

The function for GDP is N(t) = t^2 + 6t + 300. To find the rate of change of GDP, compute the derivative: \[N'(t) = \frac{d}{dt}(t^2 + 6t + 300) = 2t + 6\]

Step 3: Evaluate the derivative at the start and end of the interval

Calculate N'(8.25) and N'(8.5): \[N'(8.25) = 2(8.25) + 6 = 16.5 + 6 = 22.5\]\[N'(8.5) = 2(8.5) + 6 = 17 + 6 = 23\]

Step 4: Compute the average rate of change during the interval

Average the values from Step 3: \[\text{Average rate of change} = \frac{22.5 + 23}{2} = 22.75\]

Step 5: Approximate the change in GDP during the second quarter

The approximate change can be calculated by multiplying the average rate of change by the length of the quarter (0.25 years): \[\text{Change in GDP} \approx 22.75 \times 0.25 = 5.6875\] billion dollars

Step 6: Calculate the initial GDP at the start of the second quarter

Find N(8.25): \[N(8.25) = (8.25)^2 + 6(8.25) + 300 = 68.0625 + 49.5 + 300 = 417.5625\] billion dollars

Step 7: Compute the percentage change

Use the initial GDP from Step 6 and the change from Step 5: \[\text{Percentage change} = \left(\frac{5.6875}{417.5625}\right) \times 100 \approx 1.36%\]

Key Concepts

GDP Growth

Gross Domestic Product (GDP) is a measure of a country's economic performance. It represents the total value of all goods and services produced over a specific time period. When we talk about GDP growth, we're referring to the increase in this value from one period to the next. This growth is a critical indicator of economic health, as it often correlates with improvements in living standards and economic prosperity.

• More GDP typically means more jobs and higher incomes.
• Governments and economists closely monitor GDP trends to inform policy decisions.
• Understanding how to predict GDP changes helps in planning and investment.

By using calculus, which involves the study of rates of change, we can make more precise predictions about future GDP values.

Rate of Change

The rate of change is a fundamental concept in calculus. It measures how a quantity changes over time. In the context of GDP, it helps us understand how quickly the economy is growing. We use the derivative of a function to calculate the rate of change. In the given problem, the GDP function is given by \(N(t) = t^2 + 6t + 300\).
The derivative, \(N'(t)\), gives us the rate at which GDP is changing at any point in time:

• The derivative of \(t^2\) is \(2t\).
• The derivative of \(6t\) is \(6\).
• The derivative of a constant (300) is \(0\).

Thus, \(N'(t) = 2t + 6\). Evaluating this derivative at specific points helps us understand the rate of change in GDP growth during different quarters or any other time frames.

Derivative Calculation

A derivative represents an instantaneous rate of change. For continuous functions, it can be visualized as the slope of the tangent line to the curve at a given point. In our example with the GDP function, by taking the derivative, we identify how the GDP is evolving at a precise moment.

To compute the derivative of \(N(t) = t^2 + 6t + 300\):

• Use basic rules of differentiation: the power rule and the constant rule.
• Apply the power rule to \(t^2\), resulting in \(2t\).
• The derivative of \(6t\) is \(6\) due to the constant multiplier rule.
• Constants differentiate to zero, so the constant 300 vanishes in this process.

As such, the derivative, which is \(N'(t) = 2t + 6\), helps us find how fast the GDP is increasing or decreasing at any time t.

Average Rate of Change

The average rate of change gives us a more general picture than the instantaneous rate provided by the derivative. It shows how much a quantity changes on average over a given interval. For the GDP problem, the computation involved two specific points: the beginning and the end of the second quarter in 2012.

We calculated:

• \(N'(8.25) = 22.5\)
• \(N'(8.5) = 23\)

Averaging these values, we find the overall rate of change for the quarter:
\( \text{Average Rate of Change} = \frac{22.5 + 23}{2} = 22.75 \)
This helps us understand the average pace of economic growth during that period.

Percentage Change Calculation

Percentage change is essential for comparing changes across different scales. It standardizes the change relative to the initial value, making it easier to understand and communicate. To compute the percentage change in GDP during the second quarter of 2012:

First, calculate the GDP at the start of the quarter: \(N(8.25) = 417.5625\) billion dollars. Then, use the previously calculated change: \(5.6875\) billion dollars.

The percentage change is determined by:

• \( \frac{5.6875}{417.5625} \times 100 \)

This calculation yields approximately 1.36%, indicating a relatively modest increase in the GDP over the quarter. Understanding percentage changes is crucial for interpreting economic data and making informed decisions.