Problem 23 - Applied Calculus

Question

Table 1.19 shows the total US labor force, \(L .\) Find the average rate of change between 1940 and \(2000 ;\) between 1940 and \(1960 ;\) between 1980 and \(2000 .\) Give units and interpret your answers in terms of the labor force. \(^{37}\) $$\begin{array}{c|c|c|c|c} \hline \text { Year } & 1940 & 1960 & 1980 & 2000 \\ \hline L & 47,520 & 65,778 & 99,303 & 136,891 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The average rates of change are 1,489.52 (1940-2000), 912.9 (1940-1960), and 1,879.4 (1980-2000) people per year.

Step 1: Understand the Formula for Average Rate of Change

The average rate of change of a function between two points is given by the difference in function values divided by the difference in points. Mathematically, for points \((x_1, f(x_1))\) and \((x_2, f(x_2))\), the average rate of change is \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\). Here, \(x\) represents years and \(f(x)\) represents the labor force, \(L\).

Step 2: Calculate Average Rate of Change (1940-2000)

Identify the values from the table for the years 1940 and 2000. Here, \(L_{1940} = 47,520\) and \(L_{2000} = 136,891\). The difference in years is \(2000 - 1940 = 60\). Calculate the rate as follows:\[ \frac{136,891 - 47,520}{2000 - 1940} = \frac{89,371}{60} \approx 1,489.52 \text{ people per year} \]

Step 3: Calculate Average Rate of Change (1940-1960)

Use the table values for 1940 and 1960, \(L_{1940} = 47,520\) and \(L_{1960} = 65,778\). The time difference is \(1960 - 1940 = 20\). Calculate:\[ \frac{65,778 - 47,520}{1960 - 1940} = \frac{18,258}{20} = 912.9 \text{ people per year} \]

Step 4: Calculate Average Rate of Change (1980-2000)

Consider the table data for 1980 and 2000, \(L_{1980} = 99,303\) and \(L_{2000} = 136,891\). The difference in years is \(2000 - 1980 = 20\). Calculate:\[ \frac{136,891 - 99,303}{2000 - 1980} = \frac{37,588}{20} = 1,879.4 \text{ people per year} \]

Step 5: Interpretation of Results

Between 1940 and 2000, the labor force increased at an average rate of about 1,489.52 people per year. From 1940 to 1960, the rate was 912.9 people per year, while between 1980 and 2000, it increased to 1,879.4 people per year. This indicates a faster growth in the labor force in the latter period.

Key Concepts

US Labor Force

The US labor force consists of individuals aged 16 and over who are either working or actively seeking employment. It is a vital aspect of the national economy as it reflects the number of people available to contribute to production and services across the nation. The labor force is influenced by a variety of factors including population growth, economic conditions, and policy changes. During 1940 to 2000, the US labor force experienced significant change.

• In 1940, the labor force stood at 47,520 thousand individuals.
• By 2000, this number had increased to 136,891 thousand.

This growth can be attributed to several factors such as increases in population size, higher levels of female workforce participation, and economic expansions. Understanding the US labor force helps us to analyze economic health and predict future labor market trends.

Calculating Rates

Calculating the average rate of change is essential in understanding how quickly something is increasing or decreasing over time. In this context, we calculate the average rate of change of the US labor force over different time periods. The average rate of change formula is given by:\[\frac{f(x_2) - f(x_1)}{x_2 - x_1}\]Where:

• \(f(x_1)\) and \(f(x_2)\) are the labor force values at two different years, \(x_1\) and \(x_2\).
• \(x_1\) and \(x_2\) represent the years between which you are measuring the change.

For example, between 1940 and 2000, the average rate of change of the labor force was calculated to be 1,489.52 people per year. Each of these calculations follows the same method depending on the years in question, showing how the labor force grew over those specific periods.

Interpretation of Results

Interpreting the results helps to derive meaningful insights from raw data. When we analyze the average rate of change in the US labor force over different decades, the results tell us more than just numbers.

• From 1940 to 1960, the labor force increased at an average rate of 912.9 people per year. This reflects changes post-World War II, when many entered the workforce.
• Between 1980 and 2000, the labor force grew much faster at 1,879.4 people per year. This was a period of technological advancement and globalization, which opened up more jobs and opportunities.

These periods show not just how the workforce size changed, but they also highlight underlying socio-economic conditions. Understanding these trends can help policymakers and economists make informed decisions to support and sustain labor force growth.