Problem 23 - Applied Calculus

Question

The following table shows the number of hours worked in a week, \(f(t),\) hourly earnings, \(g(t),\) in dollars, and weekly earnings, \(h(t),\) in dollars, of production workers as functions of \(t\), the year. \(^{6}.\) (a) Indicate whether each of the following derivatives is positive, negative, or zero: \(f^{\prime}(t), g^{\prime}(t), h^{\prime}(t) .\) Interpret each answer in terms of hours or earnings. (b) Estimate each of the following derivatives, and interpret your answers: (i) \(f^{\prime}(1970)\) and \(f^{\prime}(1995)\) (ii) \(g^{\prime}(1970)\) and \(g^{\prime}(1995)\) (iii) \(h^{\prime}(1970)\) and \(h^{\prime}(1995)\) $$\begin{array}{c|c|c|c|c|c|c|c}\hline t & 1970 & 1975 & 1980 & 1985 & 1990 & 1995 & 2000 \\\\\hline f(t) & 37.0 & 36.0 & 35.2 & 34.9 & 34.3 & 34.3 & 34.3 \\\\\hline g(t) & 3.40 & 4.73 & 6.84 & 8.73 & 10.09 & 11.64 & 14.00 \\\\\hline h(t) & 125.80 & 170.28 & 240.77 & 304.68 & 349.29 & 399.53 & 480.41 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

(a) \(f'(t)\) is negative; \(g'(t)\) and \(h'(t)\) are positive. (b) 1970: \(f'(t) = -0.2\), \(g'(t) = 0.266\), \(h'(t) = 8.896\); 1995: \(f'(t) = 0\), \(g'(t) = 0.472\), \(h'(t) = 16.176\).

Step 1: Understanding the Derivatives

A derivative, such as \( f'(t) \), \( g'(t) \), and \( h'(t) \), indicates the rate of change over time. It illustrates whether a function is increasing, decreasing, or stable over the years. A positive derivative indicates an increase, a negative derivative indicates a decrease, and a zero derivative indicates no change.

Step 2: Analyzing \( f'(t) \)

The function \( f(t) \) represents hours worked per week. By observing the table, \( f(t) \) decreased from 37.0 in 1970 to 34.3 in 2000. Therefore, \( f'(t) \) is negative overall, implying a general decrease in the number of hours worked over time.

Step 3: Analyzing \( g'(t) \)

The function \( g(t) \) stands for hourly earnings. The values increase from 3.40 in 1970 to 14.00 in 2000. Hence, \( g'(t) \) is positive, indicating hourly earnings are rising over the years.

Step 4: Analyzing \( h'(t) \)

The function \( h(t) \) indicates weekly earnings. Weekly earnings grow from 125.80 in 1970 to 480.41 in 2000. As such, \( h'(t) \) is positive, reflecting an increase in weekly earnings.

Step 5: Estimating \( f'(1970) \) and \( f'(1995) \)

The approximate derivative \( f'(t) \) at 1970 can be estimated as \( \frac{f(1975) - f(1970)}{5} = \frac{36 - 37}{5} = -0.2 \). Similarly, at 1995: \( \frac{f(2000) - f(1995)}{5} = \frac{34.3 - 34.3}{5} = 0 \). Hours worked decrease at 1970 and stabilize by 1995.

Step 6: Estimating \( g'(1970) \) and \( g'(1995) \)

Calculate \( g'(1970) \) as \( \frac{g(1975) - g(1970)}{5} = \frac{4.73 - 3.40}{5} = 0.266 \). Calculate \( g'(1995) \) as \( \frac{g(2000) - g(1995)}{5} = \frac{14 - 11.64}{5} = 0.472 \). Hourly earnings increase at a higher rate in 1995 than in 1970.

Step 7: Estimating \( h'(1970) \) and \( h'(1995) \)

Estimate \( h'(1970) \): \( \frac{h(1975) - h(1970)}{5} = \frac{170.28 - 125.80}{5} = 8.896 \). For 1995: \( \frac{h(2000) - h(1995)}{5} = \frac{480.41 - 399.53}{5} = 16.176 \). Weekly earnings increase faster in 1995 than in 1970.

Key Concepts

Rates of Change

Rates of change fundamentally describe how a quantity changes with respect to a change in another variable, often time. In calculus, the rate of change is represented by the derivative. For the given problem, three key derivatives are analyzed: \( f'(t) \), \( g'(t) \), and \( h'(t) \). Each derivative provides insight into whether the respective function—hours worked, hourly earnings, and weekly earnings—is increasing, decreasing, or remaining constant over time.

• If the derivative is positive, it indicates that the function is increasing, meaning the quantity it measures is growing over time.
• If the derivative is negative, it signals a decrease, showing that the particular quantity is reducing over the years.
• A zero derivative suggests no change from one measured point in time to the next.

In the context of this exercise, we found:
\( f'(t) \) is negative overall (indicating a decrease in weekly hours worked), \( g'(t) \) is positive (demonstrating an increase in hourly earnings), and \( h'(t) \) is also positive (highlighting a rise in weekly earnings). The interpretation of these derivatives tells us how the working dynamics in terms of hours and earnings have evolved over time.

Estimating Derivatives

Estimating the derivative at specific years, like 1970 and 1995, helps us understand the changes that occur during those years more precisely. We do this by calculating the average rate of change over a defined interval using the formula:\[ f'(a) \approx \frac{f(b) - f(a)}{b - a}\]For example, estimating \( f'(1970) \), which represents the change in hours worked, involves taking the difference in function values at two years and dividing by the time period. The calculation for \( f'(1970) \) was approximately -0.2, showing a decrease in work hours at that time, whereas \( f'(1995) \) was 0, indicating no change.
Similarly, estimating derivatives for \( g(t) \) and \( h(t) \) at the same points highlighted:

• \( g'(1970) = 0.266 \) and \( g'(1995) = 0.472 \), showing hourly earnings increased at both times, but at a faster rate by 1995.
• \( h'(1970) = 8.896 \) and \( h'(1995) = 16.176 \), meaning weekly earnings grew, particularly at a higher rate in 1995.

These estimates are particularly useful in identifying periods of economic growth or stagnation and indicate how quickly these changes are occurring.

Function Analysis

To thoroughly analyze how these functions behave over time, it's important to consider the trend indicated by their derivatives. Observing the functions \( f(t) \), \( g(t) \), and \( h(t) \):

• \( f(t) \): Highlights a downward trend where the hours worked decline from 37 to 34.3 over three decades, primarily driven by the needs of efficiency and possibly technological advancements reducing workload.
• \( g(t) \): This showcases significant growth from \(3.40 to \)14.00, an increase in hourly earnings likely influenced by inflationary pressures and enhancements in productivity.
• \( h(t) \): Displays a steady increase in weekly earnings from \(125.80 to \)480.41, which can be seen as a result of increasing hourly rates and possibly the addition of economic bonuses over time.

Breaking down these functions into intervals and examining changes in their derivatives provides a layered understanding of economic and labor trends, making it clearer how and why these variables interact as they do. Understanding these patterns helps in making informed predictions about future labor statistics.