Problem 41 - Calculus and its applications

Question

The number of nonfarm proprietorships, in thousands, in the United States can be modeled by \(N(t)=8400 \ln t-10,500,\) where \(t\) is the number of years after \(1970 .\) The graph of this function is given below. a) Using this model, predict the number of nonfarm proprietorships in the United States in \(2020 .\) b) Find \(N^{\prime}(45)\). c) Interpret the meaning of \(N^{\prime}(45)\).

Step-by-Step Solution

Answer

In 2020, the number of nonfarm proprietorships is about 12376.8 thousand. In 2015, the rate of change is 186.67 thousand per year.

Step 1: Understand the problem

We are given a model for the number of nonfarm proprietorships, where \( N(t) = 8400 \ln t - 10,500 \) and \( t \) is the number of years after 1970. We need to make predictions about the year 2020, which corresponds to \( t = 50 \) (since 2020 - 1970 = 50).

Step 2: Calculate N(t) for 2020

Plug \( t = 50 \) into the function \( N(t) = 8400 \ln t - 10,500 \).\[N(50) = 8400 \ln 50 - 10,500\]Calculate \( \ln 50 \) and then find \( N(50) \).

Step 3: Compute \( \ln 50 \)

Using a calculator, \( \ln 50 \approx 3.9120 \).

Step 4: Calculate N(50)

Substitute \( \ln 50 \approx 3.9120 \) into the expression for \( N(50) \):\[N(50) = 8400 \times 3.9120 - 10,500 = 22876.8 - 10,500 = 12376.8\]Thus, the number of nonfarm proprietorships in 2020 is approximately 12376.8 thousand.

Step 5: Differentiate the function N(t)

Find the derivative, \( N'(t) \), of \( N(t) = 8400 \ln t - 10,500 \). Use the derivative of \( \ln t \), which is \( \frac{1}{t} \).\[N'(t) = 8400 \cdot \frac{1}{t} = \frac{8400}{t}\]

Step 6: Calculate N'(45)

Substitute \( t = 45 \) into the derivative found in Step 5.\[N'(45) = \frac{8400}{45}\]Simplify this expression by dividing 8400 by 45.

Step 7: Simplify N'(45)

\( N'(45) = \frac{8400}{45} \approx 186.67 \). This represents the rate of change of nonfarm proprietorships for the year corresponding to \( t = 45 \) (or the year 2015).

Step 8: Interpret N'(45)

\( N'(45) \approx 186.67 \) means that in the year 2015, the model predicts the number of nonfarm proprietorships was increasing at a rate of approximately 186.67 thousand proprietorships per year.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function changes at any given point. When you differentiate a function, you calculate its derivative. The derivative of a function provides a way to determine how quickly the output value changes as the input value changes. This is like finding the 'slope' of a curve at a particular point.

To differentiate a function like the one given in the exercise, which is in the form of a natural logarithm, you employ rules specific to logarithmic functions:

• The derivative of ln(t) is 1/t.
• For a constant multiplied by ln(t), such as 8400 ln(t), the derivative is simply the constant 8400 multiplied by the derivative of ln(t), so it becomes 8400/t.

Differentiation is an essential tool for analyzing and interpreting mathematical models, as it helps reveal trends and behaviors of the system being studied.

In the context of this exercise, differentiating the function gives us insight into how the number of nonfarm proprietorships is expected to change over time.

Natural logarithm

The natural logarithm, denoted as ln, is a logarithm to the base of the mathematical constant e, where e is approximately equal to 2.71828. It is called 'natural' because it arises naturally in various mathematical contexts, particularly in growth processes and calculus.

The natural logarithm has several useful properties in calculus and mathematical modeling:

• It converts multiplication into addition, which simplifies complex multiplicative relationships.
• The derivative of ln(t) is 1/t, making it manageable to work with when differentiating logarithmic functions.
• It often appears in models of exponential growth or decay, such as in the field of economics or population studies.

In the exercise, the natural logarithm is used in the function N(t) to model the number of nonfarm proprietorships. The inclusion of ln(t) indicates a relationship that might exhibit slow growth or gradual changes over time, capturing the nature of real-world expansions in business sectors.

Mathematical modeling

Mathematical modeling is the process of using mathematics to represent, analyze, and predict real-world phenomena. In this context, the model is expressed through a mathematical function that defines how the number of nonfarm proprietorships changes over time. This model allows you to make predictions about future trends based on historical data.

Key aspects of mathematical modeling involve:

• Identifying the variables involved, like time and number of proprietorships in the problem.
• Formulating equations or functions that describe the observed behaviors or trends.
• Using these formulas to make predictions or to understand underlying processes.

In the exercise, the model for nonfarm proprietorships is given by a logarithmic function. This choice of function is suitable for capturing behaviors that grow steadily or are governed by more complex, non-linear dynamics.

This mathematical model not only provides an estimate for specific years but, through differentiation, also gives insight into how yearly changes occur, aiding in strategic planning and policy making in economic contexts.