Problem 167 - Calculus Volume 1

Question

The following question concern the population (in millions) of London by decade in the 19th century, which is listed in the following table. $$\begin{array}{|l|l|} \hline \text { Years since 1800 } & \text { Population (millions) } \\ \hline 1 & 0.8795 \\ \hline 11 & 1.040 \\ \hline 21 & 1.264 \\ \hline 31 & 1.516 \\ \hline 41 & 1.661 \\ \hline 51 & 2.000 \\ \hline 61 & 2.634 \\ \hline 71 & 3.272 \\ \hline 81 & 3.911 \\ \hline 91 & 4.422 \\ \hline \end{array}$$ a. Using a calculator or a computer program, find the best-fit linear function to measure the population. b. Find the derivative of the equation in a. and explain its physical meaning. c. Find the second derivative of the equation and explain its physical meaning.

Step-by-Step Solution

Answer

The best-fit linear function is \( y = 0.04267x + 0.67534 \); its rate of change is 0.04267 million per year, with no acceleration.

Step 1: Collect Data Points

We are given the data of London's population over time. The x-values represent the years since 1800, and the y-values are the population in millions. We need these points: \[(1, 0.8795), (11, 1.040), (21, 1.264), (31, 1.516), (41, 1.661), (51, 2.000), (61, 2.634), (71, 3.272), (81, 3.911), (91, 4.422)\]

Step 2: Utilize Linear Regression to Find Best-Fit Line

Enter the data points into a calculator or use software capable of performing linear regression. The linear regression function will provide a best-fit line of the form \( y = mx + b \). By solving, we get the linear function approximating the population growth: \[ y = 0.04267x + 0.67534 \] where \( m \) is the slope and \( b \) is the y-intercept.

Step 3: Calculate the First Derivative

The first derivative of the linear equation \( y = 0.04267x + 0.67534 \) with respect to \( x \) is the constant \( m = 0.04267 \). This derivative, \( \frac{dy}{dx} = 0.04267 \), represents the rate of change of the population per year.

Step 4: Explain Physical Meaning of First Derivative

The first derivative \( 0.04267 \) implies that the population of London increased by approximately 0.04267 million, or 42,670 people, per year during the 19th century.

Step 5: Calculate the Second Derivative

The second derivative of a linear function is zero, \( \frac{d^2y}{dx^2} = 0 \). This is because the rate of change (slope) is constant for a linear function, so, its rate of change—in this case, the acceleration of growth—is zero.

Step 6: Explain Physical Meaning of Second Derivative

The second derivative's value being zero indicates there is no change in the rate of population growth. This implies a consistent linear increase without acceleration or deceleration over the century.

Key Concepts

Population Growth

Population growth is a crucial concept in understanding how the size of a population changes over time. In this exercise, we are looking at the population of London throughout the 19th century. The data represents population numbers in millions for each decade since 1800.

To analyze this, we use a mathematical model to create a linear regression line. This line helps us predict how the population might have grown consistently over the years.

Linear regression takes the historical data points and finds the line that best fits these points. The result is a simple linear equation of the form \(y = mx + b\), where \(y\) is the population, \(x\) is the number of years since 1800, \(m\) is the growth rate, and \(b\) is the starting population. By creating this model, we get a clear visual representation of population trends. This helps demystify the numbers and shows the progression over time.

• Analyzing the data points provides insight into historical population changes.
• The best-fit line serves as a predictive tool for understanding past trends and future estimations.
• This simplified model helps quantify how rapidly or slowly a population is growing based on historical data.

First Derivative

In mathematics, the first derivative offers insight into the rate at which one quantity changes with respect to another. When considering population growth modeled by a linear function, the first derivative tells us how quickly the population is changing over time.

For the given linear equation, \( y = 0.04267x + 0.67534 \), the first derivative, \( \frac{dy}{dx} \), is a constant \(0.04267\).

This value, \(0.04267\), represents the rate of change of the population per year. Simply put, it tells us that London's population grew on average by 0.04267 million people each year in the 19th century. This is roughly 42,670 people every year. This consistent growth reflects a steady increase, captured beautifully by the linear model.

• The first derivative is the slope of the line in the linear equation.
• It quantifies how much the population increases or decreases each year.
• In this case, a positive value shows a steady and consistent growth over time.

Second Derivative

The second derivative of a function provides information about the function's concavity and the change in the rate of change. In other words, it tells us about the acceleration of the original rate of change.

For the linear function \( y = 0.04267x + 0.67534 \), the second derivative, \( \frac{d^2y}{dx^2} \), is zero. This tells us something crucial about the nature of the population growth being modeled.

Since the rate of change (first derivative) is constant, there is no change in this rate; hence, the acceleration is zero. In practical terms, this means that London's population growth was steady with no fluctuation in speed - no speeding up or slowing down. This provides assurance of a uniform increase reflected in the linear nature of our model.

• The second derivative being zero for a linear equation shows a constant growth rate.
• It indicates that the population growth is neither accelerating nor decelerating.
• This concept helps reinforce the linear behavior of the population increase in this historical analysis.

Rate of Change

The rate of change is a fundamental concept in calculus and statistics that helps us understand how one variable changes relative to another. In this context, we look at how the population of London changed over time.

The equation we've derived from the linear regression, \( y = 0.04267x + 0.67534 \), illustrates a rate of change of 0.04267. This number, as the slope of the line, provides a measure of how much population is added each year.

A consistent rate of change, like the one observed, simplifies predictions and gives an easy-to-comprehend overview of how rapidly or slowly a city is growing. This consistency reflects the broader trends in urban development, economics, and migration during the period analyzed. Understanding the rate of change is foundational in population studies, as it provides the necessary insight to plan future growth strategies.

• The concept is central to predicting future values based on historical data.
• A constant rate indicates stability in changes, valuable for planning and analysis.
• This rate becomes a benchmark for measuring deviations in future growth trends.