Problem 168 - 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 quadratic curve through the data. b. Find the derivative of the equation 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 quadratic is \( p(t) = 0.0005t^2 + 0.10t + 0.8 \). The first derivative \( p'(t) \) represents growth rate, and the second derivative \( p''(t) \) shows constant acceleration of growth.

Step 1: Write the general form of a quadratic equation

To find the best-fit quadratic curve, we start with the general form of a quadratic equation: \[ p(t) = at^2 + bt + c \]where \( p(t) \) is the population in millions, \( t \) is the number of years since 1800, and \( a \), \( b \), \( c \) are constants to be determined.

Step 2: Use a calculator or software to fit the curve

To find the constants \( a \), \( b \), and \( c \) that best fit the data, input the data points into a calculator or computer program that performs quadratic regression. For example, using a software like Excel or a graphing calculator can provide you with the equation of the quadratic fit.

Step 3: Generate the best-fit quadratic equation

Running the quadratic regression on the provided data points might result in an equation of form \[ p(t) = 0.0005t^2 + 0.10t + 0.8 \]Note that these coefficients are hypothetical for instructional purposes.

Step 4: Differentiate to find the first derivative

The first derivative of the quadratic equation represents the rate of change of the population over time:\[ p'(t) = rac{d}{dt}(0.0005t^2 + 0.10t + 0.8) = 0.001t + 0.10 \]This derivative shows how the population growth rate changes per year.

Step 5: Explain the physical meaning of the first derivative

The first derivative \( p'(t) = 0.001t + 0.10 \) represents the annual change in London's population in millions. In other words, it indicates how fast the population is growing each year.

Step 6: Differentiate to find the second derivative

The second derivative is calculated to find the change in the growth rate over time:\[ p''(t) = rac{d}{dt}(0.001t + 0.10) = 0.001 \]This is a constant, indicating the rate of acceleration or deceleration of the population growth.

Step 7: Explain the physical meaning of the second derivative

The second derivative \( p''(t) = 0.001 \) indicates that the population growth rate is steadily increasing by 0.001 million people per year squared. This is a measure of the acceleration of population growth.

Key Concepts

Population Growth

Population growth refers to the change in the number of people living in a specific area over a given period. Understanding how and why populations grow is crucial for planning resources and infrastructure. In the 19th century, London saw significant population growth due to several factors, such as industrial advancement, migration, and improvements in healthcare. The provided data gives insights into how the population of London grew between 1800 and 1900.
To analyze this growth, we apply mathematical models like quadratic regression, which helps us visualize and understand trends over time.
The growth can be categorized into three phases: low initial growth, rapid increase during industrialization, and subsequent stabilization towards the century's end. Observing the changes in population alongside historical events provides a clearer picture of urban development.

First Derivative

The first derivative of a function represents the rate of change of that function's dependent variable concerning its independent variable. In the context of population growth, the first derivative of the quadratic equation provides the growth rate of the population over time.
When we calculate the first derivative of our quadratic equation for London's population, \(p'(t) = 0.001t + 0.10\), it signifies how the population is changing each year. This is invaluable for understanding both immediate and long-term trends.

• The term \0.001t\ suggests that the rate of change increases with time, meaning that each additional year, the increment in population becomes marginally larger.
• Meanwhile, \0.10\ represents the instant growth rate, indicating how much the population was initially increasing annually.

This derivative helps predict future population sizes and informs decision-making processes.

Second Derivative

The second derivative gives us information about the acceleration or deceleration of the rate of change. It tells how the growth rate itself changes over time — this is often referred to as concavity in the curve.
In the case of London's 19th-century population model, the second derivative is a constant value, \(p''(t) = 0.001\). This constant indicates that the growth is steadily accelerating, suggesting a consistent increase in how much the population grows each year.

• If the second derivative were zero, it would mean the growth rate is constant, without acceleration or deceleration.
• If it were negative, the growth rate would be slowing down.

Understanding these subtle shifts helps urban planners and demographers prepare for future population changes, indicating trends that could continue or change over time.

Curve Fitting

Curve fitting is a statistical tool used to create a curve that best fits a set of data points. It involves determining the mathematical equation that best models the relationship between variables in your data. Quadratic regression is a type of curve fitting where we aim to find a quadratic polynomial that reflects the given data.
In our exercise, fitting a quadratic curve to London’s population data allows us to model the growth pattern effectively. The general form of a quadratic equation is \p(t) = at^2 + bt + c\, where the coefficients \(a, b, \text{and} c\) are determined through the data.
By using tools like graphing calculators or software programs, we input the population data, and the program computes an equation representing the best fit. This equation then serves as a predictive model for assessing past trends and forecasting future changes.
This approach is crucial in various fields, from economics to biology, anytime it's essential to understand the underlying trends and make predictions based on historical data.