Question

There are many situations where the polynomial models we have considered so far are not appropriate. However, sometimes we can use a simple transformation to linearize the data. For example, if the points \((x, y)\) lie on an exponential curve, then the points \((x, \ln y)\) should lie on a straight line. To see this, assume that \(y=C e^{k x}\) and take the logarithm of both sides of the equation: $$ \begin{aligned} y &=C e^{k x} \\ \ln y &=\ln C e^{k x} \\ &=\ln C+\ln e^{k x} \\ &=k x+\ln C \end{aligned} $$ Make the change of variables \(Y=\ln y\) and \(A=\ln C\). Then we have a linear function of the form $$ Y=k x+A $$ We can find the line that best fits the \((x, Y)\) -data and then use inverse transformations to obtain the exponential model we need: $$ y=C e^{k x} $$ where $$ C=e^{A} $$ Consider the following world population data \(^{4}\) : $$ \begin{array}{c|c|c} x=\text { year } & y=\text { population (in millions) } & Y=\ln y \\ \hline 1900 & 1650 & 7.4085 \\ 1910 & 1750 & \\ 1920 & 1860 & \\ 1930 & 2070 & \\ 1940 & 2300 & \\ 1950 & 2525 & \\ 1960 & 3018 & \\ 1970 & 3682 & \\ 1980 & 4440 & \\ 1990 & 5310 & \\ 2000 & 6127 & \\ 2010 & 6930 & \\ & \end{array} $$ (a) Fill in the blanks in the table with the values for \(\ln y\). Note that in Octave, the \(\log (\mathrm{x})\) command is used for the natural logarithm. Make a scatter plot of \(x\) vs. \(Y\). This is called a semi-log plot. Is the trend approximately linear? (b) Use the polyfit function to find the best-fitting line for the \((x, Y)\) -data and add the graph of the line to your scatter plot from part (a). Save or print the plot. Your plot should have labeled axes and include a legend. Note that the vertical axis is the logarithm of the population. Give the plot the title "Semi-log plot." (c) Use the data from part (b) to determine the exponential model \(y=C e^{k x}\). Plot the original data and the exponential function on the same set of axes. Save or print the plot. Your plot should have labeled axes and include a legend. Give the plot the title "Exponential plot." (d) Use the model from part (c) to estimate the date when the global population reached 7 billion. (e) Make a projection about when the global population will reach 10 billion.

Short answer

Estimates indicate the global population reached 7 billion in 2011 and will reach 10 billion around 2055.

Step-by-step solution

Calculate Natural Logarithms

Calculate the natural logarithm (ln) of each given y-value (population in millions) using the formula \( Y = \ln y \). Fill in the missing \( Y \) values in the table:\[ \begin{array}{c|c|c}\text{Year } (x) & \text{Population } (y, \, \text{millions}) & Y = \ln y \\hline1910 & 1750 & 7.4690 \1920 & 1860 & 7.5263 \1930 & 2070 & 7.6340 \1940 & 2300 & 7.7407 \1950 & 2525 & 7.8349 \1960 & 3018 & 8.0102 \1970 & 3682 & 8.2119 \1980 & 4440 & 8.3986 \1990 & 5310 & 8.5754 \2000 & 6127 & 8.7207 \2010 & 6930 & 8.8456 \\end{array} \]

Scatter Plot and Linear Trend Check

Create a scatter plot of the calculated \( (x, Y) \) data points. On this plot, the x-axis represents the year and the y-axis represents \( \ln y \). Verify if the plotted points form an approximately straight line. Typically, they should if the data can be modeled by an exponential relationship.

Find Best-Fitting Line Using Polyfit

Use the polyfit function in a computational tool (like Python, MATLAB, or Octave) to find the coefficients of the line \( Y = kx + A \) that best fits the \( (x, Y) \) data. This function provides values for \( k \) and \( A \). Add this line to your scatter plot to visually compare it with the data points.

Determine Exponential Model Parameters

Using the linear model from Step 3, where \( Y = kx + A \), determine the exponential model. The relationship translates to \( y = Ce^{kx} \) with \( C = e^{A} \). Calculate \( C \) using the value of \( A \) obtained from the polyfit function.

Plot Exponential Model with Original Data

Create a plot showing both the original \( (x, y) \) world population data and the exponential model \( y = Ce^{kx} \). Make sure axes are labeled and the plot includes a title 'Exponential plot' and a legend explaining both the data points and the model.

Estimate Date for 7 Billion Population

Use the exponential model \( y = Ce^{kx} \) to solve for the year \( x \) in which \( y = 7000 \) million (7 billion). Set up the equation \( 7000 = Ce^{kx} \) and solve for \( x \).

Project Date for 10 Billion Population

Similar to Step 6, use the exponential model to find the year when the population is projected to reach 10 billion. Solve \( 10000 = Ce^{kx} \) to find the appropriate year \( x \).

Key concepts

Linearization

Linearization is a mathematical technique used to transform complex, nonlinear relationships into simpler, straight-line forms. This is done by applying specific transformations to the data, which often involve some form of logarithmic or exponential manipulation.
For example, if you're dealing with exponential growth data, you can take the natural logarithm of your dependent variable (such as population size) to linearize the data. In our exercise, we observed that if the relationship between the year and population growth is exponential, we can represent it as \( y = Ce^{kx} \). By taking the natural logarithm, the relationship is simplified to a linear form: \( Y = kx + A \).
This allows us to use linear techniques to analyze data, making it easier to spot trends and make predictions. In practice, this involves plotting the transformed data (like \( x, \ln(y) \)) and finding a line that best fits this linearized data.

Semi-log Plot

A semi-log plot is used to display data that follows an exponential trend. It involves plotting one of the axes logarithmically, typically the y-axis, while the x-axis remains linear. This transformation is useful in identifying exponential relationships as it linearizes the graph.
In our exercise, we graphed the years against the natural logarithm of population. This semi-log plot was instrumental in confirming the linear trend of the data over time. By observing the scatter plot, if the data points generally form a straight line, it suggests that the relationship of the data is exponential.
This visualization technique helps us better understand and interpret data trends, providing a clearer picture of growth patterns that may otherwise be obscured in a typical Cartesian plot. Semi-log plots are particularly helpful in fields where exponential growth is common, such as biology, economics, and demography.

Polyfit Function

The polyfit function is a computational tool used to find the best-fitting polynomial curve for a given set of data points. In this exercise, we used this function to determine the optimal line that fits the transformed \((x, Y)\) data, where \(Y = \ln(y)\).
This function operates by minimizing the sum of squared differences between the observed and predicted values, providing us with coefficients for the line, specifically the slope \(k\) and the intercept \(A\). These coefficients are crucial as they allow us to build a mathematical model of the data that can depict trends or predict future values with accuracy.
For our population data, the polyfit function helps establish the parameters of our exponential model \(y = Ce^{kx}\). It's an essential tool in computational data analysis, simplifying the process of model fitting and validation in scenarios involving linearized data.

Population Growth Estimation

Estimating population growth involves predicting future populations based on existing data and models. In our exercise, we used the exponential model \(y = Ce^{kx}\) derived from the previous steps to estimate when the world population would reach certain milestones, such as 7 and 10 billion.
By solving the equation \(7000 = Ce^{kx}\) for \(x\), we estimate the year when the population reaches 7 billion. Similarly, solving \(10000 = Ce^{kx}\) provides the projected year for 10 billion. These calculations require us to transform our understanding of past trends into forecasts about the future.
This method of using exponential models for estimation is valuable in fields such as environmental science, public policy, and urban planning, where understanding and preparing for future population changes are crucial.