Question

For the next set of exercises, use the following table, which shows the population of San Francisco during the 19 th century. $$ \begin{array}{|l|l|} \hline \text { Years since } 1850 & \text { Population (thousands) } \\ \hline 0 & 21.00 \\ \hline 10 & 56.80 \\ \hline 20 & 149.5 \\ \hline 30 & 234.0 \\ \hline \end{array} $$ Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?

Short answer

The second derivative is constant, indicating an increasing rate of growth, reflecting economic growth.

Step-by-step solution

Find First Differences

Calculate the differences in population between consecutive decades. - From 1850 to 1860: 56.80 - 21.00 = 35.80 - From 1860 to 1870: 149.5 - 56.80 = 92.70 - From 1870 to 1880: 234.0 - 149.5 = 84.50

Find Second Differences

Find the differences between the first differences calculated in Step 1. - From 1850-1860 to 1860-1870: 92.70 - 35.80 = 56.90 - From 1860-1870 to 1870-1880: 84.50 - 92.70 = -8.20

Model the Population Growth

Assume that the population data can be fitted with a polynomial function of degree 2 (quadratic function) since the second differences are not constant but suggest some quadratic behavior. Define the quadratic equation as: \[ P(t) = at^2 + bt + c \] where \( P(t) \) is the population (in thousands) and \( t \) is the time in years since 1850.

Determine the Coefficients

Set up a system of equations using the population values from the table and solve for \( a \), \( b \), and \( c \). - For \( t = 0 \): \[ c = 21.00 \] - For \( t = 10 \): \[ 100a + 10b + c = 56.80 \] Substitute \( c \): \[ 100a + 10b = 35.80 \]- For \( t = 20 \): \[ 400a + 20b + c = 149.5 \] Substitute \( c \): \[ 400a + 20b = 128.50 \] Solve the above system of linear equations for \( a \) and \( b \).

Find the Second Derivative

First, find the first derivative of \( P(t) \): \[ P'(t) = 2at + b \] Then, find the second derivative:\[ P''(t) = 2a \] This tells us the rate of change of the population growth is constant, as it does not depend on \( t \).

Determine the Increase Region

Since the second derivative \( P''(t) = 2a \) is a constant, determine the sign of \( 2a \). If \( 2a > 0 \), the graph is concave up, indicating that the rate of population growth is increasing. Use numeric solutions from Step 4 to ascertain the exact sign.

Interpret the Meaning

If \( P''(t) = 2a > 0 \), it means the population growth rate accelerated during the 19th century in San Francisco. This indicates positive economic growth and increased attractiveness of the city for settlers during this period.

Key concepts

Second Derivative

The concept of the second derivative is crucial when analyzing how a function's rate of change itself changes over time. In calculus, for any function \( f(x) \), the second derivative, denoted as \( f''(x) \), provides information about the curvature or concavity of the graph of the function.

When handling quadratic functions such as population growth models, the second derivative is particularly important because it tells us whether a graph is concave up or concave down. A positive second derivative \( f''(x) > 0 \) indicates the graph is concave up, suggesting that the function's rate of increase is accelerating over time. Conversely, a negative second derivative \( f''(x) < 0 \) suggests the function's rate of increase is slowing down or decelerating.

In the context of our exercise on San Francisco's population, computing the second derivative of the quadratic function modeling the population growth gives insights into whether the city's population growth rate was increasing or decreasing during the 19th century.

Quadratic Functions

Quadratic functions appear frequently in mathematics, especially in modeling real-world situations like acceleration, trajectory, and population changes. A quadratic function is typically expressed in the form \( P(t) = at^2 + bt + c \), where \( a \), \( b \), and \( c \) are constants.

The beauty of quadratic functions lies in their simplicity and ability to model processes where the rate of change itself changes at a constant rate. In practical terms, this means that the function's graph, a parabola, either opens upwards if \( a > 0 \) or downwards if \( a < 0 \).

In the population modeling exercise, after collecting data from different years, we fit the data to a quadratic model. We solved for the coefficients \( a \), \( b \), and \( c \) using a system of linear equations derived from population records at different points in time.

• The term \( at^2 \) captures the acceleration in population growth.
• The term \( bt \) provides the linear growth component.
• The constant \( c \) represents the initial population when \( t = 0 \).

Finding these coefficients allows us to predict the population in any given year within the scope of our dataset and also to calculate the second derivative, providing deeper insights into the population dynamics during that period.

Population Modeling

Population modeling is a mathematical approach to understanding and predicting changes in a population over time. Using mathematical models, such as quadratic functions, we can capture the dynamics of population growth and make predictions about future trends.

In historical contexts, population models help us understand past demographic trends. For instance, in the exercise with San Francisco's 19th-century population data, we use a quadratic function to model the growth. This model fits the population data points collected from various years and helps to illustrate how fast the population was growing.

Population models serve several purposes:

• They help policymakers and demographers understand trends and patterns.
• They aid in planning resources such as infrastructure, healthcare, and education.
• They illustrate the effects of economic, environmental, and social changes on population growth.

By analyzing the second derivative of our quadratic population model, we see whether the growth rate is increasing or decreasing, providing insights into the accelerating or decelerating trends within San Francisco. This kind of modeling is vital for strategic planning and historical understanding, making it a powerful tool in both governmental and academic settings.