Problem 374 - Calculus Volume 1

Question

For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century. $$ \begin{array}{|l|l|} \hline \begin{array}{l} \text { Years since } \\ \mathbf{1 8 5 0} \end{array} & \begin{array}{l} \text { Population } \\ \text { (thousands) } \end{array} \\ \hline 0 & 21.00 \\ \hline 10 & 56.80 \\ \hline 20 & 149.5 \\ \hline 30 & 234.0 \\ \hline \end{array} $$ [T] The best-fit exponential curve to the data of the form \(P(t)=a e^{b t}\) is given by \(P(t)=35.26 e^{0.06407 t}\). Use a graphing calculator to graph the data and the exponential curve together.

Step-by-Step Solution

Answer

Plot the data points and the best-fit exponential curve on a graph using a calculator.

Step 1: Understand the Problem

The problem is to plot the population data from the table and the best-fit exponential curve on a graph, using a graphing calculator. The exponential curve model is given by \(P(t) = 35.26 e^{0.06407 t}\).

Step 2: Gather the Data Points

Identify the data points that we need to plot. From the table, we have the following points: \((0, 21.00), (10, 56.80), (20, 149.5), (30, 234.0)\) where the first number in each pair is 'years since 1850' and the second is 'population in thousands'.

Step 3: Enter the Data into Graphing Calculator

Input the data points \((0, 21.00), (10, 56.80), (20, 149.5), (30, 234.0)\) into the 'stat' or 'data' part of the graphing calculator.

Step 4: Define the Exponential Function

Enter the exponential equation \(P(t) = 35.26 e^{0.06407 t}\) into your graphing calculator's function editor. Ensure this is input as an equation for one of the graph functions, usually called \( Y= \) or \( f(x)= \) in many calculators.

Step 5: Graph the Data Points and the Function

Use the graphing feature on the calculator to plot the data points on the graph. Then, plot the exponential function on the same axes. This will display both the data points and the best-fit curve.

Step 6: Analyze the Graph

After plotting, observe whether the exponential curve closely follows the plotted data points. This helps in assessing the accuracy of the fit of the function to the historical population data.

Key Concepts

Best-fit Curve

A best-fit curve is a mathematical tool used to represent data points on a graph with a smoothed line that approximates the trend of the data. In this exercise, we're focusing on an exponential best-fit curve for population data. The exponential function has the form:

• \( P(t) = ae^{bt} \)

where \( a \) and \( b \) are constants, \( e \) represents Euler's number, and \( t \) is the time variable. Here, the equation \( P(t) = 35.26 e^{0.06407t} \) is our best-fit curve. It was chosen because it passes closely through the core trend of the given data points. By graphing this curve alongside the data, we can visually assess how well the model represents the scheduled historical populations.

Finding a best-fit curve is essential, as it allows us to make predictions about future data points based on the observed pattern of growth. It is a fundamental step for scientific and economic forecasting and a vital tool for interpreting trends in data over time.

Population Modeling

Population modeling is the practice of using mathematical formulas to understand and predict changes in population sizes over time. The exponential growth model is particularly appropriate when a population grows proportionally to its current size, which is often seen in real populations during unrestricted growth phases.

For San Francisco's 19th-century population, the exponential model \( P(t) = 35.26 e^{0.06407t} \) is used to depict how the city's population increased dramatically over just a few decades. In this model:

• \( a = 35.26 \) represents the initial population multiplier.
• \( b = 0.06407 \) indicates the growth rate.

This specific approach is referred to as continuous growth modeling. The rate \( 0.06407 \) implies that for each year, the population increased by approximately 6.4% of its current size. Population models such as these can be used to project future population growth or decline, helping urban planners and policy-makers make informed decisions.

Graphing Calculator Steps

When learning how to graph a best-fit curve using a graphing calculator, it's important to follow a series of steps to ensure accurate representation. Here's a straightforward way to approach it:

• **Enter Data Points:** Input the data points \((0, 21.00), (10, 56.80), (20, 149.5), (30, 234.0)\) into your calculator. Use the statistical or data entry option for this step.
• **Define Exponential Function:** In the function editor, input the exponential equation \( P(t) = 35.26 e^{0.06407t} \), usually in the \( Y= \) or \( f(x)= \) menu.
• **Plot and Compare:** Use the graphing function to plot both the data and the curve on the same graph. This visual comparison allows you to see how well the curve fits the data points.

The graphical display will give insights into how accurately your model represents real-world scenarios. Such steps are crucial for students and professionals aiming to utilize calculators for interpreting and analyzing growth trends effectively.