Problem 65 - Calculus for Biology and Medicine

Question

In Problems , graph the line \(N_{t+1}=R N_{t}\) in the \(N_{t}-N_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0\), 1\. and 2. for the given value of \(N_{0}\). $$ R=\frac{1}{3}, N_{0}=81 $$

Step-by-Step Solution

Answer

Points to plot are (81, 27), (27, 9), and (9, 3); graph is a line with slope 1/3.

Step 1: Understand the Equation

The equation given is \(N_{t+1} = R N_{t}\). This is a linear equation in the form of \(y = mx + b\), where \(N_{t}\) and \(N_{t+1}\) are analogous to \(x\) and \(y\), respectively, and \(R\) represents the slope of the line.

Step 2: Plot the Line

Substitute \(R = \frac{1}{3}\) into the equation to get \(N_{t+1} = \frac{1}{3} N_{t}\). This means for every increase in \(N_{t}\), \(N_{t+1}\) will be \(\frac{1}{3}\) of that value. This line passes through the origin (0,0) and has a slope of \(\frac{1}{3}\).

Step 3: Calculate Points for t=0, 1, 2

- For \(t=0\): \(N_{0} = 81\). Thus, \(N_{1} = \frac{1}{3} \times 81 = 27\).- For \(t=1\): \(N_{1} = 27\). Thus, \(N_{2} = \frac{1}{3} \times 27 = 9\).- For \(t=2\): \(N_{2} = 9\). Thus, \(N_{3} = \frac{1}{3} \times 9 = 3\).

Step 4: Locate the Points on the Plane

The points corresponding to \((N_{t}, N_{t+1})\) for \(t = 0, 1,\) and \(2\) are:- Point for \(t=0\): \((81, 27)\)- Point for \(t=1\): \((27, 9)\)- Point for \(t=2\): \((9, 3)\).

Step 5: Draw the Graph

On a graph with the horizontal axis as \(N_{t}\) and the vertical axis as \(N_{t+1}\), draw a line with a slope of \(\frac{1}{3}\) that passes through these points: \((81, 27)\), \((27, 9)\), and \((9, 3)\). The line will descend from the point \((81, 27)\) to \((9, 3)\).

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics and are used to describe relationships between variables. In the exercise, the equation given is a type of linear equation, represented by \(N_{t+1} = R N_{t}\). Here, \(N_{t}\) and \(N_{t+1}\) are variables analogous to \(x\) and \(y\) in the standard linear equation form \(y = mx + b\). The variable \(R\) serves as the slope \(m\), while there is no constant term \(b\), indicating that the line passes through the origin (0,0).

• Linear equations have a constant rate of change, represented by the slope.
• The slope determines how steep the line is. A slope of \(\frac{1}{3}\) means that for every unit increase in \(N_{t}\), \(N_{t+1}\) increases by one-third of that amount.
• These equations are used in various fields such as physics, economics, and biology to model linear relationships.

Understanding linear equations helps in predicting one variable based on another, making them essential for modeling real-world situations.

Graphing Lines

Graphing lines from linear equations is a practical skill that involves plotting points on a coordinate plane to visually understand the relationship between the variables. In our exercise, graphing the line \(N_{t+1} = \frac{1}{3} N_{t}\) involves plotting the points calculated for different values of \(t\).

• Each point on the graph, such as \((81, 27)\), corresponds to a specific time \(t\) and shows the relationship between \(N_{t}\) and \(N_{t+1}\).
• The line is a depiction of the relationship where the slope \(\frac{1}{3}\) indicates a decrease in \(N_{t+1}\) as \(N_{t}\) increases; hence, the line descends.
• Graphing helps to validate the mathematical model by seeing if calculated points align on a linear path.

Using graphing, one can easily interpret and analyze the changes in data, making it a valuable tool for visual learners and those dealing with large datasets.

Mathematical Modeling

Mathematical modeling uses mathematical concepts to create representations of real-world scenarios, allowing predictions and insights into complex situations. The equation \(N_{t+1} = \frac{1}{3} N_{t}\) serves as a simple model that predicts the next generation's population size based on the current size, with a growth rate \(R\) of \(\frac{1}{3}\).

• Such models help in understanding and forecasting behaviors in systems like population dynamics, economics, and environmental studies.
• They simplify complex relations by representing them through mathematical language, allowing clearer insights and actionable predictions.
• By adjusting parameters (like \(R\) in the exercise), one can assess different scenarios and their potential outcomes.

Mathematical modeling, therefore, bridges theoretical mathematics with practical applications, facilitating informed decision-making and strategic planning.