Problem 62 - 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 \(N_{0}\). $$ R=4, N_{0}=2 $$

Step-by-Step Solution

Answer

Plot the points (2, 8) and (8, 32) on the \(N_{t}-N_{t+1}\) plane and draw a line through them with a slope of 4.

Step 1: Understanding the Equation

The given equation is a form of a recursive sequence: \(N_{t+1} = R N_{t}\). Here \(R\) is a constant (given as 4), and \(N_{0}\) is the initial value (given as 2). This means that each successive term \(N_{t+1}\) is obtained by multiplying the previous term \(N_{t}\) by 4.

Step 2: Calculating Further Terms

Starting with \(N_{0} = 2\), calculate \(N_{1}\) and \(N_{2}\):- Compute \(N_{1} = 4 \times N_{0} = 4 \times 2 = 8\).- Compute \(N_{2} = 4 \times N_{1} = 4 \times 8 = 32\).

Step 3: Plotting the Points

Now, plot the points on the \(N_{t}-N_{t+1}\) plane. The points are:- \((N_{0}, N_{1}) = (2, 8)\)- \((N_{1}, N_{2}) = (8, 32)\)

Step 4: Drawing the Line

Based on the equation \(N_{t+1} = 4N_{t}\), draw a straight line through the origin (since for \(N_{t} = 0\), \(N_{t+1} = 0\)). This line represents the relationship between \(N_{t}\) and \(N_{t+1}\), with a slope of 4.

Key Concepts

Graphing

Graphing is a powerful tool that helps us visualize mathematical relationships and understand patterns in data. When graphing a recursive sequence such as \(N_{t+1} = R N_{t}\), we specifically plot the relationship between the terms of the sequence. In this case, the sequential terms \((N_t, N_{t+1})\) are plotted on a coordinate plane.

• The horizontal axis represents \(N_t\) (the current term),
• while the vertical axis represents \(N_{t+1}\) (the next term in the sequence).

For the exercise, beginning at \(N_0 = 2\), we calculate successive terms: \(N_1 = 8\) and \(N_2 = 32\). These calculations give us the points to plot: \((2, 8)\) and \((8, 32)\).
Drawing these points on the graph allows us to see not only individual values but also the line they form, which illustrates the growth pattern as determined by the sequence. By connecting these points with a line passing through the origin, we emphasize the linear relationship dictated by the multiplier.

Constant Multiplier

The concept of a constant multiplier is key in sequences like \(N_{t+1} = R N_{t}\). Here, \(R = 4\) is a constant multiplier, meaning each term in the sequence is a multiple of the previous term, scaled consistently by this factor. Knowing this simple fact sheds light on the entire behavior of the sequence.
Why is the multiplier important?

• It determines the rate or factor by which the sequence grows (or shrinks, if \(R < 1\)).
• With \(R = 4\), the sequence grows quite rapidly, quadrupling each term from its prior value.

This constant multiplier simplifies predicting long-term behavior, enabling us to quickly compute or pattern future terms beyond arduous calculation. The beauty of using a constant like \(R\) is it allows you to anticipate the sequence progression easily—note how each step was just another application of multiplying by 4.

Linear Equations

Despite this sequence being a recursive process, identifying the underlying linear relationship can be very insightful. A linear equation often describes a straight line and follows the general form \(y = mx + b\). In our recursive sequence, while we continually multiply the previous term, plotting \(N_{t+1} = R N_{t}\) creates a line in the plane indicating a direct proportionality between \(N_t\) and \(N_{t+1}\).

• Here, \(m=4\), representing the slope of the line or rate of change, emphasizing how much \(N_{t+1}\) changes per unit of \(N_t\).
• The line passes through the origin (since there’s no constant offset \(b\), given \(b=0\)).

This straight line mirrors the recursive relationship visually, making it easier to see how consistent scaling not surprisingly yields linear progression. As such, linear equations and expressions are not just about predictable straight paths; they help understand recursive relationships when visualized as growth or decay patterns on a graph.