Problem 49 - Calculus for Biology and Medicine

Question

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ l i n e ~ \(\boldsymbol{N}_{t+1}=\boldsymbol{R} N_{t}\) in the \(\boldsymbol{N}_{t}-\boldsymbol{N}_{t+1}\) 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

Plot points (81, 27) and (27, 9) on the graph.

Step 1: Understanding the Equation

The equation given is \( N_{t+1} = R N_{t} \). This represents a linear relationship where \( N_{t+1} \) is a function of \( N_{t} \) scaled by the factor \( R \). For this problem, \( R = \frac{1}{3} \) and \( N_{0} = 81 \). This means to calculate \( N_{t+1} \), you multiply \( N_{t} \) by \( \frac{1}{3} \).

Step 2: Calculating \( N_{1} \)

Substitute \( N_{0} = 81 \) into the equation: \( N_{1} = \frac{1}{3} \times 81 = 27 \). The first point to plot is \((N_{0}, N_{1}) = (81, 27)\).

Step 3: Calculating \( N_{2} \)

Use \( N_{1} = 27 \) to find \( N_{2} \): \( N_{2} = \frac{1}{3} \times 27 = 9 \). The second point to plot is \((N_{1}, N_{2}) = (27, 9)\).

Step 4: Visualizing the Graph

Plot the calculated points \((81, 27)\), \((27, 9)\) on the \( N_{t}-N_{t+1} \) plane. The line graph will show a decrease in \( N_{t} \) values, reflecting the reduction factor \( R = \frac{1}{3} \).

Key Concepts

Graphing in Mathematics

Graphing is a fundamental skill in mathematics that involves plotting data points on a coordinate plane. This exercise uses the linear difference equation \( N_{t+1} = R N_{t} \), where \( R = \frac{1}{3} \) and \( N_{0} = 81 \). Here, we explore how these variables create a graph that shows a relationship between consecutive terms of a sequence.

To begin graphing, identify the initial point \( (N_0, N_1) = (81, 27) \). This point is derived by multiplying \( N_0 \) by \( R \). For each subsequent point, the current value \( N_t \) is multiplied by \( R \) to find \( N_{t+1} \).

• Point 1: \( (81, 27) \)
• Point 2: \( (27, 9) \)

This relationship is graphed on the \( N_{t}-N_{t+1} \) plane. As you plot each point, you will see a line emerge. This line represents the decay in values due to the multiplication by \( R \), which makes the values shrink over time. It helps visually track how sequences change, showing a step-by-step reduction.

Sequence and Series Analysis

In mathematics, sequence and series analysis is a method used to evaluate the behavior of numbers arranged in a particular order. Here we observe a linear difference sequence as described by the equation \( N_{t+1} = R N_{t} \). This insight helps us understand how series behave over time, particularly with changes in terms.

For the given values, \( R = \frac{1}{3} \), which acts as a reduction factor affecting the entire sequence. Starting from \( N_0 = 81 \), each term in the sequence is found by multiplying the previous term by \( \frac{1}{3} \). As seen in our example:

• \( N_1 = 27 \)
• \( N_2 = 9 \)

This sequence shows exponential decay because each value is a third of the predecessor. Analyzing sequences like this helps build a deeper understanding of series in mathematics, providing insights into long-term trends, stability, and the rate of change.

Exponential Decay in Biology

Exponential decay is a critical concept in biology, describing processes that decrease rapidly over time. The scientific phenomenon can be modeled mathematically by equations similar to the one explored: \( N_{t+1} = R N_{t} \).

In this context, \( R = \frac{1}{3} \) represents a decay factor, leading to a decline in population or concentration levels with each time step. For example, if modeling the decay of a substance or population, \( N_0 = 81 \) might represent an initial concentration or number of individuals. Over time, this value reduces as shown:

• From 81 to 27 (in one time step)
• Then from 27 to 9

This pattern is a signature of exponential decay, where each subsequent value is smaller than the last by a consistent factor, here a third of the previous value. Understanding exponential decay equips students with tools to work in biological settings where such changes need to be anticipated or measured.