Problem 44 - 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 \(N_{0}\) $$ R=2, N_{0}=3 $$

Step-by-Step Solution

Answer

Plot points (3,6), (6,12), (12,24) and draw a line through them.

Step 1: Write the equation

The given equation is \( N_{t+1} = R \cdot N_{t} \). For \( R = 2 \), it becomes: \( N_{t+1} = 2 \cdot N_{t} \). This is the equation of the line we need to graph.

Step 2: Calculate \(N_1\)

Since \( N_0 = 3 \), substitute \( N_0 \) into the equation to find \( N_1 \):\[ N_1 = 2 \cdot 3 = 6. \]

Step 3: Calculate \(N_2\)

Using the value of \( N_1 \) we found, substitute it back into the equation to find \( N_2 \): \[ N_2 = 2 \cdot 6 = 12. \]

Step 4: List calculated points

Based on the given initial value and calculated subsequent values, list the points \((N_t, N_{t+1})\) for \( t = 0, 1, \) and \( 2 \): \((3, 6)\), \((6, 12)\), \((12, 24)\).

Step 5: Plot the points and graph the line

On the \(N_t - N_{t+1}\) plane, plot the points \((3, 6)\), \((6, 12)\), and \((12, 24)\). Draw a line through these points to represent the relationship \(N_{t+1} = 2N_t\) for \(R = 2\). This line should pass through the origin since for \(N_t = 0\), \(N_{t+1} = 0\) as well.

Key Concepts

Graphing Linear Functions

Linear functions are foundational in algebra because they describe relationships with a constant rate of change. Here, we focus on the equation \( N_{t+1} = 2N_t \). This is a linear function since it can be rewritten in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In our example, the equation becomes \( N_{t+1} = 2N_t \), making the slope \( 2 \) and the y-intercept \( 0 \). The slope tells us that for every unit increase in \( N_t \), \( N_{t+1} \) increases by 2 units.
In graphing, we mark the y-intercept at the origin \((0, 0)\) and use the slope to determine other points by moving 1 unit to the right on the \( N_t \)-axis and 2 units up on the \( N_{t+1} \)-axis. The points calculated, such as \((3, 6)\), \((6, 12)\), and \((12, 24)\), help to draw the line which represents the function. This visual representation provides insight into how \( N_{t+1} \) changes relative to \( N_{t} \).

Iteration Process

The iteration process is key in understanding how values evolve in discrete dynamical systems. With the equation \( N_{t+1} = 2N_t \), iteration involves calculating a sequence of values starting from an initial condition. Here, the given initial value is \( N_0 = 3 \).

• **Step-by-step calculation**: Substitute \( N_0 \) into the equation to find \( N_1 \). For \( N_0 = 3 \), \( N_1 = 2 \times 3 = 6 \).
• Use the result from the previous step to find the next value. Thus, \( N_2 = 2 \times 6 = 12 \).

This iterative pattern continues indefinitely, producing a sequence of values that is predictable. This method is simple yet powerful, illustrating how initial conditions and rules determine the future behavior of the system. Each iteration is like taking a step further along the path established by the equation.

Exponential Growth

Exponential growth is a pattern where quantities increase at a rate proportional to their current value. In our exercise, the equation \( N_{t+1} = 2N_t \) manifests as exponential growth because each new term is a multiple of the previous one.
Here, the factor of increase is \( R = 2 \). This means that starting from \( N_0 = 3 \), each successive value doubles: \( 6, 12, \) and then \( 24 \), etc. The constant \( R \) greater than 1 leads to rapid increases, characterized by the formula:\[N_t = N_0 \times R^t\]

• At \( t = 0 \), \( N_0 = 3 \).
• At \( t = 1 \), \( N_1 = 3 \times 2^1 = 6 \).
• At \( t = 2 \), \( N_2 = 3 \times 2^2 = 12 \).

Exponential growth continues as long as the system permits, potentially leading to very large values as seen in many natural processes and models.