Problem 37 - Calculus for Biology and Medicine

Question

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=\frac{1}{2} N_{t} \text { with } N_{0}=640 $$

Step-by-Step Solution

Answer

Population sizes: \(N_0=640\), \(N_1=320\), \(N_2=160\), \(N_3=80\), \(N_4=40\), \(N_5=20\); \(N_t = 640 \cdot (\frac{1}{2})^t\).

Step 1: Understanding the Recursion

The recursive formula given is \(N_{t+1} = \frac{1}{2} N_{t}\) with an initial population \(N_0 = 640\). Each step involves calculating the next population size as half of the current population size.

Step 2: Calculate Population Size for t=1

Using the recursion formula, calculate \(N_1\): \[N_1 = \frac{1}{2} \times 640 = 320.\]

Step 3: Calculate Population Size for t=2

Calculate \(N_2\) using \(N_1\): \[N_2 = \frac{1}{2} \times 320 = 160.\]

Step 4: Calculate Population Size for t=3

Calculate \(N_3\) using \(N_2\): \[N_3 = \frac{1}{2} \times 160 = 80.\]

Step 5: Calculate Population Size for t=4

Calculate \(N_4\) using \(N_3\): \[N_4 = \frac{1}{2} \times 80 = 40.\]

Step 6: Calculate Population Size for t=5

Calculate \(N_5\) using \(N_4\): \[N_5 = \frac{1}{2} \times 40 = 20.\]

Step 7: Write the Equation for \(N_t\) as a Function of \(t\)

The population decreases by half each time period, so the population at time \(t\) can be expressed as: \[N_t = 640 \times \left(\frac{1}{2}\right)^t.\]

Key Concepts

Recursive Formula

A recursive formula allows us to determine the next term in a sequence from the previous term. In our example, the recursive formula given is \(N_{t+1} = \frac{1}{2} N_{t}\). This means each term in the sequence is half of the previous term. Understanding recursive formulas is important in predicting how sequences and systems change over time.

Key points about recursive formulas:

• They depend on an initial value, which is \(N_0 = 640\) in this problem.
• Each subsequent value is defined using the previous value.
• They are widely used in many areas such as population dynamics, finance, and computer science.

By starting with a known value and applying the formula, we can easily generate the rest of the sequence.

Population Size Calculation

Calculating the population size at different time periods using recursive formulas gives us insights into how populations evolve. In the exercise, we're specifically calculating population size for \(t=0, 1, 2, \ldots, 5\).

Here's how you can consistently calculate each term:

• Start with the initial population size \(N_0 = 640\).
• Apply the recursive formula to find each subsequent term.
• For instance, the population at \(t=1\) is \(N_1 = \frac{1}{2} \times N_0 = 320\).
• Continue using the same process for \(t\) up to 5 to get each population value.

These calculations demonstrate how the population decreases over time, which can be very helpful in predicting future trends.

Exponential Decay

Exponential decay describes a process where the quantity decreases at a consistent rate over time. In our problem, the population halves each period, exemplifying exponential decay.

Essential elements include:

• The base rate of decay, here indicated by \(\frac{1}{2}\), meaning the population shrinks by 50% at each time step.
• Starting population \(N_0 = 640\).
• The function \(N_t = 640 \times \left(\frac{1}{2}\right)^t\) provides a quick way to find population size at any time \(t\).

This function shows how exponential decay can be both smooth and predictable, making it a powerful tool for modeling real-world scenarios such as radioactivity, cooling of substances, and shrinking populations.