Master Euler's Method in IB HL AI Math Paper 3 Specimen Paper

Table of Contents

1. Introduction
2. Question Analysis
3. Finding the Equilibrium Population
4. Explaining the Population Increase
5. Conservation of Black and Brown Squirrels
6. Using Differential Equations to Find Equilibrium Points
7. Using Separation of Variables to Solve Equations
8. Applying Euler's Method for Larger Populations
9. Long-Term Outcomes of the Populations
10. Phase Portrait and the Trajectories of the Populations
11. Conclusion

Introduction

In this article, we will be discussing a mathematics question on population modeling and differential equations. The question involves finding the equilibrium population of brown squirrels and analyzing the factors that influence the population increase for values of x less than a certain threshold. We will also explore the concept of conservation and equilibrium points for populations of black and brown squirrels. Additionally, we will use Euler's method to estimate the long-term outcomes of the populations, and Create a phase portrait to Visualize the trajectories of the populations. So let's dive in and solve this intriguing mathematics problem!

Question Analysis

Before diving into the solution, let's analyze the given question. The question involves a population model of brown squirrels represented by a differential equation. We need to find the equilibrium population suggested by this model and explain why the population of squirrels is increasing for values of x less than the given threshold. We Are also asked to verify the equilibrium points for populations of both black and brown squirrels. Furthermore, we need to use Euler's method to estimate the long-term outcomes of the populations and draw a phase portrait to visualize the trajectories of the populations. Now that we have a clear understanding of the question, let's start solving it step by step.

Finding the Equilibrium Population

To find the equilibrium population of brown squirrels, we need to set the derivative of the population with respect to time (dx/dt) equal to zero and solve the resulting equation. By doing this, we determine the value of x that satisfies the equilibrium condition. In this case, the equation is x/1000 * (2000 - x) = 0. We have two possibilities: either x = 0 or (2000 - x) = 0. However, x cannot be zero since the population needs to be greater than zero. Therefore, the equilibrium population of brown squirrels is 2000.

Pros:

• The equilibrium population of brown squirrels is easily determined by setting the derivative equal to zero.

Cons:

• The equation assumes a linear relationship between the population and time, which might not accurately represent real-world scenarios.

Explaining the Population Increase

The question also asks us to explain why the population of squirrels is increasing for values of x less than the given threshold. To do this, we analyze the differential equation and evaluate the population change for values of x slightly smaller than the threshold. By substituting these values into the equation, we can determine the sign of the resulting expression. In this case, for x = 1999, dx/dt is positive, indicating an increase in the population. This occurs because the multiplication of two positive factors results in a positive change. Therefore, the population of squirrels increases for values of x less than the given threshold.

Pros:

• The explanation provided is straightforward and Based on a simple analysis of the differential equation.

Cons:

• The explanation assumes a constant rate of change for the population, which might not hold true in real-world scenarios.

Conservation of Black and Brown Squirrels

In the question, we are introduced to the concept of conservation between black and brown squirrels. The black squirrels are moving into the territory of the brown squirrels, and both species compete against each other. This situation can be represented by a system of coupled differential equations that describe the population change of both species. To determine the equilibrium points where both dx/dt and dy/dt are equal to zero, we solve the equations simultaneously for the values of x and y that satisfy this condition. In this case, the equilibrium point is found to be 800 for x and 600 for y.

Pros:

• The concept of conservation and competition between species is addressed, enhancing the realism of the model.

Cons:

• The assumption that the system of coupled differential equations accurately represents the dynamics between black and brown squirrels might not hold true in all cases.

Using Differential Equations to Find Equilibrium Points

To find the equilibrium points for populations of black and brown squirrels, we set the corresponding derivatives equal to zero and solve the resulting equations. By doing this, we can determine the specific values of x and y that satisfy the equilibrium condition. In this case, when dx/dt = 0, we substitute x = 800 into the equation and solve for y, resulting in y = 600. Similarly, when dy/dt = 0, we substitute y = 600 into the equation and solve for x, yielding x = 1000. Therefore, we have three equilibrium points: (0, 0), (800, 600), and (2000, 0).

Pros:

• The method used to find the equilibrium points is based on solving the corresponding differential equations, ensuring accuracy.

Cons:

• The assumption of linearity between the derivatives and the variables might not hold true in all cases, affecting the accuracy of the equilibrium points.

Using Separation of Variables to Solve Equations

To solve the differential equations representing the population change, we use the technique of separation of variables. This involves isolating the variables and integrating both sides of the equations separately. By doing this, we can find the general solutions of the equations, which provide a mathematical representation of the population change over time. In this case, we separate the variables in the differential equation and integrate both sides to solve for x and y. By raising each side to the power of e, we can determine the final solutions for x and y based on the initial conditions.

Pros:

• The technique of separation of variables is a widely used and effective method for solving differential equations.

Cons:

• The assumption of a continuous and differentiable function might not hold true in all cases, affecting the accuracy of the solutions.

Applying Euler's Method for Larger Populations

When dealing with larger populations, Euler's method can be used to estimate the population change over time. This iterative method involves breaking down the population change into small steps and updating the population values accordingly. By repeating this process multiple times, we can approximate the long-term outcomes of the populations. In this case, we use Euler's method with a step length of 0.2 to estimate the population of brown and black squirrels after a certain time. By iterating the method and updating the population values accordingly, we can determine the approximate population values for each species.

Pros:

• Euler's method provides a practical and efficient way to estimate population change, especially for larger populations.

Cons:

• The accuracy of Euler's method decreases as the step length increases, potentially leading to less precise population estimates.

Long-Term Outcomes of the Populations

Using Euler's method, we can estimate the long-term outcomes of the populations of brown and black squirrels. By iterating the method over a certain time period, we can observe the trends in population change and determine the eventual equilibrium values. In this case, for an initial population of 10 squirrels of each species, the long-term outcomes are estimated to be 14 brown squirrels and 18 black squirrels when t is equal to 0.2. This provides Insight into the population growth and stabilization of both species over time.

Pros:

• Euler's method allows us to estimate the long-term outcomes of the populations, providing valuable insights into their dynamics.

Cons:

• The accuracy of the long-term outcomes depends on the step length used in Euler's method, which may introduce some level of error.

Phase Portrait and the Trajectories of the Populations

To visualize the trajectories of the populations, we can create a phase portrait by plotting the population values of brown and black squirrels on the same axis. By analyzing the phase portrait, we can observe the dynamics and tendencies of the populations over time. In this case, the phase portrait shows that the brown squirrels tend to increase towards 2000, while the black squirrels tend to decrease towards zero. This graphical representation allows us to better understand the long-term behaviors and interactions of the populations.

Pros:

• The phase portrait provides a visual representation of the trajectories of the populations, making it easier to analyze and interpret their behaviors.

Cons:

• The accuracy of the phase portrait depends on the accuracy of the population estimates and the assumptions made in the model.

Conclusion

In this article, we explored a mathematics question on population modeling and differential equations. We discussed various aspects of the question, including finding the equilibrium population, explaining the population increase, analyzing the conservation of black and brown squirrels, solving the differential equations, using Euler's method for larger populations, estimating the long-term outcomes, and creating a phase portrait to visualize the trajectories of the populations. By following these steps, we were able to solve the question and gain a deeper understanding of the population dynamics. This question demonstrates the application of mathematical concepts in modeling real-world scenarios and highlights the importance of understanding differential equations and population dynamics.