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Summary Math Analysis and Approaches - Using differential calculus to find the Best model that expresses growth of a given population
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This document contains an example of a high-scoring internal assignment of Mathematics Analysis and Approaches (Standard level). The mark received was 19/20. The topic discussed is Using differential calculus to find the best model that expresses growth of a given population.

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  • May 27, 2023
  • 19
  • 2020/2021
  • Summary

Subjects

  • math aa
  • math ia
  • internal assignment
  • math sl
  • international baccalaureate
  • math analysis and approaches

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  • Secondary school
  • Math AA
  • 5
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IA - Math analysis and approaches SL

Using differential calculus to find the best model that expresses growth of a given population.

,Table of contents:

Introduction: ……………………………………………………………………………………………...page 2

Data Used For The Investigation: ……………………………………………………………………….page 3

Hypothesis: ..……………………………………………………………………………………………....page 4

Part A: Expressing The Model In Terms Of An Exponential Equation: ..……………………………page 4

- Variables and constants…………………………………………………………………………….page 4

- Method: …………………………………………………………………………………………....page 4

- Graph: Population Vs Year………………………………………………………………………....page 6

𝑑𝑓 1
- Graph: Change in 𝑑𝑡
× 𝑓
with change in population:…………...………………………………....page

8

- Graph: Population Growth And Exponential Approximation…………...…………………………page 9

Part B: Expressing the model in terms of a logistic equation: ……………..………………………….page 10

- Variables and constants…………………………………………………………………………….page 10

Method: …………………………………………………………………………………………....page 11

𝑑𝑃 1
- Graph: Change in 𝑑𝑡
× 𝑃
with change in population:…………...………………………………..page

12

- Graph: Population growth and logistic approximation…………...………………………………..page 14

Part C: Evaluation And Conclusion: ………………………………………...………………………….page 15

Bibliography: ……………………………………………………..……………………………………....page 18




1

, Introduction:

As the name of this investigation suggests, this investigation will aim to use differential calculus to find a model

that best fits the data of a given population.

As I studied calculus in mathematics class, I was fascinated when I read an article that made me learn that

calculus isn’t just significant in maths and has an essential presence in the world of ecosystems and populations.

After undergoing further research I settled on a topic which is about calculating the population growth of

different species using differential calculus. As an IB biology and mathematics student, I decided to focus this

investigation on modelling growth which combines two subjects that I love, with those two subjects being the

ones that I am going to pursue in university as an undergraduate biomedical science student.



One important dynamical process in biology is the behaviour of organisms and their population, ultimately

influencing the growth rate. The growth rate differs from one species to another, but in every species,

reproduction is crucial for surviving. In fact, an important indicator of success is the growth of the species

population. This investigation will be fully focused on data from my local human population. The human

population growth is quite complicated when comparing it to other species/kingdoms. While the growth of some

bacterial populations occurs according to an exponential model, where for example one cell gives rise to 2, 2

cells give rise to 4, 4 cells give rise to 8, the human population does not follow a certain structure. Human

population growth does not occur at a normal growth rate and may be influenced by numerous factors such as

immigration and emigration, environmental factors, birth and death rate. Therefore, it does not always follow a

particular trend. For example, during the industrial revolution, the mortality rates started decreasing, thus, the

human population started growing rapidly. (Population Reference Bureau, 2009) However, in 2003, the human

population declined from 88 million people to 74 million people. (Roser et al., 2013). This indicates that Human

population growth does not occur at a stable rate.




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