MAT097 S27 Group 5 PDF

Preview

Summary

MATGROUP PROJECT REPORTTASK 1: HUMAN POPULATIONS IN INDIATASK 2: NEWTON-RAPHSON ITERATIONACKNOWLEDGMENTWe would like to express our thanks to all those people who encouraged and guide us to do this group project for MAT097 especially to our beloved lecturer, Madam Saiyidatul Nadiah Binti Idris, who ...

Description

MAT097 GROUP PROJECT REPORT TASK 1: HUMAN POPULATIONS IN INDIA TASK 2: NEWTON-RAPHSON ITERATION

ACKNOWLEDGMENT

We would like to express our thanks to all those people who encouraged and guide us to do this group project for MAT097 especially to our beloved lecturer, Madam Saiyidatul Nadiah Binti Idris, who giving us endless guidance throughout the process to complete this project. Without her guidance, we could not accomplish our goal to finish this assignment perfectly. We also would like to thank our team members who gave us countless support and cooperation throughout the process to finish up this assignment. Without them, this assignment could not finish on time. Thank you, you guys are the best! Last but not least, we would like to send our thanks to our classmates for inspired us and gave us variety of input especially when we are out of ideas. Thus, we would like to acknowledge all of them that have played the role throughout our journey in accomplishing this assignment.

1

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS

1

TABLE OF CONTENTS

2

LIST OF TABLES

3

LIST OF FIGURES

3

ASSIGNMENTS 1

Task 1 (Human Populations in India) 1.1 Introduction

2

1.1.1 Background of study

4

1.1.2 Problem Statement

4

1.1.3 Objectives of the study

4

1.1.4 Scope and Limitations

4

1.2 Implementation

5

1.3 Data Analysis and Discussion

7

Task 2 (Newton-Raphson Iteration) 2.1Introduction

11

2.2 Implementation

13

2.3 DataAnalysis and Discussion

14

CONCLUSION

17

BIBLIOGRAPHY

17

APPENDICES

2

LIST OF TABLES Page Table 1: Human Populations in India

5

Table 2: Actual and Estimation of Human Populations in India (2010-2018)

7

Table 3: Computational steps using Excel

LIST OF FIGURES Page

Figure 1: Actual Data of Human Populations in India from 2010 until 2018

7

Figure 2: Estimation of Human Populations in India from 2010 until 2018

8

Figure 3: Newton-Raphson Iteration Graph

14

Figure 4: Newton-Raphson Iteration Graph

15

Figure 5: Newton-Raphson Iteration Graph

16

3

1 TASK 1 (HUMAN POPULATIONS IN INDIA) 1.1

Introduction

This task is carried out efficiently to fulfill the objectives of MAT097 project assignment which are to achieve the objectives of CLO2, PLO3, CLO4 and PLO7. In this project, all of us are required to study the applications of First Order Differential Equations.

1.1.1 Background of Study According to Wikipedia, population growth is the increase in the number of individuals in a population. India, country that occupies the greater part of South Asia that is a constitutional republic consisting of 29 states with the area of 3,287,260 km2 (1,269,218 mi2). India is the 7th largest country by area and the 26th by population density. According to current estimates, India’s population will peak in the early 2060s at 1.7 billion. Thus, population estimation is essential for India as it will put additional pressure on the environment and natural resources of that country. 1.1.2 Problem Statement To date, India is stretched to its limit due to overpopulation that will soon affect India in negative aspects. There are many kind of ways to overcome the impact of overpopulation in India’growth but what is the actual reasons of the increasing human population in India. 1.1.3

Objectives of the task



To study the Malthusian theory of population.



To determine the number of human populations in India from 2010 to 2018.



To predict the past and future value using particular solution and compare with historical data. 1.1.3 Scope and Limitations

This study was mainly about the coverage of the understanding of Malthusian model by using human population India for 9 years that is from 2010 to 2018. Moreover, this research focus on the human population that exhibits exponential growth, which is when the increase is

4

proportional to the amount already present. From this task, we acknowledged that P= total populations, k= constant of proportionality and A=ec. To carry out the task successfully, the graphs were plotted by using microsoft excel and also desmos. 1.2 Implementation The set of data of human populations in India were obtained from https://data.worldbank.org/. We took the data from year 2010 until 2018. Then, we calculated the particular solutions using the Malthusian model in order to calculate the growth problem. Table 1: Human Populations in India (2010-2018) Year

Population

2010

1234281170

2011

1250288729

2012

1265782780

2013

1280846129

2014

1295604184

2015

1310152403

2016

1324509589

2017

1338658835

2018

1352617328

First of all, we have to calculate the particular solution using the Malthusian model. Let P(t) = the number of human populations in India present at time t (year) Since the rate of growth is proportional to the number of human populations in India P(t), Hence,

Or

5

Which k is the constant of proportionality (Zill and Wright, 2013)

By using the approach of separable variable by first separating the variable P and t,

Then the equation will be integrated on both sides, ∫

∫ =

Hence, we can simplify the equation by letting = A , where

Firstly, to find A By using the initial condition t=0 (2010), P=1234281170,

‫؞‬

Secondly, to find k We could find the constant k by using t=1 (2011), P=1250288729,

P (t )  1234281170e 0.0129t

soc oi lo ecnh ini e thni Heht eec eH

6

From the particular solutions, we could calculate the estimated number of populations. Hence,

Table 2: Actual and Estimation of Human Populations in India (2010-2018) Year

Actual

Estimate

2010

1234281170

1234281170

2011

1250288729

1250306538

2012

1265782780

1266539973

2013

1280846129

1282984176

2014

1295604184

1299641883

2015

1310152403

1316515867

2016

1324509589

1333608935

2017

1338658835

1350923931

2018

1352617328

1368463739

1.3 Data Analysis and Discussion

1.3.1 Data Analysis After we obtained the particular solution, we could plot the graph of actual data and the data from the particular solution. ACTUAL DATA

7

HUMAN POPULATIONS I N INDIA (2010 2018) POPULATION

1.4E+09 1.35E+09 1.3E+09 1.25E+09 1.2E+09 1.15E+09 2010

2011

2012

2013

2014

2015

2016

2017

2018

YEARS

Figure 1: Actual Data of Human Populations in India from 2010 until 2018 Figure 1 shows the graph of actual data of human populations in India. As we can see from the figure above, the human populations are increasingly rapidly. If we compared between 2010 and 2018, it shows a big difference in number. In 2010, the number of populations is lower than 2018 due to some factors. The birth rate in 2010 is lower than 2018 while the death rate is rapidly increasing in that year which causes by the medical facilities that has been provided by the government is not enough for the people in India. However, as it across year by year, the number of populations is increasing due to the improvement of medical facilities. Hence, the birth rate increasing and the death rate became decreasing.

ESTIMATION OF HUMAN POPULATIONS IN INDIA (2010-2018)

8

Figure 2: Estimation of Human Populations in India from 2010 until 2018 Figure 2 above shows the estimation of human populations in India from 2010 until 2018. As we can see, from the graph, the number of populations are increasing year by year but the predicted values are slightly higher than the actual data. By comparing the data from this graph, 2018 has higher number of population than other years. It is mainly caused by several factors such as birth rate, death rate, medical facilities, cultural effect and also education. However, we can conclude that the populations are always growth proportionally across the year as proven by the Malthusian model.

From the particular solutions we could also obtain the future values and past values of human populations in India. Hence, To determine the number of human populations in India in year 2025 (future), t = 15

To determine the number of human populations in India in year 2008 (past), t = -2

1.3.2 Discussion First of all, the values that we obtained from the particular solution is not accurate with the historical values. There are slightly differences between the actual data and the data that we obtained from particular solution, which is about 0.18% of percentage differences. This is

9

because the particular solution that we calculated is only a prediction while the actual data is the accurate one. India is a developing country where its population keep increasing, unlike the developed countries such as Spain where the population decreased. Based on the graph, it can be seen that the population growth of human in India from 2010 to 2018 keep increasing from 1234281170 people in 2010 to 1352617328 in 2018. The changes in population growth of human in India in the past is affected by some reasons. First, the increase in human population growth in India in the past is because the increase in birth rate. India currently faces approximately 33 births a minute, 2000 an hour and 48000 a day which calculates nearly 12 million a year. One of the most vital reason why populations explosion happened in India is due to poverty. The people who have to struggle end up produce more children since more children means more earning hand. The children did not drain the family income, they add to it. At a very young age, they are able to earn money to support the family and help look after the elderly. The increase in birth rate is also caused by the tradition and cultural norms in India where a girl is married at an early age at the age of 14 and 15. Since they married at an early age, they can have more children throughout their lifetime and most of the girls do not have the opportunity to get a proper and better education. Therefore, they remain uneducated and passed down the cultural norms to the next generations and the cycle keep on repeating resulting in the increase of human population. Next, India medical facilities keep on improving so as the country. The improvement of medical facilities such as the medicine and sanitation help to decrease the number of deaths caused by epidemics such as plague and other diseases. This development is considered good for the economy of India, but in term of population, this advanced has further led to unprecedented increase in population growth. Last but not least, the increase in population growth is due to lack of education and awareness lead to ignorance. They do not have plan in their life and unable to realize the main reasons of scarcity of food, water and other life necessities.

10

Population, if continues to increase at the present rate it will destroy the country. Government must take this matter seriously as the higher the human population growth, the lower the life necessities and resources available. The population growth in India in the future are still affected by the same reasons as what affected the population growth in the past which is the number of birth rate, poverty, medical facilities and education. First, the population growth in the future is affected by the number of birth rate. This reason is actually related to poverty as well. In future, as the economy of India keep on developing, so there will be more job opportunity for the Indian and low birth rate since most of them produce more child just to increase the earning hand in the family. Next, the medical facilities will also keep on improving and cure more incurable disease from the past. As the country keep on developing, so as the education. Due to this, more Indian will get a proper education especially in the rural areas and help them to plan their life well. Since they have a better education and knowledge, they will know the importance of family planning to avoid the overpopulation. It cannot be denied that although the Indian become more educated and have a better salary but the population growth in the future will increase at first and at one point then, it will slowly remain stable.

2 TASK 2 (NEWTON-RAPHSON ITERATION) 2.1 Introduction For our assignment this time, we were required to use Newton’s method to solve e 6 x  3(ln 2) 2 e 2x  1 with interval given [-1,1].Newton's method, also known as the Newton–

Raphson method, named after Isaac Newton and Joseph Raphson.It was first published in 1685 in A Treatise of Algebra both Historical and Practical by John Wallis.It is used for polynomials of degree 3 or more, finding roots of

becomes more complicated. Although formulas exist for

third- and fourth-degree polynomials, they are quite complicated. Also, if

is a polynomial of

degree 5 or greater, it is known that no such formulas exist. Similar difficulties exist for nonpolynomial functions

.Newton-Raphson method is one of the ways to quickly

11

find a good approximation for the root of a real-valued function f(x)=0. It uses the idea that a continuous and differentiable function can be approximated by a straight-line tangent to it. In order to find the root of continuous and differentiable function, we need to know that the root is near to the point x=x0. Newton's method tells us that a better approximation for the root is;

This process may be repeated as many times as necessary to get the desired accuracy. In general, for any x-value x n , the next value is given by;

A graph should be drawn according to Newton’s method;

A tangent is drawn to the curve at the point x=xn. This line has slope f′(xn) and goes through the point (xn,f(xn)). Now, we find the root of this tangent line by setting y=f′(xn)(x−xn)+f(xn) for our new approximation. Solving this equation gives us our new approximation, which is xn 1  xn 

f (x n) f ' ( xn ) .

Newton's method may not work if there are points of inflection, local maxima or minima around x 0 or the rootThis caused the graph to look like this;

12

This graph has a local maximum, a local minimum and a point of inflection around x=0. Imagine choosing a point at random between x = -0.19and x = 0.19 and drawing a tangent line to the function at that point. That tangent line will have a negative slope, and therefore will intersect the y-axis at a point that is farther away from the root. In a situation like this, it will help to get an even closer starting point, where these critical points will not interfere. Objective of Newton-Raphson method:  To solve nonlinear equation  To solve non-polynomial equation  To find a good approximation for the root of a real-valued function f(x)=0 2.2 Implementation The Newton-Raphson method is a powerful technique for solving equations numerically. Like so much of the differential calculus, it is based on the simple idea of linear approximation. The Newton Method, properly used, usually homes in on a root with devastating efficiency.A function has a root when it crosses the x-axis, root, when there are multiple values for

. A function can have more than one

that satisfy this condition. In this method we applied

iteration method is a process wherein a set of instructions or structures are repeated in a sequence a specified number of times or until a condition is met. , [-1,1] The first thing is we have to derive the equation given to find f’(x), f (x )  e6 x  3(ln 2)2 e2 x  1 f ' (x )  6e 6 x  2(3(ln 2) 2 e 2 x )

13

 6e 6 x  6(ln 2) 2 e 2 x  6e 2x (e 4x  (ln 2) 2 )

We made an estimation,x 0  0.05 since the interval for the following function is -1≤x≤1.The value of x1 can be calculated using this formula;

x1  x 0 

f (x 0 ) f ' ( x0 )

 e 6(0.05)  3(ln 2) 2 e 2(0.05)  1  x1  0.05   2 ( 0 .05 ) (e 4(0.05)  (ln 2) 2 )   6e  0. 122158

2.3 Data Analysis and Discussion After calculating the value of the first x, the next value of x is calculated continuously using iteration method until constant x value obtained as below:

As we can see from the calculation above, after 5 times of calculation to obtained the constant value of x which is -0.283528

14

Figure 3: Newton-Raphson Iteration Graph

Figure 4: Newton-Raphson Iteration Graph In this task, we start with an estimate of root with

. A tangent is then drawn to

the curve at . The next point where the tangents intersects the x-axis at , which is ,0), is a better root approximation when compared to as shown is the graph above. In this graph, the tangent intersects at

. From , a new estimate of can be produced by

drawing another tangent to the curve at .This method is repeated to find , where a tangent

15

is then drawn to the curve at . The process is continued until the value is close enough to the root or until it cannot proceed any further. The whole process stated is called iterative, which can be defined as the repetition of an action. Based on the calculation made,

is

close enough to the true root of the function. When the number of iteration increase, the accuracy of the root increases as well. In other words, the latest estimate of root will be the closest to the true root. To show clearly, say, a simpler function, at

. When

, the true root can be calculated

. Say, the number of iterations is 4, the plotted intersection is at . However, it is not close enough to the true root. Next, if 7 iterations are made,

the plotted intersection is at

. All the same, it is still not as accurate to the

true root. If the number or iteration is increased to 8 iterations, the plotted root is at . To prove this accuracy, even at 50th iteration, the root achieved remains the most accurate to the true root. This proves that the accuracy of the root increases as number of iteration increases.

16

Figure 5: Newton-Raphson Iteration Graph

CONCLUSION In the conclusion, we found out that both task 1 and task 2 brings a lot of benefits to us. From task 1, we learned how to find out the populations in certain area with the Malthusian model and we managed to predict the past values and future values correctly. From task 2, we learned a new method to find roots for a complicated function by using Newton-Raphson iteration method. Although we found it really hard to co...