SEKOLAH MENENGAH KEBANGSAAN PUTRA PERDANA

JALAN PUTRA PERDANA 3A, TAMAN PUTRA PERDANA, 47130 SELANGOR

ADDITIONAL MATHEMATICS PROJECT WORK 2014

TITLE : CALCULUS IN OUR LIFE

NAME : MUHD SYAHRIL SHAHIRAN BIN ROZALI

FORM : 5 SCIENCE 2

I/C NUMBER : 970531-12-5919

TEACHER : ENCIK FAIZAL

Contents

CONTENTS PAGES

Appreciation

Objective

Preface

Introduction

Part 1

Part 2

Part 3

Further Exploration

Conclusion

Reflection

APPRECIATION

My name is Muhd Syahril Shahiran Bin Rozali. I am thankful that this Additional Mathematics Project can be done just in time. For this, I would like to seize the opportunity to express my sincere gratitude for those who had been helping me during my work.

First and foremost, I would like to say a big thank you to my Additional Mathematics sir, Faizal for giving me information about my project work. On the other hand, I would also like to thank my dear principle, Puan Tumijah Ponimin for giving me the permission to carry out this project.

Also, I would like to thank my parents. They had brought me the things that I needed during the project work was going on. Not only that, they also provided me with the nice suggestion on my project work so that I had not meet the dead and throughout this project.

In addition, I would like to thank to my last year senior for giving support and advices to me while I’m struggling trying to keep focus on completing this project.

Lastly, I would like to say thank you to my friends and the modern access in our daily life. All of my relevant information come from my friends and the internet. I managed to use all these access in our daily life, such as: computer to finish my Additional Mathematics.

OBJECTIVEWe students taking Additional Mathematics are required to carry out a project work while we are in form 5. This year the Curriculum Development Division, Ministry of Education has prepared four task for us. We are to choose and complete ONE task based on our area of interest. This project can be done in groups or individually, but each of us are expected to submit an individually written report. Upon completion of the Additional Mathematics project work, we are to gain valuable and experiences and able to :

Apply and adapt a variety of problem solving strategies to solve routine and non-routine problems.

Experience classroom environments which are challenging, interesting and meaningful and hence improve their thinking skills.

Experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems

Experience classroom environments where expressing ones mathematical thinking, reasoning and communication are highly encouraged and expected

Acquire effective mathematical communication through oral and writing, and to use the language of mathematics to express mathematical ideas correctly and precisely

Enhance acquisition of mathematical knowledge and skills through problem-solving in ways that increase interest and confidence

Prepare ourselves for the demand of our future undertakings and in workplace

Realise that mathematics is an important and powerful tool in solving real-life problems and hence develop positive attitude towards mathematics.

Train ourselves not only to be independent learners but also to collaborate, to cooperate, and to share knowledge in an engaging and healthy environment

Train ourselves to appreciate the intrinsic values of mathematics and to become more creative and innovative

Realize the importance and the beauty of mathematics

PREFACE This project is prepared by the students of class 5 (2014) and it is based on the Additional Mathematics textbooks, internet search and reference books.

Additional Mathematics is an effective subject in secondary school. Each student who takes this subject has to carry out a project work on the given tasks. The project work for the year 2014 is about calculus.

The aim of doing this project is to improve the skills in using Mathematics for students. Working on this also gives a chance for students to apply their skills on what they had learnt to solve an assigned project. Therefore, every student stand a chance to improve their thinking skills, usage of languages and grammar as well as Mathematics skills throughout the project.

After doing this project, the student will be able to master and understand more on the applications of Additional Mathematics that they learnt in their school syllabus. The student can also learnt some values during the completion of the project such as to learn how to work together or to be cooperative, improving their communication skills, responsibility and also not to give up easily on the task given.

INTRODUCTIONHISTORY OF CALCULUS

Calculus, historically known as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals,

and infinite series. Ideas leading up to the notions of function, derivative, and integral were developed throughout the 17th century, but the decisive step was

made by Isaac Newton and Gottfried Leibniz.

The discovery of calculus is often attributed to two men, Isaac Newton and Gottfried Leibniz, who independently developed its foundations. Although they both were instrumental in its creation, they thought of the fundamental concepts in very different ways. While Newton considered variables changing with time, Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a defining property. On the other hand, Newton used quantities x' and y', which were finite velocities, to compute the tangent. Of course neither Leibniz nor Newton thought in terms of functions, but both always thought in terms of graphs. For Newton the calculus was geometrical while Leibniz took it towards analysis.

It is interesting to note that Leibniz was very conscious of the importance of good notation and put a lot of thought into the symbols he used. Newton, on the other hand, wrote more for himself than anyone else. Consequently, he tended to use whatever notation he thought of on that day. This turned out to be important in later developments. Leibniz's notation was better suited to generalizing calculus to multiple variables and in addition it highlighted the operator aspect of the derivative and integral. As a result, much of the notation that is used in Calculus today is due to Leibniz.

The development of Calculus can roughly be described along a timeline which goes through three periods: Anticipation, Development, and Rigorization. In the Anticipation stage techniques were being used by mathematicians that involved infinite processes to find areas under curves or maximize certain quantities. In the Development stage Newton and Leibniz created the foundations of Calculus and brought all of these techniques together under the umbrella of the derivative and integral. However, their methods were not always

logically sound, and it took mathematicians a long time during the Rigorization stage to justify them and put Calculus on a sound mathematical foundation.

In their development of the calculus both Newton and Leibniz used "infinitesimals", quantities that are infinitely small and yet nonzero. Of course, such infinitesimals do not really exist, but Newton and Leibniz found it convenient to use these quantities in their computations and their derivations of results. Although one could not argue with the success of calculus, this concept of infinitesimals bothered mathematicians. Lord Bishop Berkeley made serious criticisms of the calculus referring to infinitesimals as "the ghosts of departed quantities".

Berkeley's criticisms were well founded and important in that they focused the attention of mathematicians on a logical clarification of the calculus. It was to be over 100 years, however, before Calculus was to be made rigorous. Ultimately, Cauchy, Weierstrass, and Riemann reformulated Calculus in terms of limits rather than infinitesimals. Thus the need for these infinitely small (and nonexistent) quantities was removed, and replaced by a notion of quantities being "close" to others. The derivative and the integral were both reformulated in terms of limits. While it may seem like a lot of work to create rigorous justifications of computations that seemed to work fine in the first place, this is an important development. By putting Calculus on a logical footing, mathematicians were better able to understand and extend its results, as well as to come to terms with some of the more subtle aspects of the theory.

When we first study Calculus we often learn its concepts in an order that is somewhat backwards to its development. We wish to take advantage of the hundreds of years of thought that have gone into it. As a result, we often begin by learning about limits. Afterward we define the derivative and integral developed by Newton and Leibniz. But unlike Newton and Leibniz we define them in the modern way - in terms of limits. Afterward we see how the derivative and integral can be used to solve many of the problems that precipitated the development of Calculus. 

Here is Archimedes' diagram

Archimedes used the method of exhaustion to compute the area inside a circle

Here is Barrow's differential triangle

Part 1

John Wallis (Mathematician)

Part 3

(a)

Since the parabolic satellite disc is symmetrical at y-axis,the curve y = f(x) can be written as y = ax2 +c,It can be seen that the curve y = f(x) cuts the y-axis at the point (0,4).Substitute (0,4) into y = ax2+c,and you will get y = ax2+4.

y

8m

1m

4 x Y=f(x)

0

Substitute the point (4,5) into y = ax2+4

5 = a(42)+4

a =

y = f(x) is now written as y=0.0625(x2)+4.

So, f(x) = +4

(b)(i)

When X = 0, F(0) = + 4

F(0) = 4

When X = 0,F(0.5) = + 4

= 4.0156

When X = 1,F(1) = + 4

= 4.0625

When X = 1.5,F(1.5) = 4.1406

When X = 2, F(2) = 4.25

When X = 2.5 , F(2.5) = 4.3906

When X = 3,F(3) = 4.5625

4

4.7656

4.39

06 4.56

25

4.1406

4.0625

4.254.0156

0.5 1 1.5 2 2.5 3 3.5 4

5

Y= f(x)

Y= + 4

f(x) = + 4

X

Y

When X = 3.5 , F(3.5) = 4.7656

Area of the first strip = 4m x 0.5m

= 2m²

Area of the second strip = 4.0156m x 0.5m

= 2.0078 m²

Area of the third strip = 4.0625m x 0.5m

= 2.0313 m²

Area of the fourth strip = 4.1406m x 0.5m

= 2.0703 m²

Area of the fifth strip = 4.25m x 0.5m

= 2.125 m²

Area of the sixth strip = 4.3906m x 0.5m

= 2.1953 m²

Area of the seventh strip = 4.5625m x 0.5m

= 2.2813 m²

Area of the eight strip = 4.7656m x 0.5m

= 2.3828 m²

Total Area = 17.0938 m²

3(b)(ii)

Area of the first strip = 4.0156m x 0.5m

= 2.0078 m²

Area of the second strip = 4.0625m x 0.5m

= 2.0313 m²

Area of the third strip = 4.1406m x 0.5m

= 2.0703 m ²

Area of the forth strip = 4.25m x 0.5m

= 2.125 m ²

Area of the fifth strip = 4.3906m x 0.5m

= 2.1953 m ²

Area of the sixth strip = 4.5625m x 0.5m

= 2.2813 m ²

Area of the seventh strip = 4.7656m x 0.5m

= 2.3828 m²

Area of the eighth strip = 5m x 0.5m = 2.5 m²

4

4.7656

4.39

06 4.56

25

4.1406

4.0625

4.254.0156

0.5 1 1.5 2 2.5 3 3.5 4

5

Y= f(x)

Y= + 4

f(x) = + 4

Total area = 17.5938 m²

X

Y

4

3(b)(iii)

Area of the first and second strips = 4.0156m x 1m

= 4.0156 m²

Area of the third and fourth strips = 4.1406m x 1m

= 4.1406 m²

Area of the fifth and sixth strips = 4.3906m x 1m

= 4.3906 m²

Area of the seventh and eighth strips = 4.7656m x 1m

= 4.7656 m ²

4

4.7656

4.39

06 4.56

25

4.1406

4.0625

4.25

4.0156

0.5 1 1.5 2 2.5 3 3.5 4

5

Y= f(x)

Y= + 4

f(x) = + 4

Total area = 17.5938 m²

X

Y

4

(C) (i) The area of the curve using integration.

Area =

=

=

= -

=17 m2

(ii) The diagram 3(iii) gives the best approximate area,which is 17.3124 m2.

(iii) We can improve the value in c(ii) by having more strips from X = 0 to X = 4

(d) y = +4

X2 = 16(y-4)

X2 = 16y-64

Volume =

=

= = 8 m3

Further Exploration

When x = 0.2, y = 1.2 -5(0.2)2

Y = 1

Y = 1.2-5x2

Y2 = (1.2-5x2)2

Y2 = 1.44+25X4-12X2

Volume of gold needed….

=

= - ] -

= ] -

= 0.5152 – 0.4

=0.36191 cm

X

Y

f(x) = 1.2-5x²

let f(x) = y

0.2-0.2

(0.2,1)

Y=1

Gold density is 19.3 g cm-3.

Let k be the weight of the gold in g 0.36191 cm³___ kg

So, k = 6.9849

Finally,(the price of 1 g of gold) x 6.9849

=the cose of the gold ring

CONCLUSION

 After doing research, answering questions, drawing graphs and some problem solving, I saw that the usage of calculus is important in daily life.It is not just widelyused in science, economics but also inengineering. Inconclusion, calculus is a daily life nessecities. Without it, marvelous buildings cant be built, human beings will not lead a luxurious life andmany more. So, we should be thankful of the people who contribute in the idea of calculus.

REFLECTION

After spending countless hours, days and night to finish this project in this mid year holiday, there are several things that I can say :

Additional Mathematics...From the day I born...From the day I was able to holding pencil...From the day I start learning...And...From the day I heard your name...I always thought that you will be my greatestobs t ac l e and r i va l i n e xce l l i ng in m y l i f e …

But after countless of hours...Countless of days...Countless of nights... After sacrificing my precious time just for you... Sacrificing my Twitter...Sacrificing my Internet...Sacrificing my K-Drama...Sacrificing my social media...I realized something really important in you...

I really love you...You are my real friend...You my partner...You are my soul mate…

I LOVE YOU ADDITIONAL MATHEMATICS.