33289888 Additional Ma Thematic Project Work 1 2010

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CONTENT

NO.CONTENT PAGE NUMBER

1ACKNOWLEDGE

1

2INTRODUCTION

2-3

3 OBJECTIVES 4

4PROCEDURE AND FINDING

5-8

5

FURTHER EXPLORATION

9-15

6

CONCLUSION

16

7

REFLECTION

17

8

REFERENCE

18



ACKNOWLEDGE

After weeks of struggle and hard work to complete assignment given to us by our teacher ,Mr.

Saini Bin Bakar. I finally did it within 2 weeks with satisfaction and senses of success because I

have understood more deeply about the interest and investment more than before. I have to be

grateful and thankful to all parties who have helped me in the process of completing my

assignment. It was a great experience for me as I have learnt to be more independent and to work

as group. For this, I would like to take this opportunity to express my thankfulness once again to

all parties concerned.

Firstly, I would like to thanks my Additional Mathematics· teacher, Mr. Saini bin Bakar

for patiently explained to us the proper and precise way to complete this assignment. With his

help and guidance, many problems I have encountered had been solved.

Beside that, I would like to thanks my parents for all their support and encouragement

they have given to me. In addition, my parents had given me guidance on the methods to account

for investment which have greatly enhanced my knowledge on particular area. Last but not least,

I would like to express my thankfulness to my cousin and friends, who have patiently explained

to me and did this project with me in group.

1



INTRODUCTION

Calculus is a branch in mathematics focused on limits, functions, derivatives, integrals,

and infinite series. This subject constitutes a major part of modern mathematics education. It has

two major branches, differential calculus and integral calculus, which are related bythe fundamental theorem of calculus. Calculus is the study of change, in the same way

that geometry is the study of shape and algebra is the study of operations and their application to

solving equations. A course in calculus is a gateway to other, more advanced courses in

mathematics devoted to the study of functions and limits, broadly called mathematical analysis.

Calculus has widespread applications in science, economics, and engineering and can solve

many problems for which algebra alone is insufficient.

Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus".

More generally, calculus may refer to any method or system of calculation guided by the

symbolic manipulation of expressions. Some examples of other well-known calculi

are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.

2



History

The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical

functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve

problems of mathematical physics. In his publications, Newton rephrased his ideas to suit the

mathematical idiom of the time, replacing calculations with infinitesimals by equivalent

geometrical arguments which were considered beyond reproach. He used the methods of

calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the

oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems

discussed in his Principia Mathematica. In other work, he developed series expansions for

functions, including fractional and irrational powers, and it was clear that he understood the

principles of the Taylor series.

These ideas were systematized into a true calculus of infinitesimals by Gottfried Wilhelm

Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an

independent inventor of and contributor to calculus. His contribution was to provide a clear set of

rules for manipulating infinitesimal quantities, allowing the computation of second and higher

derivatives, and providing the product rule and chain rule, in their differential and integral forms.

Unlike Newton, Leibniz paid a lot of attention to the formalism ± he often spent days

determining appropriate symbols for concepts.

Leibniz and Newton are usually both credited with the invention of calculus. Newton was the

first to apply calculus to general physics and Leibniz developed much of the notation used in

calculus today. The basic insights that both Newton and Leibniz provided were the laws of

differentiation and integration, second and higher derivatives, and the notion of an approximating

polynomial series. By Newton's time, the fundamental theorem of calculus was known.

3



Objectives

The aims of carrying out this project work are:-

to apply and adapt a variety of problem-solving strategies to solve problems;

to improve thinking skills;

to promote effective mathematical communication;

to develop mathematical knowledge through problem solving in a way that

increases students interest and confidence;

to use the language of mathematics to express mathematical ideas precisely;

to provide learning environment that stimulates and enhances effective

learning;

to develop positive attitude towards mathematics.

4



Procedure and Findings

Solut ion:

(a)Function 1



Function 2

6



Function 3

Maximum point (2, 4.5) and pass through point (0, 4)

7



(b)

Area to be painted

= Area of rectangle - Area under the curve

8



Further Exploration

Solution:

(a)(i) Structure 1

9



Structure 2

10



Structure 3

11



Structure 4

(ii) As the president of the Arts Club, I will decide Structure 4 as the shape of the gate to be

constructed. It is because Structure 4 will cost the minimum and it is easier to be constructed

compared to Structure 1 which is a curve.

12



(b) (i)

(ii) There is a pattern in the area to be painted.

The area to be painted decreases as the k increases 0.25m and form a series of

numbers:

3, 2.9375, 2.875, 2.8125, 2.75, 2.6875, 2.625, 2.5625, 2.5

We can see that the difference between each term and the next term is the

same.

13

k (m) Area to be painted( )

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00



We can deduce that this series of numbers is an Arithmetic Progression

(AP), with a common difference,

In conclusion, when k increases 0.25m, the area to be painted decreases by

-0.0625

(c)

14



The shape of the concrete structure will be a rectangle with length 4m and

breadth 0.5m, which may look like this:

15



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 widely used

in science, economics but also in engineering. In conclusion, calculus is a daily life

nessecities. Without it, marvelous buildings can¶t be built, human beings will not lead a

luxurious life and many more. So, we should be thankful of the people who contribute in

the idea of calculus.

16

33289888 Additional Ma Thematic Project Work 1 2010

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