additional mathematics project work Kelantan 2/2012

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PROJECT

WORK FOR

ADDITIONAL

MATHEMATICS

2012

NAME : AFIF MURSYIDI BIN MOHD HURI

CLASS : 5 BUHTURI

SCHOOL : SMK(A) WATANIAH

TEACHER : Pn. ZURAIFAH BT

NO. I/C : 951104-29-5055

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Prefaces………3

Introduction……….5

Objective……….11

Method Investigation..……….13

Task 1……….15

Task 2……….19

Task 3……….21

Task 4………..24

Conclusion………..26

Reflection………28

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In the Name Of Allah , the Most Gracious and the Most Merciful …

First of all, I would like to say thank you, for giving

me the strength to do this project work. Not forgotten my

parents for providing everything, such as money, to buy

anything that are related to this project work and their

advise, support which are the most needed for this project.

Internet, books, computers and all that. They also supported

mean encouraged me to complete this task so that I will not

procrastinate in doing it.

Then I would like to thank my teacher, Pn.Zuraifah for

guiding me and my friends throughout this project. We had

some difficulties in doing this task, but she taught us

patiently until we knew what to do. She tried and tried to

teach us until we understand what we supposed to do with the

project work. Last but not least, my friends who were doing

this project with me and sharing our ideas. They were

helpful that when we combined and discussed together, we had

this task done.

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INTRODUCTION

The purpose of add math project work is to provide an opportunity for

students to apply mathematical concepts and skills in problem-solving that they

have learnt in classroom. This project work can help students to understand add

math more easily and aid students in visualising certain mathematical concepts

which

are

difficult

to

show

clearly

through

pen

and

paper

In addition, this project work is essential to help students to learn how to cope

with future challenges using their mathematical abilities. It also aims to foster

moral

values

in

line

with

a

student's

academic

development.

besides this, the students are encouraged to do their own research. As a result

students become more independent. It also makes the learning process more fun

and

effective.

This project work also makes add math an enjoyable and exciting subject

encouraging the students to learn math more skills in more heuristic manner.

Therefore it is beneficial to all students who take add math

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History of differentiation

The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid (c. 300 BC), Archimedes (c. 287–212 BC) and Apollonius of Perga (c. 262–190 BC).[1] Archimedes also introduced the use of infinitesimals, although these were primarily used to study areas and volumes rather than derivatives and tangents; see Archimedes' use of infinitesimals.

The use of infinitesimals to study rates of change can be found in Indian mathematics, perhaps as early as 500 AD, when the astronomer and mathematician Aryabhata (476–550) used infinitesimals to study the motion of the moon.[2] The use of infinitesimals to compute rates of change was developed significantly by Bhāskara II (1114–1185); indeed, it has been argued[3] that many of the key notions of differential calculus can be found in his work, such as "Rolle's theorem".[4] The Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), was the first to discover the derivative of cubic polynomials, an important result in differential calculus;[5] his

Treatise on Equations developed concepts related to differential calculus, such as the derivative

function and the maxima and minima of curves, in order to solve cubic equations which may not have positive solutions.[6]

The modern development of calculus is usually credited to Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716), who provided independent[7] and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes,[8] which had not been significantly extended since the time of Ibn al-Haytham (Alhazen).[9] For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as Isaac Barrow (1630–1677), René Descartes (1596–1650), Christiaan Huygens (1629–1695), Blaise Pascal (1623–1662) and John Wallis (1616–1703). Isaac Barrow is generally given credit for the early development of the derivative.[10] Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to theoretical physics, while Leibniz systematically developed much of the notation still used today. Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Augustin Louis Cauchy (1789–1857), Bernhard Riemann (1826–1866), and Karl Weierstrass (1815–1897). It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.

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History of arithmetic progression

Euler stated that every arithmetic progression beginning with 1 contains an infinite number of primes. The theorem in the above form was first conjectured by Legendre in his attempted unsuccessful proofs of quadratic reciprocity and proved by Dirichlet in (Dirichlet 1837) with Dirichlet L-series. The proof is modeled on Euler's earlier work relating the Riemann zeta function to the distribution of primes. The theorem represents the beginning of rigorous analytic number theory.

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 3, 5, 7, 9, 11, 13, … is an arithmetic progression with common difference of 2.

If the initial term of an arithmetic progression is and the common difference of successive members is d, then the nth term of the sequence is given by:

and in general

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.

The behavior of the arithmetic progression depends on the common difference d. If the common difference is:

 Positive, the members (terms) will grow towards positive infinity.

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Sum

This section is about Finite arithmetic series. For Infinite arithmetic series, see Infinite arithmetic series.

The sum of the members of a finite arithmetic progression is called an arithmetic series. Expressing the arithmetic series in two different ways:

Adding both sides of the two equations, all terms involving d cancel: Dividing both sides by 2 produces a common form of the equation:

An alternate form results from re-inserting the substitution: :

In 499 AD Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave this method in the Aryabhatiya (section 2.18) .[1]

So, for example, the sum of the terms of the arithmetic progression given by an = 3 + (n-1)(5) up

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Product

The product of the members of a finite arithmetic progression with an initial element a1, common

differences d, and n elements in total is determined in a closed expression

where denotes the rising factorial and denotes the Gamma function. (Note however that the formula is not valid when is a negative integer or zero.)

This is a generalization from the fact that the product of the progression is given by the factorial and that the product

for positive integers and is given by

Taking the example from above, the product of the terms of the arithmetic progression given by

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OBJECTIVES

We 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 two tasks for us. We need to choose and complete only 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 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 waysin solving real-life problems

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

 Experience classroom environments that stimulates and enhances effective learning.

 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

 Use technology especially the ICT appropriately and effectively

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

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Method Investigation

In solving and finishing this project work done,some method is used :-

1.Communication

 Discussion with teacher and friend help in solving problem.The information from

this discussion used as a reference materials to success this project.

2.Reference

 Additional of information from various of reference material help me to find the

method to solve the problem.For this Additional Mathematics project,I can get the reference from library,internet,my friends,my teacher and many more.

3.Lesson session

 The lesson session in the class help me in solve problem by using heuristics what I

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TASK 1

The problem of sufficient water supply has become a main issue among

the countries around the world. Proper planning is very important to make

sure the water supply is sufficient. Carry out a simple study to solve the

problem. You are suggested to use various resources such as internet,

printed materials, ground study and others. Discuss on how additional

mathematics can be used to solve the problem.

For this coursework, I have carried out a study on water supply problem in the district inconclusive. Total of 1000 respondents villagers set. The following are water supply problems that occurred in the district Machang

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I've been using the knowledge of Additional Mathematics to solve the problems

on the water needs.

Problem: The size of a small tank

Methods: Differentiation

If the volume of the tank to be constructed is 50 m3.

Minimum surface area required to build the tank was calculated using the

'differentiation'

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Task 2

You are assigned as an engineer of a construction firm. You are

responsible to buld an enclosed water tank for a housing estate which

consist of 60 houses. The rate of water entering every house is different

and the rates are 1000cm

3

min

-1

for the first house , 990 cm

3

min

-1

for the

third house and so on.

By using at least 3 different methods, calculate the total volume of water

used by all the houses in 1 minute, if all the housing estate are using the

water at the same time.

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