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CONTENTS

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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 by the 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.

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 solveproblems 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.

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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.

The diagram below shows the gate of an art gallery. A concrete structure is built at the upper part

of the gate and the words ART GALLERY is written on it. The top of the concrete structure is

flat whereas at the bottom is parabolic in shape. The concrete structure is supported by two

vertical pillars at both ends.

The distance between the two pillars is 4 metres and the height of the pillar is 5 metres. The

height of the concrete structure is 1 metre. The shortest distance from point A of the concrete

structure to point B, that is the highest point on the parabolic shape, is 0.5 metres.

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OBJECTIVE

The aims of carrying out this project work are:-

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

o to improve thinking skills;

o to promote effective mathematical communication;

o to develop mathematical knowledge through problem solving in a way that increases students

interest and confidence;

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

o to provide learning environment that stimulates and enhances effective learning;

o to develop positive attitude towards mathematics

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PROCEDURE AND FINDINGS

(a) The parabolic shape of the concrete structure can be represented by various functions depending on

the point of reference. Based on different points of reference, obtain at least three different functions

which can be used to represent the curve of this concrete structure.

(b) The front surface of this concrete structure will be painted before the words ART GALLERY is

written on it. Find the area to be painted.

Solution:

(a) Function 1

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Function 2

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Function 3

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(b)

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Solution:

(a) (i)

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Structure 4 is the cheapest to construct, costing RM 840

(ii) As the president of the Arts Club, I would decide to choose Structure 4 as the shape of the

gate to be constructed. This is because Structure 4 would cost the least to be built and it is

easier to be constructed compared to Structure 1 which is a curve.

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(b) The following questions refer to the concrete structure in the diagram below. If the value

ofkincreases with a common difference of 0.25 m;

(i) Complete Table 1 by finding the values of k and the corresponding areas of the

concrete structure to be painted.

(ii) Observe the values of the area to be painted from Table 1. Do you see any pattern?

Discuss.

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

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The area to be painted decreases as the kincreases 0.25m and a form of series of numbers:

We can see that the difference between each term is the same.

We can deduce that this series of numbers is an Arithmetic Progression (AP), with a

common difference,

In conclusion, when kincreases 0.25m, the area to be painted decreases by -0.0625 .

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(c) Express the area of the concrete structure to be painted in terms ofk. Find the area when k

approaches the value of 4 and predict the shape of the concrete structure.

The shape of the concrete structure will be a rectangle with length 4m and breadth 0.5m,which may look like this:

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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 necessity. Without it,

marvellous buildings cant 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.

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REFLECTION

While I was conducting the project, I had learned many moral values. This project has taught

me to be more confident when doing something especially the homework given by teachers

and also to be more patient when working on difficult additional mathematic equations which

took me many tries to get it right. I also learned to be a more disciplined student who is always

sharp while doing work and to complete work by myself by researching the information from

the Internet. I also enjoyed myself when I was completing this project during the time given to

me.

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SEKOLAH AGAMA MENENGAH TINGGI SULTAN

HISAMUDDIN,KLANG