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Calculus (Addmath)

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Page 1: Calculus (Addmath)





Page 2: Calculus (Addmath)


No Contents Page1 Introduction2 Appreciation 3 History of Calculus4 Procedure and Findings5 Further Exploration6 Conclusion7 Reflection


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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 four tasks for us. We are to choose and complete only ONE task based on our area of interest. This

project can be done in groups or individually, and I gladly choose to do this individually.

The aim of carrying out this project work are:-

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

To promote effective mathematical communication

To improve thinking skills

To develop mathematical knowledge through problem solving in way that increases student’s interest and confidence

To provide learning environment that stimulates and enhances effective learning

To use language of mathematics to express mathematical ideas precisely

To develop positive attitude towards mathematics


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Alhamdullilah, thank you to Allah SWT for giving the will to me to complete this Additional Mathematics project.

First of all, I would like to say thanks to my parents for providing everything such as money to buy anything that are related to this project work. They also supported me and encouraged me to complete this task.

Secondly, I would like to thank the principal of Sekolah Menengah Kebangsaan Agama Nurul Ittifaq, Madame. Hjh. Wan Sabriah binti Wan Bakar for giving me the permission to do my task of Additional Mathematics Project Work.

I also like to thank my Additional Mathematics teacher, Pn. Masriza binti Said for the guide and giving useful and important information for me to complete this project work.

What is ‘CALCULUS’?????

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 study of change, in the same way that geometry is the study of shape and algebra is the study of operation s and their application to solving

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equations. A course in calculus is a gateway to other, more advance 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 propositional calculus, variational calculus, lambda calculus, pi calculus and join calculus.


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, many other problems discussed in his Principia Mathematica. In other work, he developed series expensions for functions,

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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.

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Procedure & finding


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 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.

5 m

4 m

1 m B


0.5 m

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(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.


Function 1

Maximum point ( 0 , 0.5 ) and pass through point ( 2 , 0 )y = α (b)² + c

b = 0 , c = 0.5

y = α (²

y = α²

substitute ( 2 , 0 ) into (1)

0 = α(2)² + 0.5

0 = 4 α + 0.5

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4 α = - 0.5

α = - 0.5 4

α = - 0.125

y = - 0.125²

Function 2

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

y = α(b)² + c

b = 2, c = 4.5

y = α(²

Substitute ( 0 , 4 ) into (2)

4 = α (0 – 2)² + 4.54 = α (-2)² + 4.54 = 4 α + 4.54 α = 4 – 4.54 α = - 0.5 α = - 0.5


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α = - 0.025

y = - 0.125(²

Function 3

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

y = α(b)² + c

b = 0 , c = 4.5

y = α (² + 4.5

y = α² + 4.5 (3)

Substitute ( 2 , 4 ) into (3)

4 = α (2)² + 4.54 = 4 α + 4.54 α = 4 – 4.54 α = - 0.5 α = - 0.5


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α = - 0.125

y = - 0.125² + 4.5

(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.


Area to be painted= Area of rectangle – Area under the curve=

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Further exploration

(a) You are given four different shapes of concrete structures as shown in the

diagrams below. All the structures have the same thickness of 40 cm and

are symmetrical.

Structure 1 Structure 2

Structure 3 Structure 4

5 m

1 m0.5 m

4 m

5 m

1 m 1 m

0.5 m

5 m2

4 m

1 m 2 m

0.5 m

4 m

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(i) Given that the cost to construct 1 cubic metre of concrete is RM840.00,

determine which structure will cost the minimum to construct.

Solution :

Structure 1

Area =

Thickness = 40 cm

= 0.4 m

Volume = Area x Thickness

= m² x 0.4 m

= m³

Cost = m³ x RM 840

= RM

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Structure 2

Area = Area of Rectangle – Area of Triangle




Volume = Area x Thickness



Cost =


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Structure 3

Area = Area of Rectangle – Area of Trapezium =



Volume = Area x Thickness



Cost =


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Structure 4

Area = Area of Rectangle – Area of Trapezium




Volume = Area x Thickness =


Cost =


Structure will cost the minimum to construct, that is RM .

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(ii) As the president of the Arts Club, you are given the opportunity to

decide on the shape of the gate to be constructed. Which shape

would you choose? Explain and elaborate on your reasons for

choosing the shape.


As the president of Arts Club, I will decide Structure as the shape f the gate to be constructed. It is because structure will cost the minimum and it is easier to be constructed compared to structure which is a curve.

(b) The following questions refer to the concrete structure in the diagram


If the value of k increases 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.


0.5 m1 m

4 m

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Answer :

k(m)Area to be painted (m2)

(correct to 4 decimal places)

0.00 (4 x 1) - 0 + 4 x 0.5 = 3 2

0.25 (4 x 1) - 0.25 + 4 x 0.5 = 2.9375 2

0.50 (4 x 1) - 0.50 + 4 x 0.5 = 2.875 2

0.75 (4 x 1) - 0.75 + 4 x 0.5 = 2.8125 2

1.00 (4 x 1) - 1.00 + 4 x 0.5 = 2.75 2

1.25 (4 x 1) - 1.25 + 4 x 0.5 = 2.6875 2

1.50 (4 x 1) - 1.50 + 4 x 0.5 = 2.625 2

1.75 (4 x 1) – 1.75 + 4 x 0.5 = 2.5625 2

2.00 (4 x 1) - 2.00 + 4 x 0.5 = 2.5 2

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(ii) observe the values of the area to be painted from Table 1. Do you

see any pattern? Discuss.

Answer :

The area to be paint decreases as the k increases 0.25 m 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.

2.9375 – 3 = -0.06252.875 – 2.9375 = -0.06252.8125 – 2.875 = -0.06252.75 – 2.8125 = -0.06252.6875 – 2.75 = -0.06252.625 – 2.6875 = -0.06252.5625 – 2.625 = -0.06252.5 – 2.5625 = -0.0625

we can deduce that is series of numbers in an Arithmetic Progression (AP), with a common difference, d = -0.0625In conclusion, when k increases 0.25m, the area to be painted decreases by -0.0625m².

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

Find the area a k approaches the value of 4 and predict the shape of the

concrete structure.

Answer :The area of the concrete structure to be painted




Area of concrete structure to be painted = =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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After doing research, answering question, 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. 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.

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REFLECTIONWhile you were conducting the project, what have you learnt? What moral values

did you practise? Represent your opinions or feelings creatively through usage of

symbols, illustrations, drawings or even in a song.

When things go wrong,as they sometimes will,When the road you’re tudging seems all uphill,When the funds are low and the dept are high,And you want to smile but you have to sigh,When care is pressing you down a bit,Rest,if you must, but DON’T QUIT!!

Success is failure turned inside out,The silver tint of the cloud of doubt,And you never can tell how close you’re, It may be near when it seems a far.

So,stick to the fight,When you are hardest hit,It’s when things go wrong,THAT I MUST NOT QUIT!!!!!It’s all about you add math……Love you, my dear…..