add math project 2014

Beholding

Beyond

CalculusName : Ahmad Amin bin Zainol

Abidin

Class : 506 Khawarizmi

Teacher : Cik Mariah bt Abdul

Rahman

Index

0 | Page

2014Beholding Beyond Calculus

Pag

e

Foreword 2

Objective 3

Introduction 4

Project Proposal 6

Part 1 7

Part 2 11

Part 3 15

Further Exploration 21

Conclusion 24

Reflection 25

Foreword

Firstly, I would like to say “Alhamdulillah” because of Him, and His Guidance, I was

able to finish the additional mathematic project with full requirement.

1 | Page


I would also love to express my gratitude to my mother, Mrs. Anida bt Abdul Rahim

who has always supported me in any way possible during and after the making of

this project.

I would also like to point it out that, without my additional mathematics teacher, Cik

Mariah binti Abdul Rahman I would not be able to complete or even do my project.

Her teachings and advices guide my classmates and I during the making of this

project and all year long.

Also thanks to my siblings and my friends especially my team, Nur Faris bin Nazli

and Izzudin bin Zainal for the support, ideas and for all the time of burning the

midnight oil for the sole purpose.

Objective

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

involving area under the graph, integration and real-life situation.

2 | Page


II. To improve cognitive skills such as memories, thinking skills and concept

application.

III. To promote effective mathematical communication.

IV. To develop mathematical knowledge through problem solving in a way that

increase confidence and interest.

V. To use the language of mathematics to express and deliver mathematical ideas

precisely.

VI. To provide learning environment that stimulates and enhance effective learning

VII. To develop positive attitude towards mathematics

VIII. To enhance the communication skills during group discussion and ideas

interpretation

Introduction

Calculus is the mathematical 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. It has two major branches, differential

calculus (concerning rates of change and slopes of curves), and integral

3 | Page


calculus (concerning accumulation of quantities and the areas under and between

curves); these two branches are related to each other by the fundamental theorem of

calculus. Both branches make use of the fundamental notions of

convergence of infinite sequences and infinite series to a well-defined limit. Generally

considered to have been founded in the 17th century by Isaac Newton and Gottfried

Leibniz,today calculus has wide spread uses

in science , engineering and economics and can solve many problems

that algebra alone cannot.

Calculus is a part of modern mathematics education. 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 historically

been called "the calculus of infinitesimals", or "infinitesimal calculus". The word

"calculus" comes from Latin (calculus) and refers to a small stone used for counting.

More generally, calculus (plural calculi) refers to any method or system of calculation

guided by the symbolic manipulation of expressions. Some examples of other well-

known calculi are propositional calculus, calculus of variations, lambda calculus,

and process calculus.

The idea of calculus had been developed earlier in Egypt, Greece, China,

India, Iraq, Persia and Japan. The use of calculus began in Europe, during the 17th

century, when Isaac Newton and Gottfried Wilheim Leibniz built on the work of earlier

mathematics to introduce the basic principles. The development of calculus was built

on earlier concepts of instantaneous motion and area under the curve.

4 | Page


Application of differential calculus include computations involving velocity and

acceleration, the slope of curve and optimization. Applications of integral calculus

include computations involving area, volume, arc length, centre of mass, work and

pressure. Calculus is also used to gain a more precise understanding of the nature of

space, time and motion. And this project will unravel more about a pioneer and the

application of calculus in everyday life.

Project’s Proposal

In this Project, I will present four findings which is,

1. A biography of a pioneer in calculus.

2. Problems involving a velocity-time graph and its solutions.

5 | Page


3. Problems involving integration and its calculations.

4. An example of the use of calculus in everyday life

For Part 1 which is a research on a pioneer in calculus, I choose one pioneer and

used the Internet as a source to gather information about his background, his life

story and his biography.

In part 2, a problem involving a velocity-time graph is given and I will find its

solution using as many mathematical solutions as possible. I will also create a story

to depict the velocity-time graph given.

In part 3, I was given problems involving integration and was asked to solve it.

I will use appropriate mathematical approach for this task. Also, I will do a compare

and contrast on ways to find the area under a curve and then relate it with the

concept of integration.

Lastly in ‘Further Exploration’, I will demonstrate an example of the use of

calculus (integration) in daily life.

Part 1

Choose one pioneer of modern calculus that you like and write about his

background history. Hence present your findings using one or more i-Think maps.

For this project, I search the net for the pioneer of modern calculus and I found

out that the 17th century is the start of the modern calculus and some of the pioneers

6 | Page


are Isaac Barrow, René Descartes, Pierre de Fermat, Blaise Pascal, John Wallis,

Isaac Newton and Wilheim Leibniz. And that I realized that most of the pioneers that I

searched for, most of them concentrated on the Western Europe.

History has a way of focusing credit for any invention or discovery on one or

two individuals in one time and place. The truth is not as neat. Most of the time, this

doesn't matter, but in calculus, it does. When we convey the impression that Newton

and Leibniz created calculus out of whole cloth, we do ourselves a disservice. We

present mathematicians as creatures of an entirely different level of mental ability.

Newton and Leibniz were brilliant, but not even they were capable of inventing or

discovering calculus.

We also miss out on some great stories. The body of mathematics we know

as calculus developed over many centuries in many different parts of the world, not

just Western Europe but also ancient Greece, the Middle East, India, China, and

Japan. Newton and Leibniz drew on a vast body of knowledge about topics in both

differential and integral calculus. The subject would continue to evolve and develop

long after their deaths. What marks Newton and Leibniz is that they were to the first

to state, understand, and effectively use the Fundamental Theorem of Calculus. Use

it effectively they certainly did. No two people have moved our understanding of

calculus as far or as fast. But the problems that we study in calculus -- areas and

volumes, related rates, position/velocity/acceleration, infinite series, differential

equations -- had been solved before Newton or Leibniz was born.

The expression of these solutions was awkward and progress was painfully

slow. It took some 1,250 years to move from the integral of a quadratic to that of a

7 | Page


fourth degree polynomial. But awareness of this struggle can be a useful reminder for

us. The grand sweeping results that solve so many problems so easily (integration of

a polynomial being a prime example) hide a long conceptual struggle. When we jump

too fast to the magical algorithm, when we fail to acknowledge the effort that went

into its creation, we risk dragging ourselves past that conceptual understanding.

So instead of choosing one of the Western European pioneers, I took the

liberty to make a project about the famous well debated pioneer of the Middle East,

Alhazen.

Abu Ali al-Hasan ibn al-Haytham was one of the great Arab mathematicians.

He was born in Basra, Persia, now in southeastern Iraq. Sometime after 996, he

moved to Cairo, Egypt where he became associated with the University of Al-Azhar,

founded in 970. He was prolific, writing over 90 books, and is most famous for his

work in astronomy and optics. His interest in mathematics ranged over algebra,

calculus, geometry, and number theory. I focus on him because he is the first person

I know of to have integrated a fourth-degree polynomial. And that he has a very

interesting way on his journey as mathematician.

His journey begins,

8 | Page


9 | Page

Born c. 965 in Basra, which was then part of the Buyid

emirate, to an Arab family, he lived mainly in Cairo, Egypt,

During the Islamic Golden Age, Basra was a "key beginning of learning", and he was educated there

and in Baghdad, the capital of the Abbasid Caliphate, and the focus of the "high point of

Islamic civilization". During his time in Basra, he trained for government work and became Minister

for the area.

One account of his career has him called to Egypt by Al-Hakim bi-Amr

Allah, ruler of the Fatimid Caliphate, to regulate the flooding of the Nile, a

task requiring an early attempt at building a dam at the present site of

the Aswan Dam.

After deciding the scheme was impractical and fearing the caliph's anger, he feigned

madness. He was kept under house arrest from 1011 until al-Hakim's death in

1021

During this time, he wrote his influential Book of Optics

After his house arrest ended, he wrote scores of other treatises

on physics, astronomyand mathematics

He later traveled to Islamic Spain. During this period, he had ample time

for his scientific pursuits, which included optics, mathematics,

physics, medicine, and practical experiments.

Some biographers have claimed that Alhazen fled to Syria, ventured into Baghdad later in his life, or was in Basra when he pretended to be

insane. In any case, he was in Egypt by 1038

Accounts state that during his time in Cairo he lived near and

studied at the Al-Azhar Mosque.

He died at March 6, 1040 in Cairo, Egypt


Another reason that I choose Alhazen is because I was fairly impressed and

attracted to some of his characteristics and his personality as a mathematician and

one of the credible knowledgeable scholars at the time.

10 | Page


Charachteristic of Alhazen

Perceptive

Time-efficient

Seeing good things in almost

everything

Able to make and do

unfamous decisions

Appreciate knowledge in

any forms from any

places

Part 2

A car travels a road and its velocity-time function is illustrated in Diagram 1. The

straight line PQ is parallel to the straight line RS.

(a) From the graph, find

(i) the acceleration of the car in the first hour.

Based on the graph the equation of the function for the first hour

isv=60 t+20. Since the acceleration of a velocity time graph is the

gradient of the straight line, the acceleration is60 kmh−2.

v=60 t+20

y=m( x)+c

m=60

11 | Page


m=a=60 kmh−2

(ii) The average speed of the car in the first two hours.

Average speed=Total distancetravelledTime

Average speed=110 km2 h

Average speed=55 kmh−1

(b) What is the significance of the position of the graph

(i) above the t-axis?

The position of the graph above the t-axis indicates that the car is

moving away from its initial position or its displacement. The slope or

the gradient still shows the acceleration of the car. Positive gradient

shows acceleration, zero gradients means zero acceleration or constant

velocity and negative gradient displays that the car is decelerating or

slowing down.

(ii) below the t-axis?

The position of the graph below the t-axis is the opposite of that

above the t-axis in terms of displacement which indicates that the car is

moving toward its initial position or its displacement. In other words, the

car is turning back. The slope or the gradient shows the opposite

12 | Page


acceleration of the car as in the position above the t-axis. Positive

gradient shows deceleration, zero gradients means zero acceleration or

constant velocity and negative gradient displays that the car is

accelerating

(c) Using two different methods, find the total distance travelled by the car.

(i) Area under the graph

= [ 12

(80 + 20) + 12

(0.5 + 1)(80) ] +│[ 12

(0.5)(-80) + 12

(0.5 + 1)(-80) ]│

= 110 + │(-80)│

= 190 km

(ii) Integration

= ∫0

1

60 t +20+ ∫1

1.5

80 + ∫1.5

2

−160 t+320 + │( ∫2.5

3

−160 t+400 ) +

(∫3

3.5

−80) + ( ∫3.5

4

160 t−640)│

= [30t 2+ 20t¿01 + [80t¿1

1.5 + [-80t 2+320t¿1.52 +│[-80t 2+ 400t¿2.5

3 +[-80t¿33.5 + [80t 2-

640t¿3.54 │

= 50 + 40 + 20 + │(-20) + (-40) + (-20)│

= 110 km + 80 km

= 190 km

13 | Page


Note: The area below the x-axis is added the modulus to get the total

distance travelled.

(d) Based on the above graph, write an interesting story of the journey not

more than 100 words.

A private spy, Sean King was going to a place where he taught a criminal

is going to be there. He planned on catching the culprit red-handed. As he was

fairly late, he speed his car at 60kmh−2 for an hour from his house. Upon reaching

a speed at 80kmh−1 he kept his pace for half an hour because he taught he will be

there soon. Sure enough, he saw the grey tower looming overhead and he slow

down his car at -160kmh−2 for half an hour until he stopped.

When he was inspecting the surrounding, without notice a thug came out t

beat him. He knew instantly that this was a trap. He take down the man and

dashed to his car to make a quick escape. He was delayed at the tower for half

an hour.

He sped his car at 160kmh−2 as a precaution if the thug is after him. After

half an hour later, he steadied his speed at 80kmh−1 for another half an hour.

When he almost reached his house, he slowed down at -160kmh−2 for half an hour

until he finally reached his house.

14 | Page


PART 3

Diagram 2 shows a parabolic satellite disc which is symmetrical at the y-axis. Given

that the diameter of the disc is 8 m and the depth is 1 m.

(a) Find the equation of the curve y = f(x).

y = a(x - p)2 + q

Minimum point = (0 , 4)

y = a(x - 0)2 + 4

y = ax2 + 4

At point = (4 ,5)

5 = a(4)2 + 4

5 – 4 = 16a

15 | Page


1

16 = a

∴ f(x) = 1

16x2 + 4

(b) To find the approximate area under a curve, we can divide the region into

several vertical strips, then we add up the areas of all the strips.

Using a scientific calculator or any suitable computer software, estimate

the area bounded by the curve y = f(x) at (a), the x-axis, x = 0 and x = 4.

(i)

16 | Page


Diagram 3 (i)

0 1 2 3 4x

y

y = f(x)

x-coordinate y-coordinate width area (m2)

0 4 0.5 2

0.5 4.015625 0.5 2.007813

1 4.0625 0.5 2.03125

1.5 4.140625 0.5 2.070313

2 4.25 0.5 2.125

2.5 4.390625 0.5 2.195313

3 4.5625 0.5 2.28125

3.5 4.765625 0.5 2.382813

Total 17.09375

(ii)

x-coordinate y-coordinate width area(m2)

0.5 4.015625 0.5 2.0078125

1 4.0625 0.5 2.03125

1.5 4.140625 0.5 2.0703125

2 4.25 0.5 2.125

2.5 4.390625 0.5 2.1953125

3 4.5625 0.5 2.28125

3.5 4.765625 0.5 2.3828125

4 5 0.5 2.5

Total Area 17.59375

17 | Page


0 1 2 3 4x

yy = f(x)

Diagram 3 (ii)

(iii)

(c) (i) Calculate the area under the curve using integration.

= ∫0

41

16x2+4 dx

= [ 1

48 x3+4x¿0

4

= 1.333 + 16

= 17.333 m2

18 | Page


Diagram 3(iii)

0 1 2 3 4x

y

y = f(x)

x-coordinate y-coordinate width area(m2)

0.5 4.015625 1 4.015625

1.5 4.140625 1 4.140625

2.5 4.390625 1 4.390625

3.5 4.765625 1 4.765625

Total Area 17.3125

(c) (ii) Compare your answer in c (i) with the values obtained in (b).

Hence, discuss which diagram gives the best approximate area.

Based on the calculation in (b);

Diagram 3 (i) = 17.09375

Diagram 3 (ii) = 17.59375

Diagram 3 (iii) = 17.3125

Since the area by integration is 17.333m2, the best diagram which gives the

best approximate area is Diagram 3 (iii)

(c) (iii) Explain how you can improve the value in c (ii).

I observed that the closer the value of x to zero, the more accurate the

calculation become. So, some of the ways to improve the value in (c) (ii) is to

lesser the value of the width of the strips which also means to add more strips

to the area under the graph.

19 | Page


(d) Calculate the volume of the satellite disc.

The volume of the satellite disc is;

y = 1

16 x2 + 4

16y = x2 + 64

16y – 64 = x2

volume =𝛑 ∫b

a

x2 dy

=𝛑 ∫4

5

¿¿ dy

=𝛑 ∫4

5

16 y−64 dy

=𝛑 [ 8y2- 64y ¿45

=𝛑 [ [ 8(25) – 64(5)] – [ 8(16) – 64(4)] ]

= 8𝛑m3 or =2517

m3

20 | Page


FURTHER EXPLORATION

A gold ring in Diagram 4(a) has the same volume as the solid of revolution obtained

when the shaded region in Diagram 4 (b) is rotated 360o about the x-axis.

Find

(a) the volume of gold needed,

y = 1.2 – 5x2

y2 = (1.2 – 5x2)2

= 1.44 + 25x4 – 12x2

21 | Page


Diagram 4(a)

0 x

y

0.2 -0.2

f ( x )=1 .2−5 x2

Diagram 4(b)

0x

y

∴ volume = π ∫−0.2

0.2

y2 dx

= 227

∫−0.2

0.2

1.44+25 x4−12 x2

= 1.619

(b) the cost of gold needed for the ring.

(Gold density is 19.3 gcm-3. The price of gold can be obtained from the

goldsmith)

Density = Mass

Volume

19.3 = Mass1.619

Mass = 19.3 ×1.1619

= 31.25g

On 29 May 2014, 1 gram of gold worth RM 155.00

∴ Cost of gold needed for the ring = Mass of gold × RM 155

= 31.25g × RM 155

= RM 4843.75

22 | Page


Conclusion

Throughout this Project, I found that;

In Part 1 which is a research on a pioneer in calculus, Alhazen was indeed a

pioneer in the field of calculus. I was amazed by how he managed to master not only

the field of calculus but astronomy and optics as well. Though he is well known for

his contribution by the “Book of Optics”, he was the first person to have integrated a

fourth-degree polynomial. It was amazing how calculus had developed even before

Newton or Leibniz had born.

In part 2, I have calculated the average speed and acceleration based on a

velocity-time graph. I also had explained the significance of the position of the graph

above and below the t-axis. I had also write a story to portray the graph in a real-life

situation.

In part 3, I discovered the concept beyond integration. How it is hard to find an

area that includes curve than it is with straight lines. I have done a compare and

23 | Page


contrast to find the area under the curve using strip with three different methods. I

have also calculated the area and the volume generated by rotation of the curve.

Lastly in ‘Further Exploration’, I calculated the volume of the gold and using

the formula Density = Mass

Volume , I calculated the mass of the gold and thus find the

cost of the gold needed for the gold ring.

Reflection

It is hard to express what I felt and the experiences that I have went through

during the making of this Project. But for me, teamwork is one of the most important

things. I discovered that no one, not even Alhazen can do everything on his own.

There will be time where we will need someone else to have our back and in return

they will have ours’.

Professionalism is one essential thing that I’ve learnt while finishing this

project. True, that we will have to help other but there will a limit as beyond that point,

we might destroying instead of helping. Also I’ve learnt a lot about time-management.

When I handle this project, with all other assignments and stuff, this project help me

to prioritize priorities.

Lastly, knowledge is power. As I learn more and more about calculus and

integration, I gained self-confidence when I was asked to present my findings in front

of the class. I was also taught on how to handle questions and how to answer them.

Moreover, I was also showed how to discuss and compare answers to get the best

result.

24 | Page


All in all, this project does add some values in my life for me to overcome the

obstacles on this journey called life.

25 | Page


add math project 2014

Documents

calculus ofinfinitesimals

infinitesimal calculus

word calculus

mathematical ideas

additional mathematic

communication skills

cik mariah binti abdul

study of operations