Additional

Mathematics

Project Work

2014

Name: Nabila Syuhada binti Mohd Kamal Azmy

IC no.: 970222-02-5690

Matrix no.: M000337

Class: 5K2

Instructor’s name: Puan Nurul Nadiah binti Md Lani

CONTENTS

Introduction 1

Part 1 2

Part 2 3

Part 3 5

Further Exploration 7

Reflection 8

INTRODUCTION

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 Wilhelm 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.

Applications 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.

The problem of finding the tangent to a curve has been studied by many

mathematicians since Archimedes explored the question in Antiquity. The first attempt at

determining the tangent to a curve that resembled the modern method of the Calculus came

from Gilles Persone de Roberval during the 1630's and 1640's. At nearly the same time as

Roberval was devising his method, Pierre de Fermat used the notion of maxima and the

infinitesimal to find the tangent to a curve. Some credit Fermat with discovering the

differential, but it was not until Leibniz and Newton rigorously defined their method of

tangents that a generalized technique became accepted.

The path to the development of the integral is a branching one, where similar

discoveries were made simultaneously by different people. The history of the technique that

is currently known as integration began with attempts to find the area underneath curves. The

foundations for the discovery of the integral were first laid by Cavalieri, an Italian

Mathematician, in around 1635. Cavalieri’s work centred around the observation that a curve

can be considered to be sketched by a moving point and an area to be sketched by a moving

line.

Brook

Taylor

PART 1

Born:

18 August

1685

Edmonton,

Middlesex,

England

Died:

29 December

1731

Somerset

House,

London,

England

Musician

and painter

St John's

College

Cambridge:

3 April 1703

Graduated

with an LL.B.

in 1709 First

mathematics

paper: 1708

Published in

1714

Solve problem

in Kepler's

second law of

planetary motion

Solve problem

of the centre of

oscillation of a

body

The Philosophical

Transactions of

the Royal Society

Secretary to

the Royal

Society:

1714

Methodus

incrementor

um directa

et inversa

(Book)

Linear

Perspective

(Book)

Invented

integration

by parts New branch of

maths: Calculus

of finite

differences

Discovered

Taylor's

expansion

14 January

1714 - 21

October

1718

PART 2

(a) (i)

(ii)

(b)

(i) Moving in positive direction or positive velocity.

(ii) Moving in negative direction or negative velocity.

(c)

First method: Graphical solution

Second method: Integration

(d)

The car is moving with an initial velocity of 20kmh-1

and accelerating constantly until it

reaches the final velocity of 80kmh-1

in the first hour. Then, the car is moving with a constant

velocity of 80kmh-1

for half an hour to reach point P. The car starts to decelerate constantly

for another half an hour and stops at point Q. The car rests for half an hour. At R, the car

takes a turn and accelerates constantly for half an hour until it reaches a velocity of 80kmh-1

at point S. Then, the car moves with the velocity of 80kmh-1

for half an hour and decelerates

constantly for half an hour until it stops exactly at the fourth hour.

PART 3

(a) (0, 4), (4, 5) and (-4, 5)

1.

2.

3.

Equation 2 – equation 3

Equation 2 – equation 1

(b)

(i) Area

(ii) Area

(iii) Area

(c) (i)

(ii) Diagram 3(iii) gives the best approximate area.

(iii) The value can be improvised by dividing the region into more vertical strips before

adding the area of all the strips. Thus, more accurate value of area under a curve will be

obtained.

(d)

FURTHER EXPLORATION

(a)

(b) Gold price

REFLECTION

All knowledge requires a lot of wise and diligent people to expand a certain principle, so do

calculus. Humans are created to complete and correct each other. Concepts in calculus help

us to understand and acknowledge situations in daily activities. For example, with the help of

a velocity graph, we can understand a journey accurately without long explanation. Also,

when conducting this project, I realized that concepts in calculus are widely implemented

throughout our lives indirectly. Therefore, I think knowledge of calculus is indeed very

important. Every person who is given opportunity to learn this subject should be grateful and

make full use of the knowledge by practicing it in daily lives. However, some people just do

not notice the importance of those concepts, similar to peoples described in Sophie

Doomknuckles’ poem, ‘See’.

You can't see them

but I can

They're everywhere

Your fingers just traced eight of them

Unnoticed by you

but not me

I count them every day

even though

you can't see them

Lastly, knowledge in this subject helps us to keep doing a lot of things in our lives. In my

opinion, Matthew Miklavcic’s poem, ‘Invincible’, metaphorically reflects the importance of

the subject.

I stand next to you,

Watching and learning.

I move things,

When you need them moved.

I tell you an idea,

When you are looking for one.

I help you with things,

When you need it,

But even this goes unnoticed.

I hold the door,

To let you in.

I give you my coat,

When you are cold.

I help you with problems,

You can’t figure out,

But this even goes unnoticed.

I know I ask for a lot,

But all I want,

Is to know,

I’m not invisible.