Download - projek addmath 2014

8/12/2019 projek addmath 2014

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ADDITIONAL

MATHEMATICS

PROJECT WORK 2014

Name : Mohammmed Huzaifah bin

M.Yusoff

I/C Number : 97072-29-5021

Class : 5 Cerdas

Teachers Name : Puan Siti Rafidah Bt. Mohd

Rejab

School : SMK Seksyen 4 Kota Damansara


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ACKNOWLEGMENT

First of all, I would like to say Alhamdulillah, for giving me the strength and health 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, which is the most needed for this project.

Internet, books, computers and all that. They also supported me and encouraged me to

complete this task so that I will not procrastinate in doing it.

Then I would like to thank my teacher, Puan Siti Rafidah Bt. Mohd Rejab 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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OBJECTIVES

The aims of carrying out this project work are :

To apply and adapt a variety of problem-solving strategies to solve problems. To improve thinking skills. To promote effective mathematical communication. To develop mathematical knowledge through problem solving in away that increases

students interest and confidence.

To use the language of mathematics to express mathematical ideas precisely. To provide learning environment that stimulates and enhances effective learning. To develop positive attitude towards mathematics

.


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INTRODUCTION

Calculus

Calculusis themathematical study of change, in the same way thatgeometry is the study of

shape andalgebra 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),

andintegral 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 ofinfinite

sequences andinfinite series to a well-definedlimit. Generally considered to have been

founded in the 17th century byIsaac Newton andGottfried Leibniz, today calculus has

widespread uses inscience,engineering andeconomics and can solve many problems

thatalgebra alone cannot.

Calculus is a part of modernmathematics education. A course in calculus is a gateway to

other, more advanced courses in mathematics devoted to the study offunctions and limits,

broadly calledmathematical analysis. Calculus has historically been called "the calculus

ofinfinitesimals", or "infinitesimal calculus". The word "calculus" comes

fromLatin (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 ofexpressions. Some examples of other well-known calculi

arepropositional calculus,calculus of variations,lambda calculus.
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Differential_calculushttp://en.wikipedia.org/wiki/Integral_calculushttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Convergence_(mathematics)http://en.wikipedia.org/wiki/Infinite_sequencehttp://en.wikipedia.org/wiki/Infinite_sequencehttp://en.wikipedia.org/wiki/Series_(mathematics)http://en.wikipedia.org/wiki/Limit_(mathematics)http://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Elementary_algebrahttp://en.wikipedia.org/wiki/Mathematics_educationhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Latinhttp://en.wiktionary.org/wiki/en:calculus#Latinhttp://en.wiktionary.org/wiki/en:calculus#Latinhttp://en.wiktionary.org/wiki/en:calculus#Latinhttp://en.wikipedia.org/wiki/Expression_(mathematics)http://en.wikipedia.org/wiki/Propositional_calculushttp://en.wikipedia.org/wiki/Calculus_of_variationshttp://en.wikipedia.org/wiki/Lambda_calculushttp://en.wikipedia.org/wiki/Lambda_calculushttp://en.wikipedia.org/wiki/Calculus_of_variationshttp://en.wikipedia.org/wiki/Propositional_calculushttp://en.wikipedia.org/wiki/Expression_(mathematics)http://en.wiktionary.org/wiki/en:calculus#Latinhttp://en.wikipedia.org/wiki/Latinhttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Mathematics_educationhttp://en.wikipedia.org/wiki/Elementary_algebrahttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Limit_(mathematics)http://en.wikipedia.org/wiki/Series_(mathematics)http://en.wikipedia.org/wiki/Infinite_sequencehttp://en.wikipedia.org/wiki/Infinite_sequencehttp://en.wikipedia.org/wiki/Convergence_(mathematics)http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Integral_calculushttp://en.wikipedia.org/wiki/Differential_calculushttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Mathematics


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Newton kept his discovery tohimself. However, enough was

known of his abilities to effect his

appointment in 1669 as Lucasian

Professor of Mathematics at the

University of Cambridge.

Newtonpublished a

detailed

exposition of his

fluxional method

in 1704

He generalized the methods that werebeing used to draw tangents to curves and

to calculate the area swept by curves, and

he recognized that the two procedures were

inverse operations. By joining them in

what he called the fluxional method,

Newton developed in the autumn of 1666 a

AlthoughNewton was its

inventor, he did

not introducecalculus into

European

mathematics.

Achievement of

Issac Newton incalculus

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.


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PART 2

A car travels along 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.

v = 60t + 20a == 60t +20

When t = 1, = 60 ms-2

v = 60(1) + 20

= 80

When t = 0,

v = 60(0) + 20

= 20

v = 60t + 20

v = -160t+320

P

Q R

S

v (km/h)

t (h)1.0 1.5 2.0 2.5 3.0 3.5 4.0

Diagram 1


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

Average speed =

Total distance = Area under the graph

Area of A = Area of BC =

()

= 30 = 60

Area of D = 1.0

= 20

Total distance = 30 + 60 + 20

=110

The average speed of the car in the first two hour =

= 55 km/h.

A B

D

C

(1.0 , 80) (1.5 , 80)

(2.0 , 0)

(0 , 20)

0 1.0 1.5 2.0


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Method 2

At region A, v = 60t + 20 At region D, equation PQ = v = -160t + 320

a = 60 mPQ= -160

s = mRS= -160= 50 yy1= m(x

x1)

Thus, area = 50 km v0 = -160(t2.5)

v = -160t + 400

At region B, v = 80 s = s = = -20

= 40 Thus, area = 20 km

Thus, area = 40 km

At region C, v = -160t + 320 At region E, when t = 3.0,

a = -160 v = -160t + 400

s = = -160(3.0) + 400= 20 = -80

Thus, area = 20 km s = = -40

Thus, area = 40 km

v = 80

v = -80

0 1.0 1.5 2.0 2.5 3.0 3.5 4.0

v = 60t +20

v = -160t +320

A B C

D E F

P

Q

R

S


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At region F, gradient = m =

= 160

yy1= m(xx1)

v0 = 160(t4.0)

v = 160t640

s = = -20

Thus, area = 20 km

Total distance travelled = Sum of all areas= 50 + 40 + 20 + 20 + 40 + 20

= 190 km

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

than 100 words.

Ramli was in his journey to join a convoy from Johor to Kelantan. On that day, Ramli

was late and drove his car accelerating from 20 km/h to 80 km/h. After the first hour, Ramli

found his convoys members that are moving together on the highway. He then followed

them with a constant velocity, 80 km/h for half an hour. The group then decided to take a rest

at any R&R, so they reduced their velocity for 30 minutes before they reached there. At that

moment ,realize Ramli that he forgot to bring his wallet so he headed home and decided to

turn back home to take his wallet.

He took the opposite way and drove directly to his home with increasing acceleration

from 0 km/h to 80 km/h. Unfortunately, there was a traffic jam that forces him to drive at a

constant velocity, 80 km/h for 30 minutes. He arrived his homel half and hour later with a

reduced velocity from 80 km/h to 0 km/h.


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PART 3

Diagram 2 shows a parabolic satellite disc which is symmetrical at the yaxis. Given that

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

Diagram 2

(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

xaxis,

= 4 yaxis, 4 + 1 = 5

At point = (4 ,5)

5 = a(4)2+ 4

54 = 16a

= a

f(x) = x2+ 4

y

8 m

1 m

4

0

y =f(x)

x


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(ii)

Area of region = (value ofxvalue of y) ( )

Based on the values of y obtained in the calculations at Diagram 3 (i),

Area of region A, = 0.5 4= 2

Area of region B, =() ()()= 2

Area of region C, =

( )

()()

= 2.01

Area of region D, = () ()()=2.04

Area of region E, = () ()()= 2.09

Area of region F, = () ()()

= 2.16

Area of region G, = () ()()= 2.23

Area of region H, = () ()()= 2.34

Total area under the curve = Sum of all areas= 2 + 2 + 2.01 + 2.04 + 2.09 + 2.16 + 2.23 + 2.34

= 16.87 m2

Diagram 3 (ii)

y =f(x)


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(iii)

Based on the values of y obtained in the calculations at Diagram 3 (i),

Area of region A = 0.5 4= 2

Area of region B, = ()+ ()()= 2.02

Area of region C, = ( ) ()()= 2.01

Area of region D, = ()+ ()()= 2.10

Area of region E, = () ()()= 2.09

Area of region F, = ()+ ()()

= 2.24

Area of region G, = () ()()= 2.23

Area of region H, = ()+ ()()= 2.34

Total area under the curve = Sum of all areas= 2 + 2.02 + 2.01 + 2.10 + 2.09 + 2.24 + 2.23 + 2.46= 17.15 m2

x0 0.5 1 1.5 2 2.5

A B C D E F G H

y

Diagram 3(iii)

Area of region = ( )+ ( )

y =f(x)


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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 360 about thex-axis.

Find

(a) the volume of gold needed.

y = 1.25x2

y2 = (1.25x2)2

= 1.44 + 25x412x2

Volume =

=

= 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 =

19.3 =

mass = 19.3 1.1619= 31.25g

On 29thMay 2014, 1g of gold costs RM 155.00

The cost of gold needed for the ring = mass of gold RM 155= 31.25 RM 155= RM 4843.75

y

x

0

Diagram 4 (a)

x

y

-0.2 0 0.2

Diagram 4 (b)

f(x)= 1.25x2


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Download - projek addmath 2014

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