additional mathematics folio form 5

8/6/2019 Additional Mathematics Folio Form 5

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NAQIB BIN AHMAD BATROD

ADDITIONAL MATHEMATICS 2011

Additional

MathematicsProject Work 2

Written by:Naqib bin Ahmad Batrod

Class:5 DarussalamI.C Number:940505-08-5493





Contents

No. Question Page

1.

Acknowledge 3

Introduction of project 4

Introduction of integration 5

Definition of integration 6

History of integration 7

2. Part 1 8

Part 2 9

Part 3 12

Part 4 13

Part5 15

3. Further Exploration 18

4. Conclusion 20

5. Reflection 21





ACKNOWLEDGE

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 for guiding me and my friends

throughout this project. We had some difficulties in doing this task, but he

taught us patiently until we knew what to do. He 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.





INTRODUCTION OF ADDITIONAL

MATHEMATICS PROJECT WORK1/2011

The aims of carrying out this project work areto enable students to :

a)Apply mathematics to everyday situations

and appreciate the importance and the

beauty of mathematics in everyday lives

b)Improve problem-solving skills, thinking

skills , reasoning and mathematical

communication

c) to develop mathematical knowledge

through problem solving in a way that

increases students interest and confidence

d)Stimulate learning environment that

enhances effective learning inquiry-base and

teamwork

e)Develop mathematical knowledge in a way

which increase students interest andconfidence.





Introduction of integrationIn mathematics,integration is a technique of finding afunction g ( x) the derivative of which, Dg ( x), is equal to a given

function f ( x). This is indicated by the integral sign ³�,´ as in� f ( x), usually called the indefinite integral of the function. (Thesymbol dx is usually added, which merely identifies x as thevariable.) The definite integral, written

with a and b called the limits of integration, is equalto g (b) í g (a), where Dg ( x) = f ( x).Some antiderivatives can becalculated by merely recalling which function has a given

derivative, but the techniques of integration mostly involveclassifying the functions according to which types of manipulations will change the function into a form theantiderivative of which can be more easily recognized. For example, if one is familiar with derivatives, the function 1/( x +1) can be easily recognized as the derivative of loge( x + 1). Theantiderivative of ( x2 + x + 1)/( x + 1) cannot be so easilyrecognized, but if written as x( x + 1)/( x + 1) + 1/( x + 1) = x +

1/( x + 1), it then can be recognized as the derivative of x2

/2 +loge( x + 1). One useful aid for integration is the theoremknown as integration by parts. In symbols, the rule is� fDg = fg í � gDf. That is, if a function is the product of twoother functions, f and one that can be recognized as thederivative of some function g , then the original problem can besolved if one can integrate the product gDf. For example,if f = x, and Dg = cos x, then � x·cos x = x·sin x í

�sin x = x·sin x í cos x + C . Integrals are used to evaluate suchquantities as area, volume, work, and, in general, any quantitythat can be interpreted as the area under a curve.





Definition

The process of finding a function, given its derivative,

is called anti-differentiation (or integration). If F '

( x) = f ( x), we say F ( x) is an anti-derivative of f ( x).

Examples

F ( x) =cos x is an anti-derivative of sin x, and e x is ananti-derivative of e x.

Note that if F ( x) is an anti-derivative of f ( x) then F ( x) + c, where c is a constant (called the constant of

integration) is also an anti-derivative of F ( x), as thederivative of a constant function is 0. In fact they arethe only anti-derivatives of F ( x).

We write f ( x) dx = F ( x) + c.

if F '( x) = f ( x) . We call this the indefinite integralof f ( x) .

Thus in order to find the indefinite integral of afunction, you need to be familiar with the techniquesof differentiation.





HISTORY

Over 2000 years ago, Archimedes (287-212 BC) found formulas for the surface areas andvolumes of solids such as the sphere, the cone, and the paraboloid. His method of integration wasremarkably modern considering that he did not have algebra, the function concept, or even the

decimal representation of numbers.

Leibniz (1646-1716) and Newton (1642-1727) independently discovered calculus. Their key ideawas that differentiation and integration undo each other . Using this symbolic connection, theywere able to solve an enormous number of important problems in mathematics, physics, andastronomy.

Fourier (1768-1830) studied heat conduction with a series of trigonometric terms to representfunctions. Fourier series and integral transforms have applications today in fields as far apart asmedicine, linguistics, and music.

Gauss (1777-1855) made the first table of integrals, and with many others continued to applyintegrals in the mathematical and physical sciences. Cauchy (1789-1857) took integrals to thecomplex domain. Riemann (1826-1866) and Lebesgue (1875-1941) put definite integration on afirm logical foundation.

Liouville (1809-1882) created a framework for constructive integration by finding out whenindefinite integrals of elementary functions are again elementary functions. Hermite (1822-1901) found an algorithm for integrating rational functions. In the 1940s Ostrowski extended thisalgorithm to rational expressions involving the logarithm.

In the 20th century before computers, mathematicians developed the theory of integration andapplied it to write tables of integrals and integral transforms. Among these mathematicians were

Watson, Titchmarsh, Barnes, Mellin, Meijer, Grobner, Hofreiter, Erdelyi, Lewin, Luke, Magnus,Apelblat, Oberhettinger, Gradshteyn, Ryzhik, Exton, Srivastava, Prudnikov, Brychkov, andMarichev.

In 1969 Risch made the major breakthrough in algorithmic indefinite integration when he published his work on the general theory and practice of integrating elementary functions. Hisalgorithm does not automatically apply to all classes of elementary functions because at the heartof it there is a hard differential equation that needs to be solved. Efforts since then have beendirected at handling this equation algorithmically for various sets of elementary functions. Theseefforts have led to an increasingly complete algorithmization of the Risch scheme. In the 1980ssome progress was also made in extending his method to certain classes of special functions .

The capability for definite integration gained substantial power in M athematica, first released in1988. Comprehensiveness and accuracy have been given strong consideration in the developmentof M athematica and have been successfully accomplished in its integration code. Besides beingable to replicate most of the results from well-known collections of integrals (and to find scoresof mistakes and typographical errors in them), M athematica makes it possible to calculatecountless new integrals not included in any published handbook .





Part 1

Route 1.1 2.1 1.2 2.2 1.3 2.3 Distance 131km 24km 109km 307km 85km 104km

Bearing Goestonorth

Goes to east N73.3 º N27.9ºW N77.5ºE N78.7ºE

CoordinatesPossibleDangers

Coralreef

Shark,infestedwater

Coral,reef,sunkenship

Shark,infestedwater,sunkenship,thunderstorm

Giantoctopus

Giantoctopus,thunderstorm

Time For route 1(1.1,1.2,1.3)=55minutes 59secondsFor route 2(2.1,2.2,2.3)=1hour 31minutes36seconds

Judging from the possible dangers & possibilities of intruding into the preserved and conservation aresas and the time taken to reach the offshoreoil rig,route 1 is the recommended option





Part 2

a) Starting position

Vresultant=V boat+Vcurrent

=+

Vresultant=

=

=

Vcurrent=

v=36sin a

V=60sin a _________ v=36cos a-15 ______

From,,we get a=22.4º,v=22.55km/h

Time taken=

hour

=0.4375hour





b) From

Vresultant=

=

V boat=

Vcurrent=

Vresultant=V boat+Vcurrent

=

+

By using the similar concept as shown in step,B=54.6 º,v=29.915km/h

Time taken=

hour=20.3416





c) From

Vresultant=V boat+VcurrentVcurrent=

V boat=

Vresultant=

=

Similary,by working it out youself,C=20.3 º,v=22.548km/h

Time taken=

=0.48hour d) Time to reach the wind ±

farm=10.00a.m+26minutes15seconds+20minutes28seconds

=10:46:43+2hours+28minutes48seconds=13:15:31a.m





Part 3

a) P=cAu2

C=

=

=

=5.917

b)(1)E=

50000000=

=

=10000tt=5000seconds

(2)500000000=

=

=

=

=

t=1850.6seconds





Part 4

a)v=R 2h=

Vfull=1000000000

= R 2hR 2 (3000)=100000000

R 2

=

v= R 2h=

=

__________

3000metres=(10x365x24)hours

=

=

=

=

barrels per hour





b)

V=r 2h=(0.25)2h

=0.0625h

=0.0625_______

Vfull=(0.25)2(1) =0.0625

Tfull=(5x60)seconds

=

________

=

x

= 0.0625

=

=20cms-1





Part 5

Oil Reserves - Top 20 Nations (% of Global)

Saudi Arabia has 261,700,000,000 barrels (bbl) of oil, fully 25% of theworld's oil. The United States has 22,450,000,000 bbl.

The United States government recently declared Alberta's oil sands tobe 'proven oil reserves.' Consequently, the U.S. upgraded its global oil

estimates for Canada from five billions to 175 billion barrels. OnlySaudi Arabia has more oil. The U.S. ambassador to Canada has said

the United States needs this energy supply and has called for a morestreamlined regulatory process to encourage investment and facilitate

development.- CBC Television - the nature of things - when is enough enough





Oil Production & Consumption, Top 20 Nations by Production

(% of Global)

Here are the top 20 nations sorted by production, and their production

and consumption figures. Saudi Arabia produces the most at

8,711,000.00 bbl per day, and the United States consumes the mostat 19,650,000.00 bbl per day, a full 25% of the world's oil

consumption.





Exports & Imports

Here's export and imports for all the nations listed in the CIA World

Factbook, sorted alphabetically as having exports and imports.

Conspicuously missing is the United States, but I can tell you that weconsume 19,650,000.00 bbl per day, and produce 8,054,000.00,leaving a discrepancy of 11,596,000.00 bbl per day.

This compares to the European Union, which produces 3,244,000.00bbl per day and consumes 14,480,000.00 bbl per day for a

discrepancy of 11,236,000.00 per day. Basically, about the same.

World Oil Market and Oil Price Chronologies: 1970 - 2003





Further Exploration Petroleum engineers work in the technical profession that involves

extracting oil in increasinglydifficult situations as the world's oil fields are

found and depleted. Petroleum engineers searchthe world for reservoirs

containing oil or natural gas. Once these resources are

discovered, petroleum engineers work with geologists and other specialists

to understand the geologicformation and properties of the rock containing

the reservoir, determine the drilling methods to be used, and monitor

drilling and production operations.Low-end Salary:

$58,600/yr

Median Salary: $108,910/yr

High-end Salary:

$150,310/yr EDUCATION:

Engineers typically enter the occupation with a bachelors degree in

mathematics or anengineering specialty, but some basic research

positions may require a graduate degree. Mostengineering programs

involve a concentration of study in an engineering specialty, along

withcourses in both mathematics and the physical and life sciences.Engineers offering their servicesdirectly to the public must be licensed.

Continuing education to keep current with rapidlychanging technology

is important for engineers.

MATH REQUIRED:

College Algebra,Geometry, Trigonometry, Calculus I and II

Linear Algebra,Differential, Equations,Statistics

WHEN MATH IS USED: Improvements in mathematical computer modeling, materials and the

application of statistics, probability analysis, and new technologies like

horizontal drilling and enhanced oil recovery,have drastically improved the

toolbox of the petroleum engineer in recent decades.





POTENTIAL EMPLOYERS:

About 37 percent of engineering jobs are found in manufacturing industries

and another 28 percent in professional, scientific, and technical services,

primarily in architectural, engineering,and related services. Many engineersalso work in the construction, telecommunications, andwholesale trade

industries. Some engineers also work for Federal, State, and local

governmentsin highway and public works departments. Ultimately, the

type of engineer determines the typeof potential employer.

FACTS:

Engineering diplomas accounted for 12 of the 15 top-paying majors, with

petroleum engineeringearning the highest average starting salary of

$83,121.





Conclusion

I have done many researches throughout the

internet anddiscussing with a friend who havehelped me a lot in completing this project.

Through the completion of this project, I have

learned many skills and techniques. This project

really helps me to understand more about the uses

of progressions in our daily life.

This project also helped expose the techniques of

application of additional mathematics in real life

situations. While conducting this project, a lot of

information that I found.

Apart from that, this project encourages the student

to work together and share their knowledge. It is

also encourage student to gather information from

the internet, improve thinking skills and promote

effective mathematical communication.

Last but not least, I proposed this project should be

continue because it brings a lot of moral values to

the student and also test the students understanding

in Additional Mathematics.





R eflectionAfter spending countless hours,day and night to finish thisAdditional Mathematics Project,here is what I got to say:

TEAM WORK IS IMPORTANT BE HELPFUL

ALWAYS READY TO LEARN NEW THINGS BE A HARDWORKING STUDENT

BE PATIENT ALWAYS CONFIDENT





Doing this project makes me realize how

important Additional Mathematicsis.Also, completing this project makes

me realize how fun it is and likable is

Additional Mathematics.

I used to hate Additional Mathematics

It always makes me wonder why this subject is so difficult

I always tried to love every part of it

It always an absolute obstacle for me

Throughout day and night

I sacrificed my precious time to have fun

From..

Monday,Tuesday,Wednesday,Thursday,Friday

And even the weekend that I always looking forward to

1 28ve 980ADDITIONAL

MATHEMATICS(Cover the top part of the phrase 1 28ve 980From now on, I will do my best on every second that I will learn

Additional Mathematics.

additional mathematics folio form 5

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