8/6/2019 Add Math Folio 1123123

1/25

SMK Bandar SP

ADDITIONAL MATHEMATICS

PROJECTYEAR 2011

FORM5

NAME : SAW KUAN CHE

FORM : 5 Science 1

NO I/C : 940503-89-5011

TITLE : Integration and vector space

TEACHER : Mr. Chong Hooi Yong

8/6/2019 Add Math Folio 1123123

2/25

8/6/2019 Add Math Folio 1123123

3/25

CONTENTS

A. IntroductionIntroduction of projectIntroduction of integrationIntroduction of vector

B. Task SpecificationC. Problem SolvingC1.Procedure

Part1Part2Part3Part4

Part5C2.ConjectureD. Further Investigation

D1.Further ExplorationD2.Conclusion

E. Reflection

8/6/2019 Add Math Folio 1123123

4/25

A. IntroductionINTRODUCTION OF ADDITIONAL MATHEMATICS PROJECT WORK1/2011

The aims of carrying out this project work are to enable students to:a) Apply mathematics to everyday situations and appreciate the importance and the beauty of mathematics ineveryday livesb) Improve problem-solving skills, thinking skills, reasoning and mathematical communication

c) Develop positive attitude and personalities and instead mathematical values such as accuracy, confidence andsystematic reasoningd) Stimulate learning environment that enhances effective learning inquiry-base and teamworke) Develop mathematical knowledge in a way which increases students interest and confidence.

Introduction of integration

In mathematics, integration is a technique of finding a functiong(x) the derivative of which,Dg(x), is

equal to a given functionf(x).This is indicated by the integral sign , as in f(x), usually called

the indefinite integral of the function.(The symbol dx is usually added, which merely identifiesx as the

variable.) The definite integral, written

with a and b called the limits of integration, is equal tog(b) g(a), whereDg(x) =f(x).Someanti derivatives can be calculated by merely recalling which function has a given derivative, but thetechniques of integration mostly involve classifying the functions according to which types ofmanipulations will change the function into a form the anti derivative of which can be more easilyrecognized.For example, if one is familiar with derivatives, the function 1/(x + 1) can be easilyrecognized as the derivative of loge(x + 1).The ant derivative of(x

2 +x + 1)/(x + 1) cannot be so easilyrecognized, but if written asx(x + 1)/(x + 1) + 1/(x + 1) =x + 1/(x + 1), it then can be recognized as thederivative ofx2/2 + loge(x + 1). One useful aid for integration is the theorem known as integration byparts. In symbols, the rule is fDg=fg gDf. That is, if a function is the product of two otherfunctions,fand one that can be recognized as the derivative of some functiong, then the original problemcan be solved if one can integrate the productgDf. For example, iff=x, andDg= cosx, thenxcosx =xsinx sinx =xsinx cosx + C. Integrals are used to evaluate such quantities as area,volume, work, and, in general, any quantity that can be interpreted as the area under a curve.

8/6/2019 Add Math Folio 1123123

5/25

Definition

The process of finding a function, given its derivative, is called anti-differentiation (or integration). IfF'(x)=f(x), we say F(x) is an anti-derivative off(x).

Examples

F(x) =cosx is an anti-derivative of sinx, and ex is an anti-derivative of ex.

Note that ifF(x) is an anti-derivative off(x) then F(x) + c, where c is a constant (called the constant ofintegration) is also an anti-derivative ofF(x), as the derivative of a constant function is 0. In fact they arethe only anti-derivatives ofF(x).

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

IfF'(x) =f(x). We call this the indefinite integral off(x).

Thus in order to find the indefinite integral of a function, you need to be familiar with the techniques

of differentiation.

8/6/2019 Add Math Folio 1123123

6/25

HISTORY

Over 2000 years ago, Archimedes (287-212 BC) found formulas for the surface areas and volumes ofsolids such as the sphere, the cone, and the parabolic. His method of integration was remarkably modernconsidering that he did not have algebra, the function concept, or even the decimal representation ofnumbers.

Leibniz (1646-1716) and Newton (1642-1727) independently discovered calculus.Their key idea was thatdifferentiation and integration undo each other. Using this symbolic connection, they were able to solvean enormous number of important problems in mathematics, physics, and astronomy.

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

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

Liouville (1809-1882) created a framework for constructive integration by finding out when indefiniteintegrals of elementary functions are again elementary functions. Hermite (1822-1901) found analgorithm for integrating rational functions. In the 1940s Ostrowski extended this algorithm to rationalexpressions involving the logarithm.

In the 20th century before computers, mathematicians developed the theory of integration and applied it towrite 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, and Marichev.

In 1969 Risch made the major breakthrough in algorithmic indefinite integration when he published hiswork on the general theory and practice of integrating elementary functions. His algorithm does notautomatically apply to all classes of elementary functions because at the heart of it there is a harddifferential equation that needs to be solved. Efforts since then have been directed at handling thisequation algorithmically for various sets of elementary functions.These efforts have led to anincreasingly complete algorithmization of the Risch scheme. In the 1980s some progress was also made inextending his method to certain classes of special functions.

The capability for definite integration gained substantial power in Mathematics, first released in 1988.Comprehensiveness and accuracy have been given strong consideration in the developmentofMathematics and have been successfully accomplished in its integration code. Besides being able to

replicate most of the results from well-known collections of integrals (and to find scores of mistakes andtypographical errors in them), Mathematics makes it possible to calculate countless new integrals notincluded in any published handbook.

8/6/2019 Add Math Folio 1123123

7/25

Introduction of vector

First example: arrows in the plane

The concept of vector space will first be explained by describing two particular examples.The firstexample of a vector space consists of arrows in a fixed plane, starting at one fixed point.This is used inphysics to describe forces or velocities. Given any two such arrows, v and w, the parallelogram spanned

by these two arrows contains one diagonal arrow that starts at the origin, too.This new arrow is called thesum of the two arrows and is denoted v + w. Another operation that can be done with arrows is scaling:given any positive real number a, the arrow that has the same direction as v, but is dilated or shrunk bymultiplying its length by a, is called multiplication of v by a. It is denoted a v. When a is negative, a vis defined as the arrow pointing in the opposite direction, instead.

The following shows a few examples: if a = 2, the resulting vector a w has the same direction as w, butis stretched to the double length of w (right image below). Equivalently 2 w is the sum w + w. Moreover,(1) v = v has the opposite direction and the same length as v (blue vector pointing down in the rightimage).

Second example: ordered pairs of numbers

A second key example of a vector space is provided by pairs of real numbers x and y.(The order of thecomponents x and y is significant, so such a pair is also called an ordered pair.) Such a pair is written as (x,y).The sum of two such pairs and multiplication of a pair with a number is defined as follows:

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)

and

a (x, y) = (ax, ay).

8/6/2019 Add Math Folio 1123123

8/25

Definition

A vector space over a field F is a set V together with two binary operators that satisfy 8 axioms listedbelow. Elements of V are called vectors. Elements ofF are called scalars. In this article, vectors aredifferentiated from scalars by boldface.[nb 1] In the two examples above, our set consists of the planararrows with fixed starting point and of pairs of real numbers, respectively, while our field is the realnumbers.The first operation, vector addition, takes any two vectors v and w and assigns to them a third

vector which is commonly written as v + w, and called the sum of these two vectors.The secondoperation takes any scalar a and any vector v and gives another vector av. In view of the first example,where the multiplication is done by rescaling the vector v by a scalar a, the multiplication is called scalarmultiplication of v by a.

History

Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-dimensional space. Around 1636, Descartes and Fermat founded analytic geometry by equating solutionsto an equation of two variables with points on a plane curve.To achieve geometric solutions withoutusing coordinates, Bolzano introduced, in 1804, certain operations on points, lines and planes, which are

predecessors of vectors.This work was made use of in the conception of barycentric coordinates byMbius in 1827.The foundation of the definition of vectors was Bellavitis' notion of the bipoint, anoriented segment one of whose ends is the origin and the other one a target. Vectors were reconsideredwith the presentation of complex numbers by Argand and Hamilton and the inception of quaternions andbiquaternions by the latter.They are elements in R2, R4, and R8; treating those using linear combinationsgoes back to Laguerre in 1867, who also defined systems of linear equations.

In 1857, Cayley introduced the matrix notation which allows for a harmonization and simplification oflinear maps. Around the same time, Grassmann studied the barycentric calculus initiated by Mbius. Heenvisaged sets of abstract objects endowed with operations. In his work, the concepts of linearindependence and dimension, as well as scalar products are present. Actually Grassmann's 1844 work

exceeds the framework of vector spaces, since his considering multiplication, too, led him to what aretoday called algebras.Peano was the first to give the modern definition of vector spaces and linear mapsin 1888.

An important development of vector spaces is due to the construction of function spaces by Lebesgue.This was later formalized by Banach and Hilbert, around 1920. At that time, algebra and the new field offunctional analysis began to interact, notably with key concepts such as spaces of p-integrable functionsand Hilbert spaces. Vector spaces, including infinite-dimensional ones, then became a firmly establishednotion, and many mathematical branches started making use of this concept.

8/6/2019 Add Math Folio 1123123

9/25

8/6/2019 Add Math Folio 1123123

10/25

C. Problem Solving

Part 1

Route 0.1 0.2 0.3

Distance 10km 104km 117kmBearing N53.1 E N11.3 E N56.3ECoordinates

Possibledangers

Coral reef Coral reef,sunken ship

Giant octopus

Time For route 0(0.1,0.2,0.3) = 53minutes 10seconds

8/6/2019 Add Math Folio 1123123

11/25

Route 1.1 1.2 1.3Distance 13km 109km 85kmBearing N0.0 N73.3 E N77.5ECoordinates

Possibledangers

Coral reef Coral reef,sunken ship

Giant octopus

Time For route 1(1.1,1.2,1.3) = 55minutes 59seconds

8/6/2019 Add Math Folio 1123123

12/25

Route 2.1 2.2 2.3Distance 24km 307km 104kmBearing N90.0 E N27.9W N78.7ECoordinates

Possibledangers SharkInfestedwater

Shark infestedwater, sunken ship,thunderstorm

Giant octopus,thunderstorm

Time For route 2(2.1,2.2,2.3) = 1hour31minutes36seconds

Judging from the possible dangers & possibilities of intruding into the preserved andconservation areas and the time taken to reach the offshore oil rig, route 0 is therecommended option.

8/6/2019 Add Math Folio 1123123

13/25

Part 2

a)Starting position

Vresultant=Vboat+Vcurrent

=

+

Vresultant= =

=

Vcurrent= v=36sin aV=60sin a _________v=36cos a-15______

From , , we get a=22.4,v=22.855km/hTime taken= hour

=0.4375hour

8/6/2019 Add Math Folio 1123123

14/25

b)From

Vresultant= =

Vboat=

Vcurrent=

Vresultant=Vboat+Vcurrent

=+

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

Time taken= hour=20.3416

8/6/2019 Add Math Folio 1123123

15/25

c)From

Vresultant=Vboat+VcurrentVcurrent= Vboat=Vresultant=

= Similarly, by working it out yourself,C=20.3,v=22.548km/h

Time taken= =0.48hour

d)Time to reach the windfarm=10.00a.m+26minutes15seconds+20minutes28seconds

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

8/6/2019 Add Math Folio 1123123

16/25

Part 3

a) P=cAu2C=

= =

=5.917

b) (1)E= 50000000=

= =10000t

t=5000seconds

(2)500000000=

=

=

=

=

t=1850.6seconds

8/6/2019 Add Math Folio 1123123

17/25

Part 4

a) v=R2h

=

Vfull=1000000000= R2h

R2(3000) =100000000R2=

v= R2h=

=

__________3000metres= (10x365x24) hours

=

=

= = barrels per hour

8/6/2019 Add Math Folio 1123123

18/25

b)

V=r2h= (0.25)2h=0.0625h

=0.0625_______Vfull= (0.25)

2(1)=0.0625

Tfull= (5x60) seconds

= ________

=

x

= 0.0625

=

=20cms-1

8/6/2019 Add Math Folio 1123123

19/25

Part 5

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. SaudiArabia produces the most at 8,711,000.00 bbl per day, and the United States consumes the most at

19,650,000.00 bbl per day, a full 25% of the world's oil consumption.

8/6/2019 Add Math Folio 1123123

20/25

Oil Exports & Imports

Here's export and imports for all the nations listed in the CIA World Fact book, sorted alphabetically ashaving exports and imports.

Conspicuously missing is the United States, but I can tell you that we consume 19,650,000.00 bbl per day,and produce 8,054,000.00, leaving a discrepancy of11,596,000.00 bbl per day.

This compares to the European Union, which produces 3,244,000.00 bbl per day and consumes14,480,000.00 bbl per day for a discrepancy of11,236,000.00 per day. Basically, about the same.

8/6/2019 Add Math Folio 1123123

21/25

Oil Reserves - Top 20 Nations (% of Global)

Saudi Arabia has 261,700,000,000 barrels (bbl) of oil, fully 25% of the world's oil.The United States has22,450,000,000 bbl.

The United States government recently declared Alberta's oil sands to be 'proven oil reserves.'Consequently, the U.S. upgraded its global oil estimates forCanada from five billions to 175 billionbarrels. Only Saudi Arabia has more oil.The U.S. ambassador to Canada has said the United States needsthis energy supply and has called for a more streamlined regulatory process to encourage investment andfacilitate development.- CBCTelevision - the nature of things - when is enough

8/6/2019 Add Math Folio 1123123

22/25

World Oil Market and Oil Price Chronologies: 1970 - 2003

From the Energy Information Administration.

Petroleum engineers work in the technical profession that involves extracting oil in increasingly difficultsituations as the world's oil fields are found and depleted.Petroleum engineers search the world forreservoirs containing oil or natural gas. Once these resources are discovered, petroleum engineers workwith geologists and other specialists to understand the geologic formation and properties of the rockcontaining the reservoir, determine the drilling methods to be used, and monitor drilling and productionoperations.

C3. Conjecture

My conjecture is that the suggested route is far more safe than the 2 other possible route.Thesuggested has the shortest distance among 3 routes and it takes the shortest time to travel from startingposition to wind farm and oil rig.

8/6/2019 Add Math Folio 1123123

23/25

D. Further Investigation

D1. Further ExplorationPetroleum engineers work in the technical profession that involves extracting oil in increasingly difficultsituations as the world's oil fields are found and depleted.Petroleum engineers search the world forreservoirs containing oil or natural gas. Once these resources are discovered, petroleum engineers workwith geologists and other specialists to understand the geologic formation and properties of the

rock containing the reservoir, determine the drilling methods to be used, and monitor drilling andproduction 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 an engineeringspecialty, but some basic research positions may require a graduate degree. Most engineering programsinvolve a concentration of study in an engineering specialty, along with courses in both mathematics andthe physical and life sciences. Engineers offering their services directly to the public must be licensed.Continuing education to keep current with rapidly changing 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, probabilityanalysis, and new technologies like horizontal drilling and enhanced oil recovery, have drasticallyimproved 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 inprofessional, scientific, and technical services, primarily in architectural, engineering, and relatedservices. Many engineers also work in the construction, telecommunications, and whole sale trade

industries. Some engineers also work forFederal, State, and local governments in highway and publicworks departments. Ultimately, the type of engineer determines the type of potential employer.

FACTS:

Engineering diplomas accounted for12 of the 15 top-paying majors, with petroleum engineering earningthe highest average starting salary of $83,121.

8/6/2019 Add Math Folio 1123123

24/25

D2. Conclusion

I have done many researches throughout the internet and discussing witha friend who have helped me a lot in completing this project.Through thecompletion of this project, I have learned many skills and techniques.This

project really helps me to understand more about the uses of progressions inour daily life.

This project also helped expose the techniques of application ofadditional mathematics in real life situations. While conducting this project, alot of information that I found .Apart from that, this project encourages thestudent to work together and share their knowledge. It is also encourage studentto gather information from the internet, improve thinking skills and promoteeffective mathematical communication. Last but not least, I proposed this

project should be continue because it brings a lot of moral values to the studentand also test the students understanding in Additional Mathematics.

8/6/2019 Add Math Folio 1123123

25/25

E.ReflectionWhile I was conducting the project, I learnt that flexible and critical thinking is very

important in solving mathematical problems or even in dealing with daily life.To obtainsatisfactory results, I strived to think out of the box as much as possible and to considersomething in a way which ordinary people would not even have considered about .

I learnt that there are many choices and alternatives in our life.There are manydifferent solutions to solve the same question. I learnt a lesson that I just have to make my

boldest step forward and I will able to face whatever challenges in my life.Last, I learnt to be independent and persevere on whatever task undertaken. I did my

best to think rationally and logically to complete this additional mathematics project.