Add Math Project Work 1 2010

8/9/2019 Add Math Project Work 1 2010

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Additional Mathematics Project Work 1

2010

Important ofCalculus

inour Daily Life

Name : Seum Hon Loon

IC Number : 931116 - 08 - 6471

Class : 5 Sains Iktisas

School : SMK Dato Bendahara C.M. Yusuf

Teacher : Pn. Rohidah Binti Ngah @ Salleh


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Contents

Title Page

Appreciation 2

Objectives 3

IntroductionHistory

4

6

Procedure and FindingsFurther exploration

9

12

Conclusion 23

Reflection 24


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Appreciation

First of all, I would like to thank my teacher, Pn. Rohidah Binti Ngah @ Salleh 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.

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.

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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ObjectivesThe aims of carrying out this project are :

To learn the way to apply the formulas of mathematics in our daily life accurately. To widen our prospective view on mathematics.

To improve our mastering skills.

To use the correct language to express our mathematical ideas properly.

To develop positive attitude towards mathematics

to improve our way of thinking


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Introduction

Integration is an important concept inmathematicsand, together with differentiation, is one of the two main operations incalculus. Given a function of a realvariablexand aninterval[a, b] of thereal line, the definite integral

is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of, thex-axis, and thevertical linesx= a andx= b.

The term integralmay also refer to the notion of antiderivative, a function Fwhose derivativeis the given function . In thiscase it is called an indefinite integral, while the integrals discussed in this article are termed definite integrals. Some authorsmaintain a distinction between antiderivatives and indefinite integrals.

The principles of integration were formulated independently by Isaac Newtonand Gottfried Leibniz in the late 17th century.Through the fundamental theorem of calculus, which they independently developed, integration is connected withdifferentiation: if is a continuous real-valued function defined on a closed interval[a, b], then, once an antiderivative Fofis known, the definite integral of over that interval is given by

Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. Arigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure whichapproximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenthcentury, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain overwhich the integration is performed has been generalised. Aline integralis defined for functions of two or three variables, andthe interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. In a surfaceintegral, the curve is replaced by a piece of a surfacein the three-dimensional space. Integrals of differential forms play afundamental role in moderndifferential geometry. These generalizations of integral first arose from the needs ofphysics, andthey play an important role in the formulation of many physical laws, notably those ofelectrodynamics. There are manymodern concepts of integration, among these, the most common is based on the abstract mathematical theory known asLebesgue integration, developed by Henri Lebesgue.
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Interval_(mathematics)http://en.wikipedia.org/wiki/Interval_(mathematics)http://en.wikipedia.org/wiki/Interval_(mathematics)http://en.wikipedia.org/wiki/Real_linehttp://en.wikipedia.org/wiki/Real_linehttp://en.wikipedia.org/wiki/Area_(geometry)http://en.wikipedia.org/wiki/Graph_of_a_functionhttp://en.wikipedia.org/wiki/Antiderivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Differential_calculushttp://en.wikipedia.org/wiki/Differential_calculushttp://en.wikipedia.org/wiki/Closed_intervalhttp://en.wikipedia.org/wiki/Closed_intervalhttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Curvilinearhttp://en.wikipedia.org/wiki/Curvilinearhttp://en.wikipedia.org/wiki/Curvilinearhttp://en.wikipedia.org/wiki/Line_integralhttp://en.wikipedia.org/wiki/Line_integralhttp://en.wikipedia.org/wiki/Line_integralhttp://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Surface_integralhttp://en.wikipedia.org/wiki/Surface_integralhttp://en.wikipedia.org/wiki/Surface_integralhttp://en.wikipedia.org/wiki/Surfacehttp://en.wikipedia.org/wiki/Surfacehttp://en.wikipedia.org/wiki/Differential_formhttp://en.wikipedia.org/wiki/Differential_geometryhttp://en.wikipedia.org/wiki/Differential_geometryhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Classical_electromagnetismhttp://en.wikipedia.org/wiki/Classical_electromagnetismhttp://en.wikipedia.org/wiki/Classical_electromagnetismhttp://en.wikipedia.org/wiki/Lebesgue_integrationhttp://en.wikipedia.org/wiki/Henri_Lebesguehttp://en.wikipedia.org/wiki/Henri_Lebesguehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Interval_(mathematics)http://en.wikipedia.org/wiki/Real_linehttp://en.wikipedia.org/wiki/Area_(geometry)http://en.wikipedia.org/wiki/Graph_of_a_functionhttp://en.wikipedia.org/wiki/Antiderivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Differential_calculushttp://en.wikipedia.org/wiki/Closed_intervalhttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Curvilinearhttp://en.wikipedia.org/wiki/Line_integralhttp://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Surface_integralhttp://en.wikipedia.org/wiki/Surface_integralhttp://en.wikipedia.org/wiki/Surfacehttp://en.wikipedia.org/wiki/Differential_formhttp://en.wikipedia.org/wiki/Differential_geometryhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Classical_electromagnetismhttp://en.wikipedia.org/wiki/Lebesgue_integrationhttp://en.wikipedia.org/wiki/Henri_Lebesguehttp://en.wikipedia.org/wiki/Mathematics


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History

The product ruleandchain rule, the notion ofhigher derivatives,Taylor series, andanalytical functionswere introduced by

Isaac Newtonin an idiosyncratic notation which he used to solve problems ofmathematical physics. In his publications,

Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent

geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of

planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a

cycloid, and many other problems discussed in his Principia Mathematica. In other work, he developed series expansions for

functions, including fractional and irrational powers, and it was clear that he understood the principles of theTaylor series.

These ideas were systematized into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused

ofplagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution was

to provide a clear set of rules for manipulating infinitesimal quantities, allowing the computation of second and higher

derivatives, and providing theproduct rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz paid

a lot of attention to the formalism he often spent days determining appropriate symbols for concepts.

Leibnizand Newtonare usually both credited with the invention of calculus. Newton was the first to apply calculus to general

physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz

provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating

polynomial series. By Newton's time, the fundamental theorem of calculus was known.
http://en.wikipedia.org/wiki/Product_rulehttp://en.wikipedia.org/wiki/Product_rulehttp://en.wikipedia.org/wiki/Chain_rulehttp://en.wikipedia.org/wiki/Chain_rulehttp://en.wikipedia.org/wiki/Chain_rulehttp://en.wikipedia.org/wiki/Higher_derivativehttp://en.wikipedia.org/wiki/Higher_derivativehttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Analytical_functionhttp://en.wikipedia.org/wiki/Analytical_functionhttp://en.wikipedia.org/wiki/Analytical_functionhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Mathematical_physicshttp://en.wikipedia.org/wiki/Mathematical_physicshttp://en.wikipedia.org/wiki/Mathematical_physicshttp://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematicahttp://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematicahttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibnizhttp://en.wikipedia.org/wiki/Plagiarismhttp://en.wikipedia.org/wiki/Product_rulehttp://en.wikipedia.org/wiki/Product_rulehttp://en.wikipedia.org/wiki/Chain_rulehttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Product_rulehttp://en.wikipedia.org/wiki/Chain_rulehttp://en.wikipedia.org/wiki/Higher_derivativehttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Analytical_functionhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Mathematical_physicshttp://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematicahttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibnizhttp://en.wikipedia.org/wiki/Plagiarismhttp://en.wikipedia.org/wiki/Product_rulehttp://en.wikipedia.org/wiki/Chain_rulehttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Physics


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INTEGRATION BY PARTS

One of very common mistake students usually do is

To convince yourself that it is a wrong formula, takef(x) =x andg(x)=1. Therefore, one may wonder what to do in this case.

A partial answer is given by what is called Integration by Parts. In order to understand this technique, recall the formula

which implies

Therefore if one of the two integrals and is easy to evaluate, we can use it to getthe other one. This is the main idea behind Integration by Parts. Let us give the practical steps how to perform this

technique:

1

Write the given integral

where you identify the two functions f(x) and g(x). Note that if you are given only one function, then set the second

one to be the constant function g(x)=1.

2

Introduce the intermediary functions u(x) and v(x) as:

Then you need to make one derivative (of f(x)) and one integration (of g(x)) to get

Note that at this step, you have the choice whether to differentiate f(x) or g(x). We will discuss this in little more

details later.

3

Use the formula

4

Take care of the new integral .

The first problem one faces when dealing with this technique is the choice that we encountered in Step 2. There is no

general rule to follow. It is truly a matter of experience. But we do suggest not to waste time thinking about the best choice,

just go for any choice and do the calculations. In order to appreciate whether your choice was the best one, go to Step 3: if

the new integral (you will be handling) is easier than the initial one, then your choice was a good one, otherwise go back to

Step 2 and make the switch. It is after many integrals that you will start to have a feeling for the right choice.


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In the above discussion, we only considered indefinite integrals. For the definite integral , we have two ways

to go:

1

Evaluate the indefinite integral which gives

2

Use the above steps describing Integration by Parts directly on the given definite integral. This is how it goes:(i)

Write down the given definite integral

where you identify the two functionsf(x) andg(x).(ii)

Introduce the intermediary functions u(x) and v(x) as:

Then you need to make one derivative (off(x)) and one integration (ofg(x)) to get

(iii)

Use the formula

(iv)

Take care of the new integral .

The following examples illustrate the most common cases in which you will be required to use Integration by Parts:


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EXAMPLEEvaluate

First let us point out that we have a definite integral. Therefore the final answer will be a number not a function of x! Since

the derivative or the integral of lead to the same function, it will not matter whether we do one operation or the other.

Therefore, we concentrate on the other function . Clearly, if we integrate we will increase the power. This suggests thatwe should differentiate and integrate . Hence

After integration and differentiation, we get

The integration by parts formula gives

It is clear that the new integral is not easily obtainable. Due to its similarity with the initial integral, we will

use integration by parts for a second time. The same discussion as before leads to

which implies

The integration by parts formula gives

Since , we get

which finally implies

Easy calculations give

From this example, try to remember that most of the time the integration by parts will not be enough to give you the answer

after one shot. You may need to do some extra work: another integration by parts or use other techniques,....


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Procedure and FindingsSolution:

(a)Function 1

Maximum point (0,4.5) and pass through point (2,4)

y=a(x-b)+ c b=0, c=4.5y=a(x-0)+ 4.5y=ax+ 4.5---(1)

Substitute (2,4) into (1)4=a(2)+ 4.5

4a= -0.5a= -0.125

y= -0.125x+ 4.5

Function 2

Maximum point (0, 0.5) and pass through point (2, 0)y=a(x-b)+ cb=0, c=0.5y=a(x-0)+ 0.5y=ax+ 0.5---(2) Substitute(2,0) into (2) 0=a(2)+ 0.5

4a= -0.5a= -0.125

y= -0.125x+ 0.5

Function 3


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Maximum point (2, 4.5) and pass through point (0, 4)y=a(x-b)+ c

b=2, c=4.5

y=a(x-2)+ 4.5---(3) Substitute

(0,4) into (3) 4=a(0-2)+ 4.5

4a= -0.5

a= -0.125

y= -0.125(x-2)+ 4.5


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(b)

Area to be painted

= Area of rectangle - Area under the curve= ??

Further Exploration

Solution:

(a)(i) Structure 1

Area = ??? (223m)

Volume = Area x Thickness=223m x 0.4 m

= 1615m

Cost = 1615m x RM840= RM896


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Structure 2

Area=Area of Rectangle-Area of Triangle

=1m x 4m-12 x 4m x 0.5m

=4m- 1m=3m

Volume =Area x Thickness=3m x 0.4m=1.2m7

Cost =1.2m x RM840=RM1008

Structure 3

Area=Area of Rectangle-Area of Trapezium=1m x 4m-(4m+1m) x 0.5m=4m- 54m=2.75m

Volume =Area x Thickness=2.75m x 0.4m=1.1m

Cost =1.1m x RM840=RM924


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Structure 4

Area=Area of Rectangle-Area of Trapezium=1m x 4m-(2m+4m)2 x 0.5m

=4m2- 1.5m2=2.5m2

Volume =Area x Thickness=2.5m2 x 0.4m=1m3

Cost=1m3 x RM840=RM840

Structure 4 will cost the minimum to construct, that is RM 840.

(ii) As the president of the Arts Club, I will decide Structure 4 as the shape of the gate

to be constructed. It is because Structure 4 will cost the minimum and it is easier

to be constructed compared to Structure 1 which is a curve.


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(b) (i)

k(m) Area to be painted(m)0.00 4 x 1- 0+42 x 0.5= 3

0.25 4 x 1- 0.25+42 x 0.5= 2.93750.50 4 x 1- 0.5+42 x 0.5= 2.875

0.75 4 x 1- 0.75+42 x 0.5= 2.81251.00 4 x 1- 1+42 x 0.5= 2.75

1.25 4 x 1- 1.25+42 x 0.5= 2.68751.50 4 x 1- 1.5+42 x 0.5= 2.625

1.75 4 x 1- 1.75+42 x 0.5= 2.56252.00 4 x 1- 2+42 x 0.5= 2.5

(ii) There is a pattern in the area to be painted.

The area to be painted decreases as the kincreases 0.25m and form a series of numbers:

3, 2.9375, 2.875, 2.8125, 2.75, 2.6875, 2.625, 2.5625, 2.5

We can see that the difference between each term and the next term is the same.

2.9375-3 = -0.06252.875- 2.9375 = -0.06252.8125- 2.875 = -0.06252.75- 2.8125 = -0.06252.6875- 2.75 = -0.06252.625- 2.6875 = -0.06252.5625- 2.625 = -0.06252.5- 2.5625 = -0.06257

We can deduce that this series of numbers is an Arithmetic Progression (AP), with a commondifference,

d= -0.0625

By using K= 0 m, the structure may be like this:

When the value ofKgradually increases, the area will gradually decreases.


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By using K= 0.25 m, the structure may be like this:

WhenKapproaches more to 4, the shape of concrete structure will be something like:

In conclusion, when kincreases 0.25m, the area to be painted decreases by -0.0625m

(c) The area of the concrete structure to be painted

Area of concrete structure to be painted 3-12m

The shape of the concrete structure will be a rectangle with length 4m and breadth 0.5m, whichmay look like this:


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Conclusion

As I doing this project, I notice that quadratic function and integration can be so close in our daily life.

There are many shape of gate outside there. Different shapes of the gate have different cost. From quadratic

function and integration, we can know area of the gate. From the area we can get volume og the gate. As the

result, we can know cost of the gate by times volume with RM840 (price for 1m)

After doing research, answering questions, drawing graphs and some problem solving, I saw that the

usage of calculus is important in daily life.It is not just widely used in science, economics but also in

engineering. In conclusion, calculus is a daily life nessecities. Without it, marvelous buildings cant be built,

human beings will not lead a luxurious life and many more. So, we should be thankful of the people who

contribute in the idea of calculus.

ReflectionAfter spending countless hours,days and night to finish this project and also sacrificing my time for chatting

and movies in this mid year holiday,there are several things that I can say...

Additional Mathematics...

From the day I born...

From the day I was able to holding pencil...

From the day I start learning...

And...

From the day I heard your name...

I always thought that you will be my greatest obstacle and rival in excelling in my life...

But after countless of hours...Countless of days...

Countless of nights...

After sacrificing my precious time just for you...

Sacrificing my play Time..

Sacrificing my Chatting...

Sacrificing my Facebook...

Sacrificing my internet...

Sacrifing my Anime...

Sacrificing my Movies...

I realized something really important in you...

I really love you...

You are my real friend

You my partner...

You are my soulmate...

I LOVE U ADDITIONAL MATHEMATIC
http://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Engineering

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