esei teori terbaru fluxion and product rule

8/8/2019 Esei Teori Terbaru Fluxion and Product Rule

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1.0 HISTORY CONTRIBUTIONS BY SIR ISAAC NEWTON AND GOTTFRIED

WILHELM LEIBNIZ IN CALCULUS

Gottfried Wilhelm Leibniz and Isaac Newton were geniuses who lived quitedifferent lives and invented quite different versions of the infinitesimal calculus, each tosuit his own interests and purposes. Newton discovered his fundamental ideas in16641666, while a student at Cambridge University. During a good part of theseyears the University was closed due to the plague, and Newton worked at his familyhome in Woolsthorpe, Lincolnshire. However, his ideas were not published until 1687.Leibniz, in France and Germany, on the other hand, began his own breakthroughs in

1675, publishing in 1684. The importance of publication is illustrated by the fact thatscientific communication was still sufficiently uncoordinated that it was possible for thework of Newton and Leibniz to proceed independently for many years withoutreciprocal knowledge and input. Disputes about the priority of their discoveries ragedfor centuries, fed by nationalistic tendencies in England and Germany.

Newton made contributions to all branches of mathematics then studied, but isespecially famous for his solutions to the contemporary problems in analytical

geometry of drawing tangents to curves (differentiation) and defining areas boundedby curves (integration). Not only did Newton discover that these problems wereinverse to each other, but he discovered general methods of resolving problems of curvature, embraced in his "method of fluxions" and "inverse method of fluxions",respectively equivalent to Leibniz's later differential and integral calculus. Newtonused the term "fluxion" (from Latin meaning "flow") because he imagined a quantity"flowing" from one magnitude to another. Fluxions were expressed algebraically, asLeibniz's differentials were, but Newton made extensive use also (especially in thePrincipia ) of analogous geometrical arguments.

Fluxions were introduced in the middle of 1665, perhaps inspired by Barrow's lectureson motion of the previous year. The point of using motion to define fluxions probablywas to give a better foundation than the one based on infinitesimals. But infinitesimals


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were not excluded. He was quite explicit in using and accepting infinitesimals: InOctober 1666 he wrote:

He nc e I obse rv e. F ir st y't those te rm es e v e r vani sh w' c h ar e not mul t ipl yed by o, they

be ing y'e pr o poun ded eq ua t i on . Se c ond l y those te rm es al so vani sh in w' c h o i s of

mor e y' n one d ime ns i on , be cau se they ar e inf ini te l y l esse y' n those in w' c h o i s b u t of

one d ime ns i on . Th ir d l y y'e st ill r e maining te rm es, be ing d ivi ded by o w ill hav e y't fo rm

w' c h (...) they sho ul d h av e (...)

Here, he uses the word "infinitely" in a very uncritical way. In June 1669 he mentioned:

Nor am I afraid to talk of a unity in points or infinitely small lines in as much asg eo mete r s now c ons i de r pr o por t i ons in these wh il e us ing ind ivi s i b l e m ethods.

Thus he takes comfort in the other geometers' habits (talking of infinitely small lines),and uses that as an excuse for having these habits himself. As late as 1671--2, theinfinitesimals are still there: in the following, I will use the notation x* instead of Newton's dotted x . The moments of the fluent quantities (that is, their indefinitely smallparts, by addition of which they increase during each infinitely small period of time) areas their speeds of flow. We see clearly that at this point infinitely small quantitiesplayed an important part in Newton's method of fluxions.

In 1680, in hisGeo met ria C urvilin e a , Newton started to look at fluxions in a newway, in an attempt to avoid infinitesimals:

Those who h av e t ak e n the me a sur e of curvilin e ar f igur es h av e us uall y vi ewed the m a s

ma de up of inf ini te l y man y inf ini te l y-s mall par ts. I, in f ac t, sh all c ons i de r the m a s

g e ne ra ted by gr ow ing , arguing tha t they ar e gr e a te r , eq ual or l ess acc or d ing a s they

gr ow mor e sw i ft l y, eq uall y sw i ft l y o r mor e s l ow l y f r om the ir be ginning . And th i s

sw i ft ness of gr owth I sh all call the f lu x i on of a quan t i ty .

This is not different from his previous definitions. But earlier, he had used infinitelysmall quantities to find these fluxions. Now he tried to do without them:

F lu x i ons of q uan t i t i es ar e in the f ir st ra t i o of the ir na s c e n t par ts o r , wh a t ex ac t l y the

s am e i s, in the la st ra t i o of those par ts a s they vani sh by def lu x i on .


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Newton's motive seems clear: he wants to find a ``more geometrical'' - meaning morerigorous - method. But what are these "last sums and ratios of vanishing quantities"?Newton saw that this could be difficult, and tried to explain. However, it is difficult tounderstand Newton's point. The exact speed with which the body reaches its last

position has to be zero - otherwise it would continue beyond this last position.Newtonhad some idea of a limit concept here, but the difference between an idea and a fullyexplained and understood concept is large.

The theory of fluxions yielded the heuristic methods of the calculus. Those methodswere to be justified rigorously by the theory of ultimate ratios. The theory of infinitesimals was to abbreviate the rigorous proof, and Newton thought that he hadshown the abbreviations to be permissible. Rather than competing for the same

position, the three theories were designed for quite distinct tasks.

Gottfried Wilhelm Leibniz was the one who discover product rule. It was during thisperiod in Paris that Leibniz developed the basic features of his version of the calculus.In 1673 he was still struggling to develop a good notation for his calculus and his firstcalculations were clumsy. On 21 November 1675 he wrote a manuscript using the f ( x ) dx notation for the first time. In the same manuscript the product rule for differentiation is given. By autumn 1676 Leibniz discovered the familiar d ( x n) = n x n -1dx for both integral and fractionaln . He introduced several notations used to this day, for instance the integral sign representing an elongated S, from the Latin words umma and the d used for differentials, from the Latin wordd i ffer e n t ia . This cleverlysuggestive notation for the calculus is probably his most enduring mathematicallegacy. Leibniz did not publish anything about his calculus until 1684. The product ruleof differential calculus is still called "Leibniz's law". In addition, the theorem that tellshow and when to differentiate under the integral sign is called the Leibniz integral rule.

Leibnizs mathematical investigations were thus merely part of a truly grandplan, and this explains his focus on developing useful new notation and theoreticalmethods, rather than specific results. Indeed, it is his notation and language for the


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calculus that we use today, rather than Newtons. He sought and found a calculus for infinitesimal geometry based on new symbols and rules. Second, Leibniz studied therelationship between difference sequences and sums, and then an infinitesimalversion helped suggest to him the essential features of the calculus.

Most modern historians believe that Newton and Leibniz developed infinitesimalcalculus independently, although with very different notations. Occasionally it hasbeen suggested that Newton published almost nothing about it until 1693, and did notgive a full account until 1704, while Leibniz began publishing a full account of hismethods in 1684. (Leibniz's notation and "differential Method", nowadays recognizedas much more convenient notations, were adopted by continental Europeanmathematicians, and by British mathematicians after 1820)

Newton's work on pure mathematics was virtually hidden from all but hiscorrespondents until 1704, when he published, withO pt ick s , a tract on the quadratureof curves (integration) and another on the classification of the cubic curves. HisCambridge lectures, delivered from about 1673 to 1683, were published in 1707.

Newton wrote a tract on fluxions in October 1666. This was a work which wasnot published at the time but seen by many mathematicians and had a major influence

on the direction the calculus was to take. Newton thought of a particle tracing out acurve with two moving lines which were the coordinates. The horizontal velocity x ' andthe vertical velocityy ' were the fluxions of x and y associated with the flux of time. Thefluents or f l ow ing quan t i t i es were x and y themselves. With this fluxion notationy '/ x 'was the tangent to f ( x , y ) = 0.

In his 1666 tract Newton discusses the converse problem, given therelationship between x and y '/ x ' find y . Hence the slope of the tangent was given for each x and when y '/ x ' = f ( x ) then Newton solves the problem by antidifferentiation. Healso calculated areas by antidifferentiation and this work contains the first clear statement of the F un d am e n t al Theo r e m of the C alculu s .

Newton had problems publishing his mathematical work. Barrow was in someway to blame for this since the publisher of Barrow's work had gone bankrupt and


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publishers were, after this, wary of publishing mathematical works. Newton's work on Anal ys i s w i th inf ini te se ri es was written in 1669 and circulated in manuscript. It wasnot published until 1711. Similarly hisMethod of f lu x i ons an d inf ini te se ri es was writtenin 1671 and published in English translation in 1736. The Latin original was not

published until much later.

In these two works Newton calculated the series expansion for sin x and cos x and the expansion for what was actually the exponential function, although thisfunction was not established until Euler introduced the present notatione x .

Leibniz learnt much on a European tour which led him to meet Huygens inParis in 1672. He also met Hooke and Boyle in London in 1673 where he bought

several mathematics books, including Barrow's works. Leibniz was to have a lengthycorrespondence with Barrow. On returning to Paris Leibniz did some very fine work onthe calculus, thinking of the foundations very differently from Newton.

Newton considered variables changing with time. Leibniz thought of variables x ,y as ranging over sequences of infinitely close values. He introduced dx and dy asdifferences between successive values of these sequences. Leibniz knew that dy /dx gives the tangent but he did not use it as a defining property.

For Newton integration consisted of finding fluents for a given fluxion so the factthat integration and differentiation were inverses was implied. Leibniz used integrationas a sum, in a rather similar way to Cavalieri. He was also happy to use 'infinitesimals'dx and dy where Newton used x ' and y ' which were finite velocities. Of course neither Leibniz nor Newton thought in terms of functions, however, but both always thought interms of graphs. For Newton the calculus was geometrical while Leibniz took ittowards analysis.

Leibniz was very conscious that finding a good notation was of fundamentalimportance and thought a lot about it. Newton, on the other hand, wrote more for himand, as a consequence, tended to use whatever notation he thought of on the day.Leibniz's notation of d and highlighted the operator aspect which proved important inlater developments. By 1675 Leibniz had settled on the notation


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y dy = y 2/2

written exactly as it would be today. His results on the integral calculus were publishedin 1684 and 1686 under the name ' calculu s s umma torius ', the name integral calculuswas suggested by Jacob Bernoulli in 1690.

Leibniz and Newton had very different views of calculus in that Newtons wasbased on limits and concrete reality, while Leibniz focused more on the infinite and theabstract (Struik, 1948). However, regardless of the divergent paths these two scholarschose to venture down, the question of who took the first step remained the primaryissue of debate.

In the 1690s Newton's friends proclaimed the priority of Newton's methods of fluxions. Supporters of Leibniz asserted that he had communicated the differentialmethod to Newton, although Leibniz had claimed no such thing. Newtonians thenasserted, rightly, that Leibniz had seen papers of Newton's during a London visit in1676; in reality, Leibniz had taken no notice of material on fluxions. A violent disputesprang up, part public, part private, extended by Leibniz to attacks on Newton's theoryof gravitation and his ideas about God and creation; it was not ended even byLeibniz's death in 1716. The dispute delayed the reception of Newtonian science on

the Continent, and dissuaded British mathematicians from sharing the researches of Continental colleagues for a century.

Unaware that Newton was reported to have discovered similar methods,Leibniz discovered his calculus in Paris between 1673 and 1676 (Ball, 1908). By1676, Leibniz realized that he was onto something big; he just didnt realize thatNewton was on to the same big discovery because Newton was remaining somewhattight lipped about his breakthroughs. In fact, it was actually the delayed publication of Newtons findings that caused the entire controversy. Leibniz published the firstaccount of differential calculus in 1684 and then published the explanation of integralcalculus in 1686 (Boyer, 1968 ).


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Newton did not publish his findings until 1687. Yet evidence shows thatNewton discovered his theories of fluxional calculus in 1665 and 1666, after havingstudied the work of other mathematicians such as Barrows and Wallis (Struik, 1948).Evidence also shows that Newton was the first to establish the general method called

the "theory of fluxions" was the first to state the fundamental theorem of calculus andwas also the first to explore applications of both integration and differentiation in asingle work (Struik, 1948). Newton, who was often reluctant to publish his writing, wasfinally coaxed into printing up his work with the urging of his mathematician friends.This proved too little too late for his most beloved creation, calculus. Although he haddiscovered calculus in 1666, he did not publish its description until the year 1693.During that time, a German mathematician named Leibniz had created an identicalmathematical work to calculus and published these results in Germany in 1684. As aresult, Leibniz was referred to as calculus' creator, and when this news came toEngland Newton was enraged. While the debate raged on and both sides about whohonestly claimed the rights to calculus, all communications broke down betweenGermany's mathematicians and England's mathematicians. As a result France usedthe work done by Newton and Leibniz and perfected calculus and advancedmathematics in their country.

Starting in 1699, other members of the Royal Society (of which Newton was amember) accused Leibniz of plagiarism, and the dispute broke out in full force in 1711.Newton's Royal Society proclaimed in a study that it was Newton who was the truediscoverer and labeled Leibniz a fraud. This study was cast into doubt when it waslater found that Newton himself wrote the study's concluding remarks on Leibniz. Thusbegan the bitter Newton vs Leibniz calculus controversy, which marred the lives of both Newton and Leibniz until the latter's death in 1716.

Leibniz shortly before his death admitted in a letter to Conti that in 1676 Collinshad shown him some Newtonian papers, but implied that they were of little or novalue, - presumably he referred to Newton's letters of June 13 and Oct. 24, 1676, andto the letter of Dec. 10, 1672, on the method of tangents, extracts from whichaccompanied the letter of June 13, - but it is remarkable that, on the receipt of these


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letters, Leibniz should have made no further inquiries, unless he was already awarefrom other sources of the method followed by Newton (Ball, 1908).

In 1715, just a year before Leibniz death, the Royal Society handed down their verdict crediting Sir Isaac Newton with the discovery of calculus. It was also statedthat Leibniz was guilty of plagiarism because of certain letters he was supposed tohave seen (Ball, 1908). It later became known that these accusations were false, andboth men were then given credit, but not until after Leibniz had already died. In fact,the controversy over who really deserved the credit for discovering calculus continuedto rage on long after Leibniz death in 1716 (Struik, 1948). Newton and his associateseven tried to get the ambassadors of the London diplomatic corps to review his oldmanuscripts and letters, in the hopes that they would endorse the finding of the Royal

Society that Leibniz had plagiarized his findings regarding calculus. Another argumenton the side promoting the idea of Leibniz as a plagiarist was the fact that he used analternate set of symbols. Leibniz specifically set out to develop a more meticulousnotation system than Newtons, and he developed the integral sign ( ) and the 'd'sign, which are still used today (OConnor, 1996) However this action was argued bymany to be merely a way for Leibniz to cover his tracks so as not to get accused of stealing Newtons material (Boyer, 1968). The fact that the method was more efficient

was considered to be an ancillary benefit. The fact is that Leibniz sent letters toNewton outlining his own presentation of his own methods, and these letters focusedquite stringently upon the subject of tangents and curves. Because Newton had beenapproaching calculus primarily in regards to its applications to physics, he purportedcurves to be the creation of the motion of points while perceiving velocity to be theprimary derivative. Conversely, the calculus of Leibniz was applied more todiscoveries in geometry made by scholars such as Descartes and Pascal. Since"Leibniz' approach was geometrical," the notation of the differential calculus and manyof the general rules for calculating derivatives are still used today, while Newton'sapproach, which has in many aspects, fallen by the wayside, was "primarilycinematical" (Struik, 1948).

With modern controversies covering such volatile topics as abortion and guncontrol, a debate over who discovered calculus may seem somewhat trivial by


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contemporary standards. However at the time, this was a serious issue that not onlyinvolved matters of mathematical discovery but also matters of national pride andallegiance. What is important to keep in perspective is that no matter who actuallydiscovered calculus first, both Newton and Leibniz made great contributions to the

advancement of mathematical processes, and both deserve credit for that.


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2 .0 THEORY OF DIFFERENTIATION AND INTEGRATION BY SIR ISAAC NEWTON

AND GOTTFRIED WILHELM LEIBNIZ

Sir Isaac Newton y Fluxions - calculus and the fundamental theorem

Newton developed algorithms for calculating fluxions defined in modern terms as

To solve the problems: a) Find the speed of notion of any fluent.

b) Given the speed find the length of space at any time t .

He assumes a form f ( x ,y ) = 0 and produces the differential equation using theprocedure of Hudde.

His method builds into it the product rule for derivatives. Then, he justifies this rule bydefining themoment

substituting and resolving the terms la Fermat. Note the term o is viewed as infinitelysmall.


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To resolve the length of space question, Newton reverses the procedure if possible.This is an antiderivative approach. Otherwise he resorts to power series.

Example. Consider the equation

is resolved as

Applying the binomial theorem we get for the plus root

Hence one solution is


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Gottfried Wilhelm Leibniz

y P roduct rule ( Leibnizs Law )

Gottfried Leibniz is credited with the discovery of this rule which he called Leibniz'sLaw. He demonstrated it using differentials. Product rule (also called Leibniz's law;see derivation) is a formula used to find the derivatives of products of functions. Hereis Leibniz's argument. Letu( x ) and v ( x ) be two differentiable functions of x . Then thedifferential of uv is

Since the term d u d v is "negligible" (compared tod u and d v ), Leibniz concluded that

and this is indeed the differential form of the product rule. If we divide through by the

differentialdx , we obtain

which can also be written in "prime notation " as

Or


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Ge ne raliz ed Le i bniz R ul e

is known as the Leibniz rule. Below are various ways it can be generalized.

Higher derivatives

Let be real (or complex) functions defined on an open interval of . If and

are times differentiable, then

where is the binomial coefficient G eneralized Leibniz rule for more functions

Let be real (or complex) valued functions that are defined on an open

interval of . If are times differentiable, then

where is the multinomial coefficient.

Leibniz rule for multi-indices

If are smooth functions defined on an open set of , and isa multi-index, then

where is a multi-index.


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3 .0 A PP LICATION OF DIFFERENTIATION AND INTEGRATION

When we learn about differentiation and integration, we might not realize howmuch these two theories can be applied and used in everyday life. Some of us arewondering why we should learn about differentiation and integration in our syllabus. Actually there are a lot of things that we have to used differentiation and integrationespecially in engineering and science field.

In the era when Isaac Newton lived, one of the biggest problems they had toface was poor navigation at sea.Before calculus was developed, the stars were vital for navigation. Shipwrecks

occurred because the ship was not where the captain thought it should be. That wasbecause they had not a good enough understanding of how the earth, stars andplanets moved with respect to each other. Differentiation and integration wasdeveloped to improve this understanding.

Newton made numerous and remarkable applications of his method of fluxion.

He determined maxima and minima, tangents to curves, curvature of curves, points of

inflection, convexity and concavity of curves using the method of fluxion. After

conceiving the method of fluxions, Newton adapted them to the quadrature of curves.

Furthermore, Newton makes it quite clear when he would find our conventionalview of gravity to be absurd and notional. As far as he is concerned the Earth has nogravity. Today's conventional view of gravity lacks a mechanism and continues toignore that fact that you can't have something in a vacuum. By postulating the fluxion,Newton dispels the notion of outer space as a vacuum and the absurdity of matter

emitting gravity into infinity.

Mathematicians likes to generalize things, and there are several generalizationsof the product rule, each in its own context. When Leibniz introduce about productrule, it is used to def ine what is called a derivation, not vice versa. The product rulecan be extends to scalar multiplication, dot products, and cross products of vector


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functions. This rule tells us how to compute the derivative of a product of two functionsin terms of the original functions and their derivatives. Besides, he apply this rule tocompute the derivative of a product of polynomials without having to first evaluate theproduct of the polynomials, which can often save us a lot of work. We can also use the

product rule to extend the power rule to negative powers of x.

Differential calculus can be considered as mathematics of motion, growth andchange where there is a motion, growth and change. Whenever there are variableforces producing acceleration, differential calculus is the right mathematics to apply. Applications of derivatives are used to represent and interprets the rate at whichquantities change with respect to another variable.

Apart from that, differentiation can help us solve many types of real-worldproblems for example to determine the maximum and minimum values of particular functions where we can use these values to calculate the cost, strength, amount of material used in a building and their profit and loss in making the building. However,before we find the maximum and minimum values, we have to find the gradient inorder to determine the region of increase and the region of decrease of a function. After acquiring the knowledge of gradients and convexity of a curve, then we can find

the turning points.

Besides, using the theory of differentiation, we can find the rate of changes for example in an oil tanker, the profit of a company each year and the rate of water

leaking from an inverted cone. Therefore, the meaning that is the rate of change of

y with respect to x should be taught in engineering field as it is very important for theengineers to calculate the rate of changes in their plan of buildings. We can also use

the rate of changes to calculate the rate of changes of the slope of the originalfunction for the moving objects for example cars and motorcycles. An economist mayalso use the rate of changes if he wants to study how investment changes withvariation in interest rates.


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Differentiation also can be used to find the approximation values of a function

close to special values and estimate simple error in the form by differentials. Approximation values are very important especially to find the volumes of different

kind of shapes and in the process of manufacturing mechanical clocks where error can be made in producing the length of the pendulum of the clock.

Furthermore, in primary school, we learned how to find areas of shapes withstraight sides for example to find the area of a triangle or rectangle. But how we canfind the areas when the sides are curved? In order to solve this problem, we use theformula of integration to find the area under a curves and area between two curves.Besides, to find the volume of an object with curved sides for example to make wine

barrels we use integration to find the volume of solid of revolution. Many solid objects,especially those made on a lathe, have a circular cross-section and curved sides.When we learn about calculus, we will know how to find the volume of such objectsusing integration.

All of us also knew that electric charges have a force between them that variesdepending on the amount of charge and the distance between the charges. In thiscase, we use integration to calculate the work done when charges are separated. Wecan also use integration to find the force of the liquid pressure where the force byliquid pressure varies depending on the shape of the object and its depth. Besides,average value of a curve can be calculated using integration.

From all of these cases above, we can solve the problem by considering thesimple case first. Usually this means the area or volume has straight sides. Then weextend the straight-sided case to consider curved sides cases. At this time, we need touse integration because we have curved sides and cannot use the simple formulasany more.

Thus, from these situations we know how much the theory of differentiation andintegration are useful in our everyday life. Without these two theories, we can not builda building, reduces the efficiency of hard drives and other computer components andmake it difficult to run a great company. Therefore, we have to be thankful to the two


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genius mathematicians, Sir Isaac Newton and Gottfried Wilhelm Leibniz for bringingthese two amazing theories in our life and make our life become easier and morecomfortable.


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INDIVIDUAL REFLECTION

By : Syarafina binti Mohd SalimIC Number : 900917-06-5660

When I first read the instruction given by Madam Farida on 9 August 2010, I amvery excited to do the task. This is because I never know even a little about the historyof calculus especially in the integration and differentiation. Since my high school, myteachers only teach me how to answer the exam question and the concept or theformula. They never touched about the history of the differentiation and integration.However, sometimes I also get curious where they get such this interestingmathematics topic. How can they know to calculate it? So, I believe after finished this

task, my question will be answered.

First and foremost, my group and I went to the library to get the informationabout integration and differentiation. We need to know at least two mathematicianswho contribute in calculus, their history, theory and the application in daily life. We tryto find calculus book. Unfortunately, there is no calculus book in the rack. We gotpuzzled. We try to find in general mathematics book but there is no information aboutmathematician. As a solution, we search from the websites first as it is the most easy

way to find the information. From that, we know there were many mathematicians whocontribute in calculus. But the most famous mathematicians were Sir Isaac Newtonand Leibniz. Then, we search for the information in the red spot since there was nocalculus book in the outside rack. We were lucky because there were a lot of information we got in the red spot.

Besides that, we also got some problem in sketching graph since I am notunderstood well about graph sketching. We had tried to do the question but the

answer was wrong. So, we decided to study about the topics and ask madam Faridahthe things we dont know. After struggled almost one day, we finally managed tosketch the right graph. I am very happy because we got it with our own try.

In this task, I had identified that our strength is from the sources we got to makethe report. We had the information from many type of resources such as


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encyclopedias, books and websites. For our weakness, I found that we cannot make agood chronology about the history of the contributions of the mathematicians detailed.I only stated generally topic they discover even though I believed there is a chronologywhen they got the ideas and they expand it until it becomes easier like we use today.

However, I know this little weakness can be improve if we were given a little bit moretimes since we dont have much time to do this assignment because we have aprogram.

In a nutshell, I am satisfied with this task especially the moment I when the taskwas in progress. I got a lot of new knowledge and experience. This task had teach methat we cant only study the things, but we must know where is there come from. i ampleasure to get other knowledgeable task like this in the future. Thank you.


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By : NorJamilah Ab RahmanIC Number : 910902-11-5070

On 9th August 2010, I was given the briefing of this Mathematics assignment. Itwas an assignment which must do in group of three. For this assignment, we need tomake research about mathematicians. We need to find at least two mathematiciansregards the invention especially in differentiation and integration. In this assignment,my group members and I need to explore the history, theory and application of differentiation and integration that was introduced by the mathematicians. Beside that,we are also given one question about graph sketching.

First and foremost, praise to God for giving my group members and I a goodhealth and safety while finishing this Mathematics assignment for this semester.Finally, after 2 weeks, my group members and I had finished completely our Mathematics assignment. During finishing this assignment, I found many problems,weakness and also strength during doing it.

I had face many problems when do this assignment. First, I cant get anysuitable idea which mathematicians should be chosen. There are manymathematicians that contributed to differentiation and integration. So, I overcome thisproblem by discuss it with my group members to decide who we will choose. After Imade a discussion with my group members, finally my group members and I manageto find suitable mathematicians who are Sir Isaac Newton and Gottfried LeibnizWilhelm.

Besides that, in the beginning to do this assignment, I found that my groupmembers and I do not know how to manage our time . But, due to our cooperation andour tolerance, I had managed the time very well. From this situation, it teaches me toappreciate the time that we have. My group members and I had organized andplanned well about when and how to do this assignment properly.

Although our lecturer, Madam Farida had guide us how to do this assignment,but we still need to find another information about the Sir Isaac Newton and Gottfried


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Leibniz Wilhelm. So, my group members and I find the information of thesemathematicians from different sources like books, encyclopedia and internet. We gotmany pages that related to these two mathematicians from internet but, we need tochoose the suitable pages regarding to Sir Isaac Newton and Gottfried Wilhelm

Leibniz. Then, we found that there are not much books about calculus (differentiationand integration) at library. We just manage to find a few books but it does not havemuch information in it. So, we also try to get some information about these twomathematicians from encyclopedia.

Absolutely there is not only the problem and weaknesses that I found whendoing this assignment. I also noticed that, through this assignment, it has a benefitsand strength. Through this assignment; research about history, theory and application

of differentiation and integration by Sir Isaac Newton and Gottfried Leibniz Wilhelmmade me more understand about the differentiation and integration. Before this, thetopic of differentiation and integration are my favourite topics because it is veryinteresting topics to me. Furthermore, before this I have learned about it at school.When make the research about these two mathematicians, I found that there are lotsof differentiation and integrations concept that was introduced by them. Before this, Idid not know about all these concept but now all this knowledge of integration anddifferentiation is very useful for me and I can use it in my daily life.

Then the sweetest thing during finishing this assignment is my relationshipbetween my group members and I had become closer because we had spent manytimes together to discuss about our assignment. We also discuss together aboutsketching the graph. The graph is quite difficult for us. So, we discuss it together ingroup.

In conclusion, I found that from this assignment; research about differentiation

and integration by Isaac and Leibniz, it helps me to improve my knowledge aboutthese topics. Then, I can know who are the creator differentiation and integration.Lastly, thank you to all people that help me directly or indirectly during I finishing myassignment. Thanks a lot to my parents, friends, and especially to my belovedlecturer, Madam Farida that give me support all the time.


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By : Hazirah Mohd Abdul WahidIC Number : 910721-11-5338

In my final semester of PPISMP Math, I think I was been given a different kindof coursework for my core subject, Mathematics 2 from my previous semesters. Thefirst task asks me to make a report about two mathematicians that contribute to theinvention of calculus especially according to my syllabus that are integration anddifferentiation. My impressions when I read the task, I thought it was simpler than mylast semesters assignment when we had to do the experiment to show the theory of probability.

When I went for the library to find the resources and make a research throughthe internet, then I realized it was quite difficult because there is not much historyabout how the mathematicians found the theory and their application. Usually, therewas only small part in those books where they talk about the mathematicians and of course it is not good enough for us. However, my group members and I keep onlooking and try to find as many information as we can so that we can make a greatreport about their contribution.

Finally, we decided to choose Sir Isaac Newton and Gottfried Wilhelm Leibnizas our two mathematicians because of their amazing theories and major contributionsin the introduction of calculus. Before I done this task, actually I dont know anythingabout them and the facts that what I learnt in my class today about calculus wasintroduce by these two great mathematicians. So, I am very glad and excited as I got alot of knowledge from this coursework where I had the chance to know both of them

closer from many aspects.From my opinion, the history of their live, the theory they had invented and the

application from their theory should be published among the students either in theschool or in university. I am sure this will inspire the students to know how thesepeople can found such a great theories that had made our life becomes easier. Just


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take a look everything around us and they will realize how important the calculus to uswhere we even can not build a building without using calculus.

My second task is to do the graph sketching from the question given. This taskis tougher as we cant find the right solution for the question because it is quitedifferent from the questions we usually done in the class. We had to do research,learn about it from the internet and asked experienced people like our Mathematics 2lecturer. Besides, we also made a discussion among the groups and try to share our knowledge together about this graph sketching.

In order to make a good graph sketching, we applied all the formula and ways

that our lecturer had been teaching us in the class. Finally, we come out with a graphthat we had done it all the best as we can. I gain knowledge again in the process of making a complicated graph that I will use this knowledge when I am learning or teaching my students in the future.

As a conclusion, I think we should be grateful because our two mathematiciansSir Isaac Newton and Gottfried Wilhelm Leibniz who had come with the great ideasand theories which make our life journey become smooth, easier and interesting. In

order to get the right theories, they had to go through a lot of problems and sacrificemany things. Besides, these theories took a very long time and had to face manychanges to become as simple as what we learn today. Therefore, we need tounderstand about calculus further and applied in our daily life such as when we wantto find the areas, volume or the profit in a company so that the knowledge will not bewasted.

esei teori terbaru fluxion and product rule

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