1. SIR ISAAC NEWTON BIOGRAPHY CONTRIBUTIONS NOTATIONS

2. GOTTFRIED WILHELM LEIBNIZ BIOGRAPHY CONTRIBUTIONS NOTATIONS

3. THE CALCULUS CONTROVERSY

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Sir Isaac Newton

Born [OS: 25 December 1642] Woolsthorpe-by-Colsterworth,Lincolnshire, England

Died31 March 1727 (aged 84)[OS: 20 March 1727] Kensington, Middlesex, England

Residence EnglandCitizenship EnglishNationality English (British from 1707)

Fieldsphysics, mathematics, astronomy, natural philosophy, alchemy, theology

InstitutionsUniversity of CambridgeRoyal SocietyRoyal Mint

Alma mater Trinity College, Cambridge

Academic advisors

Isaac BarrowBenjamin Pulleyn

Notable students

Roger CotesWilliam Whiston

Known for

Newtonian mechanicsUniversal gravitationCalculusOptics

Influences Henry More

InfluencedNicolas Fatio de DuillierJohn Keill

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1) BIOGRAPHY OF ISAAC NEWTON

Isaac was born on Christmas day in a village in Lincolnshire, England on 31

March 1727. He was prematurely born, and, was so small at his birth. Newton's father

had died several months before his birth. Isaac barely maintained average grades and

often lacked attention in school. However, the first hints of Newton's brilliance could be

found in his boyhood inventions. He was responsible for creating sundials, an accurate

wooden clock and water wheels. One of his most practical adolescent inventions was a

mill, which mechanically ground wheat into flour via mouse power. Newton's uncle saw

the potential of his nephew's scientific talents, and enrolled him in Cambridge

University. It was here that Newton was first exposed to the world of mathematics.

Having come across Euclid's Elements in a bookstore, Newton was able to follow

quickly the work, although he had little mathematical background to begin with. Having

found the work easy reading, Newton becomes fascinated by mathematics and he

quickly mastered Descartes' difficult work, Geometry. Newton quickly earned the

respect of his peers and professors at Cambridge. For instance, at the end of his

second year, Newton had taken the place of his professor, Dr. Isaac Barrow, who

resigned in recognition of Newton's superior mathematical skills. According to Newton's

inner circle, Newton had worked out his method years before Leibniz, yet he published

almost nothing about it until 1693, and did not give a full account until 1704. Isaac

Newton died on 20 March 1727, in Kensington, Middlesex, England.

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1.1) ISAAC NEWTON’S CONTRIBUTIONS

Mathematics

In mathematics, early brilliance appeared in Newton's student notes. He may

have learnt geometry at school, though he always spoke of himself as self-taught;

certainly he advanced through studying the writings of his compatriots William Oughtred

and John Wallis, and of Descartes and the Dutch school. Newton made contributions to

all branches of mathematics then studied, but is especially famous for his solutions to

the contemporary problems in analytical geometry of drawing tangents to curves

(differentiation) and defining areas bounded by curves (integration). Not only did

Newton discover that these problems were inverse 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. Newton used the term "fluxion" (from Latin meaning

"flow") because he imagined a quantity "flowing" from one magnitude to another.

Fluxions were expressed algebraically, as Leibniz's differentials were, but Newton made

extensive use also (especially in the Principia) of analogous geometrical arguments.

Late in life, Newton expressed regret for the algebraic style of recent mathematical

progress, preferring the geometrical method of the Classical Greeks, which he regarded

as clearer and more rigorous.Newton's work on pure mathematics was virtually hidden

from all but his correspondents until 1704, when he published, with Opticks, a tract on

the quadrature of curves (integration) and another on the classification of the cubic

curves. His Cambridge lectures, delivered from about 1673 to 1683, were published in

1707.

Calculus

Newton had made his first important contribution by advancing the binomial

theorem, which he had extended to include fractional and negative exponents. He

succeeded in expanding the applicability of the binomial theorem by applying the

algebra of finite quantities in an analysis of infinite series. He showed a willingness to

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view infinite series not only as approximate devices, but also as alternative forms of

expressing a term.

Many of Newton’s critical insights occurred during his plague-induced isolation

that the first written conception of Fluxionary Calculus was recorded in the unpublished

De Analysi per Aequationes Numero Terminorum Infinitas. In this paper, Newton

determined the area under a curve by first calculating a momentary rate of change and

then extrapolating the total area. He began by reasoning about an indefinitely small

triangle whose area is a function of x and y. He then reasoned that the infinitesimal

increase in the abscissa will create a new formula where x = x + o (importantly, o is the

letter, not the digit 0). He then recalculated the area with the aid of the binomial

theorem, removed all quantities containing the letter o and re-formed an algebraic

expression for the area. Significantly, Newton would then “blot out” the quantities

containing o because terms “multiplied by it will be nothing in respect to the rest”.

At this point Newton had begun to realize the central property of inversion. He

had created an expression for the area under a curve by considering a momentary

increase at a point. In effect, the fundamental theorem of calculus was built into his

calculations. While his new formulation offered incredible potential, Newton was well

aware of its logical limitations. He attempted to avoid the use of the infinitesimal by

forming calculations based on ratios of changes. In the Methodus Fluxionum he defined

the rate of generated change as a fluxion, which he represented by a dotted letter, and

the quantity generated he defined as a fluent.

1.2) NOTATION,

For example, if x and y are fluent, then and are their respective fluxions. Essentially,

the ultimate ratio is the ratio as the increments vanish into nothingness. Importantly,

Newton explained the existence of the ultimate ratio by appealing to motion.Newton's

name for it was "the science of fluent and fluxions".The work of both Newton and

Leibniz is reflected in the notation used today. Newton introduced the notation for the

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derivative of a function f. Leibniz introduced the symbol for the integral and wrote the

derivative of a function y of the variable x as

both of which are still in use.

Western Philosophy

17th-century philosophy

Gottfried Wilhelm Leibniz

Full name Gottfried Wilhelm Leibniz

Born1 July 1646

Leipzig, Electorate of Saxony

Died14 November 1716

Hanover, Electorate of Hanover

Main interest Metaphysics, Mathematics, Theodicy

Notable ideas Infinitesimal calculus, Calculus, Monadology, Theodicy,

Optimism

Leibniz formula for pi

Leibniz harmonic triangle

Leibniz formula for determinants

Leibniz integral rule

Principle of sufficient reason

Leibniz differential

Diagrammatic reasoning

Notation for differentiation

Differential calculus

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Proof of Fermat's little theorem

Kinetic energy

Entscheidungsproblem

2) BIOGRAPHY OF GOTTFRIED LEIBNIZ

Gottfried Leibniz was born on 1 July 1646 in Leipzig to Friedrich Leibniz and

Catherina Schmuck. The Leibniz family was of Slavonic origin, but had been

established in the Leipzig area for more than two hundred years. His father died when

he was six, so he learned his religious and moral values from his mother. When Leibniz

was six years old, his father, a Professor of Moral Philosophy at the University of

Leipzig, died, leaving a personal library to which Leibniz was granted free access from

age seven onwards. While his schoolwork focused on a small canon of authorities, his

father's library enabled him to study a wide variety of advanced philosophical and

theological works that he would not have otherwise been able to read until his university

studies. He entered his father's university at age 14 and completed a Bachelor's degree

in philosophy on 2 December 1662. He soon after took a Master's degree in philosophy

on 7 February 1664. After two years of legal studies, he was awarded a Bachelor's

degree in law on 28 September 1665. He was died in Hanover on November 14, 1716.

According to Leibniz's notebooks, a critical breakthrough occurred on 11 November

1675, when he employed integral calculus for the first time to find the area under a

function y = ƒ(x). He introduced several notations used to this day, for instance the

integral sign ∫ representing an elongated S, from the Latin word summa and the d used

for differentials, from the Latin word differentia. This ingenious and suggestive notation

for the calculus is probably his most enduring mathematical legacy. Leibniz did not

publish anything about his calculus until 1684. The product rule of differential calculus

is still called "Leibniz's law". In addition, the theorem that tells how and when to

differentiate under the integral sign is called the Leibniz integral rule.

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2.1) GOTTFRIED LEIBNITZ’S CONTRIBUTIONS

Mathematician

Although the mathematical notion of function was implicit in trigonometric and

logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to

employ it explicitly, to denote any of several geometric concepts derived from a curve,

such as abscissa, ordinate, tangent, chord, and the perpendicular. In the 18th century,

"function" lost these geometrical associations.

Leibniz was the first to see that the coefficients of a system of linear equations

could be arranged into an array, now called a matrix, which can be manipulated to find

the solution of the system, if any. This method was later called Gaussian elimination.

Leibniz's discoveries of Boolean algebra and of symbolic logic, also relevant to

mathematics, are discussed in the preceding section. A comprehensive scholarly

treatment of Leibniz's mathematical writings has yet to be written.

Calculus

Leibniz is credited, along with Isaac Newton, with the discovery of infinitesimal

calculus. According to Leibniz's notebooks, a critical breakthrough occurred on 11

November 1675, when he employed integral calculus for the first time to find the area

under a function y = ƒ(x). He introduced several notations used to this day, for instance

the integral sign ∫ representing an elongated S, from the Latin word summa and the d

used for differentials, from the Latin word differentia. This ingenious and suggestive

notation for the calculus is probably his most enduring mathematical legacy. Leibniz did

not publish anything about his calculus until 1684. The product rule of differential

calculus is still called "Leibniz's law". In addition, the theorem that tells how and when to

differentiate under the integral sign is called the Leibniz integral rule.

Leibniz's approach to the calculus fell well short of later standards of rigor (the

same can be said of Newton's). We now see a Leibniz "proof" as being in truth mostly a

heuristic hodgepodge mainly grounded in geometric intuition. Leibniz also freely invoked

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mathematical entities he called infinitesimals, manipulating them in ways suggesting

that they had paradoxical algebraic properties. George Berkeley, in a tract called The

Analyst and elsewhere, [citation needed] ridiculed this and other aspects of the early

calculus, pointing out that natural science grounded in the calculus required just as big

of a leap of faith as theology grounded in Christian revelation.

From 1711 until his death, Leibniz's life was envenomed by a long dispute with

John Keill, Newton, and others, over whether Leibniz had invented the calculus

independently of Newton, or whether he had merely invented another notation for ideas

that were fundamentally Newton's.

Modern, rigorous calculus emerged in the 19th century, thanks to the efforts of

Augustin Louis Cauchy, Bernhard Riemann, Karl Weierstrass, and others, who based

their work on the definition of a limit and on a precise understanding of real numbers.

Their work discredited the use of infinitesimals to justify calculus. Yet, infinitesimals

survived in science and engineering, and even in rigorous mathematics, via the

fundamental computational device known as the differential. Beginning in 1960,

Abraham Robinson worked out a rigorous foundation for Leibniz's infinitesimals, using

model theory. The resulting nonstandard analysis can be seen as a belated vindication

of Leibniz's mathematical reasoning.

1.2) NOTATIONS, dy/dx & ∫

Very interested in Sums and Differences. Basis for FTC. Calculus discovered while looking at infinite series

Sum of consecutive differences = (the last term – the first term)

d= the difference between two consecutive numbers

a= a number in the series

d1 + d2 + …+ dn= an – a1

Led to integration. Integration to differentiation (dy & dx differences in consecutive values

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NEWTON VS LEIBNIZ; THE CALCULUS CONTROVERSY

Like most discoveries, calculus was the culmination of centuries of work rather

than an instant epiphany. Mathematicians all over the world contributed to its

development, but the two most recognized discoverers of calculus are Isaac Newton

and Gottfried Wilhelm Leibniz. Although the credit is currently given to both men, there

was a time when the debate over which of them truly deserved the recognition was both

heated and widespread.

As the renowned author of Principia (1687) as well as a host of equally esteemed

published works, it appears that Newton not only went much further in exploring the

applications of calculus than Leibniz did, but he also ventured down a different road.

Leibniz and Newton had very different views of calculus in that Newton’s was based on

limits and concrete reality, while Leibniz focused more on the infinite and the abstract

(Struik, 1948). However, regardless of the divergent paths these two scholars chose to

venture down, the question of who took the first step remained the primary issue of

debate.

Unaware that Newton was reported to have discovered similar methods, Leibniz

discovered “his” calculus in Paris between 1673 and 1676 (Ball, 1908). By 1676, Leibniz

realized that he was onto something “big”; he just didn’t realize that Newton was on to

the same big discovery because Newton was remaining somewhat tight lipped about his

breakthroughs. In fact, it was actually the delayed publication of Newton’s findings that

caused the entire controversy. Leibniz published the first account of differential calculus

in 1684 and then published the explanation of integral calculus in 1686 (Boyer, 1968).

Newton did not publish his findings until 1687. Yet evidence shows that Newton

discovered his theories of fluxional calculus in 1665 and 1666, after having studied 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 and was also the first

to explore applications of both integration and differentiation in a single work (Struik,

1948). However, since Leibniz was the first to publish a dissertation on calculus, he was

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given the total credit for the discovery for a number of years. This later led, of course, to

accusations of plagiarism being hurled relentlessly in the direction of Leibniz.

There was speculation that Leibniz may have gleaned some of his insights from

two of Newton's manuscripts on fluxions, and that that is what sparked his

understanding of calculus. Many believed that Leibniz used Newton's unpublished

ideas, created a new notation and then published it as his own, which would obviously

constitute plagiarism. The rumor that Leibniz may have seen some of Newton's

manuscripts left little doubt in most people’s minds as to whether or not Leibniz arrived

at his conclusions independently. The rumor was, after all, believable because Newton

had admittedly bounced his ideas off a handful of colleagues, some of who were also in

close contact with Leibniz (Boyer, 1968).

It is also known that Leibniz and Newton corresponded by letter quite regularly,

and they most often discussed the subject of mathematics (Boyer, 1968). In fact,

Newton first described his methods, formulas and concepts of calculus, including his

binomial theorem, fluxions and tangents, in letters he wrote to Leibniz (Ball, 1908).

However an examination of Leibniz' unpublished manuscripts provided evidence that

despite his correspondence with Newton, he had come to his own conclusions about

calculus already. The letters may then, have merely helped Leibniz to expand upon his

own initial ideas.

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1. FUNCTION 12. GRAPH 13. FUNCTION 24. GRAPH 25. FUNCTION 36. GRAPH 3

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1. DISCUSSION2. CONCLUSION

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Given the function is

f(x) = 2x6 + 3x5 + 3x3 – 2x2

Therefore,

f’(x) = 12x5 + 15x4 + 9x2 – 4x

From Graph 2, we get the coordinate of turning points are

(0.20, - 0.34) and (- 1.2, 19)

Next,

f’’(x) = 60x4 + 60x3 + 18x – 4

When x = 0.20,

f’’(x) = 0.176 > 0

(0.20, - 0.34) is a local minimum point.

When x = -1.2

f’’(x) = - 4.864 < 0

(- 1.2, 19) is a local maximum point.

Since the point is not located in the graph, we assume that the maximum point is

infinity.

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From the second derivative, we can know that the graph has maximum and minimum

point, by substitute the value of x into second derivative. If we get the positive value, the

point is minimum and if we get the negative value, the point is maximum. From our

graph, (0.20, - 0.34) is minimum point while (- 1.2, 19) is maximum point.

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BIBLIOGRAPHY

http://mally.stanford.edu/leibniz.html Access on 21 August 2009

http://en.wikipedia.org/wiki/Calculus Access on 7 September 2009

http://en.wikipedia.org/wiki/Gottfried_Leibniz Access on 21 August 2009

http://en.wikipedia.org/wiki/Isaac_Newton Access on 21 August 2009

http://en.wikipedia.org/wiki/Leibniz_and_Newton_calculus_controversy Access on 7

September 2009

http://www.angelfire.com/md/byme/mathsample.html Access on 21 August 2009

http://www.blupete.com/Literature/Biographies/Science/Newton.htm Access on 25

August 2009

http://www.economicexpert.com/a/Gottfried:Leibniz.html Access on 8 September 2009

http://www-groups.mcs.st-andrews.ac.uk/~history/Biographies/Leibniz.html Access on 8

September 2009

http://www.maths.tcd.ie/pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html

Access on 25 August 2009

http://www.mathematicianspictures.com/Mathematicians/Leibniz.htm Access on 25

August 2009

http://www.mtsd.k12.wi.us/MTSD/Homestead/ordinans/Leibniz.htm Access on 25

August 2009

http://www.spiritus-temporis.com/gottfried-leibniz/the-calculus.html Access on 7

September 2009

http://www.stewartcalculus.com/data/ESSENTIAL%20CALCULUS/upfiles/projects/

eclt_wp_0404_stu.pdf Access on 7 September 2009

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http://www1.umn.edu/ships/9-1/calculus.htm Access on 8 September 2009

Ong Beng Sim (2005), Mathematics for STPM Pure Mathematics, Penerbitan Fajar

Bakti Sdn Bhd, Selangor Darul Ehsan

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INDIVIDUAL REFLECTION

Prepared by: Nur Afifah bt Mohamed Salleh

We are given an assignment to do for the subject Mathematics in this semester.

With my partner Siti Khirnie Kasbolah, we have to do 3 tasks. Task 1 search the

biography and contribution of Sir Isaac Newton and Gottfried Wilhelm Leibniz, task 2

isbfind the maximum and minimum points using Geometry Sketchpad and last task is

solving the question of function.

For the first discussion, we divide the work. I have to find information about

Leibniz while Khirnie has to find about Sir Isaac Newton. Then we discuss together to

complete the second and third questions which are find the maximum and minimum

point and solving the question of function

Our strength of our group is, before we start this task, we planning the draft first.

After that we divide the work with my partner so that our task is done on time. We also

have a good discussion. My partner gave full commitment in doing this task.

Besides, our group also has some weaknesses where we do not have so much

time because there are a lot of works to do. So we overcome this weakness by dividing

the works. We also has problem to do a group discussion because everyone busy with

their own works. We overcome this problem by find the suitable time to discuss and use

our time properly.

And thank God, after face so many problems , we can submit the task on time.

We also enjoy doing this task because we can fully use our idea to create a creative

brochure. We also learn how to conduct a campaign by our own. I hope with this

experience, we can apply it in our daily life.

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INDIVIDUAL REFLECTION

Prepared by: Siti Khirnie binti Kasbolah

The task had given on 14 August 2009 by our lecturer, Pn Farida bt Mohamad

Dahari. As my class have been learned Basic Calculus for this semester, we were

asked to work in pair so that we can have a lot of discussion and sharing lot of ideas. I

had chosen to work with Nur Afifah. There are three parts in our assignment. The first

part was about to make a research on calculus, which developed by Isaac Newton and

Gottfried Leibniz. The second part is about to manage the function by using Graph

Skecth Pad (GSP) software. And the third part was about to relate the graph with the

calculus.

Actually this task was interesting because it need my friend and I to explore the

one of the main chapter Mathematics which is we know it is really an interesting topic.

Besides that, I also learned how to create graphs by explore the GSP software by

myself. After that, I taught my partner and some other friends. It is really enjoying

because I do not have to draw or make a draft to draw the graph. I had just inserted the

function and with a single click, the graph is there. GSP Software is helpful in making

graph for the second part of this assignment.

The strength that I can found in my work is I have no problem with my partner.

Besides that, we can share our opinion because we were respecting each other. I was

also satisfied with the cooperation from my partner because she knew what she should

do especially when we had divided the work. She managed to complete her part

successfully. Furthermore, we have no problem to make research through the internet

because we can online during our class. We have so many materials from the websites.

Despite of strength, I was also detected some weaknesses in myself while doing

this work. The main problem is time. My partner and I have to do this assignment

separately since we do not have enough time to make a lot of discussion. This is

because we have other assignment to be completed and actually, this problem is the

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routine problem for every time we had to complete our assignment. I thought maybe I

have problem with my own time management. Therefore, I will try to work on it and

improve myself by making a timetable to make sure this problem does not occur again.

As the conclusion, even this assignment is quite difficult, I enjoyed doing this

assignment because from this assignment I knew Isaac Newton is also a Mathematician

because I had only knew that he is a scientist before. Besides that, I realize that it is not

easy for the ancient people to make research to create their own theory or formula.

They need to do so many research and experiments. There was also controversy in

order to stand with their idea. I feel so grateful because I have chance to learn

Mathematics without any difficulty to quarrel or to find something new. Everything

(theories, formulas) is there. All I need to do is just learn and maybe I can manipulate

the theory for other necessary.

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