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Before you read Discuss these questions with your partner. What do you know about ? Can you think of a problem calculus could be used to solve?

В A Vocabulary Complete the definitions below with words from the box.

slope approximation embrace acceleration

diverse indispensable sphere cube

rectangle

1 If something is a(n) it В Reading 1 isn't exact. 2 An increase in speed is called Calculus 3 If something is you can't Calculus is the branch of that deals manage without it. with the rates of change of quantities as well as the length, area and volume of objects. It grew 4 If you an idea, you accept it. out of and algebra. There are two 5 A is a three-dimensional, divisions of calculus - and square shape. calculus. Differential calculus is the form 6 Something which is is concerned with the rate of change of quantities. different or of many kinds. This can be illustrated by slopes of curves. Integral calculus is used to study length, area 7 If you place two squares side by side, you and volume. form a(n) The earliest examples of a form of calculus date 8 A is a three-dimensional back to the ancient Greeks, with Eudoxus surface, all the points of which are the same developing a mathematical method to work out distance from a fixed point. area and volume. Other important contributions 9 A is also known as a fall. were made by the famous and , Archimedes. In India, over the

98 Macmillan Guide to Science Unit 21 Mathematics

4 course of many years - from 500 AD to the 14th Pronunciation guide century - calculus was studied by a number of . In fact, the first text on calculus Archimedes /a:ki'mi:di:z/ was written in India. However, it was not until embrace /imbreis/ the end of the 1600s that calculus was taken up Eudoxus /ju'doksss/ ^indispensable /mdi spensab l/ in Europe. There was much scientific activity at У the time, and calculus was able to answer many questions, particularly in the field of . В В Comprehension Many great mathematicians of the time embraced Read the text and decide if the following calculus and furthered its development, including statements are true or false. Rene Descartes and Pierre de Fermat, but the most important contributions were made by 1 Calculus and geometry led to T • Gottfried Leibniz and . Newton was the development of algebra. F • the first to use calculus in his studies of physics 2 India was using calculus T П and Leibniz developed many of the symbols that before Europe. F we use in calculus. • 3 In the , calculus was T • The starting point of calculus is the idea that you applied to physics. F • can use an approximation and keep increasing 4 Integral calculus can calculate the rate T •О the accuracy until an exact answer is found. An at which a population increases. F example of this would be to calculate the volume • or area of a sphere by using shapes such as 5 Differential calculus has to do with T • rectangles or cubes that become increasingly systems that are undergoing change. F • smaller until the exact volume or area is determined. In calculus, this final result is called Before you listen a limit. Discuss these questions with your partner. Differential calculus describes processes that are Do you know who coined the term calculus? in flux - which means they are constantly Can you think of another word that comes from changing. Examples of this are temperature the word calculus? variations or the speed of a moving object. By using differential calculus, it is possible to determine the rate at which the temperature И С Listening *))) changes and the rate of acceleration of the Listen to a teacher and students moving body. Integral calculus begins with a discussing some of the . known rate of change and, working backwards, Then answer the questions in your finds certain values. For example, if you know the own words. rate of acceleration of a car, you can find out its 1 What was the original meaning of the speed at a given point. word calculus? Today, both forms are used in every area of 2 Who named the branch of mathematics science and knowledge. Fields as diverse as known as calculutf? engineering, medicine, business and economics make use of calculus. Calculus is such an 3 Why did Newton name it the science indispensable tool that it is applied whenever of fluxions? we have a problem that can be solved 4 What did Newton accuse Leibniz of doing? by mathematics. 5 Leibniz and Newton had different starting points in their work on calculus - what were they?

Macmillan Gui de to Science Unit 16 99 Mathematics Before you read Discuss these questions with your partner. Who do you think were the greatest mathematicians in history? Which kinds of mathematics were they involved in? What do you know about Pierre de Fermat?

В D Vocabulary a. Match these words with their definitions. 1 devotion A don't pay attention to 2 astounding В somebody working in the same field 3 councillor С top judge 4 fellow D amazing 5 neglect E commitment 6 chief magistrate F member of an administrative body

b. Complete the sentences below with words from the box. с N • devotion • astounding v • fellow • neglect • councillor В Reading 2 V 1 She shows great to her family. Pierre de Fermat 2 Please don't to pay the phone bill. Pierre de Fermat was born in , th th 3 Meet Peter. He's a(n) film on 17 August, 1601, and died on 12 January, enthusiast! We love watching films! 1665. lie came from a wealthy family, and he studied law in Orleans. After graduating, he 4 What a(n) speech he gave began to practise law and later he became a at the conference. It was amazing! councillor in parliament. By 1652, he had 5 He's been a town for over become the chief magistrate of the criminal five years now. court - a very important and highly respected position.

c. Match the words to make phrases. In 17th century France, magistrates spent large 1 practise A theory amounts of time on their own. It was during this time that de Fermat worked in the field of 2 criminal В coordinates mathematics. In fact, his devotion to this 3 remarkable С law science was so great that he spent as much free 4 number D geometry time as he could working on mathematical 5 E achievements problems and solutions. Although de Fermat published very little in his lifetime, he is still 6 Cartesian F court considered to be one of the greatest 2 analytic G recognition mathematicians of all time. His achievements in 8 receive H theory mathematics are quite astounding.

100 Macmillan Guide to Science Unit 21 Mathematics

De Fermat's most important work was done in Pronunciation guide the development of modern , which was one of his favourite areas of magistrate Amaed3istreit/ mathematics, and which had an important Orleans hi\i?azl impact on the study of calculus. Sir Isaac Toulouse /tu:lu:z/ Newton said that his own invention of calculus - differential calculus in particular - was based BE Comprehension in large part on the work of de Fermat, who had Read the text and choose the done his studies on calculus well before Isaac correct answer. Newton and Gottfried Leibniz were born. In 1654, wrote a letter to de Fermat 1 Despite having little of his work published asking about the latter's views on probability. while alive, de Fermat Thus began a correspondence that became A had a successful career in law. the foundation of , with В is seen as a great mathematician. de Fermat and Pascal considered to be the С held an important position. founders of this theory. Rene Descartes is 2 De Fermat's work on modern numbers famous for his invention of Cartesian A was influenced by Newton's work. coordinates and his important work La В helped him develop a branch of calculus. Geometrie. De Fermat had independently come С had an effect on calculus. up with his own three-dimensional , which was more complicated and 3 Pascal and de Fermat advanced than Descartes'; Descartes' work A are recognised as the fathers of became more popular, however, because its Probabality Theory. notation was more convenient. Today, both В communicated about analytical geometry. are seen as the fathers of С produced work more advanced analytic geometry. than Descartes'. 4 De Fermat's methods and results De Fermat also made contributions in the field A were never proven in his lifetime. of , formulating a law on the way light В were ahead of their time. travels. His methods were so advanced that С suffered as he never provided proof. many of his results were not proved for a century after his death, and de Fermat's Last 5 We know about de Fermat's work Theorem took more than three hundred years nowadays because to prove. De Fermat rarely provided his proofs, A his work was published. that is, evidence or procedures for reaching В learned people researched his claims. conclusions, to explain how he got his answers. С he left behind a correspondence. In his letters to fellow mathematicians, he stated theorems but neglected the proofs, which was very annoying for them. Since he never Before you listen wanted anything to be published (as he Discuss these questions with your partner. considered mathematics to be his hobby), there What do you know about de Fermat's was nowhere for scholars to check his claims Last Theorem? and consequently during his lifetime, he Do mathematical puzzles interest you? received very little recognition as a mathematician. If the people he wrote to had What do squared and cubed mean? not saved his papers and letters, we may never have heard of de Fermat and his remarkable achievements. В F Listening H))) Listen to a teacher talking to a class about Fermat's Last Theorem. Then complete the sentences below. 1 Fermat's Last Theorem was the world's most puzzling problem.

Macmillan Gui de to Science Unit 16 101 Mathematics

2 Fermat left no of the proof Remember to: he had found. • use simple examples that everyone 3 Fermat claimed there was no solution for can understand any equation beyond the • speak slowly and clearly 4 Proof is a line of reasoning that consists of many Speaking tips 5 published a proof to in • Speak to your audience, not your notes. • Pause between sections. 0 G Speaking • Speak calmly, but not too softly. Discuss these questions with В H Writing your partner. Write a composition on the topic: 'The Is calculus only useful in mathematics and greatest mathematical puzzle of them all.' sciences such as physics? Use the information from text 2 as well How can calculus be used in other branches as your notes from exercise F. You can of knowledge? listen to the class discussion again if you want to take notes.There is a plan here Task to guide you. Prepare a short presentation to answer PARAGRAPH 1 the question: 'What is calculus?' Use Introduction Say what this puzzle is, and the information in text I. give a very brief outline of its history Talk about: (include information about how long it took to solve it). • what calculus deals with Vocabulary: Briefly ... / in brief ... • the two divisions of calculus • who developed calculus PARAGRAPH 2 Discuss what de Fermat claimed to have • how it is used today discovered (the proof for a difficult equation); say where he wrote this claim. First, complete these notes. Vocabulary: Basically the theorem was ... Use them in your presentation. PARAGRAPH 3 Calculus: The study of There are two divisions of calculus: Say why mathematicians were so fascinated by the problem (it looked simple, but was very Calculus is about rates of difficult; they believed that de Fermat had found the proof, so they wanted to find it too). Integral calculus is about Vocabulary: Despite appearing ... Appearances were deceptive because ... Two mathematicians made the biggest contributions: PARAGRAPH 4 Isaac Newton, who mainly dealt with Conclusion Say when it was solved, and who solved it. Say something about the fact Gottfried Leibniz, who mainly dealt with that de Fermat's proof has not been discovered yet. Some modern-day applications are Vocabulary: After his death ... .Not until ...

V у Write 200-250 words.

102 Macmillan Guide to Science Unit 21