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UNIT 9

Limit and continuity

INTRODUCTION

Exercise 1. Fill the gaps below with words of suggested meanings. In some cases the initial letters of required words are given.

Calculus is the study of how things change. It provides a _f___________________________1 (a basic structure underlying a system, concept, or text) for modelling systems in which there is change, and a way to _d___________________________2 (to form an opinion or reach a conclusion through reasoning) the predictions of such models. The ____________________________3 (the origin or main part of something) of calculus lie in some of the oldest geometry problems on record. The Egyptian Rhind papyrus (c. 1650 BC) gives rules for finding the area of a circle and the volume of a ____________________________4 (having an apex or end removed by a plane intersection) pyramid. Ancient Greek geometers investigated finding tangents to curves, the centre of gravity of plane and solid figures, and the volumes of objects formed by ____________________________5 (turning in a circle around a central point) various curves about a fixed axis. The discovery of calculus is often attributed to two men, Isaac Newton and Gottfried Leibniz, who independently developed its foundations. In their development of the calculus, both Newton and Leibniz used "infinitesimals", quantities that are infinitely small and yet nonzero. Newton and Leibniz found it _c___________________________6

(easy to use or suitable for a particular purpose) to use these quantities in their computations and their derivations of results. Ultimately, Cauchy, Weierstrass, and Riemann _________________________6

(formulate again or differently) calculus in terms of limits rather than infinitesimals. Calculus is now the basic entry point for anyone wishing to study physics, chemistry, biology, economics, finance, or actuarial science. Calculus makes it possible to solve problems as diverse as tracking the position of a space shuttle or predicting the ____________________________7 (continuous physical force exerted on or against an object by something in contact with it) building up behind a dam as the water rises. Computers have become a _v________________________8 (extremely useful or important) tool for solving calculus problems that were once considered impossibly difficult.

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Exercise 2. Choose from the listed derived words of the term “limit” and fill them in the gaps.

limitedly unlimited limit limited limitless limitation

1) The ____________________________ of 1/x is zero as x approaches infinity. 2) The system does have its ____________________________. 3) Home Telecom delivers ____________________________ high-speed Internet for one flat fee,

with the latest technology, exceeding speeds delivered by other providers. 4) Ltd. is a suffix that follows the name of a company, indicating that it is a private

___________________________ company. 5) It could also open up ____________________________ (infinite) opportunities to influence

human evolution by manipulating genetic codes. 6) More environmentally friendly options are ____________________________ available and are

considerably more expensive.

TERMINOLOGY

Exercise 3. Match the following mathematical terms with their definitions/descriptions below.

calculus (2x) differentiation integral (2x) limit (2x) derivation

1) ____________________ either a numerical value equal to the area under the graph of a function for some interval or a new function the derivative of which is the original function

2) ____________________ the branch of mathematics, incorporating algebra, geometry, and trigonometry

3) ____________________ a number that a function approaches as the independent variable of the function approaches a given value

4) ____________________ a measurement of how a function changes when the values of its inputs change

5) ____________________ the process of finding the derivative of a function at any point 6) ____________________ the study of change and motion 7) ____________________ assigns numbers to functions in a way that can describe displacement,

area, volume, and other concepts that arise by combining infinitesimal data

8) ____________________ a concept based on the idea of closeness, used primarily to assign values to certain functions at points where no values are defined

Exercise 4. Read the following text on limits and fill in the missing prepositions.

The origin ______________1 calculus goes back to ______________2 least 2500 years, when the ancient Greeks found areas using the method of exhaustion. The Greek method of exhaustion was to inscribe polygons ______________3 the figure and circumscribe polygons ______________4 the figure and then let the number ______________5 sides of the polygon increase: Let An be the area of inscribed polygon with n sides. As n increases, it appears that An becomes closer ______________6 the area of the circle. We say that the area of a circle is the limit of the areas of inscribed polygons and we write A = lim → An.

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Exercise 5. Complete the definition of a limit using the following verbs in the appropriate form. Some of them can be used more than once.

to get to approach to give to phrase to be

A limit of a function ____________________________1 the value that function ____________________________2 as the independent variable of the function ____________________________3 a ____________________________4 value. The equation limx→c f (x) = t ____________________________5 equivalent to the statement "The limit of f as x goes to c is t ." Another way to ____________________________6 this equation is "As x ____________________________7 c , the value of f ____________________________8 arbitrarily close to t ."

Exercise 6. Read the text and replace the underlined terms with their antonyms – the first letters are given. Cross out the original terms.

The stability c______________1 of temperature during the day is a good example of a fictitious n______________2 phenomenon changing continuously. We can represent the temperature at no a______________3 time during the period from midnight to 12:00 am by plotting these points and connecting them with a smooth curve, as illustrated in the figure. Notice that this curve can be drawn without dropping l______________4 a pen off the paper. Informally, we say that this is a continuous curve. We drew the curve in this manner because our intuition tells us that temperature stays v______________5 continuously with time. Most graphs of natural phenomena (temperature, growth, decay, etc.) vary brokenly c______________6 with time. Many functions have discontinuities, i.e. places where they can be evaluated. Give parts of “Definition of Continuity at a Point” into the correct order. if / conditions / A function / are / continuous / the following / at a point / satisfied: / x = a / three / f(x) / is / _______________________________________________________________________________________________________ : a) f(a) is defined b lim → 𝑓(𝑥) c) lim → 𝑓(𝑥) = 𝑓(𝑎)

Exercise 7. When is a graph continuous/discontinuous? To check your answers, fill in the gaps with the following words in their appropriate forms.

to illustrate line domain to lift value change graph to correspond

Generally, a graph is continuous if there is corresponding _____________________1 y for any value of x. It is the case when the _____________________2 illustrates some _____________________3 which is not abrupt. The graph is a continuous _____________________4 (smooth or broken). In other words, there are no gaps in the _____________________5 of the function. Graphically, a graph of a function is continuous if, when plotting it, it is drawn without _____________________6 a pen off the paper. A graph is discontinuous if there is no _____________________7 value y for one or more values of x. It is, for example, the case when the graph _____________________8 some change (process, development) which occurs in abrupt steps (the change is not gradual).

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Exercise 8. Fill in the gaps with correct prepositions

_________ 1 calculus, the derivative is a measure _________ 2 how a function changes as its input changes. Loosely speaking, a derivative can be thought _________ 3 as how much one quantity is changing in response to changes in some other quantity; for example, the derivative _________ 4 the position _________ 5 a moving object _________ 6 respect to time is the object's instantaneous velocity. (Conversely, integrating a car's velocity _________ 7 time yields the distance travelled.)

Def.: For y = f(x) we define the derivative _________ 8 f _________ 9 x, denoted _________ 10 f '(x) to be:

f '(x) = lim∆ → ( ) – ( )

if the limit exists.

If f '(x) exists, then f is said to be a differentiable function ________ 11 x. The process _________ 12

finding the derivative _________ 13 a function is called differentiation. That is, the derivative _________ 14 a function is obtained _________ 15 differentiating the function. Differentiating a function f creates a new function f' that gives, _________ 15 other things, the instantaneous rate _________ 16 change of y = f(x) and the slope _________ 17 the tangent line to the graph of y = f (x) _________ 18 each x. The domain of f' is a subset _________ 19 the domain of f.

Exercise 9. Read the text and answer the questions below. The graph could help you.

The maximum and minimum of a function, known collectively as extrema, are the largest and smallest values that the function takes at a point either within a given neighbourhood (local extremum or relative extremum) or on the whole domain of a function (global extremum or absolute extremum).

1) What is the point where the graph of a continuous function changes from (a) rising to falling ________________________ (b) falling to rising ____________________________

2) What is f(c) if there exists an interval (m, n) containing c such that f(x) f(c) for all x in (m, n) ?

Exercise 10. Read the following text and fill in the gaps (in the part defining six properties of definite integrals) with the following terms in the appropriate form.

adjacent to affect to break up the same to interchange constant a limit

Finding the area under a curve is a useful tool in a large number of problems in many areas of science, engineering, and business. Mathematically, this is integration. For a curve produced by a function, you may be able to integrate the function from a to b and calculate the area under the curve in that way. However, for curves produced from data, or for curves that are produced by some complicated functions, analytical integration may not be possible. In these cases, the most common way to find the answer is to perform the integration "numerically". This can be done in a number of ways. In general, we can prove the following: Area under a curve: If f is continuous and f(x) 0 over the interval [a, b], then the area bounded by y = f(x), the x axis (y = 0), and the vertical lines x = a and x = b is given exactly by

𝐴 = ∫ 𝑓(𝑥)𝑑𝑥

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1) We can ______________________ the limits on any definite integral; all that we need to do is tack a minus sign onto the integral.

2) If the upper and lower limits are______________________ then there is no work to do, the integral is zero.

3) where c is any number. So, as with limits, derivatives, and indefinite integrals, we can factor out a ______________________ .

4) We can ______________________ definite integrals across a sum or difference.

5) where c is any number. This property is more important than we might realize at first. One of the main uses of this property is to tell us how we can integrate a function over the ______________________ intervals, [a, c] and [c, b]. Note, however, that c doesn’t need to be between a and b.

6) The point of this property is to notice that as long as the function and ______________________ are the same, the variable of integration that we use in the definite integral will not ______________________ the answer.

GRAMMAR

Use of articles

Exercise 11. a/ Sort out the following nouns into two categories C/U

progress supply device position evidence research information recommendation

COUNTABLE UNCOUNTABLE

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b/ Match the nouns from the exercise 10. with their C/U counterparts below and complete the table.

status availability

advice improvement

academic paper fact

equipment proof

Exercise 12. Quantifiers: which of the following expressions are used with countable (C) nouns, uncountable (U) nouns or with both (C + U)?

many much (a) few (a) little

every a great deal of a large number of a large/small amount

several some a lot of a majority of

Exercise 13. Choose the correct alternative to complete the following sentences.

1) Such feedbacks are/feedback is vital when analysing the queries. 2) There is too few/too little information to make a decision. 3) The newly employed marketing manager has much/many experience in both the public

and private sectors. 4) The news are/is on in a few minutes. 5) There are few/is little knowledge about the best way to do this. 6) How much advice/many advices were they given before starting a new research project? 7) This is an equipment /a piece of equipment that controls the speed of rotation. 8) There is little/few hope of finding a solution to this problem.

Definite and indefinite articles

Unless they are uncountable, all nouns need an article when used in the singular. The article can be either a/an or the.

Compare the following sentences fill the gaps below

A. Research is an important activity in universities. B. The research begun by Dr Stewart was continued by Professor Mathews. C. An interesting piece of research was conducted among 200 patients in the clinic.

In _______ a specific piece of research is identified. In _______ research, which is usually uncountable, is being used in a general sense. In _______ the research is being mentioned for the first time

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Exercise 14. Use the sentences below to complete the following table with rules

Rule Example

Use "a" / "an"

to introduce new information, i.e. something is mentioned for the first time

to refer to a “member of a class of objects”, to show the person/ thing is one of a group

when the person or thing we are talking about is not specific in front of an adjective which is intended to mean “having this particular quality”

to refer to profession instead of the number “one”

Use "the"

when talking about something which is already known to the listener or which has been previously mentioned or discussed

When it is clear from the situation/ context which thing or person we mean, or it is shared knowledge

when referring to a single, uniquely determined object , i.e. there is only one such thing.

in the plural when referring to all the elements of the class

before superlatives and ordinal numbers

in the construction: the + property (or another characteristic) + of +object

Other: playing instrument, job titles and official titles, rivers and canals, oceans and seas, theatre and galleries, hotels …

'Zero article'

in the plural— to talk about things in general

nouns referring to activities in front of numbered objects the names of mathematical and other disciplines Other: For years, months, days , means of transport (in general), cities, countries, streets, most companies …

A. We sometimes use this notation not to specify the universe of discourse but to restrict attention to a subset of the universe.

B. The first chapter deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers.

C. The CEO of Versa-Net is coming to our meeting. D. The meeting will begin at eight. E. The existence of test functions is not evident. F. Can you pass me a paper clip from that box by your side? G. A remarkable feature of the solution should be stressed. H. Calculators are useful. I. The meeting will be held on Thursday. J. Mr. Whong is staying at the Tower Hotel. K. Order of elements in a set can be changed and the set remains the same. L. We should arrange a meeting to talk about this issue. M. Section 4 then gives a very short summary of the problem.

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N. The natural numbers are simply the counting numbers you first learned as a child. O. The four centres lie in a plane. P. About half of this course is devoted to arithmetic and algebra. Q. Before the age of twenty-five Gauss was famous as a mathematician and astronomer. R. This problem can be solved only by a mathematician.

Exercise 15. Read the following interview with mathematician Peter Cameron and fill in the blanks with “the“, “a /an“ and (-) to show “zero“ article.

Peter Cameron has been ________1 professor of ________2 mathematics at Queen Mary, University of London for nearly 25 years. Jacob Aron, ________3 writer for the Maths Careers website, talked to him about what it means to be ________4 professional mathematician. Jacob Aron: What was it that first got you interested in ________5 maths?

Peter Cameron: I've always been interested in ________6 numbers, ever since I was ________7 small child. I grew up in ________8 Australia, and I didn't realise that you could actually be ________9 mathematician until I went to university to be interviewed, and I thought to myself “those people sitting on ________10 other side of ________11 table, they're mathematicians, that's their job, I could be one of those!” From that time on there was no doubt that was what I wanted to do, so I did my first degree there, got ________12 scholarship that brought me to ________13 Britain, and I've stayed ever since. Jacob Aron: So what does ________14 professional mathematician actually do? Peter Cameron: Basically my job is teaching and ________15 research with a little bit of ________16 administration thrown in, though I try to minimise ________17 bureaucracy as much as I can. Jacob Aron: What does mathematical research involve? Peter Cameron: People say "how do you do ________18 research in mathematics, hasn't it all been discovered?" or they say "do you just do bigger and bigger sums?" or something like that - no! If you buy the newspaper and turn to ________19 Sudoku page it says "use ________20 reasoning and ________21 logic to solve ________22 puzzle, no mathematics is required". That always annoys me, because ________23 mathematics is nothing if it isn't ________24 reasoning and ________25 logic. ________26 Mathematics is simply applying reasoning and ________27 logic in new situations, where it hasn't been applied before. Jacob Aron: What should people do if they want to become a professional mathematician? Peter Cameron: Do ________28 mathematics degree first, learn as much ________29 mathematics as you can. Then probably you'll go on to a masters and a PhD - it's more of the same but with ________30 difference. In a masters you do some courses, you learn some things, but at the same time you're expected to take some piece of mathematics and explain it in a very long project, in your own words, and if possible to improve on ________31 presentation, or perhaps even to shorten ________32 proofs or come up with some new ideas. Jacob Aron: Do you have any advice for people currently studying maths? Peter Cameron: ________33 most important advice is, if you enjoy it, stick to it, because that will take you far further than anything else. If you think you are ________34 mathematician, give it a fair run, try it out, do lots of ________35 mathematics, and if you're still enjoying it then that's obviously ________36 thing for you, go for it. Jacob Aron: And if you don't enjoy it? Peter Cameron: Then well maybe you could become a footballer or something else!

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VOCABULARY

Exercise 16. Fill the gaps below with words of suggested meanings.

Calculus in the Real World

Isaac Newton introduced calculus as a __________________________1 (an action or system by which a result is achieved; a method) of studying gravity - perhaps you have even heard of Newton's Law of Motion. Simply put, the Law of Motion surmises that: (1) an object will not move unless it is forced to do so; (2) the formula F = ma represents the relationship between __________________________2 (increasing speed) and applied force; (3) every action has a counteraction. The Law of Motion has been extended to include other areas that Newton himself likely never considered. Calculus helps explain how things change over time and it is used to gather and analyse information in engineering, science, medicine and advanced mathematics. Some of the most __________________________3 (not regarded as likely to happen) activities require complex formulas used in calculus. Every time we used our credit card, filled a prescription or pumped gasoline, a calculus formula was used to calculate it. __________________________4 (experts in economics) use calculus to predict such things as potential earnings, stock market changes and expected business profits. __________________________5 (people who purchases goods and services for personal use) rely on such forecasts, or predictions, to make decisions about spending. For example, if economists expect the housing market percentage rates to increase within the next five years, a consumer may decide that buying a house now is in his or her best interest. For this reason, forecasts need to be as __________________________6 (precise) as possible with little room for error.

Exercise 17. Discuss in pairs. If you possess a large amount of money, what are the advantages and disadvantages of the following?

putting it under the mattress buying a lottery ticket buying bonds putting it in a bank

buying gold buying shares buying a Van Gogh painting investing in property or real estate

The Stock Market

Exercise 18. Read the text and answer the questions below.

The stock market is the platform through which shares – or pieces of ownership of a company - are bought and sold by investors, investors who own shares of a company are referred to as shareholders. Thus, the stock exchange allows investors to potentially improve their worth (provided the stock price of their investments increases, or provided they receive dividends, or small, pre-planned payments from a company paid to shareholders), and companies to have the benefit of being publically operated, and also, for company founders to cash-in

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on stock (by selling their shares of the company once it goes public). Trading shares is a relatively straightforward process. Through a licensed stockbroker, brokerage firm, or trading website, one simply places an order for the desired number of stock in a designated company, a small fee is usually paid to the party responsible for performing the trade (be it a person, firm, or website). There is always another individual looking to sell or buy a particular stock, given the magnitude of the exchange, and there are therefore almost never delays in the process. There are also a number of other, more complex stock purchase and sale types for buyers and sellers to choose from. Bonds are a different story. They are essentially loans the company takes out from bondholders, who can be retail investors – the little guy, you and me – or whoever else: pension funds, central banks and sovereign wealth funds are big bond buyers. Bonds don't give their holders an ownership stake; they represent a debt owed by the company, which makes interest or „coupon“ payments until the bond has matured – expired, essentially. Then the company pays back the face value. (This is a generic example; the exact terms vary.) When things are running smoothly, shareholders have more clout than bondholders, due to their voting power. When a company goes into bankruptcy, however, the bondholders (or "creditors") get first dibs on the company's assets, while the once-mighty owners receive their cut of whatever's left, if anything. Anyone who owns stock in a company owns a piece of its assets relative to their share count. For example, a company with a stock limit (which is determined during an IPO, or initial public offering, wherein a company’s initial price and stock count are set before it debuts on the exchange), of 100 (hypothetically speaking, of course) would be 25 % owned by an individual who possessed 25 shares.

1. What’s traded on the stock market? a) money, from investor to investor b) shares, or pieces of publically traded companies c) property and other physical assets d) privately owned companies

2. How can stock be purchased by an investor?

a) through a licensed stock-trading website b) through a licensed stockbroker c) through a licensed stock brokerage firm (as opposed to an individual broker) d) all of the above

3. How can each stock be bought and sold at any time; how are there so many different customers?

a) stocks that nobody wants are sold into thin air b) certain stocks cannot be bought and sold at one’s convenience c) the stock exchange is a massive international platform that bases it stocks‘ prices on

demand, and there are therefore always buyers and sellers available d) some companies buy their own stock back

4. What is an IPO (initial public offering)?

a) any company’s scheduled, fixed-amount payout to investors b) the trading price of a company that’s making its stock exchange debut c) the amount a publicly held company pays to become privately traded d) a company’s value

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5. When can a bondholder get money back? a) when the bond matures b) when the company’s stock rises c) at any time d) when stock market declines by 10 %

Exercise 19. Complete the passage about a transaction on the stock exchange.

Janis Williams has decided to invest a part of her savings on the ___________________1. She contacts a ___________________2 for further advice about how to do this, and he recommends a selection of different companies‘___________________3 that she might be interested in buying. He also explains how these companies have performed in the past and how much she can expect to receive in ___________________4. He tells her how the transaction will be carried out and lets her know how much ___________________5 she will have to pay for the service. Once Janis has agreed to the terms, he contacts his representative on the stock exchange, who arranges the transaction. At the end of the day the transaction has been completed and Janis has become a ___________________6.

Exercise 20. Classify the following sentences, according to whether you think the verb or expression means:

(A) to rise after previously falling (B) to rise a little (C) to rise a lot

(D) to fall a little (E) to fall a lot

1) _____ Volkswagen shares rocketed after the revelation that Porsche has upped its stake in the company to 74 %.

2) _____ The Sensex index of the Bombay Stock Exchange crashed on Monday on fears of a recession in the US.

3) _____ Visa shot up yesterday on the NYSE on its first trading day, rising as high as $69 a share.

4) _____ The Footsie revived a little in London in the afternoon, gaining 30 points in late trading.

5) _____ After the strong gains of last week, Asian shares slipped on fears of a looming recession.

6) _____ In Milan, the S&P/MIB index plummeted, after the unions called for a three-day general strike next week.

7) _____ Leading shares were slightly weaker in Switzerland, the Swiss Market Index losing 20 points.

8) _____ Share prices recovered in Hong Kong today, the Hang Seng finishing up ten points. 9) _____ On the São Paulo Exchange, the Bovespa Index advanced a little, up 12 points. 10) _____ Chinese shares jumped after a two-thirds cut in a securities trading tax. 11) _____ Even after the government bailout, Citigroup is continuing to plunge, now down to

$1.95. 12) _____ Most shares were a little stronger in Madrid this morning, when the exchange

reopened after yesterday’s public holiday.

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TEXT PRACTICE

Exercise 21. Read the following text and put the paragraphs in the correct order.

The New York Stock Exchange

A. The New York Stock Exchange uses two methods of trading, brokers and all-electronic. Regardless of the method of exchange, all stock transactions are an auction.

B. Most of the 10 billion transactions occur electronically. A computer acts as the dealer, matching up buyers and sellers. Even the brokers and dealers get their information and trade electronically.

C. The New York Stock Exchange (NYSE) is the world’s largest securities exchange. It provides a Marketplace for buying and selling 9.3 million corporate stocks and other securities a day. It lists 82 % of the S&P 500, 90 % of the Dow Jones Industrial Average, and 70 of the world’s largest corporations.

D. Brokers actively trade stocks on the floor of the NYSE. Buyers and sellers auction securities for the highest price. Brokers represent the entity buying the stock, whether it’s for a retail brokerage company or institutional investors such as pension funds. The brokers set the bid“ price, which is the price you’re willing to pay for the stock. When your stockbroker executes your order to sell, it is not completed until one of the dealers on the floor of the NYSE finds another broker to buy it.

E. Before trading, brokers and dealers must get approved by the NYSE and hold a trading license. The dealers match up the brokers with the stock sellers, who submit an „ask“ price. It’s usually higher than the bid price. In this way, it’s like selling a home. The dealer is like the real estate agent, who puts the buyer and seller together. Dealers get to pocket the difference between the ask and bid price (minus fees and expenses) for their work.

Exercise 22. In the previous text find the words which are being defined below.

1) ________________ (paragraph A) an individual or firm that charges a fee or commission for executing buy and sell orders submitted by an investor

2) ________________ (paragraph A) a system where potential buyers place competitive bids on assets and services

3) ________________ (paragraph B) an occasion when someone buys or sells something, or when money is exchanged or the activity of buying or selling something

4) ________________ (paragraph C) to keep a record of short pieces of information 5) ________________ (paragraph D) to do or perform something, especially in a planned way 6) ________________ (paragraph D) a person or firm in the business of buying and selling

securities for their own account, whether through a broker or otherwise 7) ________________ (paragraph E) an amount of money paid for a particular piece of work

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