Speech exercises

I. Read and reproduce the text.

The equation of a locus

We shall now study the problem of determining the equation of a locus as the analytic interpretation of the geometric condition or conditions defining the locus.

Definition. The equation of a plane locus is an equation

f (x, y) = 0 (1)

all whose corresponding real solutions for x and y are the coordinates of those points, and only those points, satisfying the given geometric condition or conditions which define the locus.

Note that this definition expresses a necessary and sufficient condition that (1) be the equation of a locus. Accordingly, the procedure in obtaining the equation of a locus is essentially as follows:

1. Let the point P with coordinates (x, y) be any point satisfying the given condition or conditions, and hence a point on the locus.

2. Express the given geometric condition or conditions analytically by means of an equation or equations in the variable coordinates x and y.

3. Simplify, if necessary, the equation obtained in Step 2 so that it assumes the form (1).

4. Conversely, let (x₁, y₁) be the coordinates of any point satisfying (1) so that the equation

f (x₁, y₁) = 0 (2)

holds. If (2) leads to the analytic expression for the given geometric condition or conditions, as applied to the point (x₁, y₁), then (1) is the required equation of the locus.

In practice, Step 4 may usually be omitted, since retracing the work from Step 3 back to Step 2 is generally immediate. Note in Step 1 that, by considering P as any point on the locus, we are thereby considering every point on the locus.

II. Topics for discussion:

1. The equation of a plane locus.

2. The procedure in obtaining the equation of a locus.

T E X T V

Grammar: the infinitive constructions (complex subject, complex object and for-phrases) exercises

I. Translate the following sentences with the Complex Object:

1. 1. I know your colleagues to work hard. 2. They expected him to be invited to the party. 3. The commission considers the project to have some drawbacks. 4. The teacher believes his student group to pass the exam in algebra. 5. She assumed his design to be the best one. 6. We expected him to be late. 7. I suppose the new film to be exciting.

2. 1. Everyone wanted him to win the race. 2. He would like me to come early. 3. I don’t want anyone to know about my leaving the job. 4. They wish the article to be published next month. 5. The teacher required us to take an examination as soon as possible. 6. The city authorities want all of us to celebrate the anniversary of its foundation.

3. 1. I didn’t hear you come in. 2. She suddenly felt someone touch her on the shoulder. 3. Did you notice anyone go out? 4. The children saw the ball break the window. 5. I have never heard her speak English. 6. During the experiment we saw the temperature fall rapidly. 7. The boy saw the sun rise above the sea.

4. 1. He doesn’t allow anyone to smoke in his house. 2. I wouldn’t recommend you to stay in this hotel. 3. Hot weather makes me feel uncomfortable. 4. Tom let me drive his car yesterday. 5. Circumstances caused them to repeat the experiment. 6. I cannot get her to finish her lessons. 7. The professor permitted his student to be helped with the solution of the problem.

II. Translate into English using the Complex Object:

1. Моя сестра хотела бы, чтобы он объяснил ей эту теорему. 2. Вы слышали когда-нибудь, как они поют? 3. Никто не ожидал, что они вернутся через два дня. 4. Я считаю, что он опытный инженер. 5. Мы видели, как она перешла улицу и села в автобус. 6. Родители ожидали, что он приедет на каникулы. 7. Все почувствовали, что вагон начал двигаться. 8. Он позволил, чтобы его материалы были использованы в моем докладе. 9. Мы считаем, что эти предложения будут учтены при решении этого вопроса. 10. Все ожидали, что делегация приедет в субботу. 11. Я знаю, что они выучили 20 новых слов к уроку.

III. Translate the following sentences with the Complex Subject:

1. 1. He is thought to be a great artist. 2. The delegation is expected to arrive tomorrow. 3. The experiments are believed to give good results. 4. They are said to have finished their work. 5. The data were supposed to be exact. 6. She was seen to leave the house. 7. The train was found to have been an hour late. 8. The results were found to have been obtained long before the experiment was completed. 9. The importance of mathematics for all sciences is known to be growing rapidly. 10. He is believed to work hard at his pronunciation. 11. They were supposed to speak about the reconstruction of the plant. 12. This method is considered to be effective.

2. 1. He seems to know physics well. 2. His friend seems to have been working as an engineer since 1980. 3. You didn’t appear to be surprised at this news. 4. She appears to have learned many poems by Robert Burns. 5. They proved to be experienced researchers. 6. This work may prove to be of great importance. 7. This law doesn’t seem to hold for all cases. 8. There seem to be some confusion of terms in the paper.

3. 1. The expedition is likely to return the day after tomorrow. 2. The plan is unlikely to be approved at the meeting. 3. They are likely to have taken part in the discussion. 4. She was certain to have passed her exams after the winter term. 5. The weather is certain to be changing. 6. He is sure to come to the lecture. 7. The plane is sure to be the quickest means of transport.

IV. Translate the sentences using the Complex Subject.

1. Оказывается, что этот закон выполняется для деления. 2. В древности считали, что кубическое уравнение неразрешимо. 3. Считают, что су- ществует несколько решений этой задачи. 4. Оказывается, что правила арифметики для действительных чисел распространяются на комплекс- ные числа. 5. Оказалось, что многие элементарные понятия в теории вероятности такие же, как и понятия в теории множеств, но с другими названиями. 6. Поскольку рассматриваемые величины, вероятно, имеют положительные и отрицательные значения, сумма членов может быть либо положительной, либо отрицательной.

V. Compare the two Infinitive Constructions: Complex Object and Complex Subject.

1. We know her to be his best friend. 2. She is known to be his best friend. 3. I expect him to come late. 4. He is expected to come late. 5. The teacher considered her to be a good student. 6. She was considered to be a good student. 7. They believed her to come tomorrow. 8. They were believed to come to their father’s birthday.

VI. Translate the following sentences with for-phrases.

1. The only conclusion for him to make was the following. 2. For a force to exist there must be two objects involved. 3. It is for him to decide. 4. Here is one more important point for a speaker to explain. 5. It is possible for a figure to have more than one axis of symmetry. 6. For the law to hold two conditions must be observed. 7. The students waited for the bell to ring. 8. Much experimental work is needed for the phenomena to be explained. 9. The temperature outside was too high for us to go to the beach. 10. It is necessary for the theorem to be proved.

VII. Translate into English using both Complex Object and Complex Subject Constructions.

1. Сообщают, что делегация уехала вчера из Москвы. 2. Говорят, что он окончил Московский университет три года тому назад. 3. Известно, что он – автор многих статей в этой области математики. 4. Несомненно, что эта статья будет напечатана в следующем номере журнала. 5. Все ожи- дают, что новый метод даст хороший результат. 6. Она, вероятно, удовлетворена своей работой. 7. Эта задача оказалась трудной. 8. Маловероятно, что она пойдет с нами на концерт. 9. Может оказаться, что они не выполнят свою работу в срок. 10. Несомненно, кто-нибудь согласится принять участие в этой дискуссии. 11. Мне бы хотелось, чтобы эта задача была решена другим способом. 12. Обстоятельства вынудили его провести лето в городе. 13. Тебе было интересно смотреть этот фильм? 14. Преподаватель подождал, пока студенты закончат переводить текст. 15. Мы поедем в отпуск в Турцию или в Египет? – Тебе решать.

VIII. Read the word and give their Russian equivalents:

function (n) ['fANkSn] formula (n) ['fO:mjulq]

graph (n) ['grxf] distance (n) ['dIstqns]

real (a) ['rIql] temperature (n) ['temprItSq]

condition (n) [kqn'dISn] reflect (v) [rI'flekt]

quality (v) ['kwOlIfaI] assistance (n) [q'sIstqns]

physical (a) ['fIzIkql] classify (v) ['klxsIfaI]

special (a) ['speSql] coordinate (n) [kou'O:dnIt]

IX. Read and learn the following words:

approach (v,n) [q'proutS] приближаться, пoдходить; подход

assign (v) [q'saIn] ставить в соответствие

assume (v) [q'sjHm] предполагать, допускать

concept (n) ['kOnsept] логическое понятие

convex (a) ['kOn'veks] выпуклый

correspond (v) ["kOrIs'pOnd] соответствовать

describe (v) [dIs'kraIb] описывать, изображать, начертить

domain (n) [dq'meIn] область (определения)

dimension (n) [dI'menSn] размерность

emerge (v) [I'mWG] появляться, выходить, выяснять

feed (v) ['fJd] подавать, питать

graph (n) ['grxf] диаграмма, график

instance (n) ['Instqns] случай, пример

mapping (n) ['mxpIN] отображение

modify (v) ['mOdIfaI] (видо)изменять, модифицировать

notion (n) ['nouSn] понятие, определение

original (a) [O'rIGInql] первоначальный

obey (v) [q'beI] удовлетворять условиям

qualify (v) ['kwOlIfaI] определять, квалифицировать

regardless (a) [rI'gRdlIs] независимо от, несмотря на

require (v) [rI'kwaIq] нуждаться, требовать

restrict (v) [rI'strIkt] ограничивать, заключать

satisfy (v) ['sxtIsfaI] выполнять, удовлетворять

statement (n) ['steItmqnt] утверждение, формулировка

suitably (adj) ['sjHtqblI] соответственно, подходяще

whereas (cj) [wFqr'xz] тогда как, поскольку

unique (a) [jH'nIk] единственный, однозначный

yield (v) ['jJld] производить, вырабатывать

NOTES

to bear in mind помнить

according to согласно

to refer to …as называть

side by side рядом

by analogy по аналогии

an ordered pair упорядоченная пара

at least по крайней мере

FUNCTIONS AND GRAPHS

The notion of function is essentially the same as that of correspondence. A numerical-valued function f assigns to each point p in its domain of definition a single real numer f(p) called the value of f at p. The rule of correspondence may be described by a formula such as

f(p)=x² – 3xy, when p=(x,y)

or by several formulas, such as

f(p) = x when x › y

x² +y y ≤ x

or by geometric description.

f(p) is the distance from p to the point (4,7)

or even by an assumed physical relationship:

f(p) is the temperature at the point p.

In all of these instances, it is important to bear in mind that the rule of correspondence is the function f, whereas f(p) is the numerical value which f assigns to p. A function may be thought of as a machine into which specific points may be fed, while the corresponding values emerge at the other end.

Real-valued functions are often classified according to the dimension of their domain of definition. If f(p) is defined for all p є S and S is a subset of the plane, then we may write p as (x,y) and f (p) as f (x,y) and refer to f as a function of two real variables. Similarly, when S is a set in 3-space, we may write f (x, y, z) for f (p) and say that f is a function of three real variables. When S is a set on the line, we usually write f (x) d call f a function of one real variable. In all these cases, however, f can still be thought of as a function defined for the single variable point p.

Other cases also arise. A function f may be defined only for points p which lie on a certain curve C in space.

Side by side with the notion of a function as a correspondence or mapping between two sets (e.g. points and numbers), we have the concept of graph. If f is a function of one real variable, the graph of f is the set f points (x, y) in the plane for which y = f(x). If f is a function of two real variables, the graph of f is the set of points (x, y, z) in 3-space for which z = F (x, y). Conversely, it is possible to base the notion of function on that of graph. Let A and B be any two sets, and let E be any set composed of ordered pairs (a, b) with a є A and b є B. By analogy, (a, b) may be called the “point” in a A x B space having coordinates a and b, regardless of the nature of the sets A and B. Any such set E can be called a graph or relation, and those that have the special property of being single-valued are called functions.

Many special properties of a function are reflected in simple geometrical properties of its graph. A function f defined on the line is said to be monotonic increasing if f (x) ≤ f (x^') whenever x < x^'; this means that the graph of f “rises” as we move along it from left to right. Again, a function of two variables is said to be convex if it obeys the condition

f (p₁) + f (p₂) ≤ 2 f ;

this says that Σ, the graph of f, is a surface with the property that if A and B are any two points on Σ, their mid – point lies on or below Σ.

Sometimes it is said that equation in x and y defines y as a function of x. This must both be explained and qualified. What is meant is that, given an equation E (x, y) = 0, one is generally able (at least in theory) to “solve for y”, getting y = f (x). Again, solution of the equation for y seldom gives a unique answer, while in writing y = f (x), we require that exactly one value of y correspond to a given value of x. We must therefore modify the original statement and say that if the function E is suitably restricted, the equation E (x, y) = 0 defines a set of functions (possibly just one) such that if f is one of these, then E (x, F (x)) = 0 for all x in the domain of definition of f. The equation x² + y² – 16 = 0 yields two functions, f (x) = √ 16 – x² and g (x) = -√16 – x².