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III. Give the Russian equivalents of the following words and word
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- IV. Give the English equivalents оf the following words and word combinations
- Read and translate the following sentences, write 3-4 special questions to eaсh of them.
- CHECKING VOCABULARY IN ADVANCED OPERATIONS HIGHER MATHEMATICS I. Choose the appropriate answer.
- II. Give the English equivalents of the following words and word combinations
- Translate the text without using a dictionary. INTEGRAL EQUAТIONS
- UNIT 5 GEOMETRY BASIC TERMINOLOGY
- TEXT I. THE MEANING OF GEOMETRY
- Read and decide which of the statements are true and which are false. Change the sentences so they are true.
- Match the terms from the left column and definitions from the right column
- Read the sentences and think of a word which best fits each space. a)
- For ideas
- TEXT II. ANGLES An angle is a configuration
- Acute angle
- Read and decide which of the statements are true and which are false. Change the sentences so they are true.
- III. Give the Russian equivalents of the following words and word combinations
- Give the English equivalents of the following words and word combinations
III. Give the Russian equivalents of the following words and word combinations: nоn-negative integer, general term, upper limit, finite sequence, divergent series, lower limit, finite series, infinite sequence, domain, infinite series, summation, convergent, limiting value, convergent series. IV. Give the English equivalents оf the following words and word combinations: cходящийся ряд, предельное значение, конечная последовательность, область определения, суммирование, расходящийся ряд, бесконечная последовательность, конечный ряд, бесконечный ряд, сходящаяся последовательность, верхний предел, общий член, нижний предел, неотрицательное целое число.
questions to eaсh of them. 1. Тhе terms аге ordered bу matching them оnе bу оnе with the positive integers. 2. А finite sequence has а finite number of terms. 3. Аn infinite sequence can approach а limiting value. 4. Uppеr and lower limits indicate the values of the variable. 5. Аn infinite series can be either a convergent or a divergent series.
VI. Translate the text into English. Ряды Выражение вида и₁ + и₂ + и₃ + .........+ иռ + .......... где и₁, и₂ , и₃ - члены некоторой бесконечной последовательности, называется бесконечным рядом или просто рядом. Член и называется общим членом ряда. Обозначим сумму n первых членов ряда через Sռ, т.е. Sռ= и₁ + и₂ + u₃ +... + иռ Сумма Sռ называется частичной суммой ряда. При изменении п меняется и Sռ; при этом возможны два случая: 1). Величина Sռ при n → ∞ имеет предел S, т.е. 1imSռ = S n → ∞ 2) Величина Sռ при n → ∞ предела не имеет или предел ее равен. В первом случае ряд называется сходящимся, а число S = lim Sռ его
суммой. Во втором случае ряд называется расходящимся. Такой ряд суммы не имеет. CHECKING VOCABULARY IN ADVANCED OPERATIONS & HIGHER MATHEMATICS I. Choose the appropriate answer. 1. А variable whose limit is zero: (А) infinitesimal (D) unknown quantity (В) derivative (Е) constant (С) absolute value (F) limit 2. А positive and negative change in а variable: (А) increment (D) derivative (В) argument (Е) infinity (С) function (F) series 3. Тhе interval which doesn't contain the end points: (А) segment (D) partly ореn interval (В) closed interval (Е) straight line (С) ореn interval (F) curve 4. Аn equation which is true for аall values оf the variable: (А) conditional equation (D)simple linear equation (В) identical equation (Е) differential equation (С) integral equation (F) quadratic equation 5. Тhе indicated sum оf the terms оf а sequence: (А) finite sequence (D) general term (В) series (Е) summation (С) infinite sequence (F) I don't know II. Give the English equivalents of the following words and word combinations: бесконечно малая величина, извлечение корня, значение степени, подкоренное выражение, возведение в степень, тело, криволинейные фигуры, касательная, бесконечность, сходящаяся последовательность.
INTEGRAL EQUAТIONS It is аn equation that involves аn integral of an unknown function. А general integral equation of the third kind has the form ь u(x)g(x)= f(x) + λ ∫ K(x,y)g(y)dy а where the functions и(х), f(x) and К(х, у) are known and g is the unknown function. Тhе function К is the kernel (1) of the integral equation and is the parameter. Тhе limits of integration mау bе constants or mау bе functions of х. If и(х) is zero, the equation becomes аn integral equation of the first kind - i.e. it can bе put in the form:
If u(x)=1, the equation becomes an integral equation of the second kind: ь g(x) = f(x) + λ ∫ К(х, y)g(y)dy а Аn equation of the second kind is said to bе homogeneous (2) if f(x) is zero.
If the limits of integration, а and b, are constants then the integral equation is а Fredholm integral equation. If а is а constant and b is the variable х, the equation is а Volterra integral equation. UNIT 5 GEOMETRY BASIC TERMINOLOGY
TEXT I. THE MEANING OF GEOMETRY 1. Geometry is a very old subject. 2. It probably began in Babylonia and Egypt. 3. Men needed practical ways for measuring their land, for building pyramids, and for defining volumes. 4. The Egyptians were mostly concerned with applying geometry to their everyday problems. 5. Yet, as the knowledge of Egyptians spread to Greece the Greeks found the ideas about geometry very intriguing and mysterious. 6. The Greeks began to ask "Why? Why is that true?" 7. In 300 B. C. all the known facts about Greek geometry were put into a logical sequence by Euclid. 8. His book, called Elements, is one of the most famous books of mathematics. 9. In recent years men have improved on Euclid's work. 10. Today geometry includes not only the study of the shape and size of the earth and all things on it, but also the study of relations between geometric objects. 11. The most fundamental idea in the study of geometry is the idea of a point. 12. We will not try to define what a point is, but instead discuss some of its properties. 13. Think of a point as an exact location in space. 14. You cannot see a point, feel a point, or move a point, because it has no dimensions. 15. There, are points (locations) on the earth, in the earth, in the sky, on the sun, and everywhere in space. 16. When writing about points, you represent the points by dots. 17. Remember the dot is only a picture of a point and not the point itself. 18. Points are commonly referred to by using capital letters. 19. The dots below mark points and are referred to as point A, point B, and point C. .B .А .C Lines and Line Segments 20. If you mark two points on your paper and, by using a ruler, draw a straight line between them, you will get a figure. 21. The figure below is a picture of a line segment. 
22. Points D and E are referred to as endpoints of the line segment. 23. The line segment includes point D, point E, and all the points between them. 24. Imagine extending the segment indefinitely. 25. It is impossible to draw the complete picture of such an extension but it can be represented as follows. 
26. Let us agree on using the word line to mean a straight line. 27. The figure above is a picture of line DE or line ED.
1. A curve can be considered as the path of a moving point. 2. There're two types of curves: algebraic curves and transcendental curves. 3. Open curves have no end points and closed curves have a lot of end points. 4. A curve that does not lie in a plane is a skew or twisted curve. 5. A curvature is the rate of change of direction of a curve at a particular point on that curve. 6. The angle δψ through which the tangent to a curve moves as the point of contact moves along an arc PQ is the total curvature of this arc. 7. We define the mean curvature of any arc taking into account both the total curvature and the arc length. 8. At any point on a surface the curvature doesn't vary with direction.
from the right column:
a) 1. The Egyptians were mostly concerned with applying … to their everyday problems. 2. In 300 B.C. all the known facts about Greek geometry were put into a logical sequence by … . 3. Today geometry includes not only the study of the … and … of the earth and all things on it, but also the study of relations between geometric … . 4. The most fundamental idea in the study of geometry is the idea of a … .
For ideas: shape, point (4), size, geometry, Euclid, object. b) 1. ... are generally studied as graphs of equations using a coordinate systems. 2. Only ... curves (or arcs) have end points. 3. A curve that does entirely in a plane is a ... curve. 4. A curve that does not lie in ... is a skew or twisted curve. 5. The rate of change of direction of a curve at a particular point on that curve is called a ... 6. The angle δψ through which the tangent to a curve moves as the point of contact moves along any arc is the ... of this arc. 7. The ... of any arc is defined as the total curvature divided by the arc length. 8. The circle of curvature at any point on a curve is the circle that is ... to the curve at that point. 9. There are two ... in which the radius of curvature has an absolute maximum and absolute minimum. 10. The principal curvatures at the point are the curvatures in two ... directions. TEXT II. ANGLES An angle is a configuration of two lines (the sides or arms) meeting at a point (the vertex). Often an angle is regarded as the measure of rotation involved in moving from one initial axis to coincide with another final axis (termed a directions angle). If the amount and sense of the rotation are specified the angle is a rotation angle, and is positive if measured in an anticlockwise sense and negative if in a clockwise sense. Angles are classified according to their measure:
Change the sentences so they are true. 1. An angle is often regarded as the measure of rotation involved in moving from one initial axis to coincide with another final axis. 2. There’re eleven types of angles in their classification according to their measure. 3. 90° - it is the measure of an acute angle. 4. An angle is positive if it is measured in a clockwise sense. 5. The measure of a reflex angle is between 180°and360°. 6. The main difference of an angle of elevation of a points and its angle of depression is the following one: in the case of the angle of elevation the point A lies above the plane and in the case of the angle of depression - below the plane.
from the right column:
III. Give the Russian equivalents of the following words and word combinations:
and word combinations: тупой угол, развёрнутый угол, нулевой угол, угол возвышения, угол понижения, прямой угол, полный угол, сторона, направление вращения, вершина, угол в пределах от 180° 360, вращение (поворот), острый угол, по часовой стрелке, против часовой стрелки, угол вращения, направляющий угол. жүктеу/скачать 1.41 Mb. Достарыңызбен бөлісу:
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