Vocabulary and grammar exercises

I. Find Russian equivalents for the word combinations:

the same concept, to assign to, a domain of definition, the rule of correspondence, to bear in mind, to refer to… as, a value assumed, a function defined, to lie on a curve, the concept of graph, ordered pairs, to be monotonic increasing, to obey a condition, without restrictions, a unique answer, the original statement, suitably restricted.

II. Find English equivalents for the word combinations in the text.

В основном тот же, область определения, правило соответствия, геометрическое описание, помнить, в соответствии с размером, упорядоченная пара, несмотря на, особое свойство, лежать на плоскости, подчиняться условиям, без ограничений, однозначный ответ, перво- начальное утверждение, отображение, соответственно ограниченный, расширенное понятие.

III. Replace the Russian words by their English equivalents according to the text:

1. Describe (область определения) of each function f defined below. 2. (Правило соответствия) may be described by a formula. 3. A function f (может быть определена) only for points p, which lie on a certain curve C in space. 4. We must modify (первоначальное утверждение). 5. Real-valued functions are often classified (в соответствии с размером) of their domain of definition. 6. It is important (помнить) that the rule of correspondence is the function f. 7. Many (особые свойства) of a function are reflected in simple geometrical (свойства) of its graph. 8. A function of two variables, (говорят, выпуклая) if it obeys the certain conditions.

IV. Analyze and translate the sentences (Complex Object):

1. We consider all equations to be exact in the sense of the classical theory of elasticity. 2. I want the theorem to be proved at once. 3. Other mathematicians supposed the invention of analytic geometry to be credited to the fourteenth century. 4. In ancient times some philosophers thought light to be the proper- ty of the eye. 5. Scientists believed matter to be composed of many very small particles obeying the laws of classical mechanics. 6. We suppose the function f (x) to be continuous and have a continuous derivative f' (x).

V. Analyze and translate the sentences (Complex Subject):

1. Two quantities differing by an infinitesimal can be considered to be equal. 2. H. Lebesque may be said to have created the first genuine theory of integration. 3. This method is believed to be effective. 4. These data must be considered to be only approximate. 5. One can prove a theorem which may seem to disprove Euclid’s theorem and his proof. 6. Since the reader is assumed to be already familiar with connection between the derivative of a function R to R^p and the slope of its graph we shall focus our attention on the mathematical aspects of the derivative. 7. The above solution seems to be complete from the theoretical view point. 8. Axiomatic method is seen to be ultimately dependent upon the particular logic and set theory employed. 9. This method is unlikely to yield good results. 10. No one prior to Dirichlet appears to find the first truly general concept of a function as any correspondence, x→ f(x) between real numbers.

VI. Answer the questions on the text:

1. What may the notion of function be compared with? 2. What can the rule of correspondence be described by? 3. In what way are real-valued functions related to the concept of graph? 4. How are the special properties of a function reflected in geometrical properties of its graph? 5. Can it be explained that any equation in x and y defines y as a function of x?

Give a short summary of the text: