Infinitesimal and infinitely large functions.

Vocabulary

 

 

1.	A is subset of B	A- лю А – подмножество В
2.	any, for any	любой, для любого
3.	exist	существует
4.	: such that	такой что
5.	x belongs to A	х принадлежит А
6.	x tends to x0	х стремится к х0
7.	mathematical analysis	математический анализ
8.	associate	ставить в соответствие
9.	the domain of function	область определения функции
10.	the range of function	множество значений функции
11.	increasing	возрастающий
12.	nondecreasing	неубывающий
13.	decreasing	убывающий
14.	nonincreasing	невозрастающая
15.	enumerated	перечисленный
16.	monotonous	монотонный
17.	strictly monotonous	строго монотонный
18.	symmetric	симметричный
19.	mutually	взаимно
20.	bounded (unbounded)	ограниченный (неограниченный)
21.	output	результат
21.	approximately	приблизительно
22.	limit	предел
23.	sufficiently	достаточно
24.	imply	влечь, иметь следствием
25.	neighborhood	окрестность
26.	quotient	частное
27.	infinitesimal	бесконечно малый
28.	infinitely large	бесконечно большой
29.	comparison	сравнение
30.	listed	перечисленные

Section III. The basic concepts of mathematical analysis.

The concept of function.

Definition 1.1: A mathematical relation such that each element of a given set (the domain of function) is associated with a unique element of another set (the range of function) is called a function and denote . is called independent variable, – dependent variable.

There are different methods of representation of functions, such that; tabular, analytical (be means of the formulas), graphical.

Definition 1.2: A set of points in the coordinate plane with coordinates is called the graph of a function .

 

Definition 1.3: If , and , then:

1) is increasing on if whenever ;

2) is nondecreasing on if, whenever ;

3) is decreasing on if whenever ;

4) is nonincreasing on if whenever .

Functions listed in 1) – 4) are called monotonous; 1), 3) are called strictly monotonous.

 

Definition 1.4: For any and , if , then a function is called an even function, if – an odd function.

Definition 1.5: Inverse function is a function obtained by expressing the dependent variable of one function as independent variable of another; functions and are mutually inverse functions if and .

Definition 1.6: Function defined on some set is called bounded on this set, if there is a positive number such that for any belonging to the set the inequality holds . Otherwise, the function is said to be unbounded.

 

Limit of a function.

Definition 2.1: The number is said to be the limit of the function as , if for all values , lying sufficiently close to , the corresponding values of the function are arbitrary close to the number

.

Definition 2.2: The number is called the limit of the function as , if for any positive number exists the positive number such, that for all , different from , and satisfying the inequality , implies the inequality .

 

Algebra of limits.

The follow rules hold if and .

1. Sum rule: ;

2. Difference rule: ;

3. Product rule: ;

4. Quotient rule: , if .

 

Let to prove the first rule. To show that we must show that , such that for all satisfying inequality implies

 

 

From

 

From

Let . It means that and hold for all such that

and we get

.

Rule is proved.

 

Infinitesimal and infinitely large functions.

y

Definition 4.1: Function is called an infinitesimal as , if This means that (however small) such that for all satisfying inequality , implies .

Definition 4.2: The inverse of an infinitesimal, that is, is an infinity large function. In this case we write .

It is possible from limit rules for functions to draw the corresponding propositions for infinitesimal functions:

1. A sum of any finite number of infinitesimals is an infinitesimal

2. The product of a bounded function and an infinitesimal is an infinitesimal.

3. The product of any finite number of infinitesimals is an infinitesimal