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.
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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