Содержание
- 2. Literature: [1], v. І, p. 174-180, [2], part І, p. 40-57, [3], p. 174-181. Theme: The
- 3. 1.The main definitions 2.The different ways of representing of the functions 3. The main characteristics of
- 4. Given two sets X and Y Definition. A function is a rule which assigns to each
- 5. Examples. D(y) - ? 1) f(x)=x3-4x+2 (polynomial of the third power) 2) 3) . 4) .
- 6. The most important ways of representing of the functions: - the analytic method; - the tabular
- 7. The analytic method: The function y=f(x) is represented analytically if the variables x and y are
- 8. 5) The demand function: q - price, p – demand 6) Cost function V(x), income function
- 9. The tabular method:
- 10. The graphical method:
- 11. The main ways of the graph transformations 1) Right-left translation: Example: y=(x-1)2
- 12. 2) Up-down translation: Example: Sketch the graph y=x2+4x+1 - ?
- 13. 3) Changing scale: stretching and shrinking Example: Sketch the graph
- 14. Example: Sketch the graph:
- 15. The main characteristics of behavior of the function monotonic function (increasing or decreasing): - increasing: -
- 16. The basic elementary functions: The power function; The exponential function; The logarithmic function; The trigonometric functions
- 17. 1) The power function: Some particular cases: , . - even; - decreasing on [-∞;0]; -
- 18. : , . - odd; - increasing on ;
- 19. : - even; - increasing on (-∞; 0), - decreasing on (0; +∞);
- 20. : , . - odd; - decreasing on and .
- 21. 2) The exponential function: (a>1) (0 , , . If 0 if a>1 then the function
- 22. 3) The logarithmic function: y=logax , . If 0 if a>1 then the function is increasing.
- 23. 4) The trigonometric functions: , . The function is odd and periodic, period T=2π
- 24. b) y=cosx , . The function is even and periodic, period T=2π
- 25. c) y=tgx , . The function is odd and periodic, period T=π
- 26. d) y=сtgx , . The function is odd and periodic, period T=π
- 27. 5) The inverse trigonometric functions: а) y=arcsinx , . The function is odd and increasing
- 28. b) y=arccosx , . The function is decreasing
- 29. c) y=arctgx, . The function is odd and increasing
- 30. d) y=arcсtgx, . The function is decreasing
- 31. Examples 3)
- 32. y=f(u), u=g(x) y=f(g(x)) – the composite function u – the intermediate variable, f(u) - external function,
- 33. An elementary function is a function of one variable built from a finite number of the
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