Multivariable Functions of Several Their Derivatives

1. Multivariable Functions of Several Their Derivatives Dr. Ching I Chen

2. 13.1 Functions of Several Variables (1)Functions and Variables

3. 13.1 Functions of Several Variables (2)Functions and Variables DefinitionsMultivariable Functions Suppose D is a set of n real number (x1, x2,… xn). A real valued functionf on D is a rule that assigns a real number w = f(x1, x2,… xn) to each element in D. D : domain w-value : function range x1, x2,… xn: n independent (input) variables of f w : dependent (output) variable of f

4. 13.1 Functions of Several Variables (3)Functions and Variables (Example 1)

5. 13.1 Functions of Several Variables (4)Domain and range (Exploration 1)

6. 13.1 Functions of Several Variables (5)Domain and range Exercises 1: To specify the function domain and its range

7. 13.1 Functions of Several Variables (6)Domain and range

8. 13.1 Functions of Several Variables (7)Domain and range

9. 13.1 Functions of Several Variables (8)Domain and range (Example 2)

10. 13.1 Functions of Several Variables (9)Domain and range

11. y=-4 to 4 x=-4 to 4 13.1 Functions of Several Variables (10)Graphs and Level Curves of Functions of Two Variables There are two standard methods to find the values of function f(x,y) • To sketch the surface z = f(x,y) in space, corresponding to each (x,y,f(x,y)). Domain is a region in the xy -plane. The output is a surface.

12. 13.1 Functions of Several Variables (11) Graphs and Level Curves of Functions of Two Variables • To draw and level curve in the domain on which f has a constant value. For a function z = f(x,y), find the solution to a suitable c such that f(x,y) = c. For example

13. graph surface level curve 13.1 Functions of Several Variables (12)Graphs and Level Curves of Functions (Example 3) Definitions Level Curve, Graph, Surface The set of points in the plane where a function f(x,y) has a constant value f(x,y) = c is a level curve of f. The set of points (x,y,f (x,y)) is space for (x,y) in the domain of f is called the graph of f. The graph of f is also called the surfacez = f (x,y)

14. The contour line f(x,y) = y2-x in plane z = 0 The level curve f(x,y) = y2-x in parabola in the xy- plane plane z = 0 13.1 Functions of Several Variables (13) Graphs and Level Curves of Functions of Two Variables z = f(x,y) = y2-x

15. 13.1 Functions of Several Variables (14)Graphs and Level Curves of Functions of Two Variables

16. 13.1 Functions of Several Variables (15) Graphs and Level Curves of Functions of Two Variables

17. 13.1 Functions of Several Variables (16)Level Surfaces of Function of Three Variables Definition Level Surface The set of points (x, y, z) in space where a function of three independent variables has a constant value f(x,y, z) = c is a level surface of f.

18. 13.1 Functions of Several Variables (17)Level Surfaces of Function of Three Variables (Ex. 4)

19. 13.1 Functions of Several Variables (18)Level Surfaces of Function of Three Variables (Ex. 5)

20. 13.2 Limits and Continuity in Higher Dimensions(2)Limits

21. 13.2 Limits and Continuity in Higher Dimensions(3)Limits

22. 13.2 Limits and Continuity in Higher Dimensions(4)Limits

23. 13.2 Limits and Continuity in Higher Dimensions(5)Limits (Theorem1)

24. 13.2 Limits and Continuity in Higher Dimensions(6)Limits (Theorem1)

25. 13.2 Limits and Continuity in Higher Dimensions(7)Limits (Theorem 1)

26. 13.2 Limits and Continuity in Higher Dimensions(8)Limits (Example 1)

27. 13.2 Limits and Continuity in Higher Dimensions(9)Limits (Example 2)

28. 13.2 Limits and Continuity in Higher Dimensions(10)Continuity

29. 13.2 Limits and Continuity in Higher Dimensions(11)Continuity (Example 3)

30. 13.2 Limits and Continuity in Higher Dimensions(12)Continuity

31. 13.2 Limits and Continuity in Higher Dimensions(13)Continuity (Exploration 1)

32. 13.2 Limits and Continuity in Higher Dimensions(14)Continuity (Exploration 1)

33. 13.2 Limits and Continuity in Higher Dimensions(15)Continuity (Exploration 1)

34. 13.3 Partial Derivatives (1)Definitions and Notation

35. 13.3 Partial Derivatives (2)Definitions and Notation

36. 13.3 Partial Derivatives (3)Definitions and Notation

37. 13.3 Partial Derivatives (4)Definitions and Notation Partial derivative To perform partial derivative with respect to one variable, one has to keep all other variables constant. For example z = f(x,y) partial derivative of function f with respect to x

38. 13.3 Partial Derivatives (5)Definitions and Notation

39. 13.3 Partial Derivatives (6)Definitions and Notation

40. 13.3 Partial Derivatives (7)Definitions and Notation Partial derivative To perform partial derivative with respect to one variable, one has to keep all other variables constant. For example z = f(x,y) partial derivative of function f with respect to y

41. 13.3 Partial Derivatives (8)Definitions and Notation

42. 13.3 Partial Derivatives (9)Definitions and Notation z = f(x,y) (a) To compute z/x, use the laws of ordinary differentiation while treating y as constant. (b) To compute z/y, use the laws of ordinary differentiation while treating x as constant. z = f(x1,x2,…xn) (a) To compute f/xi, use the laws of ordinary differentiation while treating all xk (ki) as constant.

43. 13.3 Partial Derivatives (10)Definitions and Notation (Example 1)

44. 13.3 Partial Derivatives (11)Definitions and Notation (Example 2)

45. 13.3 Partial Derivatives (12)Definitions and Notation (Example 3)

46. 13.3 Partial Derivatives (13)Definitions and Notation (Example 4)

47. 13.3 Partial Derivatives (14)Definitions and Notation (Exercise)

48. 13.3 Partial Derivatives (15, Exercise 1)Definitions and Notation Exercises 1 : Partial derivative w.r.t each variable

49. 13.3 Partial Derivatives (17)Functions of More Than Two Variables (Example 6)

50. z L1 L2 1 y x 13.3 Partial Derivatives (18)Functions of More Than Two Variables (Example 7)