ESSENTIAL CALCULUS CH11 Partial derivatives

1. ESSENTIAL CALCULUSCH11 Partial derivatives

2. In this Chapter: • 11.1 Functions of Several Variables • 11.2 Limits and Continuity • 11.3 Partial Derivatives • 11.4 Tangent Planes and Linear Approximations • 11.5 The Chain Rule • 11.6 Directional Derivatives and the Gradient Vector • 11.7 Maximum and Minimum Values • 11.8 Lagrange Multipliers Review

3. DEFINITION A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D a unique real number denoted by f (x, y). The set D is the domain of f and its range is the set of values that f takes on, that is, . Chapter 11, 11.1, P593

4. We often write z=f (x, y) to make explicit the value taken on by f at the general point (x, y) . The variables x and y are independent variables and z is the dependent variable. Chapter 11, 11.1, P593

5. Chapter 11, 11.1, P593

6. Domain of Chapter 11, 11.1, P594

7. Domain of Chapter 11, 11.1, P594

8. Domain of Chapter 11, 11.1, P594

9. DEFINITION If f is a function of two variables with domain D, then the graph of is the set of all points (x, y, z) in R3 such that z=f (x, y) and (x, y) is in D. Chapter 11, 11.1, P594

10. Chapter 11, 11.1, P595

11. Graph of Chapter 11, 11.1, P595

12. Graph of Chapter 11, 11.1, P595

13. Chapter 11, 11.1, P596

14. DEFINITION The level curves of a function f of two variables are the curves with equations f (x, y)=k, where k is a constant (in the range of f). Chapter 11, 11.1, P596

15. Chapter 11, 11.1, P597

16. Chapter 11, 11.1, P597

17. Chapter 11, 11.1, P598

18. Contour map of Chapter 11, 11.1, P598

19. Contour map of Chapter 11, 11.1, P598

20. The graph of h (x, y)=4x2+y2 is formed by lifting the level curves. Chapter 11, 11.1, P599

21. Chapter 11, 11.1, P599

22. Chapter 11, 11.1, P599

23. DEFINITION Let f be a function of two variables whose domain D includes points arbitrarily close to (a, b). Then we say that the limit of f (x, y) as (x, y) approaches (a ,b) is L and we write • if for every number ε> 0 there is a corresponding number δ> 0 such that • If and then Chapter 11, 11.2, P604

24. Chapter 11, 11.2, P604

25. Chapter 11, 11.2, P604

26. Chapter 11, 11.2, P604

27. If f( x, y)→L1 as (x, y)→ (a ,b) along a path C1 and f (x, y) →L2 as (x, y)→ (a, b) along a path C2, where L1≠L2, then lim (x, y)→ (a, b) f (x, y) does not exist. Chapter 11, 11.2, P605

28. 4. DEFINITION A function f of two variables is called continuous at (a, b) if We say f is continuous on D if f is continuous at every point (a, b) in D. Chapter 11, 11.2, P607

29. 5.If f is defined on a subset D of Rn, then lim x→a f(x) =L means that for every number ε> 0 there is a corresponding number δ> 0 such that If and then Chapter 11, 11.2, P609

30. 4, If f is a function of two variables, its partial derivatives are the functions fx and fy defined by Chapter 11, 11.3, P611

31. NOTATIONS FOR PARTIAL DERIVATIVES If Z=f (x, y) , we write Chapter 11, 11.3, P612

32. RULE FOR FINDING PARTIAL DERIVATIVES OF z=f (x, y) • To find fx, regard y as a constant and differentiate • f (x, y) with respect to x. • 2. To find fy, regard x as a constant and differentiate f (x, y) with respect to y. Chapter 11, 11.3, P612

33. FIGURE 1 The partial derivatives of f at (a, b) are the slopes of the tangents to C1 and C2. Chapter 11, 11.3, P612

34. Chapter 11, 11.3, P613

35. Chapter 11, 11.3, P613

36. The second partial derivatives of f. If z=f (x, y), we use the following notation: Chapter 11, 11.3, P614

37. CLAIRAUT’S THEOREM Suppose f is defined on a disk D that contains the point (a, b) . If the functions fxy and fyx are both continuous on D, then Chapter 11, 11.3, P615

38. FIGURE 1 The tangent plane contains the tangent lines T1 and T2 Chapter 11, 11.4, P619

39. 2. Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z=f (x, y) at the point P (xo ,yo ,zo) is Chapter 11, 11.4, P620

40. The linear function whose graph is this tangent plane, namely 3. is called the linearization of f at (a, b) and the approximation 4. is called the linear approximation or the tangent plane approximation of f at (a, b) Chapter 11, 11.4, P621

41. 7. DEFINITION If z= f (x, y), then f is differentiable at (a, b) if ∆z can be expressed in the form where ε1 and ε2→ 0 as (∆x, ∆y)→(0,0). Chapter 11, 11.4, P622

42. 8. THEOREM If the partial derivatives fx and fy exist near (a, b) and are continuous at (a, b), then f is differentiable at (a, b). Chapter 11, 11.4, P622

43. Chapter 11, 11.4, P623

44. For a differentiable function of two variables, z= f (x ,y), we define the differentials dx and dy to be independent variables; that is, they can be given any values. Then the differential dz, also called the total differential, is defined by Chapter 11, 11.4, P623

45. Chapter 11, 11.4, P624

46. For such functions the linear approximation is and the linearization L (x, y, z) is the right side of this expression. Chapter 11, 11.4, P625

47. If w=f (x, y, z), then the increment of w is The differential dw is defined in terms of the differentials dx, dy, and dz of the independent variables by Chapter 11, 11.4, P625

48. 2. THE CHAIN RULE (CASE 1) Suppose that z=f (x, y) is a differentiable function of x and y, where x=g (t) and y=h (t) and are both differentiable functions of t. Then z is a differentiable function of t and Chapter 11, 11.5, P627

49. Chapter 11, 11.5, P628

50. 3. THE CHAIN RULE (CASE 2) Suppose that z=f (x, y) is a differentiable function of x and y, where x=g (s, t) and y=h (s, t) are differentiable functions of s and t. Then Chapter 11, 11.5, P629