1. ESSENTIAL CALCULUS CH11 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. is formed by lifting the level curves.
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. The partial derivatives of f at (a, b) are
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
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. The tangent plane contains the tangent lines T1 and T2
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
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, namely3.
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. where ε1 and ε2→ 0 as (∆x, ∆y)→(0,0).
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

51. Chapter 11, 11.5, P630

52. Chapter 11, 11.5, P630

53. Chapter 11, 11.5, P630

54. 4. THE CHAIN RULE (GENERAL VERSION) Suppose that u is a differentiable function of the n variables x1, x2,‧‧‧,xn and each xj is a differentiable function of the m variables t1, t2,‧‧‧,tm Then u is a function of t1, t2,‧‧‧, tm and
for each i=1,2,‧‧‧,m.
Chapter 11, 11.5, P630

55. Chapter 11, 11.5, P631

56. F (x, y)=0. Since both x and y are functions of x, we obtain
But dx /dx=1, so if ∂F/∂y≠0 we solve for dy/dx and obtain
Chapter 11, 11.5, P632

57. so this equation becomes
F (x, y, z)=0
But and
so this equation becomes
If ∂F/∂z≠0 ,we solve for ∂z/∂x and obtain the first formula in Equations 7. The formula for ∂z/∂y is obtained in a similar manner.
Chapter 11, 11.5, P632

58. Chapter 11, 11.6, P636

59. 2. DEFINITION The directional derivative of f at (xo,yo) in the direction of a unit vector u=<a, b> is
if this limit exists.
Chapter 11, 11.6, P636

60. 3. THEOREM If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector u=<a, b> and
Chapter 11, 11.6, P637

61. 8. DEFINITION If f is a function of two variables x and y , then the gradient of f is the vector function ∆f defined by
Chapter 11, 11.6, P638

62. Chapter 11, 11.6, P638

63. 10. DEFINITION The directional derivative of f at (x0, y0, z0) in the direction of a unit vector u=<a, b, c> is
if this limit exists.
Chapter 11, 11.6, P639

64. Chapter 11, 11.6, P639

65. Chapter 11, 11.6, P639

66. Chapter 11, 11.6, P640

67. 15. THEOREM Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative
Du f(x) is │▽f (x)│ and it occurs when u has the same direction as the gradient vector ▽ f(x) .
Chapter 11, 11.6, P640

68. The equation of this tangent plane as
The symmetric equations of the normal line to soot P are
Chapter 11, 11.6, P642

69. Chapter 11, 11.6, P644

70. Chapter 11, 11.6, P644

71. Chapter 11, 11.7, P647

72. 1. DEFINITION A function of two variables has a local maximum at (a, b) if f (x, y) ≤ f (a, b) when (x, y) is near (a, b). [This means that
f (x, y) ≤ f (a, b) for all points (x, y) in some disk with center (a, b).] The number f (a, b) is
called a local maximum value. If f (x, y) ≥ f (a, b) when (x, y) is near (a, b), then f (a, b) is a local minimum value.
Chapter 11, 11.7, P647

73. 2. THEOREM If f has a local maximum or minimum at (a, b) and the first order partial derivatives of f exist there, then fx(a, b)=1 and fy(a, b)=0.
Chapter 11, 11.7, P647

74. A point (a, b) is called a critical point (or stationary point) of f if fx (a, b)=0 and fy (a, b)=0, or if one of these partial derivatives does not exist.
Chapter 11, 11.7, P647

75. Chapter 11, 11.7, P648

76. If D>0 and fxx (a, b)>0 , then f (a, b) is a local minimum.
3. SECOND DERIVATIVES TEST Suppose the second partial derivatives of f are continuous on a disk with center (a, b) , and suppose that
fx (a, b) and fy (a, b)=0 [that is, (a, b) is a critical point of f]. Let
If D>0 and fxx (a, b)>0 , then f (a, b) is a local
minimum.
(b)If D>0 and fxx (a, b)<0, then f (a, b) is a local maximum.
(c) If D<0, then f (a, b) is not a local maximum or minimum.
Chapter 11, 11.7, P648

77. NOTE 1 In case (c) the point (a, b) is called a saddle point of f and the graph of f crosses its tangent plane at (a, b).
NOTE 2 If D=0, the test gives no information: f could have a local maximum or local minimum at (a, b), or (a, b) could be a saddle point of f.
NOTE 3 To remember the formula for D it’s helpful to write it as a determinant:
Chapter 11, 11.7, P648

78. Chapter 11, 11.7, P649

79. Chapter 11, 11.7, P649

80. Chapter 11, 11.7, P651

81. 4. EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO VARIABLES If f is
continuous on a closed, bounded set D in R2, then f attains an absolute maximum value f(x1,y1) and an absolute minimum value f(x2,y2) at some points (x1,y1) and (x2,y2) in D.
Chapter 11, 11.7, P651

82. 1. Find the values of f at the critical points of in D.
5. To find the absolute maximum and minimum values of a continuous function f on a closed, bounded set D:
1. Find the values of f at the critical points of in D.
2. Find the extreme values of f on the boundary of D.
3. The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.
Chapter 11, 11.7, P651

83. Chapter 11, 11.7, P652

84. Chapter 11, 11.8, P654

85. Chapter 11, 11.8, P655

86. (a) Find all values of x, y, z, and such that
METHOD OF LAGRANGE MULTIPLIERS To find the maximum and minimum values of f (x, y, z) subject to the constraint g (x, y, z)=k [assuming that these extreme values exist and ▽g≠0 on the surface g (x, y, z)=k]:
(a) Find all values of x, y, z, and such that
and
(b) Evaluate f at all the points (x, y, z) that result from step (a). The largest of these values is the maximum value of f; the smallest is the minimum value of f.
Chapter 11, 11.8, P655

87. Chapter 11, 11.8, P657

88. Chapter 11, 11.8, P657