Calculo 2

C H A P T E R 1 2 Functions of Several Variables
Section 12.1 Introduction to Functions of Several Variables . . . . . . . 76 Section 12.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . 80 Section 12.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . 83 Section 12.4 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . 88 Section 12.5 Chain Rules forFunctions of Several Variables . . . . . . . 92 Section 12.6 Directional Derivatives and Gradients . . . . . . . . . . . . 98 Section 12.7 Tangent Planes and Normal Lines . . . . . . . . . . . . . 103 Section 12.8 Extrema of Functions of Two Variables . . . . . . . . . . 109 Section 12.9 Applications of Extrema of Functions of Two Variables . 113

Section 12.10 Lagrange Multipliers . . . . . . . . .. . . . . . . . . . . 119 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

C H A P T E R 1 2 Functions of Several Variables
Section 12.1 Introduction to Functions of Several Variables
Solutions to Odd-Numbered Exercises
1. x 2z z x2 x2 4 y2 9 z2

yz y z

xy 10 10 x2

10 xyxy y

3.

1

No, z is not a function of x and y. For example, x, y 0, 0 corresponds to both z ± 1.

Yes, z is a function of x and y. x y 3 2 5 y (b) f 1, 4 x 2 xy z 2 3 9 1 0 1 2 3 0 1 4 (c) f 30, 5 (f) f 5, t 5 t 30 5 6

5. f x, y (a) f 3, 2 (d) f 5, y

(e) f x, 2

7. f x, y (a) f 5, 0

xe y 5e0 3e 2 1 2e 5e y xe 2 tet
1

9. h x, y, z 5

(b) f 3, 2 (c) f 2, (d) f 5, y (e) fx, 2 (f) f t, t

(a) h 2, 3, 9 2 e (b) h 1, 0, 1

y

11. f x, y (a) f 2,

x sin y 4 2 sin 3 sin 1 4 2

13. g x, y
x

2t
4

3 dt
4

(a) g 0, 4
0 4

2t 2t
1

3 dt 3 dt

t2 t2

3t
0 4

4 6
1

(b) f 3, 1

(b) g 1, 4

3t

15. f x, y (a) f x

x2

2y x, y x f x, y x x2 2x 2y x
2

2y x x y y
2

x2 2y x2

2y x2 2y 2y x2 2y x 2x x 2 y y x2 x

x

x (b)76 f x, y y y f x, y x2

2x 2y

x, x 2 y y

0 2, y 0

Section 12.1 17. f x, y Domain: 4 4 x2 x2 x
2

Introduction to Functions of Several Variables y y ≤ 1 2 21. f x, y ln 4 x x y y > 0

77

y2 y ≥ 0
2 2

19. f x, y

arcsin x

y ≤ 4 y2 ≤ 4

Domain: x, y : 1 ≤ x Range: 2

Domain: 4 x

y < 4 x, y : y < x 4

x, y : x 2 Range: 0 ≤ z ≤ 2 x xy x, y : x y

≤ z ≤

Range:all real numbers 1 xy x, y : x 0 and y 0

23. z

25. f x, y 0 and y 0 Domain:

ex

y

27. g x, y 0 Domain:

Domain:

x, y : y

Range: all real numbers 4x y2

Range: z > 0

Range: all real numbers except zero

29. f x, y

x2

1 (b) View where x is negative, y and z are positive: 15, 10, 20 (d) View from the line y 33. f x, y y2 x in the xy-plane: 20, 20, 0

(a) Viewfrom the positive x-axis: 20, 0, 0 (c) View from the first octant: 20, 15, 25 31. f x, y Plane: z 5 5
4

z

Since the variable x is missing, the surface is a cylinder with rulings parallel to the x-axis. The generating curve is z y 2. The domain is the entire xy-plane and the range is z ≥ 0.
2 4

2 4

z y
5

x

4

1 4 x

2

3

y

35. z

4

x2

y2
4

z

37. f x, ye

x
z
8 6 4 2

Paraboloid Domain: entire xy-plane Range: z ≤ 4
−3 3 x

2

3

y

Since the variable y is missing, the surface is a cylinder with rulings parallel to the y-axis. The generating curve is z e x. The domain is the entire xy-plane and the range is z > 0.

4 x

4

y

39. z

y2

x2

1

z

41. f x, y

x 2e
z

xy 2

Hyperbolic paraboloid Domain:entire xy-plane Range: < z <
y

x

y x

78

Chapter 12 x2
z
5 4

Functions of Several Variables y2 (c) g is a horizontal translation of f two units to the right. The vertex moves from 0, 0, 0 to 0, 2, 0 . (d) g is a reflection of f in the xy-plane followed by a vertical translation 4 units upward. (e)
z
5 4 5 4

43. f x, y (a)

z

−2 2 x 1 2

y

(b) g is a vertical...