Calculo 2
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...
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