Calculus 2nd Edition Rogawski

A110
ANSWERS TO ODD-NUMBERED EXERCISES
√ √
17. (a) h = √ 2
≈ 0.6,r = √ 1
≈ 0.43 (b) h 3π 3π r = √ 2
(c) There is no cone of volume 1 and maximal surface area.
( )
√
19. (8, − 2) 21. 48
97 , 108
97 23.
a a b b
(a+b) a+b 25.
a a b b
(a+b) a+b
31. r = 3, h = 6 33. x + y + z = 3
( )
39. √ −6
, √−3
, √30
41. (−1, 0, 2)
105 105 105
43. Minimum 138
11 ≈ 12.545 , no maximum value
47. (b) λ = 2p c
1 p 2
Chapter 15 Review
1. (a)
y
∂f
41. ∂s = 3s2 t + 4st 2 + t 3 − 2st 3 + 6s 2 t 2
∂f
∂t = 4s 2 t + 3st 2 + s 3 + 4s 3 t − 3s 2 t 2
45.
∂z
∂x = − ez − 1
xe z + e y
47. (0, 0) saddle point, (1, 1) and (−1, −1) local minima
( )
49. 12
, 1 2 saddle point
53. Global maximum f(2, 4) = 10 , global minimum
f(−2, 4) = −18
55. Maximum √ 26 , minimum − √ 26
13 13
57. Maximum √ 12 , minimum − √ 12
3 3
59. f(0.8, 0.52, − 0.32) = 0.88 and f(−0.13, 0.15, 0.99) = 3.14
( )
61. r = V2π 1/3, ( )
h = 2
V2π
1/3
−3
(b) f(3, 1) =
3.
x
x
√ )
2
3 , f(−5, − 3) = −2 (c)
(− 5 3 , 1
z
y
Vertical and horizontal traces: the line z = (c 2 + 1) − y in the
plane x = c, the parabola z = x 2 − c + 1 in the plane y = c.
5. (a) Graph (B) (b) Graph (C) (c) Graph (D) (d) Graph (A)
7. (a) Parallel lines 4x − y = ln c, c > 0, in the xy-plane
(b) Parallel lines 4x − y = e c in the xy-plane
(c) Hyperbolas 3x 2 − 4y 2 = c in the xy-plane
(d) Parabolas x = c − y 2 in the xy-plane
9. lim (xy + (x,y)→(1,−3) y2 ) = 6
11. The limit does not exist.
13. lim (2x + (x,y)→(1,−3) y)e−x+y = −e −4
17. f x = 2, f y = 2y
19. f x = e −x−y (y cos(xy) − sin(xy))
f y = e −x−y (x cos(yx) − sin(yx))
21. f xxyz = − cos(x + z) 23. z = 33x + 8y − 42
25. Estimate, 12.146; calculator value to three places, 11.996.
27. Statements (ii) and (iv) are true.
29.
d ( ) ∣ dt f(c(t)) ∣t=2 = 3 + 4e 4 ≈ 221.4
31.
d
dt
(
f(c(t))
) ∣ ∣ ∣t=1 = 4e − e 3e ≈−3469.3
33. D u f(3, − 1) = − 54 √
5
35. D u f (P ) = −
√
2e
5 37.
〈 1 √2 ,
〉
√1
, 0 2
Chapter 16
Section 16.1 Preliminary Questions
1. A = 1, the number of subrectangles is 32.
2. ∫∫ R fdA≈ S 1,1 = 0.16
∫∫
3. R 5 dA = 50
4. The signed volume between the graph z = f (x, y) and the
xy-plane. The region below the xy-plane is treated as negative volume.
5. (b) 6. (b), (c)
Section 16.1 Exercises
1. S 4,3 = 13.5 3. (A) S 3,2 = 42, (B) S 3,2 = 43.5
5. (A) S 3,2 = 60, (B) S 3,2 = 62
7. Two possible solutions are S 3,2 = 77
72 and S 3,2 = 79
72 .
9.
225
2
z
15
5
3
11. 0.19375 13. 1.0731, 1.0783, 1.0809 15. 0 17. 0 19. 40
21. 55 23.
4
3 25. 84 27. 4 29.
1858
15 (
31. 6ln6− 2ln2− 5ln5≈ 1.317 33.
4
3 19 − 5 √ )
5 ≈ 10.426
35.
1
2 (ln 3)(−2 + ln 48) ≈ 1.028 37. 6ln3≈ 6.592
( )(
39. 1 41. e 2 √ )
− 1 1 − 2
2 ≈ 1.871
43. m = 3 4 45. 2ln2− 1 ≈ 0.386
49.
e 3
3 − 1 3 − e + 1 ≈ 4.644
x
y