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APEX Calculus

Gregory Hartman, Ph.D., Sean Fitzpatrick, Ph.D. (Editor), Alex Jordan, Ph.D. (Editor), Carly Vollet, M.S. (Editor)

Appendix A Answers to Selected Exercises

1 Limits
1.1 An Introduction To Limits
1.1.3 Exercises

Terms and Concepts

1.1.3.3.
Answer.
False

Problems

1.1.3.7.
Answer.
5
1.1.3.9.
Answer.
DNE
1.1.3.11.
Answer.
−4
1.1.3.13.
Answer.
DNE
1.1.3.15.
Answer.
1
1.1.3.17.
Answer.
1
1.1.3.19.
Answer.
DNE
1.1.3.21.
Answer.
−7
1.1.3.23.
Answer.
5
1.1.3.25.
Answer.
29
1.1.3.27.
Answer.
−1

1.2 Epsilon-Delta Definition of a Limit

Exercises

Terms and Concepts
1.2.3.
Answer.
True

1.3 Finding Limits Analytically

Exercises

Problems
1.3.7.
Answer.
9
1.3.9.
Answer.
0
1.3.11.
Answer.
3
1.3.13.
Answer.
3
1.3.15.
Answer.
0
1.3.17.
Answer.
π
1.3.19.
Answer.
23
1.3.21.
Answer.
34
1.3.23.
Answer.
DNE
1.3.25.
Answer.
233
1.3.27.
Answer.
π2−4π−22π2−2π+1
1.3.29.
Answer.
14
1.3.31.
Answer.
174
1.3.33.
Answer.
49
1.3.35.
Answer.
0
1.3.37.
Answer.
1
1.3.39.
Answer.
8
1.3.41.
Answer.
1

1.4 One-Sided Limits

Exercises

Terms and Concepts
1.4.3.
Answer.
False
Problems
1.4.5.
1.4.5.a
Answer.
2
1.4.5.b
Answer.
2
1.4.5.c
Answer.
2
1.4.5.d
Answer.
1
1.4.5.e
Answer.
DNE
1.4.5.f
Answer.
4
1.4.7.
1.4.7.a
Answer.
DNE or ∞
1.4.7.b
Answer.
DNE or ∞
1.4.7.c
Answer.
DNE or ∞
1.4.7.d
Answer.
DNE
1.4.7.e
Answer.
5
1.4.7.f
Answer.
4
1.4.9.
1.4.9.a
Answer.
1
1.4.9.b
Answer.
1
1.4.9.c
Answer.
1
1.4.9.d
Answer.
1
1.4.11.
1.4.11.a
Answer.
2
1.4.11.b
Answer.
2
1.4.11.c
Answer.
2
1.4.11.d
Answer.
0
1.4.11.e
Answer.
2
1.4.11.f
Answer.
2
1.4.11.g
Answer.
2
1.4.11.h
Answer.
DNE
1.4.13.
1.4.13.a
Answer.
2
1.4.13.b
Answer.
6
1.4.13.c
Answer.
DNE
1.4.13.d
Answer.
2
1.4.15.
1.4.15.a
Answer.
9
1.4.15.b
Answer.
9
1.4.15.c
Answer.
9
1.4.15.d
Answer.
9
1.4.15.e
Answer.
126
1.4.15.f
Answer.
126
1.4.15.g
Answer.
126
1.4.15.h
Answer.
126
1.4.17.
1.4.17.a
Answer.
1−cos2⁡(a)
1.4.17.b
Answer.
sin2⁡(a)
1.4.17.c
Answer.
1−cos2⁡(a) or sin2⁡(a)
1.4.17.d
Answer.
sin2⁡(a)
1.4.19.
1.4.19.a
Answer.
−4
1.4.19.b
Answer.
−4
1.4.19.c
Answer.
−4
1.4.19.d
Answer.
−2
1.4.21.
1.4.21.a
Answer.
−1
1.4.21.b
Answer.
1
1.4.21.c
Answer.
DNE
1.4.21.d
Answer.
0

1.5 Continuity

Exercises

Terms and Concepts
1.5.5.
Answer.
False
1.5.7.
Answer.
True
1.5.9.
Answer.
False
Problems
1.5.11.
Answer.
No.
1.5.13.
Answer.
No.
1.5.15.
Answer.
Yes.
1.5.17.
Answer 1.
No.
Answer 2.
Yes.
Answer 3.
No.
1.5.19.
1.5.19.a
Answer.
Yes.
1.5.19.b
Answer.
Yes.
1.5.21.
1.5.21.a
Answer.
Yes.
1.5.21.b
Answer.
Yes.
1.5.23.
Answer.
(−∞,∞)
1.5.25.
Answer.
[−2,2]
1.5.27.
Answer.
(−∞,−1.73205],[1.73205,∞)
1.5.29.
Answer.
(−∞,∞)
1.5.31.
Answer.
(0,∞)
1.5.33.
Answer.
(−∞,1.09861]
1.5.39.
Answer.
1.23633
1.5.41.
Answer.
0.693164

1.6 Limits Involving Infinity
1.6.4 Exercises

Terms and Concepts

1.6.4.1.
Answer.
False
1.6.4.3.
Answer.
False
1.6.4.5.
Answer.
True

Problems

1.6.4.9.
1.6.4.9.a
Answer.
−∞
1.6.4.9.b
Answer.
∞
1.6.4.11.
1.6.4.11.a
Answer.
0
1.6.4.11.b
Answer.
3
1.6.4.11.c
Answer.
1.5
1.6.4.11.d
Answer.
1.5
1.6.4.13.
1.6.4.13.a
Answer.
DNE
1.6.4.13.b
Answer.
DNE
1.6.4.15.
1.6.4.15.a
Answer.
−∞
1.6.4.15.b
Answer.
∞
1.6.4.15.c
Answer.
DNE
1.6.4.17.
1.6.4.17.a
Answer.
∞
1.6.4.17.b
Answer.
∞
1.6.4.17.c
Answer.
∞
1.6.4.19.
Answer.
y=2,x=−2,x=9
1.6.4.21.
Answer.
y=0,x=0,x=4
1.6.4.23.
Answer.
NONE
1.6.4.25.
Answer.
∞
1.6.4.27.
Answer.
∞

2 Derivatives
2.1 Instantaneous Rates of Change: The Derivative
2.1.3 Exercises

Terms and Concepts

2.1.3.1.
Answer.
True

Problems

2.1.3.7.
Answer.
0
2.1.3.9.
Answer.
−3
2.1.3.11.
Answer.
3x2
2.1.3.13.
Answer.
−1x2
2.1.3.15.
Answer 1.
y=6
Answer 2.
x=−2
2.1.3.17.
Answer 1.
3x+y=4
Answer 2.
y−0.333333x=−19.3333
2.1.3.19.
Answer 1.
y−48x=−128
Answer 2.
0.0208333x+y=64.0833
2.1.3.21.
Answer 1.
0.25x+y=−1
Answer 2.
y−4x=7.5
2.1.3.23.
Answer.
5.9x+y=1.2
2.1.3.25.
Answer.
y−0.0192627x=0.0953664
2.1.3.27.
2.1.3.27.a
Answer.
−2,0,4
2.1.3.27.b
Answer.
2x
2.1.3.27.c
Answer.
−2,0,4
2.1.3.33.
Answer 1.
(−2,0)∪(2,∞)
Answer 2.
(−∞,−2)∪(0,2)
Answer 3.
{−2,0,2}
Answer 4.
(−1,1)
Answer 5.
(−∞,−1)∪(1,∞)
Answer 6.
{−1,1}
2.1.3.35.
Answer.
no

2.2 Interpretations of the Derivative
2.2.5 Exercises

Terms and Concepts

2.2.5.1.
Answer.
velocity
2.2.5.3.
Answer.
linear functions

Problems

2.2.5.5.
Answer.
−89
2.2.5.7.
Answer.
f(10.1)
2.2.5.9.
Answer.
7
2.2.5.11.
Answer.
foot per second squared
2.2.5.15.
Answer.
f is the derivative of g.
2.2.5.17.
Answer.
g is the derivative of f.

2.3 Basic Differentiation Rules
2.3.3 Exercises

Terms and Concepts

2.3.3.1.
Answer.
the power rule
2.3.3.3.
Answer.
ex
2.3.3.5.
Answer.
Choice 1, Choice 2, Choice 5, Choice 6
2.3.3.7.
Answer.
17x−205
2.3.3.9.
Answer 1.
a velocity function
Answer 2.
an acceleration function

Problems

2.3.3.11.
Answer.
−(14x+8)
2.3.3.13.
Answer.
9−(20t4+34t2)
2.3.3.15.
Answer.
3er
2.3.3.17.
Answer.
6x+9
2.3.3.19.
Answer.
sin⁡(t)−(et+cos⁡(t))
2.3.3.21.
Answer.
0
2.3.3.23.
Answer.
24x2+96x+96
2.3.3.25.
Answer.
8x+28
2.3.3.27.
Answer 1.
9x8
Answer 2.
72x7
Answer 3.
504x6
Answer 4.
3024x5
2.3.3.29.
Answer 1.
−(8t+3+et)
Answer 2.
−(8+et)
Answer 3.
−et
Answer 4.
−et
2.3.3.31.
Answer 1.
−(cos⁡(θ)−sin⁡(θ))
Answer 2.
sin⁡(θ)+cos⁡(θ)
Answer 3.
cos⁡(θ)−sin⁡(θ)
Answer 4.
−(sin⁡(θ)+cos⁡(θ))
2.3.3.33.
Answer 1.
y=20(x−2)+24
Answer 2.
y=−120(x−2)+24
2.3.3.35.
Answer 1.
y=x−1
Answer 2.
y=−(x−1)
2.3.3.37.
Answer 1.
y=2⋅12(x−π6)+−232
Answer 2.
y=−(12⋅2)(x−π6)+−232

2.4 The Product and Quotient Rules

Exercises

Terms and Concepts
2.4.1.
Answer.
False
2.4.3.
Answer.
True
2.4.5.
Answer.
False
Problems
2.4.15.
Answer.
sin⁡(y)+ycos⁡(y)
2.4.17.
Answer.
eqln⁡(q)+eq1q
2.4.19.
Answer.
t−4−(t+8)(t−4)2
2.4.21.
Answer.
−(csc⁡(y)cot⁡(y)+ey)
2.4.23.
Answer.
14q−6
2.4.25.
Answer.
(5r2+17r+10)er
2.4.27.
Answer.
3
2.4.29.
Answer.
csc⁡(z)sin⁡(z)−csc⁡(z)cot⁡(z)(cos⁡(z)+2)(cos⁡(z)+2)2
2.4.31.
Answer.
tan⁡(r)−rsec2⁡(r)tan2⁡(r)−csc2⁡(r)r+cot⁡(r)r2
2.4.33.
Answer.
35x4ex+7x5ex−(cos⁡(x)cos⁡(x)−sin⁡(x)sin⁡(x))
2.4.35.
Answer.
(4z3ln⁡(z)+z41z)cos⁡(z)−z4ln⁡(z)sin⁡(z)
2.4.37.
Answer 1.
y=−(7x+7)
Answer 2.
y=(17)x−7
2.4.39.
Answer 1.
y=−(15(x+5)+25)
Answer 2.
y=(115)(x+5)−25
2.4.41.
Answer.
172
2.4.43.
Answer.
NONE
2.4.45.
Answer.
2cos⁡(x)−xsin⁡(x)
2.4.47.
Answer.
csc⁡(x)cot⁡(x)cot⁡(x)+csc2⁡(x)csc⁡(x)

2.5 The Chain Rule

Exercises

Terms and Concepts
2.5.1.
Answer.
True
2.5.3.
Answer.
False
2.5.5.
Answer.
True
Problems
2.5.7.
Answer.
10(4x3−x)9(12x2−1)
2.5.9.
Answer.
3(sin⁡(θ)+cos⁡(θ))2(cos⁡(θ)−sin⁡(θ))
2.5.11.
Answer.
4(ln⁡(x)−x4)3(1x−4x3)
2.5.13.
Answer.
5(y+1y)4(1−1y2)
2.5.15.
Answer.
2sec2⁡(2q)
2.5.17.
Answer.
(6t5−3t2(t3)2)cos⁡(t6+1t3)
2.5.19.
Answer.
−3cos2⁡(y2+3y−3)(2y+3)sin⁡(y2+3y−3)
2.5.21.
Answer.
1q8⋅8q7
2.5.23.
Answer.
1.79176⋅6t
2.5.25.
Answer.
0
2.5.27.
Answer.
1.79176⋅6w(5w+6)−(6w+5)⋅1.60944⋅5w(5w+6)2
2.5.29.
Answer.
(1.60944⋅5r2⋅2r−1)⋅6r2−(5r2−r)⋅1.79176⋅6r2⋅2r(6r2)2
2.5.31.
Answer.
6(x2+4x)5(2x+4)(7x4+x)3+(x2+4x)6⋅3(7x4+x)2(28x3+1)
2.5.33.
Answer.
7cos⁡(9+7w)cos⁡(4w−5)−4sin⁡(4w−5)sin⁡(9+7w)
2.5.35.
Answer.
−6sin⁡(6r+4)(3r+1)3+9(3r+1)2cos⁡(6r+4)((3r+1)3)2
2.5.37.
Answer 1.
y=0
Answer 2.
x=0
2.5.39.
Answer 1.
y=−3(x−π2)+1
Answer 2.
y=13(x−π2)+1
2.5.41.
Answer.
1x

2.6 Implicit Differentiation
2.6.4 Exercises

Terms and Concepts

2.6.4.3.
Answer.
True

Problems

2.6.4.5.
Answer.
12w+12w(w)2
2.6.4.7.
Answer.
129+t2⋅2t
2.6.4.9.
Answer.
1.2y0.2
2.6.4.11.
Answer.
w−(w−8)12w(w)2
2.6.4.13.
Answer.
−4x32y+1
2.6.4.15.
Answer.
sin⁡(x)sec⁡(y)
2.6.4.17.
Answer.
yx
2.6.4.19.
Answer.
−2sin⁡(y)cos⁡(y)x
2.6.4.21.
Answer.
12y+2
2.6.4.23.
Answer.
1−cos⁡(x)sin⁡(y)+1
2.6.4.25.
Answer.
−(2x+y)2y+x
2.6.4.27.
2.6.4.27.a
Answer.
y=0
2.6.4.27.b
Answer.
y=−1.859(x−0.1)+0.2811
2.6.4.29.
2.6.4.29.a
Answer.
y=4
2.6.4.29.b
Answer.
y=310814(x−2)−10814
2.6.4.31.
2.6.4.31.a
Answer.
y=−13(x−72)+6+332
2.6.4.31.b
Answer.
y=3(x−(4+33))2+32
2.6.4.33.
Answer.
−((2y+1)⋅12x2−4x32(−(4x3))2y+1)(2y+1)2
2.6.4.35.
Answer.
sin2⁡(x)sec2⁡(y)tan⁡(y)+cos⁡(x)sec⁡(y)
2.6.4.37.
Answer 1.
(1+x)1x(1x(x+1)−ln⁡(1+x)x2)
Answer 2.
y=(1−2ln⁡(2))(x−1)+2
2.6.4.39.
Answer 1.
xxx+1(ln⁡(x)+1−1x+1)
Answer 2.
y=14(x−1)+12
2.6.4.41.
Answer 1.
x+1x+2(1x+1−1x+2)
Answer 2.
y=19(x−1)+23

2.7 Derivatives of Inverse Functions

Exercises

Terms and Concepts
2.7.1.
Answer.
False
Problems
2.7.9.
Answer.
17
2.7.11.
Answer.
−0.5
2.7.13.
Answer.
−254
2.7.15.
Answer.
−11−(4w)2⋅4
2.7.17.
Answer.
11+(2r)2⋅2
2.7.19.
Answer.
(sec⁡(x))2cos−1⁡(x)−11−x2tan⁡(x)
2.7.21.
Answer.
11+z2sin−1⁡(z)−11−z2tan−1⁡(z)(sin−1⁡(z))2
2.7.23.
Answer.
csc⁡(1q3)cot⁡(1q3)3q2(q3)2
2.7.29.
Answer.
y=2(x−−32)+(−π3)

3 The Graphical Behavior of Functions
3.1 Extreme Values

Exercises

Terms and Concepts
3.1.5.
Answer.
False
Problems
3.1.7.
Answer 1.
B
Answer 2.
NONE
Answer 3.
B,G
Answer 4.
C,F
3.1.9.
Answer.
0
3.1.11.
Answer 1.
0
Answer 2.
0
3.1.13.
Answer 1.
DNE
Answer 2.
0
3.1.15.
Answer.
0
3.1.17.
Answer 1.
14
Answer 2.
−2
3.1.19.
Answer 1.
−2.82843
Answer 2.
−4
3.1.21.
Answer 1.
92
Answer 2.
2.82843
3.1.23.
Answer 1.
eπ42
Answer 2.
−eπ
3.1.25.
Answer 1.
12e
Answer 2.
0

3.2 The Mean Value Theorem

Exercises

Problems
3.2.3.
Answer.
(−1,1)
3.2.5.
Answer.
−12
3.2.7.
Answer.
does not apply
3.2.9.
Answer.
does not apply
3.2.11.
Answer.
0
3.2.13.
Answer.
322
3.2.15.
Answer.
does not apply
3.2.17.
Answer.
−sec−1⁡(2π),sec−1⁡(2π)
3.2.19.
Answer.
5+776,5−776

3.3 Increasing and Decreasing Functions

Exercises

Terms and Concepts
3.3.3.
Answer.
Answers will vary; graphs should be steeper near x=0 than near x=2.
3.3.5.
Answer.
False
Problems
3.3.15.
Answer 1.
(−∞,∞)
Answer 2.
−2
Answer 3.
[−2,∞)
Answer 4.
(−∞,−2]
Answer 5.
NONE
Answer 6.
−2
3.3.17.
Answer 1.
(−∞,∞)
Answer 2.
−57,73
Answer 3.
(−∞,−0.714286],[2.33333,∞)
Answer 4.
[−0.714286,2.33333]
Answer 5.
−57
Answer 6.
73
3.3.19.
Answer 1.
(−∞,∞)
Answer 2.
5
Answer 3.
(−∞,5]
Answer 4.
[5,∞)
Answer 5.
5
Answer 6.
NONE
3.3.21.
Answer 1.
(−∞,−7)∪(−7,−5)∪(−5,∞)
Answer 2.
−5.91608,5.91608
Answer 3.
[−5.91608,−5),(−5,5.91608]
Answer 4.
(−∞,−7),(−7,−5.91608],[5.91608,∞)
Answer 5.
5.91608
Answer 6.
−5.91608
3.3.23.
Answer 1.
(−π,π)
Answer 2.
−2.35619,−0.785398,0.785398,2.35619
Answer 3.
(−3.14159,−2.35619),(−0.785398,0.785398),(2.35619,3.14159)
Answer 4.
(−2.35619,−0.785398),(0.785398,2.35619)
Answer 5.
−2.35619,0.785398
Answer 6.
−0.785398,2.35619

3.4 Concavity and the Second Derivative
3.4.3 Exercises

Terms and Concepts

3.4.3.1.
Answer.
Answers will vary.
3.4.3.3.
Answer.
Yes; Answers will vary.

Problems

3.4.3.15.
Answer 1.
NONE
Answer 2.
(−∞,∞)
Answer 3.
NONE
3.4.3.17.
Answer 1.
0
Answer 2.
[0,∞)
Answer 3.
(−∞,0]
3.4.3.19.
Answer 1.
−323,0
Answer 2.
(−∞,−10.6667],[0,∞)
Answer 3.
[−10.6667,0]
3.4.3.21.
Answer 1.
−2
Answer 2.
(−∞,∞)
Answer 3.
NONE
3.4.3.23.
Answer 1.
−0.57735,0.57735
Answer 2.
(−∞,−0.57735],[0.57735,∞)
Answer 3.
[−0.57735,0.57735]
3.4.3.25.
Answer 1.
−0.785398,2.35619
Answer 2.
(−3.14159,−0.785398],[2.35619,3.14159)
Answer 3.
[−0.785398,2.35619]
3.4.3.27.
Answer 1.
0.22313
Answer 2.
[0.22313,∞)
Answer 3.
(0,0.22313]
3.4.3.29.
Answer 1.
−7
Answer 2.
NONE
Answer 3.
−7
3.4.3.31.
Answer 1.
−1.1547,1.1547
Answer 2.
−1.1547
Answer 3.
1.1547
3.4.3.33.
Answer 1.
−4
Answer 2.
NONE
Answer 3.
−4
3.4.3.35.
Answer 1.
3
Answer 2.
NONE
Answer 3.
NONE
3.4.3.37.
Answer 1.
−9
Answer 2.
−9
Answer 3.
NONE
3.4.3.39.
Answer 1.
−2.35619,0.785398
Answer 2.
0.785398
Answer 3.
−2.35619
3.4.3.41.
Answer 1.
0.606531
Answer 2.
NONE
Answer 3.
0.606531
3.4.3.43.
Answer 1.
NONE
Answer 2.
NONE
3.4.3.45.
Answer 1.
NONE
Answer 2.
0
3.4.3.47.
Answer 1.
−283
Answer 2.
0
3.4.3.49.
Answer 1.
NONE
Answer 2.
NONE
3.4.3.51.
Answer 1.
0
Answer 2.
2
3.4.3.53.
Answer 1.
−0.785398
Answer 2.
2.35619
3.4.3.55.
Answer 1.
NONE
Answer 2.
0.22313

3.5 Curve Sketching

Exercises

Terms and Concepts
3.5.3.
Answer.
True
3.5.5.
Answer.
True

4 Applications of the Derivative
4.1 Newton’s Method

Exercises

Terms and Concepts
4.1.1.
Answer.
False
Problems
4.1.3.
Answer 1.
1.57091
Answer 2.
1.5708
Answer 3.
1.5708
Answer 4.
1.5708
Answer 5.
1.5708
4.1.5.
Answer 1.
2
Answer 2.
1.2
Answer 3.
1.01176
Answer 4.
1.00005
Answer 5.
1
4.1.7.
Answer 1.
0.613706
Answer 2.
0.913341
Answer 3.
0.996132
Answer 4.
0.999993
Answer 5.
1
4.1.9.
Answer.
{−5.15633,−0.369102,0.525428}
4.1.11.
Answer.
{−1.0134,0.988312,1.39341}
4.1.13.
Answer.
{−0.824132,0.824132}
4.1.15.
Answer.
{0}

4.2 Related Rates

Exercises

Terms and Concepts
4.2.1.
Answer.
True
Problems
4.2.3.
4.2.3.a
Answer.
0.198944 (cm(s
4.2.3.b
Answer.
0.0198944 (cm(s
4.2.3.c
Answer.
0.00198944 (cm(s
4.2.5.
Answer.
51.066 (mi(h
4.2.7.
4.2.7.a
Answer.
258.537 (rad(hr
4.2.7.b
Answer.
413.417 (rad(hr
4.2.7.c
Answer.
424 (rad(hr
4.2.9.
4.2.9.a
Answer.
0.0417029 (ft(s
4.2.9.b
Answer.
0.458349 (ft(s
4.2.9.c
Answer.
3.35489 (ft(s
4.2.9.d
Answer.
∞
4.2.11.
4.2.11.a
Answer.
19.1658 (ft(s
4.2.11.b
Answer.
0.191658 (ft(s
4.2.11.c
Answer.
0.0395988 (ft(s
4.2.11.d
Answer.
381.791 s
4.2.13.
4.2.13.a
Answer.
80 ft
4.2.13.b
Answer.
1.71499 (ft(s
4.2.13.c
Answer.
1.83829 (ft(s
4.2.13.d
Answer.
74.162 ft
4.2.15.
Answer.
0.00230973 (ft(s

4.3 Optimization

Exercises

Terms and Concepts
4.3.1.
Answer.
True
Problems
4.3.3.
Answer.
5625
4.3.5.
Answer.
DNE
4.3.7.
Answer.
1
4.3.9.
Answer 1.
3.83722 cm
Answer 2.
7.67443 cm
4.3.11.
Answer 1.
3.0456 cm
Answer 2.
12.1824 cm
4.3.13.
Answer.
10.3923 in;14.6969 in
4.3.15.
Answer 1.
0 mi
Answer 2.
$474,341.65
4.3.17.
Answer.
23.7599 ft

4.4 Differentials

Exercises

Terms and Concepts
4.4.3.
Answer.
False
Problems
4.4.7.
Answer.
81.54
4.4.9.
Answer.
32.4
4.4.11.
Answer.
5.07
4.4.13.
Answer.
4.98133
4.4.15.
Answer.
0.141593
4.4.17.
Answer.
(2x−8)dx
4.4.19.
Answer.
−24x7(3x8)2dx
4.4.21.
Answer.
(6x5+3e3x)dx
4.4.23.
Answer.
8(tan⁡(x)+5)−8xsec2⁡(x)(tan⁡(x)+5)2dx
4.4.25.
Answer.
(excot⁡(x)−excsc2⁡(x))dx
4.4.27.
Answer.
x−8−(x−6)(x−8)2dx
4.4.29.
Answer.
ln⁡(x)dx
4.4.31.
Answer.
7.53982 cm3
4.4.33.
Answer.
1.86532
4.4.35.
4.4.35.a
Answer.
297.717 ft
4.4.35.b
Answer.
62.3155 ft
4.4.35.c
Answer.
20.9%
4.4.37.
4.4.37.a
Answer.
298.868 ft
4.4.37.b
Answer.
8.66751 ft
4.4.37.c
Answer.
2.9%
4.4.39.
Answer.
1%

5 Integration
5.1 Antiderivatives and Indefinite Integration

Exercises

Terms and Concepts
5.1.7.
Answer.
velocity
Problems
5.1.9.
Answer.
(23)x9+C
5.1.11.
Answer.
(133)x3−2x+C
5.1.13.
Answer.
s+C
5.1.15.
Answer.
C−12t6
5.1.17.
Answer.
cot⁡(θ)+C
5.1.19.
Answer.
sec⁡(x)+csc⁡(x)+C
5.1.21.
Answer.
8tln⁡(8)+C
5.1.23.
Answer.
(43)t3+16t2+64t+(2563)+C
5.1.25.
Answer.
x88+C
5.1.27.
Answer.
mx+C
5.1.31.
Answer.
7ex+8
5.1.33.
Answer.
csc⁡(x)+(−9)
5.1.35.
Answer.
(32)x2+6x+7
5.1.37.
Answer.
5ex+x−8
5.1.39.
Answer.
23x412+3x1.20695−sin⁡(x)−8.91024x+4.17146

5.2 The Definite Integral

Exercises

Terms and Concepts
5.2.3.
Answer.
0
Problems
5.2.5.
5.2.5.a
Answer.
3
5.2.5.b
Answer.
4
5.2.5.c
Answer.
3
5.2.5.d
Answer.
0
5.2.5.e
Answer.
−4
5.2.5.f
Answer.
9
5.2.7.
5.2.7.a
Answer.
4
5.2.7.b
Answer.
2
5.2.7.c
Answer.
4
5.2.7.d
Answer.
2
5.2.7.e
Answer.
1
5.2.7.f
Answer.
2
5.2.9.
5.2.9.a
Answer.
π
5.2.9.b
Answer.
π
5.2.9.c
Answer.
2π
5.2.9.d
Answer.
10π
5.2.11.
5.2.11.a
Answer.
−59
5.2.11.b
Answer.
−48
5.2.11.c
Answer.
−27
5.2.11.d
Answer.
−33
5.2.13.
5.2.13.a
Answer.
4
5.2.13.b
Answer.
4
5.2.13.c
Answer.
−4
5.2.13.d
Answer.
−2
5.2.15.
5.2.15.a
Answer.
2 (ft(s
5.2.15.b
Answer.
2 ft
5.2.15.c
Answer.
1.5 ft
5.2.17.
5.2.17.a
Answer.
64 (ft(s
5.2.17.b
Answer.
64 ft
5.2.17.c
Answer.
2 s
5.2.17.d
Answer.
4.64575 s
5.2.19.
Answer.
2
5.2.21.
Answer.
16
5.2.23.
Answer.
22
5.2.25.
Answer.
0

5.3 Riemann Sums
5.3.4 Exercises

Terms and Concepts

5.3.4.1.
Answer.
limits
5.3.4.3.
Answer.
rectangles

Problems

5.3.4.5.
Answer 1.
4+9+16
Answer 2.
29
5.3.4.7.
Answer 1.
1+0+(−1)+0+1+0
Answer 2.
1
5.3.4.9.
Answer 1.
1+12+13+14+15
Answer 2.
13760
5.3.4.11.
Answer 1.
12+16+112+120+130
Answer 2.
56
5.3.4.13.
Answer.
1;6;5i
5.3.4.15.
Answer.
1;4;ii+4
5.3.4.17.
Answer.
30
5.3.4.19.
Answer.
1760
5.3.4.21.
Answer.
36030
5.3.4.23.
Answer.
3240
5.3.4.25.
Answer.
125
5.3.4.27.
Answer.
40
5.3.4.35.
Answer 1.
(n+1)24n2
Answer 2.
0.3025
Answer 3.
0.255025
Answer 4.
0.2505
Answer 5.
14
5.3.4.37.
Answer 1.
2
Answer 2.
2
Answer 3.
2
Answer 4.
2
Answer 5.
2
5.3.4.39.
Answer 1.
144−162n
Answer 2.
127.8
Answer 3.
142.38
Answer 4.
143.838
Answer 5.
144

5.4 The Fundamental Theorem of Calculus
5.4.6 Exercises

Terms and Concepts

5.4.6.3.
Answer.
True

Problems

5.4.6.5.
Answer.
32
5.4.6.7.
Answer.
0
5.4.6.9.
Answer.
1
5.4.6.11.
Answer.
(312425)ln⁡(5)
5.4.6.13.
Answer.
6
5.4.6.15.
Answer.
1223
5.4.6.17.
Answer.
454
5.4.6.19.
Answer.
45
5.4.6.21.
Answer.
12
5.4.6.23.
Answer.
14
5.4.6.25.
Answer.
18
5.4.6.27.
Answer.
0
5.4.6.31.
Answer.
2.3094
5.4.6.33.
Answer.
0.541325
5.4.6.35.
Answer.
1π2−0⋅3.14159π
5.4.6.37.
Answer.
2
5.4.6.39.
Answer.
54
5.4.6.41.
Answer.
−784 ft
5.4.6.43.
Answer.
16 ft
5.4.6.45.
Answer.
1 ft
5.4.6.47.
Answer.
−160 (ft(s
5.4.6.49.
Answer.
492 (ft(s
5.4.6.55.
Answer.
5x4−1x5−x
5.4.6.57.
Answer.
4x3(x4−7)−(x−7)
5.4.6.59.
Answer.
2xsin⁡(−2x4)

5.5 Numerical Integration
5.5.6 Exercises

Terms and Concepts

5.5.6.1.
Answer.
False

Problems

5.5.6.5.
5.5.6.5.a
Answer.
0.75
5.5.6.5.b
Answer.
0.666667
5.5.6.5.c
Answer.
0.666667
5.5.6.7.
5.5.6.7.a
Answer.
1.89612
5.5.6.7.b
Answer.
2.00456
5.5.6.7.c
Answer.
2
5.5.6.9.
5.5.6.9.a
Answer.
38.5781
5.5.6.9.b
Answer.
36.75
5.5.6.9.c
Answer.
36.75
5.5.6.11.
5.5.6.11.a
Answer.
0
5.5.6.11.b
Answer.
0
5.5.6.11.c
Answer.
0
5.5.6.13.
5.5.6.13.a
Answer.
0.900628
5.5.6.13.b
Answer.
0.904523
5.5.6.15.
5.5.6.15.a
Answer.
13.9604
5.5.6.15.b
Answer.
13.9066
5.5.6.17.
5.5.6.17.a
Answer.
1.17029
5.5.6.17.b
Answer.
1.18728
5.5.6.19.
5.5.6.19.a
Answer.
1.08025
5.5.6.19.b
Answer.
1.07699
5.5.6.21.
5.5.6.21.a
Answer.
161
5.5.6.21.b
Answer.
12
5.5.6.23.
5.5.6.23.a
Answer.
994
5.5.6.23.b
Answer.
62
5.5.6.25.
Answer 1.
30.8667 cm2
Answer 2.
308667 ft2

6 Techniques of Antidifferentiation
6.1 Substitution
6.1.5 Exercises

Terms and Concepts

6.1.5.1.
Answer.
the Chain Rule

Problems

6.1.5.3.
Answer.
17(x4+9)7+C
6.1.5.5.
Answer.
112(x2+6)6+C
6.1.5.7.
Answer.
12ln⁡(|2x+7|)+C
6.1.5.9.
Answer.
23(x+10)x−5+C
6.1.5.11.
Answer.
2ex+C
6.1.5.13.
Answer.
C−14(1x2+4)2
6.1.5.15.
Answer.
(sin⁡(x))1010+C
6.1.5.17.
Answer.
cos⁡(2−8x)8+C
6.1.5.19.
Answer.
14ln⁡(|sec⁡(4x)+tan⁡(4x)|)+C
6.1.5.21.
Answer.
16sin⁡(x6)+C
6.1.5.23.
Answer.
ln⁡(|sin⁡(x)|)+C
6.1.5.25.
Answer.
12e2x−3+C
6.1.5.27.
Answer.
12e(x−5)2+C
6.1.5.29.
Answer.
ln⁡(ex+5)+C
6.1.5.31.
Answer.
77x13.6214+C
6.1.5.33.
Answer.
ln2⁡(x)2+C
6.1.5.35.
Answer.
(32)(ln⁡(x))2+C
6.1.5.37.
Answer.
x22−3x−7ln⁡(|x|)+C
6.1.5.39.
Answer.
13(x+1)3+(32)(x+1)2+3(x+1)+C
6.1.5.41.
Answer.
3(x+9)2−103(x+9)+433ln⁡(|x+9|)+C
6.1.5.43.
Answer.
1.41421tan−1⁡(x1.41421)+C
6.1.5.45.
Answer.
9sin−1⁡(x2.23607)+C
6.1.5.47.
Answer.
(75)sec−1⁡(|x|5)+C
6.1.5.49.
Answer.
0.258199tan−1⁡(x+315)+C
6.1.5.51.
Answer.
5sin−1⁡(x+74)+C
6.1.5.53.
Answer.
C−13(x3+2)
6.1.5.55.
Answer.
(18)4+8x2+C
6.1.5.57.
Answer.
C−23(cos⁡(x))(32)
6.1.5.59.
Answer.
ln⁡(|x+5|)+C
6.1.5.61.
Answer.
(32)x2−x+ln⁡(|x2−9x+6|)+C
6.1.5.63.
Answer.
5ln⁡(|−(5x2+7x+3)|)+C
6.1.5.65.
Answer.
110tan−1⁡(x25)+C
6.1.5.67.
Answer.
sec−1⁡(|7x|)+C
6.1.5.69.
Answer.
(32)ln⁡(|x2+14x+53|)+3tan−1⁡(x+72)+C
6.1.5.71.
Answer.
x+13.0639tan−1⁡(x−72.44949)+7ln⁡(|x2−14x+55|)+C
6.1.5.73.
Answer.
12x2+6x+(32)ln⁡(|x2−6x+20|)+7.23627tan−1⁡(x−33.31662)+C
6.1.5.75.
Answer.
tan−1⁡(sin⁡(x))+C
6.1.5.77.
Answer.
3x2−10x+18+C
6.1.5.79.
Answer.
ln⁡((14))
6.1.5.81.
Answer.
0
6.1.5.83.
Answer.
12(e−e4)
6.1.5.85.
Answer.
π2

6.2 Integration by Parts

Exercises

Terms and Concepts
6.2.1.
Answer.
True
Problems
6.2.5.
Answer.
sin⁡(x)−xcos⁡(x)+C
6.2.7.
Answer.
−x2cos⁡(x)+2xsin⁡(x)+2cos⁡(x)+C
6.2.9.
Answer.
12ex2+C
6.2.11.
Answer.
−12xe−2x−e−2x4+C
6.2.13.
Answer.
15e2x(sin⁡(x)+2cos⁡(x))+C
6.2.15.
Answer.
(110)e5x(sin⁡(5x)+cos⁡(5x))+C
6.2.17.
Answer.
1−x2+xsin−1⁡(x)+C
6.2.19.
Answer.
0.5x2tan−1⁡(x)−x2+0.5tan−1⁡(x)+C
6.2.21.
Answer.
0.5x2ln⁡(x)−x24+C
6.2.23.
Answer.
12x2ln⁡(x+3)−14(x+3)2+3x−(92)ln⁡(x+3)+C
6.2.25.
Answer.
0.333333x3ln⁡(x)−x39+C
6.2.27.
Answer.
2(x+5)+(x+5)(ln⁡(x+5))2−2(x+5)ln⁡(x+5)+C
6.2.29.
Answer.
ln⁡(|sin⁡(x)|)−xcot⁡(x)+C
6.2.31.
Answer.
13(x2−3)(32)+C
6.2.33.
Answer.
xsec⁡(x)−ln⁡(|sec⁡(x)+tan⁡(x)|)+C
6.2.35.
Answer.
x2(sin⁡(ln⁡(x))+cos⁡(ln⁡(x)))+C
6.2.37.
Answer.
2cos⁡(x)+2xsin⁡(x)+C
6.2.39.
Answer.
2xex−2ex+C
6.2.41.
Answer.
π
6.2.43.
Answer.
0
6.2.45.
Answer.
72
6.2.47.
Answer.
(−94)e−8−(−74)e−6
6.2.49.
Answer.
0.2(eπ+e−π)

6.3 Trigonometric Integrals
6.3.4 Exercises

Terms and Concepts

6.3.4.1.
Answer.
False
6.3.4.3.
Answer.
False

Problems

6.3.4.5.
Answer.
−0.2cos5⁡(x)+C
6.3.4.7.
Answer.
112(cos⁡(x))12−110(cos⁡(x))10+C
6.3.4.9.
Answer.
18(sin⁡(x))8−13(sin⁡(x))6+14(sin⁡(x))4+C
6.3.4.11.
Answer.
x8−0.03125sin⁡(4x)+C
6.3.4.13.
Answer.
C−((18)cos⁡(4x)+(112)cos⁡(6x))
6.3.4.15.
Answer.
116πsin⁡(8πx)−120πsin⁡(10πx)+C
6.3.4.17.
Answer.
6πcos⁡(π12πx)+65πcos⁡(5π12πx)+C
6.3.4.19.
Answer.
tan5⁡(x)5+tan3⁡(x)3+C
6.3.4.21.
Answer.
17(tan⁡(x))7+C
6.3.4.23.
Answer.
111(sec⁡(x))11−29(sec⁡(x))9+17(sec⁡(x))7+C
6.3.4.25.
Answer.
0.25tan⁡(x)sec3⁡(x)+0.375(sec⁡(x)tan⁡(x)+ln⁡(|sec⁡(x)+tan⁡(x)|))+C
6.3.4.27.
Answer.
0.25tan⁡(x)sec3⁡(x)−0.125(sec⁡(x)tan⁡(x)+ln⁡(|sec⁡(x)+tan⁡(x)|))+C
6.3.4.29.
Answer.
0
6.3.4.31.
Answer.
0
6.3.4.33.
Answer.
25

6.4 Trigonometric Substitution

Exercises

Terms and Concepts
6.4.1.
Answer.
backward
6.4.3.
Answer 1.
tan2⁡(θ)+1=sec2⁡(θ)
Answer 2.
4sec2⁡(θ)
Problems
6.4.5.
Answer.
12(xx2+1+ln⁡(x2+1+x))+C
6.4.7.
Answer.
12sin−1⁡(x)+x21−x2+C
6.4.9.
Answer.
12xx2−1−12ln⁡(|x+x2−1|)+C
6.4.11.
Answer.
x29x2+1+16ln⁡(3x+9x2+1)+C
6.4.13.
Answer.
x225x2−1−110ln⁡(|5x+25x2−1|)+C
6.4.15.
Answer.
6sin−1⁡(x1.41421)+C
6.4.17.
Answer.
x2−14−3.74166sec−1⁡(x3.74166)+C
6.4.19.
Answer.
x2−2+C
6.4.21.
Answer.
C−1x2+16
6.4.23.
Answer.
(12)x+4x2+8x+17+(12)tan−1⁡(x+4)+C
6.4.25.
Answer.
C−(10−x27x+17sin−1⁡(x3.16228))
6.4.27.
Answer.
π2
6.4.29.
Answer.
22+2ln⁡(1+12)
6.4.31.
Answer.
9sin−1⁡((13))+22

6.5 Partial Fraction Decomposition

Exercises

Terms and Concepts
6.5.1.
Answer.
rational
6.5.3.
Answer.
Ax+Bx+6
6.5.5.
Answer.
Ax−2+Bx+2
Problems
6.5.7.
Answer.
7ln⁡(|x+4|)+6ln⁡(|x−5|)+C
6.5.9.
Answer.
ln⁡(|x+5|)−ln⁡(|x−5|)+C
6.5.11.
Answer.
3ln⁡(|x−4|)−5x−4+C
6.5.13.
Answer.
6ln⁡(|x|)+2ln⁡(|x−8|)+1x−8+C
6.5.15.
Answer.
C−((23)ln⁡(|3x−1|)+(13)ln⁡(|9x+2|)+(12)2x+2)
6.5.17.
Answer.
12x2−9x−9ln⁡(|x+3|)+72ln⁡(|x+6|)+C
6.5.19.
Answer.
(14)ln⁡(|x|)−(18)ln⁡(x2−2x+4)+0.144338tan−1⁡(x−11.73205)+C
6.5.21.
Answer.
ln⁡(|8x2+7x+1|)+ln⁡(|x−6|)+C
6.5.23.
Answer.
(28353)ln⁡(|x+7|)−(953)ln⁡(x2+4)−(139106)tan−1⁡(x2)+C
6.5.25.
Answer.
2ln⁡(|x+5|)+ln⁡(x2+4x+11)−0.377964tan−1⁡(x+22.64575)+C
6.5.27.
Answer.
ln⁡((256375))
6.5.29.
Answer.
ln⁡((54))+−π4−tan−1⁡(−3)

6.6 Hyperbolic Functions
6.6.3 Exercises

Problems

6.6.3.11.
Answer.
2cosh⁡(2x)
6.6.3.13.
Answer.
sech2⁡(x2)⋅2x
6.6.3.15.
Answer.
cosh⁡(x)cosh⁡(x)+sinh⁡(x)sinh⁡(x)
6.6.3.17.
Answer.
−1x21−(x2)2⋅2x
6.6.3.19.
Answer.
1(2x2)2−1⋅4x
6.6.3.21.
Answer.
−11−cos2⁡(x)sin⁡(x)
6.6.3.23.
Answer.
1(x−0)+0
6.6.3.25.
Answer.
0.36(x−(−1.09861))+(−0.8)
6.6.3.27.
Answer.
1(x−0)+0
6.6.3.29.
Answer.
0.5ln⁡(cosh⁡(2x))+C
6.6.3.31.
Answer.
0.5sinh2⁡(x)+C
6.6.3.33.
Answer.
xcosh⁡(x)−sinh⁡(x)+C
6.6.3.35.
Answer.
cosh−1⁡x/3+C=ln⁡(x+x2−9)+C
6.6.3.37.
Answer.
cosh−1⁡(x22)+C
6.6.3.39.
Answer.
−0.0625tan−1⁡(x2)+0.03125ln⁡(|x−2|)−0.03125ln⁡(|x+2|)+C
6.6.3.41.
Answer.
tan−1⁡(ex)+C
6.6.3.43.
Answer.
xtanh−1⁡(x)+0.5ln⁡(|x2−1|)+C
6.6.3.45.
Answer.
0
6.6.3.47.
Answer.
0.761594

6.7 L’Hospital’s Rule
6.7.4 Exercises

Terms and Concepts

6.7.4.3.
Answer.
False

Problems

6.7.4.9.
Answer.
3
6.7.4.11.
Answer.
−1
6.7.4.13.
Answer.
5
6.7.4.15.
Answer.
0.666667
6.7.4.17.
Answer.
∞
6.7.4.19.
Answer.
0
6.7.4.21.
Answer.
0
6.7.4.23.
Answer.
∞
6.7.4.25.
Answer.
0
6.7.4.27.
Answer.
−2
6.7.4.29.
Answer.
0
6.7.4.31.
Answer.
0
6.7.4.33.
Answer.
∞
6.7.4.35.
Answer.
∞
6.7.4.37.
Answer.
0
6.7.4.39.
Answer.
1
6.7.4.41.
Answer.
1
6.7.4.43.
Answer.
1
6.7.4.45.
Answer.
1
6.7.4.47.
Answer.
1
6.7.4.49.
Answer.
2
6.7.4.51.
Answer.
−∞
6.7.4.53.
Answer.
0

6.8 Improper Integration
6.8.4 Exercises

Terms and Concepts

6.8.4.5.
Answer.
p>1

Problems

6.8.4.7.
Answer.
e52
6.8.4.9.
Answer.
13
6.8.4.11.
Answer.
1ln⁡(2)
6.8.4.13.
Answer.
∞
6.8.4.15.
Answer.
1
6.8.4.17.
Answer.
∞
6.8.4.19.
Answer.
∞
6.8.4.21.
Answer.
∞
6.8.4.23.
Answer.
1
6.8.4.25.
Answer.
0
6.8.4.27.
Answer.
−14
6.8.4.29.
Answer.
∞
6.8.4.31.
Answer.
1
6.8.4.33.
Answer.
12
6.8.4.35.
Answer 1.
Limit Comparison Test
Answer 2.
diverges
Answer 3.
1x
6.8.4.37.
Answer 1.
Limit Comparison Test
Answer 2.
diverges
Answer 3.
1x
6.8.4.39.
Answer 1.
Direct Comparison Test
Answer 2.
converges
Answer 3.
e−x
6.8.4.41.
Answer 1.
Direct Comparison Test
Answer 2.
converges
Answer 3.
1x2−1
6.8.4.43.
Answer 1.
Direct Comparison Test
Answer 2.
converges
Answer 3.
1ex

7 Applications of Integration
7.1 Area Between Curves

Exercises

Terms and Concepts
7.1.1.
Answer.
True
Problems
7.1.5.
Answer.
22.436
7.1.7.
Answer.
3.14159
7.1.9.
Answer.
0.5
7.1.11.
Answer.
0.721354
7.1.13.
Answer.
4.5
7.1.15.
Answer.
0.429204
7.1.17.
Answer.
0.166667
7.1.19.
Answer.
All enclosed regions have the same area, with regions being the reflection of adjacent regions. One region is formed on [π/4,5π/4], with area 22.
7.1.21.
Answer.
1
7.1.23.
Answer.
4.5
7.1.25.
Answer.
0.514298
7.1.27.
Answer.
1
7.1.29.
Answer.
4
7.1.31.
Answer.
262800 ft2

7.2 Volume by Cross-Sectional Area; Disk and Washer Methods

Exercises

Terms and Concepts
7.2.1.
Answer.
T
Problems
7.2.5.
Answer.
175π/3 units3
7.2.7.
Answer.
π/6 units3
7.2.9.
Answer.
35π/3 units3
7.2.11.
Answer.
2π/15 units3
7.2.13.
7.2.13.a
Answer.
512π/15
7.2.13.b
Answer.
256π/5
7.2.13.c
Answer.
832π/15
7.2.13.d
Answer.
128π/3
7.2.15.
7.2.15.a
Answer.
104π/15
7.2.15.b
Answer.
64π/15
7.2.15.c
Answer.
32π/5
7.2.17.
7.2.17.a
Answer.
8π
7.2.17.b
Answer.
8π
7.2.17.c
Answer.
16π/3
7.2.17.d
Answer.
8π/3
7.2.19.
Answer.
250π/3
7.2.21.
Answer.
187.5

7.3 The Shell Method

Exercises

Terms and Concepts
7.3.1.
Answer.
T
7.3.3.
Answer.
F
Problems
7.3.5.
Answer.
9π/2 units3
7.3.7.
Answer.
π2−2π units3
7.3.9.
Answer.
48π3/5 units3
7.3.11.
Answer.
π2/4 units3
7.3.13.
7.3.13.a
Answer.
4π/5
7.3.13.b
Answer.
8π/15
7.3.13.c
Answer.
π/2
7.3.13.d
Answer.
5π/6
7.3.15.
7.3.15.a
Answer.
4π/3
7.3.15.b
Answer.
π/3
7.3.15.c
Answer.
4π/3
7.3.15.d
Answer.
2π/3
7.3.17.
7.3.17.a
Answer.
2π(2−1)
7.3.17.b
Answer.
2π(1−2+sinh−1⁡(1))

7.4 Arc Length and Surface Area
7.4.3 Exercises

Problems

7.4.3.3.
Answer.
2
7.4.3.5.
Answer.
103
7.4.3.7.
Answer.
1573
7.4.3.9.
Answer.
125
7.4.3.11.
Answer.
−ln⁡(2−3)≈1.31696
7.4.3.13.
Answer.
∫011+4x2dx
7.4.3.15.
Answer.
∫1e1+1x2dx
7.4.3.17.
Answer.
∫0π/21+sin2⁡(x)dx
7.4.3.19.
Answer.
1.4790
7.4.3.21.
Answer.
2.1300
7.4.3.23.
Answer.
1.00013
7.4.3.25.
Answer.
2π∫012x5dx=2π5
7.4.3.27.
Answer.
2π∫01x1+4x2dx=π/6(55−1)
7.4.3.29.
Answer.
∫011+14xdx
7.4.3.31.
Answer.
∫−331+x281−9x2dx
7.4.3.33.
Answer.
2π∫011−x21+x/(1−x2)dx=4π

7.5 Work
7.5.4 Exercises

Terms and Concepts

7.5.4.1.
Answer.
In SI units, it is one joule, i.e., one newton–meter, or kg·m⁄s2m In Imperial Units, it is ft–lb.
7.5.4.3.
Answer.
Smaller.

Problems

7.5.4.5.
7.5.4.5.a
Answer.
500 ft–lb
7.5.4.5.b
Answer.
100−502≈29.29 ft
7.5.4.7.
7.5.4.7.a
Answer.
12⋅d⋅l2 ft–lb
7.5.4.7.b
Answer.
75 %
7.5.4.7.c
Answer.
ℓ(1−2/2)≈0.2929ℓ
7.5.4.9.
7.5.4.9.a
Answer.
756 ft–lb
7.5.4.9.b
Answer.
60,000 ft–lb
7.5.4.9.c
Answer.
Yes, for the cable accounts for about 1% of the total work.
7.5.4.11.
Answer.
575 ft–lb
7.5.4.13.
Answer.
0.05 J
7.5.4.15.
Answer.
5/3 ft–lb
7.5.4.17.
Answer.
f⋅d/2 J
7.5.4.19.
Answer.
5 ft–lb
7.5.4.21.
7.5.4.21.a
Answer.
52,929.6 ft–lb
7.5.4.21.b
Answer.
18,525.3 ft–lb
7.5.4.21.c
Answer.
When 3.83 ft of water have been pumped from the tank, leaving about 2.17 ft in the tank.
7.5.4.23.
Answer.
212,135 ft–lb
7.5.4.25.
Answer.
187,214 ft–lb
7.5.4.27.
Answer.
4,917,150 J

7.6 Fluid Forces

Exercises

Terms and Concepts
7.6.1.
Answer.
Answers will vary.
Problems
7.6.3.
Answer.
499.2 lb
7.6.5.
Answer.
6739.2 lb
7.6.7.
Answer.
3920.7 lb
7.6.9.
Answer.
2496 lb
7.6.11.
Answer.
602.59 lb
7.6.13.
Answer.
  1. 2340 lb
  2. 5625 lb
7.6.15.
Answer.
  1. 1597.44 lb
  2. 3840 lb
7.6.17.
Answer.
  1. 56.42 lb
  2. 135.62 lb
7.6.19.
Answer.
5.1 ft

8 Differential Equations
8.1 Graphical and Numerical Solutions to Differential Equations
8.1.4 Exercises

Terms and Concepts

8.1.4.1.
Answer.
An initial value problems is a differential equation that is paired with one or more initial conditions. A differential equation is simply the equation without the initial conditions.
8.1.4.3.
Answer.
Substitute the proposed function into the differential equation, and show the the statement is satisfied.
8.1.4.5.
Answer.
Many differential equations are impossible to solve analytically.

Problems

8.1.4.7.
Answer.
Answers will vary.
8.1.4.9.
Answer.
Answers will vary.
8.1.4.11.
Answer.
C=2
8.1.4.13.
Answer.
Graph showing slope field for the given differential equation.
The x and y axes are uncalibrated.In the first quadrant in the top left, the field lines are north-east facing and in the bottom right they are southeast facing. In the second quadrant the field lines are all north-east facing. In the third quadrant like in the first quadrant in the top left the field lines are northeast facing and in the bottom right they are southeast facing. In the fourth quadrant all lines are southeast facing.
8.1.4.15.
Answer.
Graph showing slope field for the given differential equation.
The x and y axes are uncalibrated. There are five instances where the field lines run parallel to the x axis. One of them is on the x axis itself, other two pairs of such field lines are above and below the x axis. In between the x axis and the first horizontal field line for some positive y value, the field lines are all northeast facing. Above the horizontal field line for some y value until another with a higher y value, the field lines in between are southeast facing.
Similarly below the x axis till the first horizontal line with some negative y value, the field lines in between are southeast facing. In between this horizontal line and another horizontal line with a higher negative y value, the field lines are northeast facing.
8.1.4.17.
Answer.
b
8.1.4.19.
Answer.
d
8.1.4.21.
Answer.
Graph showing slope field for the given differential equation.
The x and y axes are uncalibrated, the field lines in the first quadrant are shown. The field lines very close to the y axis are almost north facing for higher values of y and almost east facing for lower values of y. With smaller values of x, the field lines, from left to right the lines first face northeast then east and southeast after for greater values of x.
A curve is drawn that starts at a point for some small value of x and a high value of y. The curve has a positive slope at first after reaching a peak it declines almost close to the x axis.
8.1.4.23.
Answer.
Graph showing slope field for the given differential equation.
The x and y axes are uncalibrated, the field lines in the first quadrant are shown. There are two instances where the field lines are parallel to the x axis. From under the x axis to the first such line the field lines transition from almost north facing to northeast facing. Between the horizontal field line for a small y value and a greater y value the field lines are facing southeast. Above the line with a higher y value the field lines transition from northeast facing to north facing.
8.1.4.25.
Answer.
xiyi0.001.00000.251.50000.502.31250.753.59381.005.5781
8.1.4.27.
Answer.
xiyi0.02.00000.22.40000.42.91970.63.58160.84.41081.05.4364
8.1.4.29.
Answer.
x 0.0 0.2 0.4 0.6 0.8 1.0
y(x) 1.0000 1.0204 1.0870 1.2195 1.4706 2.0000
h=0.2 1.0000 1.0000 1.0400 1.1265 1.2788 1.5405
h=0.1 1.0000 1.0100 1.0623 1.1687 1.3601 1.7129

8.2 Separable Differential Equations
8.2.2 Exercises

Problems

8.2.2.1.
Answer.
Separable. 1y2−ydy=dx
8.2.2.3.
Answer.
Not separable.
8.2.2.5.
Answer.
{y=1+Ce2x1−Ce2x,y=−1}
8.2.2.7.
Answer.
y=Cx4
8.2.2.9.
Answer.
(y−1)ey=−e−x−13e−3x+C
8.2.2.11.
Answer.
{arcsin⁡2y−arctan⁡(x2+1)=C,y=±12}
8.2.2.13.
Answer.
sin⁡(y)+cos⁡(x)=2
8.2.2.15.
Answer.
12y2−ln⁡(1+x2)=8
8.2.2.17.
Answer.
12y2−y=12((x2+1)ln⁡(x2+1)−(x2+1))+12
8.2.2.19.
Answer.
2tan⁡(2y)=2x+sin⁡(2x)

8.3 First Order Linear Differential Equations
8.3.2 Exercises

Problems

8.3.2.1.
Answer.
y=32+Ce2x
8.3.2.3.
Answer.
y=−12x+Cx
8.3.2.5.
Answer.
y=sec⁡x+C(csc⁡x)
8.3.2.7.
Answer.
y=Ce3x−(x+1)e2x
8.3.2.9.
Answer.
y=(x2+2)ex
8.3.2.11.
Answer.
y=1−2x+2−e1−xx2
8.3.2.13.
Answer.
y=x2+1x+1e−x
8.3.2.15.
Answer.
y=(x−2)(x+1)x−1
8.3.2.17.
Answer.
Both; y=−5ex+13x3
8.3.2.19.
Answer.
linear; y=x3−3x−63(x−1)
8.3.2.21.
Answer.
Graph showing slope field for the given differential equation.
The x and y axes are uncalibrated, the field lines in the first quadrant are shown. On the bottom right the field lines are facing northeast. On the top left the field lines transition from southeast facing to east facing moving downwards. A curve is shown that almost represents a straight line with a positive slope.
The solution will increase and begin to follow the line y=x−1.
y=x−1+e−x

8.4 Modeling with Differential Equations
8.4.3 Exercises

Problems

8.4.3.1.
Answer.
y=10+Ce−kx
8.4.3.3.
Answer.
4.43 days
8.4.3.5.
Answer.
x={ab(1−e(a−b)kt)b−ae(a−b)kt if a≠ba2kt1+akt if a=b
8.4.3.7.
Answer.
y=60−3.69858e−14t+43.69858e−0.0390169t
8.4.3.9.
Answer.
y=8(1−e−12t) g/cm2
8.4.3.11.
Answer.
11.00075 g

9 Sequences and Series
9.1 Sequences

Exercises

Terms and Concepts
9.1.1.
Answer.
Answers will vary.
9.1.3.
Answer.
Answers will vary.
Problems
9.1.5.
Answer.
2,83,83,3215,6445
9.1.7.
Answer.
−13,−2,−815,−5123,−156257
9.1.9.
Answer.
an=3n+1
9.1.11.
Answer.
an=10⋅2n−1
9.1.13.
Answer.
1/7
9.1.15.
Answer.
0
9.1.17.
Answer.
diverges
9.1.19.
Answer.
converges to 0
9.1.21.
Answer.
diverges
9.1.23.
Answer.
converges to e
9.1.25.
Answer.
converges to 0
9.1.27.
Answer.
converges to 2
9.1.29.
Answer.
bounded
9.1.31.
Answer.
bounded
9.1.33.
Answer.
neither bounded above or below
9.1.35.
Answer.
monotonically increasing
9.1.37.
Answer.
never monotonic

9.2 Infinite Series
9.2.4 Exercises

Terms and Concepts

9.2.4.1.
Answer.
Answers will vary.
9.2.4.5.
Answer.
F

9.3 Integral and Comparison Tests
9.3.4 Exercises

Terms and Concepts

9.3.4.1.
Answer.
continuous, positive and decreasing

Problems

9.3.4.5.
Answer.
Converges
9.3.4.7.
Answer.
Diverges
9.3.4.9.
Answer.
Converges
9.3.4.11.
Answer.
Converges

9.4 Ratio and Root Tests
9.4.3 Exercises

Terms and Concepts

9.4.3.1.
Answer.
algebraic, or polynomial.
9.4.3.3.
Answer.
Integral Test, Limit Comparison Test, and Root Test

Problems

9.4.3.5.
Answer.
Converges
9.4.3.7.
Answer.
Converges
9.4.3.9.
Answer.
The Ratio Test is inconclusive; the p-Series Test states it diverges.
9.4.3.11.
Answer.
Converges
9.4.3.13.
Answer.
Converges; note the summation can be rewritten as ∑n=1∞2nn!3nn!, from which the Ratio Test or Geometric Series Test can be applied.
9.4.3.15.
Answer.
Converges
9.4.3.17.
Answer.
Converges
9.4.3.19.
Answer.
Diverges
9.4.3.21.
Answer.
Diverges. The Root Test is inconclusive, but the nth-Term Test shows divergence. (The terms of the sequence approach e−2, not 0, as n→∞.)
9.4.3.23.
Answer.
Converges

9.5 Alternating Series and Absolute Convergence

Exercises

Terms and Concepts
9.5.3.
Answer.
Many examples exist; one common example is an=(−1)n/n.

9.7 Taylor Polynomials

Exercises

Terms and Concepts
9.7.3.
Answer.
6+3x−4x2
Problems
9.7.5.
Answer.
1−x+0.5x2−0.166667x3
9.7.7.
Answer.
x+x2+0.5x3+0.166667x4+0.0416667x5
9.7.9.
Answer.
1+2x+2x2+1.33333x3+0.666667x4
9.7.11.
Answer.
1−x+x2−x3+x4
9.7.13.
Answer.
1+0.5(x−1)−0.125(x−1)2+0.0625(x−1)3−0.0390625(x−1)4
9.7.15.
Answer.
0.707107−0.707107(x−π4)−0.353553(x−π4)2+0.117851(x−π4)3+0.0294628(x−π4)4−0.00589256(x−π4)5−0.000982093(x−π4)6
9.7.17.
Answer.
0.5−0.25(x−2)+0.125(x−2)2−0.0625(x−2)3+0.03125(x−2)4−0.015625(x−2)5
9.7.19.
Answer.
0.5+0.5(x+1)+0.25(x+1)2
9.7.31.
Answer.
The nth term is: when n even, 0; when n is odd, (−1)(n−1)/2n!xn.

10 Curves in the Plane
10.1 Conic Sections
10.1.4 Exercises

Problems

10.1.4.19.
Answer.
(x+1)29+(y−2)24=1; foci at (−1±5,2); e=5/3
10.1.4.29.
Answer.
x2−y23=1
10.1.4.31.
Answer.
(y−3)24−(x−1)29=1
10.1.4.45.
Answer.
The sound originated from a point approximately 31m to the right of B and 1390m above or below it. (Since the three points are collinear, we cannot distinguish whether the sound originated above/below the line containing the points.)

10.2 Parametric Equations
10.2.4 Exercises

Terms and Concepts

10.2.4.1.
Answer.
True
10.2.4.3.
Answer.
rectangular

Problems

10.2.4.5.
Answer.
Sketch of the parametric curve in this exercise.
The sketch for this exercise is a curve that lies mostly in the fourth quadrant. It resembles part of a slingshot orbit for a comet passing around the sun: the curve passes through the origin from below, turns quickly in the second quadrant, crossing the y axis at (0,1), and then the x axis at (2,0), where it returns to the fourth quadrant.
10.2.4.7.
Answer.
The horizontal line y=2, marked with two arrows.
The horizontal line y=2. On the line there are two arrows pointing in opposite directions. These indicate that the direction of travel is to the left when t<0, and to the right when t>0.
10.2.4.9.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
A curve resembling a check mark, with a cusp at the origin. Direction of travel is from the second quadrant toward the cusp, and then up from the cusp to a y intercept at (0,4), and then into the first quadrant.
10.2.4.11.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
The curve is an ellipse, centered at the origin, with counter-clockwise direction of travel.
10.2.4.13.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
The curve resembles a parabola, with vertex at (0,−1). The direction of travel is from right to left.
10.2.4.15.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
The curve resembles one branch of a hyperbola, opening to the right, with a vertex at (2,0). The direction of travel is that of increasing y value.
10.2.4.17.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
A flower-shaped curve, with 7 “petals”. Each petal is an arc that loops around and intersects itself before continuing to the next arc.
10.2.4.19.
10.2.4.19.a
Answer.
Traces the parabola y=x2, moves from left to right.
10.2.4.19.b
Answer.
Traces the parabola y=x2, but only from −1≤x≤1; traces this portion back and forth infinitely.
10.2.4.19.c
Answer.
Traces the parabola y=x2, but only for 0<x. Moves left to right.
10.2.4.19.d
Answer.
Traces the parabola y=x2, moves from right to left.
10.2.4.21.
Answer.
3x+2y=17
10.2.4.25.
Answer.
y−2x=3
10.2.4.35.
Answer 1.
t+116
Answer 2.
t2−9712
Answer 3.
(2,−8)
Answer 4.
6x−11
Answer 5.
1
10.2.4.37.
Answer 1.
cos−1⁡(t)
Answer 2.
1−t2
Answer 3.
(0,0)
Answer 4.
cos⁡(x)
Answer 5.
1
10.2.4.39.
Answer 1.
−1,1
Answer 2.
(3,−2)
10.2.4.51.
Answer.
3cos⁡(2πt)+1;3sin⁡(2πt)+1

10.3 Calculus and Parametric Equations
10.3.4 Exercises

Terms and Concepts

10.3.4.1.
Answer.
False
10.3.4.3.
Answer.
False

Problems

10.3.4.15.
Answer 1.
−0.5
Answer 2.
(0.75,−0.25)
10.3.4.21.
Answer 1.
0
Answer 2.
0
10.3.4.27.
Answer 1.
−4(2t−1)3
Answer 2.
(−∞,0.5]
Answer 3.
[0.5,∞)
10.3.4.33.
Answer.
6π
10.3.4.35.
Answer.
234

10.4 Introduction to Polar Coordinates
10.4.4 Exercises

Terms and Concepts

10.4.4.1.
Answer.
Answers will vary.
10.4.4.3.
Answer.
True

Problems

10.4.4.5.
Answer.
The four points plotted in this exercise.
On a polar grid, four points are plotted. The point A is at the intersection of the initial ray and the circle of radius 2. Points B and D are both on the circle of radius 1. The point B is on the same line as the initial ray, but in the opposite direction. The point D lies above the initial ray, making an angle of π/4. Finally, the point C is at the bottom of the circle of radius 2.
10.4.4.7.
Answer.
A=P(2.5,π/4) and P(−2.5,5π/4);
B=P(−1,5π/6) and P(1,11π/6);
C=P(3,4π/3) and P(−3,π/3);
D=P(1.5,2π/3) and P(−1.5,5π/3);
10.4.4.9.
Answer 1.
(2,2)
Answer 2.
(2,−2)
Answer 3.
(5,tan−1⁡(−12))
Answer 4.
(5,π+tan−1⁡(−12))
10.4.4.11.
Answer.
Portion of the circle of radius 2, centered at the origin, in the first quadrant.
An arc of the circle r=2 is shown, for 0≤θ≤π/2. This is the quarter of a circle of radius 2, centered at the origin, that lies in the first quadrant.
10.4.4.13.
Answer.
A cardioid, symmetric about the x axis, with x intercepts at -2 and 0.
The curve is a cardioid that is symmetric about the x axis. The cusp is at the origin, and the other x intercept is at (−2,0). (It is in the opposite direction of the example in the gallery of polar curves.)
10.4.4.15.
Answer.
A convex limaçon, symmetric about the y axis.
The curve is a convex limaçon. This is the fourth type of limaçon in the gallery of polar curves. In this case, the limaçon is symmetric about the y axis, with the flattened part of the curve at the top.
10.4.4.17.
Answer.
A limaçon with an inner loop, symmetric about the y axis.
The curve is a limaçon with an inner loop. It is symmetric about the y axis. The inner loop lies above the x axis, with y intercepts at y=0 and y=1. The outer loop has its other y intercept at y=3.
10.4.4.19.
Answer.
A rose curve with three loops, symmetric about the y axis.
A rose curve with three loops that all pass through the origin. One loop is along the negative y axis, with a y intercept at (−1,0). The other two loops lie in the first and second quadrants.
10.4.4.21.
Answer.
An elaborate rose curve with many self-intersections and four primary loops.
This is a more complicated curve. It passes several times through the origin, and has eight other points of self-intersection. The largest loops in the curve are similar to cardioids; there are four of these passing through the origin, with a second intercept at one of the four points (±1,0), (0,±1). As these loops intersect each other, they create four other loops of intermediate size, and four smaller loops in the center.
10.4.4.23.
Answer.
A circle passing through the origin with its center on the positive y axis.
A circle of radius 3/2 with its center at (0,3/2). It passes through the origin and the point (0,3).
10.4.4.25.
Answer.
A four-leaf rose with one petal in each quadrant.
The curve is a four-leafed rose that lies within the circle r=1/2. One leaf lies in each of the four quadrants.
10.4.4.27.
Answer.
A straight line with positive slope.
The curve is a straight line with x intercept at (−3,0) and y intercept (0,3/5).
10.4.4.29.
Answer.
A vertical line with x=3.
The curve is the vertical line x=3.
10.4.4.31.
Answer.
(x−3)2+y2=9
10.4.4.33.
Answer.
(x−0.5)2+(y−0.5)2=0.5
10.4.4.35.
Answer.
x=3
10.4.4.39.
Answer.
x2+y2=4
10.4.4.41.
Answer.
θ=π4
10.4.4.43.
Answer.
r=5sec⁡(θ)
10.4.4.45.
Answer.
r=cos⁡(θ)sin2⁡(θ)
10.4.4.47.
Answer.
r=7
10.4.4.49.
Answer.
P(32,π6),P(0,π2),P(−32,5π6)
10.4.4.51.
Answer.
P(0,0),P(2,π4)

10.5 Calculus and Polar Functions
10.5.5 Exercises

Problems

10.5.5.3.
Answer 1.
−cot⁡(θ)
Answer 2.
y=−(x−22)+22
Answer 3.
y=x
10.5.5.7.
Answer 1.
θcos⁡(θ)+sin⁡(θ)cos⁡(θ)−θsin⁡(θ)
Answer 2.
y=−2πx+π2
Answer 3.
y=π2x+π2
10.5.5.9.
Answer 1.
4sin⁡(θ)cos⁡(4θ)+sin⁡(4θ)cos⁡(θ)4cos⁡(θ)cos⁡(4θ)−sin⁡(θ)sin⁡(4θ)
Answer 2.
y=53(x+34)−34
Answer 3.
y=−153(x+34)−34
10.5.5.19.
Answer.
π12
10.5.5.21.
Answer.
3π2
10.5.5.23.
Answer.
2π+3⋅1.732052
10.5.5.25.
Answer.
1
10.5.5.29.
Answer.
4π
10.5.5.31.
Answer.
2π
10.5.5.33.
Answer.
2.2592 or 2.22748

11 Vectors
11.1 Introduction to Cartesian Coordinates in Space
11.1.7 Exercises

11.2 An Introduction to Vectors

Exercises

Problems
11.2.7.
11.2.7.a
Answer.
⟨1,6⟩
11.2.7.b
Answer.
i+6j
11.2.9.
11.2.9.a
Answer.
⟨6,−1,6⟩
11.2.9.b
Answer.
6i−j+6k
11.2.11.
11.2.11.a
Answer.
u→+v→=⟨2,−1⟩; u→−v→=⟨0,−3⟩; 2u→−3v→=⟨−1,−7⟩.
11.2.11.c
Answer.
x→=⟨1/2,2⟩.
11.2.17.
Answer 1.
5
Answer 2.
13
Answer 3.
26
Answer 4.
10
11.2.19.
Answer 1.
5
Answer 2.
35
Answer 3.
25
Answer 4.
45
11.2.23.
Answer.
⟨0.6,0.8⟩
11.2.25.
Answer.
⟨13,−13,13⟩
11.2.27.
Answer.
⟨−12,32⟩

11.3 The Dot Product
11.3.2 Exercises

Terms and Concepts

11.3.2.1.
Answer.
Scalar

Problems

11.3.2.5.
Answer.
−22
11.3.2.7.
Answer.
3
11.3.2.9.
Answer.
not defined
11.3.2.11.
Answer.
Answers will vary.
11.3.2.13.
Answer.
cos−1⁡(310)
11.3.2.15.
Answer.
π4
11.3.2.17.
Answer 1.
⟨−7,4⟩
Answer 2.
⟨4,7⟩
11.3.2.19.
Answer 1.
⟨1,0,−1⟩
Answer 2.
⟨1,1,1⟩
11.3.2.21.
Answer.
⟨−510,1510⟩
11.3.2.23.
Answer.
⟨−12,−12⟩
11.3.2.25.
Answer.
⟨1414,2814,4214⟩
11.3.2.27.
Answer 1.
⟨−510,1510⟩
Answer 2.
⟨1510,510⟩
11.3.2.29.
Answer 1.
⟨−12,−12⟩
Answer 2.
⟨−52,52⟩
11.3.2.31.
Answer 1.
⟨1414,2814,4214⟩
Answer 2.
⟨014,4214,−2814⟩
11.3.2.33.
Answer.
1.96lb
11.3.2.35.
Answer.
141.42ft–lb
11.3.2.37.
Answer.
500ft–lb
11.3.2.39.
Answer.
500ft–lb

11.4 The Cross Product
11.4.3 Exercises

Terms and Concepts

11.4.3.1.
Answer.
vector
11.4.3.3.
Answer.
“Perpendicular” is one answer.
11.4.3.5.
Answer.
Torque

Problems

11.4.3.7.
Answer.
⟨12,−15,3⟩
11.4.3.9.
Answer.
⟨−5,−31,27⟩
11.4.3.11.
Answer.
⟨0,−2,0⟩
11.4.3.13.
Answer.
u→×v→=⟨0,0,ad−bc⟩
11.4.3.15.
Answer.
−j
11.4.3.17.
Answer.
Answers will vary.
11.4.3.19.
Answer.
5
11.4.3.21.
Answer.
0
11.4.3.23.
Answer.
14
11.4.3.25.
Answer.
3
11.4.3.27.
Answer.
522
11.4.3.29.
Answer.
1
11.4.3.31.
Answer.
7
11.4.3.33.
Answer.
2
11.4.3.35.
Answer.
⟨0.408248,0.408248,−0.816497⟩ or ⟨−0.408248,−0.408248,0.816497⟩
11.4.3.37.
Answer.
⟨0,1,0⟩ or ⟨0,−1,0⟩
11.4.3.39.
Answer.
87.5ft–lb
11.4.3.41.
Answer.
200/3≈66.67ft–lb

11.5 Lines
11.5.4 Exercises

Terms and Concepts

11.5.4.1.
Answer.
A point on the line and the direction of the line.
11.5.4.3.
Answer.
parallel, skew

Problems

11.5.4.11.
Answer 1.
(7,2,−1)+t⟨1,−1,2⟩
Answer 2.
x=7+t,y=2−t,z=−1+2t
Answer 3.
x−7=2−y=z+12
11.5.4.15.
Answer.
parallel
11.5.4.19.
Answer.
skew
11.5.4.23.
Answer.
41/3
11.5.4.25.
Answer.
52/2
11.5.4.27.
Answer.
3/2

11.6 Planes
11.6.2 Exercises

Terms and Concepts

11.6.2.1.
Answer.
A point in the plane and a normal vector (i.e., a direction orthogonal to the plane).

Problems

11.6.2.3.
Answer.
Answers will vary.
11.6.2.5.
Answer.
Answers will vary.
11.6.2.17.
Answer 1.
x−5+y−7+z−3=0
Answer 2.
x+y+z=15
11.6.2.19.
Answer 1.
3(x+4)+8(y−7)−10(z−2)=0
Answer 2.
3x+8y−10z=24
11.6.2.27.
Answer.
5/7
11.6.2.29.
Answer.
1/3

12 Vector Valued Functions
12.1 Vector-Valued Functions
12.1.4 Exercises

Terms and Concepts

12.1.4.1.
Answer.
parametric equations
12.1.4.3.
Answer.
displacement

Problems

12.1.4.15.
Answer.
Graph of the vector valued function from the example.
Graph of the function r→(t)=⟨cos⁡(t),sin⁡(t),sin⁡(t)⟩ on [0,2π]. The graph of the function is an oval lying in the plane coming from rotating the xy plane 45 degrees towards the z-axis. The oval lying in this plane has a horizontal width of 2 and a height of 1. Ignoring the z coordinate, the curve is a unit circle in the xy plane. Similarly ignoring the y coordinate, the curve is a unit circle in the xz plane. If we now ignore the x coordinate, the resulting curve is a diagonal line given by z=y in the yz plane. This line turns back on itself, which can be seen in the image of the oval when considering all three coordinate axes.
12.1.4.17.
Answer.
|t|1+t2
12.1.4.19.
Answer.
4+t2
12.1.4.21.
Answer.
⟨2cos⁡(t)+1,2sin⁡(t)+2⟩
12.1.4.25.
Answer.
⟨t+2,5t+3⟩
12.1.4.27.
Answer.
Specific forms may vary, though most direct solutions are
r→(t)=⟨1,2,3⟩+t⟨3,3,3⟩ and
r→(t)=⟨3t+1,3t+2,3t+3⟩.
12.1.4.29.
Answer.
⟨2cos⁡(t),2sin⁡(t),2t⟩
12.1.4.31.
Answer.
⟨1,0⟩
12.1.4.33.
Answer.
⟨0,0,1⟩

12.2 Calculus and Vector-Valued Functions
12.2.5 Exercises

Terms and Concepts

12.2.5.1.
Answer.
component

Problems

12.2.5.5.
Answer.
⟨11,74,sin⁡(5)⟩
12.2.5.7.
Answer.
⟨1,e⟩
12.2.5.9.
Answer.
(−∞,0)∪(0,∞)
12.2.5.11.
Answer.
⟨−sin⁡(t),et,1t⟩
12.2.5.13.
Answer.
⟨2tsin⁡(t)+t2cos⁡(t),6t2+10t⟩
12.2.5.15.
Answer.
⟨−1,cos⁡(t)−2t,6t2+10t+2+cos⁡(t)−sin⁡(t)−tcos⁡(t)⟩
12.2.5.21.
Answer.
⟨2+3t,t⟩
12.2.5.23.
Answer.
ℓ(t)=⟨−3,0,π⟩+t⟨0,−3,1⟩
12.2.5.33.
Answer.
⟨14t4,sin⁡(t),tet−et⟩+C→
12.2.5.35.
Answer.
⟨−2,0⟩
12.2.5.37.
Answer.
⟨t22+2,−cos⁡(t)+3⟩
12.2.5.39.
Answer.
⟨t412+t+4,t36+2t+5,t22+3t+6⟩
12.2.5.41.
Answer.
2⋅3.60555π
12.2.5.43.
Answer.
154(2232−8)

12.3 The Calculus of Motion
12.3.3 Exercises

Problems

12.3.3.7.
Answer.
v→(t)=⟨2,5,0⟩, a→(t)=⟨0,0,0⟩
12.3.3.19.
Answer 1.
|sec⁡(t)|tan2⁡(t)+sec2⁡(t)
Answer 2.
0
Answer 3.
π4
12.3.3.39.
12.3.3.39.a
Answer.
0.013 radians
12.3.3.39.b
Answer.
11.7 ft

12.4 Unit Tangent and Normal Vectors
12.4.4 Exercises

Terms and Concepts

12.4.4.1.
Answer.
1
12.4.4.3.
Answer.
T→(t) and N→(t).

Problems

12.4.4.5.
Answer.
T→(t)=⟨4t20t2−4t+1,2t−120t2−4t+1⟩; T→(1)=⟨4/17,1/17⟩
12.4.4.9.
Answer.
(2,0)+t⟨417,117⟩
12.4.4.13.
Answer.
T→(t)=⟨−sin⁡(t),cos⁡(t)⟩; N→(t)=⟨−cos⁡(t),−sin⁡(t)⟩
12.4.4.15.
Answer.
T→(t)=⟨−sin⁡(t)4cos2⁡(t)+sin2⁡(t),2cos⁡(t)4cos2⁡(t)+sin2⁡(t)⟩; N→(t)=⟨−2cos⁡(t)4cos2⁡(t)+sin2⁡(t),−sin⁡(t)4cos2⁡(t)+sin2⁡(t)⟩

12.5 The Arc Length Parameter and Curvature
12.5.4 Exercises

Terms and Concepts

12.5.4.1.
Answer.
time and/or distance
12.5.4.3.
Answer.
Answers may include lines, circles, helixes
12.5.4.5.
Answer.
κ

Problems

12.5.4.15.
Answer 1.
less than
Answer 2.
|2cos⁡(t)cos⁡(2t)+4sin⁡(t)sin⁡(2t)|(4cos2⁡(2t)+sin2⁡(t))32
Answer 3.
14
Answer 4.
8
12.5.4.23.
Answer.
254,−254
12.5.4.25.
Answer.
14

13 Functions of Several Variables
13.2 Limits and Continuity of Multivariable Functions
13.2.5 Exercises

Problems

13.2.5.7.
Answer.
  1. Answers will vary. interior point: (1,3) boundary point: (3,3)
  2. S is a closed set
  3. S is bounded
13.2.5.11.
Answer.
  1. D={(x,y)|9−x2−y2≥0}.
  2. D is a closed set.
  3. D is bounded.
13.2.5.13.
Answer.
  1. D={(x,y)|y>x2}.
  2. D is an open set.
  3. D is unbounded.

13.3 Partial Derivatives
13.3.7 Exercises

Terms and Concepts

13.3.7.3.
Answer.
f_x

Problems

13.3.7.19.
Answer 1.
2y24xy2+1
Answer 2.
4xy4xy2+1
Answer 3.
−4y4(4xy2+1)3
Answer 4.
−8xy3(4xy2+1)3+4y4xy2+1
Answer 5.
−8xy3(4xy2+1)3+4y4xy2+1
Answer 6.
−16x2y2(4xy2+1)3+4x4xy2+1

13.5 The Multivariable Chain Rule
13.5.3 Exercises

Terms and Concepts

13.5.3.5.
Answer.
F

Problems

13.5.3.7.
Answer.
  1. dzdt=3(2t)+4(2)=6t+8.
  2. At t=1, dzdt=14.
13.5.3.9.
Answer.
  1. dzdt=5(−2sin⁡(t))+2(cos⁡(t))=−10sin⁡(t)+2cos⁡(t)
  2. At t=π/4, dzdt=−42.
13.5.3.11.
Answer.
  1. dzdt=2x(cos⁡(t))+4y(3cos⁡(t)).
  2. At t=π/4, x=2/2, y=32/2, and dzdt=19.
13.5.3.21.
Answer 1.
2xcos⁡(t)+2ysin⁡(t)
Answer 2.
−2xssin⁡(t)+2yscos⁡(t)
Answer 3.
4
Answer 4.
0

13.6 Directional Derivatives
13.6.3 Exercises

Terms and Concepts

13.6.3.3.
Answer.
j

Problems

13.6.3.13.
13.6.3.13.a
Answer.
2/5
13.6.3.13.b
Answer.
−2/5
13.6.3.15.
13.6.3.15.a
Answer.
0
13.6.3.15.b
Answer.
22/9
13.6.3.17.
13.6.3.17.a
Answer.
0
13.6.3.17.b
Answer.
0
13.6.3.19.
13.6.3.19.a
Answer.
∇f(2,1)=⟨−2,2⟩
13.6.3.19.b
Answer.
8
13.6.3.19.c
Answer.
⟨2,−2⟩
13.6.3.19.d
Answer.
u→=⟨1/2,1/2⟩
13.6.3.21.
13.6.3.21.a
Answer.
∇f(1,1)=⟨−2/9,−2/9⟩
13.6.3.21.b
Answer.
22/9
13.6.3.21.c
Answer.
⟨2/9,2/9⟩
13.6.3.21.d
Answer.
u→=⟨1/2,−1/2⟩
13.6.3.23.
13.6.3.23.a
Answer.
No such direction
13.6.3.23.b
Answer.
0
13.6.3.23.c
Answer.
No such direction
13.6.3.23.d
Answer.
All directions
13.6.3.25.
13.6.3.25.a
Answer.
∇F(x,y,z)=⟨6xz3+4y,4x,9x2z2−6z⟩
13.6.3.25.b
Answer.
113/3
13.6.3.27.
13.6.3.27.a
Answer.
∇F(x,y,z)=⟨2xy2,2y(x2−z2),−2y2z⟩
13.6.3.27.b
Answer.
0

13.7 Tangent Lines, Normal Lines, and Tangent Planes
13.7.5 Exercises

Terms and Concepts

13.7.5.3.
Answer.
True

13.8 Extreme Values
13.8.3 Exercises

Terms and Concepts

13.8.3.1.
Answer.
False
13.8.3.3.
Answer.
True

Problems

13.8.3.15.
Answer 1.
3
Answer 2.
(0,1)
Answer 3.
34
Answer 4.
(0,−12)

14 Multiple Integration
14.1 Iterated Integrals and Area
14.1.4 Exercises

Terms and Concepts

14.1.4.3.
Answer.
curve to curve, then from point to point

Problems

14.1.4.5.
14.1.4.5.a
Answer.
18x2+42x−117
14.1.4.5.b
Answer.
−108
14.1.4.7.
14.1.4.7.a
Answer.
x4/2−x2+2x−3/2
14.1.4.7.b
Answer.
23/15
14.1.4.9.
14.1.4.9.a
Answer.
sin2⁡(y)
14.1.4.9.b
Answer.
π/2

14.3 Double Integration with Polar Coordinates

Exercises

Problems
14.3.3.
Answer.
4π
14.3.5.
Answer.
16π

14.5 Surface Area

Exercises

Problems
14.5.7.
Answer.
SA=∫02π∫02π1+cos2⁡(x)cos2⁡(y)+sin2⁡(x)sin2⁡(y)dxdy
14.5.9.
Answer.
SA=∫−11∫−111+4x2+4y2dxdy

14.6 Volume Between Surfaces and Triple Integration
14.6.4 Exercises

Problems

14.6.4.9.
Answer.
dzdydx: ∫03∫01−x/3∫02−2x/3−2ydzdydx
dzdxdy: ∫01∫03−3y∫02−2x/3−2ydzdxdy
dydzdx: ∫03∫02−2x/3∫01−x/3−z/2dydzdx
dydxdz: ∫02∫03−3z/2∫01−x/3−z/2dydxdz
dxdzdy: ∫01∫02−2y∫03−3y−3z/2dxdzdy
dxdydz: ∫02∫01−z/2∫03−3y−3z/2dxdydz
V=∫03∫01−x/3∫02−2x/3−2ydzdydx=1.
14.6.4.11.
Answer.
dzdydx: ∫02∫−20∫y2/2−ydzdydx
dzdxdy: ∫−20∫02∫y2/2−ydzdxdy
dydzdx: ∫02∫02∫−2z−zdydzdx
dydxdz: ∫02∫02∫−2z−zdydxdz
dxdzdy: ∫−20∫y2/2−y∫02dxdzdy
dxdydz: ∫02∫−2z−z∫02dxdydz V=∫02∫02∫−2z−zdydzdx=4/3.
14.6.4.13.
Answer.
dzdydx: ∫02∫1−x/21∫02x+4y−4dzdydx
dzdxdy: ∫01∫2−2y2∫02x+4y−4dzdxdy
dydzdx: ∫02∫02x∫z/4−x/2+11dydzdx
dydxdz: ∫04∫z/22∫z/4−x/2+11dydxdz
dxdzdy: ∫01∫04y∫z/2−2y+22dxdzdy
dxdydz: ∫04∫z/41∫z/2−2y+22dxdydz V=∫04∫z/41∫z/2−2y−22dxdydz=4/3.
14.6.4.15.
Answer.
dzdydx: ∫01∫01−x2∫01−ydzdydx
dzdxdy: ∫01∫01−y∫01−ydzdxdy
dydzdx: ∫01∫0x∫01−x2dydzdx+∫01∫x1∫01−z2dydzdx
dydxdz: ∫01∫0z∫01−z2dydxdz+∫01∫z1∫01−x2dydxdz
dxdzdy: ∫01∫01−y∫01−ydxdzdy
dxdydz: ∫01∫01−z2∫01−ydxdydz Answers will vary. Neither order is particularly “hard.” The order dzdydx requires integrating a square root, so powers can be messy; the order dydzdx requires two triple integrals, but each uses only polynomials.

14.7 Triple Integration with Cylindrical and Spherical Coordinates
14.7.3 Exercises

Problems

14.7.3.11.
Answer.
∫θ1θ2∫r1r2∫z1z2h(r,θ,z)rdzdrdθ
14.7.3.19.
Answer.
Describes the portion of the unit ball that resides in the first octant.

15 Vector Analysis
15.1 Introduction to Line Integrals
15.1.4 Exercises

Terms and Concepts

15.1.4.1.
Answer.
When C is a curve in the plane and f is a function defined over C, then ∫Cf(s)ds describes the area under the spatial curve that lies on f, over C.
15.1.4.3.
Answer.
The variable s denotes the arc-length parameter, which is generally difficult to use. The Key Idea allows one to parametrize a curve using another, ideally easier-to-use, parameter.

Problems

15.1.4.5.
Answer.
122
15.1.4.7.
Answer.
40π
15.1.4.9.
Answer.
Over the first subcurve of C, the line integral has a value of 3/2; over the second subcurve, the line integral has a value of 4/3. The total value of the line integral is thus 17/6.
15.1.4.11.
Answer.
∫01(5t2+2t+2)(4t+1)2+1dt≈17.071
15.1.4.13.
Answer.
∮02π(10−4cos2⁡t−sin2⁡t)cos2⁡t+4sin2⁡tdt≈74.986
15.1.4.15.
Answer.
726/3
15.1.4.17.
Answer.
8π3
15.1.4.19.
Answer.
M=82π2; center of mass is (0,−1/(2π),8π/3).

15.2 Vector Fields
15.2.3 Exercises

Terms and Concepts

15.2.3.1.
Answer.
Answers will vary. Appropriate answers include velocities of moving particles (air, water, etc.); gravitational or electromagnetic forces.
15.2.3.3.
Answer.
Specific answers will vary, though should relate to the idea that the vector field is spinning clockwise at that point.

Problems

15.2.3.9.
Answer.
divF→=1+2y
curlF→=0
15.2.3.11.
Answer.
divF→=xcos⁡(xy)−ysin⁡(xy)
curlF→=ycos⁡(xy)+xsin⁡(xy)
15.2.3.13.
Answer.
divF→=3
curlF→=⟨−1,−1,−1⟩
15.2.3.15.
Answer.
divF→=1+2y
curlF→=0
15.2.3.17.
Answer.
divF→=2y−sin⁡(z)
curlF→=0→

15.3 Line Integrals over Vector Fields
15.3.4 Exercises

Terms and Concepts

15.3.4.1.
Answer.
False. It is true for line integrals over scalar fields, though.
15.3.4.3.
Answer.
True.
15.3.4.5.
Answer.
We can conclude that F→ is conservative.

Problems

15.3.4.7.
Answer.
11/6. (One parametrization for C is r→(t)=⟨3t,t⟩ on 0≤t≤1.)
15.3.4.9.
Answer.
0. (One parametrization for C is r→(t)=⟨cos⁡(t),sin⁡(t)⟩ on 0≤t≤π.)
15.3.4.11.
Answer.
12. (One parametrization for C is r→(t)=⟨1,2,3⟩+t⟨3,1,−1⟩ on 0≤t≤1.)
15.3.4.13.
Answer.
5/6 joules. (One parametrization for C is r→(t)=⟨t,t⟩ on 0≤t≤1.)
15.3.4.15.
Answer.
24 ft-lbs.
15.3.4.17.
Answer.
  1. f(x,y)=xy+x
  2. curlF→=0.
  3. 1. (One parametrization for C is r→(t)=⟨t,t−1⟩ on 0≤t≤1.)
  4. 1 (with A=(0,1) and B=(1,0), f(B)−f(A)=1.)
15.3.4.19.
Answer.
  1. f(x,y)=x2yz
  2. curlF→=0→.
  3. 250.
  4. 250 (with A=(1,−1,0) and B=(5,5,2), f(B)−f(A)=250.)

15.4 Flow, Flux, Green’s Theorem and the Divergence Theorem
15.4.4 Exercises

Terms and Concepts

15.4.4.1.
Answer.
along, across
15.4.4.3.
Answer.
the curl of F→, or curlF→
15.4.4.5.
Answer.
curlF→

Problems

15.4.4.7.
Answer.
12
15.4.4.9.
Answer.
−2/3
15.4.4.11.
Answer.
1/2
15.4.4.13.
Answer.
The line integral ∮CF→⋅dr→, over the parabola, is 38/3; over the line, it is −10. The total line integral is thus 38/3−10=8/3. The double integral of curlF→=2 over R also has value 8/3.
15.4.4.15.
Answer.
Three line integrals need to be computed to compute ∮CF→⋅dr→. It does not matter which corner one starts from first, but be sure to proceed around the triangle in a counterclockwise fashion.
From (0,0) to (2,0), the line integral has a value of 0. From (2,0) to (1,1) the integral has a value of 7/3. From (1,1) to (0,0) the line integral has a value of −1/3. Total value is 2.
The double integral of curlF→ over R also has value 2.
15.4.4.17.
Answer.
Any choice of F→ is appropriate as long as curlF→=1. When F→=⟨−y/2,x/2⟩, the integrand of the line integral is simply 6. The area of R is 12π.
15.4.4.19.
Answer.
Any choice of F→ is appropriate as long as curlF→=1. The choices of F→=⟨−y,0⟩, ⟨0,x⟩ and ⟨−y/2,x/2⟩ each lead to reasonable integrands. The area of R is 16/15.
15.4.4.21.
Answer.
The line integral ∮CF→⋅n→ds, over the parabola, is −22/3; over the line, it is 10. The total line integral is thus −22/3+10=8/3. The double integral of divF→=2 over R also has value 8/3.
15.4.4.23.
Answer.
Three line integrals need to be computed to compute ∮CF→⋅n→ds. It does not matter which corner one starts from first, but be sure to proceed around the triangle in a counterclockwise fashion.
From (0,0) to (2,0), the line integral has a value of 0. From (2,0) to (1,1) the integral has a value of 1/3. From (1,1) to (0,0) the line integral has a value of 1/3. Total value is 2/3.
The double integral of divF→ over R also has value 2/3.

15.5 Parametrized Surfaces and Surface Area
15.5.3 Exercises

Terms and Concepts

15.5.3.1.
Answer.
Answers will vary, though generally should meaningfully include terms like “two sided”.

Problems

15.5.3.3.
Answer.
  1. r→(u,v)=⟨u,v,3u2v⟩ on −1≤u≤1, 0≤v≤2.
  2. r→(u,v)=⟨3vcos⁡(u)+1,3vsin⁡(u)+2,3(3vcos⁡(u)+1)2(3vsin⁡(u)+2)⟩, on 0≤u≤2π, 0≤v≤1.
  3. r→(u,v)=⟨u,v(2−2u),3u2v(2−2u)⟩ on 0≤u,v≤1.
  4. r→(u,v)=⟨u,v(1−u2),3u2v(1−u2)⟩ on −1≤u≤1, 0≤v≤1.
15.5.3.5.
Answer.
r→(u,v)=⟨0,u,v⟩ with 0≤u≤2, 0≤v≤1.
15.5.3.7.
Answer.
r→(u,v)=⟨3sin⁡(u)cos⁡(v),2sin⁡(u)sin⁡(v),4cos⁡(u)⟩ with 0≤u≤π, 0≤v≤2π.
15.5.3.9.
Answer.
Answers may vary.
For z=12(3−x): r→(u,v)=⟨u,v,12(3−u)⟩, with 1≤u≤3 and 0≤v≤2.
For x=1: r→(u,v)=⟨1,u,v⟩, with 0≤u≤2, 0≤v≤1
For y=0: r→(u,v)=⟨u,0,v/2(3−u)⟩, with 1≤u≤3, 0≤v≤1
For y=2: r→(u,v)=⟨u,2,v/2(3−u)⟩, with 1≤u≤3, 0≤v≤1
For z=0: r→(u,v)=⟨u,v,0⟩, with 1≤u≤3, 0≤v≤2
15.5.3.11.
Answer.
Answers may vary.
For z=2y:r→(u,v)=⟨u,v(4−u2),2v(4−u2)⟩ with −2≤u≤2 and 0≤v≤1.
For y=4−x2:r→(u,v)=⟨u,4−u2,2v(4−u2)⟩ with −2≤u≤2 and 0≤v≤1.
For z=0: r→(u,v)=⟨u,v(4−u2),0⟩ with −2≤u≤2 and 0≤v≤1.
15.5.3.13.
Answer.
Answers may vary.
For x2+y2/9=1: r→(u,v)=⟨cos⁡(u),3sin⁡(u),v⟩ with 0≤u≤2π and 1≤v≤3.
For z=1: r→(u,v)=⟨vcos⁡(u),3vsin⁡(u),1⟩ with 0≤u≤2π and 0≤v≤1.
For z=3: r→(u,v)=⟨vcos⁡(u),3vsin⁡(u),3⟩ with 0≤u≤2π and 0≤v≤1.
15.5.3.15.
Answer.
Answers may vary.
For z=1−x2: r→(u,v)=⟨u,v,1−u2⟩ with −1≤u≤1 and −1≤v≤2.
For y=−1: r→(u,v)=⟨u,−1,v(1−u2)⟩ with −1≤u≤1 and 0≤v≤1.
For y=2: r→(u,v)=⟨u,2,v(1−u2)⟩ with −1≤u≤1 and 0≤v≤1.
For z=0: r→(u,v)=⟨u,v,0⟩ with −1≤u≤1 and −1≤v≤2.
15.5.3.17.
Answer.
S=214.
15.5.3.19.
Answer.
S=43π.
15.5.3.21.
Answer.
S=∫03∫02πv2+4v4dudv=(3737−1)π/6≈117.319.
15.5.3.23.
Answer.
S=∫01∫−11(5u2−5)2+2(1−u2)2dudv=43≈6.9282.

15.6 Surface Integrals
15.6.3 Exercises

Terms and Concepts

15.6.3.1.
Answer.
curve; surface
15.6.3.3.
Answer.
outside

Problems

15.6.3.5.
Answer.
2403
15.6.3.7.
Answer.
24
15.6.3.9.
Answer.
0
15.6.3.11.
Answer.
−1/2
15.6.3.13.
Answer.
0; the flux over S1 is −45π and the flux over S2 is 45π.

15.7 The Divergence Theorem and Stokes’ Theorem
15.7.4 Exercises

Terms and Concepts

15.7.4.1.
Answer.
Answers will vary; in Section 15.4, the Divergence Theorem connects outward flux over a closed curve in the plane to the divergence of the vector field, whereas in this section the Divergence Theorem connects outward flux over a closed surface in space to the divergence of the vector field.
15.7.4.3.
Answer.
Curl.

Problems

15.7.4.5.
Answer.
Outward flux across the plane z=2−x/2−2y/3 is 14; across the plane z=0 the outward flux is −8; across the planes x=0 and y=0 the outward flux is 0.
Total outward flux: 14.
∬DdivF→dV=∫04∫03−3x/4∫02−x/2−2y/3(2x+2y)dzdydx=14.
15.7.4.7.
Answer.
Outward flux across the surface z=xy(3−x)(3−y) is 252; across the plane z=0 the outward flux is −9.
Total outward flux: 243.
∬DdivF→dV=∫03∫03∫0xy(3−x)(3−y)12dzdydx=243.
15.7.4.9.
Answer.
Circulation on C: ∮CF→⋅dr→=−π
∬S(curlF→)⋅n→dS=−π.
15.7.4.11.
Answer.
Circulation on C: The flow along the line from (0,0,2) to (4,0,0) is 0; from (4,0,0) to (0,3,0) it is −6, and from (0,3,0) to (0,0,2) it is 6. The total circulation is 0+(−6)+6=0.
∬S(curlF→)⋅n→dS=∬S0dS=0.
15.7.4.13.
Answer.
128/225
15.7.4.15.
Answer.
8192/105≈78.019
15.7.4.17.
Answer.
5/3
15.7.4.19.
Answer.
23π
15.7.4.21.
Answer.
Each field has a divergence of 1; by the Divergence Theorem, the total outward flux across S is ∬D1dS for each field.
15.7.4.23.
Answer.
Answers will vary. Often the closed surface S is composed of several smooth surfaces. To measure total outward flux, this may require evaluating multiple double integrals. Each double integral requires the parametrization of a surface and the computation of the cross product of partial derivatives. One triple integral may require less work, especially as the divergence of a vector field is generally easy to compute.
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