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  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Checkpoint

2.1


This figure is a graph of the coordinate system. There is a line segment beginning at the ordered pair (3, -1). Also, this point is labeled “S.” The line segment ends at the ordered pair (-2, 3) and is labeled “T.” There is an arrowhead at point “T,” representing a vector. The line segment is labeled “ST.”
2.2


This figure is a triangle formed by having vector 2w on one side and vector -v adjacent to 2w. The terminal point of 2w is the initial point of -v. The third side is labeled “2w – v.”
2.3

Vectors a,a, b,b, and ee are equivalent.

2.4

3,73,7

2.5

a. a=52,a=52, b. b=−4,−3,b=−4,−3, c. 3a4b=37,153a4b=37,15

2.7

v=−5,53v=−5,53

2.8

4585,10854585,1085

2.9

a=16i11j,a=16i11j, b=22i22jb=22i22j

2.10

Approximately 516516 mph

2.12

5252

2.13

z=−4z=−4

2.14

(x+2)2+(y4)2+(z+5)2=52(x+2)2+(y4)2+(z+5)2=52

2.15

x2+(y2)2+(z+2)2=14x2+(y2)2+(z+2)2=14

2.16

The set of points forms the two planes y=−2y=−2 and z=3.z=3.

This figure is the 3-dimensional coordinate system. It has two intersecting planes drawn. The first is the x z-plane. The second is parallel to the y z-plane at the value of z = 3. They are perpendicular to each other.
2.17

A cylinder of radius 4 centered on the line with x=0andz=2.x=0andz=2.

This figure is the 3-dimensional coordinate system. It has a cylinder parallel to the y-axis and centered around the y-axis.
2.18

ST=−1,−9,1=i9j+kST=−1,−9,1=i9j+k

2.19

1310,5310,83101310,5310,8310

2.20

v=162,122,202v=162,122,202

2.21

7

2.22

a. (r·p)q=12,−12,12;(r·p)q=12,−12,12; b. p2=53p2=53

2.23

θ0.22θ0.22 rad

2.24

x=5x=5

2.25

a. α1.04α1.04 rad; b. β2.58β2.58 rad; c. γ1.40γ1.40 rad

2.26

Sales = $15,685.50; profit = $14,073.15

2.27

v=p+q,v=p+q, where p=185i+95jp=185i+95j and q=75i145jq=75i145j

2.28

21 knots

2.29

150 ft-lb

2.30

i9j+2ki9j+2k

2.31

Up (the positive z-direction)

2.32

ii

2.33

kk

2.34

1616

2.35

4040

2.36

8i35j+2k8i35j+2k

2.37

−3194,−13194,4194−3194,−13194,4194

2.38

613613

2.39

1717

2.40

88 units3

2.41

No, the triple scalar product is −40,−40, so the three vectors form the adjacent edges of a parallelepiped. They are not coplanar.

2.42

2020 N

2.43

Possible set of parametric equations: x=1+4t,y=−3+t,z=2+6t;x=1+4t,y=−3+t,z=2+6t;

related set of symmetric equations: x14=y+3=z26x14=y+3=z26

2.44

x=−17t,y=3t,z=62t,0t1x=−17t,y=3t,z=62t,0t1

2.45

107107

2.46

These lines are skew because their direction vectors are not parallel and there is no point (x,y,z)(x,y,z) that lies on both lines.

2.47

−2(x1)+(y+1)+3(z1)=0−2(x1)+(y+1)+3(z1)=0 or −2x+y+3z=0−2x+y+3z=0

2.48

15211521

2.49

x=t,y=73t,z=42tx=t,y=73t,z=42t

2.50

1.441.44 rad

2.51

930930

2.53

The traces parallel to the xy-plane are ellipses and the traces parallel to the xz- and yz-planes are hyperbolas. Specifically, the trace in the xy-plane is ellipse x232+y222=1,x232+y222=1, the trace in the xz-plane is hyperbola x232z252=1,x232z252=1, and the trace in the yz-plane is hyperbola y222z252=1y222z252=1 (see the following figure).

This figure has four images. The first image is an ellipse centered at the origin of a rectangular coordinate system. It intersects the x axis at -3 and 3. It intersects the y axis at -2 and 2. The second image is the graph of a hyperbola. It is two curves one opening in the negative x direction and a symmetric one in the positive x direction. The third image is the graph of a hyperbola in the y z plane. It is opening in the negative y direction and a symmetric curve opening in the positive y direction. The fourth image is a 3-dimensional surface. It top and bottom cross sections would be circular. A vertical intersection would be a hyperbola.
2.54

Hyperboloid of one sheet, centered at (0,0,1)(0,0,1)

2.55

The rectangular coordinates of the point are (532,52,4).(532,52,4).

This figure is the 3-dimensional coordinate system. There is a point labeled “(5, pi/6, 4).” The point is located above a line segment in the x y-plane labeled r = 5 that is pi/6 degrees from the x-axis. The distance from the x y-plane to the point is labeled “z = 4.”
2.56

(82,3π4,−7)(82,3π4,−7)

2.57

This surface is a cylinder with radius 6.6.

This figure is a right circular cylinder. It is upright with the z-axis through the center. It is on top of the x y plane.
2.58


This figure is of the 3-dimensional coordinate system. It has a point. There is a line segment from the origin to the point. The angle between this line segment and the z-axis is phi. There is a line segment in the x y-plane from the origin to the shadow of the point.The angle between the x-axis and rho is theta.


Cartesian: (32,12,3),(32,12,3), cylindrical: (1,5π6,3)(1,5π6,3)

2.59

a. This is the set of all points 1313 units from the origin. This set forms a sphere with radius 13.13. b. This set of points forms a half plane. The angle between the half plane and the positive x-axis is θ=2π3.θ=2π3. c. Let PP be a point on this surface. The position vector of this point forms an angle of φ=π4φ=π4 with the positive z-axis, which means that points closer to the origin are closer to the axis. These points form a half-cone.

2.60

(4000,151°,124°)(4000,151°,124°)

2.61

Spherical coordinates with the origin located at the center of the earth, the z-axis aligned with the North Pole, and the x-axis aligned with the prime meridian

Section 2.1 Exercises

1.

a. PQ=2,2;PQ=2,2; b. PQ=2i+2jPQ=2i+2j

3.

a. QP=−2,−2;QP=−2,−2; b. QP=−2i2jQP=−2i2j

5.

a. PQ+PR=0,6;PQ+PR=0,6; b. PQ+PR=6jPQ+PR=6j

7.

a. 2PQ2PR=8,−4;2PQ2PR=8,−4; b. 2PQ2PR=8i4j2PQ2PR=8i4j

9.

a. 12,12;12,12; b. 12i+12j12i+12j

11.

35,4535,45

13.

Q(0,2)Q(0,2)

15.

a. a+b=3i+4j,a+b=3i+4j, a+b=3,4;a+b=3,4; b. ab=i2j,ab=i2j, ab=1,−2;ab=1,−2; c. Answers will vary; d. 2a=4i+2j,2a=4i+2j, 2a=4,2,2a=4,2, b=i3j,b=i3j, b=−1,−3,b=−1,−3, 2ab=3ij,2ab=3ij, 2ab=3,−12ab=3,−1

17.

1515

19.

λ=−3λ=−3

21.

a. a(0)=1,0,a(0)=1,0, a(π)=−1,0;a(π)=−1,0; b. Answers may vary; c. Answers may vary

23.

Answers may vary

25.

v=215,285v=215,285

27.

v=213434,353434v=213434,353434

29.

u=3,1u=3,1

31.

u=0,5u=0,5

33.

u=−53,5u=−53,5

35.

θ=7π4θ=7π4

37.

Answers may vary

39.

a. z0=f(x0)+f(x0);z0=f(x0)+f(x0); b. u=11+[f(x0)]21,f(x0)u=11+[f(x0)]21,f(x0)

43.

D(6,1)D(6,1)

45.

60.62,3560.62,35

47.

The horizontal and vertical components are 750750 ft/sec and 1299.041299.04 ft/sec, respectively.

49.

The magnitude of resultant force is 94.7194.71 lb; the direction angle is 13.42°.13.42°.

51.

The magnitude of the third vector is 60.0360.03 N; the direction angle is 259.38°.259.38°.

53.

The new ground speed of the airplane is 572.19572.19 mph; the new direction is N41.82E.N41.82E.

55.

T1=30.13lb,T1=30.13lb, T2=38.35lbT2=38.35lb

57.

v1=750v1=750 lb, v2=1299v2=1299 lb

59.

The two horizontal and vertical components of the force of tension are 2828 lb and 4242 lb, respectively.

Section 2.2 Exercises

61.

a. (2,0,5),(2,0,0),(2,3,0),(0,3,0),(0,3,5),(0,0,5);(2,0,5),(2,0,0),(2,3,0),(0,3,0),(0,3,5),(0,0,5); b. 3838

63.

A union of two planes: y=5y=5 (a plane parallel to the xz-plane) and z=6z=6 (a plane parallel to the xy-plane)

This figure is the first octant of the 3-dimensional coordinate system. It has two planes drawn. The first plane is parallel to the x y-plane and is at z = 6. The second plane is parallel to the x z-plane and is at y = 5. The planes are perpendicular.
65.

A cylinder of radius 11 centered on the line y=1,z=1y=1,z=1

This figure is the first octant of the 3-dimensional coordinate system. It has a cylinder drawn. The axis of the cylinder is parallel to the x-axis.
67.

z=1z=1

69.

z=−2z=−2

71.

(x+1)2+(y7)2+(z4)2=16(x+1)2+(y7)2+(z4)2=16

73.

(x+3)2+(y3.5)2+(z8)2=294(x+3)2+(y3.5)2+(z8)2=294

75.

Center C(0,0,2)C(0,0,2) and radius 11

77.

a. PQ=−4,−1,2;PQ=−4,−1,2; b. PQ=−4ij+2kPQ=−4ij+2k

79.

a. PQ=6,−24,24;PQ=6,−24,24; b. PQ=6i24j+24kPQ=6i24j+24k

81.

Q(5,2,8)Q(5,2,8)

83.

a+b=−6,4,−3,a+b=−6,4,−3, 4a=−4,−8,16,4a=−4,−8,16, −5a+3b=−10,28,−41−5a+3b=−10,28,−41

85.

a+b=−1,0,−1,a+b=−1,0,−1, 4a=0,0,−4,4a=0,0,−4, −5a+3b=−3,0,5−5a+3b=−3,0,5

87.

uv=38,uv=38, −2u=229−2u=229

89.

uv=2,uv=2, −2u=213−2u=213

91.

a=35i45ja=35i45j

93.

262i762j+362k262i762j+362k

95.

26,16,1626,16,16

97.

Equivalent vectors

99.

u=7059,1059,3059u=7059,1059,3059

101.

u=45sint,45cost,25u=45sint,45cost,25

103.

5154,15154,601545154,15154,60154

105.

α=7,α=7, β=15β=15

111.

a. F=30,40,0;F=30,40,0; b. 53°53°

113.

D=10kD=10k

115.

F4=−20,−7,−3F4=−20,−7,−3

117.

a. F=−19.6k,F=−19.6k, F=19.6F=19.6 N; b. T=19.6k,T=19.6k, T=19.6T=19.6 N

119.

a. F=−294kF=−294k N; b. F1=4933,49,−98,F1=4933,49,−98, F2=4933,−49,−98,F2=4933,−49,−98, and F3=9833,0,−98F3=9833,0,−98 (each component is expressed in newtons)

121.

a. v(1)=−0.84,0.54,2v(1)=−0.84,0.54,2 (each component is expressed in centimeters per second); v(1)=2.24v(1)=2.24 (expressed in centimeters per second); a(1)=−0.54,−0.84,0a(1)=−0.54,−0.84,0 (each component expressed in centimeters per second squared);

b.

This figure is of the 3-dimensional coordinate system above the xy-plane. It has a spiral drawn resembling a spring. The spiral is around the z-axis. The spiral starts on the x-axis at x = 1.

Section 2.3 Exercises

123.

6

125.

0

127.

(a·b)c=−11,−11,11;(a·b)c=−11,−11,11; (a·c)b=−20,−35,5(a·c)b=−20,−35,5

129.

(a·b)c=1,0,−2;(a·b)c=1,0,−2; (a·c)b=1,0,−1(a·c)b=1,0,−1

131.

a. θ=2.82θ=2.82 rad; b. θθ is not acute.

133.

a. θ=π4θ=π4 rad; b. θθ is acute.

135.

θ=π2θ=π2

137.

θ=π3θ=π3

139.

θ=2θ=2 rad

141.

Orthogonal

143.

Not orthogonal

145.

a=4α3,α,a=4α3,α, where α0α0 is a real number

147.

u=αi+αj+βk,u=αi+αj+βk, where αα and ββ are real numbers such that α2+β20α2+β20

149.

α=−6α=−6

151.

a. OP=4i+5j,OP=4i+5j, OQ=5i7j;OQ=5i7j; b. 105.8°105.8°

153.

68.33°68.33°

155.

u and v are orthogonal; v and w are orthogonal.

161.

a. cosα=23,cosβ=23,cosα=23,cosβ=23, and cosγ=13;cosγ=13; b. α=48°,α=48°, β=48°,β=48°, and γ=71°γ=71°

163.

a. cosα=130,cosβ=530,cosα=130,cosβ=530, and cosγ=230;cosγ=230; b. α=101°,α=101°, β=24°,β=24°, and γ=69°γ=69°

167.

a. w=8029,3229;w=8029,3229; b. compuv=1629compuv=1629

169.

a. w=2413,0,1613;w=2413,0,1613; b. compuv=813compuv=813

171.

a. w=2425,1825;w=2425,1825; b. q=5125,6825,q=5125,6825, v=w+q=2425,1825+5125,6825v=w+q=2425,1825+5125,6825

173.

a. 22;22; b. 109.47°109.47°

175.

17N·m17N·m

177.

1175 ft·lbft·lb

179.

4330.13 ft-lbft-lb

181.

a. F1+F2=52.9F1+F2=52.9 lb; b. The direction angles are α=74.5°,α=74.5°, β=36.7°,β=36.7°, and γ=57.7°.γ=57.7°.

Section 2.4 Exercises

183.

a. u×v=0,0,4;u×v=0,0,4;
b.

This figure is the first octant of the 3-dimensional coordinate system. On the x-axis there is a vector labeled “u.” In the x y-plane there is a vector labeled “v.” On the z-axis there is the vector labeled “u cross v.”
185.

a. u×v=6,−4,2;u×v=6,−4,2;
b.

This figure is the first octant of the 3-dimensional coordinate system and shows three vectors. The first vector is labeled u and has components <2, 3, 0>. The second vector is labeled v and has components <0, 1, 2>.” The third vector is labeled u cross v and has components <6, -4, 2>.”
187.

−2j4k−2j4k

189.

w=136i736j236kw=136i736j236k

191.

w=421i221j121kw=421i221j121k

193.

α=10α=10

197.

−3i+11j+2k−3i+11j+2k

199.

w=−1,et,etw=−1,et,et

201.

−26i+17j+9k−26i+17j+9k

203.

72°72°

209.

77

211.

a. 56;56; b. 562;562; c. 56595659

213.

a. 2;2; b. 22

215.

v·(u×w)=−1,v·(u×w)=−1, w·(u×v)=1w·(u×v)=1

217.

a=1,2,3,a=1,2,3, b=0,2,5,b=0,2,5, c=8,9,2;c=8,9,2; a·(b×c)=−9a·(b×c)=−9

219.

a. α=1;α=1; b. h=1,h=1,

This figure is the first octant of the 3-dimensional coordinate system. There is a parallelepided drawn. From the origin there are three vectors to vertices on the parallelepiped. They are vectors to the points A (2, 1, 0); B (1, 2, 0); and C (0, 1, alpha).
225.

Yes, AD=αAB+βAC,AD=αAB+βAC, where α=−1α=−1 and β=1.β=1.

227.

kk

229.

0,±45,250,±45,25

233.

w=w31,w3+1,w3,w=w31,w3+1,w3, where w3w3 is any real number

235.

8.66 ft-lb

237.

250 N

239.

F=4.8×10−15kNF=4.8×10−15kN

241.

a. B(t)=2sint5,2cost5,15;B(t)=2sint5,2cost5,15;
b.

This figure is the first octant of the 3-dimensional coordinate system. There is a curve sketched that is increasing. On the curve is a point labeled “P.” At P there is a tangent vector to the curve labeled “v(1).” Also from P there is a vector towards the inside of the curve labeled “a(1).” Finally, there is a vector from P labeled “B(1)” pointing towards the z-axis.

Section 2.5 Exercises

243.

a. r=−3,5,9+t7,−12,−7,r=−3,5,9+t7,−12,−7, t;t; b. x=−3+7t,y=512t,z=97t,x=−3+7t,y=512t,z=97t, t;t; c. x+37=y5−12=z9−7;x+37=y5−12=z9−7; d. x=−3+7t,y=512t,z=97t,x=−3+7t,y=512t,z=97t, t[0,1]t[0,1]

245.

a. r=−1,0,5+t5,0,−2,r=−1,0,5+t5,0,−2, t;t; b. x=−1+5t,y=0,z=52t,x=−1+5t,y=0,z=52t, t;t; c. x+15=z5−2,y=0;x+15=z5−2,y=0; d. x=−1+5t,y=0,z=52t,x=−1+5t,y=0,z=52t, t[0,1]t[0,1]

247.

a. x=1+t,y=−2+2t,z=3+3t,x=1+t,y=−2+2t,z=3+3t, t;t; b. x11=y+22=z33;x11=y+22=z33; c. (0,−4,0)(0,−4,0)

249.

a. x=3+t,y=1,z=5,x=3+t,y=1,z=5, t;t; b. y=1,z=5;y=1,z=5; c. The line does not intersect the xy-plane.

251.

a. P(1,3,5),P(1,3,5), v=1,1,4;v=1,1,4; b. 33

253.

223223

255.

a. Parallel; b. 2323

259.

(−12,6,−4)(−12,6,−4)

261.

The lines are skew.

263.

The lines are equal.

265.

a. x=1+t,y=1t,z=1+2t,x=1+t,y=1t,z=1+2t, t;t; b. For instance, the line passing through AA with direction vector j:x=1,z=1;j:x=1,z=1; c. For instance, the line passing through AA and point (2,0,0)(2,0,0) that belongs to LL is a line that intersects; L:x1−1=y1=z1L:x1−1=y1=z1

267.

a. 3x2y+4z=0;3x2y+4z=0; b. 3x2y+4z=03x2y+4z=0

269.

a. (x1)+2(y2)+3(z3)=0;(x1)+2(y2)+3(z3)=0; b. x+2y+3z14=0x+2y+3z14=0

271.

a. n=4i+5j+10k;n=4i+5j+10k; b. (5,0,0),(5,0,0), (0,4,0),(0,4,0), and (0,0,2);(0,0,2);
c.

This figure is the first octant of the 3-dimensional coordinate system. It has a triangle drawn with vertices on the x, y, and z axes.
273.

a. n=3i2j+4k;n=3i2j+4k; b. (0,0,0);(0,0,0);
c.

This figure is the 3-dimensional coordinate system represented in a box. It has a tilted parallelogram inside the box representing a plane.
275.

(3,0,0)(3,0,0)

277.

x=−2+2t,y=13t,z=3+t,x=−2+2t,y=13t,z=3+t, tt

281.

a. −2y+3z1=0;−2y+3z1=0; b. 0,−2,3·x1,y1,z1=0;0,−2,3·x1,y1,z1=0; c. x=0,y=−2t,z=3t,x=0,y=−2t,z=3t, tt

283.

a. Answers may vary; b. x11=z6−1,y=4x11=z6−1,y=4

285.

2x5y3z+15=02x5y3z+15=0

287.

The line intersects the plane at point P(−3,4,0).P(−3,4,0).

289.

16141614

291.

a. The planes are neither parallel nor orthogonal; b. 62°62°

293.

a. The planes are parallel.

295.

1616

297.

a. 1829;1829; b. P(5129,13029,6229)P(5129,13029,6229)

299.

4x3y=04x3y=0

301.

a. v(1)=cos1,sin1,2;v(1)=cos1,sin1,2; b. (cos1)(xsin1)(sin1)(ycos1)+2(z2)=0;(cos1)(xsin1)(sin1)(ycos1)+2(z2)=0;
c.

This figure is the first octant of the 3-dimensional coordinate system. It has a parallelogram grid drawn representing a plane. There is a curve from y = 1 increasing. The curve intersects the plane. At the point the curve intersects the plane, there is a vector labeled “v(1).” It is upward parallel to the z-axis.

Section 2.6 Exercises

303.

The surface is a cylinder with the rulings parallel to the y-axis.

This figure is a circular cylinder inside of a box. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
305.

The surface is a cylinder with rulings parallel to the y-axis.

This figure is a surface inside of a box. Its cross section parallel to the x z plane would be a cosine curve. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
307.

The surface is a cylinder with rulings parallel to the x-axis.

This figure is a surface inside of a box. Its cross section parallel to the y z plane would be an upside down parabola. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
309.

a. Cylinder; b. The x-axis

311.

a. Hyperboloid of two sheets; b. The x-axis

313.

b.

315.

d.

317.

a.

319.

x29+y214+z214=1,x29+y214+z214=1, hyperboloid of one sheet with the x-axis as its axis of symmetry

321.

x2103+y22z210=1,x2103+y22z210=1, hyperboloid of two sheets with the y-axis as its axis of symmetry

323.

y=z25+x25,y=z25+x25, hyperbolic paraboloid with the y-axis as its axis of symmetry

325.

x215+y23+z25=1,x215+y23+z25=1, ellipsoid

327.

x240+y28z25=0,x240+y28z25=0, elliptic cone with the z-axis as its axis of symmetry

329.

x=y22+z23,x=y22+z23, elliptic paraboloid with the x-axis as its axis of symmetry

331.

Parabola y=x24,y=x24,

This figure is the graph of an upside down parabola with its highest point at the origin of a rectangular coordinate system.
333.

Ellipse y24+z2100=1,y24+z2100=1,

This figure is the graph of an ellipse centered at the origin of a rectangular coordinate system.
335.

Ellipse y24+z2100=1,y24+z2100=1,

This figure is the graph of an ellipse centered at the origin of a rectangular coordinate system.
337.

a. Ellipsoid; b. The third equation; c. x2100+y2400+z2225=1x2100+y2400+z2225=1

339.

a. (x+3)216+(z2)28=1;(x+3)216+(z2)28=1; b. Cylinder centered at (−3,2)(−3,2) with rulings parallel to the y-axis

341.

a. (x3)24+(y2)2(z+2)2=1;(x3)24+(y2)2(z+2)2=1; b. Hyperboloid of one sheet centered at (3,2,−2),(3,2,−2), with the z-axis as its axis of symmetry

343.

a. (x+3)2+y24z23=0;(x+3)2+y24z23=0; b. Elliptic cone centered at (−3,0,0),(−3,0,0), with the z-axis as its axis of symmetry

345.

x24+y216+z2=1x24+y216+z2=1

347.

(1,−1,0)(1,−1,0) and (133,4,53)(133,4,53)

349.

x2+z2+4y=0,x2+z2+4y=0, elliptic paraboloid

351.

(0,0,100)(0,0,100)

355.

a. x=2z22,y=±z24z2,x=2z22,y=±z24z2, where z[−2,2];z[−2,2];
b.

This figure is a surface inside of a box. It is a sphere with a figure eight curve on the side of the sphere. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
357.


This figure is a surface inside of a box. It is a solid oval with an elliptical cylinder vertically intersecting. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.


two ellipses of equations x22+y292=1x22+y292=1 in planes z=±22z=±22

359.

a. x239632+y239632+z239502=1;x239632+y239632+z239502=1;
b.

This figure is a surface inside of a box. It is a sphere. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

;
c. The intersection curve is the ellipse of equation x239632+y239632=(2950)(4950)39502,x239632+y239632=(2950)(4950)39502, and the intersection is an ellipse.; d. The intersection curve is the ellipse of equation 2y239632+z239502=1.2y239632+z239502=1.

361.

a.

This figure is a surface inside of a box. It is a heart. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.


b. The intersection curve is (x2+z21)3x2z3=0.(x2+z21)3x2z3=0.

This figure is a curve on a rectangular coordinate system. It is the shape of a heart centered about the y-axis.

Section 2.7 Exercises

363.

(23,2,3)(23,2,3)

365.

(−23,−2,3)(−23,−2,3)

367.

(2,π3,2)(2,π3,2)

369.

(32,π4,7)(32,π4,7)

371.

A cylinder of equation x2+y2=16,x2+y2=16, with its center at the origin and rulings parallel to the z-axis,

This figure is a right circular cylinder, vertical. It is inside of a box. The edges of the box represent the x, y, and z axes.
373.

Hyperboloid of two sheets of equation x2+y2z2=1,x2+y2z2=1, with the y-axis as the axis of symmetry,

This figure is a elliptic cone surface that is horizontal. It is inside of a box. The edges of the box represent the x, y, and z axes.
375.

Cylinder of equation x22x+y2=0,x22x+y2=0, with a center at (1,0,0)(1,0,0) and radius 1,1, with rulings parallel to the z-axis,

This figure is a right circular cylinder, vertical. It is inside of a box. The edges of the box represent the x, y, and z axes.
377.

Plane of equation x=2,x=2,

This figure is a vertical parallelogram where x = 2 and parallel to the y z-plane. It is inside of a box. The edges of the box represent the x, y, and z axes.
379.

z=3z=3

381.

r2+z2=9r2+z2=9

383.

r=16cosθ,r=0r=16cosθ,r=0

385.

(0,0,−3)(0,0,−3)

387.

(6,−6,2)(6,−6,2)

389.

(4,0,90°)(4,0,90°)

391.

(3,90°,90°)(3,90°,90°)

393.

Sphere of equation x2+y2+z2=9x2+y2+z2=9 centered at the origin with radius 3,3,

This figure is a sphere. It is inside of a box. The edges of the box represent the x, y, and z axes.
395.

Sphere of equation x2+y2+(z1)2=1x2+y2+(z1)2=1 centered at (0,0,1)(0,0,1) with radius 1,1,

This figure is a sphere of radius 1 centered in a box. The center of the sphere is the point (0, 0, 1).
397.

The xy-plane of equation z=0,z=0,

This figure is a parallelogram representing a plane. It is parallel to the x y-plane at z = 0. It is inside of a box. The edges of the box represent the x, y, and z axes.
399.

φ=π3φ=π3 or φ=2π3;φ=2π3; Elliptic cone

401.

ρcosφ=6;ρcosφ=6; Plane at z=6z=6

403.

(10,π4,0.3218)(10,π4,0.3218)

405.

(32,π2,π4)(32,π2,π4)

407.

(2,π4,0)(2,π4,0)

409.

(8,π3,0)(8,π3,0)

411.

Cartesian system, {(x,y,z)|0xa,0ya,0za}{(x,y,z)|0xa,0ya,0za}

413.

Cylindrical system, {(r,θ,z)|r2+z29,r3cosθ,0θ2π}{(r,θ,z)|r2+z29,r3cosθ,0θ2π}

415.

The region is described by the set of points {(r,θ,z)|0r1,0θ2π,r2zr}.{(r,θ,z)|0r1,0θ2π,r2zr}.

This figure is a paraboloid, vertical. It is inside of a box. The edges of the box represent the x, y, and z axes.
417.

(4000,77°,51°)(4000,77°,51°)

419.

43.17°W,43.17°W, 22.91°S22.91°S

421.

a. ρ=0,ρ=0, ρ+R2r22Rsinφ=0;ρ+R2r22Rsinφ=0;
c.

This figure is a torus. It is inside of a box. The edges of the box represent the x, y, and z axes.

Chapter Review Exercises

423.

True

425.

False

427.

a. 24,−5;24,−5; b. 85;85; c. Can’t dot a vector with a scalar; d. −29−29

429.

a=±2a=±2

431.

114,214,314114,214,314

433.

2727

435.

x=13t,y=3+3t,z=58t,r(t)=(13t)i+3(1+t)j+(58t)kx=13t,y=3+3t,z=58t,r(t)=(13t)i+3(1+t)j+(58t)k

437.

x+3y+8z=43x+3y+8z=43

439.

x=kx=k trace: k2=y2+z2k2=y2+z2 is a circle, y=ky=k trace: x2z2=k2x2z2=k2 is a hyperbola (or a pair of lines if k=0),k=0), z=kz=k trace: x2y2=k2x2y2=k2 is a hyperbola (or a pair of lines if k=0).k=0). The surface is a cone.

This figure is an elliptical cone on its side. It is inside of a box. The edges of the box represent the x, y, and z axes.
441.

Cylindrical: z=r21,z=r21, spherical: cosφ=ρsin2φ1ρcosφ=ρsin2φ1ρ

443.

x22x+y2+z2=1,x22x+y2+z2=1, sphere

445.

331 N, and 244 N

447.

15J15J

449.

More, 59.0959.09 J

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