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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

4.1

The domain is the shaded circle defined by the inequality 9x2+9y236,9x2+9y236, which has a circle of radius 22 as its boundary. The range is [0,6].[0,6].

A circle of radius two with center at the origin. The equation x2 + y2 ≤ 4 is given.
4.2

The equation of the level curve can be written as (x3)2+(y+1)2=25,(x3)2+(y+1)2=25, which is a circle with radius 55 centered at (3,−1).(3,−1).

An circle of radius 5 with center (3, –1).
4.3

z=3(x1)2.z=3(x1)2. This function describes a parabola opening downward in the plane y=3.y=3.

4.4

domain(h)={(x,y,t)3|y4x24}domain(h)={(x,y,t)3|y4x24}

4.5

(x1)2+(y+2)2+(z3)2=16(x1)2+(y+2)2+(z3)2=16 describes a sphere of radius 44 centered at the point (1,−2,3).(1,−2,3).

4.6

lim(x,y)(5,−2)x2yy2+x13=32lim(x,y)(5,−2)x2yy2+x13=32

4.7

If y=k(x2)+1,y=k(x2)+1, then lim(x,y)(2,1)(x2)(y1)(x2)2+(y1)2=k1+k2.lim(x,y)(2,1)(x2)(y1)(x2)2+(y1)2=k1+k2. Since the answer depends on k,k, the limit fails to exist.

4.8

lim(x,y)(5,−2)29x2y2lim(x,y)(5,−2)29x2y2

4.9
  1. The domain of ff contains the ordered pair (2,−3)(2,−3) because f(a,b)=f(2,−3)=162(2)2(−3)2=3f(a,b)=f(2,−3)=162(2)2(−3)2=3
  2. lim(x,y)(a,b)f(x,y)=3lim(x,y)(a,b)f(x,y)=3
  3. lim(x,y)(a,b)f(x,y)=f(a,b)=3lim(x,y)(a,b)f(x,y)=f(a,b)=3
4.10

The polynomials g(x)=2x2g(x)=2x2 and h(y)=y3h(y)=y3 are continuous at every real number; therefore, by the product of continuous functions theorem, f(x,y)=2x2y3f(x,y)=2x2y3 is continuous at every point (x,y)(x,y) in the xy-plane.xy-plane. Furthermore, any constant function is continuous everywhere, so g(x,y)=3g(x,y)=3 is continuous at every point (x,y)(x,y) in the xy-plane.xy-plane. Therefore, f(x,y)=2x2y3+3f(x,y)=2x2y3+3 is continuous at every point (x,y)(x,y) in the xy-plane.xy-plane. Last, h(x)=x4h(x)=x4 is continuous at every real number x,x, so by the continuity of composite functions theorem g(x,y)=(2x2y3+3)4g(x,y)=(2x2y3+3)4 is continuous at every point (x,y)(x,y) in the xy-plane.xy-plane.

4.11

lim(x,y,z)(4,−1,3)13x22y2+z2=2lim(x,y,z)(4,−1,3)13x22y2+z2=2

4.12

fx=8x+2y+3,fy=2x2y2fx=8x+2y+3,fy=2x2y2

4.13

fx=(3x26xy2)sec2(x33x2y2+2y4)fy=(−6x2y+8y3)sec2(x33x2y2+2y4)fx=(3x26xy2)sec2(x33x2y2+2y4)fy=(−6x2y+8y3)sec2(x33x2y2+2y4)

4.14

Using the curves corresponding to c=−2andc=−3,c=−2andc=−3, we obtain

fy|(x,y)=(0,2)f(0,3)f(0,2)32=−3(−2)32·3+23+2=32−3.146.fy|(x,y)=(0,2)f(0,3)f(0,2)32=−3(−2)32·3+23+2=32−3.146.

The exact answer is

fy|(x,y)=(0,2)=(−2y|(x,y)=(0,2)=−22−2.828.fy|(x,y)=(0,2)=(−2y|(x,y)=(0,2)=−22−2.828.

4.15

fx=4x8xy+5z26,fy=−4x2+4y,fz=10xz+3fx=4x8xy+5z26,fy=−4x2+4y,fz=10xz+3

4.16

fx=2xysec(x2y)tan(x2y)3x2yz2sec2(x3yz2)fy=x2sec(x2y)tan(x2y)x3z2sec2(x3yz2)fz=−2x3yzsec2(x3yz2)fx=2xysec(x2y)tan(x2y)3x2yz2sec2(x3yz2)fy=x2sec(x2y)tan(x2y)x3z2sec2(x3yz2)fz=−2x3yzsec2(x3yz2)

4.17

2fx2=−9sin(3x2y)cos(x+4y)2fxy=6sin(3x2y)4cos(x+4y)2fyx=6sin(3x2y)4cos(x+4y)2fy2=−4sin(3x2y)16cos(x+4y)2fx2=−9sin(3x2y)cos(x+4y)2fxy=6sin(3x2y)4cos(x+4y)2fyx=6sin(3x2y)4cos(x+4y)2fy2=−4sin(3x2y)16cos(x+4y)

4.19

z=7x+8y3z=7x+8y3

4.20

L(x,y)=62x+3y,L(x,y)=62x+3y, so L(4.1,0.9)=62(4.1)+3(0.9)=0.5L(4.1,0.9)=62(4.1)+3(0.9)=0.5 f(4.1,0.9)=e52(4.1)+3(0.9)=e−0.50.6065.f(4.1,0.9)=e52(4.1)+3(0.9)=e−0.50.6065.

4.21

f(−1,2)=−19,fx(−1,2)=3,fy(−1,2)=−16,E(x,y)=−4(y2)2.f(−1,2)=−19,fx(−1,2)=3,fy(−1,2)=−16,E(x,y)=−4(y2)2.

lim(x,y)(x0,y0)E(x,y)(xx0)2+(yy0)2=lim(x,y)(−1,2)−4(y2)2(x+1)2+(y2)2lim(x,y)(−1,2)−4((x+1)2+(y2)2)(x+1)2+(y2)2=lim(x,y)(2,−3)4(x+1)2+(y2)2=0.lim(x,y)(x0,y0)E(x,y)(xx0)2+(yy0)2=lim(x,y)(−1,2)−4(y2)2(x+1)2+(y2)2lim(x,y)(−1,2)−4((x+1)2+(y2)2)(x+1)2+(y2)2=lim(x,y)(2,−3)4(x+1)2+(y2)2=0.

4.22

dz=0.18Δz=f(1.03,−1.02)f(1,−1)=0.180682dz=0.18Δz=f(1.03,−1.02)f(1,−1)=0.180682

4.23

dzdt=fxdxdt+fydydt=(2x3y)(6cos2t)+(−3x+4y)(−8sin2t)=−92sin2tcos2t72(cos22tsin22t)=−46sin4t72cos4t.dzdt=fxdxdt+fydydt=(2x3y)(6cos2t)+(−3x+4y)(−8sin2t)=−92sin2tcos2t72(cos22tsin22t)=−46sin4t72cos4t.

4.24

zu=0,zv=−21(3sin3v+cos3v)2zu=0,zv=−21(3sin3v+cos3v)2

4.25

wu=0wv=1533sin3v+6cos3v(3+2cos3vsin3v)2wu=0wv=1533sin3v+6cos3v(3+2cos3vsin3v)2

4.26

wt=wxxt+wyytwu=wxxu+wyyuwv=wxxv+wyyvwt=wxxt+wyytwu=wxxu+wyyuwv=wxxv+wyyv

A diagram that starts with w = f(x, y). Along the first branch, it is written ∂w/∂x, then x = x(t, u, v), at which point it breaks into another three subbranches: the first subbranch says t and then ∂w/∂x ∂x/∂t; the second subbranch says u and then ∂w/∂x ∂x/∂u; and the third subbranch says v and then ∂w/∂x ∂x/∂v. Along the second branch, it is written ∂w/∂y, then y = y(t, u, v), at which point it breaks into another three subbranches: the first subbranch says t and then ∂w/∂y ∂y/∂t; the second subbranch says u and then ∂w/∂y ∂y/∂u; and the third subbranch says v and then ∂w/∂y ∂y/∂v.
4.27

dydx=2x+y+72yx+3|(3,−2)=2(3)+(−2)+72(−2)(3)+3=114dydx=2x+y+72yx+3|(3,−2)=2(3)+(−2)+72(−2)(3)+3=114
Equation of the tangent line: y=114x+254y=114x+254

4.28

Duf(x,y)=(6xy4y34)(1)2+(3x212xy2+6y)32Duf(3,4)=7225642+(27576+24)32=−9452532Duf(x,y)=(6xy4y34)(1)2+(3x212xy2+6y)32Duf(3,4)=7225642+(27576+24)32=−9452532

4.29

f(x,y)=2x2+2xy+6y2(2x+y)2ix2+12xy+3y2(2x+y)2jf(x,y)=2x2+2xy+6y2(2x+y)2ix2+12xy+3y2(2x+y)2j

4.30

The gradient of gg at (−2,3)(−2,3) is g(−2,3)=i+14j.g(−2,3)=i+14j. The unit vector that points in the same direction as g(−2,3)g(−2,3) is g(−2,3)g(−2,3)=1197i+14197j=197197i+14197197j,g(−2,3)g(−2,3)=1197i+14197j=197197i+14197197j, which gives an angle of θ=arcsin((14197)/197)1.499rad.θ=arcsin((14197)/197)1.499rad. The maximum value of the directional derivative is g(−2,3)=197.g(−2,3)=197.

4.31

f(x,y)=(2x2y+3)i+(−2x+10y2)jf(x,y)=(2x2y+3)i+(−2x+10y2)j
f(1,1)=3i+6jf(1,1)=3i+6j
Tangent vector: 6i3j6i3j or −6i+3j−6i+3j

A rotated ellipse with equation f(x, y) = 8. At the point (1, 1) on the ellipse, there are drawn two arrows, one tangent vector and one normal vector. The normal vector is marked ∇f(1, 1) and is perpendicular to the tangent vector.
4.32

f(x,y,z)=2x2+2xy+6y28xz2z2(2x+y4z)2ix2+12xy+3y224yz+z2(2x+y4z)2j+4x212y24z2+4xz+2yz(2x+y4z)2k.f(x,y,z)=2x2+2xy+6y28xz2z2(2x+y4z)2ix2+12xy+3y224yz+z2(2x+y4z)2j+4x212y24z2+4xz+2yz(2x+y4z)2k.

4.33

Duf(x,y,z)=313(6x+y+2z)+1213(x4y+4z)413(2x+4y2z)Duf(0,−2,5)=38413Duf(x,y,z)=313(6x+y+2z)+1213(x4y+4z)413(2x+4y2z)Duf(0,−2,5)=38413

4.34

(2,−5)(2,−5)

4.35

(43,13)(43,13) is a saddle point, (32,38)(32,38) is a local maximum.

4.36

The absolute minimum occurs at (1,0):(1,0): f(1,0)=−1.f(1,0)=−1.

The absolute maximum occurs at (0,3):(0,3): f(0,3)=63.f(0,3)=63.

4.37

ff has a maximum value of 976976 at the point (8,2).(8,2).

4.38

A maximum production level of 1389013890 occurs with 56255625 labor hours and $5500$5500 of total capital input.

4.39

f(33,33,33)=33+33+33=3f(33,33,33)=333333=3.f(33,33,33)=33+33+33=3f(33,33,33)=333333=3.

4.40

f(2,1,2)=9f(2,1,2)=9 is a minimum.

Section 4.1 Exercises

1.

17,7217,72

3.

20π.20π. This is the volume when the radius is 22 and the height is 5.5.

5.

All points in the xy-planexy-plane

7.

x<y2x<y2

9.

All real ordered pairs in the xy-planexy-plane of the form (a,b)(a,b)

11.

{z|0z4}{z|0z4}

13.

The set

15.

y2x2=4,y2x2=4, a hyperbola

17.

4=x+y,4=x+y, a line; x+y=0,x+y=0, line through the origin

19.

2xy=0,2xy=−2,2xy=2;2xy=0,2xy=−2,2xy=2; three lines

21.

xx+y=−1,xx+y=0,xx+y=2xx+y=−1,xx+y=0,xx+y=2

23.

exy=12,exy=3exy=12,exy=3

25.

xyx=−2,xyx=0,xyx=2xyx=−2,xyx=0,xyx=2

27.

e−2x2=y,y=x2,y=e2x2e−2x2=y,y=x2,y=e2x2

29.

The level curves are parabolas of the form y=cx22.y=cx22.

31.

z=3+y3,z=3+y3, a curve in the zy-planezy-plane with rulings parallel to the x-axisx-axis

A planar version of the function y3 + 3 with results in the z axis and nothing mattering from the x axis.
33.

x225+y241x225+y241

35.

x29+y24+z236<1x29+y24+z236<1

37.

All points in xyz-spacexyz-space

39.


An upward facing, gently increasing paraboloid.
41.


A twisted plane with corners at (1, –1, –1), (–1, –1, –1), (–1, 1, 0.5), and (1, 1, 0.5).
43.


A downward facing, gently decreasing paraboloid.
45.


A hemisphere in the center with edges then swooping up at the four corners.
47.

The contour lines are circles.

49.

x2+y2+z2=9,x2+y2+z2=9, a sphere of radius 33

51.

x2+y2z2=4,x2+y2z2=4, a hyperboloid of one sheet

53.

4x2+y2=1,4x2+y2=1,

55.

1=exy(x2+y2)1=exy(x2+y2)

57.

T(x,y)=kx2+y2T(x,y)=kx2+y2

59.

x2+y2=k40,x2+y2=k40, x2+y2=k100.x2+y2=k100. The level curves represent circles of radii 10k/2010k/20 and k/10k/10

Section 4.2 Exercises

61.

2.0

63.

2323

65.

11

67.

1212

69.

1212

71.

e−32e−32

73.

11.011.0

75.

1.01.0

77.

The limit does not exist because when xx and yy both approach zero, the function approaches ln0,ln0, which is undefined (approaches negative infinity).

79.

every open disk centered at (x0,y0)(x0,y0) contains points inside RR and outside RR

81.

0.00.0

83.

0.000.00

85.

The limit does not exist.

87.

The limit does not exist. The function approaches two different values along different paths.

89.

The limit does not exist because the function approaches two different values along the paths.

91.

The function ff is continuous in the region y>x.y>x.

93.

The function ff is continuous at all points in the xy-planexy-plane except at (0,0).(0,0).

95.

The function is continuous at (0,0)(0,0) since the limit of the function at (0,0)(0,0) is 0,0, the same value of f(0,0).f(0,0).

97.

The function is discontinuous at (0,0).(0,0). The limit at (0,0)(0,0) fails to exist and g(0,0)g(0,0) does not exist.

99.

Since the function arctanxarctanx is continuous over (,),(,), g(x,y)=arctan(xy2x+y)g(x,y)=arctan(xy2x+y) is continuous where z=xy2x+yz=xy2x+y is continuous. The inner function zz is continuous on all points of the xy-planexy-plane except where y=x.y=x. Thus, g(x,y)=arctan(xy2x+y)g(x,y)=arctan(xy2x+y) is continuous on all points of the coordinate plane except at points at which y=x.y=x.

101.

All points P(x,y,z)P(x,y,z) in space

103.

The graph increases without bound as xandyxandy both approach zero.

On xyz space, there is a shape drawn that decreases to 0 as x and y increase or decrease but that increases greatly closer to the origin. It increases to such an extent that the graph is cut off above z = 10, which coincides with a circle of radius 0.6 around (0, 0, 10).
105.

a.

A series of five concentric circles, with radii 3, 2.75, 2.5, 2.2, and 1.75. The areas between the circles are colored, with the darkest color between the circles of radii 3 and 2.75.


b. The level curves are circles centered at (0,0)(0,0) with radius 9c.9c. c. x2+y2=9cx2+y2=9c d. z=3z=3 e. {(x,y)2|x2+y29}{(x,y)2|x2+y29} f. {z|0z3}{z|0z3}

107.

1.01.0

109.

f(g(x,y))f(g(x,y)) is continuous at all points (x,y)(x,y) that are not on the line 2x5y=0.2x5y=0.

111.

2.02.0

Section 4.3 Exercises

113.

zy=−3x+2yzy=−3x+2y

115.

The sign is negative.

117.

The partial derivative is zero at the origin.

119.

zy=−3sin(3x)sin(3y)zy=−3sin(3x)sin(3y)

121.

zx=6x5x6+y4;zy=4y3x6+y4zx=6x5x6+y4;zy=4y3x6+y4

123.

zx=yexy;zy=xexyzx=yexy;zy=xexy

125.

zx=2sec2(2xy),zy=sec2(2xy)zx=2sec2(2xy),zy=sec2(2xy)

127.

fx(2,−2)=14=fy(2,−2)fx(2,−2)=14=fy(2,−2)

129.

zx=cos(1)zx=cos(1)

131.

fx=0,fy=0,fz=0fx=0,fy=0,fz=0

133.

a. V(r,h)=πr2hV(r,h)=πr2h b. Vr=2πrhVr=2πrh c. Vh=πr2Vh=πr2

135.

fxy=1(xy)2fxy=1(xy)2

137.

2zx2=2,2zy2=42zx2=2,2zy2=4

139.

fxyy=fyxy=fyyx=0fxyy=fyxy=fyyx=0

141.

d2zdx2=12(eyey)sinxd2zdy2=12(eyey)sinxd2zdx2+d2zdy2=0d2zdx2=12(eyey)sinxd2zdy2=12(eyey)sinxd2zdx2+d2zdy2=0

143.

fxyz=6y2x18yz2fxyz=6y2x18yz2

145.

(14,12),(1,1)(14,12),(1,1)

147.

(0,0),(0,2),(3,−1),(3,−1)(0,0),(0,2),(3,−1),(3,−1)

149.

2zx2+2zy2=exsin(y)exsiny=02zx2+2zy2=exsin(y)exsiny=0

151.

c22zx2=etcos(xc)c22zx2=etcos(xc)

153.

fy=−2x+7fy=−2x+7

155.

fx=ycosxyfx=ycosxy

159.

Fθ=6,Fx=433Fθ=6,Fx=433

161.

δfδxδfδx at (500,1000)=172.36,(500,1000)=172.36, δfδyδfδy at (500,1000)=36.93(500,1000)=36.93

Section 4.4 Exercises

163.

(145145)(12ik)(145145)(12ik)

165.

Normal vector: i+j,i+j, tangent vector: ijij

167.

Normal vector: 7i17j,7i17j, tangent vector: 17i+7j17i+7j

169.

−1.094i0.18238j−1.094i0.18238j

171.

−36x6yz=−39−36x6yz=−39

173.

z=0z=0

175.

5x+4y+3z22=05x+4y+3z22=0

177.

4x5y+4z=04x5y+4z=0

179.

2x+2yz=02x+2yz=0

181.

−2(x1)+2(y2)(z1)=0−2(x1)+2(y2)(z1)=0

183.

x=20t+2,y=−4t+1,z=t+18x=20t+2,y=−4t+1,z=t+18

185.

x=0,y=0,z=tx=0,y=0,z=t

187.

x1=2t;y2=−2t;z1=tx1=2t;y2=−2t;z1=t

189.

The differential of the function z(x,y)=dz=fxdx+fydyz(x,y)=dz=fxdx+fydy

191.

Using the definition of differentiability, we have exyxx+y.exyxx+y.

193.

Δz=2xΔx+3Δy+(Δx)2.Δz=2xΔx+3Δy+(Δx)2. (Δx)20(Δx)20 for small ΔxΔx and zz satisfies the definition of differentiability.

195.

Δz1.185422Δz1.185422 and dz1.108.dz1.108. They are relatively close.

197.

1616 cm3

199.

Δz=Δz= exact change =0.6449,=0.6449, approximate change is dz=0.65.dz=0.65. The two values are close.

201.

13%or0.1313%or0.13

203.

0.0250.025

205.

0.3%0.3%

207.

2x+14y12x+14y1

209.

12x+y+14π1212x+y+14π12

211.

37x+27y+67z37x+27y+67z

213.

z=0z=0

A curved surface is shown with tangent plane at (0, 0, 0). The curved surface looks like the middle part of the bottom of a boat, and the tangent plane is z = 0.

Section 4.5 Exercises

215.

dwdt=ycosz+xcosz(2t)xysinz1t2dwdt=ycosz+xcosz(2t)xysinz1t2

217.

ws=−30x+4y,ws=−30x+4y, wt=10x16ywt=10x16y

219.

fr=rsin(2θ)fr=rsin(2θ)

221.

dfdt=2t+4t3dfdt=2t+4t3

223.

dfdt=−1dfdt=−1

225.

dfdt=1dfdt=1

227.

dwdt=2e2tdwdt=2e2t in both cases

229.

22t+2π=dudt22t+2π=dudt

231.

dydx=3x2+y22xydydx=3x2+y22xy

233.

dydx=yxx+2y3dydx=yxx+2y3

235.

dydx=yx3dydx=yx3

237.

dydx=yexyxexy+ey(1+y)dydx=yexyxexy+ey(1+y)

239.

dzdt=42t13dzdt=42t13

241.

dzdt=103t7/3×e1t10/3dzdt=103t7/3×e1t10/3

243.

zu=−2sinu3sinvzu=−2sinu3sinv and zv=−2cosucosv3sin2vzv=−2cosucosv3sin2v

245.

zr=3e3,zr=3e3, zθ=(243)e3zθ=(243)e3

247.

wt=cos(xyz)×yz×(−3)cos(xyz)xze1t+cos(xyz)xy×4wt=cos(xyz)×yz×(−3)cos(xyz)xze1t+cos(xyz)xy×4

249.

f(tx,ty)=t2x2+t2y2=t1f(x,y),f(tx,ty)=t2x2+t2y2=t1f(x,y), fy=x12(x2+y2)1/2×2x+y12(x2+y2)1/2×2y=1f(x,y)fy=x12(x2+y2)1/2×2x+y12(x2+y2)1/2×2y=1f(x,y)

251.

34π334π3

253.

dVdt=1066π3cm3/mindVdt=1066π3cm3/min

255.

dAdt=12in.2/mindAdt=12in.2/min

257.

2°C/sec2°C/sec

259.

ur=ux(xwwr+xttr)+uy(ywwr+yttr)+uz(zwwr+zttr)ur=ux(xwwr+xttr)+uy(ywwr+yttr)+uz(zwwr+zttr)

Section 4.6 Exercises

261.

−33−33

263.

−1−1

265.

2626

267.

33

269.

−1.0−1.0

271.

22252225

273.

2323

275.

2(x+y)2(x+2y)22(x+y)2(x+2y)2

277.

ex(y+3)2ex(y+3)2

279.

1+232(x+2y)1+232(x+2y)

281.

5,4,35,4,3

283.

−320−320

285.

311311

287.

3125531255

289.
The top of half of an ellipse centered at the origin with major axis horizontal and of length 4 and minor axis 2. The point (–2, 0) is marked, and there is an arrow pointing out from it to the left marked –4i.
291.

43i3j43i3j

293.

2i+2j+2k2i+2j+2k

295.

1.6(1019)1.6(1019)

297.

52995299

299.

2,1,22,1,2

301.

132,−3,−2132,−3,−2

303.

a. x+y+z=3,x+y+z=3, b. x1=y1=z1x1=y1=z1

305.

a. x+yz=1,x+yz=1, b. x1=y=zx1=y=z

307.

a. 323,323, b. 38,6,12,38,6,12, c. 24062406

309.

u,v=πcos(πx)sin(2πy),2πsin(πx)cos(2πy)u,v=πcos(πx)sin(2πy),2πsin(πx)cos(2πy)

Section 4.7 Exercises

311.

(23,4)(23,4)

313.

(0,0)(0,0) (115,115)(115,115)

315.

Maximum at (4,−1,8)(4,−1,8)

317.

Relative minimum at (0,0,1)(0,0,1)

319.

The second derivative test fails. Since x2y2>0x2y2>0 for all x and y different from zero, and x2y2=0x2y2=0 when either x or y equals zero (or both), then the absolute minimum occurs at (0,0).(0,0).

321.

f(−2,32)=−6f(−2,32)=−6 is a saddle point.

323.

f(0,0)=0;f(0,0)=0; (0,0,0)(0,0,0) is a saddle point.

325.

f(0,0)=9f(0,0)=9 is a local maximum.

327.

Relative minimum located at (2,6).(2,6).

329.

(1,−2)(1,−2) is a saddle point.

331.

(2,1)(2,1) and (−2,1)(−2,1) are saddle points; (0,0)(0,0) is a relative minimum.

333.

(−1,0)(−1,0) is a relative maximum.

335.

(0,0)(0,0) is a saddle point.

337.

The relative maximum is at (40,40).(40,40).

339.

(14,12)(14,12) is a saddle point and (1,1)(1,1) is the relative minimum.

341.

A saddle point is located at (0,0).(0,0).

A complicated graph that starts near (1, 1, 1), and decreases significantly along the x and y axes, so much so along the y axis that it is cut off. The rest of the graph stays near 0.
343.

There is a saddle point at (π,π),(π,π), local maxima at (π2,π2)and(3π2,3π2),(π2,π2)and(3π2,3π2), and local minima at (π2,3π2)and(3π2,π2).(π2,3π2)and(3π2,π2).

A series of hills and holes alternating through a space with amplitude 1.
345.

(0,1,0)(0,1,0) is the absolute minimum and (0,−2,9)(0,−2,9) is the absolute maximum.

347.

There is an absolute minimum at (0,1,−1)(0,1,−1) and an absolute maximum at (0,−1,1).(0,−1,1).

349.

(5,0,0),(5,0,0)(5,0,0),(5,0,0)

351.

18 by 36 by 18 in.18 by 36 by 18 in.

353.

(4724,4712,23524)(4724,4712,23524)

355.

x=3x=3 and y=6y=6

357.

V=64,000π20,372V=64,000π20,372 cm3

Section 4.8 Exercises

359.

maximum: 233,233, minimum: −233−233

361.

maximum: (22,0,2),(22,0,2), minimum: (22,0,2)(22,0,2)

363.

maximum: 32,32, minimum = 11

365.

maxima: f(322,22)=24,f(322,22)=24, f(322,−22)=24;f(322,−22)=24; minima: f(322,22)=−24,f(322,22)=−24, f(322,−22)=−24f(322,−22)=−24

367.

maximum: 211211 at f(211,611,−211);f(211,611,−211); minimum: −211−211 at f(−211,−611,211)f(−211,−611,211)

369.

2.02.0

371.

192192

373.

(123,−123)(123,−123)

375.

f(1,2)=5f(1,2)=5

377.

f(13,13,13)=13f(13,13,13)=13

379.

minimum: f(2,3,4)=29f(2,3,4)=29

381.

The maximum volume is 44 ft3. The dimensions are 1×2×21×2×2 ft.

383.

(1,12,−3)(1,12,−3)

385.

1.01.0

387.

33

389.

(25,195)(25,195)

391.

1212

An alternating series of hills and holes of amplitude 1 across xyz space.
393.

Roughly 3365 watches at the critical point (80,60)(80,60)

A series of curves in the first quadrant, with the first starting near (2, 120), decreasing sharply to near (20, 20), and then decreasing slowly to (120, 5). The next curve starts near (10, 120), decreases sharply to near (40, 40), and then decreases slowly to (120, 20). The next curve starts near (20, 120), decreases sharply to near (60, 60), and then decreases slowly to (120, 40). The next curve starts near (40, 120), decreases to near (80, 80), and then decreases a little slowly to (120, 60). The last curve starts near (60, 120) and decreases rather evenly through (100, 100) to (120, 90).

Chapter Review Exercises

395.

True, by Clairaut’s theorem

397.

False

399.

Answers may vary

401.

Does not exist

403.

Continuous at all points on the x,y-plane,x,y-plane, except where x2+y2>4.x2+y2>4.

405.

ux=4x33y,ux=4x33y, uy=−3x,uy=−3x, ut=2,ut=2, ut=3t2,ut=3t2, ut=8x36y9xt2ut=8x36y9xt2

407.

hxx(x,y,z)=6xe2yz,hxx(x,y,z)=6xe2yz, hxy(x,y,z)=6x2e2yz,hxy(x,y,z)=6x2e2yz, hxz(x,y,z)=3x2e2yz2,hxz(x,y,z)=3x2e2yz2, hyx(x,y,z)=6x2e2yz,hyx(x,y,z)=6x2e2yz, hyy(x,y,z)=4x3e2yz,hyy(x,y,z)=4x3e2yz, hyz(x,y,z)=2x3e2yz2,hyz(x,y,z)=2x3e2yz2, hzx(x,y,z)=3x2e2yz2,hzx(x,y,z)=3x2e2yz2, hzy(x,y,z)=2x3e2yz2,hzy(x,y,z)=2x3e2yz2, hzz(x,y,z)=2x3e2yz3hzz(x,y,z)=2x3e2yz3

409.

z=x2y+5z=x2y+5

411.

dz=4dxdy,dz=4dxdy, dz(0.1,0.01)=0.39,dz(0.1,0.01)=0.39, Δz=0.432Δz=0.432

413.

385,27,6385,27,6

415.

f(x,y)=x+2y22x2yi+(1x1xy2)jf(x,y)=x+2y22x2yi+(1x1xy2)j

417.

maximum: 1633,1633, minimum: 16331633

419.

2.32282.3228 cm3

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