zill calc catalog
TRANSCRIPT
Jones and Bartlett PublishersPhone: 1-800-832-0034 | Web: www.jbpub.com
Look to Dennis Zill’sAccessible Calculus Series!
Qualified Instructors: Request Your Complimentary Review Copy Today
Don’t Let YourStudents Struggle WithEarly Transcendentals...
4 Easy Ways to Order
1. Toll Free: 1-800-832-00342. Fax: 978-443-80003. Mail: 40 Tall Pine Drive, Sudbury, MA 017764. Web: www.jbpub.com
Textbook Examination Copies
Complimentary* review copies are available for qualified instructors who wish to consider a text for course adoption. For fastest service, make your request online at www.jbpub.com or, let our knowledgeable publisher’s representatives help you find the text that best meets your course needs.
*Jones and Bartlett Publishers reserves the right to evaluate requests for complimentary review copies.
Jones and Bartlett Publishers is a world-leading provider of instructional, assessment, and learning-performance management solutions for secondary, post-secondary, and professional markets. We endeavor to develop educational programs and services that improve learning outcomes, and enhance student achievement by uniquely combining authoritative content written by respected authors with innovative, proven, and engaging technology applications that meet the diverse needs of today’s instructors, students, and professionals.
Our learning solutions are used in the following content areas:
Science, Computing, Engineering & Mathematics – From human biology to complex analysis, Jones and Bartlett is transforming scientific and technical learning with tools designed to enrich the learning experience and improve course outcomes.
Career Education & Trades – With market-leading brands such as CDX Automotive—an innovative training solution for automotive service technicians—Jones & Bartlett is redefining how skill-based education is delivered for the career education market.
Health & Medicine – Working directly with many of the world’s leading health science authors, thought-leaders, and professional associations, Jones & Bartlett produces market-leading college textbooks, electronic reference materials, drug reference handbooks, and patient education materials for consumers of healthcare services.
EMS, Fire & Safety – From emergency medical services and fire training, to first aid and CPR, construction safety, and law enforcement, Jones & Bartlett is the leading provider of education and assessment resources for public safety professionals.
For more information or to review our online product catalog, visit us on the web at www.jbpub.com.
Sign Up to Receive Updates and Special Offers by E-mail
www.jbpub.com/eUpdates
Sign Up to Receive Updates and Special Offers: www.jbpub.com/eupdates
CalculusEarly TranscendentalsFourth Edition
Dennis G. Zill,Loyola Marymount UniversityWarren S. Wright,Loyola Marymount University
ISBN-13: 978-0-7637-5995-7 Hardcover • 994 Pages • © 2011
Appropriate for the traditional three-term college calculus course, Calculus: Early Transcendentals, Fourth Edition provides the student-friendly presentation and robust examples and problem sets for which Dennis Zill is known. This outstanding revision incorporates all of the exceptional learning tools that have made Zill’s many texts a resounding success. He carefully blends the theory and application of important concepts while offering modern applications and numerous problem-solving skills.
Table of Contents
Chapter 1: FunctionsChapter 2: Limit of a FunctionChapter 3: The DerivativeChapter 4: Applications of the DerivativeChapter 5: IntegralsChapter 6: Applications of the IntegralChapter 7: Techniques of IntegrationChapter 8: First-Order Differential EquationsChapter 9: Sequences and SeriesChapter 10: Conics and Polar CoordinatesChapter 11: Vectors and 3-SpaceChapter 12: Vector-Valued FunctionsChapter 13: Partial DerivativesChapter 14: Multiple IntegralsChapter 15: Vector Integral CalculusChapter 16: Higher-Order DifferentialEquations
Key Features
• The Test Yourself section is a self-test consisting of 56 questions on four broad areas of precalculus, and encourages students to review essential prerequisites.
• Each chapter opens with its own table of contents and an introduction to the material covered in that chapter.
• Provides a straightforward exposition at a level accessible to today’s college students.
• Includes examples and applications ideal for science and engineering students.
• Includes over 7300 problems varying in degree of difficulty.
• Concise reasoning behind every calculus concept is presented.
• Notes from the Classroom sections are informal discussions that are aimed at the student and discuss common algebraic, procedural, and notational errors.
Note: Non-bolded chapter titles indicate chapters found in Single Variable Calculus: Early Transcendentals. Bolded Chapter indicate chapters found if Multivariable Calculus. Please note that chapters 9and 10 appear in both volumes. For more information on the split volumes go to Page 6.
Available with WebAssign
Visit Page 6 for a complete list of Student and Instructor supplements.
3
40 Tall Pine Drive | Sudbury, MA | 01776 | 978-443-5000 | www.jbpub.com
See For YourselfSample Chapters are Available Online at www.jbpub.com!
Marginal Figures help students understand problems and concepts throughout the text.
Dennis Zill is known for his strongexercise sets and this edition is packed
with more than 7300 Problems!
Think About It exercises deal with conceptual aspects of the material
covered in that section and are suitable for assignment or for classroom discussion.
Fundamentals exercises allow students to solve problems based on key concepts from the section. Solutions to odd-numbered problems are included as an appendix.
Revolution about a Line The next example shows how to find the volume of a solid of rev-olution when a region is revolved about an axis that is not a coordinate axis.
EXAMPLE 6 Axis of Revolution not a Coordinate AxisFind the volume V of the solid that is formed by revolving the region given in Example 2about the line
Solution The domed-shaped solid of revolution is shown in FIGURE 6.3.13. From inspectionof the figure we see that a horizontal rectangular element of width that is perpendicularto the vertical line generates a solid disk when revolved about that axis. The radius rof that disk is
,
and so its volume is then
To express x in terms of y we use to obtain Therefore,
.
This leads to the integral
� p a16y �83
y3 �15
y5b d 20
�25615
p.
� p�2
0
(16 � 8y2 � y4) dy
V � p�2
0
(4 � y2)2 dy
Vk � p(4 � (y*k )2)2 ¢yk
x*k � (y*k )2.y � 1x
Vk � p(4 � x*k )2 ¢yk.
r � (right-most x-value) � (left-most x-value) � 4 � x*k
x � 4¢yk
x � 4.
338 CHAPTER 6 Applications of the Integral
Exercises 6.3 Answers to selected odd-numbered problems begin on page ANS-20.
Fundamentals
In Problems 1 and 2, use the slicing method to find the vol-ume of the solid if its cross sections perpendicular to a diam-eter of a circular base are as given. Assume that the radius ofthe base is 4.
1. 2.
3. The base of a solid is bounded by the curves andin the xy-plane. The cross sections perpendicular to
the x-axis are rectangles for which the height is four timesthe base. Find the volume of the solid.
x � 4x � y2
4. The base of a solid is bounded by the curve and the x-axis. The cross sections perpendicular to the x-axisare equilateral triangles. Find the volume of the solid.
5. The base of a solid is an isosceles triangle whose base is4 ft and height is 5 ft. The cross sections perpendicular tothe altitude are semicircles. Find the volume of the solid.
6. A hole of radius 1 ft is drilled through the middle of thesolid sphere of radius Find the volume of theremaining solid. See FIGURE 6.3.16.
r � 1
FIGURE 6.3.16 Hole through spherein Problem 6
r � 2 ft.
y � 4 � x2
4 �x*k
�yk
x � 4
x
y
2
xy
FIGURE 6.3.14 Cross sectionsare equilateral triangles
xy
FIGURE 6.3.15 Cross sections are semicircles
FIGURE 6.3.13 Solid of revolution inExample 6
59957_CH06a_321-378.qxd 11/6/09 5:00 PM Page 338
7. The base of a solid is a right isosceles triangle that isformed by the coordinate axes and the line Thecross sections perpendicular to the y-axis are squares. Findthe volume of the solid.
8. Suppose the pyramid shown in FIGURE 6.3.17 has height hand a square base of area B. Show that the volume of thepyramid is given by [Hint: Let b denote thelength of one side of the square base.]
In Problems 9–14, refer to FIGURE 6.3.18. Use the disk orwasher method to find the volume of the solid of revolutionthat is formed by revolving the given region about the indi-cated line.
9. R1 about OC 10. R1 about OA
11. R2 about OA 12. R2 about OC
13. R1 about AB 14. R2 about AB
In Problems 15–40, use the disk or washer method to find thevolume of the solid of revolution that is formed by revolvingthe region bounded by the graphs of the given equations aboutthe indicated line or axis.
15. x-axis
16. y-axis
17. y-axis
18. x-axis
19. x-axis
20. y-axis
21. x-axis
22. first quadrant; y-axisy � 1 � x2, y � x2 � 1, x � 0,
y � 4 � x2, y � 1 � 14 x2;
y � (x � 1)2, x � 0, y � 0;
y � (x � 2)2, x � 0, y � 0;
y �1x
, x �12
, x � 3, y � 0;
y �1x
, x � 1, y �12
;
y � x2 � 1, x � 0, y � 5;
y � 9 � x2, y � 0;
yC
R2
R1
y � x2
AO
B (1, 1)
x
FIGURE 6.3.18 Regions forProblems 9–14
x
y
FIGURE 6.3.17 Pyramid in Problem 8
A � 13hB.
x � y � 3.23. y-axis
24. x-axis
25.
26.
27.
28.
29. y-axis
30. x-axis
31. y-axis
32. y-axis
33. x-axis
34. x-axis
35.
36. x-axis
37. x-axis
38. x-axis
39. x-axis
40. first quadrant; x-axis
Think About It
41. Reread Problems 68–70 in Exercises 6.2 on Cavalieri’sPrinciple. Then show that the circular cylinders in FIGURE 6.3.19have the same volume.
42. Consider the right circular cylinder of radius a shown inFIGURE 6.3.20. A plane inclined at an angle to the base of thecylinder passes through a diameter of the base. Find thevolume of the resulting wedge cut from the cylinder when(a) (b) .
a
FIGURE 6.3.20 Cylinder and wedgein Problem 42
u � 60°u � 45°
u
hh
r r
FIGURE 6.3.19 Cylinders in Problem 41
y � sin x, y � cos x, x � 0,
y � tan x, y � 0, x � p>4;
y � sec x, x � �p>4, x � p>4, y � 0;
y � 0cos x 0 , y � 0, 0 � x � 2p;
y � ex, y � 1, x � 2;
y � e�x, x � 1, y � 1; y � 2
y � x3 � 1, x � 1, y � 0;
y � x3 � x, y � 0;
y � x3 � 1, x � 0, y � 9;
x � y2, y � x � 6;
y � x2 � 6x � 9, y � 9 � 12 x2;
x2 � y2 � 16, x � 5;
x � �y2 � 2y, x � 0; x � 2
y � x1>3, x � 0, y � 1; y � 2
x � y2, x � 1; x � 1
y � 1x � 1, x � 5, y � 0; x � 5
x � y � 2, x � 0, y � 0, y � 1;
y � x, y � x � 1, x � 0, y � 2;
6.3 Volumes of Solids: Slicing Method 339
59957_CH06a_321-378.qxd 11/6/09 5:01 PM Page 339
Revolution about a Line The next example shows how to find the volume of a solid of rev-olution when a region is revolved about an axis that is not a coordinate axis.
EXAMPLE 6 Axis of Revolution not a Coordinate AxisFind the volume V of the solid that is formed by revolving the region given in Example 2about the line
Solution The domed-shaped solid of revolution is shown in FIGURE 6.3.13. From inspectionof the figure we see that a horizontal rectangular element of width that is perpendicularto the vertical line generates a solid disk when revolved about that axis. The radius rof that disk is
,
and so its volume is then
To express x in terms of y we use to obtain Therefore,
.
This leads to the integral
� p a16y �83
y3 �15
y5b d 20
�25615
p.
� p�2
0
(16 � 8y2 � y4) dy
V � p�2
0
(4 � y2)2 dy
Vk � p(4 � (y*k )2)2 ¢yk
x*k � (y*k )2.y � 1x
Vk � p(4 � x*k )2 ¢yk.
r � (right-most x-value) � (left-most x-value) � 4 � x*k
x � 4¢yk
x � 4.
338 CHAPTER 6 Applications of the Integral
Exercises 6.3 Answers to selected odd-numbered problems begin on page ANS-20.
Fundamentals
In Problems 1 and 2, use the slicing method to find the vol-ume of the solid if its cross sections perpendicular to a diam-eter of a circular base are as given. Assume that the radius ofthe base is 4.
1. 2.
3. The base of a solid is bounded by the curves andin the xy-plane. The cross sections perpendicular to
the x-axis are rectangles for which the height is four timesthe base. Find the volume of the solid.
x � 4x � y2
4. The base of a solid is bounded by the curve and the x-axis. The cross sections perpendicular to the x-axisare equilateral triangles. Find the volume of the solid.
5. The base of a solid is an isosceles triangle whose base is4 ft and height is 5 ft. The cross sections perpendicular tothe altitude are semicircles. Find the volume of the solid.
6. A hole of radius 1 ft is drilled through the middle of thesolid sphere of radius Find the volume of theremaining solid. See FIGURE 6.3.16.
r � 1
FIGURE 6.3.16 Hole through spherein Problem 6
r � 2 ft.
y � 4 � x2
4 �x*k
�yk
x � 4
x
y
2
xy
FIGURE 6.3.14 Cross sectionsare equilateral triangles
xy
FIGURE 6.3.15 Cross sections are semicircles
FIGURE 6.3.13 Solid of revolution inExample 6
59957_CH06a_321-378.qxd 11/6/09 5:00 PM Page 338
7. The base of a solid is a right isosceles triangle that isformed by the coordinate axes and the line Thecross sections perpendicular to the y-axis are squares. Findthe volume of the solid.
8. Suppose the pyramid shown in FIGURE 6.3.17 has height hand a square base of area B. Show that the volume of thepyramid is given by [Hint: Let b denote thelength of one side of the square base.]
In Problems 9–14, refer to FIGURE 6.3.18. Use the disk orwasher method to find the volume of the solid of revolutionthat is formed by revolving the given region about the indi-cated line.
9. R1 about OC 10. R1 about OA
11. R2 about OA 12. R2 about OC
13. R1 about AB 14. R2 about AB
In Problems 15–40, use the disk or washer method to find thevolume of the solid of revolution that is formed by revolvingthe region bounded by the graphs of the given equations aboutthe indicated line or axis.
15. x-axis
16. y-axis
17. y-axis
18. x-axis
19. x-axis
20. y-axis
21. x-axis
22. first quadrant; y-axisy � 1 � x2, y � x2 � 1, x � 0,
y � 4 � x2, y � 1 � 14 x2;
y � (x � 1)2, x � 0, y � 0;
y � (x � 2)2, x � 0, y � 0;
y �1x
, x �12
, x � 3, y � 0;
y �1x
, x � 1, y �12
;
y � x2 � 1, x � 0, y � 5;
y � 9 � x2, y � 0;
yC
R2
R1
y � x2
AO
B (1, 1)
x
FIGURE 6.3.18 Regions forProblems 9–14
x
y
FIGURE 6.3.17 Pyramid in Problem 8
A � 13hB.
x � y � 3.23. y-axis
24. x-axis
25.
26.
27.
28.
29. y-axis
30. x-axis
31. y-axis
32. y-axis
33. x-axis
34. x-axis
35.
36. x-axis
37. x-axis
38. x-axis
39. x-axis
40. first quadrant; x-axis
Think About It
41. Reread Problems 68–70 in Exercises 6.2 on Cavalieri’sPrinciple. Then show that the circular cylinders in FIGURE 6.3.19have the same volume.
42. Consider the right circular cylinder of radius a shown inFIGURE 6.3.20. A plane inclined at an angle to the base of thecylinder passes through a diameter of the base. Find thevolume of the resulting wedge cut from the cylinder when(a) (b) .
a
FIGURE 6.3.20 Cylinder and wedgein Problem 42
u � 60°u � 45°
u
hh
r r
FIGURE 6.3.19 Cylinders in Problem 41
y � sin x, y � cos x, x � 0,
y � tan x, y � 0, x � p>4;
y � sec x, x � �p>4, x � p>4, y � 0;
y � 0cos x 0 , y � 0, 0 � x � 2p;
y � ex, y � 1, x � 2;
y � e�x, x � 1, y � 1; y � 2
y � x3 � 1, x � 1, y � 0;
y � x3 � x, y � 0;
y � x3 � 1, x � 0, y � 9;
x � y2, y � x � 6;
y � x2 � 6x � 9, y � 9 � 12 x2;
x2 � y2 � 16, x � 5;
x � �y2 � 2y, x � 0; x � 2
y � x1>3, x � 0, y � 1; y � 2
x � y2, x � 1; x � 1
y � 1x � 1, x � 5, y � 0; x � 5
x � y � 2, x � 0, y � 0, y � 1;
y � x, y � x � 1, x � 0, y � 2;
6.3 Volumes of Solids: Slicing Method 339
59957_CH06a_321-378.qxd 11/6/09 5:01 PM Page 339
4
13.7 Tangent Planes and Normal Lines 727
FIGURE 13.7.7 Tangent plane inExample 4
z
y
x
�F(1, �1, 5)
5
(1, �1, 0)
NOTES FROM THE CLASSROOM
Water flowing down a hill chooses a path in the direction of the greatest change in altitude.FIGURE 13.7.8 shows the contours, or level curves, of a hill. As shown in the figure, a streamstarting at point P will take a path that is perpendicular to the contours. After readingSections 13.7 and 13.8 you should be able to explain why.
�f
FIGURE 13.7.8 Stream flowing downhill
contours of a hill30
4060
stream
80100
P
Exercises 13.7 Answers to selected odd-numbered problems begin on page ANS-42.
Fundamentals
In Problems 1–12, sketch the level curve or surface passingthrough the indicated point. Sketch the gradient at the point.
1.
2.
3.
4.
5.
6. f (x, y) �y2
x; (2, 2)
f (x, y) �x2
4�
y2
9; (�2, �3)
f (x, y) � x2 � y2; (�1, 3)
f (x, y) � y � x2; (2, 5)
f (x, y) �y � 2x
x; (1, 3)
f (x, y) � x � 2y; (6, 1)
7.
8.
9.
10.
11.
12.
In Problems 13 and 14, find the points on the given surface atwhich the gradient is parallel to the indicated vector.
13.
14. x3 � y2 � z � 15; 27i � 8j � k
z � x2 � y2; 4i � j � 12 k
F(x, y, z) � x2 � y2 � z; (0, �1, 1)
F(x, y, z) � 2x2 � y2 � z2; (3, 4, 0)
f (x, y, z) � x2 � y2 � z; (1, 1, 3)
f (x, y, z) � y � z; (3, 1, 1)
f (x, y) �y � 1sin x
; Ap>6, 32Bf (x, y) � (x � 1)2 � y2; (1, 1)
EXAMPLE 4 Equation of a Tangent PlaneFind an equation of the tangent plane to the graph of the paraboloid at
Solution Define so that the level surface of F passingthrough the given point is or Now, and
so that
Hence, from (5) the desired equation is
See FIGURE 13.7.7.
Normal Line Let be a point on the graph of where is not 0.The line containing that is parallel to is called the normal line to thesurface at P. The normal line is perpendicular to the tangent plane to the surface at P.
§F(x0, y0, z0)P(x0, y0, z0)§FF(x, y, z) � cP(x0, y0, z0)
(x � 1) � (y � 1) � (z � 5) � 0 or �x � y � z � 3.
§F(x, y, z) � xi � yj � k and §F(1, �1, 5) � i � j � k.
Fz � �1Fy � y,Fx � x,F(x, y, z) � 0.F(x, y, z) � F(1, �1, 5)
F(x, y, z) � 12 x2 � 1
2 y2 � z � 4
(1, �1, 5).z � 1
2 x2 � 1
2 y2 � 4
EXAMPLE 5 Normal LineFind parametric equations for the normal line to the surface in Example 4 at
Solution A direction vector for the normal line at is It follows from (4) of Section 11.5 that parametric equations for the normal line are
Expressed as symmetric equations the normal line to a surface at is given by
In Example 5, you should verify that symmetric equations of the normal line at are
x � 1 �y � 1
�1�
z � 5�1
.
(1, �1, 5)
x � x0
Fx (x0, y0, z0)
�y � y0
Fy(x0, y0, z0)�
z � z0
Fz(x0, y0, z0).
P(x0, y0, z0)F(x, y, z) � c
z � 5 � t.y � �1 � t,x � 1 � t,
§F(1, �1, 5) � i � j � k.(1, �1, 5)
(1, �1, 5).
59957_CH13c_681-748.qxd 10/28/09 9:40 AM Page 727
Sign Up to Receive Updates and Special Offers: www.jbpub.com/eupdates
Marginal Annotations and guidance annotations provide students with tips or important asides.
Vivid 3-Dimentional Drawingsallow students to visualizeimportant calculus concepts.
Notes from the Classroom are informal discussions that range from warnings about common algebraic, procedural, and notational errors; to misinterpretations of theorems; to advice; to questions asking the student to think about and extend the ideas presented.
684 CHAPTER 13 Partial Derivatives
EXAMPLE 7 Level CurvesThe level curves of the polynomial function are the family of curves definedby As shown in FIGURE 13.1.7, when or a member of this family ofcurves is a hyperbola. For we obtain the lines
In most instances the task of graphing level curves of a function of two variablesis formidable. A CAS was used to generate the surfaces and corresponding level
curves in FIGURE 13.1.8 and FIGURE 13.1.9.z � f (x, y)
y
z � y2 � x2z
x(a)
c � 1 c � 1
c � �1
c � 0
(b)
x
y
FIGURE 13.1.7 Surface and level curves in Example 7
y � x and y � �x.c � 0,c 6 0,c 7 0y2 � x2 � c.
f (x, y) � y2 � x2
2
1
0
�1
�2 0
1
2
12
�2
�1�2�1
0
(a)
z
x
y
2
2
1
1
0
0
�1
�1�2
�2(b)
FIGURE 13.1.8 Graph of in (a); level curves in (b)f (x, y) � 2 sin xy
Thus, as shown in FIGURE 13.1.5, the curves of equipotential are concentric circles surrounding thecharge. Note that in Figure 13.1.5 we can get a feeling for the behavior of the function U, specif-ically where it is increasing (or decreasing), by observing the direction of increasing c.
Level Curves In general, if a function of two variables is given by then the curvesdefined by for suitable c, are called the level curves of f. The word level arises fromthe fact that we can interpret as the projection onto the xy-plane of the curve of inter-section, or trace, of and the (horizontal or level) plane See FIGURE 13.1.6.
(a)
y
xƒ(x, y) � c
surfacez � ƒ(x, y)
z
planez � c
(b)x
increasingvalues of ƒ
y
FIGURE 13.1.6 Surface in (a) and level curves in (b)
z � c.z � f (x, y)f (x, y) � c
f (x, y) � c,z � f (x, y),
FIGURE 13.1.5 Equipotential curves inExample 6
increasingpotential
x
y
c � 1
c � 12
59957_CH13a_681-748.qxd 11/6/09 6:17 PM Page 684
Poiseuille’s law states that the discharge rate, or rate of flow, of a viscous fluid (such asblood) through a tube (such as an artery) is
where k is a constant, R is the radius of the tube, L is its length, and are the pres-sures at the ends of the tube. This is an example of a function of four variables.
Note: Since it would take four dimensions, we cannot graph a function of three variables.
p1 and p2
Q � k
R4
L (p1 � p2),
686 CHAPTER 13 Partial Derivatives
EXAMPLE 8 Domain of a Function of Four VariablesThe domain of the rational function of three variables
is the set of points that satisfy In other words, the domain of f isall of 3-space except the points that lie on the surface of a sphere of radius 2 centered at theorigin.
Level Surfaces For a function of three variables, the surfaces defined bywhere c is a constant, are called level surfaces for the function f.f (x, y, z) � c,
w � f (x, y, z),
x2 � y2 � z2 � 4.(x, y, z)
f (x, y, z) �2x � 3y � z
4 � x2 � y2 � z2
An unfortunate, but standard, choiceof words, since level surfaces areusually not level.
EXAMPLE 9 Some Level Surfaces(a) The level surfaces of the polynomial are a family of paral-
lel planes defined by See FIGURE 13.1.12.(b) The level surfaces of the polynomial are a family of con-
centric spheres defined by See FIGURE 13.1.13.(c) The level surfaces of the rational function are given by
or A few members of this family of paraboloids aregiven in FIGURE 13.1.14.
x2 � y2 � cz.(x2 � y2)>z � cf (x, y, z) � (x2 � y2)>zx2 � y2 � z2 � c, c 7 0.
f (x, y, z) � x2 � y2 � z2x � 2y � 3z � c.
f (x, y, z) � x � 2y � 3z
FIGURE 13.1.12 Level surfacesin (a) of Example 9
FIGURE 13.1.13 Level surfacesin (b) of Example 9
FIGURE 13.1.14 Level surfacesin (c) of Example 9
z
x
yy
z
x
z
y
x
c � 1
c � 2
c � �2
c � �1
Exercises 13.1 Answers to selected odd-numbered problems begin on page ANS-40.
Fundamentals
In Problems 1–10, find the domain of the given function.
1. 2.
3. 4. f (x, y) � x2 � y214 � yf (x, y) �y2
y � x2
f (x, y) � (x2 � 9y2)�2f (x, y) �xy
x2 � y2
5. 6.
7. 8.
9.
10. f (x, y, z) �225 � x2 � y2
z � 5
H(u, y, w) � 2u2 � y2 � w2 � 16
g(u, f) �tan u � tan f
1 � tan u tan fg(r, s) � e2r2s2 � 1
f (u, y) �u
ln (u2 � y2)f (s, t) � s3 � 2t2 � 8st
59957_CH13a_681-748.qxd 11/6/09 6:17 PM Page 686
684 CHAPTER 13 Partial Derivatives
EXAMPLE 7 Level CurvesThe level curves of the polynomial function are the family of curves definedby As shown in FIGURE 13.1.7, when or a member of this family ofcurves is a hyperbola. For we obtain the lines
In most instances the task of graphing level curves of a function of two variablesis formidable. A CAS was used to generate the surfaces and corresponding level
curves in FIGURE 13.1.8 and FIGURE 13.1.9.z � f (x, y)
y
z � y2 � x2z
x(a)
c � 1 c � 1
c � �1
c � 0
(b)
x
y
FIGURE 13.1.7 Surface and level curves in Example 7
y � x and y � �x.c � 0,c 6 0,c 7 0y2 � x2 � c.
f (x, y) � y2 � x2
2
1
0
�1
�2 0
1
2
12
�2
�1�2�1
0
(a)
z
x
y
2
2
1
1
0
0
�1
�1�2
�2(b)
FIGURE 13.1.8 Graph of in (a); level curves in (b)f (x, y) � 2 sin xy
Thus, as shown in FIGURE 13.1.5, the curves of equipotential are concentric circles surrounding thecharge. Note that in Figure 13.1.5 we can get a feeling for the behavior of the function U, specif-ically where it is increasing (or decreasing), by observing the direction of increasing c.
Level Curves In general, if a function of two variables is given by then the curvesdefined by for suitable c, are called the level curves of f. The word level arises fromthe fact that we can interpret as the projection onto the xy-plane of the curve of inter-section, or trace, of and the (horizontal or level) plane See FIGURE 13.1.6.
(a)
y
xƒ(x, y) � c
surfacez � ƒ(x, y)
z
planez � c
(b)x
increasingvalues of ƒ
y
FIGURE 13.1.6 Surface in (a) and level curves in (b)
z � c.z � f (x, y)f (x, y) � c
f (x, y) � c,z � f (x, y),
FIGURE 13.1.5 Equipotential curves inExample 6
increasingpotential
x
y
c � 1
c � 12
59957_CH13a_681-748.qxd 11/6/09 6:17 PM Page 684
5
40 Tall Pine Drive | Sudbury, MA | 01776 | 978-443-5000 | www.jbpub.com
Single Variable CalculusEarly TranscendentalsFourth Edition
Dennis G. Zill, Loyola Marymount UniversityWarren S. Wright, Loyola Marymount University
ISBN-13: 978-0-7637-4965-1 Hardcover • 673 Pages • © 2011
Dennis Zill’s mathematics texts are renowned for their student-friendly presentation and robust examples and problem sets. The Fourth Edition of Single Variable Calculus: Early Transcendentals is no exception. This outstanding revision incorporates all of the exceptional learning tools that have made Zill’s texts a resounding success. Appropriate for the first two terms in the college calculus sequence, students are provided with a solid foundation in important mathematical concepts and problem solving skills, while maintaining the level of rigor expected of a Calculus course.
Multivariable CalculusFourth Edition
Dennis G. Zill, Loyola Marymount UniversityWarren S. Wright, Loyola Marymount University
ISBN-13: 978-0-7637-4966-8 Hardcover • 469 Pages • © 2011
Appropriate for the third semester in the college calculus sequence, the Fourth Edition of Multivarible Calculus maintains student-friendly writing style and robust exercises and problem sets that Dennis Zill is famous for. Ideal as a follow-up companion to Zill’s first volume, or as a stand-alone text, this exceptional revision presents the topics typically covered in the traditional third course, including Vector-valued Functions, Differential Calculus of Functions of Several Variables, Integral Calculus of Functions of Several Variables, Vector Integral Calculus, and an Introduction to Differential Equations.
Supplements:
Contact Your Publisher’s Representative to learn more about money saving bundling opportunities!
For Instructors:
Online Instructor’s ToolKit includes a computerized TestBank and PowerPoint Figure Slides featuring all labeled figures from the text.
A Complete Solutions Manual contains detailed solutions to every problem in the text.
WebAssign: Developed by instructors for instructors, WebAssign is the premier independent online teaching and learning environment.
For Students:
The Student Resource Manual is divided into four sections and includes: Essays, Topics in Precalculus, Use of a Calculator, and Selected Solutions. This valuable resource can significantly increase student success in their calculus course.
WebAssign Access Code Card.
6
Sign Up to Receive Updates and Special Offers: www.jbpub.com/eupdates
Advanced Engineering MathematicsFourth EditionDennis G. Zill, Loyola Marymount UniversityWarren S. Wright, Loyola Marymount University
ISBN-13: 978-0-7637-7966-5 Hardcover • 1008 pages • © 2011
Precalculus with Calculus PreviewsExpanded Volume Fourth EditionDennis G. Zill, Loyola Marymount UniversityJacqueline M. Dewar, Loyola Marymount University
ISBN-13: 978-0-7637-6631-3 Hardcover • 528 Pages • © 2009
A First Course in Complex Analysis with ApplicationsSecond EditionDennis G. Zill, Loyola Marymount UniversityPatrick D. Shanahan, Loyola Marymount University
ISBN-13: 978-0-7637-5772-4 Hardcover • 480 • © 2009
Also Available From Dennis Zill Contact Your Publishers Representative:
1-800-832-0034 | www.jbpub.com
7
Kirstie Mason1-800-832-0034 ext. [email protected], DC, DE, MA, MO, NH, NY, PA, TX, VT, WV, UT
Stephanie Leighs1-800-832-0034 ext. [email protected], AZ, CO, CT, GA, HI, ID, IL, KS, KY, LA, ND, NE, NJ, NM, NV, OK, SD, TN VA
Laura Pagluica1-800-832-0034 ext. [email protected], CA, FL, IA, IN, MS, MD, ME, MI, MN, MT, NC, OH, OR, RI, SC, WA, WI, WY
Linda McGarveySpecial Markets Manager1-800-832-0034 ext. [email protected] Contact for information on bulk sales or custom publishing opportunities
Robert RosenitschDirector of Sales & Marketing: [email protected]
Career Proprietary Schools:
Jenn Solomon1-800-832-0034 ext. 8118 [email protected], CO, CT, DC, DE, KS, MA, MD, ME, MO, NH, OK, PA, RI, TX, UT, VT, WV
Amy [email protected], AR, FL, GA, KY, LA, MS, NC, SC, TN, VA
Bridgette Hunt1-800-396-0531 [email protected], CA, HI, ID, MT, ND, NM, NV, OR, SD, WA, WY
Chris [email protected], IL, IN, MI, MN, NE, NJ, NY, OH, WI, & Canada
Contact your Publisher’sRepresentative regarding money-saving
bundling opportunities.
Available with WebAssign
Available with WebAssign
Jon
es a
nd
Bar
tlet
t P
ub
lish
ers
40 T
all P
ine
Dri
veSu
db
ury
, MA
017
76Ph
on
e: 1
-800
-832
-003
4Fa
x: 1
-978
-443
-800
0E-
mai
l: in
fo@
jbp
ub.
com
Web
: ww
w.jb
pu
b.co
m
PRSR
T ST
DU
.S. P
ost
age
PAID
Perm
it N
o. 6
Hu
dso
n, M
A
Sou
rce
Co
de:
Cal
c201
0
Cle
ar, M
od
ern
Tex
tsfo
r To
day
’sC
alcu
lus
Stu
den
ts!
Qu
alifi
ed In
stru
cto
rs: R
equ
est Y
ou
r Co
mp
limen
tary
Rev
iew
Co
py
Tod
ay