when actions conflict with established laws or legal norms. The challenge arises in balancing both
ethical and legal considerations when making decisions.### 2. **Ethical Issues in
Healthcare**Healthcare is one of the sectors where ethical and legal issues are particularly
pronounced. Medical professionals regularly face situations that require them to make complex
decisions that not only have life-altering consequences for patients but also raise profound moral
questions. These issues include:#### 2.1 **Patient Autonomy and Informed Consent**One of the
fundamental ethical principles in healthcare is respect for patient autonomy—the right of patients to
make decisions about their own bodies and medical
INSTRUCTOR’S
SOLUTIONS MANUAL
MULTIVARIABLE
WILLIAM ARDIS
Collin County Community College
THOMAS’ CALCULUS
TWELFTH EDITION
BASED ON THE ORIGINAL WORK BY
George B. Thomas, Jr.
Massachusetts Institute of Technology
AS REVISED BY
Maurice D. Weir
Naval Postgraduate School
Joel Hass
University of California, Davis
, ISBN-13: 978-0-321-60072-1
ISBN-10: 0-321-60072-X
1 2 3 4 5 6 BB 14 13 12 11 10
when actions conflict with established laws or legal norms. The challenge arises in balancing
both ethical and legal considerations when making decisions.### 2. **Ethical Issues in
Healthcare**Healthcare is one of the sectors where ethical and legal issues are particularly
pronounced. Medical professionals regularly face situations that require them to make
complex decisions that not only have life-altering consequences for patients but also raise
profound moral questions. These issues include:#### 2.1 **Patient Autonomy and Informed
Consent**One of the fundamental ethical principles in healthcare is respect for patient
autonomy—the right of patients to make decisions about their own bodies and medical
REFACE TO THE INSTRUCTOR
This Instructor's Solutions Manual contains the solutions to every exercise in the 12th Edition of THOMAS' CALCULUS
by Maurice Weir and Joel Hass, including the Computer Algebra System (CAS) exercises. The corresponding Student's
Solutions Manual omits the solutions to the even-numbered exercises as well as the solutions to the CAS exercises (because
the CAS command templates would give them all away).
In addition to including the solutions to all of the new exercises in this edition of Thomas, we have carefully revised or
rewritten every solution which appeared in previous solutions manuals to ensure that each solution
● conforms exactly to the methods, procedures and steps presented in the text
● is mathematically correct
● includes all of the steps necessary so a typical calculus student can follow the logical argument and algebra
● includes a graph or figure whenever called for by the exercise, or if needed to help with the explanation
● is formatted in an appropriate style to aid in its understanding
Every CAS exercise is solved in both the MAPLE and MATHEMATICA computer algebra systems. A template showing
an example of the CAS commands needed to execute the solution is provided for each exercise type. Similar exercises within
the text grouping require a change only in the input function or other numerical input parameters associated with the problem
(such as the interval endpoints or the number of iterations).
For more information about other resources available with Thomas' Calculus, visit http://pearsonhighered.com.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
, TABLE OF CONTENTS
10 Infinite Sequences and Series 569
10.1 Sequences 569
10.2 Infinite Series 577
10.3 The Integral Test 583
10.4 Comparison Tests 590
10.5 The Ratio and Root Tests 597
10.6 Alternating Series, Absolute and Conditional Convergence 602
10.7 Power Series 608
10.8 Taylor and Maclaurin Series 617
10.9 Convergence of Taylor Series 621
10.10 The Binomial Series and Applications of Taylor Series 627
Practice Exercises 634
Additional and Advanced Exercises 642
when actions conflict with established laws or legal norms. The challenge arises in balancing both ethical and
legal considerations when making decisions.### 2. **Ethical Issues in Healthcare**Healthcare is one of the
sectors where ethical and legal issues are particularly pronounced. Medical professionals regularly face
situations that require them to make complex decisions that not only have life-altering consequences for patients
but also raise profound moral questions. These issues include:#### 2.1 **Patient Autonomy and Informed
Consent**One of the fundamental ethical principles in healthcare is respect for patient autonomy—the right of
patients to make decisions about their own bodies and medical
11 Parametric Equations and Polar Coordinates 647
11.1 Parametrizations of Plane Curves 647
11.2 Calculus with Parametric Curves 654
11.3 Polar Coordinates 662
11.4 Graphing in Polar Coordinates 667
11.5 Areas and Lengths in Polar Coordinates 674
11.6 Conic Sections 679
11.7 Conics in Polar Coordinates 689
Practice Exercises 699
Additional and Advanced Exercises 709
12 Vectors and the Geometry of Space 715
12.1 Three-Dimensional Coordinate Systems 715
12.2 Vectors 718
12.3 The Dot Product 723
12.4 The Cross Product 728
12.5 Lines and Planes in Space 734
12.6 Cylinders and Quadric Surfaces 741
Practice Exercises 746
Additional Exercises 754
13 Vector-Valued Functions and Motion in Space 759
13.1 Curves in Space and Their Tangents 759
13.2 Integrals of Vector Functions; Projectile Motion 764
13.3 Arc Length in Space 770
13.4 Curvature and Normal Vectors of a Curve 773
13.5 Tangential and Normal Components of Acceleration 778
13.6 Velocity and Acceleration in Polar Coordinates 784
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
, Practice Exercises 785
Additional Exercises 791
14 Partial Derivatives 795
14.1 Functions of Several Variables 795
14.2 Limits and Continuity in Higher Dimensions 804
14.3 Partial Derivatives 810
14.4 The Chain Rule 816
14.5 Directional Derivatives and Gradient Vectors 824
14.6 Tangent Planes and Differentials 829
14.7 Extreme Values and Saddle Points 836
14.8 Lagrange Multipliers 849
14.9 Taylor's Formula for Two Variables 857
14.10 Partial Derivatives with Constrained Variables 859
Practice Exercises 862
Additional Exercises 876
when actions conflict with established laws or legal norms. The challenge arises in balancing both ethical and
legal considerations when making decisions.### 2. **Ethical Issues in Healthcare**Healthcare is one of the
sectors where ethical and legal issues are particularly pronounced. Medical professionals regularly face
situations that require them to make complex decisions that not only have life-altering consequences for patients
but also raise profound moral questions. These issues include:#### 2.1 **Patient Autonomy and Informed
Consent**One of the fundamental ethical principles in healthcare is respect for patient autonomy—the right of
patients to make decisions about their own bodies and medical
15 Multiple Integrals 881
15.1 Double and Iterated Integrals over Rectangles 881
15.2 Double Integrals over General Regions 882
15.3 Area by Double Integration 896
15.4 Double Integrals in Polar Form 900
15.5 Triple Integrals in Rectangular Coordinates 904
15.6 Moments and Centers of Mass 909
15.7 Triple Integrals in Cylindrical and Spherical Coordinates 914
15.8 Substitutions in Multiple Integrals 922
Practice Exercises 927
Additional Exercises 933
16 Integration in Vector Fields 939
16.1 Line Integrals 939
16.2 Vector Fields and Line Integrals; Work, Circulation, and Flux 944
16.3 Path Independence, Potential Functions, and Conservative Fields 952
16.4 Green's Theorem in the Plane 957
16.5 Surfaces and Area 963
16.6 Surface Integrals 972
16.7 Stokes's Theorem 980
16.8 The Divergence Theorem and a Unified Theory 984
Practice Exercises 989
Additional Exercises 997
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
ethical and legal considerations when making decisions.### 2. **Ethical Issues in
Healthcare**Healthcare is one of the sectors where ethical and legal issues are particularly
pronounced. Medical professionals regularly face situations that require them to make complex
decisions that not only have life-altering consequences for patients but also raise profound moral
questions. These issues include:#### 2.1 **Patient Autonomy and Informed Consent**One of the
fundamental ethical principles in healthcare is respect for patient autonomy—the right of patients to
make decisions about their own bodies and medical
INSTRUCTOR’S
SOLUTIONS MANUAL
MULTIVARIABLE
WILLIAM ARDIS
Collin County Community College
THOMAS’ CALCULUS
TWELFTH EDITION
BASED ON THE ORIGINAL WORK BY
George B. Thomas, Jr.
Massachusetts Institute of Technology
AS REVISED BY
Maurice D. Weir
Naval Postgraduate School
Joel Hass
University of California, Davis
, ISBN-13: 978-0-321-60072-1
ISBN-10: 0-321-60072-X
1 2 3 4 5 6 BB 14 13 12 11 10
when actions conflict with established laws or legal norms. The challenge arises in balancing
both ethical and legal considerations when making decisions.### 2. **Ethical Issues in
Healthcare**Healthcare is one of the sectors where ethical and legal issues are particularly
pronounced. Medical professionals regularly face situations that require them to make
complex decisions that not only have life-altering consequences for patients but also raise
profound moral questions. These issues include:#### 2.1 **Patient Autonomy and Informed
Consent**One of the fundamental ethical principles in healthcare is respect for patient
autonomy—the right of patients to make decisions about their own bodies and medical
REFACE TO THE INSTRUCTOR
This Instructor's Solutions Manual contains the solutions to every exercise in the 12th Edition of THOMAS' CALCULUS
by Maurice Weir and Joel Hass, including the Computer Algebra System (CAS) exercises. The corresponding Student's
Solutions Manual omits the solutions to the even-numbered exercises as well as the solutions to the CAS exercises (because
the CAS command templates would give them all away).
In addition to including the solutions to all of the new exercises in this edition of Thomas, we have carefully revised or
rewritten every solution which appeared in previous solutions manuals to ensure that each solution
● conforms exactly to the methods, procedures and steps presented in the text
● is mathematically correct
● includes all of the steps necessary so a typical calculus student can follow the logical argument and algebra
● includes a graph or figure whenever called for by the exercise, or if needed to help with the explanation
● is formatted in an appropriate style to aid in its understanding
Every CAS exercise is solved in both the MAPLE and MATHEMATICA computer algebra systems. A template showing
an example of the CAS commands needed to execute the solution is provided for each exercise type. Similar exercises within
the text grouping require a change only in the input function or other numerical input parameters associated with the problem
(such as the interval endpoints or the number of iterations).
For more information about other resources available with Thomas' Calculus, visit http://pearsonhighered.com.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
, TABLE OF CONTENTS
10 Infinite Sequences and Series 569
10.1 Sequences 569
10.2 Infinite Series 577
10.3 The Integral Test 583
10.4 Comparison Tests 590
10.5 The Ratio and Root Tests 597
10.6 Alternating Series, Absolute and Conditional Convergence 602
10.7 Power Series 608
10.8 Taylor and Maclaurin Series 617
10.9 Convergence of Taylor Series 621
10.10 The Binomial Series and Applications of Taylor Series 627
Practice Exercises 634
Additional and Advanced Exercises 642
when actions conflict with established laws or legal norms. The challenge arises in balancing both ethical and
legal considerations when making decisions.### 2. **Ethical Issues in Healthcare**Healthcare is one of the
sectors where ethical and legal issues are particularly pronounced. Medical professionals regularly face
situations that require them to make complex decisions that not only have life-altering consequences for patients
but also raise profound moral questions. These issues include:#### 2.1 **Patient Autonomy and Informed
Consent**One of the fundamental ethical principles in healthcare is respect for patient autonomy—the right of
patients to make decisions about their own bodies and medical
11 Parametric Equations and Polar Coordinates 647
11.1 Parametrizations of Plane Curves 647
11.2 Calculus with Parametric Curves 654
11.3 Polar Coordinates 662
11.4 Graphing in Polar Coordinates 667
11.5 Areas and Lengths in Polar Coordinates 674
11.6 Conic Sections 679
11.7 Conics in Polar Coordinates 689
Practice Exercises 699
Additional and Advanced Exercises 709
12 Vectors and the Geometry of Space 715
12.1 Three-Dimensional Coordinate Systems 715
12.2 Vectors 718
12.3 The Dot Product 723
12.4 The Cross Product 728
12.5 Lines and Planes in Space 734
12.6 Cylinders and Quadric Surfaces 741
Practice Exercises 746
Additional Exercises 754
13 Vector-Valued Functions and Motion in Space 759
13.1 Curves in Space and Their Tangents 759
13.2 Integrals of Vector Functions; Projectile Motion 764
13.3 Arc Length in Space 770
13.4 Curvature and Normal Vectors of a Curve 773
13.5 Tangential and Normal Components of Acceleration 778
13.6 Velocity and Acceleration in Polar Coordinates 784
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
, Practice Exercises 785
Additional Exercises 791
14 Partial Derivatives 795
14.1 Functions of Several Variables 795
14.2 Limits and Continuity in Higher Dimensions 804
14.3 Partial Derivatives 810
14.4 The Chain Rule 816
14.5 Directional Derivatives and Gradient Vectors 824
14.6 Tangent Planes and Differentials 829
14.7 Extreme Values and Saddle Points 836
14.8 Lagrange Multipliers 849
14.9 Taylor's Formula for Two Variables 857
14.10 Partial Derivatives with Constrained Variables 859
Practice Exercises 862
Additional Exercises 876
when actions conflict with established laws or legal norms. The challenge arises in balancing both ethical and
legal considerations when making decisions.### 2. **Ethical Issues in Healthcare**Healthcare is one of the
sectors where ethical and legal issues are particularly pronounced. Medical professionals regularly face
situations that require them to make complex decisions that not only have life-altering consequences for patients
but also raise profound moral questions. These issues include:#### 2.1 **Patient Autonomy and Informed
Consent**One of the fundamental ethical principles in healthcare is respect for patient autonomy—the right of
patients to make decisions about their own bodies and medical
15 Multiple Integrals 881
15.1 Double and Iterated Integrals over Rectangles 881
15.2 Double Integrals over General Regions 882
15.3 Area by Double Integration 896
15.4 Double Integrals in Polar Form 900
15.5 Triple Integrals in Rectangular Coordinates 904
15.6 Moments and Centers of Mass 909
15.7 Triple Integrals in Cylindrical and Spherical Coordinates 914
15.8 Substitutions in Multiple Integrals 922
Practice Exercises 927
Additional Exercises 933
16 Integration in Vector Fields 939
16.1 Line Integrals 939
16.2 Vector Fields and Line Integrals; Work, Circulation, and Flux 944
16.3 Path Independence, Potential Functions, and Conservative Fields 952
16.4 Green's Theorem in the Plane 957
16.5 Surfaces and Area 963
16.6 Surface Integrals 972
16.7 Stokes's Theorem 980
16.8 The Divergence Theorem and a Unified Theory 984
Practice Exercises 989
Additional Exercises 997
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
