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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

INSTRUCTOR’S

SOLUTIONS MANUAL

SINGLE VARIABLE

WILLIAM ARDIS

Collin County Community College

608070 _ISM_ThomasCalc_WeirHass_ttl.qxd:harsh_569709_ttl 9/3/09 3:11 PM Page 1

PREFACE 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 ì conforms exactly to the methods, procedures and steps presented in the text ì is mathematically correct ì includes all of the steps necessary so a typical calculus student can follow the logical argument and algebra ì includes a graph or figure whenever called for by the exercise, or if needed to help with the explanation ì 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.

TABLE OF CONTENTS

• 1 Functions
  • 1.1 Functions and Their Graphs
  • 1.2 Combining Functions; Shifting and Scaling Graphs
  • 1.3 Trigonometric Functions
  • 1.4 Graphing with Calculators and Computers
    • Practice Exercises
    • Additional and Advanced Exercises
• 2 Limits and Continuity
  • 2.1 Rates of Change and Tangents to Curves
  • 2.2 Limit of a Function and Limit Laws
  • 2.3 The Precise Definition of a Limit
  • 2.4 One-Sided Limits
  • 2.5 Continuity
  • 2.6 Limits Involving Infinity; Asymptotes of Graphs
    • Practice Exercises
    • Additional and Advanced Exercises
• 3 Differentiation
  • 3.1 Tangents and the Derivative at a Point
  • 3.2 The Derivative as a Function
  • 3.3 Differentiation Rules
  • 3.4 The Derivative as a Rate of Change
  • 3.5 Derivatives of Trigonometric Functions
  • 3.6 The Chain Rule
  • 3.7 Implicit Differentiation
  • 3.8 Related Rates
  • 3.9 Linearizations and Differentials
    • Practice Exercises
    • Additional and Advanced Exercises
• 4 Applications of Derivatives
  • 4.1 Extreme Values of Functions
  • 4.2 The Mean Value Theorem
  • 4.3 Monotonic Functions and the First Derivative Test
  • 4.4 Concavity and Curve Sketching
  • 4.5 Applied Optimization
  • 4.6 Newton's Method
  • 4.7 Antiderivatives
    • Practice Exercises
    • Additional and Advanced Exercises
• 5 Integration
  • 5.1 Area and Estimating with Finite Sums
  • 5.2 Sigma Notation and Limits of Finite Sums
  • 5.3 The Definite Integral
  • 5.4 The Fundamental Theorem of Calculus
  • 5.5 Indefinite Integrals and the Substitution Rule
  • 5.6 Substitution and Area Between Curves
    • Practice Exercises
    • Additional and Advanced Exercises
• 6 Applications of Definite Integrals
  • 6.1 Volumes Using Cross-Sections
  • 6.2 Volumes Using Cylindrical Shells
  • 6.3 Arc Lengths
  • 6.4 Areas of Surfaces of Revolution
  • 6.5 Work and Fluid Forces
  • 6.6 Moments and Centers of Mass
    • Practice Exercises
    • Additional and Advanced Exercises
• 7 Transcendental Functions
  • 7.1 Inverse Functions and Their Derivatives
  • 7.2 Natural Logarithms
  • 7.3 Exponential Functions
  • 7.4 Exponential Change and Separable Differential Equations
  • 7.5 Indeterminate Forms and L'Hopital's Rule^
  • 7.6 Inverse Trigonometric Functions
  • 7.7 Hyperbolic Functions
  • 7.8 Relative Rates of Growth
    • Practice Exercises
    • Additional and Advanced Exercises
• 8 Techniques of Integration
  • 8.1 Integration by Parts
  • 8.2 Trigonometric Integrals
  • 8.3 Trigonometric Substitutions
  • 8.4 Integration of Rational Functions by Partial Fractions
  • 8.5 Integral Tables and Computer Algebra Systems
  • 8.6 Numerical Integration
  • 8.7 Improper Integrals
    • Practice Exercises
    • Additional and Advanced Exercises
• 9 First-Order Differential Equations
  • 9.1 Solutions, Slope Fields and Euler's Method
  • 9.2 First-Order Linear Equations
  • 9.3 Applications
  • 9.4 Graphical Solutions of Autonomous Equations
  • 9.5 Systems of Equations and Phase Planes
    • Practice Exercises
    • Additional and Advanced Exercises
• 10 Infinite Sequences and Series
  • 10.1 Sequences
  • 10.2 Infinite Series
  • 10.3 The Integral Test
  • 10.4 Comparison Tests
  • 10.5 The Ratio and Root Tests
  • 10.6 Alternating Series, Absolute and Conditional Convergence
  • 10.7 Power Series
  • 10.8 Taylor and Maclaurin Series
  • 10.9 Convergence of Taylor Series
  • 10.10 The Binomial Series and Applications of Taylor Series
    • Practice Exercises
    • Additional and Advanced Exercises
• 10 Infinite Sequences and Series TABLE OF CONTENTS
  • 10.1 Sequences
  • 10.2 Infinite Series
  • 10.3 The Integral Test
  • 10.4 Comparison Tests
  • 10.5 The Ratio and Root Tests
  • 10.6 Alternating Series, Absolute and Conditional Convergence
  • 10.7 Power Series
  • 10.8 Taylor and Maclaurin Series
  • 10.9 Convergence of Taylor Series
  • 10.10 The Binomial Series and Applications of Taylor Series - Practice Exercises - Additional and Advanced Exercises
• 11 Parametric Equations and Polar Coordinates
  • 11.1 Parametrizations of Plane Curves
  • 11.2 Calculus with Parametric Curves
  • 11.3 Polar Coordinates
  • 11.4 Graphing in Polar Coordinates
  • 11.5 Areas and Lengths in Polar Coordinates
  • 11.6 Conic Sections
  • 11.7 Conics in Polar Coordinates - Practice Exercises - Additional and Advanced Exercises
• 12 Vectors and the Geometry of Space - 12.1 Three-Dimensional Coordinate Systems - 12.2 Vectors - 12.3 The Dot Product - 12.4 The Cross Product - 12.5 Lines and Planes in Space - 12.6 Cylinders and Quadric Surfaces - Practice Exercises - Additional Exercises
• 13 Vector-Valued Functions and Motion in Space - 13.1 Curves in Space and Their Tangents - 13.2 Integrals of Vector Functions; Projectile Motion - 13.3 Arc Length in Space - 13.4 Curvature and Normal Vectors of a Curve - 13.5 Tangential and Normal Components of Acceleration - 13.6 Velocity and Acceleration in Polar Coordinates - Practice Exercises - Additional Exercises
• 14 Partial Derivatives
  • 14.1 Functions of Several Variables
  • 14.2 Limits and Continuity in Higher Dimensions
  • 14.3 Partial Derivatives
  • 14.4 The Chain Rule
  • 14.5 Directional Derivatives and Gradient Vectors
  • 14.6 Tangent Planes and Differentials
  • 14.7 Extreme Values and Saddle Points
  • 14.8 Lagrange Multipliers
  • 14.9 Taylor's Formula for Two Variables
  • 14.10 Partial Derivatives with Constrained Variables
    • Practice Exercises
    • Additional Exercises
• 15 Multiple Integrals
  • 15.1 Double and Iterated Integrals over Rectangles
  • 15.2 Double Integrals over General Regions
  • 15.3 Area by Double Integration
  • 15.4 Double Integrals in Polar Form
  • 15.5 Triple Integrals in Rectangular Coordinates
  • 15.6 Moments and Centers of Mass
  • 15.7 Triple Integrals in Cylindrical and Spherical Coordinates
  • 15.8 Substitutions in Multiple Integrals
    • Practice Exercises
    • Additional Exercises
• 16 Integration in Vector Fields
  • 16.1 Line Integrals
  • 16.2 Vector Fields and Line Integrals; Work, Circulation, and Flux
  • 16.3 Path Independence, Potential Functions, and Conservative Fields
  • 16.4 Green's Theorem in the Plane
  • 16.5 Surfaces and Area
  • 16.6 Surface Integrals
  • 16.7 Stokes's Theorem
  • 16.8 The Divergence Theorem and a Unified Theory
    • Practice Exercises
    • Additional Exercises

2 Chapter 1 Functions

15. The domain is a
    _ß b. 16. The domain is a
    ß _b.
16. The domain is a
    _ß b. 18. The domain is Ð
    ß !Ó.
17. The domain is a
    _ß !b  a !ß b. 20. The domain is a
    ß !b  a !ß _b.
18. The domain is a
    _ß
    5 b  Ð
    ß
    Ó  Ò 5 3 3, 5 Ñ  a5, _ b22. The range is 2, 3 .Ò Ñ
19. Neither graph passes the vertical line test

(a) (b)

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Section 1.1 Functions and Their Graphs 3

24. Neither graph passes the vertical line test

(a) (b)

x y 1

x y y 1 x or or x y y x

k k

Ú Þ Ú Þ

Û ß Û ß Ü à Ü à

 œ Í Í

 œ " œ

 œ
" œ
"

25. x 0 1 2 26. x 0 1 2 y 0 1 0 y 1 0 0
26. F x 28. G x

4 x , x 1 x 2x, x 1

, x 0 x, 0 x a b œ (^) œ a bœœ

2 2 x

"

29. (a) Line through a !ß !b and a "ß "b : y œ x; Line through a "ß "b and a #ß !b : y œ
    x  2

f(x) x, 0 x 1 x 2, 1 x 2 œ

œ
  Ÿ

(b) f(x)

2, x x 2 x x

œ

!ß " Ÿ  # ß # Ÿ  $ !ß $ Ÿ Ÿ %

ÚÝ

Ý

Û

ÝÝ

Ü

30. (a) Line through a !ß 2 b and a #ß !b : y œ
    x  2

Line through a 2 ß "b and a &ß !b : m œ!
"&
#^ œ
"$ œ
"$^ , so y œ
"$^ ax
2 b " œ
"$^ x&$

f(x)

x , 0 x x , x œ

œ (^)
"$ (^)  &$ #  Ÿ &

(b) Line through a
"ß !b and a !ß
$b : m œ (^)!
Ð
"Ñ
$
! œ
$, so y œ
$x
$ Line through a !ß $b and a #ß
"b : m œ
"
$#
! œ
%#œ
#, so y œ
#x  $

f(x) x , x x , x œ

œ
#  $!  Ÿ #

Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

Section 1.1 Functions and Their Graphs 5

37. Symmetric about the origin 38. Symmetric about the y-axis

Dec:
_  x  _ Dec:
_  x! Inc: nowhere Inc:!  x _

39. Symmetric about the origin 40. Symmetric about the y-axis

Dec: nowhere Dec:!  x _ Inc:
_  x ! Inc:
_  x! !  x _

41. Symmetric about the y-axis 42. No symmetry

Dec:
_  x Ÿ! Dec:
_  xŸ! Inc:!  x  _ Inc: nowhere

Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

6 Chapter 1 Functions

43. Symmetric about the origin 44. No symmetry

Dec: nowhere Dec:! Ÿ x _ Inc:
_  x  _ Inc: nowhere

45. No symmetry 46. Symmetric about the y-axis

Dec:! Ÿ x  _ Dec:
_  xŸ! Inc: nowhere Inc:!  x _

47. Since a horizontal line not through the origin is symmetric with respect to the y-axis, but not with respect to the origin, the

function is even.

48. f xa b œ x
    &^ œ (^) x" &^ and fa
    x b œ a
    x b
    &œ (^) a
    "x^ b& œ
    ˆ (^) x"&‰œ
    f x. Thus the function is odd.a b
49. Since f xa b œ x #^  " œ a
    x b # " œ
    f x. The function is even.a b
50. Since f xÒ a b œ x #^  x Ó Á Òf a
    x b œ a
    x b #^
    x and f xÓ Ò a b œ x # x Ó Á Ò
    f x a b œ
    a bx #
    x the function is neither even norÓ

odd.

51. Since g xa b œ x $^  x, g a
    x b œ
    x $^
    x œ
    a x $ x b œ
    g x. So the function is odd.a b
52. g xa b œ x %^  $x #^
    " œ a
    x b %^  $
    a x b #
    " œ g a
    x bßthus the function is even.
53. g xa b œ (^) x # "
    "^ œ (^) a
    x"b#
    "œ g a
    x. Thus the function is even.b
54. g xa b œ (^) x # x
    "^ ; g a
    x b œ
    (^) x#x
    "œ
    g x. So the function is odd.a b
55. h ta b œ (^) t
    ""^ ; h a
    t b œ (^)
    
    "t "^ ;
    h t a b œ (^) "
    "t. Since h ta b Á
    h t a b and h ta b Á h a
    t , the function is neither even nor odd.b
56. Since l t |$^ œ l
    a t b$ |, h ta b œ h a
    t band the function is even.

Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

8 Chapter 1 Functions

68. (a) From the graph, (^) x
    ^3 1  (^) x ^21 Ê x − (
    _ß
    5) 
    ß( 1 1)

(b) Case x 
1: (^) x
^3 1  (^) x ^21 Ê 3(xx
^  1 1) 2 Ê 3x  3  2x
2 Ê x 
5. Thus, x − (
ß
5) solves the inequality. Case
1  x  1: (^) x
^3 1  (^) x ^21 Ê 3(xx
^  1 1) 2 Ê 3x  3  2x
2 Ê x 
5 which is true if x 
1. Thus, x − (
ß1 1) solves the inequality. Case 1  x: (^) x
^3 1  (^) x ^21 Ê 3x  3  2x
2 Ê x 
5 which is never true if 1 x, so no solution here. In conclusion, x − (
ß
5) 
ß( 1 1).

69. A curve symmetric about the x-axis will not pass the vertical line test because the points ax, y b and ax,
    y blie on the same vertical line. The graph of the function y œ f xa b œ! is the x-axis, a horizontal line for which there is a single y-value, !, for any x.
70. price œ 40  5x, quantity œ 300
    25x Ê R xa b œ a 40  5x ba 300
    25xb
71. x 2  x 2 œ h 2 Ê x œ (^) Èh 2 œ 2 2 h ; cost œ 5 2x  10h Ê C h œ 10 2 2 h  10h œ 5h 2  2 È È a b a b Š ‹ Š È ‹
72. (a) Note that 2 mi = 10,560 ft, so there are È 800 #^  x #feet of river cable at $180 per foot and a10,560
    x bfeet of land

cable at $100 per foot. The cost is C xa b œ 180 È 800 #^  x # 100 10,560a
x .b (b) C a b! œ $"ß #!!ß !!! C a &!!b ¸ $"ß "(&ß )"# C a "!!!b ¸ $"ß ")'ß &"# C a "&!!b ¸ $"ß #"#ß !!! C a #!!!b ¸ $"ß #%$ß ($# C a #&!!b ¸ $"ß #()ß %(* C a $!!!b ¸ $"ß $"%ß )(! Values beyond this are all larger. It would appear that the least expensive location is less than 2000 feet from the point P.

1.2 COMBINING FUNCTIONS; SHIFTING AND SCALING GRAPHS

1. D :f
   _  x  , D : x (^) g 1 Ê D (^) f g œ D : xfg 1. R :f
    y  _, R : yg 0, R (^) f g: y 1, R : yfg 0
2. D : xf  1 0 Ê x
   1, D : x (^) g
   1 0 Ê x 1. Therefore D (^) f g œD : xfg 1. R (^) f œ R : yg 0, R (^) f g : y È2, R : yfg 0
3. D :f
   _  x  , D : (^) g
    x  , D (^) f gÎ :
    x  , D (^) g fÎ :
    x  _, R : y (^) f œ2, R :g y 1, R (^) f gÎ : 0  y Ÿ 2, Rg f (^) Î : "#Ÿ y _
4. D :f
   _  x  _, D : x (^) g 0 , D (^) f gÎ : x 0, D (^) g fÎ : x 0; R : yf œ 1, R :g y 1, R (^) f gÎ : 0  y Ÿ 1, R (^) g fÎ: 1 Ÿ y _
5. (a) 2 (b) 22 (c) x #^  2 (d) (x  5) #^
   3 œ x # 10x  22 (e) 5 (f)
   2 (g) x  10 (h) (x #^
   3) #^
   3 œ x %^
   6x # 6

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Section 1.2 Combining Functions; Shifting and Scaling Graphs 9

6. (a)
   " 3 (b) 2 (c) (^) x "^1
   1 œx
   x 1 (d) " x (e) 0 (f) 43 (g) x
   2 (h) " "^  œ " #^ œ  " # x 1 x 1 1 x xx
7. af g h ‰ ‰ ba bx œ f g h xa a a b bb œ f g 4a a
   x bb œ f 3 4a a
   x bb œ f 12a
   3x b œ a 12
   3xb  1 œ 13
   3x
8. af g h ‰ ‰ ba bx œ f g h xa a a b bb œ f g xa a 2 bb œ f 2 xa a 2 b
   1 b œ f 2xa 2
   1 b œ 3 2xa 2
   1 b 4 œ 6x 2  1
9. af g h ‰ ‰ ba bx œ f g h xa a a b b b œ f gˆ^ ˆ ‰^1 x^ ‰^ œ f Š 11 ‹ œ fˆ^1 x^ 4x ‰œ (^) É 1 x^ 4x  " œÉ5x 1 4x x  %^ ^ ^ 

 "

10. af g h ‰ ‰ ba bx œ f g h xa a a b b bœ f gŠ Š È 2
    x ‹‹œ f (^)   œ fˆ^ ‰œ œ

Š È ‹ Š È ‹

2 x 2 x 1

2 x 8 3x x 7 2x

2 3



$



2 2 2 x

2 x x x

$ $

11. (a) af g ‰ ba bx (b) a j g‰ ba bx (c) ag g ‰ ba bx

(d) a j j‰ ba bx (e) ag h f ‰ ‰ ba bx (f) ah j f ‰ ‰ ba bx

12. (a) af j ‰ ba bx (b) ag h ‰ ba bx (c) ah h ‰ ba bx

(d) af f ‰ ba bx (e) a j g f‰ ‰ ba bx (f) ag f h ‰ ‰ ba bx

13. g(x) f(x) (f ‰g)(x)

(a) x
7 Èx^ Èx
7 (b) x  2 3x 3(x  2) œ 3x  6 (c) x #^ Èx^
5 Èx #
5

(d) (^) x
x^1 x
x^1
1 x
(xx
1) x x x 1 x x 1 ^ œ^ œ (e) (^) x
"^1 1 "x x (f) " x^ "x x

14. (a) af g ‰ ba bx œ lg x a b l œ (^) l x
    "l".

(b) af g ‰ ba bx œ g xa bg xa b^
" œ (^) x  "x^ Ê "
(^) g xa b"^ œ (^) x  "x^ Ê "
(^) x  "x œ (^) g xa b"^ Ê (^) x  ""^ œ (^) g xa b" ß so g xa b œ x  ". (c) Since af g ‰ ba bx œ Èg xa b œ l lx , g x a b œ x .# (d) Since af g ‰ ba bx œ f ˆ^ Èx ‰œ l x , f xl a b œ x. (Note that the domain of the composite is# Ò!ß _Ñ.) The completed table is shown. Note that the absolute value sign in part (d) is optional. g x f x f g x x x x x x x x x

a b a b a ba b

È

È

l l  " l l l l

" "
" l
"l
"  "

x x x x x x

15. (a) f ga a
    1 bb œ f 1a b œ 1 (b) g f 0a a b b œ g a
    2 b œ 2 (c) f fa a
    1 bb œ f 0a b œ
    2

(d) g g 2a a b b œ g 0a b œ 0 (e) g fa a
2 bb œ g 1a b œ
1 (f) f g 1a a b b œ f a
1 bœ 0

16. (a) f g 0a a b b œ f a
    1 b œ 2
    a
    1 b œ 3, where g 0a b œ 0
    1 œ
    1

(b) g f 3a a b b œ g a
1 b œ

a 1 b œ 1, where f 3a b œ 2
3 œ
1 (c) g ga a
1 bb œ g 1a b œ 1
1 œ 0, where g a
1 b œ

a 1 bœ 1

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Section 1.2 Combining Functions; Shifting and Scaling Graphs 11

Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

12 Chapter 1 Functions

Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.