Instructors solutions manual, single variable for thomas calculus, 12e

INSTRUCTOR’S
SOLUTIONS MANUAL
SINGLE VARIABLE
Collin County Community College

WILLIAM ARDIS

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


This work is protected by United States copyright laws and is provided solely
for the use of instructors in teaching their courses and assessing student
learning. Dissemination or sale of any part of this work (including on the
World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to
students except by instructors using the accompanying text in their

classes. All recipients of this work are expected to abide by these
restrictions and to honor the intended pedagogical purposes and the needs of
other instructors who rely on these materials.

The author and publisher of this book have used their best efforts in preparing this book. These efforts
include the development, research, and testing of the theories and programs to determine their
effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard
to these programs or the documentation contained in this book. The author and publisher shall not be
liable in any event for incidental or consequential damages in connection with, or arising out of, the
furnishing, performance, or use of these programs.
Reproduced by Pearson Addison-Wesley from electronic files supplied by the author.
Copyright © 2010, 2005, 2001 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-60807-9
ISBN-10: 0-321-60807-0
1 2 3 4 5 6 BB 12 11 10 09

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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 .

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TABLE OF CONTENTS
1 Functions 1
1.1
1.2
1.3
1.4

Functions and Their Graphs 1
Combining Functions; Shifting and Scaling Graphs 8
Trigonometric Functions 19
Graphing with Calculators and Computers 26
Practice Exercises 30
Additional and Advanced Exercises 38

2 Limits and Continuity 43
2.1

2.2
2.3
2.4
2.5
2.6

Rates of Change and Tangents to Curves 43
Limit of a Function and Limit Laws 46
The Precise Definition of a Limit 55
One-Sided Limits 63
Continuity 67
Limits Involving Infinity; Asymptotes of Graphs 73
Practice Exercises 82
Additional and Advanced Exercises 86

3 Differentiation 93
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

Tangents and the Derivative at a Point 93
The Derivative as a Function 99
Differentiation Rules 109
The Derivative as a Rate of Change 114

Derivatives of Trigonometric Functions 120
The Chain Rule 127
Implicit Differentiation 135
Related Rates 142
Linearizations and Differentials 146
Practice Exercises 151
Additional and Advanced Exercises 162

4 Applications of Derivatives 167
4.1
4.2
4.3
4.4
4.5
4.6
4.7

Extreme Values of Functions 167
The Mean Value Theorem 179
Monotonic Functions and the First Derivative Test 188
Concavity and Curve Sketching 196
Applied Optimization 216
Newton's Method 229
Antiderivatives 233
Practice Exercises 239
Additional and Advanced Exercises 251

5 Integration 257
5.1
5.2

5.3
5.4
5.5
5.6

Area and Estimating with Finite Sums 257
Sigma Notation and Limits of Finite Sums 262
The Definite Integral 268
The Fundamental Theorem of Calculus 282
Indefinite Integrals and the Substitution Rule 290
Substitution and Area Between Curves 296
Practice Exercises 310
Additional and Advanced Exercises 320

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6 Applications of Definite Integrals 327
6.1
6.2
6.3
6.4
6.5
6.6

Volumes Using Cross-Sections 327
Volumes Using Cylindrical Shells 337
Arc Lengths 347
Areas of Surfaces of Revolution 353
Work and Fluid Forces 358

Moments and Centers of Mass 365
Practice Exercises 376
Additional and Advanced Exercises 384

7 Transcendental Functions 389
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8

Inverse Functions and Their Derivatives 389
Natural Logarithms 396
Exponential Functions 403
Exponential Change and Separable Differential Equations 414
^
Indeterminate Forms and L'Hopital's
Rule 418
Inverse Trigonometric Functions 425
Hyperbolic Functions 436
Relative Rates of Growth 443
Practice Exercises 447
Additional and Advanced Exercises 458

8 Techniques of Integration 461
8.1
8.2

8.3
8.4
8.5
8.6
8.7

Integration by Parts 461
Trigonometric Integrals 471
Trigonometric Substitutions 478
Integration of Rational Functions by Partial Fractions 484
Integral Tables and Computer Algebra Systems 491
Numerical Integration 502
Improper Integrals 510
Practice Exercises 518
Additional and Advanced Exercises 528

9 First-Order Differential Equations 537
9.1
9.2
9.3
9.4
9.5

Solutions, Slope Fields and Euler's Method 537
First-Order Linear Equations 543
Applications 546
Graphical Solutions of Autonomous Equations 549
Systems of Equations and Phase Planes 557
Practice Exercises 562
Additional and Advanced Exercises 567

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

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11 Parametric Equations and Polar Coordinates 647
11.1
11.2
11.3
11.4
11.5
11.6
11.7

Parametrizations of Plane Curves 647
Calculus with Parametric Curves 654
Polar Coordinates 662

Graphing in Polar Coordinates 667
Areas and Lengths in Polar Coordinates 674
Conic Sections 679
Conics in Polar Coordinates 689
Practice Exercises 699
Additional and Advanced Exercises 709

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CHAPTER 1 FUNCTIONS
1.1 FUNCTIONS AND THEIR GRAPHS
1. domain œ (c_ß _); range œ [1ß _)

2. domain œ [0ß _); range œ (c_ß 1]

3. domain œ Ịc2ß _); y in range and y œ È5x b 10   ! Ê y can be any positive real number Ê range œ Ị!ß _).
4. domain œ (c_ß 0Ĩ r Ị3, _); y in range and y œ Èx2 c 3x   ! Ê y can be any positive real number Ê range œ Ị!ß _).
5. domain œ (c_ß 3Đ r Ð3, _); y in range and y œ
Ê3 c t!Ê

4
3ct

4
3ct,

now if t  3 Ê 3 c t € ! Ê

4

3ct

€ !, or if t € 3

 ! Ê y can be any nonzero real number Ê range œ Ðc_ß 0Đ r Ð!ß _).

6. domain œ (c_ß c%Đ r Ðc4, 4Đ r Ð4, _); y in range and y œ
2

c%  t  4 Ê c16 Ÿ t c 16  ! Ê
nonzero real number Ê range œ Ðc_ß

#
c "'
c 18 Ĩ

Ÿ

2
t2 c 16

2
t2 c 16 ,

2
t2 c 16

now if t  c% Ê t2 c 16 € ! Ê
2

 !, or if t € % Ê t c 16 € ! Ê

2
t2 c 16

€ !, or if

€ ! Ê y can be any

r Ð!ß _).

7. (a) Not the graph of a function of x since it fails the vertical line test.
(b) Is the graph of a function of x since any vertical line intersects the graph at most once.
8. (a) Not the graph of a function of x since it fails the vertical line test.
(b) Not the graph of a function of x since it fails the vertical line test.
#

9. base œ x; (height)# b ˆ #x ‰ œ x# Ê height œ

È3
#

x; area is a(x) œ

"
#

(base)(height) œ

"

#

(x) Š

È3
# x‹

œ

È3
4

x# ;

perimeter is p(x) œ x b x b x œ 3x.
10. s œ side length Ê s# b s# œ d# Ê s œ

d
È2

; and area is a œ s# Ê a œ

"
#

d#

11. Let D œ diagonal length of a face of the cube and j œ the length of an edge. Then j# b D# œ d# and
D# œ 2j# Ê 3j# œ d# Ê j œ

d
È3

. The surface area is 6j# œ

6d#
3

12. The coordinates of P are ˆxß Èx‰ so the slope of the line joining P to the origin is m œ
ˆx, Èx‰ œ ˆ m"# ,

#

œ 2d# and the volume is j$ œ Š d3 ‹
Èx
x

œ

"
Èx

$Ỵ#

œ

(x € 0). Thus,

"‰
m .

13. 2x b 4y œ 5 Ê y œ c "# x b 54 ; L œ ÈÐx c 0Ñ2 b Ðy c 0Ñ2 œ Éx2 b Ðc "# x b 54 Ñ2 œ Éx2 b 4" x2 c 54 x b
œ É 54 x2 c 54 x b

25
16

œ É 20x

2

c 20x b 25
16

œ

È20x2 c 20x b 25
4

14. y œ Èx c 3 Ê y2 b 3 œ x; L œ ÈÐx c 4Ñ2 b Ðy c 0Ñ2 œ ÈÐy2 b 3 c 4Ñ2 b y2 œ ÈÐy2 c 1Ñ2 b y2
œ Èy4 c 2y2 b 1 b y2 œ Èy4 c y2 b 1

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d$
3È 3

25
16

.


2

Chapter 1 Functions

15. The domain is ac_ß _b.

16. The domain is ac_ß _b.

17. The domain is ac_ß _b.

18. The domain is Ðc_ß !Ĩ.

19. The domain is ac_ß !b r a!ß _b.

20. The domain is ac_ß !b r a!ß _b.

21. The domain is ac_ß c5b r Ðc5ß c3Ĩ r Ị3, 5Đ r a5, _b 22. The range is Ị2, 3Đ.
23. Neither graph passes the vertical line test
(a)

(b)

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Section 1.1 Functions and Their Graphs
24. Neither graph passes the vertical line test
(a)

(b)

Ú xbyœ" Þ
Ú yœ1cx Þ
or
or
kx b yk œ 1 Í Û
Í Û
ß
ß
Ü x b y œ c" à
Ü y œ c" c x à
25.

x
y

0
0

1
1

27. Faxb œ œ

2

0

26.

x
y

0
1

1
0

2
0

"
, x0
28. Gaxb œ œ x
x, 0 Ÿ x

4 c x2 , x Ÿ 1
x2 b 2x, x € 1

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 cx b 2
x, 0 Ÿ x Ÿ 1
f(x) œ œ
cx b 2, 1  x Ÿ 2
Ú
Ý 2, ! Ÿ x  "

Ý
!ß " Ÿ x  #
(b) f(x) œ Û
Ý
Ý 2ß # Ÿ x  $
Ü !ß $ Ÿ x Ÿ %
30. (a) Line through a!ß 2b and a#ß !b: y œ cx b 2
"
Line through a2ß "b and a&ß !b: m œ !& c
c# œ
cx b #, 0  x Ÿ #
f(x) œ œ "
c $ x b &$ , #  x Ÿ &

f(x) œ œ

œ c "$ , so y œ c "$ ax c 2b b " œ c "$ x b

c$ c !
! c Ðc"Ñ œ
c" c $
c%
#c! œ #

(b) Line through ac"ß !b and a!ß c$b: m œ
Line through a!ß $b and a#ß c"b: m œ

c"
$

&
$

c$, so y œ c$x c $
œ c#, so y œ c#x b $

c$x c $, c"  x Ÿ !
c#x b $, !  x Ÿ #

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3


4

Chapter 1 Functions

31. (a) Line through ac"ß "b and a!ß !b: y œ cx
Line through a!ß "b and a"ß "b: y œ "
Line through a"ß "b and a$ß !b: m œ !c"
$c" œ
Ú
cx
c" Ÿ x  !
"
!xŸ"
f(x) œ Û
Ü c "# x b $#

"x$

c"
#

(b) Line through ac2ß c1b and a0ß 0b: y œ 12 x
Line through a0ß 2b and a1ß 0b: y œ c2x b 2
Line through a1ß c1b and a3ß c1b: y œ c1
32. (a) Line through ˆ T# ß !‰ and aTß "b: m œ
faxb œ J

(b)

"c!
TcaTỴ#b

œ c "# , so y œ c "# ax c "b b " œ c "# x b

Ú

1
2x

faxb œ Û c2x b 2
Ü c1

$
#

c2 Ÿ x Ÿ 0

0xŸ1
1xŸ3

œ T# , so y œ T# ˆx c T# ‰ b 0 œ T# x c "

!, 0 Ÿ x Ÿ T#
#
T
T x c ", #  x Ÿ T

Ú
A,
Ý
Ý
Ý
cAß
faxb œ Û
Aß
Ý
Ý
Ý
Ü cAß

! Ÿ x  T#
T
# Ÿx T
T Ÿ x  $#T
$T
# Ÿ x Ÿ #T

33. (a) ÚxÛ œ 0 for x − [0ß 1)

(b) ÜxÝ œ 0 for x − (c1ß 0]

34. ÚxÛ œ ÜxÝ only when x is an integer.
35. For any real number x, n Ÿ x Ÿ n b ", where n is an integer. Now: n Ÿ x Ÿ n b " Ê cÐn b "Ñ Ÿ cx Ÿ cn. By
definition: ÜcxÝ œ cn and ÚxÛ œ n Ê cÚxÛ œ cn. So ÜcxÝ œ cÚxÛ for all x − d .
36. To find f(x) you delete the decimal or
fractional portion of x, leaving only
the integer part.

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Section 1.1 Functions and Their Graphs
37. Symmetric about the origin
Dec: c_  x  _
Inc: nowhere

38. Symmetric about the y-axis
Dec: c_  x  !
Inc: !  x  _

39. Symmetric about the origin
Dec: nowhere
Inc: c_  x  !
!x_

40. Symmetric about the y-axis

Dec: !  x  _
Inc: c_  x  !

41. Symmetric about the y-axis
Dec: c_  x Ÿ !
Inc: !  x  _

42. No symmetry
Dec: c_  x Ÿ !
Inc: nowhere

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5


6

Chapter 1 Functions

43. Symmetric about the origin
Dec: nowhere
Inc: c_  x  _

44. No symmetry
Dec: ! Ÿ x  _
Inc: nowhere

45. No symmetry

Dec: ! Ÿ x  _
Inc: nowhere

46. Symmetric about the y-axis
Dec: c_  x Ÿ !
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. faxb œ xc& œ

"
x&

and facxb œ acxbc& œ

"
ac x b&

œ cˆ x"& ‰ œ cfaxb. Thus the function is odd.

49. Since faxb œ x# b " œ acxb# b " œ cfaxb. The function is even.
50. Since Ịfaxb œ x# b xĨ Á Ịfacxb œ acxb# c xĨ and Ịfaxb œ x# b xĨ Á Ịcfaxb œ caxb# c xĨ the function is neither even nor
odd.
51. Since gaxb œ x$ b x, gacxb œ cx$ c x œ cax$ b xb œ cgaxb. So the function is odd.
52. gaxb œ x% b $x# c " œ acxb% b $acxb# c " œ gacxbß thus the function is even.
53. gaxb œ

"
x# c "

54. gaxb œ

x
x# c " ;

55. hatb œ

"
t c ";

œ

"
acxb# c"

œ gacxb. Thus the function is even.

gacxb œ c x#xc" œ cgaxb. So the function is odd.

h a ct b œ

"
ct c " ;

ch at b œ

"
" c t.

Since hatb Á chatb and hatb Á hactb, the function is neither even nor odd.

56. Since l t$ | œ l actb$ |, hatb œ hactb and the function is even.

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Section 1.1 Functions and Their Graphs
57. hatb œ 2t b ", hactb œ c2t b ". So hatb Á hactb. chatb œ c2t c ", so hatb Á chatb. The function is neither even nor
odd.
58. hatb œ 2l t l b " and hactb œ 2l ct l b " œ 2l t l b ". So hatb œ hactb and the function is even.
59. s œ kt Ê 25 œ kÐ75Ñ Ê k œ

"
3

Ê s œ 3" t; 60 œ 3" t Ê t œ 180

60. K œ c v# Ê 12960 œ ca18b2 Ê c œ 40 Ê K œ 40v# ; K œ 40a10b# œ 4000 joules
61. r œ
62. P œ

k
s

Ê6œ

k
v

k
4

Ê k œ 24 Ê r œ

Ê 14.7 œ

k
1000

24
s ;

10 œ

24
s

Ê k œ 14700 Ê P œ

Êsœ

14700
v ;

12
5

23.4 œ

14700
v

Êvœ

24500
39

¸ 628.2 in3

63. v œ f(x) œ xÐ"% c 2xÑÐ22 c 2xÑ œ %x$ c 72x# b $!)x; !  x  7Þ
64. (a) Let h œ height of the triangle. Since the triangle is isosceles, AB # b AB # œ 2# Ê AB œ È2Þ So,
#

h# b "# œ ŠÈ2‹ Ê h œ " Ê B is at a!ß "b Ê slope of AB œ c" Ê The equation of AB is
y œ f(x) œ cx b "; x − Ị!ß "Ĩ.
(b) xĐ œ 2x y œ 2xÐcx b "Đ œ c2x# b #x; x − Ị!ß "Ó.
65. (a) Graph h because it is an even function and rises less rapidly than does Graph g.
(b) Graph f because it is an odd function.
(c) Graph g because it is an even function and rises more rapidly than does Graph h.
66. (a) Graph f because it is linear.
(b) Graph g because it contains a!ß "b.
(c) Graph h because it is a nonlinear odd function.
x
#

67. (a) From the graph,
(b)

x
#

€1b

x € 0:

x
#

x  0:

x
2

4
x

c1c

Ê
4
x

x
#

€

€1b

4
x

Ê x − (c2ß 0) r (%ß _)

c 1 c 4x € 0
#
2xc8
0 Ê x c2x

€0 Ê

(xc4)(xb2)
#x

€0

(xc4)(xb2)
#x

0

Ê x € 4 since x is positive;
c1c

4
x

€0 Ê

x# c2xc8
2x

0 Ê

Ê x  c2 since x is negative;
sign of (x c 4)(x b 2)
b
b
c
ùùùùùùùùùùùùùùợ
c2
%
Solution interval: (c#ò 0) r (%ò _)

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7


8

Chapter 1 Functions
3
2
x c1  x b 1
3
2

x c1  x b 1

68. (a) From the graph,
(b) Case x  c1:

Ê x − (c_ß c5) r (c1ß 1)
Ê

3(xb1)
x c1

€2

Ê 3x b 3  2x c 2 Ê x  c5.
Thus, x − (c_ß c5) solves the inequality.
Case c1  x  1:

3
x c1



2
x b1

Ê

3(xb1)
x c1

2

Ê 3x b 3 € 2x c 2 Ê x € c5 which is true
if x € c1. Thus, x − (c1ß 1) solves the
inequality.
3
Case 1  x: xc1  xb2 1 Ê 3x b 3  2x c 2 Ê x  c5
which is never true if 1  x, so no solution here.
In conclusion, x − (c_ß c5) r (c1ß 1).
69. A curve symmetric about the x-axis will not pass the vertical line test because the points ax, yb and ax, cyb lie on the same
vertical line. The graph of the function y œ faxb œ ! is the x-axis, a horizontal line for which there is a single y-value, !,
for any x.
70. price œ 40 b 5x, quantity œ 300 c 25x Ê Raxb œ a40 b 5xba300 c 25xb
71. x2 b x2 œ h2 Ê x œ

h
È2

œ

È2 h
2 ;

cost œ 5a2xb b 10h Ê Cahb œ 10Š

È2 h
2 ‹

b 10h œ 5hŠÈ2 b 2‹

72. (a) Note that 2 mi = 10,560 ft, so there are È800# b x# feet of river cable at $180 per foot and a10,560 c xb feet of land
cable at $100 per foot. The cost is Caxb œ 180È800# b x# b 100a10,560 c xb.
(b) Ca!b œ $"ß #!!ò !!!
Ca&!!b á $"ò "(&ò )"#
Ca"!!!b á $"ò ")'ò &"#
Ca"&!!b ¸ $"ß #"#ß !!!
Ca#!!!b ¸ $"ß #%$ß ($#
Ca#&!!b ¸ $"ß #()ò %(*
Ca$!!!b á $"ò $"%ò )(!
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. Df : c_  x  _, Dg : x   1 Ê Dfbg œ Dfg : x   1. Rf : c_  y  _, Rg : y   0, Rfbg : y   1, Rfg : y   0
2. Df : x b 1   0 Ê x   c1, Dg : x c 1   0 Ê x   1. Therefore Dfbg œ Dfg : x   1.
Rf œ Rg : y   0, Rfbg : y   È2, Rfg : y   0
3. Df : c_  x  _, Dg : c_  x  _, DfỴg : c_  x  _, DgỴf : c_  x  _, Rf : y œ 2, Rg : y   1,
RfỴg : 0  y Ÿ 2, RgỴf : "# Ÿ y  _
4. Df : c_  x  _, Dg : x   0 , DfỴg : x   0, DgỴf : x   0; Rf : y œ 1, Rg : y   1, RfỴg : 0  y Ÿ 1, RgỴf : 1 Ÿ y  _
5. (a) 2
(d) (x b 5)# c 3 œ x# b 10x b 22
(g) x b 10

(b) 22
(e) 5
(h) (x# c 3)# c 3 œ x% c 6x# b 6

(c) x# b 2
(f) c2

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Section 1.2 Combining Functions; Shifting and Scaling Graphs
6. (a) c "3
(d)

(b) 2

"
x

(c)

(e) 0

(g) x c 2

(h)

(f)

"
"
xb1 b 1

"

œ

xb#
xb1

xb"
xb#

œ

"
x b1
3
4

c1œ

cx
xb1

7. af‰g‰hbaxb œ fagahaxbbb œ faga4 c xbb œ fa3a4 c xbb œ fa12 c 3xb œ a12 c 3xb b 1 œ 13 c 3x
8. af‰g‰hbaxb œ fagahaxbbb œ fagax2 bb œ fa2ax2 b c 1b œ fa2x2 c 1b œ 3a2x2 c 1b b 4 œ 6x2 b1
b"
9. af‰g‰hbaxb œ fagahaxbbb œ fˆgˆ 1x ‰‰ œ fŠ 1 b1 % ‹ œ fˆ 1 bx 4x ‰ œ É 1 bx 4x b " œ É 5x
1 b 4x
x

2

10. af‰g‰hbaxb œ fagahaxbbb œ fŠgŠÈ2 c x‹‹ œ f:

ŠÈ2 c x‹

2

ŠÈ2 c x‹

; œ fˆ $ c x ‰ œ
b1
2cx

2cx
$cx b 2
cx
3 c $2 c
x

8 c 3x
7 c 2x

œ

11. (a) af‰gbaxb
(d) a j‰jbaxb

(b) a j‰gbaxb
(e) ag‰h‰f baxb

(c) ag‰gbaxb
(f) ah‰j‰f baxb

12. (a) af‰jbaxb
(d) af‰f baxb

(b) ag‰hbaxb
(e) a j‰g‰f baxb

(c) ah‰hbaxb
(f) ag‰f‰hbaxb

g(x)

f(x)

(f ‰ g)(x)

(a)

xc7

Èx

Èx c 7

(b)

xb2

3x

3(x b 2) œ 3x b 6

(c)

x#

Èx c 5

Èx# c 5

(d)

x
xc1

x
xc1

"
xc1
"
x

1b

13.

(e)
(f)

"
x

gaxbc"
g ax b

œ

x
x c (xc1)

œx

x

"
x

x

"
lx c "l .

14. (a) af‰gbaxb œ lgaxbl œ
(b) af‰gbaxb œ

x
xc1
x
xc1 c 1

x
xb"

œ

Ê"c

"
g ax b

œ

x
xb"

Ê"c

x
xb"

œ

"
g ax b

Ê

"
xb"

œ

"
gaxb ß so

gaxb œ x b ".

(c) Since af‰gbaxb œ Ègaxb œ lxl, gaxb œ x .
(d) Since af‰gbaxb œ fˆÈx‰ œ l x l, faxb œ x# . (Note that the domain of the composite is Ị!ß _Ñ.)
#

The completed table is shown. Note that the absolute value sign in part (d) is optional.
gaxb
faxb
af‰gbaxb
"
"
lxl
xc"
lx c "l
xb"
x#
Èx

xc"
x

Èx
#

x

15. (a) fagac1bb œ fa1b œ 1
(d) gaga2bb œ ga0b œ 0

x
xb"

lxl
lxl
(b) gafa0bb œ gac2b œ 2
(e) gafac2bb œ ga1b œ c1

(c) fafac1bb œ fa0b œ c2
(f) faga1bb œ fac1b œ 0

16. (a) faga0bb œ fac1b œ 2 c ac1b œ 3, where ga0b œ 0 c 1 œ c1
(b) gafa3bb œ gac1b œ cac1b œ 1, where fa3b œ 2 c 3 œ c1
(c) gagac1bb œ ga1b œ 1 c 1 œ 0, where gac1b œ cac1b œ 1

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9


10

Chapter 1 Functions
(d) fafa2bb œ fa0b œ 2 c 0 œ 2, where fa2b œ 2 c 2 œ 0
(e) gafa0bb œ ga2b œ 2 c 1 œ 1, where fa0b œ 2 c 0 œ 2
(f) fˆgˆ "# ‰‰ œ fˆc #" ‰ œ 2 c ˆc #" ‰ œ 5# , where gˆ "# ‰ œ "# c 1 œ c "#

17. (a) af‰gbaxb œ fagaxbb œ É 1x b 1 œ É 1 bx x
ag‰f baxb œ gafaxbb œ

1
Èx b 1

(b) Domain af‰gb: Ðc_, c1Ó r Ð0, _Ñ, domain ag‰f b: Ðc1, _Ñ
(c) Range af‰gb: Ð1, _Ñ, range ag‰f b: Ð0, _Ñ
18. (a) af‰gbaxb œ fagaxbb œ 1 c 2Èx b x
ag‰f baxb œ gafaxbb œ 1 c kxk
(b) Domain af‰gb: Ị0, _Đ, domain ag‰f b: Ðc_, _Đ
(c) Range af‰gb: Ð0, _Đ, range ag‰f b: Ðc_, 1Ĩ
19. af‰gbaxb œ x Ê fagaxbb œ x Ê

g ax b
g ax b c 2

œ x Ê gaxb œ agaxb c 2bx œ x † gaxb c 2x

Ê gaxb c x † gaxb œ c2x Ê gaxb œ c 1 2x
cx œ

2x
xc1

20. af‰gbaxb œ x b 2 Ê fagaxbb œ x b 2 Ê 2agaxbb3 c 4 œ x b 2 Ê agaxbb3 œ
21. (a) y œ c(x b 7)#

(b) y œ c(x c 4)#

22. (a) y œ x# b 3

(b) y œ x# c 5

xb6
2

3 xb6
Ê gaxb œ É
2

23. (a) Position 4

(b) Position 1

(c) Position 2

(d) Position 3

24. (a) y œ c(x c 1)# b 4

(b) y œ c(x b 2)# b 3

(c) y œ c(x b 4)# c 1

(d) y œ c(x c 2)#

25.

26.

27.

28.

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Section 1.2 Combining Functions; Shifting and Scaling Graphs
29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

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11


12

Chapter 1 Functions

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

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Section 1.2 Combining Functions; Shifting and Scaling Graphs
53.

54.

55. (a) domain: [0ß 2]; range: [#ß $]

(b) domain: [0ß 2]; range: [c1ß 0]

(c) domain: [0ß 2]; range: [0ß 2]

(d) domain: [0ß 2]; range: [c1ß 0]

(e) domain: [c2ß 0]; range: [!ß 1]

(f) domain: [1ß 3]; range: [!ß "]

(g) domain: [c2ß 0]; range: [!ß "]

(h) domain: [c1ò 1]; range: [!ò "]

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13


14

Chapter 1 Functions

56. (a) domain: [0ß 4]; range: [c3ß 0]

(b) domain: [c4ß 0]; range: [!ß $]

(c) domain: [c4ß 0]; range: [!ß $]

(d) domain: [c4ß 0]; range: ["ß %]

(e) domain: [#ß 4]; range: [c3ß 0]

(f) domain: [c2ß 2]; range: [c3ß 0]

(g) domain: ["ß 5]; range: [c3ß 0]

(h) domain: [0ß 4]; range: [0ß 3]

58. y œ a2xb# c 1 œ %x# c 1

57. y œ 3x# c 3
59. y œ "# ˆ" b

"‰
x#

œ

"
#

b

"
#x#

60. y œ 1 b

"
axỴ$b#

œ1b

61. y œ È%x b 1

62. y œ 3Èx b 1

#

63. y œ É% c ˆ x# ‰ œ "# È16 c x#

64. y œ "$ È% c x#

65. y œ " c a3xb$ œ " c 27x$

66. y œ " c ˆ x# ‰ œ " c

$

*
x#

x$
)

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Section 1.2 Combining Functions; Shifting and Scaling Graphs
67. Let y œ cÈ#x b " œ faxb and let gaxb œ x"Ỵ# ,
"Ỵ#
"Ỵ#
haxb œ ˆx b " ‰ , iaxb œ È#ˆx b " ‰ , and
#

#

"Ỵ#

jaxb œ c’È#ˆx b "# ‰ “ œ faBb. The graph of

haxb is the graph of gaxb shifted left

"
#

unit; the

graph of iaxb is the graph of haxb stretched
vertically by a factor of È#; and the graph of
jaxb œ faxb is the graph of iaxb reflected across
the x-axis.
68. Let y œ È" c

x
#

œ faxbÞ Let gaxb œ acxb"Ỵ# ,

haxb œ acx b #b"Ỵ# , and iaxb
ẩ" c

x
#

"
ẩ # ac x

b #b"ẻ#

faxbị The graph of gaxb is the

graph of y œ Èx reflected across the x-axis.
The graph of haxb is the graph of gaxb shifted
right two units. And the graph of iaxb is the
graph of haxb compressed vertically by a factor
of È#.
69. y œ faxb œ x$ . Shift faxb one unit right followed by a
shift two units up to get gaxb œ ax c "b3 b #.

70. y œ a" c Bb$ b # œ cỊax c "b$ b ac#bĨ œ faxb.
Let gaxb œ x$ , haxb œ ax c "b$ , iaxb œ ax c "b$ b ac#b,
and jaxb œ cÒax c "b$ b ac#bÓ. The graph of haxb is the
graph of gaxb shifted right one unit; the graph of iaxb is
the graph of haxb shifted down two units; and the graph
of faxb is the graph of iaxb reflected across the x-axis.

71. Compress the graph of faxb œ
of 2 to get gaxb œ
unit to get haxb œ

"
#x . Then
"
#x c ".

"
x

horizontally by a factor

shift gaxb vertically down 1

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15


16

Chapter 1 Functions

72. Let faxb œ
œ

"

#

ŠxỴÈ#‹

"
x#

and gaxb œ

b"œ

#
x#

b"œ

"
#
’Š"ỴÈ#‹B“

"
#
Š B# ‹

b"

b "ị Since

ẩ# á "ị%, we see that the graph of faxb stretched
horizontally by a factor of 1.4 and shifted up 1 unit
is the graph of gaxb.

$
73. Reflect the graph of y œ faxb œ È
x across the x-axis
$
to get gaxb œ cÈx.

74. y œ faxb œ ac#xb#Ỵ$ œ Ịac"ba#bxĨ#Ỵ$
œ ac"b#Ỵ$ a#xb#Ỵ$ œ a#xb#Ỵ$ . So the graph
of faxb is the graph of gaxb œ x#Ỵ$ compressed

horizontally by a factor of 2.

75.

76.

77. *x# b #&y# œ ##& Ê

x#
&#

b

y#
$#

œ"

78. "'x# b (y# œ ""# Ê

x#
#
È
Š (‹

b

y#
%#

œ"

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Section 1.2 Combining Functions; Shifting and Scaling Graphs
79. $x# b ay c #b# œ $ Ê

x#
"#

b

a y c #b #
#

ŠÈ$‹

80. ax b "b# b #y# œ % Ê

œ"

Ê

83.

x#
"'

#

ŠÈ#‹

b

y#
*

b

#

ŠÈ$‹

b

#
#
82. 'ˆx b $# ‰ b *ˆy c "# ‰ œ &%

81. $ax c "b# b #ay b #b# œ '
ax c " b #

##

#

œ"

Ê

’xcˆc $# ‰“
$#

b

ˆy c "# ‰#
#

ŠÈ'‹

œ"

œ " has its center at a!ß !b. Shiftinig 4 units

left and 3 units up gives the center at ah, kb œ ac%ß $b.
#
b ay c3#3b œ "
4#
a y c $b #
œ ". Center, C, is ac%ß
3#

So the equation is
Ê

ax b % b #
4#

b

$b, and

major axis, AB, is the segment from ac)ß $b to a!ß $b.

84. The ellipse

x#
%

y#
#&

b

œ " has center ah, kb œ a!ß !b.

Shifting the ellipse 3 units right and 2 units down
produces an ellipse with center at ah, kb œ a$ß c#b
and an equation

ax c 3 b#
%

b

#&

œ ". Center,

C, is a3ß c#b, and AB, the segment from a$ß $b to
a$ß c(b is the major axis.

85. (a) (fg)(cx) œ f(cx)g(cx) œ f(x)(cg(x)) œ c(fg)(x), odd
(b) Š gf ‹ (cx) œ

f(cx)
g(cx)

œ

f(x)
cg(x)

œ c Š gf ‹ (x), odd

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y#
#
È
Š #‹

œ"

17


18

Chapter 1 Functions
(c) ˆ gf ‰ (cx) œ
(d)
(e)
(f)
(g)
(h)
(i)

g(cx)
f(cx)

œ

cg(x)
f(x)

œ c ˆ gf ‰ (x), odd

f # (cx) œ f(cx)f(cx) œ f(x)f(x) œ f # (x), even
g# (cx) œ (g(cx))# œ (cg(x))# œ g# (x), even
(f ‰ g)(cx) œ f(g(cx)) œ f(cg(x)) œ f(g(x)) œ (f ‰ g)(x), even
(g ‰ f)(cx) œ g(f(cx)) œ g(f(x)) œ (g ‰ f)(x), even
(f ‰ f)(cx) œ f(f(cx)) œ f(f(x)) œ (f ‰ f)(x), even

(g ‰ g)(cx) œ g(g(cx)) œ g(cg(x)) œ cg(g(x)) œ c(g ‰ g)(x), odd

86. Yes, f(x) œ 0 is both even and odd since f(cx) œ 0 œ f(x) and f(cx) œ 0 œ cf(x).
87. (a)

(b)

(c)

(d)

88.

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Section 1.3 Trigonometric Functions
1.3 TRIGONOMETRIC FUNCTIONS
1. (a) s œ r) œ (10) ˆ 451 ‰ œ 81 m
radians and

51
4

1 ‰
3. ) œ 80° Ê ) œ 80° ˆ 180°
œ

41

9

2. ) œ

s
r

œ

101
8

œ

51
4

1 ‰
(b) s œ r) œ (10)(110°) ˆ 180°
œ

c1

)

c 231

0

1

#

s
r

œ

30
50

31
4
"
È2
c È"
2

sin )

0

cos )

c1

tan )

0

È3

0

und.

c"

und.

"
È3

und.

0

c1

und.

cÈ 2

c1

c#

und.

c È23

sec )
csc )

0

"

"

0

"
und.

7. cos x œ c 45 , tan x œ c 34
9. sin x œ c

È8
3

, tan x œ cÈ8

"

6.

È2

c 3#1

)

c 1'

sin )

"

cos )

!

"
#

tan )

und.

cÈ 3

cot )

!

c È"3

sec )

und.

#

csc )

"

c È23

8. sin x œ

2
È5

10. sin x œ

12
13

13.

14.

period œ 1

c 13

È
c #3

12. cos x œ c

, cos x œ

"
È2

&1
'
"
#
È
c #3

c È"3

"

c È"3

cÈ 3

"

cÈ 3

2
È3

È2

c È23

c#

È2

#

c "#
È3
#

"
È5

, tan x œ c 12
5
È3
#

, tan x œ

"
È3

period œ 41
16.

period œ 2

m

‰ ¸ 34°
œ 0.6 rad or 0.6 ˆ 180°
1

11. sin x œ c È"5 , cos x œ c È25

15.

551
9

Ê s œ (6) ˆ 491 ‰ œ 8.4 in. (since the diameter œ 12 in. Ê radius œ 6 in.)

È
c #3
c "#

cot )

œ

ˆ 180°
‰ œ 225°
1

4. d œ 1 meter Ê r œ 50 cm Ê ) œ
5.

1101
18

period œ 4

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1
%
"
È2

19