Solucionario - Thomas cálculo,12 edición

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Published on July 7, 2016

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1. SOLUTION MANUAL

2. 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

3. 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 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 permit- ted. 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. 608070 _ISM_ThomasCalc_WeirHass_ttl.qxd:harsh_569709_ttl 9/3/09 3:11 PM Page 2

4. 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 conforms exactly to the methods, procedures and steps presented in the textì is mathematically correctì includes all of the steps necessary so a typical calculus student can follow the logical argument and algebraì includes a graph or figure whenever called for by the exercise, or if needed to help with the explanationì is formatted in an appropriate style to aid in its understandingì Every CAS exercise is solved in both the MAPLE and computer algebra systems. A template showingMATHEMATICA 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.

5. TABLE OF CONTENTS 1 Functions 1 1.1 Functions and Their Graphs 1 1.2 Combining Functions; Shifting and Scaling Graphs 8 1.3 Trigonometric Functions 19 1.4 Graphing with Calculators and Computers 26 Practice Exercises 30 Additional and Advanced Exercises 38 2 Limits and Continuity 43 2.1 Rates of Change and Tangents to Curves 43 2.2 Limit of a Function and Limit Laws 46 2.3 The Precise Definition of a Limit 55 2.4 One-Sided Limits 63 2.5 Continuity 67 2.6 Limits Involving Infinity; Asymptotes of Graphs 73 Practice Exercises 82 Additional and Advanced Exercises 86 3 Differentiation 93 3.1 Tangents and the Derivative at a Point 93 3.2 The Derivative as a Function 99 3.3 Differentiation Rules 109 3.4 The Derivative as a Rate of Change 114 3.5 Derivatives of Trigonometric Functions 120 3.6 The Chain Rule 127 3.7 Implicit Differentiation 135 3.8 Related Rates 142 3.9 Linearizations and Differentials 146 Practice Exercises 151 Additional and Advanced Exercises 162 4 Applications of Derivatives 167 4.1 Extreme Values of Functions 167 4.2 The Mean Value Theorem 179 4.3 Monotonic Functions and the First Derivative Test 188 4.4 Concavity and Curve Sketching 196 4.5 Applied Optimization 216 4.6 Newton's Method 229 4.7 Antiderivatives 233 Practice Exercises 239 Additional and Advanced Exercises 251 5 Integration 257 5.1 Area and Estimating with Finite Sums 257 5.2 Sigma Notation and Limits of Finite Sums 262 5.3 The Definite Integral 268 5.4 The Fundamental Theorem of Calculus 282 5.5 Indefinite Integrals and the Substitution Rule 290 5.6 Substitution and Area Between Curves 296 Practice Exercises 310 Additional and Advanced Exercises 320

6. 6 Applications of Definite Integrals 327 6.1 Volumes Using Cross-Sections 327 6.2 Volumes Using Cylindrical Shells 337 6.3 Arc Lengths 347 6.4 Areas of Surfaces of Revolution 353 6.5 Work and Fluid Forces 358 6.6 Moments and Centers of Mass 365 Practice Exercises 376 Additional and Advanced Exercises 384 7 Transcendental Functions 389 7.1 Inverse Functions and Their Derivatives 389 7.2 Natural Logarithms 396 7.3 Exponential Functions 403 7.4 Exponential Change and Separable Differential Equations 414 7.5 Indeterminate Forms and L'Hopital's Rule 418^ 7.6 Inverse Trigonometric Functions 425 7.7 Hyperbolic Functions 436 7.8 Relative Rates of Growth 443 Practice Exercises 447 Additional and Advanced Exercises 458 8 Techniques of Integration 461 8.1 Integration by Parts 461 8.2 Trigonometric Integrals 471 8.3 Trigonometric Substitutions 478 8.4 Integration of Rational Functions by Partial Fractions 484 8.5 Integral Tables and Computer Algebra Systems 491 8.6 Numerical Integration 502 8.7 Improper Integrals 510 Practice Exercises 518 Additional and Advanced Exercises 528 9 First-Order Differential Equations 537 9.1 Solutions, Slope Fields and Euler's Method 537 9.2 First-Order Linear Equations 543 9.3 Applications 546 9.4 Graphical Solutions of Autonomous Equations 549 9.5 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

7. 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 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 Practice Exercises 785 Additional Exercises 791 Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

8. 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 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.

9. CHAPTER 1 FUNCTIONS 1.1 FUNCTIONS AND THEIR GRAPHS 1. domain ( ); range [1 ) 2. domain [0 ); range ( 1]œ c_ß _ œ ß _ œ ß _ œ c_ß 3. domain 2 ); y in range and y 5x 10 y can be any positive real number range ).œ Òc ß _ œ b   ! Ê Ê œ Ò!ß _È 4. domain ( 0 3, ); y in range and y x 3x y can be any positive real number range ).œ c_ß Ó r Ò _ œ c   ! Ê Ê œ Ò!ß _È 2 5. domain ( 3 3, ); y in range and y , now if t 3 3 t , or if t 3œ c_ß Ñ r Ð _ œ  Ê c € ! Ê € ! €4 4 3 t 3 tc c 3 t y can be any nonzero real number range 0 ).Ê c  ! Ê  ! Ê Ê œ Ðc_ß Ñ r Ð!ß _4 3 tc 6. domain ( 4, 4 4, ); y in range and y , now if t t 16 , or ifœ c_ß c%Ñ r Ðc Ñ r Ð _ œ  c% Ê c € ! Ê € !2 2 t 16 t 16 2 2 2c c t 4 16 t 16 , or if t t 16 y can be anyc%   Ê c Ÿ c  ! Ê c Ÿ  ! € % Ê c € ! Ê € ! Ê2 22 2 t 16 t 16 # "' c c2 2 nonzero real number range ).Ê œ Ðc_ß c Ó r Ð!ß _1 8 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 x; (height) x height x; area is a(x) (base)(height) (x) x x ;œ b œ Ê œ œ œ œ# # # # # # # # # " "ˆ ‰ Š ‹x 3 3 3 4 È È È perimeter is p(x) x x x 3x.œ b b œ 10. s side length s s d s ; and area is a s a dœ Ê b œ Ê œ œ Ê œ# # # # #" # d 2È 11. Let D diagonal length of a face of the cube and the length of an edge. Then D d andœ j œ j b œ# # # D 2 3 d . The surface area is 6 2d and the volume is .# # # # # # $ $Î# œ j Ê j œ Ê j œ j œ œ j œ œd 6d d d 3 3 33 3È È # # $ Š ‹ 12. The coordinates of P are x x so the slope of the line joining P to the origin is m (x 0). Thus,ˆ ‰Èß œ œ € È È x x x " x, x , .ˆ ‰ ˆ ‰È œ " " m m# 13. 2x 4y 5 y x ; L x 0 y 0 x x x x xb œ Ê œ c b œ Ð c Ñ b Ð c Ñ œ b Ðc b Ñ œ b c b" " " # # 5 5 5 25 4 4 4 4 16 2 2 2 2 2 2È É É x xœ c b œ œÉ É5 5 25 20x 20x 25 4 4 16 16 4 2 20x 20x 252 2 c b c bÈ 14. y x 3 y 3 x; L x 4 y 0 y 3 4 y y 1 yœ c Ê b œ œ Ð c Ñ b Ð c Ñ œ Ð b c Ñ b œ Ð c Ñ bÈ È È È2 2 2 2 2 2 2 2 2 y 2y 1 y y y 1œ c b b œ c bÈ È4 2 2 4 2 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

10. 2 Chapter 1 Functions 15. The domain is . 16. The domain is .a b a bc_ß _ c_ß _ 17. The domain is . 18. The domain is .a bc_ß _ Ðc_ß !Ó 19. The domain is . 20. The domain is .a b a b a b a bc_ß ! r !ß _ c_ß ! r !ß _ 21. The domain is 5 5 3 3, 5 5, 22. The range is 2, 3 .a b a bc_ß c r Ðc ß c Ó r Ò Ñ r _ Ò Ñ 23. Neither graph passes the vertical line test (a) (b) Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

11. 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 b œ Í Í b œ " œ c b œ c" œ c" c 25. x 0 1 2 26. x 0 1 2 y 0 1 0 y 1 0 0 27. F x 28. G x 4 x , x 1 x 2x, x 1 , x 0 x, 0 x a b a bœ œœ œ c Ÿ b €  Ÿ 2 2 x " 29. (a) Line through and : y x; Line through and : y x 2a b a b a b a b!ß ! "ß " œ "ß " #ß ! œ c b f(x) x, 0 x 1 x 2, 1 x 2 œ Ÿ Ÿ c b  Ÿœ (b) f(x) 2, x x 2 x x œ ! Ÿ  " !ß " Ÿ  # ß # Ÿ  $ !ß $ Ÿ Ÿ % ÚÝÝ Û ÝÝ Ü 30. (a) Line through 2 and : y x 2a b a b!ß #ß ! œ c b Line through 2 and : m , so y x 2 xa b a b a bß " &ß ! œ œ œ c œ c c b " œ c b! c " c" " " " & & c # $ $ $ $ $ f(x) x , 0 x x , x œ c b #  Ÿ # c b #  Ÿ &œ " & $ $ (b) Line through and : m , so y xa b a bc"ß ! !ß c$ œ œ c$ œ c$ c $c$ c ! ! c Ðc"Ñ Line through and : m , so y xa b a b!ß $ #ß c" œ œ œ c# œ c# b $c" c $ c% # c ! # f(x) x , x x , x œ c$ c $ c"  Ÿ ! c# b $ !  Ÿ #œ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

12. 4 Chapter 1 Functions 31. (a) Line through and : y xa b a bc"ß " !ß ! œ c Line through and : ya b a b!ß " "ß " œ " Line through and : m , so y x xa b a b a b"ß " $ß ! œ œ œ c œ c c " b " œ c b!c" c" " " " $ $c" # # # # # f(x) x x x x x œ c c" Ÿ  ! " !  Ÿ " c b "   $ Ú Û Ü " $ # # (b) Line through 2 1 and 0 0 : y xa b a bc ß c ß œ 1 2 Line through 0 2 and 1 0 : y 2x 2a b a bß ß œ c b Line through 1 1 and 3 1 : y 1a b a bß c ß c œ c f x x 2 x 0 2x 2 0 x 1 1 1 x 3 a b Ú Û Ü œ c Ÿ Ÿ c b  Ÿ c  Ÿ 1 2 32. (a) Line through and T : m , so y x 0 xˆ ‰ ˆ ‰a bT T T T T T T# c Î# # "c! # # # ß ! ß " œ œ œ c b œ c "a b f x , 0 x x , x T a b Jœ ! Ÿ Ÿ c "  Ÿ T T T # # # (b) f x A, x A x T A T x A x T a b ÚÝÝÝ Û ÝÝÝ Ü œ ! Ÿ  c ß Ÿ  ß Ÿ  c ß Ÿ Ÿ # T T T T # # $ # $ # 33. (a) x 0 for x [0 1) (b) x 0 for x ( 1 0]Ú Û œ − ß Ü Ý œ − c ß 34. x x only when x is an integer.Ú Û œ Ü Ý 35. For any real number x, n x n , where n is an integer. Now: n x n n x n. ByŸ Ÿ b " Ÿ Ÿ b " Ê cÐ b "Ñ Ÿ c Ÿ c definition: x n and x n x n. So x x for all x .Üc Ý œ c Ú Û œ Ê cÚ Û œ c Üc Ý œ cÚ Û − d 36. To find f(x) you delete the decimal or fractional portion of x, leaving only the integer part. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

13. Section 1.1 Functions and Their Graphs 5 37. Symmetric about the origin 38. Symmetric about the y-axis Dec: x Dec: xc_   _ c_   ! Inc: nowhere Inc: x!   _ 39. Symmetric about the origin 40. Symmetric about the y-axis Dec: nowhere Dec: x!   _ Inc: x Inc: xc_   ! c_   ! x!   _ 41. Symmetric about the y-axis 42. No symmetry Dec: x Dec: xc_  Ÿ ! c_  Ÿ ! Inc: x Inc: nowhere!   _ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

14. 6 Chapter 1 Functions 43. Symmetric about the origin 44. No symmetry Dec: nowhere Dec: x! Ÿ  _ Inc: x Inc: nowherec_   _ 45. No symmetry 46. Symmetric about the y-axis Dec: x Dec: x! Ÿ  _ c_  Ÿ ! 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 x x and f x x f x . Thus the function is odd.a b a b a b a bˆ ‰œ œ c œ c œ œ c œ cc& " " "c& cx xx& && a b 49. Since f x x x f x . The function is even.a b a b a bœ b " œ c b " œ c# # 50. Since f x x x f x x x and f x x x f x x x the function is neither even norÒ œ b Ó Á Ò c œ c c Ó Ò œ b Ó Á Òc œ c c Óa b a b a b a b a b a b# ## # odd. 51. Since g x x x, g x x x x x g x . So the function is odd.a b a b a b a bœ b c œ c c œ c b œ c$ $ $ 52. g x x x x x g x thus the function is even.a b a b a b a bœ b $ c " œ c b $ c c " œ c ß% # % # 53. g x g x . Thus the function is even.a b a bœ œ œ c" " c " c c"x x# # a b 54. g x ; g x g x . So the function is odd.a b a b a bœ c œ c œ cx x x x# #c " c" 55. h t ; h t ; h t . Since h t h t and h t h t , the function is neither even nor odd.a b a b a b a b a b a b a bœ c œ c œ Á c Á c" " " c " c c " " ct t t 56. Since t | t |, h t h t and the function is even.l œ l c œ c$ $ a b a b a b Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

15. Section 1.1 Functions and Their Graphs 7 57. h t 2t , h t 2t . So h t h t . h t 2t , so h t h t . The function is neither even nora b a b a b a b a b a b a bœ b " c œ c b " Á c c œ c c " Á c odd. 58. h t 2 t and h t 2 t 2 t . So h t h t and the function is even.a b a b a b a bœ l l b " c œ l c l b " œ l l b " œ c 59. s kt 25 k 75 k s t; 60 t t 180œ Ê œ Ð Ñ Ê œ Ê œ œ Ê œ" " " 3 3 3 60. K c v 12960 c 18 c 40 K 40v ; K 40 10 4000 joulesœ Ê œ Ê œ Ê œ œ œ# # # a b a b2 61. r 6 k 24 r ; 10 sœ Ê œ Ê œ Ê œ œ Ê œk k 24 24 12 s 4 s s 5 62. P 14.7 k 14700 P ; 23.4 v 628.2 inœ Ê œ Ê œ Ê œ œ Ê œ ¸k k 14700 14700 24500 v 1000 v v 39 3 63. v f(x) x 2x 22 2x x 72x x; x 7œ œ Ð"% c ÑÐ c Ñ œ % c b $!) !   Þ$ # 64. (a) Let h height of the triangle. Since the triangle is isosceles, AB AB 2 AB 2 So,œ b œ Ê œ Þ# # # È h 2 h B is at slope of AB The equation of AB is# # # b " œ Ê œ " Ê !ß " Ê œ c" ÊŠ ‹È a b y f(x) x ; x .œ œ c b " − Ò!ß "Ó (b) A x 2x y 2x x 2x x; x .Ð Ñ œ œ Ðc b "Ñ œ c b # − Ò!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 b!ß " (c) Graph h because it is a nonlinear odd function. 67. (a) From the graph, 1 x ( 2 0) ( )x 4 x# € b Ê − c ß r %ß _ (b) 1 1 0x 4 x 4 x x# #€ b Ê c c € x 0: 1 0 0 0€ c c € Ê € Ê €x 4 x 2x 8 x 2x x (x 4)(x 2) # # c c c b# x 4 since x is positive;Ê € x 0: 1 0 0 0 c c € Ê  Ê x 4 x 2x 8 2 x 2x x (x 4)(x 2)# c c c b # x 2 since x is negative;Ê  c sign of (x 4)(x 2)c b 2 ïïïïïðïïïïïðïïïïîb b c c % Solution interval: ( 0) ( )c#ß r %ß _ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

16. 8 Chapter 1 Functions 68. (a) From the graph, x ( 5) ( 1 1)3 2 x 1 x 1c b Ê − c_ß c r c ß (b) x 1: 2Case  c  Ê €3 2 x 1 x 1 x 1 3(x 1) c b c b 3x 3 2x 2 x 5.Ê b  c Ê  c Thus, x ( 5) solves the inequality.− c_ß c 1 x 1: 2Case c    Ê 3 2 x 1 x 1 x 1 3(x 1) c b c b 3x 3 2x 2 x 5 which is trueÊ b € c Ê € c if x 1. Thus, x ( 1 1) solves the€ c − c ß inequality. 1 x: 3x 3 2x 2 x 5Case   Ê b  c Ê  c3 2 x 1 x 1c b which is never true if 1 x, so no solution here. In conclusion, x ( 5) ( 1 1).− c_ß c r c ß 69. A curve symmetric about the x-axis will not pass the vertical line test because the points x, y and x, y lie on the sama b a bc e vertical line. The graph of the function y f x is the x-axis, a horizontal line for which there is a single y-value, ,œ œ ! !a b for any x. 70. price 40 5x, quantity 300 25x R x 40 5x 300 25xœ b œ c Ê œ b ca b a ba b 71. x x h x ; cost 5 2x 10h C h 10 10h 5h 2 22 2 2 h 2 2 h 2 h 2 2b œ Ê œ œ œ b Ê œ b œ bÈ È È a b a b Š ‹ Š ‹È 72. (a) Note that 2 mi = 10,560 ft, so there are 800 x feet of river cable at $180 per foot and 10,560 x feet of landÈ a b# #b c cable at $100 per foot. The cost is C x 180 800 x 100 10,560 x .a b a bÈœ b b c# # (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 : x , D : x 1 D D : x 1. R : y , R : y 0, R : y 1, R : y 0f g f g fg f g f g fgc_   _   Ê œ   c_   _      b b 2. D : x 1 0 x 1, D : x 1 0 x 1. Therefore D D : x 1.f g f g fgb   Ê   c c   Ê   œ  b R R : y 0, R : y 2, R : y 0f g f g fgœ      b È 3. D : x , D : x , D : x , D : x , R : y 2, R : y 1,f g f g g f f gc_   _ c_   _ c_   _ c_   _ œ  Î Î R : 0 y 2, R : yf gÎ  Ÿ Ÿ  _g fÎ " # 4. D : x , D : x 0 , D : x 0, D : x 0; R : y 1, R : y 1, R : 0 y 1, R : 1 yf g f g g f f g f gc_   _       œ    Ÿ Ÿ  _Î Î Î g fÎ 5. (a) 2 (b) 22 (c) x 2# b (d) (x 5) 3 x 10x 22 (e) 5 (f) 2b c œ b b c# # (g) x 10 (h) (x 3) 3 x 6x 6b c c œ c b# # % # Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

17. Section 1.2 Combining Functions; Shifting and Scaling Graphs 9 6. (a) (b) 2 (c) 1c c œ" " c b b3 x 1 x 1 x (d) (e) 0 (f)" x 4 3 (g) x 2 (h)c œ œ" " b " b b #" b# b bx 1 x 1 x 1 x x 7. f g h x f g h x f g 4 x f 3 4 x f 12 3x 12 3x 1 13 3xa ba b a b a b a b a b a ba b a b a ba b‰ ‰ œ œ c œ c œ c œ c b œ c 8. f g h x f g h x f g x f 2 x 1 f 2x 1 3 2x 1 4 6x 1a ba b a b a b a b a b a ba b a b a ba b‰ ‰ œ œ œ c œ c œ c b œ b2 2 2 2 2 9. f g h x f g h x f g f fa ba b a b Éa ba b ˆ ‰ ˆ ‰ˆ ‰ Š ‹ ɉ ‰ œ œ œ œ œ b " œ1 1 x x 5x x 1 4x 1 4x 1 4x1 x b % b b b b " 10. f g h x f g h x f g 2 x f fa ba b a ba ba b Š ‹Š ‹È : ; ˆ ‰‰ ‰ œ œ c œ œ œ œ Š ‹È Š ‹È 2 x 2 x 1 2 x 8 3x x 7 2x 2 3 c c b c c $ c c b c 2 2 2 x 2 x x x c $c c $c 11. (a) f g x (b) j g x (c) g g xa ba b a ba b a ba b‰ ‰ ‰ (d) j j x (e) g h f x (f) h j f xa ba b a ba b a ba b‰ ‰ ‰ ‰ ‰ 12. (a) f j x (b) g h x (c) h h xa ba b a ba b a ba b‰ ‰ ‰ (d) f f x (e) j g f x (f) g f h xa ba b a ba b a ba b‰ ‰ ‰ ‰ ‰ 13. g(x) f(x) (f g)(x)‰ (a) x 7 x x 7c cÈ È (b) x 2 3x 3(x 2) 3x 6b b œ b (c) x x 5 x 5# #È Èc c (d) xx x x x 1 x 1 1 x (x 1)c c c c c x x 1 x x 1 c c œ œ (e) 1 x" " cx 1 xb (f) x" " x x 14. (a) f g x g x .a ba b a b‰ œ l l œ " l c "lx (b) f g x so g x x .a ba b a b‰ œ œ Ê " c œ Ê " c œ Ê œ ß œ b "g x g x x g x x x g x x g x x x xa b a b a b a b a b c" b " b " b " b " " " " " (c) Since f g x g x x , g x x .a ba b a b a bȉ œ œ l l œ # (d) Since f g x f x x , f x x . (Note that the domain of the composite is .)a ba b a bˆ ‰È‰ œ œ l l œ 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 b " l l l l " " c " l c "l c " b " # # x x x x x x 15. (a) f g 1 f 1 1 (b) g f 0 g 2 2 (c) f f 1 f 0 2a b a b a b a b a b a ba b a b a bc œ œ œ c œ c œ œ c (d) g g 2 g 0 0 (e) g f 2 g 1 1 (f) f g 1 f 1 0a b a b a b a b a b a ba b a b a bœ œ c œ œ c œ c œ 16. (a) f g 0 f 1 2 1 3, where g 0 0 1 1a b a b a b a ba b œ c œ c c œ œ c œ c (b) g f 3 g 1 1 1, where f 3 2 3 1a b a b a b a ba b œ c œ c c œ œ c œ c (c) g g 1 g 1 1 1 0, where g 1 1 1a b a b a b a ba bc œ œ c œ c œ c c œ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

18. 10 Chapter 1 Functions (d) f f 2 f 0 2 0 2, where f 2 2 2 0a b a b a ba b œ œ c œ œ c œ (e) g f 0 g 2 2 1 1, where f 0 2 0 2a b a b a ba b œ œ c œ œ c œ (f) f g f 2 , where g 1ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ˆ ‰" " " " " " # # # # # # #œ c œ c c œ œ c œ c5 17. (a) f g x f g x 1a ba b a ba b É É‰ œ œ b œ1 1 x x x b g f x g f xa ba b a ba b‰ œ œ 1 x 1È b (b) Domain f g : , 1 0, , domain g f : 1,a b a b‰ Ðc_ c Ó r Ð _Ñ ‰ Ðc _Ñ (c) Range f g : 1, , range g f : 0,a b a b‰ Ð _Ñ ‰ Ð _Ñ 18. (a) f g x f g x 1 2 x xa ba b a ba b ȉ œ œ c b g f x g f x 1 xa ba b a b k ka b‰ œ œ c (b) Domain f g : 0, , domain g f : ,a b a b‰ Ò _Ñ ‰ Ðc_ _Ñ (c) Range f g : 0, , range g f : , 1a b a b‰ Ð _Ñ ‰ Ðc_ Ó 19. f g x x f g x x x g x g x 2 x x g x 2xa ba b a b a b a b a ba b a b‰ œ Ê œ Ê œ Ê œ c œ † cg x g x 2 a b a b c g x x g x 2x g xÊ c † œ c Ê œ c œa b a b a b 2x 2x 1 x x 1c c 20. f g x x 2 f g x x 2 2 g x 4 x 2 g x g xa ba b a b a b a b a ba b a b a b ɉ œ b Ê œ b Ê c œ b Ê œ Ê œ3 3 x 6 x 6 2 2 b b3 21. (a) y (x 7) (b) y (x 4)œ c b œ c c# # 22. (a) y x 3 (b) y x 5œ b œ c# # 23. (a) Position 4 (b) Position 1 (c) Position 2 (d) Position 3 24. (a) y (x 1) 4 (b) y (x 2) 3 (c) y (x 4) 1 (d) y (x 2)œ c c b œ c b b œ c b c œ c c# # # # 25. 26. 27. 28. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

19. Section 1.2 Combining Functions; Shifting and Scaling Graphs 11 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

20. 12 Chapter 1 Functions 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

21. Section 1.2 Combining Functions; Shifting and Scaling Graphs 13 53. 54. 55. (a) domain: [0 2]; range: [ ] (b) domain: [0 2]; range: [ 1 0]ß #ß $ ß c ß (c) domain: [0 2]; range: [0 2] (d) domain: [0 2]; range: [ 1 0]ß ß ß c ß (e) domain: [ 2 0]; range: [ 1] (f) domain: [1 3]; range: [ ]c ß !ß ß !ß " (g) domain: [ 2 0]; range: [ ] (h) domain: [ 1 1]; range: [ ]c ß !ß " c ß !ß " Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

22. 14 Chapter 1 Functions 56. (a) domain: [0 4]; range: [ 3 0] (b) domain: [ 4 0]; range: [ ]ß c ß c ß !ß $ (c) domain: [ 4 0]; range: [ ] (d) domain: [ 4 0]; range: [ ]c ß !ß $ c ß "ß % (e) domain: [ 4]; range: [ 3 0] (f) domain: [ 2 2]; range: [ 3 0]#ß c ß c ß c ß (g) domain: [ 5]; range: [ 3 0] (h) domain: [0 4]; range: [0 3]"ß c ß ß ß 57. y 3x 3 58. y 2x 1 x 1œ c œ c œ % c# ## a b 59. y 60. y 1 1œ " b œ b œ b œ b" " " " " * # # # Î$ ˆ ‰x x xx# # ## a b 61. y x 1 62. y 3 x 1œ % b œ bÈ È 63. y 16 x 64. y xœ % c œ c œ % cÉ ˆ ‰ È Èx # # $ # " "# # 65. y 3x 27x 66. yœ " c œ " c œ " c œ " ca b ˆ ‰$ $ # ) $x x$ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

23. Section 1.2 Combining Functions; Shifting and Scaling Graphs 15 67. Let y x f x and let g x x ,œ c # b " œ œÈ a b a b "Î# h x x , i x x , anda b a bˆ ‰ ˆ ‰Èœ b œ # b" " # # "Î# "Î# j x x f . The graph ofa b a b’ “È ˆ ‰œ c # b œ B" # "Î# h x is the graph of g x shifted left unit; thea b a b " # graph of i x is the graph of h x stretcheda b a b vertically by a factor of ; and the graph ofÈ# j x f x is the graph of i x reflected acrossa b a b a bœ the x-axis. 68. Let y f x Let g x x ,œ " c œ Þ œ cÈ a b a b a bx # "Î# h x x , and i x xa b a b a b a bœ c b # œ c b #"Î# "Î#" #È f x The graph of g x is theœ " c œ ÞÈ a b a bx # graph of y x reflected across the x-axis.œ È The graph of h x is the graph of g x shifteda b a b right two units. And the graph of i x is thea b graph of h x compressed vertically by a factora b of .È# 69. y f x x . Shift f x one unit right followed by aœ œa b a b$ shift two units up to get g x x .a b a bœ c " b #3 70. y x f x .œ " c B b # œ cÒ c " b c# Ó œa b a b a b a b$ $ Let g x x , h x x , i x x ,a b a b a b a b a b a bœ œ c " œ c " b c#$ $ $ and j x x . The graph of h x is thea b a b a b a bœ cÒ c " b c# Ó$ graph of g x shifted right one unit; the graph of i x isa b a b the graph of h x shifted down two units; and the grapha b of f x is the graph of i x reflected across the x-axis.a b a b 71. Compress the graph of f x horizontally by a factora b œ " x of 2 to get g x . Then shift g x vertically down 1a b a bœ " #x unit to get h x .a b œ c "" #x Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

24. 16 Chapter 1 Functions 72. Let f x and g xa b a bœ œ b " œ b "" # " x x# # B# #Š ‹ Sinceœ b " œ b "Þ" " Î # "Î # BŠ ‹ ’Š ‹ “È Èx # # , we see that the graph of f x stretchedÈ a b# ¸ "Þ% horizontally by a factor of 1.4 and shifted up 1 unit is the graph of g x .a b 73. Reflect the graph of y f x x across the x-axisœ œa b È$ to get g x x.a b Èœ c $ 74. y f x x xœ œ c# œ Ò c" # Óa b a b a ba b#Î$ #Î$ x x . So the graphœ c" # œ #a b a b a b#Î$ #Î$ #Î$ of f x is the graph of g x x compresseda b a b œ #Î$ horizontally by a factor of 2. 75. 76. 77. x y 78. x y* b #& œ ##& Ê b œ " "' b ( œ ""# Ê b œ "# # # # & $ % ( x xy y# # # # # # # # Š ‹È Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

25. Section 1.2 Combining Functions; Shifting and Scaling Graphs 17 79. x y 80. x y$ b c # œ $ Ê b œ " b " b # œ % Ê b œ "# ## c # # " # $ # c c" a b a bx y x y# # # # # # # #a b Š ‹ Š ‹È È < ‘a b 81. x y 82. x y$ c " b # b # œ ' ' b b * c œ &%a b a b ˆ ‰ ˆ ‰# # $ " # # # # Ê b œ " Ê b œ "a b Š ‹ Š ‹ Š ‹È È < ‘a b ’ “ˆ ‰ ˆ ‰ È x y x yc " # $ ' c c# c c $ c# # # # # $ # # # " # # 83. has its center at . Shiftinig 4 unitsx y# # "' *b œ " !ß !a b left and 3 units up gives the center at h, k .a b a bœ c%ß $ So the equation is < ‘a b a bx 4 4 3 y 3c c c # # # # b œ " . Center, C, is , andÊ b œ " c%ß $a b a bx y 4 3 b % c $# # # # a b major axis, AB, is the segment from to .a b a bc)ß $ !ß $ 84. The ellipse has center h, k .x y# # % #&b œ " œ !ß !a b a b Shifting the ellipse 3 units right and 2 units down produces an ellipse with center at h, ka b a bœ $ß c# and an equation . Center,a b < ‘a bx 3 yc % #& c c## # b œ " C, is 3 , and AB, the segment from toa b a bß c# $ß $ is the major axis.a b$ß c( 85. (a) (fg)( x) f( x)g( x) f(x)( g(x)) (fg)(x), oddc œ c c œ c œ c (b) ( x) (x), oddŠ ‹ Š ‹f f g g( x) g(x) g f( x) f(x) c œ œ œ cc c c Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

26. 18 Chapter 1 Functions (c) ( x) (x), oddˆ ‰ ˆ ‰g g( x) g(x) g f f( x) f(x) fc œ œ œ cc c c (d) f ( x) f( x)f( x) f(x)f(x) f (x), even# # c œ c c œ œ (e) g ( x) (g( x)) ( g(x)) g (x), even# # # # c œ c œ c œ (f) (f g)( x) f(g( x)) f( g(x)) f(g(x)) (f g)(x), even‰ c œ c œ c œ œ ‰ (g) (g f)( x) g(f( x)) g(f(x)) (g f)(x), even‰ c œ c œ œ ‰ (h) (f f)( x) f(f( x)) f(f(x)) (f f)(x), even‰ c œ c œ œ ‰ (i) (g g)( x) g(g( x)) g( g(x)) g(g(x)) (g g)(x), odd‰ c œ c œ c œ c œ c ‰ 86. Yes, f(x) 0 is both even and odd since f( x) 0 f(x) and f( x) 0 f(x).œ c œ œ c œ œ c 87. (a) (b) (c) (d) 88. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

27. Section 1.3 Trigonometric Functions 19 1.3 TRIGONOMETRIC FUNCTIONS 1. (a) s r (10) 8 m (b) s r (10)(110°) mœ œ œ œ œ œ œ) 1 )ˆ ‰ ˆ ‰4 110 55 5 180° 18 9 1 1 1 1 2. radians and 225°) œ œ œ œs 10 5 5 180° r 8 4 4 1 1 1 1 ˆ ‰ 3. 80° 80° s (6) 8.4 in. (since the diameter 12 in. radius 6 in.)) )œ Ê œ œ Ê œ œ œ Ê œˆ ‰ ˆ ‰1 1 1 180° 9 9 4 4 4. d 1 meter r 50 cm 0.6 rad or 0.6 34°œ Ê œ Ê œ œ œ ¸) s 30 180° r 50 ˆ ‰1 5. 0 6 sin 0 0 cos 1 0 tan 0 3 0 und. cot und. und. 0 1 sec 1 und. 2 csc und. und. 2 ) 1 ) ) ) ) ) ) c c c " c c " c c" c c c# " c c " 2 3 3 4 3 2 2 3 2 3 1 1 1 # # " " " # " È È È È È È È È . sin cos tan und. 3 cot 3 3 sec und. 2 csc ) ) ) ) ) ) ) c c c " c c ! c c c " c ! c c " c # c " c c# 3 3 3 2 3 3 2 3 3 3 2 2 3 3 2 3 1 1 1 1 1 # ' % ' & # # # " " " " " # # # " " " È È È È È È È È È È È È È È È È2 # 7. cos x , tan x 8. sin x , cos xœ c œ c œ œ4 3 2 5 4 5 5È È " 9. sin x , tan x 8 10. sin x , tan xœ c œ c œ œ c È8 3 13 5 12 12È 11. sin x , cos x 12. cos x , tan xœ c œ c œ c œ" " #È È È È 5 5 3 2 3 13. 14. period period 4œ œ1 1 15. 16. period 2 period 4œ œ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

28. 20 Chapter 1 Functions 17. 18. period 6 period 1œ œ 19. 20. period 2 period 2œ œ1 1 21. 22. period 2 period 2œ œ1 1 23. period , symmetric about the origin 24. period 1, symmetric about the originœ œ1 # 25. period 4, symmetric about the s-axis 26. period 4 , symmetric about the originœ œ 1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

29. Section 1.3 Trigonometric Functions 21 27. (a) Cos x and sec x are positive for x in the interval , ; and cos x and sec x are negative for x in theˆ ‰c1 1 2 2 intervals , and , . Sec x is undefinedˆ ‰ ˆ ‰c c3 3 2 2 2 2 1 1 1 1 when cos x is 0. The range of sec x is ( 1] [ ); the range of cos x is [ 1].c_ß c r "ß _ c"ß (b) Sin x and csc x are positive for x in the intervals , and , ; and sin x and csc x are negativeˆ ‰ a bc c !3 2 1 1 1 for x in the intervals , and , . Csc x isa b ˆ ‰c !1 1 3 2 1 undefined when sin x is 0. The range of csc x is ( 1] [1 ); the range of sin x is [ ].c_ß c r ß _ c"ß " 28. Since cot x , cot x is undefined when tan x 0œ œ" tan x and is zero when tan x is undefined. As tan x approaches zero through positive values, cot x approaches infinity. Also, cot x approaches negative infinity as tan x approaches zero through negative values. 29. D: x ; R: y 1, 0, 1 30. D: x ; R: y 1, 0, 1c_   _ œ c c_   _ œ c 31. cos x cos x cos sin x sin (cos x)(0) (sin x)( 1) sin xˆ ‰ ˆ ‰ ˆ ‰c œ c c c œ c c œ1 1 1 # # # 32. cos x cos x cos sin x sin (cos x)(0) (sin x)(1) sin xˆ ‰ ˆ ‰ ˆ ‰b œ c œ c œ c1 1 1 # # # 33. sin x sin x cos cos x sin (sin x)(0) (cos x)(1) cos xˆ ‰ ˆ ‰ ˆ ‰b œ b œ b œ1 1 1 # # # 34. sin x sin x cos cos x sin (sin x)(0) (cos x)( 1) cos xˆ ‰ ˆ ‰ ˆ ‰c œ c b c œ b c œ c1 1 1 # # # 35. cos (A B) cos (A ( B)) cos A cos ( B) sin A sin ( B) cos A cos B sin A ( sin B)c œ b c œ c c c œ c c cos A cos B sin A sin Bœ b 36. sin (A B) sin (A ( B)) sin A cos ( B) cos A sin ( B) sin A cos B cos A ( sin B)c œ b c œ c b c œ b c sin A cos B cos A sin Bœ c 37. If B A, A B 0 cos (A B) cos 0 1. Also cos (A B) cos (A A) cos A cos A sin A sin Aœ c œ Ê c œ œ c œ c œ b cos A sin A. Therefore, cos A sin A 1.œ b b œ# # # # Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

30. 22 Chapter 1 Functions 38. If B 2 , then cos (A 2 ) cos A cos 2 sin A sin 2 (cos A)(1) (sin A)(0) cos A andœ b œ c œ c œ1 1 1 1 sin (A 2 ) sin A cos 2 cos A sin 2 (sin A)(1) (cos A)(0) sin A. The result agrees with theb œ b œ b œ1 1 1 fact that the cosine and sine functions have period 2 .1 39. cos ( x) cos cos sin sin x ( 1)(cos x) (0)(sin x) cos x1 1 1b œ B c œ c c œ c 40. sin (2 x) sin 2 cos ( x) cos (2 ) sin ( x) (0)(cos ( x)) (1)(sin ( x)) sin x1 1 1c œ c b c œ c b c œ c 41. sin x sin cos ( x) cos sin ( x) ( 1)(cos x) (0)(sin ( x)) cos xˆ ‰ ˆ ‰ ˆ ‰3 3 31 1 1 # # #c œ c b c œ c b c œ c 42. cos x cos cos x sin sin x (0)(cos x) ( 1)(sin x) sin xˆ ‰ ˆ ‰ ˆ ‰3 3 31 1 1 # # #b œ c œ c c œ 43. sin sin sin cos cos sin7 1 4 3 4 3 4 3 4 2 2 3 6 21 1 1 1 1 1 1 # # # # # " b œ b œ b œ b œˆ ‰ ˆ ‰Š ‹ Š ‹ Š ‹ È È È È È 44. cos cos cos cos sin sin11 2 2 2 1 4 3 4 3 4 3 4 2 2 3 2 61 1 1 1 1 1 1 # # # # # " b œ b œ c œ c c œ cˆ ‰ ˆ ‰Š ‹ Š ‹ Š ‹ È È È ÈÈ 45. cos cos cos cos sin sin1 1 1 1 1 1 1 12 3 4 3 4 3 4 2 23 1 3 2 2 œ c œ c c c œ c c œˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ Š ‹ Š ‹ Š ‹" # # # # bÈ È È È È 46. sin sin sin cos cos sin5 2 2 2 1 3 4 3 4 3 4 3 1 32 2 2 2 1 1 1 1 1 1 1 # # # # # " b œ c œ c b c œ b c c œˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰Š ‹ Š ‹ Š ‹ È È È È È 47. cos 48. cos# #b b b # # # # # b b c c1 1 8 4 1 4 1 cos 1 1 cos2 2 5 1 2 3 œ œ œ œ œ œ ˆ ‰ ˆ ‰È ÈŠ ‹2 10 8 1 2 3 1 1È È # # # 49. sin 50. sin# # # # # # # c c cc c c b1 1 1 4 8 4 1 cos 1 1 cos2 3 3 1 2 2 œ œ œ œ œ œ ˆ ‰ ˆ ‰È ÈŠ ‹2 6 1 8 3 2 1 1 # # # È È 51. sin sin , , ,2 3 2 4 5 4 2 3 3 3 3 3 ) ) )œ Ê œ „ Ê œ È 1 1 1 1 52. sin cos tan 1 tan 1 , , ,2 2 2sin cos 3 5 7 cos cos 4 4 4 4) ) ) ) )œ Ê œ Ê œ Ê œ „ Ê œ 2 2 2 2 ) ) 1 1 1 1 ) ) 53. sin 2 cos 0 2sin cos cos 0 cos 2sin 1 0 cos 0 or 2sin 1 0 cos 0 or) ) ) ) ) ) ) ) ) )c œ Ê c œ Ê c œ Ê œ c œ Ê œa b sin , , or , , , ,) ) ) )œ Ê œ œ Ê œ" # 1 1 1 1 1 1 1 1 2 2 6 6 6 2 6 2 3 5 5 3 54. cos 2 cos 0 2cos 1 cos 0 2cos cos 1 0 cos 1 2cos 1 0) ) ) ) ) ) ) )b œ Ê c b œ Ê b c œ Ê b c œ2 2 a ba b cos 1 0 or 2cos 1 0 cos 1 or cos or , , ,Ê b œ c œ Ê œ c œ Ê œ œ Ê œ) ) ) ) ) 1 ) ) 1" # 1 1 1 1 3 3 3 3 5 5 55. tan (A B)b œ œ œsin (A B) cos (A B) cos A cos B sin A sin B sin A cos B cos A cos Bb b c b bsin A cos B cos A sin B cos A cos B cos A cos B cos A cos B sin A sin B cos A cos B cos A cos Bc b cœ tan A tan B 1 tan A tan B 56. tan (A B)c œ œ œsin (A B) cos (A B) cos A cos B sin A sin B sin A cos B cos A cos Bc c b c csin A cos B cos A sin B cos A cos B cos A cos B cos A cos B sin A sin B cos A cos B cos A cos Bb c bœ tan A tan B 1 tan A tan B 57. According to the figure in the text, we have the following: By the law of cosines, c a b 2ab cos# # # œ b c ) 1 1 2 cos (A B) 2 2 cos (A B). By distance formula, c (cos A cos B) (sin A sin B)œ b c c œ c c œ c b c# # # # # cos A 2 cos A cos B cos B sin A 2 sin A sin B sin B 2 2(cos A cos B sin A sin B). Thusœ c b b c b œ c b# # # # c 2 2 cos (A B) 2 2(cos A cos B sin A sin B) cos (A B) cos A cos B sin A sin B.# œ c c œ c b Ê c œ b Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

31. Section 1.3 Trigonometric Functions 23 58. (a) cos A B cos A cos B sin A sin Ba bc œ b sin cos and cos sin) ) ) )œ c œ cˆ ‰ ˆ ‰1 1 # # Let A B) œ b sin A B cos A B cos A B cos A cos B sin A sin Ba b a b’ “ ’ “ˆ ‰ ˆ ‰ ˆ ‰b œ c b œ c c œ c b c1 1 1 1 # # # # sin A cos B cos A sin Bœ b (b) cos A B cos A cos B sin A sin Ba bc œ b cos A B cos A cos B sin A sin Ba b a b a ba bc c œ c b c cos A B cos A cos B sin A sin B cos A cos B sin A sin BÊ b œ c b c œ b ca b a b a b a b cos A cos B sin A sin Bœ c Because the cosine function is even and the sine functions is odd. 59. c a b 2ab cos C 2 3 2(2)(3) cos (60°) 4 9 12 cos (60°) 13 12 7.# # # # # " #œ b c œ b c œ b c œ c œˆ ‰ Thus, c 7 2.65.œ ¸È 60. c a b 2ab cos C 2 3 2(2)(3) cos (40°) 13 12 cos (40°). Thus, c 13 12 cos 40° 1.951.# # # # # œ b c œ b c œ c œ c ¸È 61. From the figures in the text, we see that sin B . If C is an acute angle, then sin C . On the other hand,œ œh h c b if C is obtuse (as in the figure on the right), then sin C sin ( C) . Thus, in either case,œ c œ1 h b h b sin C c sin B ah ab sin C ac sin B.œ œ Ê œ œ By the law of cosines, cos C and cos B . Moreover, since the sum of theœ œa b c a c b 2ab 2ac # # # # # # b c b c interior angles of a triangle is , we have sin A sin ( (B C)) sin (B C) sin B cos C cos B sin C1 1œ c b œ b œ b 2a b c c b ah bc sin A.œ b œ b c b c œ Ê œˆ ‰ ˆ ‰ ˆ ‰’ “ ’ “ a bh a b c a c b h h ah c 2ab 2ac b 2abc bc # # # # # # b c b c # # # # # Combining our results we have ah ab sin C, ah ac sin B, and ah bc sin A. Dividing by abc givesœ œ œ .h sin A sin C sin B bc a c bœ œ œðóóóóóóóñóóóóóóóò law of sines 62. By the law of sines, . By Exercise 61 we know that c 7. Thus sin B 0.982.sin A sin B 3 c 3/2 3 3 2 7# œ œ œ œ ¶ È È È È 63. From the figure at the right and the law of cosines, b a 2 2(2a) cos B# # # œ b c a 4 4a a 2a 4.œ b c œ c b# #" # ˆ ‰ Applying the law of sines to the figure, sin A sin B a bœ b a. Thus, combining results,Ê œ Ê œ È È2/2 a b 3/2 3 É# a 2a 4 b a 0 a 2a 4# # # # # # " c b œ œ Ê œ b c3 0 a 4a 8. From the quadratic formula and the fact that a 0, we haveÊ œ b c €# a 1.464.œ œ ¶ c b c c # # c4 4 4(1)( 8) 4 3 4È È# 64. (a) The graphs of y sin x and y x nearly coincide when x is near the origin (when the calculatorœ œ is in radians mode). (b) In degree mode, when x is near zero degrees the sine of x is much closer to zero than x itself. The curves look like intersecting straight lines near the origin when the calculator is in degree mode. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

32. 24 Chapter 1 Functions 65. A 2, B 2 , C , D 1œ œ œ c œ c1 1 66. A , B 2, C 1, Dœ œ œ œ" " # # 67. A , B 4, C 0, Dœ c œ œ œ2 1 1 " 68. A , B L, C 0, D 0œ œ œ œL 21 69-72. Example CAS commands: Maple f := x -> A*sin((2*Pi/B)*(x-C))+D1; A:=3; C:=0; D1:=0; f_list := [seq( f(x), B=[1,3,2*Pi,5*Pi] )]; plot( f_list, x=-4*Pi..4*Pi, scaling=constrained, color=[red,blue,green,cyan], linestyle=[1,3,4,7], legend=["B=1","B=3","B=2*Pi","B=3*Pi"], title="#69 (Section 1.3)" ); Mathematica Clear[a, b, c, d, f, x] f[x_]:=a Sin[2 /b (x c)] + d1 c Plot[f[x]/.{a 3, b 1, c 0, d 0}, {x, 4 , 4 }]Ä Ä Ä Ä c 1 1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

33. Section 1.3 Trigonometric Functions 25 69. (a) The graph stretches horizontally. (b) The period remains the same: period B . The graph has a horizontal shift of period.œ l l " # 70. (a) The graph is shifted right C units. (b) The graph is shifted left C units. (c) A shift of one period will produce no apparent shift. C„ l l œ ' 71. (a) The graph shifts upwards D units for Dl l € ! (b) The graph shifts down D units for Dl l  !Þ 72. (a) The graph stretches A units. (b) For A , the graph is inverted.l l  ! Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

34. 26 Chapter 1 Functions 1.4 GRAPHING WITH CALCULATORS AND COMPUTERS 1-4. The most appropriate viewing window displays the maxima, minima, intercepts, and end behavior of the graphs and has little unused space. 1. d. 2. c. 3. d. 4. b. 5-30. For any display there are many appropriate display widows. The graphs given as answers in Exercises 5 30c are not unique in appearance. 5. 2 5 by 15 40 6. 4 4 by 4 4Òc ß Ó Òc ß Ó Òc ß Ó Òc ß Ó 7. 2 6 by 250 50 8. 1 5 by 5 30Òc ß Ó Òc ß Ó Òc ß Ó Òc ß Ó Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

35. Section 1.4 Graphing with Calculators and Computers 27 9. 4 4 by 5 5 10. 2 2 by 2 8Òc ß Ó Òc ß Ó Òc ß Ó Òc ß Ó 11. 2 6 by 5 4 12. 4 4 by 8 8Òc ß Ó Òc ß Ó Òc ß Ó Òc ß Ó 13. 1 6 by 1 4 14. 1 6 by 1 5Òc ß Ó Òc ß Ó Òc ß Ó Òc ß Ó 15. 3 3 by 0 10 16. 1 2 by 0 1Òc ß Ó Ò ß Ó Òc ß Ó Ò ß Ó Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

36. 28 Chapter 1 Functions 17. 5 1 by 5 5 18. 5 1 by 2 4Òc ß Ó Òc ß Ó Òc ß Ó Òc ß Ó 19. 4 4 by 0 3 20. 5 5 by 2 2Òc ß Ó Ò ß Ó Òc ß Ó Òc ß Ó 21. by 22. byÒc"!ß "!Ó Òc'ß 'Ó Òc&ß &Ó Òc#ß #Ó 23. by 24. byÒc'ß "!Ó Òc'ß 'Ó Òc$ß &Ó Òc#ß "!Ó Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

37. Section 1.4 Graphing with Calculators and Computers 29 25. 0 03 0 03 by 1 25 1 25 26. 0 1 0 1 by 3 3Òc Þ ß Þ Ó Òc Þ ß Þ Ó Òc Þ ß Þ Ó Òc ß Ó 27. 300 300 by 1 25 1 25 28. 50 50 by 0 1 0 1Òc ß Ó Òc Þ ß Þ Ó Òc ß Ó Òc Þ ß Þ Ó 29. 0 25 0 25 by 0 3 0 3 30. 0 15 0 15 by 0 02 0 05Òc Þ ß Þ Ó Òc Þ ß Þ Ó Òc Þ ß Þ Ó Òc Þ ß Þ Ó 31. x x y y y x x .# # #b # œ % b % c Ê œ # „ c c # b )È The lower half is produced by graphing y x x .œ # c c c # b )È # 32. y x y x . The upper branch# # #c "' œ " Ê œ „ " b "'È is produced by graphing y x .œ " b "'È # Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

38. 30 Chapter 1 Functions 33. 34. 35. 36. 37. 38Þ 39. 40. CHAPTER 1 PRACTICE EXERCISES 1. The area is A r and the circumference is C r. Thus, r A .œ œ # œ Ê œ œ1 1 1# # # % #C C C 1 1 1 ˆ ‰ # 2. The surface area is S r r . The volume is V r r . Substitution into the formula forœ % Ê œ œ Ê œ1 1# $ % $ % "Î# % $ˆ ‰ ÉS V 1 1 $ surface area gives S r .œ % œ %1 1# $ % #Î$ ˆ ‰V 1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

39. Chapter 1 Practice Exercises 31 3. The coordinates of a point on the parabola are x x . The angle of inclination joining this point to the origin satisfiesa bß # ) the equation tan x. Thus the point has coordinates x x tan tan .) ) )œ œ ß œ ßx x # a b a b# # 4. tan h tan ft.) )œ œ Ê œ &!!rise h run &!! 5. 6. Symmetric about the origin. Symmetric about the y-axis. 7. 8. Neither Symmetric about the y-axis. 9. y x x x y x . Even.a b a b a bc œ c b " œ b " œ# # 10. y x x x x x x x y x . Odd.a b a b a b a b a bc œ c c c c c œ c b b œ c& $ & $ 11. y x cos x cos x y x . Even.a b a b a bc œ " c c œ " c œ 12. y x sec x tan x sec x tan x y x . Odd.a b a b a b a bc œ c c œ œ œ c œ csin x cos x cos x sin xa b a b c c c # # 13. y x y x . Odd.a b a bc œ œ œ c œ ca b a b a b c b" c c# c b" b" c b# c# x x x x x x x x x % $ % % $ $ 14. y x x sin x x sin x x sin x y x . Odd.a b a b a b a b a b a bc œ c c c œ c b œ c c œ c 15. y x x cos x x cos x. Neither even nor odd.a b a bc œ c b c œ c b 16. y x x cos x x cos x y x . Odd.a b a b a b a bc œ c c œ c œ c 17. Since f and g are odd f x f x and g x g x .Ê c œ c c œ ca b a b a b a b (a) f g x f x g x f x g x f x g x f g x f g is evena ba b a b a b a b a b a b a b a ba b† c œ c c œ Òc ÓÒc Ó œ œ † Ê † (b) f x f x f x f x f x f x f x f x f x f x f x f is odd.3 3 3 a b a b a b a b a b a b a b a b a b a b a bc œ c c c œ Òc ÓÒc ÓÒc Ó œ c † † œ c Ê (c) f sin x f sin x f sin x f sin x is odd.a b a b a b a ba b a b a b a bc œ c œ c Ê (d) g sec x g sec x g sec x is even.a b a b a ba b a b a bc œ Ê (e) g x g x g x g is evenl c l œ lc l œ l l Ê l l Þa b a b a b Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

40. 32 Chapter 1 Functions 18. Let f a x f a x and define g x f x a . Then g x f x a f a x f a x f x a g xa b a b a b a b a b a b a b a b a b a ba bc œ b œ b c œ c b œ c œ b œ b œ g x f x a is even.Ê œ ba b a b 19. (a) The function is defined for all values of x, so the domain is .a bc_ß _ (b) Since x attains all nonnegative values, the range is .l l Òc#ß _Ñ 20. (a) Since the square root requires x , the domain is ." c   ! Ðc_ß "Ó (b) Since x attains all nonnegative values, the range is .È" c Òc#ß _Ñ 21. (a) Since the square root requires x , the domain is ."' c   ! Òc%ß %Ó# (b) For values of x in the domain, x , so x . The range is .! Ÿ "' c Ÿ "' ! Ÿ "' c Ÿ % Ò!ß %Ó# #È 22. (a) The function is defined for all values of x, so the domain is .a bc_ß _ (b) Since attains all positive values, the range is .$ "ß _#cx a b 23. (a) The function is defined for all values of x, so the domain is .a bc_ß _ (b) Since e attains all positive values, the range is .# c$ß _cx a b 24. (a) The function is equivalent to y tan x, so we require x for odd integers k. The domain is given by x forœ # # Á Ák k1 1 # % odd integers k. (b) Since the tangent function attains all values, the range is .a bc_ß _ 25. (a) The function is defined for all values of x, so the domain is .a bc_ß _ (b) The sine function attains values from to , so sin x and hence sin x . Thec" " c# Ÿ # $ b Ÿ # c$ Ÿ # $ b c " Ÿ "a b a b1 1 range is 3 1 .Òc ß Ó 26. (a) The function is defined for all values of x, so the domain is .a bc_ß _ (b) The function is equivalent to y x , which attains all nonnegative values. The range is .œ Ò!ß _ÑÈ& # 27. (a) The logarithm requires x , so the domain is .c $ € ! $ß _a b (b) The logarithm attains all real values, so the range is .a bc_ß _ 28. (a) The function is defined for all values of x, so the domain is .a bc_ß _ (b) The cube root attains all real values, so the range is .a bc_ß _ 29. (a) Increasing because volume increases as radius increases (b) Neither, since the greatest integer function is composed of horizontal (constant) line segments (c) Decreasing because as the height increases, the atmospheric pressure decreases. (d) Increasing because the kinetic (motion) energy increases as the particles velocity increases. 30. (a) Increasing on 2, (b) Increasing on 1,Ò _Ñ Òc _Ñ (c) Increasing on , (d) Increasing on ,a bc_ _ Ò _Ñ" # 31. (a) The function is defined for x , so the domain is .c% Ÿ Ÿ % Òc%ß %Ó (b) The function is equivalent to y x , x , which attains values from to for x in the domain. Theœ l l c% Ÿ Ÿ % ! #È range is .Ò!ß #Ó Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

41. Chapter 1 Practice Exercises 33 32. (a) The function is defined for x , so the domain is .c# Ÿ Ÿ # Òc#ß #Ó (b) The range is .Òc"ß "Ó 33. First piece: Line through and . m y x xa b a b!ß " "ß ! œ œ œ c" Ê œ c b " œ " c! c " c" " c ! " Second piece: Line through and . m y x x xa b a b a b"ß " #ß ! œ œ œ c" Ê œ c c " b " œ c b # œ # c! c " c" # c " " f x x, x x, x a b œœ " c ! Ÿ  " # c " Ÿ Ÿ # 34. First piece: Line through and 2 5 . m y xa b a b!ß ! ß œ œ Ê œ5 5 5 2 2 2 c ! c ! Second piece: Line through 2 5 and 4 . m y x 2 5 x 10 10a b a b a bß ß ! œ œ œ c Ê œ c c b œ c b œ c! c c c 5 5 5 5 5 5x 4 2 2 2 2 2 2 f x (Note: x 2 can be included on either piece.) x, x 2 10 , 2 x 4 a b Jœ œ ! Ÿ  c Ÿ Ÿ 5 2 5x 2 35. (a) f g f g f fa ba b a b a ba b Š ‹‰ c" œ c" œ œ " œ œ "" " c" b # "È (b) g f g f g ora ba b a ba b ˆ ‰ ɉ # œ # œ œ œ" " " # b # #Þ& &2 É È" # (c) f f x f f x f x, xa ba b a ba b ˆ ‰‰ œ œ œ œ Á !" " "Îx x (d) g g x g g x ga ba b a ba b Š ‹‰ œ œ œ œ" " b # b# b # " b # b # È É É È Èx x x " b# % Èx 36. (a) f g f g f fa ba b a b a ba b ˆ ‰È‰ c" œ c" œ c" b " œ ! œ # c ! œ #$ (b) g f f g g ga ba b a b a b a ba b ȉ # œ # œ # c # œ ! œ ! b " œ "$ (c) f f x f f x f x x xa ba b a b a b a ba b‰ œ œ # c œ # c # c œ (d) g g x g g x g x xa ba b a ba b ˆ ‰È Èɉ œ œ b " œ b " b "$ $$ 37. (a) f g x f g x f x x x, x .a ba b a ba b ˆ ‰ ˆ ‰È ȉ œ œ b # œ # c b # œ c   c# # g f x f g x g x x xa ba b a b a b a ba b È È‰ œ œ # c œ # c b # œ % c# # # (b) Domain of f g: (c) Range of f g:‰ Òc#ß _ÑÞ ‰ Ðc_ß #ÓÞ Domain of g f: Range of g f:‰ Òc#ß #ÓÞ ‰ Ò!ß #ÓÞ 38. (a) f g x f g x f x x x.a ba b a ba b Š ‹È È Èɉ œ œ " c œ " c œ " c% g f x f g x g x xa ba b a ba b ˆ ‰È Èɉ œ œ œ " c (b) Domain of f g: (c) Range of f g:‰ Ðc_ß "ÓÞ ‰ Ò!ß _ÑÞ Domain of g f: Range of g f:‰ Ò!ß "ÓÞ ‰ Ò!ß "ÓÞ 39. y f x y f f xœ œ ‰a b a ba b Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

42. 34 Chapter 1 Functions 40. 41. 42. The graph of f (x) f x is the same as the It does not change the graph.# "œ a bk k graph of f (x) to the right of the y-axis. The" graph of f (x) to the left of the y-axis is the# reflection of y f (x), x 0 across the y-axis.œ  " 43. 44. Whenever g (x) is positive, the graph of y g (x) Whenever g (x) is positive, the graph of y g (x) g (x)" # " # "œ œ œ k k g (x) is the same as the graph of y g (x). is the same as the graph of y g (x). When g (x) isœ œ œk k" " " " When g (x) is negative, the graph of y g (x) is negative, the graph of y g (x) is the reflection of the" # #œ œ the reflection of the graph of y g (x) across the graph of y g (x) across the x-axis.œ œ" " x-axis. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

43. Chapter 1 Practice Exercises 35 45. 46. Whenever g (x) is positive, the graph of The graph of f (x) f x is the same as the" # "œ a bk k y g (x) g (x) is the same as the graph of graph of f (x) to the right of the y-axis. Theœ œ# " "k k y g (x). When g (x) is negative, the graph of graph of f (x) to the left of the y-axis is theœ " " # y g (x) is the reflection of the graph of reflection of y f (x), x 0 across the y-axis.œ œ  # " y g (x) across the x-axis.œ " 47. 48. The graph of f (x) f x is the same as the The graph of f (x) f x is the same as the# " # "œ œa b a bk k k k graph of f (x) to the right of the y-axis. The graph of f (x) to the right of the y-axis. The" " graph of f (x) to the left of the y-axis is the graph of f (x) to the left of the y-axis is the# # reflection of y f (x), x 0 across the y-axis. reflection of y f (x), x 0 across the y-axis.œ   œ  " " 49. (a) y g x 3 (b) y g x 2œ c b œ b ca b ˆ ‰" # # 3 (c) y g x (d) y g xœ c œ ca b a b (e) y 5 g x (f) y g 5xœ † œa b a b 50. (a) Shift the graph of f right 5 units (b) Horizontally compress the graph of f by a factor of 4 (c) Horizontally compress the graph of f by a factor of 3 and a then reflect the graph about the y-axis (d) Horizontally compress the graph of f by a factor of 2 and then shift the graph left unit." # (e) Horizontally stretch the graph of f by a factor of 3 and then shift the graph down 4 units. (f) Vertically stretch the graph of f by a factor of 3, then reflect the graph about the x-axis, and finally shift the graph up unit." 4 51. Reflection of the grpah of y x about the x-axisœ È followed by a horizontal compression by a factor of then a shift left 2 units.1 2 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

44. 36 Chapter 1 Functions 52. Reflect the graph of y x about the x-axis, followedœ by a vertical compression of the graph by a factor of 3, then shift the graph up 1 unit. 53. Vertical compression of the graph of y by aœ 1 x2 factor of 2, then shift the graph up 1 unit. 54. Reflect the graph of y x about the y-axis, thenœ 1 3Î compress the graph horizontally by a factor of 5. 55. 56. period period 4œ œ1 1 57. 58. period 2 period 4œ œ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

45. Chapter 1 Practice Exercises 37 59. 60. period 2 period 2œ œ1 1 61. (a) sin B sin b 2 sin 2 3. By the theorem of Pythagoras,œ œ œ Ê œ œ œ1 1 3 c 3 b b 3 # #Š ‹ ÈÈ a b c a c b 4 3 1.# # # # #b œ Ê œ c œ c œÈ È (b) sin B sin c . Thus, a c b (2) .œ œ œ Ê œ œ œ œ c œ c œ œ1 3 c c sin 3 b 2 2 2 4 4 4 2 3 3 3 1 3 3 Š ‹ È È ÈÈ # È ÊŠ ‹ É# # # # 62. (a) sin A a c sin A (b) tan A a b tan Aœ Ê œ œ Ê œa a c b 63. (a) tan B a (b) sin A cœ Ê œ œ Ê œb b a a a tan B c sin A 64. (a) sin A (c) sin Aœ œ œa a c c c c bÈ # #c 65. Let h height of vertical pole, and let b and c denote theœ distances of points B and C from the base of the pole, measured along the flatground, respectively. Then, tan 50° , tan 35° , and b c 10.œ œ c œh h c b Thus, h c tan 50° and h b tan 35° (c 10) tan 35°œ œ œ b c tan 50° (c 10) tan 35°Ê œ b c (tan 50° tan 35°) 10 tan 35°Ê c œ c h c tan 50°Ê œ Ê œ10 tan 35° tan 50° tan 35°c 16.98 m.œ ¸10 tan 35° tan 50° tan 50° tan 35°c 66. Let h height of balloon above ground. From the figure atœ the right, tan 40° , tan 70° , and a b 2. Thus,œ œ b œh h a b h b tan 70° h (2 a) tan 70° and h a tan 40°œ Ê œ c œ (2 a) tan 70° a tan 40° a(tan 40° tan 70°)Ê c œ Ê b 2 tan 70° a h a tan 40°œ Ê œ Ê œ2 tan 70° tan 40° tan 70°b 1.3 km.œ ¸2 tan 70° tan 40° tan 40° tan 70°b 67. (a) (b) The period appears to be 4 .1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

46. 38 Chapter 1 Functions (c) f(x 4 ) sin (x 4 ) cos sin (x 2 ) cos 2 sin x cosb œ b b œ b b b œ b1 1 1 1ˆ ‰ ˆ ‰x 4 x xb # # # 1 since the period of sine and cosine is 2 . Thus, f(x) has period 4 .1 1 68. (a) (b) D ( 0) ( ); R [ 1 1]œ c_ß r !ß _ œ c ß (c) f is not periodic. For suppose f has period p. Then f kp f sin 2 0 for allˆ ‰ ˆ ‰" " # #1 1b œ œ œ1 integers k. Choose k so large that kp 0 . But then" " " # b1 1 1b € Ê  (1/2 ) kp 1 f kp sin 0 which is a contradiction. Thus f has no period, as claimed.ˆ ‰ Š ‹" " # # b1 1b œ €(1/ ) kp CHAPTER 1 ADDITIONAL AND ADVANCED EXERCISES 1. There are (infinitely) many such function pairs. For example, f(x) 3x and g(x) 4x satisfyœ œ f(g(x)) f(4x) 3(4x) 12x 4(3x) g(3x) g(f(x)).œ œ œ œ œ œ 2. Yes, there are many such function pairs. For example, if g(x) (2x 3) and f(x) x , thenœ b œ$ "Î$ (f g)(x) f(g(x)) f (2x 3) (2x 3) 2x 3.‰ œ œ b œ b œ ba b a b$ $ "Î$ 3. If f is odd and defined at x, then f( x) f(x). Thus g( x) f( x) 2 f(x) 2 whereasc œ c c œ c c œ c c g(x) (f(x) 2) f(x) 2. Then g cannot be odd because g( x) g(x) f(x) 2 f(x) 2c œ c c œ c b c œ c Ê c c œ c b 4 0, which is a contradiction. Also, g(x) is not even unless f(x) 0 for all x. On the other hand, if f isÊ œ œ even, then g(x) f(x) 2 is also even: g( x) f( x) 2 f(x) 2 g(x).œ c c œ c c œ c œ 4. If g is odd and g(0) is defined, then g(0) g( 0) g(0). Therefore, 2g(0) 0 g(0) 0.œ c œ c œ Ê œ 5. For (x y) in the 1st quadrant, x y 1 xß b œ bk k k k x y 1 x y 1. For (x y) in the 2ndÍ b œ b Í œ ß quadrant, x y x 1 x y x 1k k k kb œ b Í c b œ b y 2x 1. In the 3rd quadrant, x y x 1Í œ b b œ bk k k k x y x 1 y 2x 1. In the 4thÍ c c œ b Í œ c c quadrant, x y x 1 x ( y) x 1k k k kb œ b Í b c œ b y 1. The graph is given at the right.Í œ c Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

47. Chapter 1 Additional and Advanced Exercises 39 6. We use reasoning similar to Exercise 5. (1) 1st quadrant: y y x xb œ bk k k k 2y 2x y x.Í œ Í œ (2) 2nd quadrant: y y x xb œ bk k k k 2y x ( x) 0 y 0.Í œ b c œ Í œ (3) 3rd quadrant: y y x xb œ bk k k k y ( y) x ( x) 0 0Í b c œ b c Í œ points in the 3rd quadrantÊ all satisfy the equation. (4) 4th quadrant: y y x xb œ bk k k k y ( y) 2x 0 x. CombiningÍ b c œ Í œ these results we have the graph given at the right: 7. (a) sin x cos x 1 sin x 1 cos x (1 cos x)(1 cos x) (1 cos x)# # # # bb œ Ê œ c œ c b Ê c œ sin x 1 cos x # Ê œ1 cos x sin x sin x 1 cos x c b (b) Using the definition of the tangent function and the double angle formulas, we have tan .# # b cˆ ‰x 1 cos xsin cos 1 cos xœ œ œ # # # # # "c # "b # # ˆ ‰ ˆ ‰ x x cos 2 x cos 2 x Š ‹Š ‹ Š ‹Š ‹ 8. The angles labeled in the accompanying figure are# equal since both angles subtend arc CD. Similarly, the two angles labeled are equal since they both subtend! arc AB. Thus, triangles AED and BEC are similar which implies a c 2a cos b b a c c c bœ ) (a c)(a c) b(2a cos b)Ê c b œ c) a c 2ab cos bÊ c œ c# # # ) c a b 2ab cos .Ê œ b c# # # ) 9. As in the proof of the law of sines of Section 1.3, Exercise 61, ah bc sin A ab sin C ac sin Bœ œ œ the area of ABC (base)(height) ah bc sin A ab sin C ac sin B.Ê œ œ œ œ œ" " " " " # # # # # 10. As in Section 1.3, Exercise 61, (Area of ABC) (base) (height) a h a b sin C# # # # # # # #" " " œ œ œ4 4 4 a b cos C . By the law of cosines, c a b 2ab cos C cos C .œ " c œ b c Ê œ" b c# # # # # # 4 2ab a b c a b # # # Thus, (area of ABC) a b cos C a b# # # # # #" " b c # # b c œ " c œ " c œ " c4 4 ab 4 4a b a b c a b a b c a b Œ 9Š ‹ Š ‹ # # # # # # # # # # # a b 4a b a b c 2ab a b c 2ab a b cœ c b c œ b b c c b c" "# # # # # # # # # # ## 16 16Š ‹a b c da b a ba b a b (a b) c c (a b) ((a b) c)((a b) c)(c (a b))(c (a b))œ b c c c œ b b b c b c c c" "# # # # 16 16c d c da b a b s(s a)(s b)(s c), where s .œ œ c c c œ< ‘ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰a b c a b c a b c a b c a b cb b c b b c b b c b b # # # # # Therefore, the area of ABC equals s(s a)(s b)(s c) .È c c c 11. If f is even and odd, then f( x) f(x) and f( x) f(x) f(x) f(x) for all x in the domain of f.c œ c c œ Ê œ c Thus 2f(x) 0 f(x) 0.œ Ê œ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

48. 40 Chapter 1 Functions 12. (a) As suggested, let E(x) E( x) E(x) E is anœ Ê c œ œ œ Êf(x) f( x) f( x) f( ( x)) f(x) f( x)b c c b c c b c # # # even function. Define O(x) f(x) E(x) f(x) . Thenœ c œ c œf(x) f( x) f(x) f( x)b c c c # # O( x) O(x) O is an odd functionc œ œ œ c œ c Êf( x) f( ( x)) f( x) f(x) f(x) f( x)c c c c c c c c # # #Š ‹ f(x) E(x) O(x) is the sum of an even and an odd function.Ê œ b (b) Part (a) shows that f(x) E(x) O(x) is the sum of an even and an odd function. If alsoœ b f(x) E (x) O (x), where E is even and O is odd, then f(x) f(x) 0 E (x) O (x)œ b c œ œ b" " " " " "a b (E(x) O(x)). Thus, E(x) E (x) O (x) O(x) for all x in the domain of f (which is the same as thec b c œ c" " domain of E E and O O ). Now (E E )( x) E( x) E ( x) E(x) E (x) (since E and E arec c c c œ c c c œ c" " " " " " even) (E E )(x) E E is even. Likewise, (O O)( x) O ( x) O( x) O (x) ( O(x))œ c Ê c c c œ c c c œ c c c" " " " " (since O and O are odd) (O (x) O(x)) (O O)(x) O O is odd. Therefore, E E and" " " " "œ c c œ c c Ê c c O O are both even and odd so they must be zero at each x in the domain of f by Exercise 11. That is," c E E and O O, so the decomposition of f found in part (a) is unique." "œ œ 13. y ax bx c a x x c a x cœ b b œ b b c b œ b c b# # # Š ‹ ˆ ‰b b b b b a 4a 4a 2a 4a # # # # (a) If a 0 the graph is a parabola that opens upward. Increasing a causes a vertical stretching and a shift€ of the vertex toward the y-axis and upward. If a 0 the graph is a parabola that opens downward. Decreasing a causes a vertical stretching and a shift of the vertex toward the y-axis and downward. (b) If a 0 the graph is a parabola that opens upward. If also b 0, then increasing b causes a shift of the€ € graph downward to the left; if b 0, then decreasing b causes a shift of the graph downward and to the right. If a 0 the graph is a

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