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Student’s Solutions Manual Part One S ’ TUDENT S S M OLUTIONS ANUAL P O ART NE ARDIS ∙ BORZELLINO ∙ BUCHANAN ∙ KOUBA MOGILL ∙ NELSON U C NIVERSITY ALCULUS Joel Hass University of California, Davis Maurice D. Weir Naval Postgraduate School George B. Thomas, Jr. Massachusetts Institute of Technology ▼▼ PEARSON Addison Wesley Boston San Francisco New York London Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Paris Cape Town Hong Kong Montreal Reproduced by Pearson Addison-Wesley from electronic files supplied by the author. Copyright © 2007 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 trans mitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, with out the prior written permission of the publisher. Printed in the United States of America. ISBN 0-321-38849-6 6 BR 09 08 PEARSON —— Addison Wesley PREFACE TO THE STUDENT The Student's Solutions Manual contains the solutions to all of the odd-numbered exercise in the 11th Edition of THOMAS' UNIVERSITY CALCULUS by Maurice Weir, Joel Hass and Frank Giordano, excluding the Computer Algebra System (CAS) exercises. We have worked each solution to ensure that it • conforms exactly to the methods, procedures and steps presented in the text • is mathematically correct • includes all of the steps necessary so you can follow the logical argument and algebra • includes a graph or figure whenever called for by the exercise, or if needed to help with the explanation • is formatted in an appropriate style to aid in its understanding How to use a solution's manual • solve the assigned problem yourself • if you get stuck along the way, refer to the solution in the manual as an aid but continue to solve the problem on your own • if you cannot continue, reread the textbook section, or work through that section in the Student Study Guide, or consult your instructor • if your answer is correct by your solution procedure seems to differ from the one in the manual, and you are unsure your method is correct, consult your instructor • if your answer is incorrect and you cannot find your error, consult your instructor Acknowledgments Solutions Writers William Ardis, Collin County Community College-Preston Ridge Campus Joseph Borzellino, California Polytechnic State University Linda Buchanan, Howard College Duane Kouba, University of California-Davis Tim Mogill Patricia Nelson, University of Wisconsin-La Crosse Accuracy Checkers Karl Kattchee, University of Wisconsin-La Crosse Marie Vanisko, California State University, Stanislaus Tom Weigleitner, VISTA Information Technologies Thanks to Rachel Reeve, Christine O'Brien, Sheila Spinney, Elka Block, and Joe Vetere for all their guidance and help at every step. TABLE OF CONTENTS 1 Functions 1 1.1 Functions and Their Graphs 1 1.2 Combining Functions; Shifting and Scaling Graphs 4 1.3 Trigonometric Functions 10 1.4 Exponential Functions 13 1.5 Inverse Functions and Logarithms 15 1.6 Graphing with Calculators and Computers 19 2 Limits and Continuity 23 2.1 Rates of Change and Tangents to Curves 23 2.2 Limit of a Function and Limit Laws 24 2.3 The Precise Definition of a Limit 29 2.4 One-Sided Limits and Limits at Infinity 32 2.5 Infinite Limits and Vertical Asymptotes 35 2.6 Continuity 39 2.7 Tangents and Derivatives at a Point 41 Practice Exercises 44 Additional and Advanced Exercises 47 3 Differentiation 51 3.1 The Derivative as a Function 51 3.2 Differentiation Rules for Polynomials, Exponentials, Products, and Quotients 54 3.3 The Derivative as a Rate of Change 57 3.4 Derivatives of Trigonometric Functions 59 3.5 The Chain Rule and Parametric Equations 63 3.6 Implicit Differentiation 69 3.7 Derivatives of Inverse Functions and Logarithms 72 3.8 Inverse Trigonometric Functions 75 3.9 Related Rates 78 3.10 Linearizations and Differentials 80 3.11 Hyperbolic Functions 82 Practice Exercises 84 Additional and Advanced Exercises 91 4 Applications of Derivatives 95 4.1 Extreme Values of Functions 95 4.2 The Mean Value Theorem 100 4.3 Monotonic Functions and the First Derivative Test 103 4.4 Concavity and Curve Sketching 110 4.5 Applied Optimization 119 4.6 Indeterminate Forms and L'HopitaΓs Rule 125 4.7 Newton's Method 128 4.8 Antiderivatives 130 Practice Exercises 133 Additional and Advanced Exercises 142 5 Integration 147 5.1 Estimating with Finite Sums 147 5.2 Sigma Notation and Limits of Finite Sums 149 5.3 The Definite Integral 151 5.4 The Fundamental Theorem of Calculus 156 5.5 Indefinite Integrals and the Substitution Rule 160 5.6 Substitution and Area Between Curves 163 5.7 The Logarithm Defined as an Inegral 171 Practice Exercises 174 Additional and Advanced Exercises 181 6 Applications of Definite Integrals 187 6.1 Volumes by Slicing and Rotation About an Axis 187 6.2 Volumes by Cylindrical Shells 191 6.3 Lengths of Plane Curves 196 6.4 Areas of Surfaces of Revolution 198 6.5 Exponential Change and Separable Differential Equations 202 6.6 Work 203 6.7 Moments and Centers of Mass 206 Practice Exercises 209 Additional and Advanced Exercises 214 7 Techniques of Integration 217 7.1 Integration by Parts 217 7.2 Trigonometric Integrals 220 7.3 Trigonometric Substitutions 222 7.4 Integration of Rational Functions by Partial Fractions 224 7.5 Integral Tables and Computer Algebra Systems 226 7.6 Numerical Integration 230 7.7 Improper Integrals 234 Practice Exercises 238 Additional and Advanced Exercises 242 8 Infinite Sequences and Series 245 8.1 Sequences 245 8.2 Infinite Series 249 8.3 The Integral Test 251 8.4 Comparison Tests 253 8.5 The Ratio and Root Tests 255 8.6 Alternating Series, Absolute and Conditional Convergence 257 8.7 Power Series 259 8.8 Taylor and Maclaurin Series 264 8.9 Convergence of Taylor Series 265 8.10 The Binomial Series 268 Practice Exercises 269 Additional and Advanced Exercises 273 9 Conic Sections and Polar Coordinates 275 9.1 Polar Coordinates 275 9.2 Graphing in Polar Coordinates 277 9.3 Areas and Lengths in Polar Coordinates 280 9.4 Conic Sections 282 9.5 Conics in Polar Coordinates 287 9.6 Conics and Parametric Equations; The Cycloid 292 Practice Exercises 295 Additional and Advanced Exercises 299 CHAPTER 1 FUNCTIONS 1.1 FUNCTIONS AND THEIR GRAPHS 1. domain = (—∞, oo); range = [1, ∞) 3. domain = (0, oo); y in range =≠> y = t > 0 => y2 = ∣ and y > 0 => y can be any positive real number => range = (0, ∞). 5. (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. 7. base = x; (height)2 + (∣)2 = x2 => height = ∕⅛ area is a(x) = ∣ (base)(height) = ∣ (x) [y x j = χ2i perimeter is p(x) = x + x + x = 3x. 9. Let D = diagonal of a face of the cube and ¢. = the length of an edge. Then + D2 = d2 and D2 = 2£2 => 3f2 — d2 / ∖ 3/2 => I = . The surface area is 6£2 = = 2d2 and the volume is (3 = = . 11. The domain is (—∞, ∞). 13. The domain is (—∞, ∞). 15. The domain is (—∞, 0) U (0, ∞). -4 -3 -2 -! ! 2 3 4 .......................'--O∙ 17. Neither graph passes the vertical line test (a) (b) 2 Chapter 1 Functions 19. i 3 — x, x ≤ 1 21. G(x) = ∣ 2x, x > 1 23. (a) Line through (0, 0) and (1, 1): y = x Line through (1, 1) and (2, 0): y — -x + 2 x, 0 ≤ x ≤ 1 + 2, 1 < x ≤ 2 2, 0 ≤ x < 1 1 < x < 2 (b) 2 ≤ x < 3 3 ≤ x ≤ 4 25. (a) Line through (—1, 1) and (0, 0): y = -x Line through (0, 1) and (1, 1): y = 1 Line through (1, 1) and (3, 0): m = j ∣ — -y -∣,so y = -∣(x -1) + 1 = -∣x + ∣ z { -x -1 ≤ x < 0 1 0 < x ≤ 1 — ∣x + | 1 < x < 3 (b) Line through (—2, —1) and (0, 0): y = ~x Line through (0, 2) and (1, 0): y = -2x + 2 Line through (1, -1) and (3, —1): y = — 1 -2 ≤ x ≤ 0 f(x) = —2x + 2 0 < x ≤ 1 -1 1 < x ≤ 3 27. (a) [xj = 0 for x ∈ [0, 1) (b) ∣^x^∣ = 0forx ∈ (-l,0] 29. For any real number x, n ≤ x ≤ n + 1, where n is an integer. Now: n ≤ x ≤ n + l= > —(n + 1) ≤ —x ≤ —n. By definition: ∣^-x] = -n and ∣xj = n => -∣xj = -n. So ∣^-x^] = -[xj for all x ∈ 3⅞. Section 1.1 Functions and Their Graphs 3 31. Symmetric about the origin 33. Symmetric about the origin Dec: —∞ < x < ∞ Dec: nowhere Inc: nowhere Inc: —∞ < x < 0 0 < x < ∞ 35. Symmetric about the y-axis 37. Symmetric about the origin Dec: —∞ < x ≤ 0 Dec: nowhere Inc: 0 < x < ∞ Inc: —∞ < x < ∞ 39. No symmetry 41. Symmetric about the y-axis Dec: 0 ≤ x < ∞ Dec: —∞ < x ≤ 0 Inc: nowhere Inc: 0 < x < ∞ 43. 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. 45. Since f(x) = x2 + 1 = (-x)2 + 1 = ~f(x)∙ The function is even. 47. Since g(x) = x3 + x, g(-x) = -x3 - x = -(x3 + x) — -g(x). So the function is odd. 49. g(x) = = (, )⅞, = g(-x )∙ Thus the function is even. x 1 51. h(t) = 4ιJ h(-t) = r⅛ 4 -h(t) = ~- . Since h(t) ≠ -h(t) and h(t) ≠ h(-t), the function isn either even nor odd. f i 53. h(t) = 2t + 1, h(-t) = -2t + 1. So h(t) ≠ h(-t). -h(t) = -2t - 1, so h(t) ≠ -h(t). The function isn either even nor odd. 4 Chapter 1 Functions 55. V = f(x) = x(14 - 2x)(22 - 2x) = 4x3 - 72x2 + 308x; 0 < x < 7. 57. (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. 59. (a) From the graph, ∣ > 1 + f => x ∈ (—2,0) U (4, ∞) (b) ∣ > 1 + f => I - 1 - -i > 0 => x > 4 since x is positive; x < 0: ∣ - 1 - * > 0 ≠> < 0 => <,⅛-4⅜t+2> < 0 => x < —2 since x is negative; sign of (x — 4)(x + 2) + . + : -2 4 Solution interval: (—2,0) U (4, ∞) 61. 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 same vertical line. The graph of the function y = f(x) = 0 is the x-axis, a horizontal line for which there is a single y-value, 0, for any x. 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 ≥ 0 f g f+g fg f g f+g fg 3. D : —∞ < x < ∞, D : —∞ < x < ∞, D : —∞ < x < ∞, D : —∞ < x < ∞, R : y = 2, R : y ≥ 1, f g f/g g/f f g Rf/ : 0 < y ≤ 2, R : ∣ ≤ y < ∞ e g/f 5. (a) 2 (b) 22 (c) x2 +2 (d) (x + 5)2 - 3 = x2 + 10x + 22 (e) 5 (f) -2 (g) x+ 10 (h) (x2 - 3)2 -3 = X4 -6 X2 + 6 7. (a) 4 s (b) ⅜ -5 (c) (*~5)2 (d) (4X-15)5 (e) 4⅛5 (f) (4x -1 5)2 9. (a) (f°g)W (b) (jog)(x) (C) (g°g)(x) (d) (j°j)W (e) (gohof)(χ) (f) (hojof)( ' x