Ebook Calculus begins with four diagnostic tests, in basic algebra, analytic geometry, functions, and trigonometry; overview of the subject and includes a list of questions to motivate the study of calculus; from the beginning, multiple representations of functions are stressed: verbal, numerical, visual, and algebraic. A discussion of mathematical models leads to a review of the standard functions, including exponential and logarithmic functions, from these four points of view.
CA L C U L U S SIXTH EDITION J A M E S S T E WA RT McMASTER UNIVERSITY AUSTRALIA N BRAZIL N C A N A DA N MEXICO N SINGAPORE N S PA I N N UNITED KINGDOM N U N I T E D S TAT E S Calculus, Sixth Edition James Stewart Publisher Bob Pirtle Assistant Editor Stacy Green Editorial Assistant Elizabeth Rodio Technology Project Manager Sam Subity Marketing Manager Mark Santee Marketing Assistant Melissa Wong Marketing Communications Manager Bryan Vann Project Manager, Editorial Production Cheryll Linthicum Creative Director Rob Hugel Art Director Vernon T Boes Print Buyer Becky Cross Permissions Editor Bob Kauser Production Service TECH·arts Text Designer Kathi Townes Photo Researcher Stephanie Kuhns Copy Editor Kathi Townes Illustrator Brian Betsill Cover Designer Irene Morris Cover Image Amelie Fear, Folkmusician.com Cover Printer R R Donnelley /Willard Compositor Stephanie Kuhns, TECH·arts Printer R R Donnelley /Willard COPYRIGHT © 2008, 2003 Thomson Brooks/Cole, a part of The Thomson Corporation Thomson, the Star logo, and Brooks/Cole are trademarks used herein under license Trademarks ExamView ® and ExamViewPro ® are registered trademarks of FSCreations, Inc Windows is a registered trademark of the Microsoft Corporation and used herein under license Macintosh and Power Macintosh are registered trademarks of Apple Computer, Inc Used herein under license Derive is a registered trademark of Soft Warehouse, Inc Maple is a registered trademark of Waterloo Maple, Inc Mathematica is a registered trademark of Wolfram Research, Inc Tools for Enriching is a trademark used herein under license N N N N N N N N N N N ALL RIGHTS RESERVED No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution, information storage and retrieval systems, or in any other manner—without the written permission of the publisher Printed in the United States of America N N N N N N N N N N N 11 10 09 08 07 For more information about our products, contact us at: Thomson Learning Academic Resource Center 1-800-423-0563 For permission to use material from this text or product, submit a request online at http://www.thomsonrights.com Any additional questions about permissions can be submitted by email to thomsonrights@thomson.com © 2008 Thomson Learning, Inc All Rights Reserved Thomson Learning WebTutor™ is a trademark of Thomson Learning, Inc Library of Congress Control Number: 2006939651 ISBN-13: 978-0-495-01160-6 ISBN-10: 0-495-01160-6 Thomson Higher Education 10 Davis Drive Belmont, CA 94002 USA CONTENTS Preface xi To the Student xxii Diagnostic Tests xxiv A PREVIEW OF CALCULUS FUNCTIONS AND MODELS 10 1.1 Four Ways to Represent a Function 1.2 Mathematical Models: A Catalog of Essential Functions 1.3 New Functions from Old Functions 1.4 Graphing Calculators and Computers Review LIMITS 11 37 46 54 60 2.1 The Tangent and Velocity Problems 2.2 The Limit of a Function 2.3 Calculating Limits Using the Limit Laws 2.4 The Precise Definition of a Limit 2.5 Continuity Review Problems Plus 24 52 Principles of Problem Solving 2 61 66 77 87 97 108 110 iii iv |||| CONTENTS DERIVATIVES 3.1 m=0 112 Derivatives and Rates of Change 113 Writing Project Early Methods for Finding Tangents 123 N m=1 m=_1 π π 3.2 The Derivative as a Function 3.3 Differentiation Formulas 123 135 Applied Project Building a Better Roller Coaster N y 3.4 Derivatives of Trigonometric Functions 3.5 The Chain Rule 148 148 155 Applied Project Where Should a Pilot Start Descent? 164 N π π 3.6 Implicit Differentiation 164 3.7 Rates of Change in the Natural and Social Sciences 3.8 Related Rates 3.9 Linear Approximations and Differentials 182 Laboratory Project Taylor Polynomials 195 N Review Problems Plus 196 200 APPLICATIONS OF DIFFERENTIATION 4.1 189 Maximum and Minimum Values 204 205 Applied Project The Calculus of Rainbows N 213 4.2 The Mean Value Theorem 214 4.3 How Derivatives Affect the Shape of a Graph 4.4 Limits at Infinity; Horizontal Asymptotes 4.5 Summary of Curve Sketching 4.6 Graphing with Calculus and Calculators 4.7 Optimization Problems 4.8 Newton’s Method 4.9 Antiderivatives Review Problems Plus 281 285 269 274 243 256 Applied Project The Shape of a Can N 230 268 250 220 170 CONTENTS INTEGRALS 288 5.1 Areas and Distances 289 5.2 The Definite Integral 300 Discovery Project Area Functions N 312 5.3 The Fundamental Theorem of Calculus 5.4 Indefinite Integrals and the Net Change Theorem 313 Writing Project Newton, Leibniz, and the Invention of Calculus N 5.5 The Substitution Rule Review Problems Plus 332 340 344 6.1 Areas Between Curves 6.2 Volumes 6.3 Volumes by Cylindrical Shells 6.4 Work 6.5 Average Value of a Function Problems Plus 324 333 APPLICATIONS OF INTEGRATION Review |||| 346 347 354 365 370 374 378 380 INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS 7.1 Inverse Functions 385 Instructors may cover either Sections 7.2–7.4 or Sections 7.2*–7.4* See the Preface 7.2 Exponential Functions and Their Derivatives 392 7.2* The Natural Logarithmic Function 421 7.3 Logarithmic Functions 405 7.3* The Natural Exponential Function 430 7.4 Derivatives of Logarithmic Functions 411 7.4* General Logarithmic and Exponential Functions 438 384 v vi |||| CONTENTS 7.5 Exponential Growth and Decay 7.6 Inverse Trigonometric Functions 447 454 Applied Project Where To Sit at the Movies 463 N 7.7 Hyperbolic Functions 463 7.8 Indeterminate Forms and L’Hospital’s Rule Writing Project The Origins of L’Hospital’s Rule 481 N Review Problems Plus 482 486 TECHNIQUES OF INTEGRATION 488 8.1 Integration by Parts 489 8.2 Trigonometric Integrals 8.3 Trigonometric Substitution 8.4 Integration of Rational Functions by Partial Fractions 8.5 Strategy for Integration 8.6 Integration Using Tables and Computer Algebra Systems 496 503 N 8.7 Approximate Integration 8.8 Improper Integrals Review Problems Plus 530 531 544 554 557 FURTHER APPLICATIONS OF INTEGRATION 9.1 Arc Length N Area of a Surface of Revolution Discovery Project Rotating on a Slant N 9.3 568 568 574 Applications to Physics and Engineering Discovery Project Complementary Coffee Cups N Applications to Economics and Biology 9.5 Probability Problems Plus 591 598 600 575 586 9.4 Review 560 561 Discovery Project Arc Length Contest 9.2 509 519 Discovery Project Patterns in Integrals 470 586 525 CONTENTS 10 DIFFERENTIAL EQUATIONS 602 10.1 Modeling with Differential Equations 10.2 Direction Fields and Euler’s Method 10.3 Separable Equations 603 608 616 Applied Project How Fast Does a Tank Drain? 624 N Applied Project Which Is Faster, Going Up or Coming Down? N 10.4 Models for Population Growth Applied Project Calculus and Baseball N 10.5 Linear Equations 10.6 Predator-Prey Systems Review Problems Plus 11 627 637 638 644 650 654 PARAMETRIC EQUATIONS AND POLAR COORDINATES 11.1 Curves Defined by Parametric Equations Laboratory Project Running Circles Around Circles N 11.2 Calculus with Parametric Curves Laboratory Project Bézier Curves N 11.4 Areas and Lengths in Polar Coordinates 11.5 Conic Sections 11.6 Conic Sections in Polar Coordinates 675 686 690 698 705 708 INFINITE SEQUENCES AND SERIES 12.1 665 675 Polar Coordinates Problems Plus 657 666 11.3 Review 12 626 Sequences 710 711 Laboratory Project Logistic Sequences N 723 12.2 Series 723 12.3 The Integral Test and Estimates of Sums 12.4 The Comparison Tests 12.5 Alternating Series 12.6 Absolute Convergence and the Ratio and Root Tests 733 741 746 750 656 |||| vii viii |||| CONTENTS 12.7 Strategy for Testing Series 757 12.8 Power Series 12.9 Representations of Functions as Power Series 759 12.10 Taylor and Maclaurin Series 764 770 Laboratory Project An Elusive Limit 784 N Writing Project How Newton Discovered the Binomial Series N 12.11 Applications of Taylor Polynomials Applied Project Radiation from the Stars Problems Plus 13 785 793 N Review 794 797 VECTORS AND THE GEOMETRY OF SPACE 13.1 Three-Dimensional Coordinate Systems 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product O 800 801 806 815 822 Discovery Project The Geometry of a Tetrahedron N 13.5 LONDON Equations of Lines and Planes N Cylinders and Quadric Surfaces Review PARIS Problems Plus 14 830 830 Laboratory Project Putting 3D in Perspective 13.6 840 840 848 851 VECTOR FUNCTIONS 852 14.1 Vector Functions and Space Curves 14.2 Derivatives and Integrals of Vector Functions 14.3 Arc Length and Curvature 14.4 Motion in Space: Velocity and Acceleration Applied Project Kepler’s Laws N Review Problems Plus 784 885 888 853 860 866 884 874 CONTENTS 15 PARTIAL DERIVATIVES 890 15.1 Functions of Several Variables 891 15.2 Limits and Continuity 15.3 Partial Derivatives 15.4 Tangent Planes and Linear Approximations 15.5 The Chain Rule 15.6 Directional Derivatives and the Gradient Vector 15.7 Maximum and Minimum Values 906 914 928 937 Applied Project Designing a Dumpster N 958 969 Discovery Project Quadratic Approximations and Critical Points N 15.8 Lagrange Multipliers 970 Applied Project Rocket Science 977 N Applied Project Hydro-Turbine Optimization 979 N Review Problems Plus 16 980 984 MULTIPLE INTEGRALS 986 16.1 Double Integrals over Rectangles 16.2 Iterated Integrals 16.3 Double Integrals over General Regions 16.4 Double Integrals in Polar Coordinates 16.5 Applications of Double Integrals 16.6 Triple Integrals 1001 1010 1016 1026 Discovery Project Volumes of Hyperspheres 16.7 987 995 N 1036 Triple Integrals in Cylindrical Coordinates 1036 Discovery Project The Intersection of Three Cylinders N 16.8 Triple Integrals in Spherical Coordinates Applied Project Roller Derby N 16.9 Problems Plus 1057 1060 1041 1041 1048 Change of Variables in Multiple Integrals Review 946 1048 969 |||| ix APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES 47 49 y _2 y 27 Parabola yϭ4 A123 29 Parabola y y xϭ2 x |||| (3, 4) x Ϫ2 x 51 x y y=1+x 31 Ellipse y x (3, 9) y=1-2x (b) ͑3.5, Ϫ3͒ 53 ͑0, Ϫ4͒ 55 (a) ͑4, 9͒ 59 y x Ϫ 61 (b) 4x Ϫ 3y Ϫ 24 EXERCISES C 33 y ͑0, 1͒ ͑x Ϫ 3͒2 ϩ ͑ y ϩ 1͒2 25 (Ϫ , 0), ͑2, Ϫ5͒, 11 Parabola 35 y x Ϫ 2x 37 x 39 y y ( 14 , Ϫ 14 ), s10͞4 13 Ellipse y x x ϩ y 65 57 ͑1, Ϫ2͒ PAGE A23 N x y x Ϫ1 x _4 x y N PAGE A32 7͞6 ͞20 5 720° 75° 11 Ϫ67.5Њ 13 3 cm 15 rad ͑120͒͞Њ _2 15 Hyperbola EXERCISES D 17 17 Ellipse 19 y y y y= x 0 x x 315° _5 x _2 x 21 _1 y=_ x 19 Parabola 3π _ y 21 Hyperbola y rad y x 1 _1 _1 x y=_ x y= x 23 sin͑3͞4͒ 1͞s2, cos͑3͞4͒ Ϫ1͞s2, tan͑3͞4͒ Ϫ1, _1 23 Hyperbola csc͑3͞4͒ s2, sec͑3͞4͒ Ϫs2, cot͑3͞4͒ Ϫ1 25 sin͑9͞2͒ 1, cos͑9͞2͒ 0, csc͑9͞2͒ 1, cot͑9͞2͒ 0, tan͑9͞2͒ and sec͑9͞2͒ undefined 25 Ellipse y x y 27 sin͑5͞6͒ , cos͑5͞6͒ Ϫs3͞2, tan͑5͞6͒ Ϫ1͞s3, csc͑5͞6͒ 2, sec͑5͞6͒ Ϫ2͞s3, cot͑5͞6͒ Ϫs3 29 cos , tan , csc , sec , cot (1, 2) x x 31 sin s5͞3, cos Ϫ , tan Ϫs5͞2, csc 3͞s5, cot Ϫ2͞s5 A124 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES 19 33 sin  Ϫ1͞s10, cos  Ϫ3͞s10, tan  , csc  Ϫs10, sec  Ϫs10͞3 35 5.73576 cm 61 15 i1 37 24.62147 cm (3 ϩ s2 ) 63 15 (4 ϩ s2 ) 73 ഛ x ഛ ͞6 and 5͞6 ഛ x ഛ 2 75 ഛ x Ͻ ͞4, 3͞4 Ͻ x Ͻ 5͞4, 7͞4 Ͻ x ഛ 2 77 n i iϩ1 21 80 ͚ 2i 15 23 3276 43 25 ͚x 19 i i1 29 n͑n ϩ 1͒ 27 61 33 n͑n ϩ 6n ϩ 11͒͞3 49 45 14 n i i0 35 n͑n ϩ 2n Ϫ n Ϫ 10͒͞4 41 (a) n (b) 100 Ϫ (c) ͚2 17 i1 31 n͑n ϩ 6n ϩ 17͒͞3 69 ͞6, ͞2, 5͞6, 3͞2 67 ͞4, 3͞4, 5͞4, 7͞4 71 0, , 2 59 65 ͞3, 5͞3 24 25 ͚ 13 nϩ1 (d) an Ϫ a0 ϩ n2 ϩ n Ϫ 97 300 y EXERCISES G 1 N Ϫ 4i π x 5π PAGE A55 13 ϩ 18i Ϫ i 11 Ϫi 19 Ϯ i 17 4i, 23 Ϫ Ϯ (s7͞2)i 79 y [ 27 5{cos tan 12 Ϫ 7i 11 13 ϩ 10 13 i 15 12 ϩ 5i, 13 13 5i 21 Ϫ1 Ϯ 2i 25 s2 ͓cos͑3͞4͒ ϩ i sin͑3͞4͔͒ ( )] ϩ i sin[tanϪ1( 43)]} Ϫ1 29 4͓cos͑͞2͒ ϩ i sin͑͞2͔͒, cos͑Ϫ͞6͒ ϩ i sin͑Ϫ͞6͒ , π 3π 2π π 5π 3π 2 x ͓cos͑Ϫ͞6͒ ϩ i sin͑Ϫ͞6͔͒ 31 s2 ͓cos͑7͞12͒ ϩ i sin͑7͞12͔͒ , (2 s2 )͓cos͑13͞12͒ ϩ i sin͑13͞12͔͒ , 14 ͓cos͑͞6͒ ϩ i sin͑͞6͔͒ 81 33 Ϫ1024 37 Ϯ1, Ϯi, (1͞s2 )͑Ϯ1 Ϯ i ͒ _π 35 Ϫ512 s3 ϩ 512i y π Im i x 2π 39 Ϯ(s3͞2) ϩ i, Ϫi Im 89 14.34457 cm EXERCISES E N Re PAGE A38 _i s1 ϩ s2 ϩ s3 ϩ s4 ϩ s5 3 ϩ ϩ 3 Ϫ1 ϩ ϩ ϩ ϩ 110 ϩ 210 ϩ 310 ϩ и и и ϩ n10 10 Ϫ ϩ Ϫ ϩ и и и ϩ ͑Ϫ1͒nϪ1 11 ͚i i1 41 i 43 ϩ (s3͞2) i 45 Ϫe 47 cos 3 cos3 Ϫ cos sin2, sin 3 cos2 sin Ϫ sin3 Re INDEX RP denotes Reference Page numbers Abel, Niels, 167 absolute maximum and minimum, 205, 959, 964 absolute maximum and minimum values, 205, 959, 964 absolute value, 17, A6, A47 absolute value function, 17 absolutely convergent series, 750 acceleration, 130, 171 acceleration of a particle, 875 components of, 878 as a vector, 875 Achilles and the tortoise, adaptive numerical integration, 540 addition formulas for sine and cosine, A28, A29 addition of vectors, 806, 809 Airy, Sir George, 764 Airy function, 764 algebraic function, 31 alternating harmonic series, 747, 750 alternating series, 746 Alternating Series Estimation Theorem, 748 Alternating Series Test, 746 analytic geometry, A10 angle, A24 between curves, 202 of deviation, 213 negative, A25 positive, A25 standard position, A25 between vectors, 815 angular momentum, 884 angular speed, 876 antiderivative, 274 antidifferentiation formulas, 276 aphelion, 703 apolune, 697 approach path of an aircraft, 164 approximate integration, 531 approximating cylinder, 356 approximating surface, 568 approximation by differentials, 191 to e, 397 by the Midpoint Rule, 532 by Newton’s method, 269 by an nth-degree Taylor polynomial, 195 quadratic, 195 by Riemann sums, 301 by Simpson’s Rule, 536, 537 tangent line, 189 by Taylor polynomials, 785 by Taylor’s Inequality, 773 by the Trapezoidal Rule, 533 approximation, linear, 189, 930, 934 to a tangent plane, 930 Archimedes’ Principle, 381, 1140 arc length, 561, 670 of a parametric curve, 669 of a polar curve, 688 of a space curve, 866, 867 arc length contest, 568 arc length formula, 562 arc length function, 564 area, 3, 289 of a circle, 505 under a curve, 289, 294, 300 between curves, 347, 350 of an ellipse, 505 by exhaustion, 3, 80 by Green’s Theorem, 1094 enclosed by a parametric curve, 668 in polar coordinates, 686 of a sector of a circle, 686 surface, 570, 671, 1111, 1113 of a surface of a revolution, 568, 574 area function, 312 Area Problem, 3, 289 argument of a complex number, A48 arithmetic-geometric mean, 722 arrow diagram, 12 astroid, 169, 665 asymptote, 244 horizontal, 232, 244 of a hyperbola, 694, A20 slant, 247 vertical, 73, 244 asymptotic curve, 249 autonomous differential equation, 611 auxiliary equation, 1148 complex roots of, 1150 real roots of, 1149 average cost function, 265 average rate of change, 117, 171 average speed of molecules, 552 average value of a function, 374, 375, 593, 992, 1036 average velocity, 5, 63, 115, 171, 874 axes, coordinate, 801, A11 axes of an ellipse, A19 axis of a parabola, 691 bacterial growth, 627, 634 Barrow, Isaac, 4, 123, 313, 332 baseball and calculus, 637 A125 A126 |||| INDEX base of a cylinder, 354 base of a logarithm, 405, 442* change of, 408, 443* basis vectors, 810, 811 Bernoulli, James, 616, 643 Bernoulli, John, 481, 616, 661, 772 Bernoulli differential equation, 643 Bessel, Friedrich, 760 Bessel function, 710, 760, 764 Bézier, Pierre, 675 Bézier curves, 660, 675 binomial coefficients, 778 binomial series, 778, 784 discovery by Newton, 784 Binomial Theorem, 778, RP1 binormal vector, 870 blackbody radiation, 793 blood flow, 176, 267, 587 boundary curve, 1129 boundary-value problem, 1152 bounded sequence, 718 bounded set, 964 Boyle’s Law, 181, 188, 421 Brache, Tycho, 880 brachistochrone problem, 661 branches of hyperbola, 665, 694, A20 Buffon’s needle problem, 601 bullet-nose curve, 51, 162 C transformation, 1049 cable (hanging), 464 calculator, graphing, 46, 250, 660, 682 calculus, invention of, 332 cancellation equations for inverse functions, 387 for inverse trigonometric functions, 455 for logarithms, 405, 430*, 442* cans, minimizing manufacturing cost of, 268 Cantor, Georg, 732 Cantor set, 732 capital formation, 590 cardiac output, 588 cardioid, 169, 679 carrying capacity, 182, 604 Cartesian coordinate system, A11 Cartesian plane, A11 CAS See computer algebra system Cassini, Giovanni, 685 catenary, 464 Cauchy, Augustin-Louis, 91, 477, 997 Cauchy’s Mean Value Theorem, 477 Cauchy-Schwarz Inequality, 822 Cavalieri’s Principle, 364 center of gravity, 578 center of mass, 578 of a lamina, 1017 of a solid, 1032 of a surface, 1119 of a wire, 1072 centripetal force, 888 centroid of a plane region, 579 centroid of a solid, 1032 Chain Rule, 155, 156 for several variables, 937, 939, 940 change of base, formula for, 408, 443* change of variables in a double integral, 1011, 1049, 1052 in integration, 334 in a triple integral, 1038, 1043, 1055 characteristic equation, 1148 charge, electric, 173, 1016, 1032 in a circuit, 1165 charge density, 1016, 1032 chemical reaction, 174 circle, area of, 505 circle, equation of, A16 circle of curvature, 871 circular cylinder, 354 circular paraboloid, 846 circulation of a vector field, 1132 cissoid of Diocles, 664, 684 Clairaut, Alexis, 921 Clairaut’s Theorem, 921, A3 clipping planes, 840 closed curve, 1084 closed interval, A3 Closed Interval Method, 209 for a function of two variables, 965 closed set, 964 closed surface, 1122 Cobb, Charles, 892 Cobb-Douglas production function, 893, 923, 976 cochleoid, 706 coefficient(s) binomial, 778 of friction, 155, 212 of inequality, 331 of a polynomial, 28 of a power series, 759 static friction, 851 combinations of functions, 41 comets, orbits of, 704 common ratio, 725 comparison properties of the integral, 309 Comparison Test, 550, 741 comparison tests for series, 741 Comparison Theorem for integrals, 550 complementary equation, 1153 Completeness Axiom, 718 complex conjugate, A47 complex exponentials, A52 complex number(s), A46 argument of, A48 division of, A47, A49 equality of, A46 imaginary part of, A46 modulus of, A47 multiplication of, A47, A49 polar form, A48 powers of, A50 principal square root of, A47 real part of, A46 roots of, A51 component function, 853, 1064 components of acceleration, 878 components of a vector, 808, 818 composition of functions, 41, 156 continuity of, 103, 911 derivative of, 157 compound interest, 451, 480, 654 compressibility, 175 computer algebra system, 69, 660 computer algebra system, graphing with, 46, 250, 682 for creating a three-dimensional scene, 840 function of two variables, 895 level curves, 900 parametric equations, 660 parametric surface, 1108 partial derivatives, 921 polar curve, 682 sequence, 716 space curve, 856 vector field, 1065 computer algebra system, integrating with, 527, 768 concavity, 224 Concavity Test, 224, A43 concentration, 174 conchoid, 662, 684 conditionally convergent series, 751 conductivity (of a substance), 1126 cone, 363, 844 parametrization of, 1109 conic section, 690, 698 directrix, 698 eccentricity, 698 focus (foci), 698 polar equations for, 700 shifted, 695, A21 vertex (vertices), 691 INDEX conjugates, properties of, A47 connected region, 1084 conservation of energy, 1088 conservative vector field, 1068, 1089 constant function, 135 constraint, 970, 974 Constant Multiple Law of limits, 78 Constant Multiple Rule, 137 consumer surplus, 586, 587 continued fraction expansion, 722 continuity of a function, 97 of a function of three variables, 912 of a function of two variables, 910 on an interval, 99 from the left, 99 from the right, 99 of a vector function, 854 continuous compounding of interest, 451, 480, 654 continuous random variable, 591 contour curves, 896 contour map, 896 convergence absolute, 750 conditional, 751 of an improper integral, 545, 548 interval of, 761 radius of, 761 of a sequence, 713 of a series, 724 convergent improper integral, 545, 548 convergent sequence, 713 convergent series, 724 properties of, 729 cooling tower, hyperbolic, 846 coordinate(s), A2 Cartesian, A11 polar, 675 rectangular, A11 coordinate axes, 801, A11 coordinate planes, 801 coordinate system cylindrical, 1037 polar, 675 spherical, 1041 three-dimensional rectangular, 802 coplanar vectors, 827 Coriolis acceleration, 887 Cornu’s spiral, 673 cosine function, A26 derivative of, 151 graph of, 32, A30 power series for, 776 cost function, 177, 261 critically damped vibration, 1163 critical point(s), 959, 969 critical number, 208 cross product, 822 direction of, 824 geometric characterization of, 825 magnitude of, 825 properties of, 826 cross-section, 354 cross-section of a surface, 840 cubic function, 28 curl of a vector field, 1098 current, 173 curvature, 674, 868 curve(s) asymptotic, 249 Bézier, 660, 675 boundary, 1129 bullet-nose, 51, 162 cissoid of Diocles, 664, 684 closed, 1084 Devil’s, 169 dog saddle, 904 epicycloid, 666 equipotential, 904 grid, 1107 helix, 854 length of, 561, 866 level, 890, 896 monkey saddle, 904 orientation of, 1075, 1091 orthogonal, 170 ovals of Cassini, 685 parametric, 657, 854 piecewise-smooth, 1071 polar, 677 simple, 1085 smooth, 561, 867 space, 854 strophoid, 689, 707 swallowtail catastrophe, 665 toroidal spiral, 856 trefoil knot, 856 trochoid, 664 twisted cubic, 856 witch of Maria Agnesi, 664 curve fitting, 25 curve-sketching procedure, 243, 244 cusp, 662 cycloid, 660 cylinder, 354 parabolic, 841 parametrization of, 1109 cylindrical coordinate system, 1037 conversion equations for, 1037 triple integrals in, 1038 cylindrical shell, 365 |||| A127 damped vibration, 1162 damping constant, 1162 decay, law of natural, 447 decay, radioactive, 449 decreasing function, 20 decreasing sequence, 717 definite integral, 300, 987 properties of, 307, 309 Substitution Rule for, 336 of a vector function, 863 definite integration by parts, 489, 492 by substitution, 336 degree of a polynomial, 28 del ( ٌ ), 949 delta (⌬) notation, 117 demand curve, 261, 586 demand function, 261, 586 De Moivre, Abraham, A50 De Moivre’s Theorem, A50 density of a lamina, 1016 linear, 173 liquid, 576 mass vs weight, 576 of a solid, 1032 dependent variable, 11, 891, 939 derivative(s), 113, 116, 123 of a composite function, 157 of a constant function, 135 of a difference, 138 directional, 946, 947, 950 domain of, 123 of exponential functions, 396, 398, 433* as a function, 123 higher, 129 higher partial, 920 of hyperbolic functions, 463, 465 of an integral, 315 of an inverse function, 456 of inverse trigonometric functions, 454, 456 left-hand, 134 of logarithmic functions, 412, 422*, 443* normal, 1105 notation, 126 notation for partial, 916 partial, 915, 916 of a power function, 135, 142 of a power series, 765 of a product, 139 of a quotient, 140 as a rate of change, 117 right-hand, 134 second, 129 second partial, 862 A128 |||| INDEX derivative(s) (continued) as the slope of a tangent, 113 of a sum, 137 third, 130 of trigonometric functions, 148, 152 of a vector function, 860 Descartes, René, A11 descent of aircraft, determining start of, 164 determinant, 822 Devil’s curve, 169 Difference Law of limits, 78 Difference Rule, 138 differentiable function, 126, 931 differential, 191, 932, 934 differential equation, 147, 276, 602, 605 autonomous, 611 Bernoulli, 643 first-order, 616 general solution of, 605 homogeneous, 1147 linear, 638 linearly independent solutions, 1148 logistic, 723 nonhomogeneous, 1147, 1153 logistic, 628, 723 order of, 605 partial, 922 second-order, 605, 1147 separable, 616 solution of, 605 differentiation, 126 formulas for, 135, 144, RP5 formulas for vector functions, 862 implicit, 164, 919, 941 logarithmic, 417, 428* partial, 914, 919, 920 of a power series, 765 term by term, 765 of a vector function, 860 differentiation operator, 126 Direct Substitution Property, 80 direction angles, 817 direction cosines, 817 direction field, 608, 609, 636 direction numbers, 831 directional derivative, 946, 947, 950 maximum value of, 951 of a temperature function, 946, 947 directrix, 691, 698 discontinuity, 97, 98 discontinuous function, 97, 98 discontinuous integrand, 547 disk method, 359 dispersion, 214 displacement, 115, 328 displacement vector, 806, 819 distance between planes, 837 between point and line in space, 829 between point and plane, 829, 836 between points in a plane, A11 between points in space, 803 between real numbers, A7 distance formula, A12 in three dimensions, 803 distance problem, 296 divergence of an improper integral, 545, 548 of an infinite series, 724 of a sequence, 713 Test for, 728 of a vector field, 1101 Divergence, Test for, 728 Divergence Theorem, 1135 divergent improper integral, 545, 548 divergent sequence, 713 divergent series, 724 division of power series, 781 DNA, helical shape of, 855 dog saddle, 904 domain of a function, 11, 891 Doppler effect, 945 dot product, 815 in component form, 815 properties of, 815 double integral, 987, 989 change of variable in, 1049, 1052 over general regions, 1001, 1002 Midpoint Rule for, 991 in polar coordinates, 1010, 1011 properties of, 994, 1006 over rectangles, 987 double Riemann sum, 990 double-angle formulas, A29 Douglas, Paul, 892 drumhead, vibration of, 760 dumpster design, minimizing construction cost, 969 dye dilution method, 588 e (the number), 397, 424* as a limit, 418, 444* as a sum of an infinite series, 775 eccentricity, 698 electric charge, 1016, 1032 electric circuit, 641 analysis of, 1165 electric field (force per unit charge), 1067 electric force, 1067 electric flux, 1125 elementary function, integrability of, 523 ellipse, 170, 692, 698 area, 527 directrix, 698 eccentricity, 698 foci, 692, 698 major axis, 693 polar equation, 700, 703 reflection property, 694 rotated, 170 vertices, 693 ellipsoid, 842, 844 elliptic paraboloid, 842, 844 empirical model, 25 end behavior of a function, 230, 242 endpoint extreme values, 205, 206 energy conservation of, 1088 kinetic, 1088 potential, 1089 epicycloid, 666 equation(s) of a circle, A17 differential See differential equation of an ellipse, 693, 700, A19 of a graph, A10, A16 heat conduction, 926 of a hyperbola, 694, 695, 700, A20 Laplace’s, 922, 1102 of a line, A12, A13, A14, A16 of a line in space, 830, 831, 832 linear, 834, A14 logistic, 604 logistic differential, 604, 636 Lotka-Volterra, 645 nth-degree, 167 of a parabola, 691, 700, A18 parametric, 667, 831, 854, 1106 of a plane, 834 point-slope, 18, A12 polar, 677, 700 predator-prey, 645 second-degree, A16 slope-intercept, A13 of a space curve, 854 of a sphere, 804 symmetric, 831 two-intercept form, A16 vector, 830 wave, 922 equilateral hyperbola, A21 equilibrium point, 646 equilibrium position, 155 equilibrium solution, 604, 645 equipotential curves, 904 equivalent vectors, 806 INDEX error in approximate integration, 557 percentage, 193 relative, 193 in Taylor approximation, 785 error bounds, 535, 539 error estimate for alternating series, 748 for the Midpoint Rule, 557 for Simpson’s Rule, 561 for the Trapezoidal Rule, 557 escape velocity, 553 estimate of the sum of a series, 736, 744, 748, 753 Eudoxus, Euler, Leonard, 734, 775 Euler’s constant, 740 Euler’s formula, A52 Euler’s Method, 611 even function, 19, 244 expected values, 1023 exponential decay, 627 exponential function(s), 33, 392, 438* with base a, 392, 416, 438* derivatives of, 396, 416, 438* graphs of, 394, 399, 440* integration of, 401, 435*, 441*, 779, 780 limits of, 395, 400, 432* power series for, 772 properties of, 395, 432*, 438* exponential graph, 399, 440* exponential growth, 399, 627, 628 exponents, laws of, 395, 432*, 438* extrapolation, 27 extreme value, 205 Extreme Value Theorem, 206, 964 family of functions, 50, 253 family of hypocycloids, 665 family of solutions, 604 fat circles, 168, 567 Fermat, Pierre, 4, 123, 207, 332 Fermat’s Principle, 266 Fermat’s Theorem, 207 Fibonacci, 722 Fibonacci sequence, 712 field conservative, 1068 electric, 1067 force, 1067 gradient, 955, 1067 gravitational, 1067 incompressible, 1102 irrotational, 1100 scalar, 1065 vector, 1063, 1064 velocity, 1063, 1066 First Derivative Test, 222 for Absolute Extreme Values, 259 first octant, 801 first-order linear differential equation, 638 first-order optics, 790 fixed point of a function, 111, 220 flash bulb, current to, 62 flow lines, 1069 fluid flow, 1066, 1102, 1124 flux, 588, 588, 1123, 1125 flux integral, 1123 FM synthesis, 253 focus of a conic section, 698 of an ellipse, 692, 698 of a hyperbola, 694 of a parabola, 691 folium of Descartes, 165, 708 force, 370 centripetal, 888 constant, 819 exerted by fluid, 575 resultant, 812 torque, 827 force field, 1063, 1067 forced vibrations, 1164 Fourier, Joseph, 179 Fourier sequence, 503 four-leaved rose, 679 fractions (partial), 509, 510 Frenet-Serret formulas, 874 Fresnel, Augustin, 317 Fresnel function, 317 frustum of a cone, 363 of a pyramid, 364 Fubini, Guido, 997 Fubini’s Theorem, 997, 1027 function(s), 11 absolute value, 17 Airy function, 764 algebraic, 31 arc length, 564, 866, 867 arrow diagram of, 12 average cost, 265 average value of, 374, 375, 593, 992, 1036 Bessel, 710, 760, 764 Cobb-Douglas production, 893, 923, 976 combinations of, 41 component, 853, 1064 composite, 41, 156, 911 constant, 135 |||| A129 continuity of, 97, 854, 910, 912 cost, 177, 261 cubic, 28 decreasing, 20 demand, 261 derivative of, 116 differentiability of, 126, 931 discontinuous, 97, 98 domain of, 11, 891 elementary, 523 even, 19, 244 exponential, 33, 392, 398 extreme values of, 205 family of, 50, 253 fixed point of, 111, 220 Fresnel, 317 Gompertz, 636 gradient of, 949, 951 graph of, 12, 894 greatest integer, 83 harmonic, 922 Heaviside, 45, 70 homogeneous, 945 hyperbolic, 463 implicit, 164 increasing, 20 integrable, 989 inverse, 385, 386 inverse hyperbolic, 466, 467 inverse trigonometric, 454, 455 joint density, 1021, 1032 limit of, 66, 88, 907, 912 linear, 24, 894 logarithmic, 405, 421*, 442* machine diagram of, 12 marginal cost, 118, 178, 261 marginal profit, 262 marginal revenue, 261 maximum and minimum values of, 205, 958, 959 of n variables, 901 natural logarithmic, 406, 421* nondifferentiable, 128 odd, 19, 244 one-to-one, 385 periodic, 244 piecewise defined, 17 polynomial, 28, 910 position, 115 potential, 1068 power, 29, 142 probability density, 591, 1021 profit, 262 quadratic, 28 ramp, 45 range of, 11, 891 A130 |||| INDEX function(s) (continued) rational, 31, 509, 910 reciprocal, 31 reflected, 38 representation as a power series, 764 representations of, 11, 13 revenue, 261 root, 30 of several variables, 891, 900 shifted, 37 sine integral, 322 step, 18 stretched, 38 of three variables, 900 of two variables, 891 transcendental, 34 transformation of, 37, 38 translation of, 37 trigonometric, 32, A26 value of, 11 vector, 853 vector-valued, 853 Fundamental Theorem of Calculus, 313, 315, 318, 320 higher-dimensional versions, 1141 for line integrals, 1082 for vector functions, 864 G (gravitational constant), 181, 374 Gabriel’s horn, 573 Galileo, 661, 669 Galois, Evariste, 167 Gause, G F., 632 Gauss, Karl Friedrich, 1135, A35 Gaussian optics, 790 Gauss’s Law, 1126 Gauss’s Theorem, 1135 geometric series, 724 geometry of a tetrahedron, 830 Gompertz function, 636 gradient, 949, 951 gradient vector, 949, 951 interpretations of, 955, 956 gradient vector field, 955, 1067 graph(s) of an equation, A10, A16 of exponential functions, 394, 399, 440* of a function, 12 of a function of two variables, 894 of logarithmic functions, 405, 408, 424*, 442*, 443* of a parametric curve, 658, 1119 polar, 677 of power functions, 30, RP3 of a sequence, 716 of trignometric functions, A30, RP2 graphing calculator, 46, 250, 660, 682 graphing device See computer algebra system gravitation law, 180, 374 gravitational acceleration, 370 gravitational field, 1067 great circle, 1047 greatest integer function, 83 Green, George, 1092, 1134 Green’s identities, 1105 Green’s Theorem, 1091, 1134 vector forms, 1102, 1103 Gregory, James, 768, 772 Gregory’s series, 768 grid curve, 1107 ground speed, 814 growth, law of natural, 447, 627 growth rate, 175 relative, 448 half-angle formulas, A29 half-life, 449 half-space, 900 hare-lynx system, 648 harmonic function, 922 harmonic series, 727 harmonic series, alternating, 747 heat conduction equation, 926 heat conductivity, 1126 heat flow, 1126 heat index, 914 Heaviside, Oliver, 70 Heaviside function, 45, 70 Hecht, Eugene, 191, 790 height of a rocket, 495 helix, 854 higher derivatives, 129 higher partial derivatives, 920 homogeneous differential equation, 1147 homogeneous function, 945 Hooke’s Law, 371, 1161 horizontal asymptote, 232 horizontal line, A13 Horizontal Line Test, 385 horizontal plane, equation of, 802 Hubble Space Telescope, 210 Huygens, Christiaan, 661 hydrostatic pressure and force, 575 hydro-turbine optimization, 979 hyperbola, 169, 694, 698, A20 asymptotes, 694, A20 branches, 694, A20 directrix, 698 eccentricity, 698 equation, 694, A20 equilateral, A21 foci, 694, 698 polar equation, 700 reflection property, 698 vertices, 694 hyperbolic function(s), 463 derivatives, 463, 465 inverse, 466, 467 hyperbolic identities, 464 hyperbolic paraboloid, 843, 844 hyperbolic substitution, 507 hypocycloid, 665, 844, 846 hypersphere, 1036 hypocycloid, 665 i (imaginary number), A46 i (standard basis vector), 810 I/D Test, 221 ideal gas law, 182, 927 image of a point, 1049 image of a region, 1049 implicit differentiation, 164, 919, 941 implicit function, 164 Implicit Function Theorem, 942 improper integral, 544 impulse of a force, 637 incompressible velocity field, 1102 increasing function, 20 increasing sequence, 717 Increasing/Decreasing Test, 221 increment, 117, 934 indefinite integrals, 324 table of, 325 independence of path, 1083 independent random variable, 1022 independent variable, 11, 891, 939 indeterminate difference, 470, 471 indeterminate forms of limits, 470, 471 indeterminate power, 476 indeterminate product, 474 index of summation, A34 inequalities, rules for, A4 inertia (moment of ), 1019, 1032, 1081 infinite discontinuity, 98 infinite interval, 544 infinite limit, 72, 94, 236 infinite sequence See sequence infinite series See series inflection point, 225 initial condition, 606 initial point of a parametric curve, 658 of a vector, 806 initial-value problem, 606, 1151 inner product, 815 instantaneous rate of change, 64, 117, 171 INDEX instantaneous rate of growth, 175 instantaneous rate of reaction, 174 instantaneous velocity, 64, 115, 171 integer, A2 integrable function, 989 integral(s) approximations to, 306 change of variables in, 334, 1047, 1052, 1055 comparison properties of, 307 conversion to cylindrical coordinates, 1038 conversion to polar coordinates, 1011 conversion to spherical coordinates, 1043 definite, 300, 863, 987 derivative of, 334 double, 987, 989 See also double integral evaluating, 303 improper, 544 indefinite, 324 iterated, 995, 996 line, 1070 See also line integral patterns in, 530 properties of, 307 surface, 1117, 1123 of symmetric functions, 337 table of, 520, RP6–10 trigonometric, 496, 498, 499 triple, 1026 See also triple integral units for, 329 Integral Test, 733, 735 integrand, 300 discontinuous, 547 integration, 300 approximate, 531 by computer algebra system, 527 of exponential functions, 401, 435*, 441* formulas, 488, 520, RP6–10 indefinite, 324 limits of, 300 numerical, 531 partial, 996 by partial fractions, 509 by parts, 489, 492 of a power series, 765 reversing order of, 998, 1006 term by term, 765 of a vector function, 863 intercept(s), 244, A19 Intermediate Value Theorem, 104 intermediate variable, 939 interpolation, 27 intersection of planes, 835 intersection of polar graphs, area of, 687 intersection of sets, A3 intersection of three cylinders, 1041 interval, A3 interval of convergence, 761 inverse function(s), 40 inverse transformation, 1049 irrational number, A2 irrotational vector field, 1100 isobars, 890, 897 isothermal compressibility, 175 isothermals, 897, 904 iterated integral, 995, 996 j (standard basis vector), 810 Jacobi, Carl, 1051 Jacobian of a transformation, 1051, 1055 jerk, 131 joint density function, 1021, 1032 joule, 370 jump discontinuity, 98 k (standard basis vector), 810 Kampyle of Eudoxus, 169 Kepler, Johannes, 880, 884 Kepler’s Laws, 880, 884 kinetic energy, 1088 Kirchhoff’s Laws, 1165 Kondo, Shigeru, 775 Lagrange, Joseph, 216, 217, 971 Lagrange multiplier, 970, 971 lamina, 579, 1016 Laplace, Pierre, 922, 1102 Laplace operator, 1102 Laplace’s equation, 922, 1102 lattice point, 203 Law of Conservation of Angular Momentum, 884 Law of Conservation of Energy, 1089 law of cosines, A33 law of gravitation, 180, 374 law of laminar flow, 176, 587 least squares method, 27, 968 least upper bound, 718 left-hand derivative, 134 left-hand limit, 71, 91 Leibniz, Gottfried Wilhelm, 4, 126, 332, 616, 784 Leibniz notation, 126 lemniscate, 169 length of a curve, 561 of a line segment, A7, A12 of a parametric curve, 669 of a polar curve, 688 of a space curve, 866 of a vector, 809 level curve(s), 890, 896 of barometric pressure, 890 |||| A131 of temperatures, 897 level surface, 901 tangent plane to, 953 l’Hospital, Marquis de, 472, 481, 482 l’Hospital’s Rule, 472, 481, 482 origins of, 481, 482 libration point, 274 limaỗon, 683 Limit Comparison Test, 743 Limit Laws, 77, 78 Limit Laws for sequences, 714 limit(s), 3, 66 calculating, 77, 233, 234 of exponential functions, 395, 400, 432* of a function, 66, 88 of a function of three variables, 912 of a function of two variables, 907 infinite, 72, 94, 236 at infinity, 230, 231, 236 of integration, 300 left-hand, 71, 91 one-sided, 70, 91 precise definitions, 88, 91, 94, 237, 238, 240 properties of, 77, 83 right-hand, 71, 91 of a sequence, 6, 291, 713 involving sine and cosine functions, 150, 151 of a vector function, 853 Limit Comparison Test, 743 Limit Laws for functions of two variables, 909 for sequences, 714 linear approximation, 189, 930, 934 linear combination, 1147 linear density, 173 linear differential equation, 638, 1147 linear equation, A14 of a plane, 834 linear function, 24, 894 linearity of an integral, 994 linearization, 189, 930 linearly independent solutions, 1148 linear model, 24 linear regression, 27 line(s) in the plane, A12 equations of, A12, A13, A14 horizontal, A13 normal, 145, 146 parallel, A14 perpendicular, A14 secant, 4, 62 slope of, A120 tangent, 4, 62, 113 A132 |||| INDEX line(s) in space normal, 954 parametric equations of, 831 skew, 833 symmetric equations of, 831 tangent, 860 vector equation of, 830, 831 line integral, 1070 with respect to arc length, 1073 Fundamental Theorem for, 1082 for a plane curve, 1070 for a space curve, 1075 of vector fields, 1077, 1078 work defined as, 1077 liquid force, 575, 576 Lissajous figure, 665 lithotripsy, 694 local maximum and minimum values, 205, 959 logarithm(s), 34, 405, 421*, 438* laws of, 405, 422* natural, 406, 421* notation for, 405, 443* logarithmic differentiation, 417, 428* logarithmic function(s), 34, 405, 421*, 442* with base a, 405 derivatives of, 412, 422*, 443* graphs of, 405, 408, 424*, 442*, 443* integration of, 417, 428* limits of, 406, 408, 424* properties of, 405, 422* logistic difference equation, 723 logistic differential equation, 604, 628 logistic model, 636 logistic sequence, 723 LORAN system, 697 Lotka-Volterra equations, 645 machine diagram of a function, 12 Maclaurin, Colin, 772 Maclaurin series, 770, 772 table of, 779 magnitude of a vector, 809 major axis of ellipse, 693 marginal cost function, 118, 178, 261 marginal profit function, 262 marginal productivity, 923 marginal propensity to consume or save, 731 marginal revenue function, 261 mass of a lamina, 1016 of a solid, 1032 of a surface, 1119 of a wire, 1072 mass, center of See center of mass mathematical induction, 731 principle of, 55, 58, A36 mathematical model, 14, 24 Cobb-Douglas, for production costs, 893, 923, 976 for vibration of membrane, 760 maximum and minimum values, 205, 958, 959 mean life of an atom, 553 mean of a probability density function, 593 Mean Value Theorem, 214, 216 for double integrals, 1059 for integrals, 375 mean waiting time, 593 median of a probability density function, 594 method of cylindrical shells, 365 method of exhaustion, 3, 80 method of Lagrange multipliers, 970, 971, 974 method of least squares, 27, 968 method of undetermined coefficients, 1154, 1158 midpoint formula, A16 Midpoint Rule, 306, 532 for double integrals, 991 error in using, 533 for triple integrals, 1034 mixing problems, 620 Möbius, August, 1121 Möbius strip, 1115, 1121 modeling, mathematical with differential equations, 603 motion of a spring, 604 population growth, 396, 603, 627, 633, 636, 652 vibration of membrane, 760 model(s), mathematical, 24 comparison of natural growth vs logistic, 632 empirical, 25 exponential, 33 Gompertz function, 636 linear, 24 logarithmic, 34 polynomial, 28 power function, 29 predator-prey, 182, 645 rational function, 31 seasonal growth, 636 trigonometric, 32, 33 von Bertalanffy, 652 modulus, A47 moment about an axis, 579, 1017 of inertia, 1019, 1032, 1081 of a lamina, 579, 1017 of a mass, 579 about a plane, 1032 polar, 1019 second, 1019 of a solid, 1031 of a system of particles, 579 momentum of an object, 637 monkey saddle, 904 monotonic sequence, 717 Monotonic Sequence Theorem, 719 motion in space, 874 motion of a spring, force affecting restoring, 1161 damping, 1162 resonance, 1165 movie theater seating, 463 multiple integrals See double integral; triple integral multiplication, scalar, of vectors, 807, 809 multiplication of power series, 781 multiplier (Lagrange), 970, 971, 974 multiplier effect, 731 natural exponential function, 398, 400, 430* derivatives of, 398, 433*, 434* graph of, 399, 432* power series for, 772 properties of, 400, 432* natural growth law, 447, 627 natural logarithm function, 406, 421* derivatives of, 413, 424*, 427* limits of, 408, 424* properties of, 405, 422* n-dimensional vector, 810 negative angle, A25 net area, 301 Net Change Theorem, 327 net investment flow, 590 newton (unit of force), 370 Newton, Sir Isaac, 4, 9, 80, 123, 313, 332, 784, 880, 884 Newton’s Law of Cooling, 450 Newton’s Law of Gravitation, 180, 374, 880, 1066 Newton’s method, 269 Newton’s Second Law of Motion, 370, 876, 880, 1161 Nicomedes, 662 nondifferentiable function, 128 nonhomogeneous differential equation, 1147, 1153 nonparallel planes, 835 INDEX normal component of acceleration, 878 normal derivative, 1105 normal distribution, 595 normal line, 145, 146, 954 normal plane, 871 normal vector, 833, 870 nth-degree equation, finding roots of, 167 nth-degree Taylor polynomial, 195, 773 number complex, A46 integer, A2 irrational, A2 rational, A2 real, A2 numerical integration, 531 octant, 801 odd function, 19, 244 one-sided limits, 70, 91 one-to-one function, 385 one-to-one transformation, 1049 open interval, A3 open region, 1084 optics first-order, 790 Gaussian, 790 third-order, 791 optimization problems, 205, 256 orbits of planets, 880, 884 order of a differential equation, 605 order of integration, reversed, 998, 1006 ordered pair, A10 ordered triple, 801 Oresme, Nicole, 728 orientation of a curve, 1075, 1091 of a surface, 1122 oriented surface, 1121, 1122 origin, A2, A10 orthogonal curves, 170 orthogonal projection, 821 orthogonal surfaces, 958 orthogonal trajectory, 170, 619 orthogonal vectors, 817 osculating circle, 871 osculating plane, 871 Ostrogradsky, Mikhail, 1135 ovals of Cassini, 685 overdamped vibration, 1163 Pappus, Theorem of, 582 Pappus of Alexandria, 582 parabola, 691, 698, A18 axis, 691 directrix, 691, 698 equation, 691, 692 focus, 691, 698 polar equation, 700 reflection property, 203 vertex, 691 parabolic cylinder, 841 paraboloid, 842, 846 paradoxes of Zeno, 6, parallel lines, A14 parallel planes, 835 parallel vectors, 807 parallelepiped, 354 volume of, 827 Parallelogram Law, 807, 822 parameter, 657, 831, 854 parametric curve, 657, 854 arc length of, 669 area under, 668 slope of tangent line to, 666 parametric equations, 657 of a line, 831 of a space curve, 854 of a surface, 1106 of a trajectory, 877 parametric surface, 1106 graph of, 1119 surface area of, 1111, 1112 surface integral over, 1117 tangent plane to, 1110 parametrization of a space curve, 856 with respect to arc length, 867 smooth, 867 paraxial rays, 191 partial derivative(s), 915, 916 of a function of more than three variables, 919 interpretations of, 917 notations for, 916 as rates of change, 916 rules for finding, 916 second, 920 as slopes of tangent lines, 917 partial differential equation, 922 partial fractions, 509, 510 partial integration, 489, 490, 996 partial sum of a series, 724 particle, motion of, 874 parts, integration by, 489, 490 path, 1083 patterns in integrals, 530 pendulum, approximating the period of, 191, 194 percentage error, 193 perihelion, 703 perilune, 697 period, 244 periodic function, 244 |||| perpendicular lines, A14 perpendicular vectors, 817 phase plane, 646 phase portrait, 646 phase trajectory, 646 piecewise defined function, 17 piecewise-smooth curve, 1071 Planck’s Law, 793 plane(s), 833 coordinate, 801 equation(s) of, 833, 834 horizontal, 802 normal, 871 osculating, 871 parallel, 835 tangent to a surface, 928, 953, 1110 vertical, 802 plane region of type I, 1002 plane region of type II, 1003 planetary motion, 880 point of inflection, 225 point(s) in space coordinates of, 801 distance between, 803 projection of, 802 point-slope equation of a line, 18, A12 Poiseuille, Jean-Louis-Marie, 176 Poiseuille’s Laws, 194, 267, 588 polar axis, 675 polar coordinate system, 675 area in, 686 conic sections in, 698 conversion equations for Cartesian coordinates, 676, 677 conversion of double integral to, 1010, 1011 polar curve, 677 arc length of, 688 graph of, 677 symmetry in, 680 tangent line to, 680 polar equation, graph of, 677 polar equation of a conic, 700 polar form of a complex number, A48 polar graph, 677 polar moment of inertia, 1019 polar rectangle, 1010 polar region, area of, 686 pole, 675 polynomial function, 28 of two variables, 910 population growth, 627 of bacteria, 176, 627, 634 of insects, 519 models, 603 world, 396, 448, 449 A133 A134 |||| INDEX position function, 115 position vector, 809 positive angle, A25 positive orientation of a boundary curve, 1129 of a closed curve, 1091 of a surface, 1122 potential, 556 potential energy, 1089 potential function, 1068 pound, 370 power consumption, approximation of, 329 power function, 29, 142 Power Law of limits, 79 Power Rule, 136, 142 power series, 759 coefficients of, 759 for cosine, 776 differentiation of, 765 division of, 781 for an exponenial function, 776 integration of, 765 interval of convergence, 761 multiplication of, 781 radius of convergence, 761 representations of functions as, 764 for sine, 776 predator, 644 predator-prey model, 182, 645 pressure exerted by a fluid, 576 prey, 645 prime notation, 116, 137, 139, 140 principal square root of a complex number, A47 principal unit normal vector, 870 principle of mathematical induction, 55, 58, A36 principle of superposition, 1156 probability, 1021 probability density function, 591, 1021 problem-solving principles, 54 producer surplus, 589 product cross, 822 See also cross product dot, 815 See also dot product scalar, 815 scalar triple, 826 triple, 826 product formulas, A29 Product Law of limits, 78 Product Rule, 139 profit function, 262 projectile, path of, 665, 877 projection, 802, 818, 819, 821 p-series, 735 quadrant, A11 quadratic approximation, 195, 969 quadratic function, 28 quadric surface(s), 841 cone, 844 cylinder, 841 ellipsoid, 842, 844 hyperboloid, 844, 846 paraboloid, 842, 846 table of graphs, 844 Quotient Law of limits, 78 Quotient Rule, 140 radian measure, 148, A24 radiation from stars, 793 radioactive decay, 449 radiocarbon dating, 454 radius of convergence, 761 radius of gyration, 1020 rainbow, formation and location of, 213 rainbow angle, 213 ramp function, 45 range of a function, 11, 891 rate of change average, 117, 171 derivative as, 117 instantaneous, 64, 117, 171 rate of growth, 175 rate of reaction, 119, 174 rates, related, 182 rational function, 31, 910 integration of, 509 rational number, A2 rationalizing substitution, 517 Ratio Test, 752 Rayleigh-Jeans Law, 793 real line, A3 real number, A2 rearrangement of a series, 755 reciprocal function, 31 rectangular coordinate system, A11 conversion to cylindrical coordinates, 1037 conversion to spherical coordinates, 1042 three-dimensional, 802 rectilinear motion, 277 recursion relation, 1170 reduction formula, 493 reflecting a function, 38 reflection property of an ellipse, 694 of a hyperbola, 698 of a parabola, 203 region connected, 1084 under a graph, 289, 295 open, 1084 plane, of type I or II, 1002, 1003 simple plane, 1092 simple solid, 1135 simply-connected, 1085 solid (of type 1, 2, or 3), 1027, 1029 between two graphs, 347 related rates, 182 relative error, 193 relative growth rate, 448, 628 relative maximum and minimum, 205 remainder estimates for the Alternating Series, 748 for the Integral Test, 737 remainder of the Taylor series, 773 removable discontinuity, 98 representations of functions, 11, 13 as power series, 764 resonance, 1165 restoring force, 1161 resultant force, 812 revenue function, 261 reversing order of integration, 998, 1006 revolution, solid of, 359 revolution, surface of, 568 Riemann, Georg Bernhard, 301 Riemann sum(s), 301 for multiple integrals, 990, 1026 right circular cylinder, 354 right-hand derivative, 134 right-hand limit, 71, 91 right-hand rule, 801, 824 RMS voltage, 502 Roberval, Gilles de, 319, 669 rocket science, 977 Rolle, Michel, 214 roller coaster, design of, 148 roller derby, 1048 Rolle’s Theorem, 214 root function, 30 root law for limits, 79 Root Test, 754 roots of a complex number, A51 roots of an nth-degree equation, 167 rubber membrane, vibration of, 760 ruled surface, 848 ruling of a surface, 840 rumors, rate of spread of, 179 saddle point, 960 sample point, 294, 988 satellite dish, parabolic, 846 scalar, 807 scalar equation of a plane, 834 scalar field, 1064 scalar multiple of a vector, 807 INDEX scalar product, 815 scalar projection, 818, 819 scalar triple product, 826 geometric characterization of, 827 scatter plot, 14 seasonal growth model, 636 secant function, A26 derivative, 152 graph, A31 secant line, 4, 62 secant vector, 860 second derivative, 129 of a vector function, 862 Second Derivative Test, 225 Second Derivatives Test, 960 second moment of inertia, 1019 second partial derivative, 920 second-order differential equation, 605 solutions of, 1147, 1152 sector of a circle, area of, 686 sensitivity, 182 separable differential equation, 616 sequence, 6, 711 bounded, 718 convergent, 713 decreasing, 717 divergent, 713 Fibonacci, 712 graph of, 716 increasing, 717 limit of, 6, 291, 713 limit laws for, 714 monotonic, 717 of partial sums, 724 terms of, 711 series, 7, 723 absolutely convergent, 750 alternating, 746 alternating harmonic, 747, 751 binomial, 778, 784 coefficients of, 759 conditionally convergent, 751 convergent, 724 divergent, 724 geometric, 724 Gregory’s, 768 harmonic, 727 infinite, 723 Maclaurin, 770, 772 p-, 735 partial sum of, 724 power, 759 rearrangement of, 755 strategy for testing, 756 sum of, 7, 724 Taylor, 770, 772 terms of, 723 trigonometric, 759 series solution of a differential equation, 1169 set, A3 set, bounded or closed, 964 serpentine, 145 shell method, 365 shift of a function, 37 shifted conics, 695, A21 shock absorber, 1162 Sierpinski carpet, 732 sigma notation, 294, A34 simple curve, 1085 simple harmonic motion, 163, 1161 simple plane region, 1092 simple solid region, 1135 simply-connected region, 1085 Simpson, Thomas, 537, 538, 985 Simpson’s Rule, 536, 538 error bounds for, 539 sine function, A26 derivative of, 151 graph of, 32, A30 power series for, 776 sine integral function, 322 sink, 1139 skew lines, 833 slant asymptote, 247 slope, A12 of a curve, 113 slope field, 609 slope-intercept equation of a line, A13 smooth curve, 561, 867 smooth function, 561 smooth parametrization, 867 smooth surface, 1111 Snell’s Law, 266 snowflake curve, 797 solid, 354 volume of, 359, 366, 1027, 1028 solid angle, 1145 solid region, 1135 solid of revolution, 359 rotated on a slant, 574 volume of, 359, 366, 574 solution curve, 608 solution of predator-prey equations, 645 source, 1139 space, three-dimensional, 801 space curve, 854 arc length of, 866, 867 speed of a particle, 118, 875 sphere equation of, 804 flux across, 1124 |||| parametrization of, 1108 surface area of, 1112 spherical coordinate system, 1041 conversion equations for, 1042 triple integrals in, 1042 spherical wedge, 1043 spherical zone, 600 spring constant, 371, 604, 1161 Squeeze Theorem, 83, A42 for sequences, 715 standard basis vectors, 810, 811 standard deviation, 595 static friction, coefficient of, 851 stationary points, 959 steady state solution, 1167 step function, 18 Stokes, Sir George, 1129, 1134 Stokes’ Theorem, 1128, 1129 strategy for integration, 519, 520 for optimization problems, 256 for problem solving, 54 for related rates, 185 for testing series, 757 for trigonometric integrals, 498, 499 streamlines, 1069 stretching a function, 38 strophoid, 689, 707 Substitution Rule, 333, 334, 336 subtraction formulas for sine and cosine, A29 sum of a geometric series, 725 of an infinite series, 724 of partial fractions, 510 Riemann, 301 telescoping, 727 of vectors, 806, 809 Sum Law of limits, 78 Sum Rule, 137 summation notation, A34 supply function, 589 surface(s), 802 closed, 1122 graph of, 1119 level, 901 oriented, 1122 parametric, 1106 positive orientation of, 1122 quadric, 841 See also quadric surface smooth, 1111 surface area, 570, 574 of a parametric surface, 671, 1111 of a sphere, 1112 of a surface z f ͑x, y͒, 1113 A135 A136 |||| INDEX surface integral, 1117 over a parametric surface, 1117 of a vector field, 1123 surface of revolution, 568 area of, 570, 574 parametric representation of, 1109 swallowtail catastrophe curve, 665 symmetric equations of a line, 831 symmetric functions, integrals of, 337 symmetry, 19, 244, 337 in polar graphs, 680 symmetry principle, 579 T and T Ϫ1 transformations, 1049 table of differentiation formulas, 144, RP5 tables of integrals, 520, RP6 –10 use of, 525 tangent function, A26 derivative, 152 graph of, 33, A31 tangent line(s), 113 to a curve, 4, 61, 113 early methods of finding, 123 to a parametric curve, 666 to a polar curve, 680 to a space curve, 860 tangent line approximation, 189 tangent plane to a level surface, 953 to a parametric surface, 1110 to a surface F͑x, y, z͒ k, 953 to a surface z f ͑x, y͒, 928 tangent plane approximation, 930 tangent problem, 4, 61, 113 tangent vector, 860 tangential component of acceleration, 878 tautochrone problem, 661 Taylor, Brook, 772 Taylor polynomial, 195, 773, 969 applications of, 785 Taylor series, 770, 772 Taylor’s Inequality, 773 techniques of integration, summary, 520 telescoping sum, 727 temperature-humidity index, 902, 914 term of a sequence, 711 term of a series, 723 term-by-term differentiation and integration, 765 terminal point of a parametric curve, 658 of a vector, 806 terminal velocity, 623 Test for Divergence, 728 tests for convergence and divergence of series Alternating Series Test, 746 Comparison Test, 741 Integral Test, 733, 735 Limit Comparison Test, 743 Ratio Test, 752 Root Test, 754 summary of tests, 757 Test for Divergence, 728 tetrahedron, 830 third derivative, 130 Thomson, Sir William (Lord Kelvin), 1092, 1129, 1134 three-dimensional coordinate system, 802 TNB frame, 871 toroidal spiral, 856 torque, 827, 884 Torricelli, Evangelista, 669 Torricelli’s Law, 180 torsion of a space curve, 874 torus, 364, 509, 1117 total differential, 932 total electric charge, 1016, 1032 total fertility rate, 197 trace of a surface, 840 trajectories, orthogonal, 170 trajectory, parametric equations for, 877 transcendental function, 34 transfer curve, 887 transformation, 1049 of a function, 37 inverse, 1049 Jacobian of, 1051, 1055 one-to-one, 1049 translation of a function, 37 Trapezoidal Rule, 533 error in, 533 tree diagram, 939 trefoil knot, 856 Triangle Inequality, 93, A8 for vectors, 822 Triangle Law, 807 trigonometric functions, 32, A26 derivatives of, 148, 152 graphs of, A30 integrals of, 496 inverse, 454, 455 limits involving, 150, 151 trigonometric identities, A28 trigonometric integrals, 496 strategy for evaluating, 498, 499 trigonometric series, 759 trigonometric substitution, 503 table of, 503 triple integral(s), 1026 applications of, 1031 in cylindrical coordinates, 1038 over a general bounded region, 1027 Midpoint Rule for, 1034 in spherical coordinates, 1043, 1044 triple product, 826 triple Riemann sum, 1026 trochoid, 664 Tschirnhausen cubic, 169, 353 twisted cubic, 856 type I or type II plane region, 1002, 1003 type 1, 2, or solid region, 1027, 1029 ultraviolet catastrophe, 793 underdamped vibration, 1163 undetermined coefficients, method of, 1154, 1158 union of sets, A3 unit normal vector, 870 unit tangent vector, 860 unit vector, 811 value of a function, 11 variable continuous random, 591 dependent, 11, 891, 939 independent, 11, 891, 939 independent random, 1022 intermediate, 939 variables, change of See change of variables variation of parameters, method of, 1158, 1159 vascular branching, 267 vector(s), 806 acceleration as, 875 addition of, 806, 809 algebraic, 808 angle between, 815 basis, 810, 811 binormal, 870 components of, 808 coplanar, 827 cross product of, 822 difference of, 807 displacement, 806, 819 dot product, 815, 816 equivalent, 806 force, 1066 geometric representations of, 808 gradient, 949, 951 i, j, and k, 811 initial point of, 806 length of, 809 magnitude of, 809 INDEX multiplication of, 807, 809 n-dimensional, 810 negative, 807 normal, 833, 870 orthogonal, 817 parallel, 807 perpendicular, 817 position, 809 principal unit normal, 870 projection of, 818, 819 properties of, 810 scalar multiple of, 807 standard basis, 811 subtraction of, 809 sum of, 806, 809 tangent, 860 terminal point of, 806 three-dimensional, 808 triple product, 826, 827 two-dimensional, 808 unit, 808 unit normal, 870 unit tangent, 860 velocity, 867 wind velocity, 800 zero, 806 vector equation of a line, 830, 831 of a line segment, 833 of a plane, 834 of a plane curve, 854 vector field, 1063, 1064 conservative, 1068 curl of, 1098 divergence of, 1101 electric flux of, 1125 force, 1063, 1067 flux of, 1123 gradient, 1067 gravitational, 1067 incompressible, 1102 irrotational, 1100 line integral of, 1077, 1078 surface integral of, 1123 velocity, 1063 vector function, 853 continuity of, 854 derivative of, 860 differentiation formulas for, 862 integration of, 863 limit of, 853 parametric equations of, 854 vector product, 822 properties of, 826 vector projection, 818, 819 vector triple product, 827 vector-valued function See vector function velocity, 4, 63, 115, 171 average, 5, 63, 115, 171 instantaneous, 64, 115, 171 velocity field, 1066 air flow, 1063 ocean currents, 1063 wind patterns, 1063 velocity gradient, 177 velocity problem, 63, 114 velocity vector, 867 velocity vector field, 1063 Verhulst, Pierre-Franỗois, 604 vertex of a parabola, 691 vertical asymptote, 73, 244 Vertical Line Test, 16 vertical plane, equation of, 802 vertical tangent line, 129 vertical translation of a graph, 37 vertices of an ellipse, 693 vertices of a hyperbola, 694 vibrating spring, 1161 vibration of a rubber membrane, 760 vibrations, 1161, 1162, 1164 viewing rectangle, 46 volume, 354 by cross-sections, 355 by cylindrical shells, 365, 366 by disks, 356, 359 |||| by double integrals, 987 of a hypersphere, 1036 of a solid, 355, 360, 989 of a solid of revolution, 359, 574 of a solid on a slant, 574 by triple integrals, 1031 by washers, 358, 359 Volterra, Vito, 645 Von Bertalanffy model, 652 Wallis, John, Wallis product, 495 washer method, 358 wave equation, 922 Weierstrass, Karl, 518 weight, 370 wind patterns in San Francisco Bay area, 1063 wind-chill index, 892 witch of Maria Agnesi, 145, 664 work, 370, 371, 819 defined as a line integral, 1077 Wren, Sir Christopher, 671 x-axis, 837, A10 x-coordinate, 837, A10 x-intercept, A19 X-mean, 1023 xy-plane, 802 xz-plane, 802 y-axis, 837, A10 y-coordinate, 837, A10 y-intercept, A19 Y-mean, 1023 yz-plane, 802 z-axis, 837 z-coordinate, 837 Zeno, Zeno’s paradoxes, zero vector, 806 zone, spherical, 600ˆ A137 ... Tools for Enriching Calculus (TEC) Modules, Visuals, and Homework Hints CONTENT N Diagnostic Tests The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Functions,... STEWART xxiii DIAGNOSTIC TESTS Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry The... calculus WHAT’S NEW IN THE SIXTH EDITION? Here are some of the changes for the sixth edition of Calculus N N N N N N N At the beginning of the book there are four diagnostic tests, in Basic Algebra,