calculus fourth edition pdf

www.EngineeringBooksPDF.com This page intentionally left blank www.EngineeringBooksPDF.com P1: OSO/OVY P2: OSO/OVY MHDQ256-FM-MAIN QC: OSO/OVY MHDQ256-Smith-v1.cls T1: OSO January 8, 2011 LT (Late Transcendental) 12:17 Calculus Fourth Edition RO B E R T T SM I T H Millersville University of Pennsylvania RO LA ND B M I N T O N Roanoke College www.EngineeringBooksPDF.com i CONFIRMING PAGES P1: OSO/OVY P2: OSO/OVY MHDQ256-FM-MAIN QC: OSO/OVY MHDQ256-Smith-v1.cls T1: OSO January 8, 2011 LT (Late Transcendental) 12:17 CALCULUS, FOURTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright c 2012 by The McGraw-Hill Companies, Inc All rights reserved Previous editions c 2008, 2002, and 2000 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper QVR/QVR ISBN 978–0–07–338311–8 MHID 0–07–338311–2 Vice President, Editor-in-Chief: Marty Lange Vice President, EDP: Kimberly Meriwether David Senior Director of Development: Kristine Tibbetts Editorial Director: Stewart K Mattson Sponsoring Editor: John R Osgood Developmental Editor: Eve L Lipton Marketing Manager: Kevin M Ernzen Lead Project Manager: Peggy J Selle Senior Buyer: Sandy Ludovissy Lead Media Project Manager: Judi David Senior Designer: Laurie B Janssen Cover Designer: Ron Bissell Cover Image: c Gettyimages/George Diebold Photography Senior Photo Research Coordinator: John C Leland Compositor: Aptara, Inc Typeface: 10/12 Times Roman Printer: Quad/Graphics All credits appearing on page or at the end of the book are considered to be an extension of the copyright page Library of Congress Cataloging-in-Publication Data Smith, Robert T (Robert Thomas), 1955Calculus / Robert T Smith, Roland B Minton.— 4th ed p cm Includes index ISBN 978–0–07–338311–8—ISBN 0–07–338311–2 (hard copy : alk paper) Transcendental functions—Textbooks Calculus—Textbooks I Minton, Roland B., 1956– II Title QA353.S649 2012 515 22—dc22 2010030314 www.mhhe.com www.EngineeringBooksPDF.com ii CONFIRMING PAGES P1: OSO/OVY P2: OSO/OVY MHDQ256-FM-MAIN QC: OSO/OVY MHDQ256-Smith-v1.cls T1: OSO January 8, 2011 12:17 LT (Late Transcendental) DE DIC AT ION To Pam, Katie and Michael To Jan, Kelly and Greg And in memory of our parents: George and Anne Smith and Paul and Mary Frances Minton www.EngineeringBooksPDF.com iii CONFIRMING PAGES P1: OSO/OVY P2: OSO/OVY MHDQ256-FM-MAIN QC: OSO/OVY MHDQ256-Smith-v1.cls T1: OSO January 8, 2011 12:17 LT (Late Transcendental) About the Authors Robert T Smith is Professor of Mathematics and Dean of the School of Science and Mathematics at Millersville University of Pennsylvania, where he has been a faculty member since 1987 Prior to that, he was on the faculty at Virginia Tech He earned his Ph.D in mathematics from the University of Delaware in 1982 Professor Smith’s mathematical interests are in the application of mathematics to problems in engineering and the physical sciences He has published a number of research articles on the applications of partial differential equations as well as on computational problems in x-ray tomography He is a member of the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics Professor Smith lives in Lancaster, Pennsylvania, with his wife Pam, his daughter Katie and his son Michael His ongoing extracurricular goal is to learn to play golf well enough to not come in last in his annual mathematicians/statisticians tournament Roland B Minton is Professor of Mathematics and Chair of the Department of Mathematics, Computer Science and Physics at Roanoke College, where he has taught since 1986 Prior to that, he was on the faculty at Virginia Tech He earned his Ph.D from Clemson University in 1982 He is the recipient of Roanoke College awards for teaching excellence and professional achievement, as well as the 2005 Virginia Outstanding Faculty Award and the 2008 George Polya Award for mathematics exposition Professor Minton’s current research program is in the mathematics of golf, especially the analysis of ShotLink statistics He has published articles on various aspects of sports science, and co-authored with Tim Pennings an article on Pennings’ dog Elvis and his ability to solve calculus problems He is co-author of a technical monograph on control theory He has supervised numerous independent studies and held workshops for local high school teachers He is an active member of the Mathematical Association of America Professor Minton lives in Salem, Virginia, with his wife Jan and occasionally with his daughter Kelly and son Greg when they visit He enjoys playing golf when time permits and watching sports events even when time doesn’t permit Jan also teaches at Roanoke College and is very active in mathematics education In addition to Calculus: Early Transcendental Functions, Professors Smith and Minton are also coauthors of Calculus: Concepts and Connections c 2006, and three earlier books for McGraw-Hill Higher Education Earlier editions of Calculus have been translated into Spanish, Chinese and Korean and are in use around the world iv www.EngineeringBooksPDF.com CONFIRMING PAGES P1: OSO/OVY P2: OSO/OVY MHDQ256-FM-MAIN QC: OSO/OVY MHDQ256-Smith-v1.cls T1: OSO January 8, 2011 12:17 LT (Late Transcendental) Brief Table of Contents CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER 10 CHAPTER 11 CHAPTER 12 CHAPTER 13 CHAPTER 14 CHAPTER 15 CHAPTER 16 APPENDIX A APPENDIX B Preliminaries Limits and Continuity 47 Differentiation 107 Applications of Differentiation 173 Integration 251 Applications of the Definite Integral 315 Exponentials, Logarithms and Other Transcendental Functions 375 Integration Techniques 421 First-Order Differential Equations 491 Infinite Series 531 Parametric Equations and Polar Coordinates 625 Vectors and the Geometry of Space 687 Vector-Valued Functions 749 Functions of Several Variables and Partial Differentiation 809 Multiple Integrals 901 Vector Calculus 977 Second-Order Differential Equations 1073 Proofs of Selected Theorems A-1 Answers to Odd-Numbered Exercises A-13 v www.EngineeringBooksPDF.com CONFIRMING PAGES P1: OSO/OVY P2: OSO/OVY MHDQ256-FM-MAIN QC: OSO/OVY MHDQ256-Smith-v1.cls T1: OSO January 10, 2011 LT (Late Transcendental) 8:35 Table of Contents Seeing the Beauty and Power of Mathematics xiii Applications Index xxiv CHAPTER Preliminaries 0.1 The Real Numbers and the Cartesian Plane The Real Number System and Inequalities The Cartesian Plane 0.2 Lines and Functions Equations of Lines Functions 0.3 Graphing Calculators and Computer Algebra Systems 21 0.4 Trigonometric Functions 27 0.5 Transformations of Functions 36 CHAPTER Limits and Continuity 47 1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve 47 1.2 The Concept of Limit 52 1.3 Computation of Limits 59 1.4 Continuity and Its Consequences 68 The Method of Bisections 1.5 Limits Involving Infinity; Asymptotes 78 Limits at Infinity 1.6 Formal Definition of the Limit 87 Exploring the Definition of Limit Graphically Limits Involving Infinity 1.7 Limits and Loss-of-Significance Errors 98 Computer Representation of Real Numbers CHAPTER Differentiation 107 2.1 Tangent Lines and Velocity 107 The General Case Velocity 2.2 The Derivative 118 Alternative Derivative Notations Numerical Differentiation vi www.EngineeringBooksPDF.com CONFIRMING PAGES P1: OSO/OVY P2: OSO/OVY MHDQ256-FM-MAIN QC: OSO/OVY MHDQ256-Smith-v1.cls T1: OSO January 8, 2011 LT (Late Transcendental) 12:17 Table of Contents 2.3 Computation of Derivatives: The Power Rule 127 The Power Rule General Derivative Rules Higher Order Derivatives Acceleration 2.4 The Product and Quotient Rules 135 Product Rule Quotient Rule Applications 2.5 The Chain Rule 142 2.6 Derivatives of Trigonometric Functions 147 Applications 2.7 Implicit Differentiation 155 2.8 The Mean Value Theorem 162 CHAPTER Applications of Differentiation 173 3.1 Linear Approximations and Newton’s Method 174 Linear Approximations Newton’s Method 3.2 Maximum and Minimum Values 185 3.3 Increasing and Decreasing Functions 195 3.4 3.5 3.6 3.7 3.8 What You See May Not Be What You Get Concavity and the Second Derivative Test 203 Overview of Curve Sketching 212 Optimization 223 Related Rates 234 Rates of Change in Economics and the Sciences 239 CHAPTER Integration 251 4.1 Antiderivatives 252 4.2 Sums and Sigma Notation 259 Principle of Mathematical Induction 4.3 Area 266 4.4 The Definite Integral 273 Average Value of a Function 4.5 The Fundamental Theorem of Calculus 284 4.6 Integration by Substitution 292 Substitution in Definite Integrals 4.7 Numerical Integration 298 Simpson’s Rule CHAPTER Error Bounds for Numerical Integration Applications of the Definite Integral 315 5.1 Area Between Curves 315 5.2 Volume: Slicing, Disks and Washers 324 Volumes by Slicing The Method of Disks The Method of Washers www.EngineeringBooksPDF.com CONFIRMING PAGES vii P1: OSO/OVY P2: OSO/OVY MHDQ256-FM-MAIN viii QC: OSO/OVY MHDQ256-Smith-v1.cls T1: OSO January 8, 2011 LT (Late Transcendental) 12:17 Table of Contents 5.3 Volumes by Cylindrical Shells 338 5.4 Arc Length and Surface Area 345 Arc Length Surface Area 5.5 Projectile Motion 352 5.6 Applications of Integration to Physics and Engineering 361 CHAPTER Exponentials, Logarithms and Other Transcendental Functions 375 6.1 The Natural Logarithm 375 Logarithmic Differentiation 6.2 Inverse Functions 384 6.3 The Exponential Function 391 Derivative of the Exponential Function 6.4 The Inverse Trigonometric Functions 399 6.5 The Calculus of the Inverse Trigonometric Functions 405 Integrals Involving the Inverse Trigonometric Functions 6.6 The Hyperbolic Functions 411 The Inverse Hyperbolic Functions CHAPTER Derivation of the Catenary Integration Techniques 421 7.1 Review of Formulas and Techniques 422 7.2 Integration by Parts 426 7.3 Trigonometric Techniques of Integration 433 Integrals Involving Powers of Trigonometric Functions Trigonometric Substitution 7.4 Integration of Rational Functions Using Partial Fractions 442 Brief Summary of Integration Techniques 7.5 Integration Tables and Computer Algebra Systems 450 Using Tables of Integrals Integration Using a Computer Algebra System 7.6 Indeterminate Forms and L’Hˆ opital’s Rule 457 Other Indeterminate Forms 7.7 Improper Integrals 467 Improper Integrals with a Discontinuous Integrand Improper Integrals with an Infinite Limit of Integration A Comparison Test 7.8 Probability 479 CHAPTER First-Order Differential Equations 491 8.1 Modeling with Differential Equations 491 Growth and Decay Problems Compound Interest 8.2 Separable Differential Equations 501 Logistic Growth www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index I-4 QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Cycloid, 640, 644 Cylinder defined, 324, 734 in cylindrical coordinates, 948–949 volume of, 665 Cylindrical coordinates, 948–953 cone equation in, 949 conversion to/from rectangular coordinates, 951–952 cylinder equation in, 948–949 defined, 948 triple integrals in, 950–951 Cylindrical shells method, 338–343 Cylindrical surface, graph of, 734 D Damping force, 1074, 1094–1095 Daubechies, Ingrid, 610 de Branges, Louis, 270 Decay constant, 494 Decay problems, 491–496 Decreasing function defined, 195 Decreasing sequence, 538 Definite integral approximation of, with Midpoint Rule, 274 computing exactly, 285 defined, 273, 903 integration by parts for, 430–431 of vector-valued function, 766 signed area and, 275–276 substitution in, 295–296 Taylor series for approximating, 601 with variable upper limit, 286 Degree of polynomial, 13 Demand elasticity of, 241–242, 245 relative change in, 241 Density linear, 243 mass density, 243 weight, 947 Density plot, 814, 816 Derivative Test First, 199 Second, 877 Derivative(s) at a point unspecified, 118–119 chain rule for, 142–145 computation of, 127–133, 151 defined, 118 directional, 864–871 computation of, 865 defined, 864, 869 finding, 866 level curves and, 867 general rules, 130–131 given, finding functions with, 167 gradient, 864–871 higher order, 131–132 notation, alternative, 121–123 numeric approximation of, 122 of cosine function, 151 of exponential functions, 393–396 of log of absolute value, 377 of sum, 131 of trigonometric functions, 147–152 inverse, 406 of vector-valued function, 761 partial, 833–840 applications, 836–837, 839 computing of, 835–836 defined, 835 from table of data, 839–840 higher-order, 837 mixed second-order, 837 of three variable functions, 838–839 power rule, 127–133 product rule, 135–140, 151 quotient rule, 135–140 rewriting functions for, 131 second, 131 implicit, 158 third, 131 undefined, 188, 189 zero, function with, at local maximum, 187 Determinant, 715 Deviation, standard, 486 Diaconis, Persi, 483 Diameter, conjugate, 768 Diameter, of section, 1032 Difference quotients, 110, 127 Difference, indefinite integral of, 256 Differentiable functions defined, 118 Differential, 175 total, 850 Differential equations as initial value problems, 503 defined, 252, 259, 492 equilibrium solutions, 505 family of solutions for, 503 first-order ordinary, 501 flow lines with, 983 general solution for, 492, 1075 higher-order, 1081 homogeneous, 1082 initial condition in, 503 linear ordinary, 989 modeling with, 491–498 nonhomogeneous, 1082–1088 power series solutions of, 1098–1106 second-order applications, 1090–1095 as first-order systems, 525 with constant coefficients, 1074–1080 separable, 501–507 systems of, 521–525 with variable coefficients, solving, 1100–1101 Differential geometry, 788 Differential operator, 122 Differentiation implicit, 155–160, 860 numerical, 122–123 of exponential functions, 395 of logarithms, 377, 380–381, 381–382 of power series, 582–583 of vector-valued functions, 761 term-by-term, 582 Diffusivity, thermal, 862 Dimensionless variables, 858 Dipole electric, 586 electrostatic field of, 986–987 Dirac delta, 298 Direct linear transformation, 972 Direction field, 510, 512 Direction of maximum increase, 869–870 Direction vectors, 711 Directional derivative, 864–871 computation of, 865 defined, 864, 869 finding, 866 level curves and, 867 Directixes, effects of, 679 Directrix, of parabola, 668 Discontinuity defined, 68, 828 removable, defined, 70 Discontinuous integrand, 278, 467–470 Discrete probability distributions, 481 Discriminant defined, 877 local extrema and, 877–878 Disk closed, 828 open, 828 Displacement vector, 710 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Distance from point, 720 minimization of, 226–227 minimum, 888 Distance formula, 6–7, 699 Divergence comparison test for, 475, 559 kth term test for, 549–550 of alternating series, 567 of geometric series, 548–549 of improper integrals, 468, 471 of infinite series, 546 of sequence, 534–535 of vector field, 1022–1029 computing, 1025 definition of, 1025 interpretation of, 1025 ratio test for, 575 Divergence Theorem, 1044–1051 applications, 1047 definition of, 1044 proof of, 1045 proving general result with, 1048 Domain in graph drawing, 212 of function, defined, 13 of three variable function, 810 of two variable function, 810 Domes, volume of, 327 Dot product, of vector, 704–711 Double integrals, 901–914 area with, 918 change of variables in, 967 defined, 906, 909 evaluation of, 910, 912 in polar coordinates, 926–931 irregular partitions and, 904 order in, 913 over general regions, 908–913 over rectangle, 903–907 volume with, 904–905, 918 Double Riemann sum, 904, 917 Doubling time, 493 Down concavity, 203 Drag coefficient, 525 E e (irrational number) as logarithmic base, 391 Taylor series for approximation of, 591 Eccentricity, 677, 678–679 Economic Order Quantity, 283, 291 Elasticity of demand, 241–242, 245 Electric dipole, 586 Electrical circuits, charge in, 1091 Electrical potential, 607 Electrostatic field, 986–987 Elementary polar regions, 926 Ellipse area enclosed by, 233, 639 axes of, 671 center of, 672 defined, 671 equation of, 671, 672–673 features of, 672 parametric equations of, 627–628 vector-valued function defining, 751 vertices of, 671 Ellipsoid equation for, 735 graph of, 735 inertia, 745 Elliptic cone, 738 Elliptical helix defined, 752 vector-valued function defining, 751–752 Endpoints, Energy conservation of, 1013 kinetic, 361, 1013 potential, 361, 1013 Energy spectrum, 620 Enzyme, allosteric, 142 Epicycloid, 641 Equal-tempered tuning, Equation(s) calculator for solving, 26 characteristic, 1075 continuity, 1052 derivation of, 1064 converting between rectangular and polar coordinates, 652 differential as initial value problems, 503 defined, 252, 259, 492 equilibrium solutions, 505 family of solutions for, 503 first-order ordinary, 501 flow lines with, 983 general solution for, 492, 1075 higher-order, 1081 homogeneous, 1082 initial condition in, 503 linear ordinary, 989 modeling with, 491–498 nonhomogeneous, 1082–1088 power series solutions of, 1098–1106 second-order, 525, 1074–1080 applications, 1090–1095 I-5 separable, 501–507 systems of, 521–525 with variable coefficients, solving, 1100–1101 Euler, 1081 Euler’s, 1064 heat, 862, 1062, 1063 Hermite’s, 1107 impulse-momentum, 283, 364 Laplace, 989 linear ordinary differential, 989 linear, in three dimensions, 729 logistic, 244, 245, 505 Maxwell’s, 1065–1066 of cone, 956 of cylinder in cylindrical coordinates, 948–949 of ellipse, parametric, 671, 672–673 of ellipsoid, 735 of hyperbola, 674 of hyperboloid, 739 of lines, 9–18 of motion, 773–776 of parabola, 669, 670 of parallel line, 12 of perpendicular line, 12 of plane, 729 of sphere, 702 of tangent line, 110, 137 of tangent plane, 846–847 parametric, 457 calculus and, 634–639 circles defined by, 627–628 conic sections in, 668–675 defined, 625 ellipses defined by, 627–628 for intersecting surfaces, 755 for line segment, 628 for x-y equations, 629 parameter of, 626 plane curves and, 625–630 projectile motion and, 626–627 sine and, 627 slope of, 635 surface area with, 641–647 symmetric, of lines, 726 wave, 843 Equiangular curve, 660 Equilibrium position, of spring-mass system, 1074 Equilibrium solutions, 505, 516–517 of systems of equations, 524 stable and unstable, 512, 522 Equipotential curves, 989 Equivalence point, 382 Error bounds, for numerical integration, 305–308 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index I-6 QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Error estimate, for integral test, 557 Error function, 598 Escalante, Jaime, 432 Escape velocity, 86 Euler equation, 1081 Euler’s constant, 554 Euler’s equation, 1064 Euler’s formula, 598 Euler’s method, 510, 512, 983–984 Improved, 520 Euler, Leonhard, 510 Evaluation points, 269, 270 Evaluation Theorem, 991, 997, 1033 Expansion Fourier series, 611–612, 614–615 of determinant, 715 Taylor series, 588 Exponent(s) fractional, 191, 200, 255 negative, 254 Exponential decay law, 492 Exponential functions base of, 391–392 defined, 391 derivatives of, 393–396 differentiation of, 395 integral of, 394 Extrapolation, 12 Extreme value theorem, 186 Extremum (extrema) absolute, 185 approximation of, 192 finding, 882–883 on closed interval, 191 at undefined derivative, 189 defined, 22 for function with fractional exponents, 191 local approximation of, 200 defined, 22 discriminant in finding, 877–878 graph of, 197, 875–876 in functions of several variables, 874 of polynomial, 189 with first derivative test, 200 of functions of several variables, 874–883 with second derivative test, 207 F Factor Theorem, 17 Factor, integrating, 989 Factorial, defined, 537 Factoring finding limit by, 61 finding zeros by, 16 Family of solutions, to differential equations, 503 Faraday’s Law of Induction, 1066 Fermat’s Theorem, 188 Fermat, Pierre de, 188 Feynman, Richard, 135 Fibonacci sequence, 1, 544 Field(s) direction, 510, 522 electrostatic, 986–987 gradient definition of, 984 finding, 984–985 potential function of, 985–986 inverse square, flux of, 1049–1050 magnetic, flux of, 1051, 1059 slope, 510 vector, 977–987 conservative, 1003–1011 determination of, 1027 curl of, 1022–1029 computing, 1023 definition of, 1023 interpretation of, 1024 definition of, 978 divergence of, 1022–1029 computing, 1025 defined, 1025 flux of, 1039–1041 graphing of, 978–979 incompressible, 1026 irrotational, 1024, 1027–1028 plotting, 978 potential function of, 1010 sink points, 1026 source points, 1026 source-free, 1026 with gradient, 1026 velocity, 981 flux of, 1061–1062 First component, of vector, 689 First Derivative Test, 199 First moment(s), 365, 944 First octant, 698 First-order ordinary differential equations, 501 Fitzgerald, Ella, 1073 Fixed graphing window, 21 Fixed point, 42 Flow lines definition of, 981 differential equations with, 983 Euler’s method in approximation of, 983–984 graphing of, 982 Flow, irrotational, 1058 Flux definition of, 1040 of inverse square field, 1049–1050 of magnetic field, 1051, 1059 of vector field, 1039–1041 of velocity field, 1061–1062 FM See Frequency modulation (FM) Focus (foci) of hyperbola, 673 of parabola, 668 Force(s) centripetal, 771 constant, work and, 361 Coulomb, 1013 damping, 1074, 1094–1095 hydrostatic, 367 Magnus, 722, 723 resultant, 694 Four-leaf rose, 656 Fourier analysis, 620 Fourier convergence theorem, 616 Fourier cosine series, 620 Fourier series, 607–618 convergence of, 616–617 defined, 608 expansion of, 610, 611–612, 614–615 music synthesizers and, 617–618 Fourier sine series, 618, 620 Fourier’s law, 989 Fourier, Jean Baptiste Joseph, 608 Fraction(s) inequality with, partial, 442–448 Fractional exponent extrema of function with, 191, 200 power rule with, 255 Freedman, Michael, 65 Frenet-Serret formulas, 798 Frequency natural, 1087 of sine wave, 31 resonant, 1096 Frequency modulation (FM), 222 Frequency response curve, 1092–1093, 1096 Friction, coefficient of, 697 Frustrum of cone, 348 Fubini’s Theorem, 907, 927, 928 Fubini, Guido, 907 Function(s) absolute extrema of, 186 antiderivative of given, 253 area, 272, 286 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index area of regions bounded by, of y, 320 average cost, 211 average value of, 279–281 Bessel, 602–603 combination of, 36 component, defined, 750 composition of, 36 continuity of, 830 finding, 37 identifying, 37 continuity of defined, 68, 828 of composite functions, 72 removable, 70 continuous, absolute extrema of, 187 cost, 211 cumulative distribution, 478 decreasing defined, 195 defined, 13 density plots and, matching of, 816 derivatives and, rewriting for, 131 differentiable, defined, 118 discontinuous, 68 domain of, 13 error, 598 exponential base of, 391–392 defined, 391 differentiation of, 395 integral of, 394 gamma, 478 generating, 586 hyperbolic, 411–417 derivative of, 412 integral involving, 413 inverse, 414–416 formula for, 415 increasing defined, 195 integrable, 273, 906 inverse defined, 384–385 finding, 386 graphing of, 387 tangent line to, 389 unknown, 387 iterations of, 42 log integral, 384 maximum rate of change of, 868 minimum rate of change of, 868 of several variables, 809–818 extrema of, 874–883 of two variables, 809–810 graph of, 811 limit of, 828 omega, 478 one-to-one, 385 periodic defined, 28, 607 fundamental period of, 28 piecewise-continuous, 278 piecewise-defined, limit of, 65–66 polynomial, defined, 13 potential, 292, 984 of vector field, 1010 power, arc length of, 346 probability density, 481, 482 standard deviation for, 486 range of, 13 rational continuity of, 69, 71 critical number of, 190 defined, 15 graph of, 23–24, 208, 213–214 integration of, 442–448 limit of, 61 with no vertical asymptotes, 25 reliability, 478 Riemann sum for, with positive and negative values, 274 Riemann-zeta, 563 root of, 14 scalar, 1026 sigmoid, 398 sine, 29 special, 602 square root, 15 chain rule with, 144 derivative of, 119–120 square-wave, 610 squeeze theorem for, 64 sum of values of, 263 table of data, defined by, 810 transformations of, 36–41 trigonometric, 26–34 derivatives of, 147–152 in powers, 294 inverse, 399–403 calculus of, 405–409 derivative of, 406 integrals involving, 407–409 simplification of, 402 limit of, 63, 91 loss of significance involving, 101 polynomial and, sum of, 220–221 vector-valued antiderivative of, 765 calculus of, 758–767 continuous, 759–760 defined, 750 definite integral of, 766 derivative of, 761 I-7 differentiation of, 761 ellipse, 751 elliptical helix, 751–752 graphing of, 750–751 increment of, 760 indefinite integral of, 766 limit of, 759 line, 752 matching to graph, 752 wave, 1098 well-defined, 290 with no inverse, 385 with second derivative test as inconclusive, 207 with vertical tangent line at inflection point, 209 with zero derivative at local maximum, 187 zeros of approximate, 25 approximating by method of bisection, 74–75 with Newton’s method, 179 Bessel functions, 603 by factoring, 16 by quadratic formula, 16 cubic polynomial, 17 determining number of, 164 finding, by method of bisections, 74–75 Fundamental period, 28 Fundamental Theorem for Line Integrals, 1007 Fundamental Theorem of Algebra, 261 Fundamental Theorem of Calculus, 284–290 Future value, 501 G Gabriel’s horn, 351, 477 Galileo, 1097 Gamma function, 478 Gauss, Karl Friedrich, 261, 269 Gauss’ Law for electricity, 1050 Gauss’ Theorem See Divergence Theorem Gaussian quadrature, 310 General solution, for differential equations, 492, 1075 General term, 532 Generating function, 586 Geometric series, 547 convergent, 547–548 divergence of, 548–549 Geometry, differential, 788 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index I-8 QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Geosynchronous orbit, 779 Gibbs phenomenon, 618 Gibbs, Josiah Willard, 717 Gini index, 272 Global behavior defined, 23, 198 Gradient applications, 871 defined, 865, 869 scalar functions with, 1026 vector fields with, 1026 Gradient derivatives, 864–871 Gradient field(s) definition of, 984 finding, 984–985 potential function of, 985–986 Granville, Evelyn, 772 Graph(s) comparing, 39 first derivatives in, 212 global behavior in, 23, 198 hidden behavior in, 198 horizontal asymptotes, 212 horizontal translation of, 38–39 intercepts in, 212 limit determination with, 54 local behavior in, 23, 198 of cardioid, 654–655 of contour plots, 815 of cubic polynomial, 23 of cylindrical surface, 734 of ellipsoid, 735 of elliptic cone, 738 of flow lines, 982 of hyperboloid, 738–739 of inverse function, 387 of lima¸cons, 653–654, 655 of line, 10 of local extrema, 197, 875–876 of paraboloid, 737 of parametric surface, 799–800 of plane curve, 626 of polar coordinates, 652 of polynomial, 212–213 of polynomial and trigonometric function sum, 220–221 of rational function, 23–24, 208, 213–214 of Taylor polynomials, 588 of three variable functions, 811 of two variable functions, 811 of vector-valued function, 750–751, 752 removing hole in, 69–70 vector fields, 978–979 vertical asymptotes, 24, 212 vertical translation of, 37–38 with difficult-to-see features, 218–219 with no inflection points, 206 with no tangent lines, 115 with two vertical asymptotes, 215–216 Graphing calculators automatic graphing window, 21 fixed graphing window, 21 generating graph, 21 intersections on, 25 pixels in, 21 solving equations on, 26 Graphing window, 21 Gravitation, 680, 794 Green’s Theorem, 1014–1021 Green, George, 1014 Gross domestic product (GDP), 272 Growth and decay problems, 491–496 Growth constant, 492 Growth, logistic, 505–507 Guess, initial, 880 H Half-life, 494 Half-open interval of convergence, 581 Halmos, Paul, 90 Hamilton, William Rowan, 697 Hardy-Weinberg law, 886 Harmonic content, 619 Harmonic motion, simple, 1079, 1098 Harmonic series, 550 alternating, 565 Hau, Lene, 708 Heat conductivity, 1041 Heat equation, 862, 1062, 1063 Heat index, 818 Heat, specific, 1063 Helix curvature of, 783 elliptical defined, 752 vector-valued function defining, 751–752 Hermite polynomial, 1107 Hermite’s equation, 1107 Higher order derivatives, 131–132 Higher order differential equations, 1081 Higher-order partial derivatives, 837 Homeomorphism, 411 Homogeneous differential equations, 1082 Hooke’s Law, 362 Horizontal asymptotes finding, 81 in graphing, 212 Horizontal component, of vector, 637 Horizontal line test, 385 Horizontal tangent, 190, 661 Horizontal translations, 38–39 Hydrostatic force, 367 Hyperbola center of, 674 defined, 673 equation of, 674 foci of, 673 Hyperbolic cosecant function, 412 Hyperbolic cosine function, 411 inverse, 414 Hyperbolic cotangent function, 412 Hyperbolic functions, 411–417 derivative of, 412 integral involving, 413 inverse, 414–416 formula for, 415 Hyperbolic paraboloid, 740–741, 801 Hyperbolic secant function, 412 Hyperbolic sine function, 411 inverse, 414 Hyperbolic tangent function, 412 inverse, 414 Hyperboloid equation for, 739 of one sheet, 738–739 of two sheets, 739, 740 Hypersurface, 928 Hypocycloid, 640 I Identities, trigonometric functions and, 32 Image, of transformation, 963 Implicit differentiation, 155–160, 860 Implicit plot, 736 Implicit solution, 504 Improper integrals, 467–476 comparison test for, 474 convergence of, 469, 471 defined, 467 divergence of, 468, 469, 471 with discontinuous integrand, 467–470 with infinite limit of integration, 470–473 Improved Euler’s method, 520 Impulse, 364 Impulse-momentum equation, 283, 364 Incompressible vector field, 1026 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Increasing function defined, 195 Increasing sequence, 538 Increment computation of, 849 defined, 848 of vector-valued functions, 760 Indefinite integrals defined, 253 evaluating, 254 of difference, 256 of sum, 256 of vector-valued function, 766 power rule for, 254 Indeterminate forms defined, 82, 457 limit of, 459 other, 461–464 simplification of, 461 Index of summation, 260 Induction assumption, 263 Induction, mathematical, 263–264 Inequality alternate method of solving, Cauchy-Schwartz, 707, 712 linear, quadratic, real number system and, 2–5 triangle, 5, 708, 712 two-sided, 3–4 with absolute value, with fraction, with sum inside absolute value, Inequality constraint, 890–891 Inertia ellipsoids, 745 Infant mortality phase, 421 Infinite products, 553 Infinite series, 266 convergent, 546 defined, 545 divergence of, 546 sums of, 545 Infinity limits at, 81–84, 93–96 Inflection points, 205, 206 Information theory, 478 Information, qualitative, 510 Initial condition, 493, 503, 1077 Initial guess, 880 Initial point, of vector, 688 Initial value problem (IVP), 503, 1077 Inner partition, 908, 926, 929 Instantaneous rate of change, 114, 132–133 Instantaneous velocity, 113 Integers, 2, 261, 1034–1035 Integrable function, defined, 273, 906 Integral Mean Value Theorem, 280 Integral test, 554–562 Integral(s) Boltzmann, 478 completing the square with, 423 definite approximation of, with Midpoint Rule, 274 computing exactly, 285 defined, 903 integration by parts for, 430–431 of vector-valued functions, 766 signed area and, 275–276 substitution in, 295–296 Taylor series for approximating, 601 with variable upper limit, 286 double, 901–914 area with, 918 change of variables in, 967 defined, 906, 909 evaluation of, 910, 912 in polar coordinates, 926–931 irregular partitions and, 904 order in, 913 over general regions, 908–913 over rectangle, 903–907 volume with, 904–905, 918 improper, 467–476 convergence of, 469, 471 defined, 467 divergent, 468, 471 with discontinuous integrand, 467–470 with infinite limit, 470–473 indefinite defined, 253, 273 evaluating, 254 of difference, 256 of sum, 256 of vector-valued functions, 766 power rule for, 254 line, 990–1001 defined, 990 determining sign of, graphically, 1000–1001 evaluation of over piecewise-smooth curve, 993–995 with Green’s Theorem, 1017–1018 with respect to arc length, 992 with Stokes’ Theorem, 1055 fundamental theorem for, 1007 Green’s Theorem and, 1017 independence of path, 1003–1011 with respect to x, 997 I-9 with respect to y, 997 work and, 999–1000 of exponential functions, 394 of inverse trigonometric functions, 407–409 Riemann-Stieltjes, 426 substitution in evaluation of, 293 surface, 1032–1041 definition of, 1032, 1040 evaluation of, 1034–1035 using complement of surface, 1062 with polar coordinates, 1035 with spherical coordinates, 1038 with Stokes’ Theorem, 1056 surface area with, 1038–1039 tangent line for, 289 Taylor series for approximation of, 602 triple, 928–946 center of mass and, 944–946 change of variables in, 970 defined, 928 in cylindrical coordinates, 950–951 in spherical coordinates, 957–960 inner partition of, 929 order of integration in, 942–943 over rectangular box, 929 over tetrahedron, 940–941 volume with, 943–944, 952–953 with first integration with respect to x, 941–942 with polar coordinates, 948 with discontinuous integrand, 278 with logarithms, 378 with powers of trigonometric functions, 433–437 with variable upper and lower limits, 288 Integrand, 253, 278, 295 discontinuous, 467–470 expansion of, 423 with even power of cosine, 435 with even power of secant, 436 with even power of sine, 435 with odd power of cosine, 434 with odd power of sine, 434 with odd power of tangent, 436 with single term, 428 Integrating factor, 989 Integration by parts, 426–431 for definite integral, 430–431 repeated, 428 by substitution, 292–296, 422, 447 constant of, 253 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index I-10 QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Integration—Cont defined, 253 generalizing rule of, 423 in engineering, 361–369 in physics, 361–369 lower limit of, 273 numerical, 298–309 error bounds for, 305–308 of power series, 583–584 of rational functions, 442–448 partial, 906 reduction formula and, 430, 451 tables, 301, 450–456 trigonometric techniques of, 433–440 with computer algebra systems, 450–456 with partial fractions, 442–448 Intercepts, 212 Interest compound, 496–498 continuous compound, 496 Interior point, 828 Intermediate Value Theorem, 74 Interpolation, linear, 177 Interval closed, absolute extremum on, 191 continuity on, 72 open, derivative on, 119 Interval of convergence, 581 half-closed, 581 Inverse cosine function defined, 400 evaluation of, 400 Inverse function defined, 384–385 finding, 386 graphing of, 387 tangent line to, 389 unknown, 387 Inverse hyperbolic cosine function, 414 Inverse hyperbolic functions, 414–416 formula for, 415 Inverse hyperbolic sine function, 414 Inverse hyperbolic tangent function, 414 Inverse problem, 375 Inverse relationship, 384 Inverse secant function defined, 401 evaluation of, 402 Inverse sine function, 399 evaluation of, 400 hyperbolic, 414 Inverse square field, flux of, 1049–1050 Inverse square law, 987 Inverse tangent function defined, 401 evaluation of, 401 hyperbolic, 414 integral related to, 408 simplification of expression with, 402–403 Inverse trigonometric functions, 399–403 calculus of, 405–409 derivatives of, 406 integrals involving, 407–409 simplification of, 402 Irrational numbers, Irregular partitions, 272, 904 Irrotational flow, 1058 Irrotational vector fields, 1024, 1027–1028 Iterates, 42 Iterations, of functions, 42 IVP See Initial value problem (IVP) J Jacobi, Carl Gustav, 966 Jacobian of transformation, 967 Jones, Vaughan, 464 Julia set, 712 Just-in-time inventory, 251 K Kepler’s laws of planetary motion, 680, 794–797 Kepler, Johannes, 194, 794 Kinetic energy, 361, 1013 Klein bottle, 803 kth term test, 549–550 L L’Hˆopital’s Rule, 459, 535–536 L’Hˆopital, Guillaume de, 459 Lagrange multiplier(s), 887–894 defined, 889 method of, 888 Lagrange points, 184, 799 Lagrange, Joseph-Louis, 888 Lambert shading, 874 Lamina, 920 center of mass of, 920, 921–922 moments of inertia of, 923 Laplace equation, 989 Laplace transform, 478 Laplace, Pierr´e-Simon, 470 Laplacian, 858, 862, 873, 1026 Least squares method, 878 Legendre polynomials, 607 Leibniz notation, 122 Leibniz, Gottfried Wilhelm, 122, 140 Level curves, 814 directional derivatives and, 867 Level surfaces, 817–818 Lima¸cons, 653–654, 655, 664 Limit approaching, 53 approaching value of, 56 at infinity, 81–84, 93–96 by factoring, 61 by rationalizing, 62–63 comparison test for, 560 computation of, 59–66 concept of, 52–57 continuity and, 822–831 describing velocity, 66 evaluating, 53 formal definition of, 87–96, 823 graphic determination of, 54, 91–93 infinite, 94 of improper integral, 470–473 L’Hˆopital’s rule and, 459 loss of significance errors, 98–103 nonexistent, 54, 55, 92, 825 of indeterminate forms, 459 of integration, 273 of natural log, 381 of nth root of polynomial, 62 of piecewise defined functions, 65–66 of polynomial, 61, 824 of product that is not product of limits, 63–64 of quotient that is not quotient of limits, 82 of rational function, 61 of sequence, 534 of trigonometric functions, 63, 80, 91 of two variable function, 828 one-sided, 53, 56, 79 precise definition of, 89 proving as correct, 89 proving existence of, 827 simple, 88 Taylor series for conjecture of value of, 600–601 variable, integral with, 288 verifying, 64–65, 88 where two factors cancel, 55 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Limit cycle, 668 Line integrals, 990–1001 defined, 990 determining sign of, graphically, 1000–1001 evaluating over piecewise-smooth curve, 993–995 with Green’s Theorem, 1017–1018 with respect to arc length, 992 with Stokes’ Theorem, 1055 fundamental theorem for, 1007 Green’s theorem and, 1017 in space, 997–998 independence of path, 1003–1011 with respect to x, 997 with respect to y, 997 work and, 999–1000 Line segments, parametric equations of, 628 Line(s) curvature of, 782 distance from point to, 720 equations of, 9–18 flow definition of, 981 differential equations with, 983 Euler’s method in approximation of, 983–984 graphing of, 982 graphing of, 10 in space, 726–731 nonintersecting but not parallel, 727 normal, 846, 870 orthogonal, 727 parabola and, intersection of, 17–18 parallel, 11, 12, 727 perpendicular, 11, 12 point-slope form, 11 secant, 108 slope of, 9, 10 slope-intercept form of, 11 symmetric equations of, 726 tangent, 47–51 equation of, 110, 137, 152 finding with implicit differentiation, 156 for function defined as integral, 289 graphical approximation of, 111 horizontal, 190 numerical representation of, 111 to inverse function, 389 to parametric curve, 635 velocity and, 107–115 vertical, 190, 209 vector-valued function defining, 752 Linear approximation defined, 175, 847, 850 finding, 175–176, 847–848 for linear interpolation, 177 of cube roots, 176 of sine function, 176 tangent planes and, 844–852 Linear convergence, 184 Linear density, 243 Linear equations, in three dimensions, 729 Linear inequality, Linear interpolation, 177 Linear momentum, 775, 778 conservation of, 778 Linear ordinary differential equations, 989 Linear polynomial, 14 Linear regression, 878, 879–880 Linear transformations, 27 direct, 972 Local behavior, in graphing, 23, 198 Local extremum approximation of, 200 defined, 22 discriminant in finding, 877–878 graph of, 197, 875–876 in functions of several variables, 874 of polynomial, 189 using first derivative test, 200 Local maximum defined, 22, 187 function with zero derivative at, 187 in functions of several variables, 874 Local minimum defined, 22, 187 function with undefined derivative at, 188 in functions of several variables, 874 Log integral function, 384 Logarithm(s) derivative of of absolute value, 377 differentiation of, 377, 380–381, 381–382 integrals involving, 378 natural defined, 376 inverse of, 391–393 limiting behavior of, 381 rewriting expressions with, 380 Logistic equation, 244, 245, 505 Logistic growth, 505–507 Lorentz curve, 272 Loss-of-significance errors, limits and, 98–103 I-11 Loss-of-significant-digits error, 100 Lower limit of integration, 273 M M-test, 586 Maclaurin series, 588, 598 Maclaurin, Colin, 555 Magnetic field, flux of, 1051, 1059 Magnitude, of vector, 688, 700 Magnus force, 356, 722, 723 Major axes, of ellipse, 671 Mandelbrot set, 712 Mandelbrot, Benoit, 286 Map, topographical, 821 Marginal cost, 239 Marginal profit, defined, 239 Mass center of, 365 of solid, 945–946 triple integrals and, 944–946 point, 365 Mass density, 243 Mathematical analysis, 87 Mathematical induction, 263–264 Matrix determinant of, 715 Maximum absolute, 185, 882 local defined, 22, 187 function with zero derivative at, 187 in functions of several variables, 874 Maximum increase, direction of, 869–870 Maximum rate of change of function, 868 Maxwell’s equations, 1065–1066 McDuff, Dusa, 159 McNulty, Kay, 515 Mean arithmetic, 478, 896 Mean Value Theorem, 162–167, 280 Median, 484 Method of bisections, finding zeros by, 74–75 Method of cylindrical shells, 338–343 Method of disks, 328–330 Method of Lagrange multipliers, 888 Method of undetermined coefficients, 1083 Method of washers, 330–335, 340 Midpoint Rule, 273, 299–300, 304 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index I-12 QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Minimum absolute, 185, 882 distance, 888 local defined, 22, 187 function with undefined derivative at, 188 in functions of several variables, 874 Minimum rate of change of function, 868 Minor axes, of ellipse, 671 Mixed second-order partial derivatives, 837 Măobius, August Ferdinand, 1039 Modeling, with differential equations, 491–498 Moment of inertia about the x-axis, 922 about the y-axis, 922 of lamina, 923 Moment(s) first, 365, 944 second, 922 with respect to x-axis, 921 with respect to y-axis, 921 Momentum angular, 775, 778 conservation of, 776, 925 impulse and, 364 linear, 775 conservation of, 778 Monotonic sequence, 538, 540 Monte Carlo method, 916 Morawetz, Cathleen, 1057 Mori, Shigefumi, 1082 Motion equations of, 773–776 in space, 769–777 Kepler’s laws of, 794–797 Newton’s second law of, 252, 770 planetary, 794–797 projectile analyzing, 771–772 in three dimensions, 776 in two dimensions, 355 initial velocity to reach given height, 354 Newton’s second law and, 352 parametric equations and, 626–627 terminal velocity in, 361 velocity at impact, 352–353 vertical motion in, 353 with air resistance, 356 simple harmonic, 1079, 1098 Moving trihedral, 788 Multiplicity, of zeros, 68, 184, 466 Multiplier effect, 553 Multipliers, Lagrange, 887–894 defined, 889 method of, 888 Music synthesizers, 617–618, 619 N Napier, John, 378 Natural frequency, 1087 Natural logarithms defined, 376 inverse of, 391–393 limiting behavior of, 381 Negative exponent, 254 Negative orientation, 1014 Newton’s Law of Cooling, 494 Newton’s method, 178–181 Newton’s second law of motion, 252, 352, 770 Newton, Isaac, 96, 178 Newton-Raphson method, 179 Nonexistent limit, 54, 55, 92, 825 Nonhomogeneous differential equations, 1082–1088 Nontrivial solution, 1098 Normal component of acceleration, 790–794 Normal distribution, 482 Normal line, 846, 870 Normal plane, 789 Normal vector, 713, 786–797 Normalization, of vector, 693 Notation derivatives, 121–123 Leibniz, 122 sigma, 259–264 summation, 260 vector, 688 Number(s) critical defined, 188 of rational function, 190 irrational, rational, real, 2–7 computer representation of, 99 sequences of, 532–542 Numerical differentiation, 122–123 Numerical integration, 298–309 error bounds for, 305–308 O Oblique asymptote, 83 Octants, 698 Octave, Omega function, 478 One-sided limits, 53, 56, 79 One-to-one functions, 385 One-to-one transformation, 963 Open disk, 828 Open interval, differential on, 119 Opposite vector, 690 Optimization, 223–230 constrained, 887–894 with inequality constraint, 890–891 with two constraints, 893–894 Orbit, geosynchronous, 779 Ordered pair, Orientable surface, 1039 Orientation of curve definition of, 990 negative, 1014 positive, 1014 vector-valued curve, 751 Orthogonal lines, 727 Orthogonal planes, 730 Orthogonal projection, 712 Orthogonal vectors, 706, 792 Orthogonality condition, 607 Osculating circle, 789 Osculating plane, 789 P p-series, 556–557 Pappus’ Theorem, 337 Parabola closest point on, 225–226 curvature of, 784 defined, 668 directrix of, 668 equation for, 669, 670 focus of, 668 line and, intersection of, 17–18 minimum distance on, 226–227 opening left, 670 reflective property of, 670 Paraboloid(s) applications of, 741–742 circular, 737 graph of, 737 hyperbolic, 740–741, 801 volume between two, 930 Parallel lines, 11, 12, 727 Parallel planes, 730 Parallel vectors, 691 Parallelepiped, volume of, 721 Parallelogram, area of, 720 Parameter, in parametric equations, 626 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Parameterizations, 628, 780 Parametric equations, 457 calculus and, 634–639 circles defined by, 627–628 conic sections in, 668–675 defined, 625 ellipses defined by, 627–628 for intersecting surfaces, 755 for line segment, 628 for x-y equations, 629 of hyperbolic paraboloid, 801 parameter of, 626 plane curves and, 625–630 projectile motion and, 626–627 sine and, 627 slope of, 635 surface area with, 641–647 Parametric plot, 736 Parametric surfaces, 799–802 graphing of, 799–800 Partial derivatives, 833–840 applications, 836–837, 839 computing of, 835–836 defined, 835 from table of data, 839–840 higher-order, 837 mixed second-order, 837 of three variable functions, 838–839 Partial differential operator, 835 Partial fraction decomposition, 442–448 Partial integration, 906 Partial sum, 546–547 error estimation in, 557 Partition(s) inner, 908, 926, 929 irregular, 272, 904 regular, 266 Parts, integration by, 426–431 repeated, 428 Pascal’s Principle, 367 Pascal, Blaise, 481 Path definition of, 1004 independence of, in line integrals, 1003–1011 of steepest ascent, 868 pdf See Probability density function (pdf) Pendulum, undamped, 1094 Perelman, Grigori, 735 Period, of periodic function, 30–31, 607 Periodic function defined, 28, 607 fundamental period of, 28 Perpendicular lines, 11, 12 Perpendicular vectors, 706 Phase portrait, 487, 522 Piecewise-continuous function, 278 Piecewise-defined functions, limit of, 65–66 Piecewise-smooth curve, line integrals over, 993–995 Piecewise-smooth surface, 1037 Pixels, 21 Planck’s law, 398, 606 Plane curves arc length of, 643 defined, 626 graph of, 626 unusual, 629 Plane, Cartesian, 6–7 Plane(s) equation of, 729 in R3 , 728–731 in space, 726–731 intersection of, 730 normal, 789 orthogonal, 730 osculating, 789 parallel, 730 tangent, 844–852 equation of, 846–847 gradient and, 870 normal line, 870 vectors in, 688–695 Planetary motion, 680, 794–797 Plot Bode, 1096 contour, 814 density, 814, 816 Point masses, 365 Point of diminishing returns, 233 Point-slope form, of line, 11 Point(s) boundary, 828 colinear, 9, 10 critical, 875, 878 derivative at, 118–119 distance from, 720 evaluation, 269, 270 fixed, 42 in three dimensions, 698 inflection, 205, 206 initial, 688 interior, 828 Lagrange, 184, 799 on parabola, closest, 225–226 saddle, 741, 876 source, 1026 I-13 Polar coordinates arc length in, 666 area in, 664, 927 calculus and, 660–666 conic sections in, 677–681 conversion to/from rectangular coordinates, 649–650 defined, 649 double integrals in, 926–931 graphing of, 652 horizontal tangent lines and, 661 intersections in, 665 plotting points in, 649 surface area in, 935 surface integral evaluation with, 1035 transformation involving, 965 triple integrals with, 948 volume in, 928 Polar form of vector, 694 Polynomial(s) coefficients of, 13 constant, 14 cubic, 14, 17 graph of, 23 defined, 13 degree of, 13 graph of, 212–213 Hermite, 1107 Legendre, 607 limit of, 61, 824 linear, 14 local extrema of, 189 quadratic, 14 quartic, 14 quintic, 14 Taylor, of degree n, 588 trigonometric function and, sum of, 220–221 Position equilibrium, 1074 estimating overall change in, 277 Position vector, 689, 692, 750 Positive orientation, 1014, 1039 Potential energy, 361, 1013 Potential function, 292 of gradient field definition of, 984 finding, 985–986 of vector field, 1010 Potential, electrical, 607 Power functions, arc length of, 346 Power functions, inside cosine, 294 Power rule for derivatives, 127–133 for indefinite integral, 254 with fractional exponent, 255 with negative exponent, 254 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index I-14 QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Power series convergence of, 580 defined, 580 differentiation of, 582–583 integration of, 583–584 solutions of differential equations, 1098–1106 Power, definition of, 863 Predator-prey systems, 298, 521–522 Present value, 496, 553 Pressure, of gas, vs volume and temperature, 157 Price vector, 713 Price, relative change in, 241 Price-to-earnings ratio, 821 Principal unit normal vector, 786 Probability density function (pdf), 481, 482 standard deviation for, 486 Probability distributions continuous, 481 discrete, 481 normal, 482 Probability, conditional, 478 Product (in chemistry), 246 Product rule(s), for derivatives, 135–140, 151 Profit defined, 245 marginal, defined, 239 Projectile motion, 265 analyzing, 771–772 in three dimensions, 776 in two dimensions, 355 initial velocity to reach given height, 354 Newton’s second law and, 352 parametric equations and, 626–627 terminal velocity in, 361 velocity at impact, 352–353 vertical motion in, 353 with air resistance, 356 Projection, of vector, 708–711 orthogonal, 712 Pythagorean comma, Q Quadratic approximation, 211 Quadratic factor, partial fractions with, 445–446 Quadratic formula, 16 Quadratic inequality, Quadratic polynomial, 14 Quadric surfaces, 735 Qualitative information, 510 Quartic polynomial, 14 Quaternions, 697 Quintic polynomial, 14 Quotient difference, 110, 127 limit of, 82 Quotient rule, for derivatives, 135–140 R Radians, 28 Radioactive decay, 494 Radius of convergence, 581, 602–603 Radius of curvature, 789 Radius of sphere, 702 Ramanujan, Srinivasa, 575 Range, of function, defined, 13 Rate of change in economics, 239–244 in sciences, 239–244 in volume, 157 instantaneous, 114, 132–133 interpreting, 114 maximum, 868 minimum, 868 Ratio test, 571–577 Rational function continuity of, 69, 71 critical number of, 190 defined, 15 graph of, 23–24, 208, 213–214 integration of, 442–448 limit of, 61 with no vertical asymptotes, 25 Rational numbers, Rationalizing, finding limit by, 62–63 Rayleigh-Jeans law, 606–607 Reactants, 246 Real line, Real number(s), 2–7 computer representation of, 99 sequences of, 532–542 Rectangle approximating area under curve with, 267 double integrals over, 903–907 Rectangular coordinates conversion to/from cylindrical coordinates, 951–952 conversion to/from polar coordinates, 649–650 conversion to/from spherical coordinates, 956 defined, 649 Recurrence relation, 1100 Reduction formula, 430, 451 Reflective property, 670 Region connected, 1004 continuity on, 829 double integrals over general, 908–913 elementary polar, 926 simply-connected, 1009, 1058 transformation of simple, 964 Regression, linear, 878, 879–880 Regular partition, 266 Related rates problems, 234–237 Relation, recurrence, 1100 Relative change in demand, 241 Relative change in price, 241 Reliability function, 478 Remainder term, 589 Removable discontinuity, defined, 70 Repeated integration by parts, 428 Residual, 879 Resonance, 1073, 1087 Resonant frequency, 1096 Restitution, coefficients of, 266, 358 Resultant force, 694 Resultant vector, 688 Revolution, solids of, 343 Riemann sum, 269, 270, 274 double, 904, 917 Riemann’s condition, 284 Riemann, Bernhard, 269 Riemann-Stieltjes integral, 426 Riemann-zeta function, 563 Right hand rule, 717 Right-handed coordinate system, 697 Rolle’s Theorem, 162, 596 Rolle, Michel, 162 Root test, 576 Root, of function, 14 Rose, four-leaf, 656 Rossmo, Ken, 388 Rudin, Mary Ellen, 911 Rule of 72, 501 S Saddle point, 741, 876 Sales vector, 713 Scalar functions, with gradient, 1026 Scalar product, 705 Scalar triple product, 721 Scalars, 688 Scatter plot, 67 Schrăodingers wave function, 1098 Secant function defined, 30 hyperbolic, 412 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index integrand with even power of, 436 inverse defined, 401 evaluation of, 402 Secant line, 108 Second component, of vector, 689 Second derivative, 131 implicit, 158 Second Derivative Test, 203–209, 877, 883 Second moment(s), 922 Second-order differential equations applications, 1090–1095 as system of first-order, 525 with constant coefficients, 1074–1080 Second-order partial derivatives, 837 Sensitivity, 246 Separable differential equations, 501–507 Sequence(s) bounded, 539–540 completeness axiom and, 541 convergence of, 532–533 decreasing, 538 defined, 532 divergence of, 534–535 factorial, 537 Fibonacci, 1, 544 general term of, 532 increasing, 538 L’Hˆopital’s Rule and, 535–536 limit of, 534 monotonic, 538, 540 of real numbers, 532–542 squeeze theorem for, 536 term of, 532 with terms of alternating signs, 537 Series alternating defined, 565 divergent, 567 harmonic, 565 sum of, 568–569 test, 565–566 binomial, 604–605 comparison test for, 559 conditional convergence of, 572 Fourier, 607–618 convergence of, 616–617 defined, 608 expansion of, 610, 611–612, 614–615 music synthesizers and, 617–618 Fourier cosine, 620 Fourier sine, 618, 620 geometric, 547 convergent, 547–548 divergence of, 548–549 harmonic, 550 alternating, 565 infinite, 266 convergence of, 546 defined, 545 divergence of, 546 sums of, 545 Maclaurin, 588, 598 p-series, 556–557 power convergence of, 580 defined, 580 differentiation of, 582–583 integration of, 583–584 solutions of differential equations, 1098–1106 Taylor applications of, 599–605 convergence of, 590 defined, 587 expansion, 588 for approximation of e, 591 for approximation of integral, 602 for approximation of sine value, 599 for definite integral, 601 new from old, 595 of sin x, 591 Set(s) Cantor, 552 Julia, 712 Mandelbrot, 712 Shading, Lambert, 874 Sigma factors, 620 Sigma notation, 259–264 Sigmoid function, 398 Signed area, 275 Simple curve, 1014 Simple harmonic motion, 1079, 1098 Simple region, transformation of, 964 Simply-connected region, 1009, 1058 Simpson’s Rule, 303–304, 310 Simpson, Thomas, 303 Sine function combined with cosine, 32 hyperbolic, 411 inverse, 414 integrand with even power of, 435 integrand with odd power of, 434 inverse, 399 evaluation of, 400 linear approximation of, 176 parametric equations and, 627 root function in, 294–295 I-15 solving equations with, 29 Taylor series for, 591, 599 Sink, in vector field, 1026 Slant asymptotes, 83 Slant height, 347 Slicing, calculation of volume by, 325–328 Slope finding, 10 for determining colinearity, 10 of curve, 49 of line, of parametric curve, 635 Slope field, 510 Slope-intercept form, 11 Smooth curves, 763, 991 Smooth surface, 1037 Solid center of mass of, 945–946 volume of, 331–332, 919 Solids of revolution, 343 Solution(s) direction field to visualize behavior of, 511–512 equilibrium, 505, 516–517, 522 family of, for differential equations, 503 general, for differential equations, 492, 1075 implicit, 504 nontrivial, 1098 power series, 1098–1106 steady-state, 1086 transient, 1086 Source, in vector field, 1026 Source-free vector field, 1026 Space line integrals in, 997–998 lines in, 726–731 motion in, 769–777 planes in, 726–731 surfaces in, 734–743 vector field graphing in, 981 vectors in, 697–702 Space curve, 750 Special functions, 602 Specific heat, 1063 Spectrum, energy, 620 Speed angular, 771 defined, 637 Sphere as quadric surface, 735 center of, 702 defined, 702 equation of, 702 radius of, 702 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index I-16 QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Spherical coordinates, 800, 956–960 cone equation in, 956 conversion to/from rectangular coordinates, 956 evaluation formula for, 970–971 surface integral evaluation with, 1038 triple integrals in, 957–960 volume with, 959–960 Spiral Archimedean, 653 Cornu’s, 648, 786 Spring constant, 362, 1074 Spring-mass system, 1074, 1078–1079 Spring(s) Hooke’s Law and, 362 work done in stretching, 362 Square root function, 15 chain rule with, 144 derivative of, 119–120 Square-wave function, Fourier series expansion, 610, 614–615 Squeeze theorem for functions, 64 for sequences, 536 Stable equilibrium solutions, 512, 517 Standard basis vectors, 693 Standard deviation, 486 Steady-state solution, 1086 Steepest ascent method, 868, 880–881 Stokes’ Theorem, 1053–1059 definition of, 1053 line integral evaluation with, 1055 proof of, 1053–1054 surface integral evaluation with, 1056 Stokes, George Gabriel, 1054 Stretching, 40 Substitute commodities, 844 Substitution for power function inside cosine, 294 in evaluating definite integrals, 293, 295–296 integral table and, 447 integrand expansion with, 295 integration by, 292–296, 422 trigonometric, 437–440, 448 Sum(s), 259–264 Cesaro, 552 computation of, 262 derivative of, 131 indefinite integral of, 256 of alternating series, 568–569 of function values, 263 of infinite series, 545 of odd integers, 261 of trigonometric function and polynomial, 220–221 partial, 546–547 error estimate in, 557 Riemann, 269, 270, 274 double, 904, 917 Summation notation, 260 Summation, index of, 260 Surface area arc length and, 345–347, 641–647 calculation of, 934 computation of, 350 defined, 347 in polar coordinates, 935 numerical approximation of, 935–936 of parametric curves, 641–647 surface integrals for, 1038–1039 with parametric equations, 646 Surface integrals, 1032–1041 computing surface area with, 1038–1039 definition of, 1032, 1040 evaluation of, 1034–1035 using complement of surface, 1062 using polar coordinates, 1035 using spherical coordinates, 1038 using Stokes’ Theorem, 1056 Surface(s) complement of, 1062 contour plots and, 815 cylindrical, 734 in space, 734–743 intersecting, parametric equation for, 755 level, 817–818 orientable, 1039 paraboloid, 737 parametric, 799–802 parametric representations of, 1035–1041 piecewise-smooth, 1037 quadric, 735 smooth, 1037 two-sided, 1039 volume beneath, 904–905 Symmetric difference quotient, 127 Symmetric equations, of line, 726 Systems, of differential equations, 521–525 T Tables, integration, 301, 450–456 Tacoma Narrows Bridge disaster, 1073 Tag-and-recapture process, 843 Tangent function defined, 30 hyperbolic, 412 inverse, 414 integrand with odd power of, 436 inverse defined, 401 evaluation of, 401 integral related to, 408 simplification of expression with, 402–403 Tangent line approximation See Linear approximation Tangent line(s), 47–51 equation of, 110, 137, 152 finding, with implicit differentiation, 156 for function defined as integral, 289 graphical approximation of, 111 horizontal, 190, 661 numerical representation of, 111 to inverse function, 389 to parametric curve, 635 velocity and, 107–115 vertical, 190, 209 Tangent planes, 844–852 equation of, 846–847 gradient and, 870 normal line, 870 Tangent vector, 764, 786–797 unit, 780, 787–788 Tangential component of acceleration, 790–794 Tautochrone, 457, 648 Tautochrone problem, 645 Taylor polynomial of degree n, 588 Taylor series applications of, 599–605 convergence of, 590 defined, 587 expansion, 588 for approximation of e, 591 for approximation of integral, 602 for approximation of sine value, 599 new from old, 595 of sin x, 591 to approximate definite integral, 601 Taylor’s Theorem, 589, 595–596 Taylor, Brook, 426 Temperature, ambient, 494 Term of sequence, 532 remainder, 589 Term-by-term differentiation, 582 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Terminal point, of vector, 688 Terminal velocity, 361, 509 Tetrahedron, triple integral over, 940–941 Thermal diffusivity, 862 Third derivatives, 131 Three body problems, 184 Three-dimensions, plotting points in, 698 Three-leaf rose, 661 Threshold, critical, 512 Thrust-time curve, 370 Timbre, 617 Time, doubling, 493 TNB frame, 788 Topographical map, 821 Torque, 721, 722, 775 Torsion, 798 Total area, defined, 275 Total differential, 850 Transform, Laplace, 478 Transformation changing variables for, 968 image of, 963 in polar coordinates, 965 Jacobian of, 967 linear, 27 direct, 972 of data, of functions, 36–41 combination, 36 composition, 36 finding, 37 identification of, 37 stretching, 40 of simple region, 964 one-to-one, 963 Transient solution, 1086 Translation(s) comparing, 39 horizontal, 38–39 vertical, 37–38 Trapezoidal Rule, 302, 304 Triangle Inequality, 708, 712 Triangle inequality, Triangular wave function, Fourier series expansion, 611–612 Trigonometric functions, 26–34 derivatives of, 147–152 in powers, 294 integrals involving powers of, 433–437 inverse, 399–403 calculus of, 405–409 derivatives of, 406 integrals involving, 407–409 simplification of, 402 limit of, 63, 80, 91 loss of significance involving, 101 polynomial and, sum of, 220–221 Trigonometric identities, 32 Trigonometric substitution, 437–440, 448 Trihedral, moving, 788 Triple integrals, 928–946 center of mass and, 944–946 change of variables in, 970 defined, 928 for volume, 952–953 in cylindrical coordinates, 950–951 in spherical coordinates, 957–960 inner partition of, 929 order of integration in, 942–943 over rectangular box, 929 over tetrahedron, 940–941 volume with, 943–944 with first integration with respect to x, 941–942 with polar coordinates, 948 Trochoid, 640 Trojan asteroids, 185 Two-sided inequality, 3–4 Two-sided surface, 1039 U Undamped pendulum, 1094 Undetermined coefficients, method of, 1083 Unit ball, 957 Unit normal vector principal, 786 Unit tangent vector, 780, 787–788 Unit vector, 693, 701 Universal law of gravitation, 680, 794 Unstable equilibrium solutions, 512, 522 Up concavity, 203 Upper limit of integration, 273 Useful life phase, 421 V Variable(s) change of in antiderivative, 968–969 in double integral, 967 in multiple integrals, 962–971 in triple integral, 970 dimensionless, 858 several, functions of, 809–818 extrema of, 874–883 two, functions of, 809–810 I-17 Vector field(s), 977–987 conservative, 1003–1011 definition of, 1009 determination of, 1010, 1027 curl of, 1022–1029 computing, 1023 definition of, 1023 interpretation of, 1024 definition of, 978 divergence of, 1022–1029 computing, 1025 definition of, 1025 flux of, 1039–1041 graphing of, 978–979 incompressible, 1026 irrotational, 1024, 1027–1028 plotting, 978 potential function of, 1010 sink points, 1026 source points, 1026 source-free, 1026 with gradient, 1026 Vector format, 627 Vector-valued curves graph of, 750–751 orientation of, 751 Vector-valued functions antiderivative of, 765 calculus of, 758–767 continuity of, 759–760 defined, 750 definite integral of, 766 derivative of, 761 differentiation of, 761 ellipse, 751 elliptical helix, 751–752 graphing of, 750–751 increment of, 760 indefinite integral of, 766 limit of, 759 line, 752 matching to graph, 752 Vector(s) acceleration, 770 addition of, 688–689 additive inverse of, 691, 700 angle between two, 706 arithmetic with, 690–691 binormal, 788 components of, 689, 708–711 cross product, 714–723 direction, 711 displacement, 710 dot product of, 704–711 first component of, 689 horizontal component of, 637, 693 in plane, 688–695 www.EngineeringBooksPDF.com CONFIRMING PAGES P1: NAI/NAI P2: NAI/NAI MHDQ256-Sub-Index I-18 QC: NAI/NAI MHDQ256-Smith-v1.cls T1: NAI January 10, 2011 11:25 LT (Late Transcendental) Subject Index Vector(s)—Cont in R3 , 699–702 in space, 697–702 initial point of, 688 magnitude of, 688, 700 normal, 712, 728, 786–797 normalization of, 693 notation, 688 opposite, 690 orthogonal, 706, 792 parallel, 691 perpendicular, 706 polar form of, 694 position, 689, 692, 750 price, 713 principal unit normal, 786 projection of, 708–711 orthogonal, 712 resultant, 688 right-handed coordinate system and, 697 sales, 713 scalar product of, 705 scalar triple product of, 721 scalars and, 688 second component of, 689 standard basis, 693 subtraction of, 689 tangent, 764, 786–797 terminal point of, 688 unit, 693, 701 unit tangent, 780, 787–788 velocity, 770 vertical component of, 693 zero, 691, 700 Velocity angular, 774 average, 112–113 escape, 86 estimating numerically, 124 from acceleration, 770 from position, 770 horizontal component of, 637 instantaneous, 113 limit describing, 66 tangent lines and, 107–115 terminal, 361, 509 vector, 770 vertical component of, 637 Velocity field, 981 flux of, 1061–1062 Verhulst, Pierre, 505 Vertex of ellipse, 671 Vertical asymptotes, 24 in graphing, 212, 215–216 Vertical component, of vector, 637 Vertical line test, 13 Vertical translation(s), 37–38 Volume beneath surface, 904–905 between two paraboloids, 930 calculation of by cross-sectional areas, 326–327 by cylindrical shells, 338–343 by slicing, 325–328 by washers, 330–335, 340 disk method for, 328–330 estimating from cross-sectional data, 327 in polar coordinates, 928 in spherical coordinates, 959–960 maximization of, 224–225 of cylinder, 325, 665 of dome, 328 of parallelepiped, 721 of solid, 331–332, 919 of solid with cavities, 331–332 rate of change of, 157 with double integrals, 918 with triple integrals, 943–944, 952–953 W Washers, method of, 330–335, 340 Watts, Robert, 357 Wave equation, 843 Wave function, 1098 Weierstrass M-test, 586 Weierstrass, Karl, 74 Weight density, 947 Well-defined functions, 290 Wiles, Andrew, 189 Witten, Edward, 790 Work calculation of, 362, 710 defined, 361 line integrals and, 999–1000 Y Yau, Shing-Tung, 837 Yoccoz, Jean-Christophe, 455 Z Zero of multiplicity, 68, 184, 466 Zero vector, 691, 700 Zero(s) of function approximate, 25 approximating by method of bisection, 74–75 by Newton’s method, 179 Bessel function, 603 by factoring, 16 by quadratic formula, 16 cubic polynomial, 17 determining number of, 164 finding, by method of bisections, 74–75 www.EngineeringBooksPDF.com CONFIRMING PAGES ... THE FOURTH EDITION Building upon the success of the Third Edition of Calculus, we have made the following revisions to produce an even better Fourth Edition: Presentation r A key goal of the Fourth. .. www.EngineeringBooksPDF.com i CONFIRMING PAGES P1: OSO/OVY P2: OSO/OVY MHDQ256-FM-MAIN QC: OSO/OVY MHDQ256-Smith-v1.cls T1: OSO January 8, 2011 LT (Late Transcendental) 12:17 CALCULUS, FOURTH EDITION Published... beauty and power of mathematics Lastly, calculus faculty told us that it is critical for a calculus text to include all the classic calculus problems Other calculus textbooks may reflect one or