Lines Distance and Midpoint Formulas Slope of line through (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ): Distance 𝐷 between (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ): √ 𝐷 = (𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 𝑚= 𝑦2 − 𝑦1 𝑥2 − 𝑥1 Point-slope equation of line through (𝑥1 , 𝑦1 ) with slope 𝑚: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) Slope-intercept equation of line with slope 𝑚 and 𝑦-intercept 𝑏: 𝑦 = 𝑏 + 𝑚𝑥 Midpoint of (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ): ( ) 𝑥1 + 𝑥2 𝑦1 + 𝑦2 , 2 Quadratic Formula If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, then √ −𝑏 ± 𝑏2 − 4𝑎𝑐 𝑥= 2𝑎 Factoring Special Polynomials 𝑥2 − 𝑦2 = (𝑥 + 𝑦)(𝑥 − 𝑦) 𝑥3 + 𝑦3 = (𝑥 + 𝑦)(𝑥2 − 𝑥𝑦 + 𝑦2 ) 𝑥3 − 𝑦3 = (𝑥 − 𝑦)(𝑥2 + 𝑥𝑦 + 𝑦2 ) Rules of Exponents 𝑎𝑥 𝑎𝑡 = 𝑎𝑥+𝑡 𝑎𝑥 = 𝑎𝑥−𝑡 𝑎𝑡 (𝑎𝑥 )𝑡 = 𝑎𝑥𝑡 Circles Definition of Natural Log Center (ℎ, 𝑘) and radius 𝑟: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2 𝑦 𝑦 = ln 𝑥 means 𝑒 = 𝑥 ex: ln = since 𝑒0 = Ellipse 𝑥2 𝑦2 + =1 𝑎2 𝑏2 𝑦 𝑥 𝑦=𝑒 𝑦 = ln 𝑥 𝑥 𝑦 𝑏 −𝑎 𝑎 Identities 𝑥 −𝑏 ln 𝑒𝑥 = 𝑥 𝑒ln 𝑥 = 𝑥 Rules of Natural Logarithms Hyperbola 𝑥2 𝑦2 − =1 𝑎2 𝑏2 𝑦 ln(𝐴𝐵) = ln 𝐴 + ln 𝐵 ( ) 𝐴 = ln 𝐴 − ln 𝐵 ln 𝐵 𝑝 ln 𝐴 = 𝑝 ln 𝐴 𝑦 = 𝑏𝑥∕𝑎 𝑎 𝑥 𝑦 = −𝑏𝑥∕𝑎 Geometric Formulas Conversion Between Radians and Degrees: 𝜋 radians = 180◦ Triangle Circle Sector of Circle 𝐴 = 𝜋𝑟2 𝐴 = 12 𝑟2 𝜃 (𝜃 in radians) 𝐴 = 12 𝑏ℎ 𝑠 = 𝑟𝜃 (𝜃 in radians) 𝐶 = 2𝜋𝑟 = 𝑎𝑏 sin 𝜃 𝑟 𝑎 𝑟 𝑠 ℎ ✛ 𝜃 𝜃 ✲ 𝑏 𝑟 Sphere 𝑉 = 43 𝜋𝑟3 𝐴 = 4𝜋𝑟2 Cylinder 𝑉 = 𝜋𝑟2 ℎ Cone 𝑉 = 13 𝜋𝑟2 ℎ Trigonometric Functions 𝑦 𝑟 𝑥 cos 𝜃 = 𝑟 𝑦 tan 𝜃 = 𝑥 sin 𝜃 tan 𝜃 = cos 𝜃 (𝑥, 𝑦) ✛ sin 𝜃 = sin(𝐴±𝐵) = sin 𝐴 cos 𝐵±cos 𝐴 sin 𝐵 𝑦 cos(𝐴±𝐵) = cos 𝐴 cos 𝐵∓sin 𝐴 sin 𝐵 𝑟 ✻ ✛ sin(2𝐴) = sin 𝐴 cos 𝐴 ✛ 𝜃 𝑥 ✲ cos(2𝐴) = cos2 𝐴−1 = 1−2 sin2 𝐴 ❄ cos2 𝜃 + sin2 𝜃 = 𝑦 𝑦 𝑦 𝑦 = tan 𝑥 𝑦 = sin 𝑥 𝑦 = cos 𝑥 𝑥 𝜋 𝑥 𝜋 2𝜋 −1 The Binomial Theorem 2𝜋 𝑥 −𝜋 𝜋 −1 𝑛(𝑛 − 1) 𝑛−2 𝑛(𝑛 − 1)(𝑛 − 2) 𝑛−3 𝑥 𝑦 + 𝑥 𝑦 + ⋯ + 𝑛𝑥𝑦𝑛−1 + 𝑦𝑛 1⋅2 1⋅2⋅3 𝑛(𝑛 − 1) 𝑛−2 𝑛(𝑛 − 1)(𝑛 − 2) 𝑛−3 𝑥 𝑦 − 𝑥 𝑦 + ⋯ ± 𝑛𝑥𝑦𝑛−1 ∓ 𝑦𝑛 (𝑥 − 𝑦)𝑛 = 𝑥𝑛 − 𝑛𝑥𝑛−1 𝑦 + 1⋅2 1⋅2⋅3 (𝑥 + 𝑦)𝑛 = 𝑥𝑛 + 𝑛𝑥𝑛−1 𝑦 + CALCULUS Seventh Edition We dedicate this book to Andrew M Gleason His brilliance and the extraordinary kindness and dignity with which he treated others made an enormous difference to us, and to many, many people Andy brought out the best in everyone Deb Hughes Hallett for the Calculus Consortium CALCULUS Seventh Edition Produced by the Calculus Consortium and initially funded by a National Science Foundation Grant Deborah Hughes-Hallett University of Arizona William G McCallum University of Arizona Andrew M Gleason Harvard University Eric Connally David Lovelock Karen Rhea Harvard University Extension University of Arizona University of Michigan Daniel E Flath Macalester College Guadalupe I Lozano University of Arizona Ayşe Şahin Wright State University Selin Kalaycıoğlu Jerry Morris Adam H Spiegler New York University Sonoma State University Loyola University Chicago Brigitte Lahme Brad G Osgood Jeff Tecosky-Feldman Sonoma State University Stanford University Haverford College Patti Frazer Lock St Lawrence University Cody L Patterson University of Texas at San Antonio Thomas W Tucker Colgate University David O Lomen Douglas Quinney Aaron D Wootton University of Arizona University of Keele University of Portland with the assistance of Otto K Bretscher David E Sloane, MD Colby College Harvard Medical School Coordinated by Elliot J Marks ACQUISITIONS EDITOR VICE PRESIDENT AND DIRECTOR Shannon Corliss Laurie Rosatone DEVELOPMENT EDITOR FREELANCE DEVELOPMENTAL EDITOR Adria Giattino Anne Scanlan-Rohrer/Two Ravens Editorial MARKETING MANAGER John LaVacca SENIOR PRODUCT DESIGNER SENIOR PRODUCTION EDITOR David Dietz Laura Abrams COVER DESIGNER COVER AND CHAPTER OPENING PHOTO Maureen Eide ©Patrick Zephyr/Patrick Zephyr Nature Photography Problems from Calculus: The Analysis of Functions, by Peter D Taylor (Toronto: Wall & Emerson, Inc., 1992) Reprinted with permission of the publisher This book was set in Times Roman by the Consortium using TEX, Mathematica, and the package ASTEX, which was written by Alex Kasman It was printed and bound by 2VBE(SBQIJDT7FSTBJMMFT The cover was printed by 2VBE(SBQIJDT7FSTBJMMFT This book is printed on acid-free paper 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www.wiley.com/go/returnlabel If you have chosen to adopt this textbook for use in your course, please accept this book as your complimentary desk copy Outside of the United States, please contact your local sales representative This material is based upon work supported by the National Science Foundation under Grant No DUE-9352905 Opinions expressed are those of the authors and not necessarily those of the Foundation ISBN-13 978-1-119-37426-8 The inside back cover will contain printing identification and country of origin if omitted from this page In addition, if the ISBN on the back cover differs from the ISBN on this page, the one on the back cover is correct Printed in the United States of America 10 PREFACE Calculus is one of the greatest achievements of the human intellect Inspired by problems in astronomy, Newton and Leibniz developed the ideas of calculus 300 years ago Since then, each century has demonstrated the power of calculus to illuminate questions in mathematics, the physical sciences, engineering, and the social and biological sciences Calculus has been so successful both because its central theme—change—is pivotal to an analysis of the natural world and because of its extraordinary power to reduce complicated problems to simple procedures Therein lies the danger in teaching calculus: it is possible to teach the subject as nothing but procedures— thereby losing sight of both the mathematics and of its practical value This edition of Calculus continues our effort to promote courses in which understanding and computation reinforce each other It reflects the input of users at research universities, four-year colleges, community colleges, and secondary schools, as well as of professionals in partner disciplines such as engineering and the natural and social sciences Mathematical Thinking Supported by Theory and Modeling The first stage in the development of mathematical thinking is the acquisition of a clear intuitive picture of the central ideas In the next stage, the student learns to reason with the intuitive ideas in plain English After this foundation has been laid, there is a choice of direction All students benefit from both theory and modeling, but the balance may differ for different groups Some students, such as mathematics majors, may prefer more theory, while others may prefer more modeling For instructors wishing to emphasize the connection between calculus and other fields, the text includes: • A variety of problems from the physical sciences and engineering • Examples from the biological sciences and economics • Models from the health sciences and of population growth • Problems on sustainability • Case studies on medicine by David E Sloane, MD Active Learning: Good Problems As instructors ourselves, we know that interactive classrooms and well-crafted problems promote student learning Since its inception, the hallmark of our text has been its innovative and engaging problems These problems probe student understanding in ways often taken for granted Praised for their creativity and variety, these problems have had influence far beyond the users of our textbook The Seventh Edition continues this tradition Under our approach, which we call the “Rule of Four,” ideas are presented graphically, numerically, symbolically, and verbally, thereby encouraging students to deepen their understanding Graphs and tables in this text are assumed to show all necessary information about the functions they represent, including direction of change, local extrema, and discontinuities Problems in this text include: • Strengthen Your Understanding problems at the end of every section These problems ask students to reflect on what they have learned by deciding “What is wrong?” with a statement and to “Give an example” of an idea • ConcepTests promote active learning in the classroom These can be used with or without personal response systems (e.g., clickers), and have been shown to dramatically improve student learning Available in a book or on the web at www.wiley.com/college/hughes-hallett • Class Worksheets allow instructors to engage students in individual or group class-work Samples are available in the Instructor’s Manual, and all are on the web at www.wiley.com/college/hughes-hallett • Data and Models Many examples and problems throughout the text involve data-driven models For example, Section 11.7 has a series of problems studying the spread of the chikungunya virus that arrived v vi Preface in the US in 2013 Projects at the end of each chapter of the E-Text (at www.wiley.com/college/hugheshallett) provide opportunities for sustained investigation of real-world situations that can be modeled using calculus • Drill Exercises build student skill and confidence Enhancing Learning Online This Seventh Edition provides opportunities for students to experience the concepts of calculus in ways that would not be possible in a traditional textbook The E-Text of Calculus, powered by VitalSource, provides interactive demonstrations of concepts, embedded videos that illustrate problem-solving techniques, and built-in assessments that allow students to check their understanding as they read The E-Text also contains additional content not found in the print edition: • Worked example videos by Donna Krawczyk at the University of Arizona, which provide students the opportunity to see and hear hundreds of the book’s examples being explained and worked out in detail • Embedded Interactive Explorations, applets that present and explore key ideas graphically and dynamically— especially useful for display of three-dimensional graphs • Material that reviews and extends the major ideas of each chapter: Chapter Summary, Review Exercises and Problems, CAS Challenge Problems, and Projects • Challenging problems that involve further exploration and application of the mathematics in many sections • Section on the 𝜖, 𝛿 definition of limit (1.10) • Appendices that include preliminary ideas useful in this course Problems Available in WileyPLUS Students and instructors can access a wide variety of problems through WileyPLUS with ORION, Wiley’s digital learning environment ORION Learning provides an adaptive, personalized learning experience that delivers easy-to-use analytics so instructors and students can see exactly where they’re excelling and where they need help WileyPLUS with ORION features the following resources: • Online version of the text, featuring hyperlinks to referenced content, applets, videos, and supplements • Homework management tools, which enable the instructor to assign questions easily and grade them automatically, using a rich set of options and controls • QuickStart pre-designed reading and homework assignments Use them as-is or customize them to fit the needs of your classroom • Intelligent Tutoring questions, in which students are prompted for responses as they step through a problem solution and receive targeted feedback based on those responses • Algebra & Trigonometry Refresher material, delivered through ORION, Wiley’s personalized, adaptive learning environment that assesses students’ readiness and provides students with an opportunity to brush up on material necessary to master Calculus, as well as to determine areas that require further review Online resources and support are also available through WebAssign WebAssign for Hughes-Hallett Calculus Seventh Edition contains a vast range of assignable and autogradable homework questions as well as an Enhanced VitalSouce e-text with embedded videos, interatives, and questions Flexibility and Adaptability: Varied Approaches The Seventh Edition of Calculus is designed to provide flexibility for instructors who have a range of preferences regarding inclusion of topics and applications and the use of computational technology For those who prefer the lean topic list of earlier editions, we have kept clear the main conceptual paths For example, • The Key Concept chapters on the derivative and the definite integral (Chapters and 5) can be covered at the outset of the course, right after Chapter INDEX 𝑆-𝐼-𝑅 model, 616 SARS, online threshold value, 618, online Δ𝑡, 277 Δ, Delta, notation, 5, 144 𝛾, Euler’s constant, online ∫ , 322 𝑏 ∫𝑎 , 284 𝜇, mean, 469 𝜎, standard deviation, 469 ∑ , Sigma, notation, 283 →, tends to, 53 𝑑∕𝑑𝑥 notation, 108 𝑒, the number, 17 as limit, 255, online property of, 32 𝑓 ′ (𝑥) notation, 100 𝑔, 330, 441 𝑖, the number, online 𝑛 factorial, 475 𝑝-series, 493 𝑟, 𝜃, polar coordinates, 420 absolute growth rate, 586 absolute value function, 124 absolutely convergent, 500, 659 acceleration, 332 average, 118 instantaneous, 119 vector, online acceleration due to gravity, 330 accuracy, online to 𝑝 decimal places, online adding sines and cosines, 637 addition of vectors, online air pressure, 183 alternating series, 499 error bounds for, 500, 543 test, 499, 660 amplitude, 41, 637 analytical solution of differential equation, 580 angular frequency, 553 antiderivative, 316, 657 computing, 317, 323 constructing with definite integral, 335 Construction Theorem for, 336, 657 differential equations and, 329 family of, 316 from graph, 317 Fundamental Theorem and, 317 most general, 322 of 0, 322 of 1∕𝑥, 324, 657 of sin 𝑥, cos 𝑥, 657 of sin 𝑥, cos 𝑥, 324 of 𝑒𝑥 , 324, 657 of 𝑒−𝑥 , 316, 336 𝑛 of 𝑥 , 323, 657 of constant function, 323 of rational functions, 369 of same function, 322 of sin𝑚 𝑥 cos𝑛 𝑥, 361 properties of, 325 reduction formulas, 358 𝑥𝑛 cos 𝑎𝑥, 358 𝑥𝑛 sin 𝑎𝑥, 358 𝑥𝑛 𝑒𝑥 , 358 cos𝑛 𝑥, 358 tables of, 360 visualizing, 316 apple orchard yield, online approximation bisection, online error in Taylor polynomial, 539, 540, 660 Euler’s method, 576 finite difference, online Fourier polynomial, 547 global vs local, 549 linear, 178, 514, 655 Newton’s method, online of definite integral, 376 Picard’s method, online quadratic, 515 tangent line, 178, 514, 655 arc length, 413, 658 circle, 40 in polar coordinates, 427 of hanging cable, online of parametric curve, 414 arccosh, online arccosine function, 49, 653 Archimedean spiral, 423 Archimedes’ Principle, 418 arcsine function, 44, 653 derivative of, 166 domain and range, 44, 653 graph of, 45 arcsinh, online For “online” material, please see www.wiley.com/college/hughes-hallett 707 arctangent function, 45, 653 derivative of, 166 domain and range, 45, 653 graph, 45 Taylor series, 531 area between curves, 305 definite integral and, 285, 295, 402 finding by slicing, 402 polar coordinates, 425 Aristotle (384-322 BC), 332 arithmetic mean, 222 Aswan Dam, 448 asymptote horizontal, 18, 53, 69, 653 of rational functions, 53, 653 vertical, 33, 53, 653 average cost, 241 average rate of change, 91, 655 average value of function, 308, 657 average velocity, 84, online slope and, 87 bakery, 12 bank deposits as series, 484 Barbra Streisand, 486 barometric pressure, 604 base of exponential function, 15 of logarithm function, 32, 652 Bay of Fundy, 161 behaves like principle, 395 bell-shaped curve, 224, 312, 468 bending of light, 537 Bernoulli, Johann (1667-1748), 605 Bessel functions, online binomial series, 525, 660 bioavailability, 359 biodiesel fuel, 11, 22 bisection, method of, online bladder, 321 Bohr radius, 472 Bohr, Niels (1885-1962), 472 Boltzmann constant, online bonds, online trading at a discount, 487 trading at a premium, 487 bound, 540 above, 307 below, 307 708 INDEX best possible, 207, online greatest lower, online least upper, online lower, 207, online upper, 207, online boundary-value problem, 636 bounded sequence, 477 Boyle’s Law, online caffeine, 37, 604 calculators round-off error, 141, online Cantor Set, online carbon dioxide level, 351 carbon-14, 16, 596, online cardioid, 428, 429 carrying capacity, 56, 118, 607, 661 Cartesian coordinates, 420 catenary, 174, 177, online Cauchy, Augustin (1789-1857) formal definition of limit, 60 cdf, 461 center of mass, 432, 658 continuous mass density, 434 moment, 432 of point masses, 433 cephalexin, antibiotic, online chain rule, 152 and units, 152 applications, 164 integration and, 342 chaos, online characteristic equation, 642 critically damped case, 643 overdamped case, 642 underdamped case, 644 Chernobyl accident, 21 chikungunya, 613 chlorofluorocarbons, 114, 123 circle parameterization of, 259 closed curves as trajectories in phase plane, 621 closed form of series, 481 closed interval, CO2 in pond water, 295 coal production in US, 299 Cobb-Douglas function, 241 code-red computer virus, 614 common ratio, 482 comparison test for improper integrals, 395 for series, 494, 659 competition, 624 Competitive Exclusion Principle of, 626 completing the square, 362, 373 complex number algebra of, online complex plane, online conjugates, online definition, online differential equations and, 641 imaginary part, online polar representation, online powers of, online real part, online roots of, online complex plane, online polar coordinates, online compliance, 114 components of vector, 262, online composite functions, 24 derivative of, 151 compound interest, 450, 598 continuous, 587 Rule of Seventy, online compressibility index, online concavity, 14 downward, 14, 116, 655 error in definite integrals and, 379 Euler’s method and, 576 inflection point and, 196 of parametric curve, 266 power functions and, 134 second derivative and, 116, 192 upward, 14, 116, 655 concentration, 601 conditionally convergent, 500, 659 conjugates, online consols, online Constant Function Theorem, 187, 190, 656 constant of proportionality, Construction Theorem for Antiderivatives, 336 consumer surplus definite integral for, 454 definition, 453 continuity, 3, 58 at a point, 58, 60 definition, 60 definition of, 655 differentiability and, 125, 656 extrema and, 203 graphically, 58 limits and, 58 numerically, 58 of composite functions, 71 of function, 59 of sums, products, quotients of functions, 71, 655 on an interval, 58, 60, 654, 655 continuous rate, 17 continuous variable, 2, convergence of 𝑝-series, 493 of alternating series, 499 of geometric series, 483 of improper integrals, 385, 390 of partial sums, 488, 659 of power series, 504 of sequence, 476, 659 bounded, monotone, 477 of Taylor series, 526, 542 radius of, 507 rearranging terms, 501 coordinates Cartesian, 420 polar, 420 relation between Cartesian and polar, 421 Coroner’s Rule of Thumb, 250 cosh 𝑥, 174, 653 Taylor polynomial, 538 cosine function, 40 addition formula, online derivative of, 158, 160 graph of, 41, 653 Taylor polynomial approximation, 518 Taylor series, 524 convergence, 542 cost average, 241 fixed, 11, 233, 328 marginal, 110, 235, 328 definition of, 235 total, 233 variable, 11 cost function, 233 coupon, online credit multiplier, online critical point, 193, 318, 656 critical value, 193 extrema and, 194, 656 cross-section, 412 cubic polynomial, 51, 653 cumulative distribution function, 460 probability and, 464 properties of, 461 curve length of, 414 parameterization, 264 damped spring equation, 640, 641 damping, 641, 642 coefficient, 641 critical, 643 INDEX over, 642 term, 641 under, 644 daylight hours as a function of latitude, online Madrid, online decreasing, decreasing function, derivative of, 101, 115, 192, 655 integral of, 277 Decreasing Function Theorem, 190 definite integral, 283, 656 odd and even functions, 306 symmetry and, 306 as antiderivative, 336 as area, 285, 295, 402, 657 in polar coordinates, 425 as area above axis minus area below axis, 286 as average, 308 as total change, 294, 657 by substitution, 346 changing limits of, 346 comparison of, 307 definition of, 284 for arc length, 414 in polar coordinates, 427 of parametric curve, 414 for average value, 308 for center of mass, 432 for consumer surplus, 454 for cumulative distribution function, 460 for electric potential, 449 for force, 444 for force of gravity, 449, online for fraction of population, 459 for kinetic energy, online for mass, 430 for mean, 468 for present/future value, 452 for probability, 464 for producer surplus, 454 for surface area, online for volume, 403 for volume of revolution, 411 for work, 441 Fundamental Theorem of Calculus, 293, 297, 325, 657 interpretation, 285, 294 limits of integration, 302 notation for, 292 of density, 430 of density function, 459 of rate of change, 294 of sums and multiples, 304 properties of, 302, 657 units, 292 degree of a polynomial, 51 degree-days, online Delta, Δ, notation, 5, 144 demand curve, 12, 453 density, 429 of earth, 439 of water, 444 probability, 459 slicing and, 430 density function, 459, 658 probability and, 464 properties of, 459 dependent variable, 2, depreciation, 10 derivative, 655 𝑒𝑥 , 655 𝑛𝑡ℎ , 137 of inverse function, 166 approximation of, online as a function, 99 definition of, 100 finding algebraically, 103 finding graphically, 100 finding numerically, 101 formulas for, 103 at a point, 91 chain rule, 152, 655 critical points, 193, 656 definition of, 92, 655 differentiability and, 123 estimating graphically, 99 estimating numerically, 94 graphical interpretation, 101 higher, 115 inflection point, 656 interpretation, 91, 109 Leibniz notation for, 108 local maxima/minima, 194 test for, 194 of 𝑎𝑥 , 142 of 𝑒𝑥 , 141 of composite functions, 151, 655 of constant functions, 103 of constant multiples of a function, 130, 655 of cosine, 159, 655 of exponential functions, 140, 141, 165, 655 of hyperbolic functions, 176 of implicit functions, 171 of inverse trigonometric functions, 166, 655 of linear functions, 103 of ln 𝑥, 165, 655 of polynomials, 134 of positive integer powers, 131 709 of power functions, 132, 164, 655 of products of functions, 145, 655 of quotients of functions, 146, 655 of sine, 160, 161, 655 of sums and differences of functions, 130, 655 of tangent function, 160, online power rule, 104 product rule, 146, 655 quotient rule, 147, 655 second derivative, 115, 655 concavity and, 116 test, 196 slope of curve, 93 in polar coordinates, 427 slope of tangent line, 93 units of, 109, 655 visualizing, 92, 567, 655 derivative function, 100 Descartes, René (1596-1650), 332 difference quotient, 5, 91 differentiability continuity and, 125, 656 determining from graph, 123 local linearity and, 179, 181, 656 differentiable, 92, 100, 123 everywhere, 100 differential equations, 562, 661 𝐿𝑅𝐶 circuits, 649 𝑆-𝐼-𝑅 model, 616 SARS, online account balance, 587 analytical solution, 580 antiderivatives and, 329 arbitrary constants, 564 compartmental analysis, 601 concentration, 601 damped spring, 641 damped spring equation, 640 damping, 642, 643, 645 decay, 586 electric circuit, 640 equilibrium solution, 592, 661 Euler’s method, 575 existence of solutions, 569 exponential decay, 581 exponential growth, 580 SARS, online exponential growth and decay, 661 first-order, 563 general solution, 563, 661 exponential equation, 581, 710 INDEX 587 growth, 586 guess-and-check spring equation, 634 Hooke’s Law, 634 implicit functions and, 569 initial conditions, 329, 563, 661 initial value problem, 329, 563, 635, 661 linear second-order, 641 logistic, 607, 609, 661 Lotka-Volterra, 619 net worth of a company, 599 Newton’s Law of Cooling (Heating), 590 nullclines, 626, 627 numerical solutions of firstorder, 575 oil prices, 606 order of, 661 particular solution, 329, 563, 661 pendulum, 639 phase plane, 617 Picard’s method, online pollution, 588 population growth, 611 predator-prey model, 620 second-order, 564, 632 characteristic equation, 642 separable, 583 separation of variables, 580, 600 slope field, 567, 661 solution(s) to, 563, 661 spring equation, 634 systems of, 616, 661 thickness of ice, 597 uniqueness of solutions, 569 differentiation implicit, 171 of series, 531 convergence, 531 radius of convergence, 531 dipole, 535 direction field, online direction of vector, online directly proportional, discrete variable, 2, disease, 616 disease incidence, 554 displacement, 432 displacement vector, online distance estimating from velocity, 272 on parametric curve, 414 visualizing on velocity graph, 273 distribution function, 457 cumulative, 460 probability and, 464 distribution of resources, online divergence of geometric series, 483 of harmonic series, 490 of improper integrals, 386, 390 of partial sums, 488, 659 of sequence, 476, 659 diverging to infinity, 63, 387 domain, restricting, dominance l’Hopital’s rule and, 254 dominates, 50 double angle formulas, 351, 358, 653, online double factorial, online doubling time, 16, 587, 652 drug desensitization, online drug dosage as series, 481, 486 Dubois formula, 56, 247 Ebbinghaus model, 605 economy of scale, 233 Einstein, Albert (1879-1955), 537 Theory of Relativity, 157 electric charge, online electric circuits, 640, 649, online resonance, 537 electric dipole, 535 electric potential, online definite integral for, 449 electron wave function, online elementary functions, 342 end behavior of a function, 53 endocrinologist, 301 energy electrical, 352 wind, online energy spectrum, 552 musical instruments, 553, 557 energy theorem, 551 epidemic, 616, 618, online equation graphing in polar coordinates, 422 equations of motion, 330 equilibrium stable, online equilibrium point, 627 predator-prey model, 621 equilibrium price/quantity, 453 equilibrium solution of differential equation, 563 of differential equation, 592, 598, 661 stable, 592 unstable, 592 erf(𝑥), 340, 359, online error, online alternating series bounds, 500 Taylor polynomial, 543 concavity and, 379 Euler’s method, 577 in approximating definite integral, 658 in approximating derivative, online in linear approximation, 178 in numerical integration, 378, 379 left and right rules, 379 Riemann sum approximation, 277 round-off, 98, 141, online tangent line approximation, 179, 180 estimation of, 179 Taylor polynomial, 539, 540 trapezoid vs midpoint rule, 380 error bound, 540 Lagrange, 540 error function, erf(𝑥), 340, 359, online escape velocity, 606 Ethanol fuel, 37 Euler’s Constant, online Euler’s formula, 645, online Euler’s method, 575, 661 accuracy of, 577 error, 577 Euler, Leonhard (1707-1783), 575, online Eulerian logarithmic integral, 359 even function, 25, 654 definite integral of, 306 expansions, Taylor, 524 explicit function, 171 exponential decay, 15, 17, 652 differential equation for, 586 half-life, 16, 587, 652 radioactive decay, 16 exponential function, 13, 15, 652 𝑦-intercept, 17 population growth, 13 as solution to differential equation, 580 base 𝑒, 17 base of, 15 compared to power functions, 50 concavity, 14 derivative of, 140, 165 domain, 15 formula for, 15, 17, 652 INDEX Padé approximant to, 537 table of values, 14 Taylor polynomial approximation, 518 Taylor series, 524 convergence, 545 exponential growth, 14, 15, 17, 652 annual, 588, 652 continuous, 588, 652 differential equation for, 586 doubling time, 16, 587, 652 growth factor, 14, 16 rate absolute, 586 annual, 588, 652 continuous, 17, 588, 652 continuous vs annual, 588 relative, 586 SARS, online exponential growth and decay differential equation, 661 extrapolation, 4, 611 extrema, 203 continuity and, 203 critical point and, 194, 656 local, 193 Extreme Value Theorem, 203, 655 factorial, 517 double, online factoring, online family of functions, 5, 224 antiderivatives, 316 exponential, 16 with limit, 226 Gompertz growth, 233 linear, logistic, 228 motion due to gravity, 226 normal density, 224 power, 49 sinusoidal, 41 surge, 231 Fermat’s Principle, 222 Fibonacci sequence, 479 finite difference approximation, online firebreaks, online first derivative inflection point and, 197 first-derivative test, 194, 656 First Fundamental Theorem of Calculus, 293 fixed cost, 11, 233, 328 foot-pound, 444 force, 333, 444, 658 between atoms, online between molecules, online definite integral for, 444 from pressure, 444 of gravity definite integral for, 449, online spring, 440 units, 440 force of gravity, 441 Four, Rule of, Fourier coefficients formulas for period 2𝜋, 547 period 𝑏, 553 justification of, 555, 558, 559 Fourier polynomial, 547 square wave, 548 Fourier series, 546, 660 period 2𝜋, 550 period 𝑏, 553 square wave, 550 Fourier, Joseph (1768–1830), 546 fraction of population from cumulative distribution function, 461 from density function, 459 frequency angular, 553 fuel consumption, 223 function, absolute value, 124 average value of, 308, 657 bounded, online Cobb-Douglas, 241 composite, 24 concave (up or down), 14, 655 cost, 233 cumulative distribution, 460 decreasing, 5, 6, 655 density, 459 distribution, 457 drug buildup, 18 elementary, 342 energy theorem, 552 even, 25 exponential, 13, 15, 652 family of, 226 Fourier series for, 549 gamma, online global approximation, 549 global behavior of, 50 graph of, hyperbolic, 653 increasing, 4, 6, 655 input, inverse, 26 inverse trigonometric, 44 invertible, 26 711 linear, 4, 5, 652 local approximation, 549 local behavior of, 50 logarithm, 32 logistic, 609 monotonic, odd, 25 output, periodic, 40, 546 piecewise linear, 124 polynomial, 51, 653 power, 49 probability density, 459, 464 profit, 234 pulse train, 550 rational, 53, 653 reflection across 𝑥-axis, 23 representation, revenue, 234 shift and stretch, 23 sinusoidal, 41, 652 surge, 231 table, Taylor series for, 524 trigonometric, 39 cosine, 40 sine, 40 tangent, 43 unbounded, online zeros of, 51, online fundamental harmonic, 550 Fundamental Theorem of Calculus, 293, 297, 317, 325, 336, 657 future value, 658 annual compounding, 450 continuous compounding, 450 definite integral for, 452 definition, 450 of income stream, 451 G H Hardy, 1847 − 1947, online Galilei, Galileo (1564-1642), 13, 596, online Dialogues Concerning Two New Sciences, 332 gamma function, online Gause yeast population model, 631, online general solution of differential equation, 329 general term, 524 general term of a series, 488 geometric mean, 222 geometric series, 480, 482, 659 as binomial series, 525 common ratio, 482 convergence of, 483 712 INDEX divergence of, 483 finite, sum of, 482, 659 infinite, 482 infinite, sum of, 483, 659 partial sum of, 483 geometry definite integral and, 410 Gini’s index of inequality, online global behavior of a function, 50, 52 global maxima/minima, 203, 656 Golden Gate Bridge, online golden ratio, 479 Gompertz equation, online Gompertz growth, 233 Grand Canyon flooding, online Grand Coulee Dam, 409 graph in polar coordinates, 422 gravitational constant, online gravitational field, online gravity acceleration due to, 139, 184, 330, 332, 334, 606, online force of, 247, 449, online Great Lakes, 588, 594 Great Pyramid of Egypt, 405, 443 greatest lower bound, online Greenland Ice Sheet, 112 Gregory, James (1638–1675), 531 Grinnell Glacier, 595 growth continuous, 17 growth factor, 14, 16 growth of children, 301 growth rate absolute, 586 annual, 588 continuous, 588 relative, 586 guess-and-check, 342 spring equation, 634 half-life, 16, 587, 652 hanging cable, 566 arc length, online harmonic series, 490 harmonics, 550 height velocity graph, 301 higher derivative, 115 histogram, 457, 458 Hooke’s Law, 440, 634 Hooke, Robert (1635–1703), 634 Hoover Dam, 445 l’Hopital, Marquis de (1661–1704), 605, online l’Hopital’s rule, 252, 656 horizontal asymptote, 18, 53 horizontal line, horizontal line test, 27 Hubbert, M King (1903–1989), 607 hybrid cars, online hydroelectric power, 11, 22 hydrogen, reduced mass of, 537 hyperbola, 176 hyperbolic cosine, 174, 653 identities, 175 sine, 174, 653 tangent, 176 inverse, online hyperbolic functions derivative of, 176 hyperinflation, 38 identities, 653 imaginary numbers, online imaginary part of complex number, online implicit differentiation, 171, 656 implicit function as solution to differential equation, 569 derivative of, 171 improper integral, 385, 658 comparing, 394, 395 with 1∕𝑥𝑝 , 396 with 𝑒−𝑎𝑥 , 396 comparison test, 395, 658 convergence/divergence, 385, 390 energy and, 388 infinite integrand, 389 income stream, 451 increasing function, 4, 6, 187 derivative of, 101, 115, 192, 655 integral of, 277 Increasing Function Theorem, 187, 656 indefinite integral, 322, 657 computing, 323 properties of, 325, 657 visualizing, 316 independent variable, 2, index in power series, 524 index in Taylor series, 524 inertia, principle of, 333 infinite series geometric, 482 sum of, 480 inflection point, 196, 318, 656 first derivative and, 197 initial condition, 329 initial value problem, 329, 563, 636, 661 mass/spring system, 635 input into a function, instantaneous velocity, 87 growth rate, 588 rate of change, 91, 655 speed, 262 velocity, 86 integral definite vs indefinite, 323 definite see definite integral, 283 equation, online improper see improper integral, 385 indefinite see indefinite integral, 322 integral test for series, 492 integrand, 284 infinite, 389 sums and multiples, 304 integration applications, 658 limits of, 284, 302 numerical methods, 376 error, 378 LEFT(𝑛), 377 left-hand sum, 284 MID(𝑛), 377 midpoint rule, 376 Riemann sums, 284, 377 RIGHT(𝑛), 377 right-hand sum, 284 SIMP(𝑛), 381 TRAP(𝑛), 378 of series, 531 reduction formulas, 358, 360 techniques by parts, 353, 354, 658 completing the square, 362, 373 factoring, 362 guess-and-check, 342 long division, 362 partial fractions, 366, 658 reduction formulas, 360 sine substitution, 369 substitution, 342, 343, 346, 657 tangent substitution, 372 trigonometric substitutions, 369 using table, 360 intercept factored form and, 51 vertical, 4, 5, 652 interest, 598 compound, 450 continuously compounded, 587 Rule of Seventy, online INDEX Intermediate Value Theorem, 60, 655 interval notation, interval of convergence, 505, 506, 660 inverse function, 26 definition, 27 derivative of, 166 domain and range, 26 for exponential function, 35 formulas for, 27 graphs of, 27 hyperbolic tangent, online inverse square law, online inverse trigonometric functions, 44 arccosine, 49, 653 arcsine, 44, 653 arctangent, 45, 653 derivative of, 166 inversely proportional, invertible function, 26 isotopes, 37 iteration, online Iwo Jima, Battle of, 625 joule, 440 Keeling Curve, 157 kinetic energy, online definite integral for, online l’Hopital’s rule, 252, 253, 656 ⋅ ∞, 254 and 𝑒, 255 limits involving ∞, 253 l’Hopital, Marquis de (1661–1704), 605, online Lagrange error bound, 540, 660 Lagrange, Joseph-Louis (1736– 1813), 540 Lake Mead, 409 Lambert function, online Lanchester, F.W (1868–1946) differential equations, 625 square law, 625 lapse rate, 111 leading coefficient, 51 leaf decomposition, 603 least upper bound, online left rule, LEFT(𝑛), 377 error in approximation, 380 LEFT(𝑛), 377 left-hand limit, 67 left-hand sum, 656 Leibniz, Gottfried Wilhelm (16461716), 108, 584 lemniscate, 429 Lennard-Jones model, 232 Leonardo di Pisa, 479 light reflection, 222 light, bending of, 537 limaỗon, 424 limit, 654 , 𝛿 definition, 654 at infinity, 68, 654 continuity and, 58 continuous function, 62 definite integral and, 283 definition, 60, 654 evaluating limits using a new variable, 78 instantaneous acceleration and, 119 instantaneous velocity and, 86 left-hand, 67 local linearity and, 252 meaning of, 60 of (1 + ℎ)1∕ℎ , 141, online of 𝑥𝑛 ∕𝑛, 543 of (sin ℎ)/ℎ, 159 of a constant function, 71 of a constant multiple, 71 of a product, 71 of a quotient, 71 of a sum, 71 of improper integrals, 385 of sequence, 476 of sum, 131 of the type 0∕0, 76 one-sided, 67, 654 properties of, 71, 654 quotients of functions, 75 right-hand, 67 round-off error and, 141 rules, 70 two-sided, 67 limit comparison test, 496, 659 limits of integration, 284, 302 substitution and, 346 line best fitting, online equation of, 5, 652 parametric equation, 261 tangent, 93 linear approximation, 178, 514, 655 linear function, 4, 5, 652 derivative of, 103 intercept of, 5, 652 slope of, 5, 652 table of, linearization, 178 Lissajous figure, 266, online Liu Hui, online loading curve, 221 local behavior of a function, 50, 52 local extrema, 193 713 local linearity, 179, 252, 656 differentiability and, 179 local linearization, 178, 655 local maxima/minima, 193, 656 tests for, 193 logarithm function, 32 derivative of, 165 domain, 32, 652 graph, 33 intercept, 32 Taylor polynomial approximation, 520 Taylor series, 527 vertical asymptote, 33 logarithms, 32 base 𝑒 (natural), 32 base ten (common), 32, 652 definition of, 32 properties, 33 rules of manipulation, 33, 652 solving equations, 33 logistic equation, 607 logistic function, 609 logistic model, 117, 228, 607, 661, online analytic solution to, 609 carrying capacity, 118, 607 equation, 612 peak oil, 606 qualitative solution to, 607 SARS, online spread of information, online long division, 362, 368 Lotka-Volterra equations, 619 lottery winnings, 455 lower bound, 207, online lung, 114 Machin’s formula for 𝜋, online Maclaurin polynomial 𝑛th , 517 first, 514 second, 516 Maclaurin series, 524 magnitude of vector, online marginal, 233, 235 costs/revenues, 110, 235, 328 Marquis de l’Hopital (1661–1704), 605, online mass center of, 432 from density, 430 of earth, 439 relativistic, 538 vs weight, 441, 444 Mass Action, Law of, online Massachusetts Turnpike, 430 Mauna Loa Observatory, 48, 157 714 INDEX maxima/minima concavity and, 195 global, 203, 656 continuity and, 203 on (𝑎, 𝑏), 204 on [𝑎, 𝑏], 204 local, 193, 238, 656 first derivative test, 656 first-derivative test, 194 second derivative test, 195, 656 Maxwell distribution, online mean, 467 arithmetic vs geometric, 222 definite integral for, 468 of normal distribution, 469 Mean Value Inequality, 190 Mean Value Theorem, 186, 656, online measles, 554 median, 466 meteorology, 111 Michelson-Morley experiment, 538 MID(𝑛), 377 midpoint rule, MID(𝑛), 376 error in approximation, 379, 381 modeling, 212 and differential equations, 562, 597 compartmental analysis, 601 competitive exclusion, 626 epidemic, 616 growth and decay, 586 logistic peak oil, 606 optimization, 214 predator-prey, 620 US population, 611 vibrating spring, 633, 640 moment, 432 monotone sequence, 477 monotonic function, 6, 115, 123, 192 Montgolfier brothers, online morphine, 604 motion 𝑥𝑦-plane, 263 damped simple harmonic, 645 parametric equations, 259 position vectors and, online simple harmonic, 636 undamped, 645 Mt Shasta, online murder, 590, 596 natural logarithm, 32 derivative of, 165 graph of, 33 Nelson, Admiral Horatio, online newton, 440, 444 Newton’s law of gravity, online Newton’s method, online chaos, online failure, online initial estimate, online Newton, Isaac (1642-1727) Law of Cooling (Heating), 157, 590 Law of Gravity, 50 laws of motion, 332 First Law, 333 Second Law, 333, 600, 634, 641 Nicotine, 595 nondecreasing function, 187 nondifferentiability, examples, 124 normal distribution, 224, 312, 468, 469 standard, 469 normal line, online nullclines, 626, 627 numerical methods accuracy, online bisection, online decimal answer, online differential equations, 575 error, online left and right rules, 379 trapezoid and midpoint rules, 380 Euler’s method, 575 finding derivative, 101 integration, 376, 658 error, 378, 379 left rule, 377 midpoint rule, 377 right rule, 377 Simpson’s rule, 381 trapezoid rule, 378 iterative, online Newton’s method, online Picard’s method, online odd function, 25, 654 definite integral of, 306 oil prices, 606 oil production worldwide, 615 Olympic pole vault, one-sided limit, 67 open interval, optimization, 203, 212 maximizing/minimizing ages, 217 constrained, 242 optimal values, 203 oscillations, 633 aver- of spring, 634 damped, 641 output of a function, ozone, 123 ozone depleting gas index, 114 Padé approximants, 537 parabola, 51 parallel lines, parameter, 6, 224 change of, 260 parameterization length of a curve and, 414 of circle, 259 of curve, 264 changing, 260 of graph of function, 265 of line, 261 speed and, 262 tangent lines and, 264 velocity, 262 components, 262 parametric curve concavity of, 266 slope of, 266 parametric equations, 259 parasite drag, 223 Pareto’s Law, online partial fractions, 366, 658 method of, 658 partial sum, 488, 659 of geometric series, 483 particular solution, 329 parts integration by, 353, 658 pascal, 444 pdf, 459 peak oil, 606 US, 610 pendulum, 139, 247, 639 differential equation for, 523 period, 41, 636, 653 angular frequency and, 553 periodic function, 40, 546 perpendicular lines, pH, 168 phase difference, 41 phase plane, 617, 620 equilibrium point, 627 nullclines, 627 trajectories, 618, 621, 629 phase shift (angle), 41, 637 vs horizontal translation, 638 phyllotaxis, 479 Picard’s method, online piecewise linear function, 125 Pisa, Tower of, 333 Planck, Max (1858-1947) INDEX radiation law, 399, online plutonium, 37 Poiseuille’s Law, 56, online polar coordinates, 420 Archimedean spiral, 423 area, 425 cardioid, 428 circle, 422 graphing equations, 422 lemniscate, 429 limaỗon, 424 negative , 423 roses, 423 slope, 427 polynomial, 51, 653 cubic, 51, 653 degree of, 51 derivative of, 134 double zero, 52 factoring, online Fourier, 547 intercepts, 51 leading coefficient, 54 quadratic, 51, 653 quartic, 51, 653 quintic, 51, 653 zeros, 52 population of Burkina Faso, 13, 33 of China, 38 of Hungary, online of India, 38, 595 of Jamaica, online of Mexico, 144, 311 of US, 36, 144, 157, 611, online of world, 21, 38, 142, 156, 247, 297, 352, 479, 604, online population growth, 611 equilibrium population, 621 exponential, 13 logistic, 611 predator-prey model, 619 position vector, online motion and, online potential energy, 538 power function, 49, 653 compared to exponential functions, 50 concavity of, 134 derivative of, 131, 164 formula for, 49 long-term behavior, 654 power rule, 104, 132 power series, 504, 660 convergence of, 504 interval of convergence, 506, 660 radius of convergence, 506 ratio test, 507, 660 Taylor, 524 predator-prey model, 620 equilibrium values, 620 prednisone, immunosuppressant, online present value, 486, 658 annual compounding, 450 continuous compounding, 450 definite integral for, 452 definition, 450 of income stream, 451 pressure, 444, 658 air, 183 barometric, 604 definite integral of, 444 units, 444 principal, online Principle of Competitive Exclusion, 626 probability, 464 cumulative distribution function, 464 definite integral for, 464 density function and, 464 histogram, 457 probability density function, 459 producer surplus, 453 definite integral for, 454 product rule, 146 three dimensional, online production and price of raw materials, online production function Cobb-Douglas, 241 profit, 234 maximizing, 237, 238 projectile motion of, 185 Properties of Continuous Functions, 71 proportional directly, inversely, pulmonologist, 114 pulse train, 550, 558 Pyramid of Egypt, 405, 443 Quabbin Reservoir, 321 quadratic approximation, 515 quadratic formula, online quadratic polynomial, 51, 653 quarantine, online quartic polynomial, 51, 653 quintic polynomial, 51, 653 quotient rule, 147 715 Racetrack Principle, 188, 189, 656 radians, 39 conversion to degrees, 40 vs degrees for calculus, 94 radius of convergence, 506 and ratio test, 507 range, rate of change, 243 absolute, online average, 91, 655 instantaneous, 91, 655 related, 244 relative, online ratio test finding radius of convergence, 507 power series, 507, 660 series of constants, 498, 659 rational functions, 53, 653 asymptotes, 53, 653 end behavior, 53 intercepts, 54 real part of complex number, online recurrence relations, 476 reduction formulas 𝑝(𝑥)cos 𝑥, 360 𝑝(𝑥)sin 𝑥, 360 𝑝(𝑥)𝑒𝑥 , 360 𝑥𝑛 ln 𝑥, 360 cos𝑛 𝑥, 360 sin𝑛 𝑥, 360 reflection coefficient, online reflection of light, 222 reflection, Law of, 222 related rate, 243, 244 relative growth rate, 586 relative rate of change, online relativistic mass, 538 Relativity, Theory of, 157, 538 rent controls, 456 repeating decimal, 486 resistance, wind, online resonance, 537 revenue marginal, 235 definition of, 236 total, 234 revenue function, 234 Richardson arms race model, 631, online Riemann sum, 284, 287 area, 402 in polar coordinates, 425 density, 430 for arc length, 413 for center of mass, 432 for consumer surplus, 454 716 INDEX for force, 444 for mass, 430 for mean, 467 for present/future value, 452 for producer surplus, 454 for volume of revolution, 411 for work, 440 slicing and, 403 right rule, RIGHT(𝑛), 377 error in approximation, 380 RIGHT(𝑛), 377, 384 right-hand limit, 67 right-hand sum, 656 rise, 5, 652 Rolle’s Theorem, 187, online root mean square, online root test, online roots, online by bisection method, online by factoring, online by Newton’s method, online by numerical methods, online by zooming, online roses, 423 round-off error, 141 Rule of Four, Rule of Seventy, online run, 5, 652 Saint Louis arch, 177 sample a signal, 479 SARS, online satellite power, online saturation level, 18, 226 scalar, online scalar multiplication, online second derivative, 115 concavity and, 116, 192 inflection point and, 196 interpretation, 117 maxima/minima, 195 test, 195, 656 Second Fundamental Theorem of Calculus, 336 second-order differential equations, 632 sensitivity of a drug, 210 separable differential equation, 583 separation of variables, 580, 600, 661 exponential growth equation, 580 justification, 583 sequence, 474, 476 bounded, 477 bounded, monotone convergence of, 477 Calkin-Wilf-Newman, 480 convergence of, 476, 659 defined recursively, 475 divergence of, 476, 659 Fibonacci, 479 general term, 474 limit of, 476 monotone, 477 of partial sums, 488 smoothing, 479 series, 480 𝑝-series, 493 general term, 524 alternating, 499 harmonic, 500 alternating series test, 499 binomial, 525, 660 closed form, 481 comparison test, 494, 659 convergence, 488 absolute vs conditional, 500, 659 convergence properties, 490, 659 divergence, 488 Fourier, 546, 549, 660 period 𝑏, 553 general term, 488 geometric, 480, 482, 659 as binomial series, 525 harmonic, 490 alternating, 500 infinite, 480 integral test, 492, 659 limit comparison test, 496, 659 of constants, 659 partial sum of, 488, 659 power, 504, 660 ratio test, 498, 659 root test, online sum of, 480 Taylor, 524, 660 terms in, 480 Si(𝑥), 338 ∑ Sigma, , notation, 283 sigmoid curve, 607 signal, 479 SIMP(𝑛), 381 simple harmonic motion, 636 Simpson’s rule approximating by quadratic, online Simpson’s rule, SIMP(𝑛), 381 sine function, 40 addition formula, online derivative of, 158, 160 graph of, 41, 653 Taylor polynomial approximation, 517 Taylor series, 524 convergence, 545 sine-integral, Si(𝑥), 338, 339, online sinh 𝑥, 174, 653 Taylor polynomial, 538 sinusoidal functions, 41, 652 slicing, 403, 430 slope, of curve, 86, 92 of line, 5, 652 of parametric curve, 266 parallel lines, perpendicular lines, polar coordinates, 427 units of, velocity and, 87 slope field, 567, 661, online smooth a sequence, 479 Snowy tree cricket, solar panels, 12 solar photovoltaic installations, 595 solar power, 47 solid of revolution, 411 surface area, online solution curve, 568 Soviet-German pipeline, online speed, 84, 414, online instantaneous, 262 parameterized curve, 263 vs velocity, 84 speed of sound, 183 spiral, Archimedean, 423 spring, 163, 633 spring constant, 634 square wave, 547 Fourier polynomial, 548 Fourier series, 550 Srinivasa Ramanujan, 1887 − 1920, online stable equilibrium, 592, online standard deviation of normal distribution, 469 Statue of Liberty, 222 Streisand, Barbra, 486 strontium-90, 22, online substitution, 657 into definite integrals, 346 into indefinite integrals, 343 into Taylor series, 530 sum left-hand, 277 of infinite series, 480 overestimate vs underestimate, 277 Riemann, 284, 287 right-hand, 277 visualizing left- and right-hand, INDEX 277, 278 sum formulas for sin/cos, 653 summation notation, 283 superposition, 641 supply curve, 12, 453 surface area, online surge function, 206, 231 surplus, consumer and producer, 453 survival time, 471 symbiosis, 619, 624 symmetry, 25 definite integral and, 306 systems of differential equations, 616, 661 table, exponential function, and, 14 linear function, and, table of integrals, 360 tangent field, online tangent function, 43 asymptotes, 44 derivative of, 160, online domain, 44 graph, 44 period, 44 tangent line, 93, 655 approximation, 178, 514, 655 error, 178–180, 185, 656 for parametric curves, 264 slope of, 93 𝑥, 176 tax cut, 487 Taylor expansions, 524 Taylor polynomial for ln 𝑥 near 𝑥 = 1, 520 about 𝑥 = 0, 517 about 𝑥 = 𝑎, 520 accuracy, 521 alternating series error bound, 543 degree 𝑛, 517, 660 degree one, 514 degree two, 516 error, 539 bound, 540, 660 for 1∕(1 − 𝑥), 519 for cos 𝑥, 518 for cosh 𝑥, 538 for sinh 𝑥, 538 for sin 𝑥, 517 for 𝑒𝑥 , 518 inequalities, online linear approximation and, 180 Taylor series, 524, 660 (1 + 𝑥)𝑝 , 525 cos 𝑥, 524, 660 sin 𝑥, 524, 660 𝑒𝑥 , 524, 660 𝑒−𝑥 , 530 general term, 524 about 𝑥 = 0, 524 about 𝑥 = 𝑎, 525 arctan 𝑥, 531 comparing, 534 condition for, 524 convergence of, 526, 541 𝑒𝑥 , 545 cos 𝑥, 541, 542 ln(1 + 𝑥), 527 sin 𝑥, 545 differentiation of, 531 convergence, 531 radius of convergence, 531 general, 524 integration of, 531 ln(1 + 𝑥), 527 manipulations of, 530 multiplying, 532 substitution, 530, 532 terminal velocity, 113, 226, 601 terms in series, 480 theorem Antiderivative Construction, 336 Constant Function, 187 Extreme Value, 203 Fundamental, of Calculus, 293, 336 Increasing Function, 187 Lagrange Error Bound, 540 Mean Value, 186 Theory of Relativity, 157, 538 Three Gorges Dam, 448 threshold value, 618, online tin pest, online Titanic, 445, 448 Torricelli’s Law, online total cost, 233 total quantity from density, 430 total revenue, 234 total utility, 121 tractrix, 606 Trafalgar, Battle of, online trajectories phase plane, 618, 621, 629 transmission coefficient, online TRAP(𝑛), 378, 384 trapezoid rule, TRAP(𝑛), 378 error in approximation, 379, 381 trapezoid, area of, 378 triangular wave, 547 trigonometric functions, 39 amplitude, 41, 637, 653 717 cosine, 40 derivatives of, 158, online inverse, 44 period, 41, 653 phase difference, 41 phase shift, 637 sine, 40 superposition of, 637 tangent, 43 trigonometric identity, 40, 653 addition formula, online double angle, 351, 358, online trigonometric substitutions, 369 method of, 369 two-sided limit, 67 unbounded function, online unit circle, 39, 41, 652 units force, 440, 444 pressure, 444 work, 440 unstable equilibrium, 592 upper bound, 207, online urologist, 321 Van der Waal’s equation, 173, 538 variable continuous, dependent, 2, discrete, discrete vs continuous, independent, 2, variable cost, 11 vector, online acceleration, online addition, online components, online direction, online displacement, online magnitude, online position, online scalar multiplication, online unit, online velocity, online velocity 𝑥𝑦-plane, 263 as derivative, 92 average, 84, online escape, 606 instantaneous, 86, 119 of a falling body, 600 parameterized curve, 263 slope and, 87 speed, 414 terminal, 113, 601 vector, online visualizing from distance graph, 718 INDEX 86 vs speed, 84 velocity vector, 262 Verhulst, P F (1804–1849), 607 Vermeer (1632–1675), 38 vertical asymptote, 33, 53 voltage, 47, online volume definite integral for, 403 finding by slicing, 403 from cross-sections, 412 of revolution, 411 surface area, online warfarin, anticoagulant, 595 water clock, online water, density of, 444 weight, 441 weight vs mass, 444 wind power, 22 wind resistance, online winning probability, online work, 439, 658 definite integral for, 441 definition, 440 units, 440 Zeno, 84 zeros, online zeros of a function, 51 continuity and, 60 double, 52 multiplicity 𝑚, online zooming, online Differentiation Formulas (𝑓 (𝑥) ± 𝑔(𝑥))′ = 𝑓 ′ (𝑥) ± 𝑔 ′ (𝑥) ′ ( = 𝑘𝑓 ′ (𝑥) (𝑘𝑓 (𝑥))) ′ 𝑓 ′ (𝑥)𝑔(𝑥) − 𝑓 (𝑥)𝑔 ′ (𝑥) 𝑓 (𝑥) = 𝑔(𝑥) (𝑔(𝑥))2 𝑑 𝑛 (𝑥 ) = 𝑛𝑥𝑛−1 𝑑𝑥 𝑑 (ln 𝑥) = 𝑑𝑥 𝑥 𝑑 12 (tan 𝑥) = 𝑑𝑥 cos2 𝑥 𝑑 (arctan 𝑥) = 14 𝑑𝑥 + 𝑥2 (𝑓 (𝑥)𝑔(𝑥))′ = 𝑓 ′ (𝑥)𝑔(𝑥) + 𝑓 (𝑥)𝑔 ′ (𝑥) (𝑓 (𝑔(𝑥)))′ = 𝑓 ′ (𝑔(𝑥)) ⋅ 𝑔 ′ (𝑥) 𝑑 𝑥 (𝑒 ) = 𝑒𝑥 𝑑𝑥 𝑑 (sin 𝑥) = cos 𝑥 11 10 𝑑𝑥 𝑑 (arcsin 𝑥) = √ 13 𝑑𝑥 − 𝑥2 𝑑 𝑥 (𝑎 ) = 𝑎𝑥 ln 𝑎 (𝑎 > 0) 𝑑𝑥 𝑑 (cos 𝑥) = − sin 𝑥 𝑑𝑥 A Short Table of Indefinite Integrals I Basic Functions ∫ 𝑥𝑛 𝑑𝑥 = 𝑛+1 𝑥 + 𝐶, 𝑛+1 𝑑𝑥 = ln |𝑥| + 𝐶 ∫ 𝑥 𝑥 𝑎𝑥 𝑑𝑥 = 𝑎 + 𝐶, ∫ ln 𝑎 𝑛 ≠ −1 ∫ 𝑎>0 ln 𝑥 𝑑𝑥 = 𝑥 ln 𝑥 − 𝑥 + 𝐶 ∫ ∫ ∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶 cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶 tan 𝑥 𝑑𝑥 = − ln | cos 𝑥| + 𝐶 II Products of 𝒆𝒙 , cos 𝒙, and sin 𝒙 10 11 12 ∫ ∫ ∫ ∫ ∫ 𝑒𝑎𝑥 [𝑎 sin(𝑏𝑥) − 𝑏 cos(𝑏𝑥)] + 𝐶 + 𝑏2 𝑒𝑎𝑥 cos(𝑏𝑥) 𝑑𝑥 = 𝑒𝑎𝑥 [𝑎 cos(𝑏𝑥) + 𝑏 sin(𝑏𝑥)] + 𝐶 𝑎 + 𝑏2 [𝑎 cos(𝑎𝑥) sin(𝑏𝑥) − 𝑏 sin(𝑎𝑥) cos(𝑏𝑥)] + 𝐶, 𝑎 ≠ 𝑏 sin(𝑎𝑥) sin(𝑏𝑥) 𝑑𝑥 = 𝑏2 − 𝑎2 [𝑏 cos(𝑎𝑥) sin(𝑏𝑥) − 𝑎 sin(𝑎𝑥) cos(𝑏𝑥)] + 𝐶, 𝑎 ≠ 𝑏 cos(𝑎𝑥) cos(𝑏𝑥) 𝑑𝑥 = 𝑏 − 𝑎2 sin(𝑎𝑥) cos(𝑏𝑥) 𝑑𝑥 = [𝑏 sin(𝑎𝑥) sin(𝑏𝑥) + 𝑎 cos(𝑎𝑥) cos(𝑏𝑥)] + 𝐶, 𝑎 ≠ 𝑏 𝑏2 − 𝑎2 𝑒𝑎𝑥 sin(𝑏𝑥) 𝑑𝑥 = 𝑎2 III Product of Polynomial 𝒑(𝒙) with ln 𝒙, 𝒆𝒙 , cos 𝒙, sin 𝒙 13 14 ∫ ∫ 𝑥𝑛 ln 𝑥 𝑑𝑥 = 𝑛+1 𝑥 ln 𝑥 − 𝑥𝑛+1 + 𝐶, 𝑛+1 (𝑛 + 1)2 𝑛 ≠ −1 1 𝑝(𝑥)𝑒𝑎𝑥 − 𝑝′ (𝑥)𝑒𝑎𝑥 𝑑𝑥 𝑎 𝑎∫ 1 = 𝑝(𝑥)𝑒𝑎𝑥 − 𝑝′ (𝑥)𝑒𝑎𝑥 + 𝑝′′ (𝑥)𝑒𝑎𝑥 − ⋯ 𝑎 𝑎 𝑎 (+ − + − …) (signs alternate) 𝑝(𝑥)𝑒𝑎𝑥 𝑑𝑥 = 1 𝑝(𝑥) sin 𝑎𝑥 𝑑𝑥 = − 𝑝(𝑥) cos 𝑎𝑥 + 𝑝′ (𝑥) cos 𝑎𝑥 𝑑𝑥 𝑎 𝑎∫ 1 = − 𝑝(𝑥) cos 𝑎𝑥 + 𝑝′ (𝑥) sin 𝑎𝑥 + 𝑝′′ (𝑥) cos 𝑎𝑥 − ⋯ 𝑎 𝑎 𝑎 (− + + − − + + …) (signs alternate in pairs after first term) 1 16 𝑝(𝑥) cos 𝑎𝑥 𝑑𝑥 = 𝑝(𝑥) sin 𝑎𝑥 − 𝑝′ (𝑥) sin 𝑎𝑥 𝑑𝑥 ∫ 𝑎 𝑎∫ 1 = 𝑝(𝑥) sin 𝑎𝑥 + 𝑝′ (𝑥) cos 𝑎𝑥 − 𝑝′′ (𝑥) sin 𝑎𝑥 − ⋯ 𝑎 𝑎2 𝑎3 (+ + − − + + − − …) (signs alternate in pairs) 15 ∫ IV Integer Powers of sin 𝒙 and cos 𝒙 17 18 19 20 21 22 23 ∫ ∫ ∫ ∫ ∫ ∫ 𝑛−1 sin𝑛 𝑥 𝑑𝑥 = − sin𝑛−1 𝑥 cos 𝑥 + sin𝑛−2 𝑥 𝑑𝑥, 𝑛 positive 𝑛 𝑛 ∫ 𝑛−1 cos𝑛 𝑥 𝑑𝑥 = cos𝑛−1 𝑥 sin 𝑥 + cos𝑛−2 𝑥 𝑑𝑥, 𝑛 positive 𝑛 𝑛 ∫ −1 cos 𝑥 𝑚−2 𝑑𝑥 = + 𝑑𝑥, 𝑚 ≠ 1, 𝑚 positive 𝑚 − sin𝑚−1 𝑥 𝑚 − ∫ sin𝑚−2 𝑥 sin𝑚 𝑥 | (cos 𝑥) − || 𝑑𝑥 = ln || +𝐶 sin 𝑥 | (cos 𝑥) + || 1 sin 𝑥 𝑚−2 𝑑𝑥 = + 𝑑𝑥, 𝑚 ≠ 1, 𝑚 positive cos𝑚 𝑥 𝑚 − cos𝑚−1 𝑥 𝑚 − ∫ cos𝑚−2 𝑥 | (sin 𝑥) + || 𝑑𝑥 = ln || +𝐶 cos 𝑥 | (sin 𝑥) − || sin𝑚 𝑥 cos𝑛 𝑥 𝑑𝑥: If 𝑚 is odd, let 𝑤 = cos 𝑥 If 𝑛 is odd, let 𝑤 = sin 𝑥 If both 𝑚 and 𝑛 are even and positive, ∫ convert all to sin 𝑥 or all to cos 𝑥 (using sin2 𝑥 + cos2 𝑥 = 1), and use IV-17 or IV-18 If 𝑚 and 𝑛 are even and one of them is negative, convert to whichever function is in the denominator and use IV-19 or IV-21 If both 𝑚 and 𝑛 are even and negative, substitute 𝑤 = tan 𝑥, which converts the integrand to a rational function that can be integrated by the method of partial fractions V Quadratic in the Denominator 𝑥 1 𝑑𝑥 = arctan + 𝐶, 𝑎 ≠ ∫ 𝑥2 + 𝑎2 𝑎 𝑎 𝑏𝑥 + 𝑐 𝑥 𝑏 𝑐 25 𝑑𝑥 = ln |𝑥2 + 𝑎2 | + arctan + 𝐶, 𝑎 ≠ ∫ 𝑥2 + 𝑎2 𝑎 𝑎 1 26 𝑑𝑥 = (ln |𝑥 − 𝑎| − ln |𝑥 − 𝑏|) + 𝐶, 𝑎 ≠ 𝑏 ∫ (𝑥 − 𝑎)(𝑥 − 𝑏) 𝑎−𝑏 𝑐𝑥 + 𝑑 27 𝑑𝑥 = [(𝑎𝑐 + 𝑑) ln |𝑥 − 𝑎| − (𝑏𝑐 + 𝑑) ln |𝑥 − 𝑏|] + 𝐶, ∫ (𝑥 − 𝑎)(𝑥 − 𝑏) 𝑎−𝑏 24 VI Integrands Involving √ √ √ 𝒂𝟐 + 𝒙𝟐 , 𝒂𝟐 − 𝒙𝟐 , 𝒙𝟐 − 𝒂𝟐 , 𝒂>𝟎 𝑥 𝑑𝑥 = arcsin + 𝐶 ∫ √ 𝑎 𝑎 −𝑥 √ | | 29 𝑑𝑥 = ln |𝑥 + 𝑥2 ± 𝑎2 | + 𝐶 √ | | ∫ 2 𝑥 ±𝑎 ( ) √ √ 1 𝑥 𝑎2 ± 𝑥2 + 𝑎2 𝑎2 ± 𝑥2 𝑑𝑥 = 𝑑𝑥 + 𝐶 30 ∫ ∫ √𝑎2 ± 𝑥2 ( ) √ √ 1 𝑥 𝑥2 − 𝑎2 − 𝑎 𝑥2 − 𝑎2 𝑑𝑥 = 31 𝑑𝑥 + 𝐶 √ ∫ ∫ 𝑥2 − 𝑎2 28 𝑎≠𝑏 WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ... Andy brought out the best in everyone Deb Hughes Hallett for the Calculus Consortium CALCULUS Seventh Edition Produced by the Calculus Consortium and initially funded by a National Science Foundation... danger in teaching calculus: it is possible to teach the subject as nothing but procedures— thereby losing sight of both the mathematics and of its practical value This edition of Calculus continues... modeled using calculus • Drill Exercises build student skill and confidence Enhancing Learning Online This Seventh Edition provides opportunities for students to experience the concepts of calculus