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Larson 100% FREE Edwards with and GO DIGITAL The NEW GO DIGITAL feature gives you access to many helpful student resources, such as instructional and proof videos, interactive examples, three-dimensional graphs, and much more. Access these valuable tools by scanning the GO DIGITAL QR Codes® that appear on the pages in the text. CalcChat.com offers you the solutions to the odd-numbered exercises from the text. When the solutions are not enough, you can chat with an online tutor for live help. Visit the website for the tutors’ availability. CalcView.com presents video solutions of selected exercises from the text. Watch calculus instructors progress step-by-step through solutions, providing guidance to help you solve the selected exercise and others like it. Access the videos by scanning the GO DIGITAL QR Codes® or by visiting CalcView.com. 12e RRoonn LLaarrssoonn BBrruuccee EEddwwaarrddss Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-322 99778800335577774499113355__CCaallcc__1122ee__ccvvrr..iinndddd 11 1100//1133//2211 99::2255 AAMM Index of Applications Engineering and Physical Explorer 1, 698 Moving shadow, 159, 160, 162, 163 Sciences Explorer 18, 745 Muzzle velocity, 761 Explorer 55, 698 Navigation, 699, 761 Acceleration, 128, 132, 160, 162, 180, Falling object, 312, 434, 437 Newton’s Law of Cooling, 419, 422 257, 910 Ferris wheel, 870 Newton’s Law of Gravitation, 1045 Air pressure, 440 Field strength, 548 Newton’s Law of Universal Gravitation, Air traffic control, 158, 750, 854 Flight control, 159 487, 492, 854 Aircraft glide path, 197 Flow rate, 290, 294, 307, 351, 1109 Oblateness of Saturn, 473 Angle of elevation, 155, 159, 160 Fluid force, 506, 507, 508, 509, 510, 513, Ohm’s Law, 241 Angular rate of change, 381 514, 546, 549 Oil leak, 294 Angular speed, 38, 381 Force, 293, 509, 774, 775, 785, 786 Orbit Apparent temperature, 903 Free-falling object, 72, 96 of Earth, 698 Archimedes’ Principle, 514 Frictional force, 862, 866 of the moon, 690 Architecture, 698 Fuel efficiency, 581 of a satellite, 698, 731, 870 Asteroid Apollo, 742 Gauss’s Law, 1107, 1109 Orbital speed, 854 Atmospheric pressure and altitude, 323, Geography, 807, 817 Parabolic reflector, 688 349, 955 Gravitational fields, 1045 Particle motion, 132, 291, 294, 295, 698, Automobile aerodynamics, 30 Gravitational force, 581 717, 827, 835, 837, 844, 853, 854, Average speed, 44, 93 Halley’s Comet, 698, 741 865 Average temperature, 988, 1039 Hanging power cables, 393, 397 Path Average velocity, 116 Harmonic motion, 142, 163, 349 of a ball, 706, 842 Beam deflection, 697 Heat equation, 901 of a baseball, 709, 841, 842, 843, 864 Beam strength, 226 Heat flux, 1128 of a football, 843 Boyle’s Law, 493, 512 Heat transfer, 332 of a projectile, 186, 716, 842, 843 Braking load, 778 Heat-seeking particle, 925, 930 of a shot, 843 Breaking strength of a steel cable, 360 Height Pendulum, 142, 241, 910 Bridge design, 698 of a Ferris wheel, 40 Planetary motion, 745 Building design, 453, 571, 1012, 1040, of a man, 581 Planetary orbits, 691 1068 rate of change of, 157 Power, 173, 910 Cable tension, 761, 769 Highway design, 173, 197, 870 Producing a machine part, 463 Carbon dating, 421 Honeycomb, 173 Projectile motion, 164, 241, 679, 709, Center of mass, 504 Hooke’s Law, 487, 491, 512 761, 840, 842, 843, 851, 853, 854, Centripetal acceleration, 854 Hydraulics, 1005 864, 868, 870, 917 Centripetal force, 854 Hyperbolic detection system, 695 Psychrometer, 844 Centroid, 502, 503, 527 Hyperbolic mirror, 699 Radioactive decay, 352, 417, 421, 430, Charles’s Law, 78 Ideal Gas Law, 883, 903, 918 440 Chemical mixture problem, 435, 437 Illumination, 226, 246 Rectilinear motion, 257 Chemical reaction, 430, 558, 967 Inductance, 910 Refraction of light, 963 Circular motion, 844, 852 Kepler’s Laws, 741, 742, 866 Resultant force, 758, 760, 761 Comet Hale-Bopp, 745 Kinetic and potential energy, 1075, 1078 Resultant velocity, 758 Construction, 158, 769 Law of Conservation of Energy, 1075 Ripples in a pond, 29, 153 Cooling superconducting magnets with Length Rotary engine, 747 liquid helium, 78 of a cable, 477, 481 Satellite antenna, 747 Cycloidal motion, 844, 853 of Gateway Arch, 482 Satellites, 131 Dissolving chlorine, 85 of pursuit, 484 Sending a space module into orbit, 488, Doppler effect, 142 of a stream, 483 575 Einstein’s Special Theory of Relativity of warblers, 585 Solar collector, 697 and Newton’s First Law of Motion, Linear vs. angular speed, 160, 163 Sound intensity, 323, 422 207 Load supports, 769 Specific gravity of water, 198 Electric circuit, 371, 414, 434, 437 Lunar gravity, 257 Speed of sound, 286 Electric force, 492 Machine design, 159 Surveying, 241, 565 Electric force fields, 1045 Machine part, 471 Suspension bridge, 484 Electric potential, 882 Magnetic field of Earth, 1054 Temperature, 18, 180, 208, 322, 340, Electrical resistance, 189, 910 Mass, 1059, 1065, 1066 413, 963 Electricity, 159, 307 on the surface of Earth, 494 normal daily maximum in Chicago, Electromagnetic theory, 581 Mechanical design, 453, 797 142 Electronically controlled thermostat, 29 Meteorology, 883 at which water boils, 323 Emptying a tank of oil, 489 Motion Temperature distribution, 882, 902, 925, Engine design, 1067 of a liquid, 1122, 1123, 1127 930, 968 Engine efficiency, 207 of a spring, 531 Theory of Relativity, 93 Escape velocity, 98, 257 Moving ladder, 93, 158 Topography, 875, 929, 930 (continued on back inside cover) Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-322 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. DERIVATIVES AND INTEGRALS s. s Basic Differentiation Rules e c uc d d d S 1. [cu]=cu′ 2. [u±v]=u′±v′ 3. [uv]=uv′+vu′ dx dx dx k r wo 4. d [u]=vu′−uv′ 5. d [c]=0 6. d [un]=nun−1u′ e dx v v2 dx dx m o d d u d u′ H 7. [x]=1 8. [∣u∣]= (u′), u≠0 9. [ln u]= r dx dx ∣u∣ dx u o f s d d u′ d d 10. [eu]=euu′ 11. [log u]= 12. [au]=(ln a)auu′ r dx dx a (ln a)u dx a C a 13. d [sin u]=(cos u)u′ 14. d [cos u]=−(sin u)u′ 15. d [tan u]=(sec2 u)u′ ul dx dx dx m r d d d Fo 16. [cot u]=−(csc2 u)u′ 17. [sec u]=(sec u tan u)u′ 18. [csc u]=−(csc u cot u)u′ dx dx dx ut o d u′ d −u′ d u′ r 19. [arcsin u]= 20. [arccos u]= 21. [arctan u]= ea dx √1−u2 dx √1−u2 dx 1+u2 T d −u′ d u′ d −u′ 22. [arccot u]= 23. [arcsec u]= 24. [arccsc u]= dx 1+u2 dx ∣u∣√u2−1 dx ∣u∣√u2−1 d d d 25. [sinh u]=(cosh u)u′ 26. [cosh u]=(sinh u)u′ 27. [tanh u]=(sech2 u)u′ dx dx dx d d d 28. [coth u]=−(csch2 u)u′ 29. [sech u]=−(sech u tanh u)u′ 30. [csch u]=−(csch u coth u)u′ dx dx dx d u′ d u′ d u′ 31. [sinh−1 u]= 32. [cosh−1 u]= 33. [tanh−1 u]= dx √u2+1 dx √u2−1 dx 1−u2 d u′ d −u′ d −u′ 34. [coth−1 u]= 35. [sech−1 u]= 36. [csch−1 u]= dx 1−u2 dx u√1−u2 dx ∣u∣√1+u2 Basic Integration Formulas ∫ ∫ ∫ ∫ ∫ 1. kf(u) du=k f(u) du 2. [f(u)±g(u)] du= f(u) du± g(u) du ∫ ∫ un+1 3. du=u+C 4. un du= +C, n≠−1 n+1 ∫ ∫ du 5. =ln∣u∣+C 6. eu du=eu +C u ∫ ∫ 1 7. au du= ( )au +C 8. sin u du=−cos u+C ln a ∫ ∫ 9. cos u du=sin u+C 10. tan u du=−ln∣cos u∣+C ∫ ∫ 11. cot u du=ln∣sin u∣+C 12. sec u du=ln∣sec u+tan u∣+C ∫ ∫ 13. csc u du=−ln∣csc u+cot u∣+C 14. sec2 u du=tan u+C ∫ ∫ 15. csc2 u du=−cot u+C 16. sec u tan u du=sec u+C g n ni ∫ ∫ r du u a e 17. csc u cot u du=−csc u+C 18. =arcsin +C e L √a2 −u2 a ag ∫ du 1 u ∫ du 1 ∣u∣ ng 19. = arctan +C 20. = arcsec +C Ce a2 +u2 a a u√u2 −a2 a a © Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-322 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. TRIGONOMETRY Definition of the Six Trigonometric Functions Right triangle definitions, where 0 < θ < π(cid:21)2. y opp hyp Hθy p oten use Opposite scions θθ==hhayydppj cssecc θθ==ohapydppj (− 22, (−2212), 233π) 23π π2 9(00,° 1) π3 (π12, 2(3)22, 22) Adjacent tan θ = opp cot θ = adj (− 23, 12) 56π 4 135°120° 60°45° 4 π6 ( 23, 21) adj opp 150° 30° Circular function definitions, where θ is any angle. 0° 0 x y y r (−1, 0) π 180° 360° 2π (1, 0) sin θ = csc θ = (x, yy) r rθ = x2 + y2 cos θ =rxr sec θ =yxr (−( 232, −12) 2)76π542π12042°π5°240° 300°351π353°704π°116π( 2( 23, −2)21) x x y x − 2 ,− 2 3 270° 32π 3 2 , − 2 tan θ = cot θ = ( 1 3) (1 3) x y −2,− 2 (0, −1) 2, − 2 Reciprocal Identities Double-Angle Formulas 1 1 1 sin 2u=2 sin u cos u sin x= cos x= tan x= csc x sec x cot x cos 2u=cos2 u−sin2 u=2 cos2 u−1=1−2 sin2 u 1 1 1 2 tan u csc x= sec x= cot x= tan 2u= sin x cos x tan x 1−tan2 u Quotient Identities Power-Reducing Formulas sin x cos x 1−cos 2u tan x= cot x= sin2 u= cos x sin x 2 1+cos 2u Pythagorean Identities cos2 u= 2 sin2 x+cos2 x=1 1−cos 2u tan2 u= 1+tan2 x=sec2 x 1+cot2 x=csc2 x 1+cos 2u Cofunction Identities Sum-to-Product Formulas sin(π −x) =cos x cos(π −x) =sin x sin u+sin v=2 sin(u+v) cos(u−v) 2 2 2 2 csc(π −x) =sec x tan(π −x) =cot x sin u−sin v=2 cos(u+v) sin(u−v) 2 2 2 2 sec(π −x) =csc x cot(π −x) =tan x cos u+cos v=2 cos(u+v) cos(u−v) 2 2 2 2 Even/Odd Identities cos u−cos v=−2 sin(u+v) sin(u−v) 2 2 sin(−x)=−sin x cos(−x)=cos x Product-to-Sum Formulas csc(−x)=−csc x tan(−x)=−tan x 1 sec(−x)=sec x cot(−x)=−cot x sin u sin v= [cos(u−v)−cos(u+v)] 2 Sum and Difference Formulas 1 g sin(u±v)=sin u cos v±cos u sin v cos u cos v= 2[cos(u−v)+cos(u+v)] rnin a 1 e cos(u±v)=cos u cos v ∓ sin u sin v sin u cos v= 2[sin(u+v)+sin(u−v)] ge L tan u±tan v a tan(u±v)= 1 ∓ tan u tan v cos u sin v= 1[sin(u+v)−sin(u−v)] eng 2 C © Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-322 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Ron Larson The Pennsylvania State University The Behrend College Bruce Edwards University of Florida Australia • Brazil • Canada • Mexico • Singapore • United Kingdom • United States Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-322 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version. Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-322 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Calculus © 2023, 2018, 2014 Cengage Learning, Inc. ALL RIGHTS RESERVED. with CalcChat® and CalcView® WCN: 02-300 Twelfth Edition No part of this work covered by the copyright herein may be reproduced Ron Larson or distributed in any form or by any means, except as permitted by Bruce Edwards U.S. copyright law, without the prior written permission of the SVP, Higher Education Product Management: Erin Joyner copyright owner. VP, Product Management, Learning Experiences: Thais Alencar Unless otherwise noted, all content is © Cengage. Product Director: Mark Santee Senior Product Manager: Gary Whalen For product information and technology assistance, contact us at Senior Product Assistant: Tim Rogers Cengage Customer & Sales Support, 1-800-354-9706 or support.cengage.com. Senior Learning Designer: Laura Gallus For permission to use material from this text or product, Content Manager: Rachel Pancare submit all requests online at www.copyright.com. 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To learn more about Cengage platforms and services, register or access your online learning solution, or purchase materials for your course, visit www.cengage.com. Printed in the United States of America Print Number: 01 Print Year: 2022 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-322 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Contents P Preparation for Calculus 1 P.1 Graphs and Models 2 P.2 Linear Models and Rates of Change 10 P.3 Functions and Their Graphs 19 P.4 Review of Trigonometric Functions 31 Review Exercises 41 P.S. Problem Solving 44 1 Limits and Their Properties 45 1.1 A Preview of Calculus 46 1.2 Finding Limits Graphically and Numerically 52 1.3 Evaluating Limits Analytically 63 1.4 Continuity and One-Sided Limits 74 1.5 Infinite Limits 87 Section Project: Graphs and Limits of Trigonometric Functions 94 Review Exercises 95 P.S. Problem Solving 98 2 Differentiation 99 2.1 The Derivative and the Tangent Line Problem 100 2.2 Basic Differentiation Rules and Rates of Change 110 2.3 Product and Quotient Rules and Higher-Order Derivatives 122 2.4 The Chain Rule 133 2.5 Implicit Differentiation 144 Section Project: Optical Illusions 151 2.6 Related Rates 152 Review Exercises 161 P.S. Problem Solving 164 3 Applications of Differentiation 165 3.1 Extrema on an Interval 166 3.2 Rolle’s Theorem and the Mean Value Theorem 174 3.3 I ncreasing and Decreasing Functions and the First Derivative Test 181 Section Project: Even Polynomial Functions of Fourth Degree 190 3.4 Concavity and the Second Derivative Test 191 3.5 Limits at Infinity 199 3.6 A Summary of Curve Sketching 209 3.7 Optimization Problems 219 Section Project: Minimum Time 228 3.8 Newton’s Method 229 3.9 Differentials 235 Review Exercises 242 P.S. Problem Solving 246 iii Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-322 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. iv Contents 4 Integration 247 4.1 Antiderivatives and Indefinite Integration 248 4.2 Area 258 4.3 Riemann Sums and Definite Integrals 270 4.4 The Fundamental Theorem of Calculus 281 4.5 Integration by Substitution 296 Section Project: Probability 308 Review Exercises 309 P.S. Problem Solving 312 Logarithmic, Exponential, and 5 Other Transcendental Functions 313 5.1 The Natural Logarithmic Function: Differentiation 314 5.2 The Natural Logarithmic Function: Integration 324 5.3 Inverse Functions 333 5.4 Exponential Functions: Differentiation and Integration 342 5.5 Bases Other than e and Applications 352 Section Project: Using a Graphing Utility to Estimate Slope 361 5.6 Indeterminate Forms and L’Hôpital’s Rule 362 5.7 Inverse Trigonometric Functions: Differentiation 373 5.8 Inverse Trigonometric Functions: Integration 382 5.9 Hyperbolic Functions 390 Section Project: Mercator Map 399 Review Exercises 400 P.S. Problem Solving 404 6 Differential Equations 405 6.1 Slope Fields and Euler’s Method 406 6.2 Growth and Decay 415 6.3 Separation of Variables and the Logistic Equation 423 6.4 First-Order Linear Differential Equations 432 Section Project: Weight Loss 438 Review Exercises 439 P.S. Problem Solving 442 7 Applications of Integration 443 7.1 Area of a Region Between Two Curves 444 7.2 Volume: The Disk Method 454 7.3 Volume: The Shell Method 465 Section Project: Saturn 473 7.4 Arc Length and Surfaces of Revolution 474 7.5 Work 485 Section Project: Pyramid of Khufu 493 7.6 Moments, Centers of Mass, and Centroids 494 7.7 Fluid Pressure and Fluid Force 505 Review Exercises 511 P.S. Problem Solving 514 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-322 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Contents v 8 Integration Techniques and Improper Integrals 515 8.1 Basic Integration Rules 516 8.2 Integration by Parts 523 8.3 Trigonometric Integrals 532 Section Project: The Wallis Product 540 8.4 Trigonometric Substitution 541 8.5 Partial Fractions 550 8.6 Numerical Integration 559 8.7 Integration by Tables and Other Integration Techniques 566 8.8 Improper Integrals 572 Review Exercises 583 P.S. Problem Solving 586 9 Infinite Series 587 9.1 Sequences 588 9.2 Series and Convergence 599 Section Project: Cantor’s Disappearing Table 608 9.3 The Integral Test and p-Series 609 Section Project: The Harmonic Series 615 9.4 Comparisons of Series 616 9.5 Alternating Series 623 9.6 The Ratio and Root Tests 631 9.7 Taylor Polynomials and Approximations 640 9.8 Power Series 651 9.9 Representation of Functions by Power Series 661 9.10 Taylor and Maclaurin Series 668 Review Exercises 680 P.S. Problem Solving 684 Conics, Parametric Equations, and 10 Polar Coordinates 685 10.1 Conics and Calculus 686 10.2 Plane Curves and Parametric Equations 700 Section Project: Cycloids 709 10.3 Parametric Equations and Calculus 710 10.4 Polar Coordinates and Polar Graphs 719 Section Project: Cassini Oval 728 10.5 Area and Arc Length in Polar Coordinates 729 10.6 Polar Equations of Conics and Kepler’s Laws 738 Review Exercises 746 P.S. Problem Solving 750 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-322 Copyright 2023 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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