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 Copyright 2010 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 R E F E R E N C E PA G E Cut here and keep for reference ALGEBRA GEOMETRY Arithmetic Operations Geometric Formulas a c ad bc 苷 b d bd a d ad b a 苷 苷 c b c bc d a共b c兲 苷 ab ac a c ac 苷 b b b Formulas for area A, circumference C, and volume V: Triangle Circle Sector of Circle A 苷 12 bh A 苷 r A 苷 12 r 2 C 苷 2 r s 苷 r 共 in radians兲 苷 ab sin a Exponents and Radicals x 苷 x mn xn xn 苷 n x x m x n 苷 x mn 共x 兲 苷 x m n mn 冉冊 x y 共xy兲n 苷 x n y n n 苷 xn yn n n x m兾n 苷 s x m 苷 (s x )m n x 1兾n 苷 s x 冑 n n n xy s xs y s n r h ă m r s ă b r Sphere V 43 r Cylinder V 苷 r 2h Cone V 苷 13 r 2h A 苷 4 r A 苷 rsr h n x x s 苷 n y sy r r h h Factoring Special Polynomials r x y 苷 共x y兲共x y兲 x y 苷 共x y兲共x xy y 2兲 x y 苷 共x y兲共x xy y 2兲 Distance and Midpoint Formulas Binomial Theorem 共x y兲2 苷 x 2xy y 共x y兲2 苷 x 2xy y Distance between P1共x1, y1兲 and P2共x 2, y2兲: d 苷 s共x x1兲2 共 y2 y1兲2 共x y兲3 苷 x 3x y 3xy y 共x y兲3 苷 x 3x y 3xy y 共x y兲n 苷 x n nx n1y 冉冊 n共n 1兲 n2 x y 冉冊 n nk k x y nxy n1 y n k n共n 1兲 共n k 1兲 n where 苷 k ⴢ ⴢ ⴢ ⴢ k Midpoint of P1 P2 : 冉 x1 x y1 y2 , 2 Lines Slope of line through P1共x1, y1兲 and P2共x 2, y2兲: Quadratic Formula m苷 If ax bx c 苷 0, then x 苷 冊 b sb 4ac 2a y2 y1 x x1 Point-slope equation of line through P1共x1, y1兲 with slope m: Inequalities and Absolute Value y y1 苷 m共x x1兲 If a b and b c, then a c Slope-intercept equation of line with slope m and y-intercept b: If a b, then a c b c If a b and c 0, then ca cb y 苷 mx b If a b and c 0, then ca cb If a 0, then ⱍxⱍ 苷 a ⱍxⱍ a ⱍxⱍ a means x 苷 a or x 苷 a means a x a means x a or x a Circles Equation of the circle with center 共h, k兲 and radius r: 共x h兲2 共 y k兲2 苷 r Copyright 2010 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 R E F E R E N C E PA G E TRIGONOMETRY Angle Measurement Fundamental Identities radians 苷 180⬚ csc 苷 sin sec 苷 cos tan 苷 sin cos cot 苷 cos sin 共 in radians兲 cot 苷 tan sin 2 ⫹ cos 2 苷 Right Angle Trigonometry ⫹ tan 2 苷 sec 2 ⫹ cot 2 苷 csc 2 sin共⫺兲 苷 ⫺sin cos共⫺兲 苷 cos tan共⫺兲 苷 ⫺tan sin 1⬚ 苷 rad 180 rad 180 ă r s 苷 r sin 苷 cos 苷 tan 苷 opp hyp csc 苷 adj hyp sec 苷 opp adj cot 苷 s r hyp opp hyp hyp adj opp ă adj adj opp cos Trigonometric Functions sin 苷 y r csc 苷 r y cos 苷 x r sec 苷 r x tan 苷 y x cot x y B a r C c ă The Law of Cosines x b a 苷 b ⫹ c ⫺ 2bc cos A b 苷 a ⫹ c ⫺ 2ac cos B y A c 苷 a ⫹ b ⫺ 2ab cos C y=tan x y=cos x 1 π ⫺ 苷 cot sin A sin B sin C 苷 苷 a b c (x, y) y y=sin x tan ⫺ 苷 cos The Law of Sines y Graphs of Trigonometric Functions y ⫺ 苷 sin 冉 冊 冉 冊 2π Addition and Subtraction Formulas 2π x _1 π 2π x π x sin共x ⫹ y兲 苷 sin x cos y ⫹ cos x sin y sin共x ⫺ y兲 苷 sin x cos y ⫺ cos x sin y _1 cos共x ⫹ y兲 苷 cos x cos y ⫺ sin x sin y y y y=csc x y y=sec x cos共x ⫺ y兲 苷 cos x cos y ⫹ sin x sin y y=cot x 1 π 2π x π 2π x π 2π x tan共x ⫹ y兲 苷 tan x ⫹ tan y ⫺ tan x tan y tan共x ⫺ y兲 苷 tan x ⫺ tan y ⫹ tan x tan y _1 _1 Double-Angle Formulas sin 2x 苷 sin x cos x Trigonometric Functions of Important Angles cos 2x 苷 cos 2x ⫺ sin 2x 苷 cos 2x ⫺ 苷 ⫺ sin 2x radians sin cos tan 0⬚ 30⬚ 45⬚ 60⬚ 90⬚ 兾6 兾4 兾3 兾2 1兾2 s2兾2 s3兾2 1 s3兾2 s2兾2 1兾2 0 s3兾3 s3 — tan 2x 苷 tan x ⫺ tan2x Half-Angle Formulas sin 2x 苷 ⫺ cos 2x cos 2x 苷 ⫹ cos 2x Copyright 2010 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 CA L C U L U S SEVENTH EDITION JAMES STEWART McMASTER UNIVERSITY AND UNIVERSITY OF TORONTO Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States Copyright 2010 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, Seventh Edition James Stewart Executive Editor: Liz Covello Assistant Editor: Liza Neustaetter Editorial Assistant: Jennifer Staller Media Editor : Maureen Ross Marketing Manager: Jennifer Jones Marketing Coordinator: Michael Ledesma Marketing Communications Manager: Mary Anne Payumo Content Project Manager: Cheryll Linthicum Art 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Corporation and used herein under license Macintosh and Power Macintosh are registered trademarks of Apple Computer, Inc Used herein under license Derive is a registered trademark of Soft Warehouse, Inc Maple is a registered trademark of Waterloo Maple, Inc Mathematica is a registered trademark of Wolfram Research, Inc Tools for Enriching is a trademark used herein under license Copyright 2010 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 K10T10 Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com Contents Preface xi To the Student xxiii Diagnostic Tests xxiv A Preview of Calculus 1 Functions and Limits 9 1.1 Four Ways to Represent a Function 1.2 Mathematical Models: A Catalog of Essential Functions 1.3 New Functions from Old Functions 36 1.4 The Tangent and Velocity Problems 44 1.5 The Limit of a Function 1.6 Calculating Limits Using the Limit Laws 1.7 The Precise Definition of a Limit 1.8 Continuity Review 23 50 62 72 81 93 Principles of Problem Solving 10 97 Derivatives 103 2.1 Derivatives and Rates of Change Writing Project N Early Methods for Finding Tangents 2.2 The Derivative as a Function 2.3 Differentiation Formulas Applied Project N 104 114 126 Building a Better Roller Coaster 2.4 Derivatives of Trigonometric Functions 2.5 The Chain Rule Applied Project 2.6 114 140 140 148 N Where Should a Pilot Start Descent? Implicit Differentiation Laboratory Project N 156 157 Families of Implicit Curves 163 iii Copyright 2010 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 2.7 Rates of Change in the Natural and Social Sciences 2.8 Related Rates 2.9 Linear Approximations and Differentials 176 Laboratory Project Review Problems Plus Taylor Polynomials N 183 189 190 194 Applications of Differentiation 197 3.1 Maximum and Minimum Values Applied Project N 198 The Calculus of Rainbows 206 3.2 The Mean Value Theorem 3.3 How Derivatives Affect the Shape of a Graph 3.4 Limits at Infinity; Horizontal Asymptotes 3.5 Summary of Curve Sketching 3.6 Graphing with Calculus and Calculators 3.7 Optimization Problems Applied Project N 3.8 Newton’s Method 3.9 Antiderivatives Review Problems Plus 164 208 213 223 237 244 250 The Shape of a Can 262 263 269 275 279 Integrals 283 4.1 Areas and Distances 284 4.2 The Definite Integral 295 Discovery Project N Area Functions 309 4.3 The Fundamental Theorem of Calculus 4.4 Indefinite Integrals and the Net Change Theorem Writing Project 4.5 N Problems Plus 321 Newton, Leibniz, and the Invention of Calculus The Substitution Rule Review 310 329 330 337 341 Copyright 2010 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 Applications of Integration 343 5.1 Areas Between Curves Applied Project The Gini Index 5.2 Volumes 5.3 Volumes by Cylindrical Shells 5.4 Work 5.5 Average Value of a Function Review Problems Plus 351 352 363 368 Applied Project N 344 N 373 Calculus and Baseball 376 378 380 Inverse Functions: 383 Exponential, Logarithmic, and Inverse Trigonometric Functions 6.1 Inverse Functions 384 Instructors may cover either Sections 6.2–6.4 or Sections 6.2*–6.4* See the Preface 6.2 Exponential Functions and Their Derivatives 391 6.2* The Natural Logarithmic Function 421 6.3 Logarithmic Functions 404 6.3* The Natural Exponential Function 429 6.4 Derivatives of Logarithmic Functions 410 6.4* General Logarithmic and Exponential Functions 437 6.5 Exponential Growth and Decay 6.6 Inverse Trigonometric Functions Applied Project N 446 453 Where to Sit at the Movies 6.7 Hyperbolic Functions 6.8 Indeterminate Forms and l’Hospital’s Rule Writing Project Review Problems Plus N 461 462 The Origins of l’Hospital’s Rule 469 480 480 485 Copyright 2010 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 v vi CONTENTS Techniques of Integration 487 7.1 Integration by Parts 7.2 Trigonometric Integrals 7.3 Trigonometric Substitution 7.4 Integration of Rational Functions by Partial Fractions 7.5 Strategy for Integration 7.6 Integration Using Tables and Computer Algebra Systems Discovery Project 502 508 518 Patterns in Integrals Approximate Integration 7.8 Improper Integrals Problems Plus 524 529 530 543 553 557 Further Applications of Integration 561 8.1 Arc Length 562 Discovery Project 8.2 8.3 N Arc Length Contest Area of a Surface of Revolution Discovery Project N 569 569 Rotating on a Slant 575 Applications to Physics and Engineering Discovery Project N Applications to Economics and Biology 8.5 Probability Problems Plus 576 Complementary Coffee Cups 8.4 Review 495 7.7 Review N 488 586 587 592 599 601 Differential Equations 603 9.1 Modeling with Differential Equations 9.2 Direction Fields and Euler’s Method 9.3 Separable Equations 604 609 618 Applied Project N How Fast Does a Tank Drain? Applied Project N Which Is Faster, Going Up or Coming Down? 9.4 Models for Population Growth 9.5 Linear Equations 627 628 629 640 Copyright 2010 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 s2 ) 12 2, (a) (1, 0, s3 ) (b) (s3 , 1, 2s3 ) 13 1, 兾4 14 2, csc 5–6 Describe in words the surface whose equation is given 苷 兾3 15 A solid lies above the cone z 苷 sx y and below the 苷 sphere x y z 苷 z Write a description of the solid in terms of inequalities involving spherical coordinates 7–8 Identify the surface whose equation is given 苷 sin sin ; 16 (a) Find inequalities that describe a hollow ball with diameter 共sin 2 sin 2 cos2兲 苷 Graphing calculator or computer required 30 cm and thickness 0.5 cm Explain how you have positioned the coordinate system that you have chosen CAS Computer algebra system required Homework Hints available at stewartcalculus.com Copyright 2010 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 1062 CHAPTER 15 MULTIPLE INTEGRALS 32 Let H be a solid hemisphere of radius a whose density at any (b) Suppose the ball is cut in half Write inequalities that describe one of the halves point is proportional to its distance from the center of the base (a) Find the mass of H (b) Find the center of mass of H (c) Find the moment of inertia of H about its axis 17–18 Sketch the solid whose volume is given by the integral and evaluate the integral 17 18 兾6 兾2 y y y 0 y y y 兾2 sin d d d 33 (a) Find the centroid of a solid homogeneous hemisphere of radius a (b) Find the moment of inertia of the solid in part (a) about a diameter of its base sin d d d 34 Find the mass and center of mass of a solid hemisphere of 19–20 Set up the triple integral of an arbitrary continuous function radius a if the density at any point is proportional to its distance from the base f 共x, y, z兲 in cylindrical or spherical coordinates over the solid shown z 19 z 20 35–38 Use cylindrical or spherical coordinates, whichever seems more appropriate 35 Find the volume and centroid of the solid E that lies above the cone z 苷 sx y and below the sphere x y z 苷 y x x 36 Find the volume of the smaller wedge cut from a sphere of y 21–34 Use spherical coordinates radius a by two planes that intersect along a diameter at an angle of 兾6 CAS 21 Evaluate xxxB 共x y z 兲 dV, where B is the ball with 37 Evaluate xxxE z dV, where E lies above the paraboloid z 苷 x y and below the plane z 苷 2y Use either the Table of Integrals (on Reference Pages 6–10) or a computer algebra system to evaluate the integral center the origin and radius 22 Evaluate xxxH 共9 x y 兲 dV, where H is the solid hemisphere x y z 9, z CAS 38 (a) Find the volume enclosed by the torus 苷 sin (b) Use a computer to draw the torus 23 Evaluate xxxE 共x y 兲 dV, where E lies between the spheres x y z 苷 and x y z 苷 24 Evaluate xxxE y dV, where E is the solid hemisphere x y z 9, y 25 Evaluate xxxE xe 39– 41 Evaluate the integral by changing to spherical coordinates s1 x 39 yy 40 y y 41 y y 0 x 2 y 2 z dV, where E is the portion of the unit ball x y z that lies in the first octant 26 Evaluate xxxE xyz dV, where E lies between the spheres 苷 and 苷 and above the cone 苷 兾3 27 Find the volume of the part of the ball a that lies between the cones 苷 兾6 and 苷 兾3 28 Find the average distance from a point in a ball of radius a to its center 29 (a) Find the volume of the solid that lies above the cone 苷 兾3 and below the sphere 苷 cos (b) Find the centroid of the solid in part (a) 30 Find the volume of the solid that lies within the sphere x y z 苷 4, above the xy-plane, and below the cone z 苷 sx y 31 (a) Find the centroid of the solid in Example (b) Find the moment of inertia about the z-axis for this solid y s2 x y a sa y a sa y 2 s4 x 2 s4