Table Of ContentUNIVERSITY
CALCULUS
EARLY TRANSCENDENTALS
HASS • HEIL • BOGACKI • WEIR • THOMAS
FOURTH EDITION
MyLab Math for
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U N I V E R S I T Y
C A L C U L U S
EARLY TRANSCENDENTALS
Fourth Edition
Joel Hass
University of California, Davis
Christopher Heil
Georgia Institute of Technology
Przemyslaw Bogacki
Old Dominion University
Maurice D. Weir
Naval Postgraduate School
George B. Thomas, Jr.
Massachusetts Institute of Technology
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Cataloging-in-Publication Data on file with the Library of Congress.
1 19
Instructor’s Edition
ISBN 13: 978-0-13-516487-7
ISBN 10: 0-13-516487-7
Student’s Edition
ISBN 13: 978-0-13-499554-0
ISBN 10: 0-13-499554-6
A01_HASS5540_15_SE_FM.indd 2 05/12/18 1:08 PM
Contents
Preface ix
1
Functions 1
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 14
1.3 Trigonometric Functions 21
1.4 Graphing with Software 29
1.5 Exponential Functions 33
1.6 Inverse Functions and Logarithms 38
Also available: A.1 Real Numbers and the Real Line, A.3 Lines and Circles
2
Limits and Continuity 51
2.1 Rates of Change and Tangent Lines to Curves 51
2.2 Limit of a Function and Limit Laws 58
2.3 The Precise Definition of a Limit 69
2.4 One-Sided Limits 78
2.5 Continuity 85
2.6 Limits Involving Infinity; Asymptotes of Graphs 97
Questions to Guide Your Review 110
Practice Exercises 111
Additional and Advanced Exercises 113
Also available: A.5 Proofs of Limit Theorems
3
Derivatives 116
3.1 Tangent Lines and the Derivative at a Point 116
3.2 The Derivative as a Function 120
3.3 Differentiation Rules 129
3.4 The Derivative as a Rate of Change 139
3.5 Derivatives of Trigonometric Functions 148
3.6 The Chain Rule 154
3.7 Implicit Differentiation 162
3.8 Derivatives of Inverse Functions and Logarithms 167
3.9 Inverse Trigonometric Functions 177
3.10 Related Rates 184
3.11 Linearization and Differentials 192
Questions to Guide Your Review 203
Practice Exercises 204
Additional and Advanced Exercises 208
iii
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iv Contents
4
Applications of Derivatives 212
4.1 Extreme Values of Functions on Closed Intervals 212
4.2 The Mean Value Theorem 220
4.3 Monotonic Functions and the First Derivative Test 228
4.4 Concavity and Curve Sketching 233
4.5 Indeterminate Forms and L’Hôpital’s Rule 246
4.6 Applied Optimization 255
4.7 Newton’s Method 266
4.8 Antiderivatives 271
Questions to Guide Your Review 281
Practice Exercises 282
Additional and Advanced Exercises 286
5
Integrals 290
5.1 Area and Estimating with Finite Sums 290
5.2 Sigma Notation and Limits of Finite Sums 300
5.3 The Definite Integral 307
5.4 The Fundamental Theorem of Calculus 320
5.5 Indefinite Integrals and the Substitution Method 332
5.6 Definite Integral Substitutions and the Area Between Curves 339
Questions to Guide Your Review 349
Practice Exercises 350
Additional and Advanced Exercises 353
6
Applications of Definite Integrals 356
6.1 Volumes Using Cross-Sections 356
6.2 Volumes Using Cylindrical Shells 367
6.3 Arc Length 375
6.4 Areas of Surfaces of Revolution 381
6.5 Work 386
6.6 Moments and Centers of Mass 392
Questions to Guide Your Review 401
Practice Exercises 402
Additional and Advanced Exercises 403
7
Integrals and Transcendental Functions 405
7.1 The Logarithm Defined as an Integral 405
7.2 Exponential Change and Separable Differential Equations 415
7.3 Hyperbolic Functions 425
Questions to Guide Your Review 433
Practice Exercises 433
Additional and Advanced Exercises 434
Also available: B.1 Relative Rates of Growth
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Contents v
8
Techniques of Integration 436
8.1 Integration by Parts 437
8.2 Trigonometric Integrals 445
8.3 Trigonometric Substitutions 451
8.4 Integration of Rational Functions by Partial Fractions 456
8.5 Integral Tables and Computer Algebra Systems 463
8.6 Numerical Integration 469
8.7 Improper Integrals 478
Questions to Guide Your Review 489
Practice Exercises 490
Additional and Advanced Exercises 492
Also available: B.2 Probability
9
Infinite Sequences and Series 495
9.1 Sequences 495
9.2 Infinite Series 508
9.3 The Integral Test 518
9.4 Comparison Tests 524
9.5 Absolute Convergence; The Ratio and Root Tests 529
9.6 Alternating Series and Conditional Convergence 536
9.7 Power Series 543
9.8 Taylor and Maclaurin Series 554
9.9 Convergence of Taylor Series 559
9.10 Applications of Taylor Series 565
Questions to Guide Your Review 574
Practice Exercises 575
Additional and Advanced Exercises 577
Also available: A.6 Commonly Occurring Limits
10
Parametric Equations and Polar Coordinates 580
10.1 Parametrizations of Plane Curves 580
10.2 Calculus with Parametric Curves 588
10.3 Polar Coordinates 598
10.4 Graphing Polar Coordinate Equations 602
10.5 Areas and Lengths in Polar Coordinates 606
Questions to Guide Your Review 611
Practice Exercises 611
Additional and Advanced Exercises 613
Also available: A.4 Conic Sections, B.3 Conics in Polar Coordinates
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vi Contents
11
Vectors and the Geometry of Space 614
11.1 Three-Dimensional Coordinate Systems 614
11.2 Vectors 619
11.3 The Dot Product 628
11.4 The Cross Product 636
11.5 Lines and Planes in Space 642
11.6 Cylinders and Quadric Surfaces 651
Questions to Guide Your Review 657
Practice Exercises 657
Additional and Advanced Exercises 659
Also available: A.9 The Distributive Law for Vector Cross Products
12
Vector-Valued Functions and Motion in Space 662
12.1 Curves in Space and Their Tangents 662
12.2 Integrals of Vector Functions; Projectile Motion 671
12.3 Arc Length in Space 678
12.4 Curvature and Normal Vectors of a Curve 682
12.5 Tangential and Normal Components of Acceleration 687
12.6 Velocity and Acceleration in Polar Coordinates 690
Questions to Guide Your Review 694
Practice Exercises 694
Additional and Advanced Exercises 696
13
Partial Derivatives 697
13.1 Functions of Several Variables 697
13.2 Limits and Continuity in Higher Dimensions 705
13.3 Partial Derivatives 714
13.4 The Chain Rule 726
13.5 Directional Derivatives and Gradient Vectors 736
13.6 Tangent Planes and Differentials 744
13.7 Extreme Values and Saddle Points 754
13.8 Lagrange Multipliers 763
Questions to Guide Your Review 773
Practice Exercises 773
Additional and Advanced Exercises 777
Also available: A.10 The Mixed Derivative Theorem and the Increment Theorem,
B.4 Taylor’s Formula for Two Variables, B.5 Partial Derivatives with Constrained Variables
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Contents vii
14
Multiple Integrals 779
14.1 Double and Iterated Integrals over Rectangles 779
14.2 Double Integrals over General Regions 784
14.3 Area by Double Integration 793
14.4 Double Integrals in Polar Form 796
14.5 Triple Integrals in Rectangular Coordinates 803
14.6 Applications 813
14.7 Triple Integrals in Cylindrical and Spherical Coordinates 820
14.8 Substitutions in Multiple Integrals 832
Questions to Guide Your Review 841
Practice Exercises 842
Additional and Advanced Exercises 844
15
Integrals and Vector Fields 847
15.1 Line Integrals of Scalar Functions 847
15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 854
15.3 Path Independence, Conservative Fields, and Potential Functions 867
15.4 Green’s Theorem in the Plane 878
15.5 Surfaces and Area 890
15.6 Surface Integrals 900
15.7 Stokes’ Theorem 910
15.8 The Divergence Theorem and a Unified Theory 923
Questions to Guide Your Review 934
Practice Exercises 934
Additional and Advanced Exercises 937
16
First-Order Differential Equations 16-1 (Online at bit.ly/2pzYlEq)
16.1 Solutions, Slope Fields, and Euler’s Method 16-1
16.2 First-Order Linear Equations 16-9
16.3 Applications 16-15
16.4 Graphical Solutions of Autonomous Equations 16-21
16.5 Systems of Equations and Phase Planes 16-28
Questions to Guide Your Review 16-34
Practice Exercises 16-34
Additional and Advanced Exercises 16-36
17
Second-Order Differential Equations 17-1 (Online at bit.ly/2IHCJyE)
17.1 Second-Order Linear Equations 17-1
17.2 Nonhomogeneous Linear Equations 17-7
17.3 Applications 17-15
17.4 Euler Equations 17-22
17.5 Power-Series Solutions 17-24
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