Thomas’ Calculus: Early Transcendentals, Fifteenth Edition
By Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir and José Luis Zuleta Estrugo
Contents:
Preface 9
1 Functions 21
1.1 Functions and Their Graphs 21
1.2 Combining Functions; Shifting and Scaling Graphs 34
1.3 Trigonometric Functions 41
1.4 Exponential Functions 49
1.5 Inverse Functions and Logarithms 54
Questions to Guide Your Review 67
Practice Exercises 67
Additional and Advanced Exercises 69
Technology Application Projects 71
2 Limits and Continuity 72
2.1 Rates of Change and Tangent Lines to Curves 72
2.2 Limit of a Function and Limit Laws 79
2.3 The Precise Definition of a Limit 90
2.4 One-Sided Limits 99
2.5 Limits Involving Infinity; Asymptotes of Graphs 106
2.6 Continuity 120
Questions to Guide Your Review 132
Practice Exercises 133
Additional and Advanced Exercises 134
Technology Application Projects 137
3 Derivatives 138
3.1 Tangent Lines and the Derivative at a Point 138
3.2 The Derivative as a Function 142
3.3 Differentiation Rules 151
3.4 The Derivative as a Rate of Change 161
3.5 Derivatives of Trigonometric Functions 170
3.6 The Chain Rule 176
3.7 Implicit Differentiation 184
3.8 Derivatives of Inverse Functions and Logarithms 189
3.9 Inverse Trigonometric Functions 200
3.10 Related Rates 206
3.11 Linearization and Differentials 214
Questions to Guide Your Review 226
Practice Exercises 227
Additional and Advanced Exercises 231
Technology Application Projects 234
4 Applications of Derivatives 235
4.1 Extreme Values of Functions on Closed Intervals 235
4.2 The Mean Value Theorem 243
4.3 Monotonic Functions and the First Derivative Test 250
4.4 Concavity and Curve Sketching 255
4.5 Indeterminate Forms and L’Hôpital’s Rule 268
4.6 Applied Optimization 277
4.7 Newton’s Method 289
4.8 Antiderivatives 294
Questions to Guide Your Review 304
Practice Exercises 305
Additional and Advanced Exercises 308
Technology Application Projects 311
5 Integrals 312
5.1 Area and Estimating with Finite Sums 312
5.2 Sigma Notation and Limits of Finite Sums 322
5.3 The Definite Integral 329
5.4 The Fundamental Theorem of Calculus 342
5.5 Indefinite Integrals and the Substitution Method 354
5.6 Definite Integral Substitutions and the Area Between Curves 361
Questions to Guide Your Review 372
Practice Exercises 372
Additional and Advanced Exercises 375
Technology Application Projects 379
6 Applications of Definite Integrals 380
6.1 Volumes Using Cross-Sections 380
6.2 Volumes Using Cylindrical Shells 391
6.3 Arc Length 399
6.4 Areas of Surfaces of Revolution 405
6.5 Work and Fluid Forces 410
6.6 Moments and Centers of Mass 420
Questions to Guide Your Review 431
Practice Exercises 432
Additional and Advanced Exercises 434
Technology Application Projects 435
7 Integrals and Transcendental Functions 436
7.1 The Logarithm Defined as an Integral 436
7.2 Exponential Change and Separable Differential Equations 447
7.3 Hyperbolic Functions 457
7.4 Relative Rates of Growth 465
Questions to Guide Your Review 470
Practice Exercises 471
Additional and Advanced Exercises 472
8 Techniques of Integration 473
8.1 Using Basic Integration Formulas 473
8.2 Integration by Parts 478
8.3 Trigonometric Integrals 486
8.4 Trigonometric Substitutions 492
8.5 Integration of Rational Functions by Partial Fractions 497
8.6 Integral Tables and Computer Algebra Systems 504
8.7 Numerical Integration 510
8.8 Improper Integrals 520
Questions to Guide Your Review 531
Practice Exercises 532
Additional and Advanced Exercises 534
Technology Application Projects 537
9 Infinite Sequences and Series 538
9.1 Sequences 538
9.2 Infinite Series 551
9.3 The Integral Test 561
9.4 Comparison Tests 567
9.5 Absolute Convergence; The Ratio and Root Tests 572
9.6 Alternating Series and Conditional Convergence 579
9.7 Power Series 586
9.8 Taylor and Maclaurin Series 597
9.9 Convergence of Taylor Series 602
9.10 Applications of Taylor Series 609
Questions to Guide Your Review 618
Practice Exercises 619
Additional and Advanced Exercises 621
Technology Application Projects 623
10 Parametric Equations and Polar Coordinates 624
10.1 Parametrizations of Plane Curves 624
10.2 Calculus with Parametric Curves 633
10.3 Polar Coordinates 642
10.4 Graphing Polar Coordinate Equations 646
10.5 Areas and Lengths in Polar Coordinates 650
10.6 Conic Sections 655
10.7 Conics in Polar Coordinates 663
Questions to Guide Your Review 669
Practice Exercises 670
Additional and Advanced Exercises 672
Technology Application Projects 674
11 Vectors and the Geometry of Space 675
11.1 Three-Dimensional Coordinate Systems 675
11.2 Vectors 680
11.3 The Dot Product 691
11.4 The Cross Product 699
11.5 Lines and Planes in Space 705
11.6 Cylinders and Quadric Surfaces 714
Questions to Guide Your Review 720
Practice Exercises 720
Additional and Advanced Exercises 722
Technology Application Projects 724
12 Vector-Valued Functions and Motion in Space 725
12.1 Curves in Space and Their Tangents 725
12.2 Integrals of Vector Functions; Projectile Motion 734
12.3 Arc Length in Space 743
12.4 Curvature and Normal Vectors of a Curve 747
12.5 Tangential and Normal Components of Acceleration 753
12.6 Velocity and Acceleration in Polar Coordinates 759
Questions to Guide Your Review 762
Practice Exercises 763
Additional and Advanced Exercises 765
Technology Application Projects 766
13 Partial Derivatives 767
13.1 Functions of Several Variables 767
13.2 Limits and Continuity in Higher Dimensions 775
13.3 Partial Derivatives 784
13.4 The Chain Rule 796
13.5 Directional Derivatives and Gradient Vectors 806
13.6 Tangent Planes and Differentials 815
13.7 Extreme Values and Saddle Points 825
13.8 Lagrange Multipliers 834
13.9 Taylor’s Formula for Two Variables 844
13.10 Partial Derivatives with Constrained Variables 848
Questions to Guide Your Review 852
Practice Exercises 853
Additional and Advanced Exercises 856
Technology Application Projects 858
14 Multiple Integrals 859
14.1 Double and Iterated Integrals over Rectangles 859
14.2 Double Integrals over General Regions 864
14.3 Area by Double Integration 873
14.4 Double Integrals in Polar Form 876
14.5 Triple Integrals in Rectangular Coordinates 883
14.6 Applications 893
14.7 Triple Integrals in Cylindrical and Spherical Coordinates 903
14.8 Substitutions in Multiple Integrals 915
Questions to Guide Your Review 924
Practice Exercises 925
Additional and Advanced Exercises 927
Technology Application Projects 929
15 Integrals and Vector Fields 930
15.1 Line Integrals of Scalar Functions 930
15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 937
15.3 Path Independence, Conservative Fields, and Potential Functions 950
15.4 Green’s Theorem in the Plane 961
15.5 Surfaces and Area 973
15.6 Surface Integrals 983
15.7 Stokes’ Theorem 993
15.8 The Divergence Theorem and a Unified Theory 1006
Questions to Guide Your Review 1018
Practice Exercises 1019
Additional and Advanced Exercises 1021
Technology Application Projects 1023
16 First-Order Differential Equations 1024
16.1 Solutions, Slope Fields, and Euler’s Method 1024
16.2 First-Order Linear Equations 1032
16.3 Applications 1038
16.4 Graphical Solutions of Autonomous Equations 1044
16.5 Systems of Equations and Phase Planes 1053
17
Questions to Guide Your Review 1059
Practice Exercises 1059
Additional and Advanced Exercises 1061
Technology Application Projects 1061
Second-Order Differential Equations (O nline)
17.1 Second-Order Linear Equations
17.2 Nonhomogeneous Linear Equations
17.3 Applications
17.4 Euler Equations
17.5 Power-Series Solutions
18 Complex Functions (Online)
18.1 Complex Numbers
18.2 Functions of a Complex Variable
18.3 Derivatives
18.4 The Cauchy-Riemann Equations
18.5 Complex Power Series
18.6 Some Complex Functions
18.7 Conformal Maps
Questions to Guide Your Review
Additional and Advanced Exercises
19 Fourier Series and Wavelets (Online)
19.1 Periodic Functions
19.2 Summing Sines and Cosines
19.3 Vectors and Approximation in Three and More Dimensions
19.4 Approximation of Functions
19.5 Advanced Topic: The Haar System and Wavelets
Questions to Guide Your Review
Additional and Advanced Exercises
Appendix A AP-1
A.1 Real Numbers and the Real Line AP-1
A.2 Graphing with Software AP-6
A.3 Mathematical Induction AP-10
A.4 Lines, Circles, and Parabolas AP-13
A.5 Proofs of Limit Theorems AP-23
A.6 Commonly Occurring Limits AP-26
A.7 Theory of the Real Numbers AP-27
A.8 Probability AP-30
A.9 The Distributive Law for Vector Cross Products AP-43
A.10 The Mixed Derivative Theorem and the Increment Theorem AP-44
Appendix B (Online)
B.1 Determinants
B.2 Extreme Values and Saddle Points for Functions of More than Two Variables
B.3 The Method of Gradient Descent
Answers to Odd-Numbered Exercises AN-1
Applications Index AI-1
Subject Index I-1
Credits C-1
A Brief Table of Integrals T-1