Calculus: Early Transcendental Functions, 8th Edition
Ron Larson and Bruce Edwards
Contents
Preparation for Calculus 1
1.1 Graphs and Models 2
1.2 Linear Models and Rates of Change 10
1.3 Functions and Their Graphs 19
1.4 Review of Trigonometric Functions 31
1.5 Inverse Functions 41
1.6 Exponential and Logarithmic Functions 52
Review Exercises 60
P.S. Problem Solving 64
Limits and Their Properties 65
2.1 A Preview of Calculus 66
2.2 Finding Limits Graphically and Numerically 72
2.3 Evaluating Limits Analytically 83
2.4 Continuity and One-Sided Limits 94
2.5 Infinite Limits 107
Section Project: Graphs and Limits of
Trigonometric Functions 114
Review Exercises 115
P.S. Problem Solving 118
Differentiation 119
3.1 The Derivative and the Tangent Line Problem 120
3.2 Basic Differentiation Rules and Rates of Change 130
3.3 Product and Quotient Rules and
Higher-Order Derivatives 143
3.4 The Chain Rule 154
3.5 Implicit Differentiation 169
Section Project: Optical Illusions 177
3.6 Derivatives of Inverse Functions 178
3.7 Related Rates 185
3.8 Newton’s Method 194
Review Exercises 200
P.S. Problem Solving 204
Applications of Differentiation 205
4.1 Extrema on an Interval 206
4.2 Rolle’s Theorem and the Mean Value Theorem 214
4.3 Increasing and Decreasing Functions and
the First Derivative Test 221
Section Project: Even Polynomial Functions of
Fourth Degree 230
4.4 Concavity and the Second Derivative Test 231
4.5 Limits at Infinity 239
4.6 A Summary of Curve Sketching 249
4.7 Optimization Problems 260
Section Project: Minimum Time 270
4.8 Differentials 271
Review Exercises 278
P.S. Problem Solving 282
Integration 283
5.1 Antiderivatives and Indefinite Integration 284
5.2 Area 294
5.3 Riemann Sums and Definite Integrals 306
5.4 The Fundamental Theorem of Calculus 317
5.5 Integration by Substitution 332
Section Project: Probability 344
5.6 Indeterminate Forms and L’Hôpital’s Rule 345
5.7 The Natural Logarithmic Function: Integration 356
5.8 Inverse Trigonometric Functions: Integration 365
5.9 Hyperbolic Functions 373
Section Project: Mercator Map 382
Review Exercises 383
P.S. Problem Solving 386
Differential Equations 387
6.1 Slope Fields and Euler’s Method 388
6.2 Growth and Decay 397
6.3 Separation of Variables 405
6.4 The Logistic Equation 417
6.5 First-Order Linear Differential Equations 424
Section Project: Weight Loss 430
6.6 Predator-Prey Differential Equations 431
Review Exercises 438
P.S. Problem Solving 442
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
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
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
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
Vectors and the Geometry of Space 751
11.1 Vectors in the Plane 752
11.2 Space Coordinates and Vectors in Space 762
11.3 The Dot Product of Two Vectors 770
11.4 The Cross Product of Two Vectors in Space 779
11.5 Lines and Planes in Space 787
Section Project: Distances in Space 797
11.6 Surfaces in Space 798
11.7 Cylindrical and Spherical Coordinates 808
Review Exercises 815
P.S. Problem Solving 818
Vector-Valued Functions 819
12.1 Vector-Valued Functions 820
Section Project: Witch of Agnesi 827
12.2 Differentiation and Integration of Vector-Valued
Functions 828
12.3 Velocity and Acceleration 836
12.4 Tangent Vectors and Normal Vectors 845
12.5 Arc Length and Curvature 855
Review Exercises 867
P.S. Problem Solving 870
Functions of Several Variables 871
13.1 Introduction to Functions of Several Variables 872
13.2 Limits and Continuity 884
13.3 Partial Derivatives 894
13.4 Differentials 904
13.5 Chain Rules for Functions of Several Variables 911
13.6 Directional Derivatives and Gradients 919
13.7 Tangent Planes and Normal Lines 931
Section Project: Wildflowers 939
13.8 Extrema of Functions of Two Variables 940
13.9 Applications of Extrema 948
Section Project: Building a Pipeline 955
13.10 Lagrange Multipliers 956
Review Exercises 964
P.S. Problem Solving 968
Multiple Integration 969
14.1 Iterated Integrals and Area in the Plane 970
14.2 Double Integrals and Volume 978
14.3 Change of Variables: Polar Coordinates 990
14.4 Center of Mass and Moments of Inertia 998
Section Project: Center of Pressure on a Sail 1005
14.5 Surface Area 1006
Section Project: Surface Area in Polar Coordinates 1012
14.6 Triple Integrals and Applications 1013
14.7 Triple Integrals in Other Coordinates 1024
Section Project: Wrinkled and Bumpy Spheres 1030
14.8 Change of Variables: Jacobians 1031
Review Exercises 1038
P.S. Problem Solving 1042
Vector Analysis 1043
15.1 Vector Fields 1044
15.2 Line Integrals 1055
15.3 Conservative Vector Fields and Independence of Path 1069
15.4 Green’s Theorem 1079
Section Project: Hyperbolic and Trigonometric Functions 1087
15.5 Parametric Surfaces 1088
15.6 Surface Integrals 1098
Section Project: Hyperboloid of One Sheet 1109
15.7 Divergence Theorem 1110
15.8 Stokes’s Theorem 1118
Review Exercises 1124
P.S. Problem Solving 1128
Additional Topics in Differential Equations (Online)*
16.1 Exact First-Order Equations
16.2 Second-Order Homogeneous Linear Equations
16.3 Second-Order Nonhomogeneous Linear Equations
Section Project: Parachute Jump
16.4 Series Solutions of Differential Equations
Review Exercises
P.S. Problem Solving
Appendices
Appendix A: Proofs of Selected Theorems A2
Appendix B: Integration Tables A3
Appendix C: Precalculus Review A7
C.1 Real Numbers and the Real Number Line A7
C.2 The Cartesian Plane A16
Appendix D: Rotation and the General Second-Degree
Equation (Online)*
Appendix E: Complex Numbers (Online)*
Appendix F: Business and Economic Applications (Online)*
Appendix G: Fitting Models to Data (Online)*
Answers to All Odd-Numbered Exercises A23
Index A143