Calculus : early transcendental functions / Robert T. Smith,Roland B. Minton

Table of contents

New Features xiii

A commitment to Accuracy xiv

Preface xv

Guided Tour xxiv

Application Index xxx

Chapter 0: Preliminaries 1
0.1 Polynomials and Rational Functions 2
0.2 Graphing Calculators and computer Algebra Systems 20
0.3 Inverse Functions 29
0.4 Trigonometric and Inverse Trigonometric Functions 36
0.5 Transformations of Functions 61

Chapter 1: Limits and continuity 73
1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve 74
1.2 The Concept of Limit 79
1.3 Computation of Limits 87
1.4 Continuity and its Consequences 97
1.5 Limits Involving Infinity 110
1.6 Formal Definition of the Limit 121
1.7 Limits and Loss-of-Significance Errors 134

Chapter 2: Differentiation 145
2.1 Tangent Lines and Velocity 150
2.2 The Derivative 146
2.3 Computation of Derivatives: The power Rule 170
2.4 The Product and Quotient Rules 180
2.5 The Chain Rule 189
2.6 Derivatives of Trigonometric Functions 196
2.7 Derivatives of Exponential and Logarithmic Functions 206
2.8 Implicit Differentiation and Inverse Trigonometric Functions 216
2.9 The Mean Value Theorem 226
2.9 The Mean Value Theorem 229

Chapter 3: Applications of Differentiation 241

3.1 Linear Approximations and L’Hopital’s Rule 242
3.2 Indeterminate Forms and L’Hopital’s Rule 255
3.3 Maximum and Minimum Values 265
3.4 Increasing and Decreasing Functions 277
3.5 Concavity and the Second Derivative Test 286
3.6 Overview of Curve Sketching 296
3.7 Optimization 308
3.8 Related Rates 321
3.9 Rates of change in Economics and the Sciences 327

Chapter 4: Integration 343

4.1 Antiderivatives 344
4.2 Sums and Sigma Notation 354
4.3 Area 362
4.4 The Definite Integral 369
4.5 The Fundamental Theorem of Calculus 383
4.6 Integration by substitution 393
4.7 Numerical Integration 402
4.8 The Natural Logarithm as an Integral 416

Chapter 5: Applications of the Definite Integral 431

5.1 Area Between Curves 432
5.2 Volume: Slicing, Disks, and washers 441
5.3 Volume by Cylindrical Shells 456
5.4 Arc Length and Surface Area 464
5.5 Projectile Motion 472
5.6 Application of Integration to Physics and Engineering 484
5.7 Probability 496

Chapter 6: Integration Techniques 509

6.1 Review of Formulas and Techniques 510
6.2 Integration by Parts 514
6.3 Trigonometric Techniques of Integration 521
6.4 Integration of Rational Functions Using Partial Fractions 530
6.5 Integration Tables and Computer Algebra Systems 538
6.6 Euler’s Method 521
6.7 Improper Integrals 546

Chapter 7: First-Order Differential Equations 565

7.1 Modeling with Differential Equations 566
7.2 Separable Differential Equations 577
7.3 Direction Fields and Euler’s Method 587
7.4 Systems of First-Order Differential Equations 599

Chapter 8: Infinite Series 611
8.1 Sequence of Real Numbers 612
8.2 Infinite Series 626
8.3 The Integral Test and Comparison Tests 636
8.4 Alternating Series 648
8.5 Absolute Convergence and the Ratio Test 656
8.6 Power Series 664
8.7 Taylor Series 672
8.8 Applications of Taylor Series 685
8.9 Fourier Series 694

Chapter 9: Parametric Equations and Polar Coordinates 715
9.1 Plane Curves and Parametric Equations 716
9.2 Calculus and Parametric Equations 726
9.3 Arc Length and Surface Area in Parametric Equations 734
9.4 Polar Coordinates 742
9.5 Calculus and Polar Coordinates 755
9.6 Conic Sections 764
9.7 Conic Sections in Polar Coordinates 774

Chapter 10: Vectors and the Geometry of Pace 785

10.1 vectors in the Plane 786
10.2 vectors in Space 796
10.3 The Dot Product 804
10.4 The Cross Product 815
10.5 Lines and Planes in Space 828
10.6 Surfaces in Space 837

Chapter 11: Vectors- Valued Functions 853
11.1 Vector-Valued Functions 854
11.2 The Calculus of Vector-Valued Functions 864
11.3 Motion in Space 876
11.4 Curvature 887
11.5 Tangent and Normal Vectors 895

Chapter 12: Functions of Several Variables and Partial Differentiation 919
12.1 Functions of Several Variables 920
12.2 Limits and Continuity 936
12.3 Partial Derivatives 949
12.4 Tangent Planes and Linear Approximations 962
12.5 The Chain Rule 973
12.6 The Gradient and Directional Derivatives 983
12.7 Extrema of Functions of Several Variables 996
12.8 Constrained Optimization and Lagrange Multipliers 1012

Chapter 13: Multiple Integrals 1029
13.1 Double Integrals 1029
13.2 Area, Volume and Center of Mass 1046
13.3 Double Integrals in Polar Coordinates 1058
13.4 Surface Area 1066
13.5 Triple Integrals 1072
13.6 Cylindrical coordinates 1084
13.7 Spherical coordinates 1092
13.8 Change of variables in Multiple Integrals 1100

Chapter 14: Vector Calculus 1115
14.1 Vector Fields 1116
14.2 Line Integrals 1130
14.3 Independence of Path and Conservative Vector Fields 1145
14.4 Green’s Theorem 1156
14.5 Curl and Divergence 1165
14.6 Surface Integrals 1176
14.7 The Divergence Theorem 1189
14.8 Stokes’ Theorem 1199
14.9 Applications of Vector Calculus 1208

Chapter 15: Second- Order Differential Equations 1221

15.1 Second- Order Equations with Constant Coefficients 1222
15.2 Nonhomogeneous Equations: Undetermined Coefficients 1231
15.3 Applications of Second-Order Equations 1240
15.4 Power Series Solutions of Differential Equations 1249

Appendix A Proofs of Select Theorems A-I
Appendix B Answers to odd- Numbered Exercises A-13
Credits C-I
Index I-I

includes appendix