chapter one FUNCTIONS 1
1.1 Functions 1
1.2 Graphing Functions Using Calculators and Computer Algebra Systems16
1.3 New Functions from Old 27
1.4 Families of Functions40
1.5 Inverse Functions; Inverse Trigonometric Functions 51
1.6 Mathematical Models 59
1.7 Parametric Equations 69

chapter two LIMITS AND CONTINUITY 84
2.1 Limits (An Intuitive Approach) 84
2.2 Computing Limits 96
2.3 Limits at Infinity; End Behavior of a Function 105
2.4 Limits (Discussed More Rigorously) 116
2.5 Continuity 125
2.6 Continuity of Trigonometric and Inverse Functions 137

chapter three THE DERIVATIVE 146
3.1 Tangent Lines, Velocity, and General Rates of Change 146
3.2 The Derivative Function 159
3.3 Techniques of Differentiation 171
3.4 The Product and Quotient Rules 179
3.5 Derivatives of Trigonometric Functions 185
3.6 The Chain Rule 190
3.7 Implicit Differentiation 198
3.8 Related Rates 206
3.9 Local Linear Approximation; Differentials 213

chapter four THE DERIVATIVE IN GRAPHING AND APPLICATIONS 225
4.1 Analysis of Functions I: Increase, Decrease, and Concavity 225
4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 234
4.3 More on Curve Sketching: Rational Functions; Curves with Cusps and Vertical  Tangent Lines; Using Technology 245
4.4 Absolute Maxima and Minima 254
4.5 Applied Maximum and Minimum Problems 262
4.6 Newton’s Method 276
4.7 Rolle’s Theorem; Mean-Value Theorem 281
4.8 Rectilinear Motion 289

chapter five INTEGRATION 302
5.1 An Overview of the Area Problem 302
5.2 The Indefinite Integral 308
5.3 Integration by Substitution 318
5.4 The Definition of Area as a Limit; Sigma Notation 324
5.5 The Definite Integral 337
5.6 The Fundamental Theorem of Calculus 347
5.7 Rectilinear Motion Revisited Using Integration 361
5.8 Evaluating Definite Integrals by Substitution 370

 

chapter six APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING 380
6.1 Area Between Two Curves 380
6.2 Volumes by Slicing; Disks and Washers 388
6.3 Volumes by Cylindrical Shells 397
6.4 Length of a Plane Curve 403
6.5 Area of a Surface of Revolution 409
6.6 Average Value of a Function and its Applications 414
6.7 Work 419
6.8 Fluid Pressure and Force 427

chapter seven EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS 435
7.1 Exponential and Logarithmic Functions 435
7.2 Derivatives and Integralsd Involving Logarithmic Functions 447
7.3 Derivatives of Inverse Functions; Derivatives and Integrals Involving  Exponential Functions 453
7.4 Graphs and Applications Involving Logarithmic and Exponential Functions 460
7.5 L’Hôpital’s Rule; Indeterminate Forms 467
7.6 Logarithmic Functions from the Integral Point of View 476
7.7 Derivatives and Integrals Involving Inverse Trigonometric Functions 488
7.8 Hyperbolic Functions and Hanging Cables 498

chapter eight PRINCIPLES OF INTEGRAL EVALUATION 514
8.1 An Overview of Integration Methods 514
8.2 Integration by Parts 517
8.3 Trigonometric Integrals 526
8.4 Trigonometric Substitutions 534
8.5 Integrating Rational Functions by Partial Fractions 5441
8.6 Using Computer Algebra Systems and Tables of Integrals 549
8.7 Numerical Integration; Simpson’s Rule 560
8.8 Improper Integrals 573

chapter 9 MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS 586
9.1 First-Order Differential Equations and Applications 586
9.2 Slope Fields; Euler’s Method 600
9.3 Modeling with First-Order Differential Equations 607
9.4 Second-Order Linear Homogeneous Differential Equations; The Vibrating Spring  616

chapter ten INFINITE SERIES 628
10.1 Sequences 628
10.2 Monotone Sequences 639
10.3 Infinite Series 647
10.4 Convergence Tests 656
10.5 The Comparison, Ratio, and Root Tests 663
10.6 Alternating Series; Conditional Convergence 670
10.7 Maclaurin and Taylor Polynomials 679
10.8 Maclaurin and Taylor Series; Power Series 689
10.9 Convergence of Taylor Series 698
10.10 Differentiating and Integrating Power Series; Modeling with Taylor Series  708

chapter eleven ANALYTIC GEOMETRY IN CALCULUS 721
11.1 Polar Coordinates 721
11.2 Tangent Lines and Arc Length for Parametric and Polar Curves 735
11.3 Area in Polar Coordinates 744
11.4 Conic Sections in Calculus 750
11.5 Rotation of Axes; Second-Degree Equations 769
11.6 Conic Sections in Polar Coordinates 775
Horizon Module: Comet Collision 787

chapter twelve THREE-DIMENSIONAL SPACE; VECTORS 790
12.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 790
12.2 Vectors 796
12.3 Dot Product; Projections 808
12.4 Cross Product 817
12.5 Parametric Equations of Lines 828
12.6 Planes in 3-Space 835
12.7 Quadric Surfaces 843
12.8 Cylindrical and Spherical Coordinates 854

chapter thirteen VECTOR-VALUED FUNCTIONS 863
13.1 Introduction to Vector-Valued Functions 863
13.2 Calculus of Vector-Valued Functions 869
13.3 Change of Parameter; Arc Length 880
13.4 Unit Tangent, Normal, and Binormal Vectors 890
13.5 Curvature 8926
13.6 Motion Along a Curve 905
13.7 Kepler’s Laws of Planetary Motion 918

chapter fourteen PARTIAL DERIVATIVES 928
14.1 Functions of Two or More Variables 928
14.2 Limits and Continuity 940
14.3 Partial Derivatives 949
14.4 Differentiability, Differentials, and Local Linearity 963
14.5 The Chain Rule 972
14.6 Directional Derivatives and Gradients 982
14.7 Tangent Planes and Normal Vectors 993
14.8 Maxima and Minima of Functions of Two Variables 1000
14.9 Lagrange Multipliers 10128

chapter fifteen MULTIPLE INTEGRALS 1022
15.1 Double Integrals 1022
15.2 Double Integrals over Nonrectangular Regions 1030
15.3 Double Integrals in Polar Coordinates 1039
15.4 Parametric Surfaces; Surface Area 1047
15.5 Triple Integrals 1060
15.6 Centroid, Center of Gravity, Theorem of Pappus 1069
15.7 Triple Integrals in Cylindrical and Spherical Coordinates 1080
15.8 Change of Variables in Multiple Integrals; Jacobians 1091

chapter sixteen  TOPICS IN VECTOR CALCULUS 1106 
16.1 Vector Fields 1106
16.2 Line Integrals 1116
16.3 Independence of Path; Conservative Vector Fields 1133
16.4 Green’s Theorem 1143
16.5 Surface Integrals 1151
16.6 Applications of Surface Integrals; Flux 1159
16.7 The Divergence Theorem 1168
16.8 Stokes’Theorem 1177
Horizon Module: Hurricane Modeling 1187

appendix  a  TRIGONOMETRY REVIEW A1 
appendix  b  SOLVING POLYNOMIAL EQUATIONS A15 
appendix  c  SELECTED PROOFS A22 

ANSWERS  A33 

PHOTOCREDITS C1 

INDEX  I-1 

web  appendix  d  REAL NUMBERS,INTERVALS, AND INEQUALITIES W1 
web  appendix  e  ABSOLUTE VALUE W11 
web  appendix  f  COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS W16 
web  appendix  g  DISTANCE, CIRCLES, AND QUADRATIC FUNCTIONS W32 
web  appendix  h  THE DISCRIMINANT W41