A Word from the Authors (Preface) 

xi  
Index of Applications 

xxxi  


1  (38) 


3  (8) 

Linear Models and Rates of Change 


11  (9) 

Functions and Their Graphs 


20  (11) 


31  (8) 


37  (2) 

Limits and Their Properties 


39  (50) 


41  (6) 

Finding Limits Graphically and Numerically 


47  (9) 

Evaluating Limits Analytically 


56  (11) 

Continuity and OneSided Limits 


67  (12) 


79  (10) 


87  (2) 


89  (64) 

The Derivative and the Tangent Line Problem 


91  (11) 

Basic Differentiation Rules and Rates of Change 


102  (12) 

The Product and Quotient Rules and HigherOrder Derivatives 


114  (10) 


124  (10) 


134  (7) 


141  (12) 


150  (3) 

Applications of Differentiation 


153  (86) 


155  (8) 

Rolle's Theorem and the Mean Value Theorem 


163  (6) 

Increasing and Decreasing Functions and the First Derivative Test 


169  (10) 

Concavity and the Second Derivative Test 


179  (8) 


187  (9) 

A Summary of Curve Sketching 


196  (9) 


205  (10) 


215  (6) 


221  (7) 

Business and Economics Applications 


228  (11) 


235  (4) 


239  (70) 

Antiderivatives and Indefinite Integration 


241  (11) 


252  (12) 

Riemann Sums and Definite Integrals 


264  (10) 

The fundamental Theorem of Calculus 


274  (13) 

Integration by Substitution 


287  (12) 


299  (10) 


306  (3) 

Logarithmic, Exponential, and Other Transcendental Functions 


309  (96) 

The Natural Logarithmic Function and Differentiation 


311  (10) 

The Natural Logarithmic function and Integration 


321  (8) 


329  (9) 

Exponential Functions: Differentiation and Integration 


338  (10) 

Bases Other than e and Applications 


348  (10) 

Differential Equations: Growth and Decay 


358  (8) 

Differential Equations: Separation of Variables 


366  (11) 

Inverse Trigonometric Functions and Differentiation 


377  (8) 

Inverse Trigonometric Functions and Integration 


385  (7) 


392  (13) 


402  (3) 

Applications of Integration 


405  (68) 

Area of a Region Between Two Curves 


407  (9) 


416  (11) 


427  (8) 

Arc Length and Surfaces of Revolution 


435  (10) 


445  (9) 

Moments, Centers of Mass, and Centroids 


454  (11) 

Fluid Pressure and Fluid Force 


465  (8) 


471  (2) 

Integration Techniques, L'Hopital's Rule, and Improper Integrals 


473  (72) 


475  (6) 


481  (9) 


490  (9) 

Trigonometric Substitution 


499  (9) 


508  (9) 

Integration by Tables and Other Integration Techniques 


517  (6) 

Indeterminate Forms and L'Hopital's Rule 


523  (10) 


533  (12) 


542  (3) 


545  (90) 


547  (11) 


558  (10) 

The Integral Test and pSeries 


568  (6) 


574  (7) 


581  (7) 


588  (8) 

Taylor Polynomials and Approximations 


596  (10) 


606  (9) 

Representation of Functions by Power Series 


615  (7) 

Taylor and Maclaurin Series 


622  (13) 


632  (3) 

Conics, Parametric Equations, and Polar Coordinates 


635  (64) 


637  (15) 

Plane Curves and Parametric Equations 


652  (10) 

Parametric Equations and Calculus 


662  (9) 

Polar Coordinates and Polar Graphs 


671  (10) 

Area and Arc Length in Polar Coordinates 


681  (8) 

Polar Equations of Conics and Kepler's Laws 


689  (10) 


696  (3) 

Vectors and the Geometry of Space 


699  (68) 


701  (11) 

Space Coordinates and Vectors in Space 


712  (8) 

The Dot Product of Two Vectors 


720  (9) 

Teh Cross Product of Two Vectors in Space 


729  (8) 

Lines and Planes in Space 


737  (11) 


748  (10) 

Cyclindrical and Spherical Coordinates 


758  (9) 


765  (2) 


767  (50) 


769  (8) 

Differentiation and Integration of VectorValued Functions 


777  (8) 

Velocity and Acceleration 


785  (9) 

Tangent Vectors and Normal Vectors 


794  (9) 


803  (14) 


815  (2) 

Functions of Several Variables 


817  (96) 

Introduction to Functions of Several Variables 


819  (12) 


831  (9) 


840  (9) 


849  (8) 

Chain Rules for Functions of Several Variables 


857  (8) 

Directional Derivatives and Gradients 


865  (12) 

Tangent Planes and Normal Lines 


877  (9) 

Extrema of Functions of Two Variables 


886  (8) 

Applications of Extrema of Functions of Two Variables 


894  (8) 


902  (11) 


910  (3) 


913  (70) 

Interated Integrals and Area in the Plane 


915  (8) 

Double Itegrals and Volume 


923  (11) 

Change of Variables: Polar Coordinates 


934  (8) 

Center of Mass and Moments of Inertia 


942  (8) 


950  (7) 

Triple Integrals and Applications 


957  (10) 

Triple Integrals in Cylindrical and Spherical Coordinates 


967  (7) 

Change of Variables: Jacobians 


974  (9) 


980  (3) 


983  (84) 


985  (11) 


996  (13) 

Conservative Vector Fields and Independence of Path 


1009  (10) 


1019  (9) 


1028  (10) 


1038  (12) 


1050  (8) 


1058  (9) 


1064  (3) 


1067  

Exact FirstOrder Equations 


1069  

FirstOrder Linear Differential Equations 


1076  

SecondOrder Homogeneous Linear Equations 


1085  

SecondOrder Nonhomogeneous Linear Equations 


1093  

Series Solutions of Differential Equations 


1101  


1104  
Appendix A Precalculus Review 

A1  

A.1 Real Numbers and the Real Line 


A1  


A10  

A.3 Review of Trigonometric Functions 


A17  
Appendix B Proofs of Selected Theorems 

A28  
Appendix C Basic Differentiation Rules for Elementary Functions 

A44  
Appendix D Integration Tables 

A45  
Appendix E Rotation and the General SecondDegree Equation 

A51  
Appendix F Complex Numbers 

A57  
Answers to OddNumbered Exercises 

A69  
Index 

A197  