New Features 

xiii  
A Commitment to Accuracy 

xiv  
Preface 

xv  
Guided Tour 

xxiv  
Application Index 

xxx  


1  (72) 

Polynomials and Rational Functions 


2  (18) 

Graphing Calculators and Computer Algebra Systems 


20  (9) 


29  (7) 

Trigonometric and Inverse Trigonometric Functions 


36  (12) 

Exponential and Logarithmic Functions 


48  (13) 





Transformations of Functions 


61  (12) 


73  (72) 

A Brief Preview of Calculus: Tangent Lines and the Length of a Curve 


74  (5) 


79  (8) 


87  (10) 

Continuity and Its Consequences 


97  (13) 



Limits Involving Infinity 


110  (11) 



Formal Definition of the Limit 


121  (13) 

Exploring the Definition of Limit Graphically 



Limits and LossofSignificance Errors 


134  (11) 

Computer Representation of Real Numbers 




145  (96) 

Tangent Lines and Velocity 


146  (13) 


159  (11) 

Numerical Differentiation 



Computation of Derivatives: The Power Rule 


170  (10) 





The Product and Quotient Rules 


180  (9) 


189  (7) 

Derivatives of Trigonometric Functions 


196  (10) 

Derivatives of Exponential and Logarithmic Functions 


206  (10) 

Implicit Differentiation and Inverse Trigonometric Functions 


216  (10) 


226  (15) 

Applications of Differentiation 


241  (102) 

Linear Approximations and Newton's Method 


242  (13) 

Indeterminate Forms and L'Hopital's Rule 


255  (10) 

Maximum and Minimum Values 


265  (12) 

Increasing and Decreasing Functions 


277  (9) 

Concavity and the Second Derivative Test 


286  (10) 

Overview of Curve Sketching 


296  (12) 


308  (13) 


321  (6) 

Rates of Change in Economics and the Sciences 


327  (16) 


343  (88) 


344  (10) 


354  (8) 

Principle of Mathematical Induction 




362  (7) 


369  (14) 

Average Value of a Function 



The Fundamental Theorem of Calculus 


383  (10) 

Integration by Substitution 


393  (9) 


402  (14) 

Error Bounds for Numerical Integration 



The Natural Logarithm as an Integral 


416  (15) 

The Exponential Function as the Inverse of the Natural Logarithm 



Applications of the Definite Integral 


431  (78) 


432  (9) 

Volume: Slicing, Disks, and Washers 


441  (15) 

Volumes by Cylindrical Shells 


456  (8) 

Arc Length and Surface Area 


464  (8) 


472  (12) 

Applications of Integration to Physics and Engineering 


484  (12) 


496  (13) 


509  (56) 

Review of Formulas and Techniques 


510  (4) 


514  (7) 

Trigonometric Techniques of Integration 


521  (9) 

Integrals Involving Powers of Trigonometric Functions 



Trigonometric Substitution 



Integration of Rational Functions Using Partial Fractions 


530  (8) 

Brief Summary of Integration Techniques 



Integration Tables and Computer Algebra Systems 


538  (8) 


546  (19) 



FirstOrder Differential Equations 


565  (46) 

Modeling with Differential Equations 


566  (11) 

Growth and Decay Problems 





Separable Differential Equations 


577  (10) 



Direction Fields and Euler's Method 


587  (12) 

Systems of FirstOrder Differential Equations 


599  (12) 




611  (104) 

Sequences of Real Numbers 


612  (14) 


626  (10) 

The Integral Test and Comparison Tests 


636  (12) 


648  (8) 

Estimating the Sum of an Alternating Series 



Absolute Convergence and the Ratio Test 


656  (8) 



Summary of Convergence Tests 




664  (8) 


672  (13) 

Representation of Functions as Power Series 



Proof of Taylor's Theorem 



Applications of Taylor Series 


685  (9) 




694  (21) 

Parametric Equations and Polar Coordinates 


715  (70) 

Plane Curves and Parametric Equations 


716  (10) 

Calculus and Parametric Equations 


726  (8) 

Arc Length and Surface Area in Parametric Equations 


734  (8) 


742  (13) 

Calculus and Polar Coordinates 


755  (9) 


764  (10) 

Conic Sections in Polar Coordinates 


774  (11) 

Vectors and the Geometry of Space 


785  (68) 


786  (10) 


796  (8) 


804  (11) 

Components and Projections 




815  (13) 

Lines and Planes in Space 


828  (9) 


837  (16) 


853  (66) 


854  (10) 

The Calculus of VectorValued Functions 


864  (12) 


876  (11) 


887  (8) 

Tangent and Normal Vectors 


895  (13) 

Tangential and Normal Components of Acceleratior 






908  (11) 

Functions of Several Variables and Partial Differentiation 


919  (110) 

Functions of Several Variables 


920  (16) 


936  (13) 


949  (13) 

Tangent Planes and Linear Approximations 


962  (11) 

Increments and Differentials 




973  (10) 

The Gradient and Directional Derivatives 


983  (13) 

Extrema of Functions of Several Variables 


996  (16) 

Constrained Optimization and Lagrange Multipliers 


1012  (17) 


1029  (86) 


1029  (17) 

Area, Volume, and Center of Mass 


1046  (12) 

Double Integrals in Polar Coordinates 


1058  (8) 


1066  (6) 


1072  (12) 




1084  (8) 


1092  (8) 

Change of Variables in Multiple Integrals 


1100  (15) 


1115  (106) 


1116  (14) 


1130  (15) 

Independence of Path and Conservative Vector Fields 


1145  (11) 


1156  (9) 


1165  (11) 


1176  (13) 


1189  (10) 


1199  (9) 

Applications of Vector Calculus 


1208  (13) 

SecondOrder Differential Equations 


1221  

SecondOrder Equations with Constant Coefficients 


1222  (9) 

Nonhomogeneous Equations: Undetermined Coefficients 


1231  (9) 

Applications of SecondOrder Equations 


1240  (9) 

Power Series Solutions of Differential Equations 


1249  
Appendix A: Proofs of Selected Theorems 

1  (12) 
Appendix B: Answers to OddNumbered Exercises 

13  
Credits 

1  (1) 
Index 

1  