To the Instructor 

xiii  
To the Student 

xxv  
Preliminaries 

1  (1) 


1  (9) 


10  (14) 


24  (7) 

Inverse Functions and Logarithms 


31  (13) 

Trigonometric Functions and Their Inverses 


44  (16) 


60  (7) 


67  (18) 

Questions to Guide Your Review 


76  (1) 


77  (3) 

Additional Exercises: Theory, Examples, Applications 


80  (5) 


85  (62) 

Rates of Change and Limits 


85  (14) 

Finding Limits and OneSided Limits 


99  (13) 

Limits Involving Infinity 


112  (11) 


123  (11) 


134  (13) 

Questions to Guide Your Review 


141  (1) 


142  (1) 

Additional Exercises: Theory, Examples, Applications 


143  (4) 


147  (78) 

The Derivative as a Function 


147  (13) 

The Derivative as a Rate of Change 


160  (13) 

Derivatives of Products, Quotients, and Negative Powers 


173  (6) 

Derivatives of Trigonometric Functions 


179  (8) 


187  (11) 


198  (9) 


207  (18) 

Questions to Guide Your Review 


216  (1) 


217  (4) 

Additional Exercises: Theory, Examples, Applications 


221  (4) 

Applications of Derivatives 


225  (88) 

Extreme Values of Functions 


225  (12) 

The Mean Value Theorem and Differential Equations 


237  (8) 


245  (12) 

Graphical Solutions of Autonomous Differential Equations 


257  (9) 

Modeling and Optimization 


266  (17) 

Linearization and Differentials 


283  (14) 


297  (16) 

Questions to Guide Your Review 


305  (1) 


305  (4) 

Additional Exercises: Theory, Examples, Applications 


309  (4) 


313  (80) 

Indefinite Integrals, Differential Equations, and Modeling 


313  (9) 

Integral Rules; Integration by Substitution 


322  (7) 

Estimating with Finite Sums 


329  (11) 

Riemann Sums and Definite Integrals 


340  (11) 

The Mean Value and Fundamental Theorems 


351  (13) 

Substitution in Definite Integrals 


364  (9) 


373  (20) 

Questions to Guide Your Review 


384  (1) 


385  (4) 

Additional Exercises: Theory, Examples, Applications 


389  (4) 

Applications of Integrals 


393  (64) 

Volumes by Slicing and Rotation About an Axis 


393  (13) 

Modeling Volume Using Cylindrical Shells 


406  (7) 


413  (8) 

Springs, Pumping, and Lifting 


421  (11) 


432  (7) 

Moments and Centers of Mass 


439  (18) 

Questions to Guide Your Review 


451  (1) 


451  (3) 

Additional Exercises: Theory, Examples, Applications 


454  (3) 

Transcendental Functions and Differential Equations 


457  (82) 


457  (9) 


466  (11) 

Derivatives of Inverse Trigonometric Functions; Integrals 


477  (8) 

FirstOrder Separable Differential Equations 


485  (14) 

Linear FirstOrder Differential Equations 


499  (8) 

Euler's Method; Population Models 


507  (13) 


520  (19) 

Questions to Guide Your Review 


530  (1) 


531  (4) 

Additional Exercises: Theory, Examples, Applications 


535  (4) 

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


539  (68) 

Basic Integration Formulas 


539  (7) 


546  (9) 


555  (10) 

Trigonometric Substitutions 


565  (5) 

Integral Tables, Computer Algebra Systems, and Monte Carlo Integration 


570  (8) 


578  (8) 


586  (21) 

Questions to Guide Your Review 


600  (1) 


601  (2) 

Additional Exercises: Theory, Examples, Applications 


603  (4) 


607  (110) 

Limits of Sequences of Numbers 


608  (11) 

Subsequences, Bounded Sequences, and Picard's Method 


619  (8) 


627  (12) 

Series of Nonnegative Terms 


639  (12) 

Alternating Series, Absolute and Conditional Convergence 


651  (9) 


660  (9) 

Taylor and Maclaurin Series 


669  (14) 

Applications of Power Series 


683  (8) 


691  (7) 

Fourier Cosine and Sine Series 


698  (19) 

Questions to Guide Your Review 


707  (1) 


708  (3) 

Additional Exercises: Theory, Examples, Applications 


711  (6) 

Vectors in the Plane and Polar Functions 


717  (70) 


717  (11) 


728  (10) 


738  (11) 

Modeling Projectile Motion 


749  (12) 

Polar Coordinates and Graphs 


761  (9) 


770  (17) 

Questions to Guide Your Review 


780  (1) 


780  (4) 

Additional Exercises: Theory, Examples, Applications 


784  (3) 

Vectors and Motion in Space 


787  (86) 

Cartesian (Rectangular) Coordinates and Vectors in Space 


787  (9) 


796  (11) 

Lines and Planes in Space 


807  (9) 

Cylinders and Quadric Surfaces 


816  (9) 

VectorValued Functions and Space Curves 


825  (13) 

Arc Length and the Unit Tangent Vector T 


838  (9) 

The TNB Frame; Tangential and Normal Components of Acceleration 


847  (10) 

Planetary Motion and Satellites 


857  (16) 

Questions to Guide Your Review 


866  (1) 


867  (3) 

Additional Exercises: Theory, Examples, Applications 


870  (3) 

Multivariable Functions and Their Derivatives 


873  (102) 

Functions of Several Variables 


873  (9) 

Limits and Continuity in Higher Dimensions 


882  (8) 


890  (12) 


902  (9) 

Directional Derivatives, Gradient Vectors, and Tangent Planes 


911  (14) 

Linearization and Differentials 


925  (11) 

Extreme Values and Saddle Points 


936  (11) 


947  (11) 

Partial Derivatives with Constrained Variables 


958  (5) 

Taylor's Formula for Two Variables 


963  (12) 

Questions to Guide Your Review 


968  (1) 


968  (4) 

Additional Exercises: Theory, Examples, Applications 


972  (3) 


975  (78) 


975  (12) 

Areas, Moments and Centers of Mass 


987  (13) 

Double Integrals in Polar Form 


1000  (7) 

Triple Integrals in Rectangular Coordinates 


1007  (10) 

Masses and Moments in Three Dimensions 


1017  (7) 

Triple Integrals in Cylindrical and Spherical Coordinates 


1024  (13) 

Substitutions in Multiple Integrals 


1037  (16) 

Questions to Guide Your Review 


1046  (1) 


1047  (2) 

Additional Exercises: Theory, Examples, Applications 


1049  (4) 

Integration in Vector Fields 


1053  (90) 


1053  (6) 

Vector Fields, Work, Circulation, and Flux 


1059  (11) 

Path Independence, Potential Functions, and Conservative Fields 


1070  (10) 

Green's Theorem in the Plane 


1080  (12) 

Surface Area and Surface Integrals 


1092  (11) 


1103  (10) 


1113  (11) 

Divergence Theorem and a Unified Theory 


1124  (19) 

Questions to Guide Your Review 


1136  (1) 


1136  (3) 

Additional Exercises: Theory, Examples, Applications 


1139  (4) 
Appendices 

1143  (40) 

A.1 Mathematical Induction 


1143  (3) 

A.2 Proofs of Limit Theorems in Section 1.2 


1146  (4) 

A.3 Proof of the Chain Rule 


1150  (1) 


1151  (11) 

A.5 Simpson's OneThird Rule 


1162  (1) 

A.6 Cauchy's Mean Value Theorem and the Stronger Form of L'Hopital's Rule 


1163  (1) 

A.7 Limits That Arise Frequently 


1164  (2) 

A.8 Proof of Taylor's Theorem 


1166  (1) 

A.9 The Distributive Law for Vector Cross Products 


1167  (2) 

A.10 Determinants and Cramer's Rule 


1169  (7) 

A.11 The Mixed Derivative Theorem and the Increment Theorem 


1176  (5) 

A.12 The Area of a Parallelogram's Projection on a Plane 


1181  (2) 
Answers 

1183  
Index 

I1  
A Brief Table of Integrals 

T1  