Preface 

xi  


1  (54) 

Functions and Their Graphs 


1  (13) 

Combining Functions; Shifting and Scaling Graphs 


14  (8) 


22  (8) 


30  (6) 

Inverse Functions and Logarithms 


36  (14) 

Graphing with Calculators and Computers 


50  (5) 


55  (70) 

Rates of Change and Tangents to Curves 


55  (7) 

Limit of a Function and Limit Laws 


62  (12) 

The Precise Definition of a Limit 


74  (10) 

OneSided Limits and Limits at Infinity 


84  (13) 

Infinite Limits and Vertical Asymptotes 


97  (6) 


103  (12) 

Tangents and Derivatives at a Point 


115  (10) 

Questions to Guide Your Review 


119  (1) 


120  (2) 

Additional and Advanced Exercises 


122  (3) 


125  (112) 

The Derivative as a Function 


125  (9) 

Differentiation Rules for Polynomials, Exponentials, Products, and Quotients 


134  (12) 

The Derivative as a Rate of Change 


146  (11) 

Derivatives of Trigonometric Functions 


157  (7) 

The Chain Rule and Parametric Equations 


164  (13) 


177  (6) 

Derivatives of Inverse Functions and Logarithms 


183  (11) 

Inverse Trigonometric Functions 


194  (7) 


201  (8) 

Linearization and Differentials 


209  (12) 


221  (16) 

Questions to Guide Your Review 


227  (1) 


228  (6) 

Additional and Advanced Exercises 


234  (3) 

Applications of Derivatives 


237  (78) 

Extreme Values of Functions 


237  (8) 


245  (9) 

Monotonic Functions and the First Derivative Test 


254  (6) 

Concavity and Curve Sketching 


260  (11) 


271  (12) 

Indeterminate Forms and L'Hopital's Rule 


283  (8) 


291  (5) 


296  (19) 

Questions to Guide Your Review 


306  (1) 


307  (4) 

Additional and Advanced Exercises 


311  (4) 


315  (76) 

Estimating with Finite Sums 


315  (10) 

Sigma Notation and Limits of Finite Sums 


325  (7) 


332  (13) 

The Fundamental Theorem of Calculus 


345  (9) 

Indefinite Integrals and the Substitution Rule 


354  (6) 

Substitution and Area Between Curves 


360  (10) 

The Logarithm Defined as an Integral 


370  (21) 

Questions to Guide Your Review 


381  (1) 


382  (4) 

Additional and Advanced Exercises 


386  (5) 

Applications of Definite Integrals 


391  (57) 

Volumes by Slicing and Rotation About an Axis 


391  (10) 

Volumes by Cylindrical Shells 


401  (7) 


408  (7) 

Areas of Surfaces of Revolution 


415  (6) 

Exponential Change and Separable Differential Equations 


421  (9) 


430  (7) 

Moments and Centers of Mass 


437  (11) 

Questions to Guide Your Review 


444  (1) 


444  (2) 

Additional and Advanced Exercises 


446  (2) 

Techniques of Integration 


448  (54) 


448  (7) 


455  (6) 

Trigonometric Substitutions 


461  (3) 

Integration of Rational Functions by Partial Fractions 


464  (7) 

Integral Tables and Computer Algebra Systems 


471  (6) 


477  (10) 


487  (15) 

Questions to Guide Your Review 


497  (1) 


497  (3) 

Additional and Advanced Exercises 


500  (2) 

Infinite Sequences and Series 


502  (75) 


502  (13) 


515  (8) 


523  (6) 


529  (4) 


533  (4) 

Alternating Series, Absolute and Conditional Convergence 


537  (6) 


543  (10) 

Taylor and Maclaurin Series 


553  (6) 

Convergence of Taylor Series 


559  (10) 


569  (8) 

Questions to Guide Your Review 


572  (1) 


573  (2) 

Additional and Advanced Exercises 


575  (2) 

Polar Coordinates and Conics 


577  (37) 


577  (5) 

Graphing in Polar Coordinates 


582  (4) 

Areas and Lengths in Polar Coordinates 


586  (4) 


590  (9) 

Conics in Polar Coordinates 


599  (7) 

Conics and Parametric Equations; The Cycloid 


606  (8) 

Questions to Guide Your Review 


610  (1) 


610  (2) 

Additional and Advanced Exercises 


612  (2) 

Vectors and the Geometry of Space 


614  (49) 

ThreeDimensional Coordinate Systems 


614  (5) 


619  (9) 


628  (8) 


636  (6) 

Lines and Planes in Space 


642  (10) 

Cylinders and Quadric Surfaces 


652  (11) 

Questions to Guide Your Review 


657  (1) 


658  (2) 

Additional and Advanced Exercises 


660  (3) 

VectorValued Functions and Motion in Space 


663  (39) 

Vector Functions and Their Derivatives 


663  (9) 

Integrals of Vector Functions 


672  (6) 


678  (5) 


683  (6) 

Tangential and Normal Components of Acceleration 


689  (5) 

Velocity and Acceleration in Polar Coordinates 


694  (8) 

Questions to Guide Your Review 


698  (1) 


698  (2) 

Additional and Advanced Exercises 


700  (2) 


702  (83) 

Functions of Several Variables 


702  (9) 

Limits and Continuity in Higher Dimensions 


711  (8) 


719  (12) 


731  (8) 

Directional Derivatives and Gradient Vectors 


739  (8) 

Tangent Planes and Differentials 


747  (9) 

Extreme Values and Saddle Points 


756  (9) 


765  (10) 

Taylor's Formula for Two Variables 


775  (10) 

Questions to Guide Your Review 


779  (1) 


780  (3) 

Additional and Advanced Exercises 


783  (2) 


785  (66) 

Double and Iterated Integrals over Rectangles 


785  (5) 

Double Integrals over General Regions 


790  (9) 

Area by Double Integration 


799  (3) 

Double Integrals in Polar Form 


802  (5) 

Triple Integrals in Rectangular Coordinates 


807  (9) 

Moments and Centers of Mass 


816  (9) 

Triple Integrals in Cylindrical and Spherical Coordinates 


825  (12) 

Substitutions in Multiple Integrals 


837  (14) 

Questions to Guide Your Review 


846  (1) 


846  (2) 

Additional and Advanced Exercises 


848  (3) 

Integration in Vector Fields 


851  (1) 


851  (5) 

Vector Fields, Work, Circulation, and Flux 


856  (11) 

Path Independence, Potential Functions, and Conservative Fields 


867  (10) 

Green's Theorem in the Plane 


877  (10) 


887  (9) 

Surface Integrals and Flux 


896  (9) 


905  (9) 

The Divergence Theorem and a Unified Theory 


914  (11) 

Questions to Guide Your Review 


925  (1) 


925  (3) 

Additional and Advanced Exercises 


928  

FirstOrder Differential Equations (online) 



Solutions, Slope Fields, and Picard's Theorem 



FirstOrder Linear Equations 







Graphical Solutions of Autonomous Equations 



Systems of Equations and Phase Planes 



SecondOrder Differential Equations (online) 



SecondOrder Linear Equations 



Nonhomogeneous Linear Equations 










1  (1) 

A.1 Real Numbers and the Real Line 


1  (6) 

A.2 Mathematical Induction 


7  (3) 

A.3 Lines, Circles, and Parabolas 


10  (9) 

A.4 Trigonometry Formulas 


19  (2) 

A.5 Proofs of Limit Theorems 


21  (4) 

A.6 Commonly Occurring Limits 


25  (1) 

A.7 Theory of the Real Numbers 


26  (3) 

A.8 The Distributive Law for Vector Cross Products 


29  (1) 

A.9 The Mixed Derivative Theorem and the Increment Theorem 


30  
Answers 

1  (1) 
Index 

1  (1) 
A Brief Table of Integrals 

1  (1) 
Credits 

1  