Preface 

ix  


1  (54) 

Real Numbers, Estimation, and Logic 


1  (7) 

Inequalities and Absolute Values 


8  (8) 

The Rectangular Coordinate System 


16  (8) 


24  (5) 

Functions and Their Graphs 


29  (6) 


35  (6) 


41  (10) 


51  (4) 

Review and Preview Problems 


54  (1) 


55  (38) 


55  (6) 


61  (7) 


68  (5) 

Limits Involving Trigonometric Functions 


73  (4) 

Limits at Infinity; Infinite Limits 


77  (5) 


82  (8) 


90  (3) 

Review and Preview Problems 


92  (1) 


93  (58) 

Two Problems with One Theme 


93  (7) 


100  (7) 

Rules for Finding Derivatives 


107  (7) 

Derivatives of Trigonometric Functions 


114  (4) 


118  (7) 


125  (5) 


130  (5) 


135  (7) 

Differentials and Approximations 


142  (5) 


147  (4) 

Review and Preview Problems 


150  (1) 

Applications of the Derivative 


151  (64) 


151  (4) 

Monotonicity and Concavity 


155  (7) 

Local Extrema and Extrema on Open Intervals 


162  (5) 


167  (11) 

Graphing Functions Using Calculus 


178  (7) 

The Mean Value Theorem for Derivatives 


185  (5) 

Solving Equations Numerically 


190  (7) 


197  (6) 

Introduction to Differential Equations 


203  (6) 


209  (6) 

Review and Preview Problems 


214  (1) 


215  (60) 


215  (9) 


224  (8) 

The First Fundamental Theorem of Calculus 


232  (11) 

The Second Fundamental Theorem of Calculus and the Method of Substitution 


243  (10) 

The Mean Value Theorem for Integrals and the Use of Symmetry 


253  (7) 


260  (10) 


270  (5) 

Review and Preview Problems 


274  (1) 

Applications of the Integral 


275  (50) 

The Area of a Plane Region 


275  (6) 

Volumes of Solids: Slabs, Disks, Washers 


281  (7) 

Volumes of Solids of Revolution: Shells 


288  (6) 


294  (7) 


301  (7) 

Moments and Center of Mass 


308  (8) 

Probability and Random Variables 


316  (6) 


322  (3) 

Review and Preview Problems 


324  (1) 


325  (58) 

The Natural Logarithm Function 


325  (6) 

Inverse Functions and Their Derivatives 


331  (6) 

The Natural Exponential Function 


337  (5) 

General Exponential and Logarithmic Functions 


342  (5) 

Exponential Growth and Decay 


347  (8) 

FirstOrder Linear Differential Equations 


355  (4) 

Approximations for Differential Equations 


359  (6) 

The Inverse Trigonometric Functions and Their Derivatives 


365  (9) 

The Hyperbolic Functions and Their Inverses 


374  (6) 


380  (3) 

Review and Preview Problems 


382  (1) 

Techniques of Integration 


383  (40) 


383  (4) 


387  (6) 

Some Trigonometric Integrals 


393  (6) 

Rationalizing Substitutions 


399  (5) 

Integration of Rational Functions Using Partial Fractions 


404  (7) 

Strategies for Integration 


411  (8) 


419  (4) 

Review and Preview Problems 


422  (1) 

Indeterminate Forms and Improper Integrals 


423  (26) 

Indeterminate Forms of Type 0/0 


423  (5) 

Other Indeterminate Forms 


428  (5) 

Improper Integrals: Infinite Limits of Integration 


433  (9) 

Improper Integrals: Infinite Integrands 


442  (4) 


446  (3) 

Review and Preview Problems 


448  (1) 


449  (60) 


449  (6) 


455  (8) 

Positive Series: The Integral Test 


463  (5) 

Positive Series: Other Tests 


468  (6) 

Alternating Series, Absolute Convergence, and Conditional Convergence 


474  (5) 


479  (5) 

Operations on Power Series 


484  (5) 

Taylor and Maclaurin Series 


489  (8) 

The Taylor Approximation to a Function 


497  (7) 


504  (5) 

Review and Preview Problems 


508  (1) 

Conics and Polar Coordinates 


509  (46) 


509  (4) 


513  (10) 

Translation and Rotation of Axes 


523  (7) 

Parametric Representation of Curves in the Plane 


530  (7) 

The Polar Coordinate System 


537  (5) 

Graphs of Polar Equations 


542  (5) 

Calculus in Polar Coordinates 


547  (5) 


552  (3) 

Review and Preview Problems 


554  (1) 

Geometry in Space and Vectors 


555  (62) 

Cartesian Coordinates in ThreeSpace 


555  (5) 


560  (6) 


566  (8) 


574  (5) 

VectorValued Functions and Curvilinear Motion 


579  (10) 

Lines and Tangent Lines in ThreeSpace 


589  (4) 

Curvature and Components of Acceleration 


593  (10) 


603  (6) 

Cylindrical and Spherical Coordinates 


609  (4) 


613  (4) 

Review and Preview Problems 


616  (1) 

Derivatives for Functions of Two or More Variables 


617  (58) 

Functions of Two or More Variables 


617  (7) 


624  (5) 


629  (6) 


635  (6) 

Directional Derivatives and Gradients 


641  (6) 


647  (5) 

Tangent Planes and Approximations 


652  (5) 


657  (9) 

The Method of Lagrange Multipliers 


666  (6) 


672  (3) 

Review and Preview Problems 


674  (1) 


675  (56) 

Double Integrals over Rectangles 


675  (5) 


680  (4) 

Double Integrals over Nonrectangular Regions 


684  (7) 

Double Integrals in Polar Coordinates 


691  (5) 

Applications of Double Integrals 


696  (4) 


700  (6) 

Triple Integrals in Cartesian Coordinates 


706  (7) 

Triple Integrals in Cylindrical and Spherical Coordinates 


713  (5) 

Change of Variables in Multiple Integrals 


718  (10) 


728  (3) 

Review and Preview Problems 


730  (1) 


731  


731  (4) 


735  (7) 


742  (7) 

Green's Theorem in the Plane 


749  (6) 


755  (9) 

Gauss's Divergence Theorem 


764  (6) 


770  (3) 


773  


1  (6) 

A.1 Mathematical Induction 


1  (2) 

A.2 Proofs of Several Theorems 


3  (4) 
Answers to OddNumbered Problems 

7  
Index 

1  (1) 
Photo Credits 

1  