A Word from the Authors 

x  
Integrated Learning System for Calculus 

xii  
Features 

xviii  


1  (40) 


2  (8) 

Linear Models and Rates of Change 


10  (9) 

Functions and Their Graphs 


19  (12) 


31  (6) 


37  (2) 


39  (2) 

Limits and Their Properties 


41  (54) 


42  (6) 

Finding Limits Graphically and Numerically 


48  (11) 

Evaluating Limits Analytically 


59  (11) 

Continuity and OneSided Limits 


70  (13) 


83  (7) 

Section Project: Graphs and Limits of Trigonometric Functions 


90  (5) 


91  (2) 


93  (2) 


95  (68) 

The Derivative and the Tangent Line Problem 


96  (11) 

Basic Differentiation Rules and Rates of Change 


107  (12) 

Product and Quotient Rules and HigherOrder Derivatives 


119  (11) 


130  (11) 


141  (7) 

Section Project: Optical Illusions 


148  (15) 


149  (9) 


158  (3) 


161  (2) 

Applications of Differentiation 


163  (84) 


164  (8) 

Rolle's Theorem and the Mean Value Theorem 


172  (7) 

Increasing and Decreasing Functions and the First Derivative Test 


179  (10) 

Section Project: Rainbows 


189  (39) 

Concavity and the Second Derivative Test 


190  (8) 


198  (11) 

A Summary of Curve Sketching 


209  (9) 


218  (10) 

Section Project: Connecticut River 


228  (19) 


229  (6) 


235  (7) 


242  (3) 


245  (2) 


247  (74) 

Antiderivatives and Indefinite Integration 


248  (11) 


259  (12) 

Riemann Sums and Definite Integrals 


271  (11) 

The Fundamental Theorem of Calculus 


282  (12) 

Section Project: Demonstrating the Fundamental Theorem 


294  (27) 

Integration by Substitution 


295  (14) 


309  (7) 


316  (3) 


319  (2) 

Logarithmic, Exponential, and Other Transcendental Functions 


321  (82) 

The Natural Logarithmic Function: Differentiation 


322  (10) 

The Natural Logarithmic Function: Integration 


332  (9) 


341  (9) 

Exponential Functions: Differentiation and Integration 


350  (10) 

Bases Other Than e and Applications 


360  (10) 

Section Project: Using Graphing Utilities to Estimate Slope 


370  (28) 

Inverse Trigonometric Functions: Differentiation 


371  (9) 

Inverse Trigonometric Functions: Integration 


380  (8) 


388  (10) 

Section Project: St. Louis Arch 


398  (5) 


399  (2) 


401  (2) 


403  (42) 

Slope Fields and Euler's Method 


404  (9) 

Differential Equations: Growth and Decay 


413  (8) 

Separation of Variables and the Logistic Equation 


421  (11) 

FirstOrder Linear Differential Equations 


432  (8) 

Section Project: Weight Loss 


440  (5) 


441  (2) 


443  (2) 

Applications of Integration 


445  (72) 

Area of a Region Between Two Curves 


446  (10) 


456  (11) 


467  (8) 


475  (20) 

Arc Length and Surfaces of Revolution 


476  (11) 


487  (8) 

Section Project: Tidal Energy 


495  (22) 

Moments, Centers of Mass, and Centroids 


496  (11) 

Fluid Pressure and Fluid Force 


507  (6) 


513  (2) 


515  (2) 

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


517  (76) 


518  (7) 


525  (9) 


534  (8) 

Section Project: Power Lines 


542  (51) 

Trigonometric Substitution 


543  (9) 


552  (9) 

Integration by Tables and Other Integration Techniques 


561  (6) 

Indeterminate Forms and L'Hopital's Rule 


567  (11) 


578  (11) 


589  (2) 


591  (2) 


593  (100) 


594  (12) 


606  (10) 

Section Project: Cantor's Disappearing Table 


616  (7) 

The Integral Test and pSeries 


617  (6) 

Section Project: The Harmonic Series 


623  (7) 


624  (6) 

Section Project: Solera Method 


630  (63) 


631  (8) 


639  (9) 

Taylor Polynomials and Approximations 


648  (11) 


659  (10) 

Representation of Functions by Power Series 


669  (7) 

Taylor and Maclaurin Series 


676  (12) 


688  (3) 


691  (2) 

Conics, Parametric Equations, and Polar Coordinates 


693  (68) 


694  (15) 

Plane Curves and Parametric Equations 


709  (9) 

Section Project: Cycloids 


718  (20) 

Parametric Equations and Calculus 


719  (10) 

Polar Coordinates and Polar Graphs 


729  (9) 

Section Project: Anamorphic Art 


738  (23) 

Area and Arc Length in Polar Coordinates 


739  (9) 

Polar Equations of Conics and Kepler's Laws 


748  (8) 


756  (3) 


759  (2) 

Vectors and the Geometry of Space 


761  (70) 


762  (11) 

Space Coordinates and Vectors in Space 


773  (8) 

The Dot Product of Two Vectors 


781  (9) 

The Cross Product of Two Vectors in Space 


790  (8) 

Lines and Planes in Space 


798  (11) 

Section Project: Distances in Space 


809  (22) 


810  (10) 

Cylindrical and Spherical Coordinates 


820  (7) 


827  (2) 


829  (2) 


831  (52) 


832  (7) 

Section Project: Witch of Agnesi 


839  (44) 

Differentiation and Integration of VectorValued Functions 


840  (8) 

Velocity and Acceleration 


848  (9) 

Tangent Vectors and Normal Vectors 


857  (10) 


867  (12) 


879  (2) 


881  (2) 

Functions of Several Variables 


883  (98) 

Introduction to Functions of Several Variables 


884  (12) 


896  (10) 


906  (9) 

Section Project: Moire Fringes 


915  (36) 


916  (7) 

Chain Rules for Functions of Several Variables 


923  (8) 

Directional Derivatives and Gradients 


931  (12) 

Tangent Planes and Normal Lines 


943  (8) 

Section Project: Wildflowers 


951  (16) 

Extrema of Functions of Two Variables 


952  (8) 

Applications of Extrema of Functions of Two Variables 


960  (7) 

Section Project: Building a Pipeline 


967  (14) 


968  (8) 


976  (3) 


979  (2) 


981  (72) 

Iterated Integrals and Area in the Plane 


982  (8) 

Double Integrals and Volume 


990  (11) 

Change of Variables: Polar Coordinates 


1001  (8) 

Center of Mass and Moments of Inertia 


1009  (7) 

Section Project: Center of Pressure on a Sail 


1016  (7) 


1017  (6) 

Section Project: Capillary Action 


1023  (18) 

Triple Integrals and Applications 


1024  (11) 

Triple Integrals in Cylindrical and Spherical Coordinates 


1035  (6) 

Section Project: Wrinkled and Bumpy Spheres 


1041  (12) 

Change of Variables: Jacobians 


1042  (6) 


1048  (3) 


1051  (2) 


1053  


1054  (11) 


1065  (14) 

Conservative Vector Fields and Independence of Path 


1079  (10) 


1089  (8) 

Section Project: Hyperbolic and Trigonometric Functions 


1097  (22) 


1098  (10) 


1108  (11) 

Section Project: Hyperboloid of One Sheet 


1119  (17) 


1120  (8) 


1128  (6) 


1134  (2) 

Section Project: The Planimeter 


1136  


1137  

Appendix A Proofs of Selected Theorems 


1  (19) 

Appendix B Integration Tables 


20  (7) 
Answers to OddNumbered Exercises 

27  (116) 
Index of Applications 

143  (4) 
Index 

147  