Preface 

xiv  
To the Student 

xxvi  
A Preview of Calculus 

2  (8) 


10  (54) 

Four Ways to Represent a Function 


11  (14) 

Mathematical Models: A Catalog of Essential Functions 


25  (13) 

New Functions from Old Functions 


38  (10) 

Graphing Calculators and Computers 


48  (16) 


55  (3) 

Principles of Problem Solving 


58  (6) 

Limits and Rates of Change 


64  (62) 

The Tangent and Velocity Problems 


65  (5) 


70  (12) 

Calculating Limits Using the Limit Laws 


82  (10) 

The Precise Definition of a Limit 


92  (10) 


102  (10) 

Tangents, Velocities, and Other Rates of Change 


112  (14) 


121  (3) 


124  (2) 


126  (96) 


127  (7) 

Writing Project Early Methods for Finding Tangents 


133  (1) 

The Derivative as a Function 


134  (11) 


145  (12) 

Rates of Change in the Natural and Social Sciences 


157  (12) 

Derivatives of Trigonometric Functions 


169  (6) 


175  (9) 


184  (6) 


190  (8) 

Applied Project Where Should a Pilot Start Descent? 


197  (1) 

Applied Project Building a Better Roller Coaster 


198  (1) 


198  (7) 

Linear Approximations and Differentials 


205  (17) 

Laboratory Project Taylor Polynomials 


212  (1) 


213  (5) 


218  (4) 

Applications of Differentiation 


222  (92) 

Maximum and Minimum Values 


223  (11) 

Applied Project The Calculus of Rainbows 


232  (2) 


234  (6) 

How Derivatives Affect the Shape of a Graph 


240  (9) 

Limits at Infinity; Horizontal Asymptotes 


249  (14) 

Summary of Curve Sketching 


263  (8) 

Graphing with Calculus and Calculators 


271  (7) 


278  (11) 

Applied Project The Shape of a Can 


288  (1) 

Applications to Business and Economics 


289  (5) 


294  (6) 


300  (14) 


308  (3) 


311  (3) 


314  (60) 


315  (11) 


326  (14) 

Discovery Project Area Functions 


339  (1) 

The Fundamental Theorem of Calculus 


340  (10) 

Indefinite Integrals and the Net Change Theorem 


350  (10) 

Writing Project Newton, Leibniz, and the Invention of Calculus 


359  (1) 


360  (14) 


368  (4) 


372  (2) 

Applications of Integration 


374  (38) 


375  (7) 


382  (11) 

Volumes by Cylindrical Shells 


393  (5) 


398  (4) 

Average Value of a Function 


402  (10) 


406  (2) 


408  (4) 

Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions 


412  (98) 


413  (8) 

Instructors may cover either Sections 7.27.4 or Sections 7.2*7.4*. See the Preface 



Exponential Functions and Their Derivatives 


421  (13) 


434  (7) 

Derivatives of Logarithmic Functions 


441  (10) 

The Natural Logarithmic Function 


451  (9) 

The Natural Exponential Function 


460  (7) 

General Logarithmic and Exponential Functions 


467  (10) 

Inverse Trigonometric Functions 


477  (9) 

Applied Project Where to Sit at the Movies 


485  (1) 


486  (7) 

Indeterminate Forms and L'Hospital's Rule 


493  (17) 

Writing Project The Origins of L'Hospital's Rule 


504  (1) 


504  (4) 


508  (2) 

Techniques of Integration 


510  (72) 


511  (7) 


518  (7) 

Trigonometric Substitution 


525  (7) 

Integration of Rational Functions by Partial Fractions 


532  (9) 


541  (6) 

Integration Using Tables and Computer Algebra Systems 


547  (7) 

Discovery Project Patterns in Integrals 


553  (1) 


554  (12) 


566  (16) 


576  (3) 


579  (3) 

Further Applications of Integration 


582  (40) 


583  (7) 

Discovery Project Arc Length Contest 


590  (1) 

Area of a Surface of Revolution 


590  (7) 

Discovery Project Rotating on a Slant 


596  (1) 

Applications to Physics and Engineering 


597  (10) 

Applications to Economics and Biology 


607  (4) 


611  (11) 


618  (2) 


620  (2) 


622  (64) 

Modeling with Differential Equations 


623  (5) 

Direction Fields and Euler's Method 


628  (9) 


637  (10) 

Applied Project How Fast Does a Tank Drain? 


645  (1) 

Applied Project Which Is Faster, Going Up or Coming Down? 


646  (1) 

Exponential Growth and Decay 


647  (12) 

Applied Project Calculus and Baseball 


658  (1) 


659  (9) 


668  (6) 


674  (12) 


680  (4) 


684  (2) 

Parametric Equations and Polar Coordinates 


686  (50) 

Curves Defined by Parametric Equations 


687  (9) 

Laboratory Project Running Circles around Circles 


695  (1) 

Calculus with Parametric Curves 


696  (9) 

Laboratory Project Bezier Curves 


705  (1) 


705  (10) 

Areas and Lengths in Polar Coordinates 


715  (5) 


720  (8) 

Conic Sections in Polar Coordinates 


728  (8) 


732  (3) 


735  (1) 

Infinite Sequences and Series 


736  (92) 


737  (12) 

Laboratory Project Logistic Sequences 


749  (1) 


749  (10) 

The Integral Test and Estimates of Sums 


759  (7) 


766  (5) 


771  (5) 

Absolute Convergence and the Ratio and Root Tests 


776  (7) 

Strategy for Testing Series 


783  (2) 


785  (5) 

Representations of Functions as Power Series 


790  (6) 

Taylor and Maclaurin Series 


796  (12) 

Laboratory Project An Elusive Limit 


808  (1) 


808  (4) 

Writing Project How Newton Discovered the Binomial Series 


812  (1) 

Applications of Taylor Polynomials 


812  (16) 

Applied Project Radiation from the Stars 


821  (1) 


822  (3) 


825  (3) 

Vectors and the Geometry of Space 


828  (56) 

ThreeDimensional Coordinate Systems 


829  (5) 


834  (9) 


843  (7) 


850  (8) 

Discovery Project The Geometry of a Tetrahedron 


858  (1) 

Equations of Lines and Planes 


858  (10) 

Laboratory Project Putting 3D in Perspective 


868  (1) 

Cylinders and Quadric Surfaces 


868  (7) 

Cylindrical and Spherical Coordinates 


875  (9) 

Laboratory Project Families of Surfaces 


880  (1) 


880  (3) 


883  (1) 


884  (38) 

Vector Functions and Space Curves 


885  (7) 

Derivatives and Integrals of Vector Functions 


892  (6) 


898  (8) 

Motion in Space: Velocity and Acceleration 


906  (16) 

Applied Project Kepler's Laws 


916  (1) 


917  (3) 


920  (2) 


922  (94) 

Functions of Several Variables 


923  (15) 


938  (7) 


945  (14) 

Tangent Planes and Linear Approximations 


959  (8) 


967  (9) 

Directional Derivatives and the Gradient Vector 


976  (13) 

Maximum and Minimum Values 


989  (12) 

Applied Project Designing a Dumpster 


999  (1) 

Discovery Project Quadratic Approximations and Critical Points 


1000  (1) 


1001  (15) 

Applied Project Rocket Science 


1008  (1) 

Applied Project HydroTurbine Optimization 


1009  (1) 


1010  (4) 


1014  (2) 


1016  (74) 

Double Integrals over Rectangles 


1017  (8) 


1025  (6) 

Double Integrals over General Regions 


1031  (8) 

Double Integrals in Polar Coordinates 


1039  (6) 

Applications of Double Integrals 


1045  (10) 


1055  (4) 


1059  (10) 

Discovery Project Volumes of Hyperspheres 


1068  (1) 

Triple Integrals in Cylindrical and Spherical Coordinates 


1069  (8) 

Applied Project Roller Derby 


1075  (1) 

Discovery Project The Intersection of Three Cylinders 


1076  (1) 

Change of Variables in Multiple Integrals 


1077  (13) 


1085  (3) 


1088  (2) 


1090  (86) 


1091  (7) 


1098  (12) 

The Fundamental Theorem for Line Integrals 


1110  (9) 


1119  (7) 


1126  (8) 

Parametric Surfaces and Their Areas 


1134  (11) 


1145  (12) 


1157  (6) 

Writing Project Three Men and Two Theorems 


1162  (1) 


1163  (7) 


1170  (6) 


1171  (3) 


1174  (2) 

SecondOrder Differential Equations 


1176  

SecondOrder Linear Equations 


1177  (6) 

Nonhomogeneous Linear Equations 


1183  (8) 

Applications of SecondOrder Differential Equations 


1191  (8) 


1199  


1203  
Appendixes 

1  (124) 

A Numbers, Inequalities, and Absolute Values 


2  (8) 

B Coordinate Geometry and Lines 


10  (6) 

C Graphs of SecondDegree Equations 


16  (8) 


24  (10) 


34  (5) 


39  (9) 


48  (8) 

H Answers to OddNumbered Exercises 


56  (69) 
Index 

125  