Preface 

xiv  
To the Student 

xxvi  
A Preview of Calculus 

2  (8) 


10  (76) 

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  (7) 


55  (8) 

Inverse Functions and Logarithms 


63  (23) 


77  (3) 

Principles of Problem Solving 


80  (6) 


86  (96) 

The Tangent and Velocity Problems 


87  (5) 


92  (12) 

Calculating Limits Using the Limit Laws 


104  (10) 

The Precise Definition of a Limit 


114  (10) 


124  (11) 

Limits at Infinity; Horizontal Asymptotes 


135  (14) 

Tangents, Velocities, and Other Rates of Change 


149  (9) 


158  (7) 

Writing Project Early Methods for Finding Tangents 


164  (1) 

The Derivative as a Function 


165  (17) 


176  (4) 


180  (2) 


182  (96) 

Derivatives of Polynomials and Exponential Functions 


183  (9) 

The Product and Quotient Rules 


192  (7) 

Rates of Change in the Natural and Social Sciences 


199  (12) 

Derivatives of Trigonometric Functions 


211  (6) 


217  (10) 


227  (9) 


236  (8) 

Applied Project Where Should a Pilot Start Descent? 


243  (1) 

Applied Project Building a Better Roller Coaster 


243  (1) 

Derivatives of Logarithmic Functions 


244  (6) 


250  (6) 


256  (6) 

Linear Approximations and Differentials 


262  (16) 

Laboratory Project Taylor Polynomials 


269  (1) 


270  (4) 


274  (4) 

Applications of Differentiation 


278  (90) 

Maximum and Minimum Values 


279  (11) 

Applied Project The Calculus of Rainbows 


288  (2) 


290  (6) 

How Derivatives Affect the Shape of a Graph 


296  (11) 

Indeterminate Forms and L'Hospital's Rule 


307  (9) 

Writing Project The Origins of L'Hospital's Rule 


315  (1) 

Summary of Curve Sketching 


316  (8) 

Graphing with Calculus and Calculators 


324  (7) 


331  (11) 

Applied Project The Shape of a Can 


341  (1) 

Applications to Business and Economics 


342  (5) 


347  (6) 


353  (15) 


361  (4) 


365  (3) 


368  (68) 


369  (11) 


380  (14) 

Discovery Project Area Functions 


393  (1) 

The Fundamental Theorem of Calculus 


394  (11) 

Indefinite Integrals and the Net Change Theorem 


405  (9) 

Writing Project Newton, Leibniz, and the Invention of Calculus 


413  (1) 


414  (8) 

The Logarithm Defined as an Integral 


422  (14) 


430  (4) 


434  (2) 

Applications of Integration 


436  (38) 


437  (7) 


444  (11) 

Volumes by Cylindrical Shells 


455  (5) 


460  (4) 

Average Value of a Function 


464  (10) 

Applied Project Where to Sit at the Movies 


468  (1) 


468  (2) 


470  (4) 

Techniques of Integration 


474  (72) 


475  (7) 


482  (7) 

Trigonometric Substitution 


489  (7) 

Integration of Rational Functions by Partial Fractions 


496  (9) 


505  (6) 

Integration Using Tables and Computer Algebra Systems 


511  (7) 

Discovery Project Patterns in Integrals 


517  (1) 


518  (12) 


530  (16) 


540  (3) 


543  (3) 

Further Applications of Integration 


546  (40) 


547  (7) 

Discovery Project Arc Length Contest 


554  (1) 

Area of a Surface of Revolution 


554  (7) 

Discovery Project Rotating on a Slant 


560  (1) 

Applications to Physics and Engineering 


561  (10) 

Applications to Economics and Biology 


571  (4) 


575  (11) 


582  (2) 


584  (2) 


586  (64) 

Modeling with Differential Equations 


587  (5) 

Direction Fields and Euler's Method 


592  (9) 


601  (10) 

Applied Project How Fast Does a Tank Drain? 


609  (1) 

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


610  (1) 

Exponential Growth and Decay 


611  (12) 

Applied Project Calculus and Baseball 


622  (1) 


623  (9) 


632  (6) 


638  (12) 


644  (4) 


648  (2) 

Parametric Equations and Polar Coordinates 


650  (50) 

Curves Defined by Parametric Equations 


651  (9) 

Laboratory Project Running Circles around Circles 


659  (1) 

Calculus with Parametric Curves 


660  (9) 

Laboratory Project Bezier Curves 


669  (1) 


669  (10) 

Areas and Lengths in Polar Coordinates 


679  (5) 


684  (8) 

Conic Sections in Polar Coordinates 


692  (8) 


696  (3) 


699  (1) 

Infinite Sequences and Series 


700  (92) 


701  (12) 

Laboratory Project Logistic Sequences 


713  (1) 


713  (10) 

The Integral Test and Estimates of Sums 


723  (7) 


730  (5) 


735  (5) 

Absolute Convergence and the Ratio and Root Tests 


740  (7) 

Strategy for Testing Series 


747  (2) 


749  (5) 

Representations of Functions as Power Series 


754  (6) 

Taylor and Maclaurin Series 


760  (12) 

Laboratory Project An Elusive Limit 


772  (1) 


772  (4) 

Writing Project How Newton Discovered the Binomial Series 


776  (1) 

Applications of Taylor Polynomials 


776  (16) 

Applied Project Radiation from the Stars 


785  (1) 


786  (3) 


789  (3) 

Vectors and the Geometry of Space 


792  (56) 

ThreeDimensional Coordinate Systems 


793  (5) 


798  (9) 


807  (7) 


814  (8) 

Discovery Project The Geometry of a Tetrahedron 


822  (1) 

Equations of Lines and Planes 


822  (10) 

Laboratory Project Putting 3D in Perspective 


832  (1) 

Cylinders and Quadric Surfaces 


832  (7) 

Cylindrical and Spherical Coordinates 


839  (9) 

Laboratory Project Families of Surfaces 


844  (1) 


844  (3) 


847  (1) 


848  (38) 

Vector Functions and Space Curves 


849  (7) 

Derivatives and Integrals of Vector Functions 


856  (6) 


862  (8) 

Motion in Space: Velocity and Acceleration 


870  (16) 

Applied Project Kepler's Laws 


880  (1) 


881  (3) 


884  (2) 


886  (94) 

Functions of Several Variables 


887  (15) 


902  (7) 


909  (14) 

Tangent Planes and Linear Approximations 


923  (8) 


931  (9) 

Directional Derivatives and the Gradient Vector 


940  (13) 

Maximum and Minimum Values 


953  (12) 

Applied Project Designing a Dumpster 


963  (1) 

Discovery Project Quadratic Approximations and Critical Points 


964  (1) 


965  (15) 

Applied Project Rocket Science 


972  (1) 

Applied Project HydroTurbine Optimization 


973  (1) 


974  (4) 


978  (2) 


980  (74) 

Double Integrals over Rectangles 


981  (8) 


989  (6) 

Double Integrals over General Regions 


995  (8) 

Double Integrals in Polar Coordinates 


1003  (6) 

Applications of Double Integrals 


1009  (10) 


1019  (4) 


1023  (10) 

Discovery Project Volumes of Hyperspheres 


1032  (1) 

Triple Integrals in Cylindrical and Spherical Coordinates 


1033  (8) 

Applied Project Roller Derby 


1039  (1) 

Discovery Project The Intersection of Three Cylinders 


1040  (1) 

Change of Variables in Multiple Integrals 


1041  (13) 


1049  (3) 


1052  (2) 


1054  (86) 


1055  (7) 


1062  (12) 

The Fundamental Theorem for Line Integrals 


1074  (9) 


1083  (7) 


1090  (8) 

Parametric Surfaces and Their Areas 


1098  (11) 


1109  (12) 


1121  (6) 

Writing Project Three Men and Two Theorems 


1126  (1) 


1127  (7) 


1134  (6) 


1135  (3) 


1138  (2) 

SecondOrder Differential Equations 


1140  

SecondOrder Linear Equations 


1141  (6) 

Nonhomogeneous Linear Equations 


1147  (8) 

Applications of SecondOrder Differential Equations 


1155  (8) 


1163  


1167  
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  (10) 


49  (8) 

H Answers to OddNumbered Exercises 


57  (68) 
Index 

125  