Preface 

xiii  
To the Student 

xxiv  
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  


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  

Applied Project Radiation from the Stars 


785  (1) 


786  (3) 


789  
Appendixes 

1  (104) 

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  (8) 


47  (8) 

H Answers to OddNumbered Exercises 


55  (50) 
Index 

105  