A PREVIEW OF CALCULUS 1
1 Functions and Models 9
1.1 Four Ways to Represent a Function 10
1.2 Mathematical Models:A Catalog of Essential Functions 23
1.3 New Functions from Old Functions 36
1.4 Graphing Calculators and Computers 44
1.5 Exponential Functions 51
1.6 Inverse Functions and Logarithms 58
Review 72
Principles of Problem Solving 75
2 Limits and Derivatives 81
2.1 The Tangent and Velocity Problems 82
2.2 The Limit of a Function 87
2.3 Calculating Limits Using the Limit Laws 99
2.4 The Precise Definition of a Limit 108
2.5 Continuity 118
2.6 Limits at Infinity;Horizontal Asymptotes 130
2.7 Derivatives and Rates of Change 143
Writing Project Early Methods for Finding Tangents 153
2.8 The Derivative as a Function 154
Review 165
Problems Plus 170
3 Differentiation Rules 173
3.1 Derivatives of Polynomials and Exponential Functions 174
Applied Project Building a Better Roller Coaster 184
3.2 The Product and Quotient Rules 184
3.3 Derivatives of Trigonometric Functions 191
3.4 The Chain Rule 198
Applied Project Where Should a Pilot Start Descent? 208
3.5 Implicit Differentiation 209
Laboratory Project Families of Implicit Curves 217
3.6 Derivatives of Logarithmic Functions 218
3.7 Rates of Change in the Natural and Social Sciences 224
3.8 Exponential Growth and Decay 237
3.9 Related Rates 244
3.10 Linear Approximations and Differentials 250
Laboratory Project Taylor Polynomials 256
3.11 Hyperbolic Functions 257
Review 264
Problems Plus 268
4 Applications of Differentiation 273
4.1 Maximum and Minimum Values 274
Applied Project The Calculus of Rainbows 282
4.2 The Mean Value Theorem 284
4.3 How Derivatives Affect the Shape of a Graph 290
4.4 Indeterminate Forms and l'Hospital's Rule 301
Writing Project The Origins of I'Hospital's Rule 310
4.5 Summary of Curve Sketching 310
4.6 Graphing with Calculus and Calculators 318
4.7 Optimization Problems 325
Applied Project The Shape of a Can 337
4.8 Newton's Method 338
4.9 Antiderivatives 344
Review 351
Problems Plus 355
5 Integrals 359
5.1 Areas and Distances 360
5.2 The Definite Integral 371
Discovery Project Area Functions 385
5.3 The Fundamental Theorem of Calculus 386
5.4 Indefinite Integrals and the Net Change Theorem 397
Writing Project Newton,Leibniz,and the Invention of Calculus 406
5.5 The Substitution Rule 407
Review 415
Problems Plus 419
6 Applications of Integration 421
6.1 Areas Between Curves 422
Applied Project The Gini Index 429
6.2 Volumes 430
6.3 Volumes by Cylindrical Shells 441
6.4 Work 446
6.5 Average Value of a Function 451
Applied Project Calculus and Baseball 455
Applied Project Where to Sit at the Movies 456
Review 457
Problems Plus 459
7 Techniques of Integration 463
7.1 Integration by Parts 464
7.2 Trigonometric Integrals 471
7.3 Trigonometric Substitution 478
7.4 Integration of Rational Functions by Partial Fractions 484
7.5 Strategy for Integration 494
7.6 Integration Using Tables and Computer Algebra Systems 500
Discovery Project Patterns in Integrals 505
7.7 Approximate Integration 506
7.8 Improper Integrals 519
Review 529
Problems Plus 533
8 Further Applications of Integration 537
8.1 Arc Length 538
Discovery Project Arc Length Contest 545
8.2 Area of a Surface of Revolution 545
Discovery Project Rotating on a Slant 551
8.3 Applications to Physics and Engineering 552
Discovery Project Complementary Coffee Cups 562
8.4 Applications to Economics and Biology 563
8.5 Probability 568
Review 575
Problems Plus 577
9 Differential Equations 579
9.1 Modeling with Differential Equations 580
9.2 Direction Fields and Euler's Method 585
9.3 Separable Equations 594
Applied Project How Fast Does a Tank Drain? 603
Applied Project Which Is Faster,Going Up or Coming Down? 604
9.4 Models for Population Growth 605
9.5 Linear Equations 616
9.6 Predator-Prey Systems 622
Review 629
Problems Plus 633
10 Parametric Equations and Polar Coordinates 635
10.1 Curves Defined by Parametric Equations 636
Laboratory Project Running Circles around Circles 644
10.2 Calculus with Parametric Curves 645
Laboratory Project Bézier Curves 653
10.3 Polar Coordinates 654
Laboratory Project Families of Polar Curves 664
10.4 Areas and Lengths in Polar Coordinates 665
10.5 Conic Sections 670
10.6 Conic Sections in Polar Coordinates 678
Review 685
Problems Plus 688