Preface　 xiv
To the Student　 xxvi
A Preview of Calculus　 2
1 Functions and Hodels　 10
1.1　　Four Ways to Represent a Function　 11
1.2　　Mathematical Models: A Catalog of Essential Functions　 25
i.3　　New Functions from Old Functions　 38
1.4　　Graphing Calculators and Computers　 48
1.5　　Exponential Functions　 55
1.6　　Inverse Functions and Logarithms　 63
Review　 77
Principles of Problem Solving　 80
2 Limils and Derivatives　 8G
2.1　　The Tangent and Velocity Problems　 87
2.2　 The Limit of a Function　 92
2.3　 Calculating Limits Using the Limit Laws　 104
2.4　　The Precise Definition of a Limit　 114
2.5　 Continuity　 124
2.6　 Limits at Infinity; Horizontal Asymptotes　 135
2.7　　Tangents， Velocities， and Other Rates of Change　 149
2.8　　Derivatives　 158
Writing Project o Early Methods for Finding Tangents　 164
2.9　　The Derivative as a Function　 165
Review　 176
Problems Plus　 180
3 Lifferenliatiun Rules　 182
3.1　　Derivatives of Polynomials and Exponential Functions　 183
3.2　　The Product and Quotient Rules　 192
3.3　　Rates of Change in the Natural and Social Sciences　 199
3.4　　Derivatives of Trigonometric Functions　 211
3.5　　The Chain Rule　 217
3.6　 Implicit Differentiation　 227
3.7　　Higher Derivatives　 236
Applied Project o Where Should a Pilot Start Descent?　 243
Applied Project o Building a Better Roller Coaster　 243
3.8　　Derivatives of Logarithmic Functions　 244
3.9　　Hyperbolic Functions　 250
3.10　 Related Rates　 256
3.11　 Linear Approximations and Differentials　 262
Laboratory Project o Taylor Polynomials　 269
Review　 270
Problems Plus　 274
4 Applications of gifferenliulion　 278
4.1　　Maximum and Minimum Values　 279
Applied Project o The Calculus of Rainbows　 288
4.2　　The Mean Value Theorem　 290
4.3　　How Derivatives Affect the Shape of a Graph　 296
4.4　　Indeterminate Forms and LHospitals Rule　 307
Writing Project o The Origins of LHospitars Rule　 315
4.5　　Summary of Curve Sketching　 316
4.6　　Graphing with Calculus and Calculators　 324
4.7　　Optimization Problems　 331
Applied Project o The Shape of a Can　 341
4.8　　Applications to Business and Economics　 342
4.9　　Newtons Method　 347
4.10　 Antiderivatives　 353
Review　 361
Problems Plus　 365
5 Inteorals　 360
5.1　　Areas and Distances　 369
5.2　 The Definite Integral　 380
Discovery Project o Area Functions　 393
5.3　　The Fundamental Theorem of Calculus　 394
5.4　　Indefinite Integrals and the Net Change Theorem　 405
Writing Project o Newton， keibniz， and the Invention of Calculus　 413
5.5　 The Substitution Rule　 414
5.6　　The Logarithm Defined as an Integral　 422
Review　 430
Problems Plus　 434
6 Applicotions of Inteoration　 436
6.1　　Areas between Curves　 437
6.2　　Volumes　 444
6.3　　Volumes by Cylindrical Shells　 455
6.4　　Work　 460
6.5　　Average Value of a Function　 464
Applied Project o Where to Sit at the Movies　 468
Review　 468
Problems Plus　 470
7 Techniques of Integration　 474
7.1　　Integration by Parts　 475
7.2　　Trigonometric Integrals　 482
7.3　　Trigonometric Substitution　 489
7.4　　Integration of Rational Functions by Partial Fractions　 496
7.5　　Strategy for Integration　 505
7.6　　Integration Using Tables and Computer Algebra Systems　511
Discovery Project o Patterns in Integrals　 517
7.7　　Approximate Integration　 518
7.8　　Improper Integrals　 530
Review　 540
Problems Plus　 543
8 Further Applications of Inteoralion　 546
8.1　　Arc Length　 547
Discovery Project o Arc Length Contest　 554
8.2　　Area of a Surface of Revolution　 554
Discovery Project o Rotating on a Slant　 560
8.3　　Applications to Physics and Engineering　 561
8.4　　Applications to Economics and Biology　 571
8.5　 Probability　575
Review　 582
Problems Plus　 584
9 Diffeiential Equations　 586
9.1　　Modeling with Differential Equations　 587
9.2　　Direction Fields and Eulers Method　 592
9.3　　Separable Equations　 601
Applied Project　How Fast Does a Tank Drain?　 609
Applied Project　Which Is Faster， Going Up or Coming Down?　 610
9.4　　Exponential Growth and Decay　 611
Applied Project　Calculus and Baseball　 622
……
10 Parametric Equations and Polar Coordinates
11 Infinite Sequences and Series
12 Vectors and the Geometru of Space
13 Vector Functions
14 Partial Derivatives
15 Multiple Integrals
16 Vector Calculus
17 Second-Order Offerential Equations
Appendixes
Index