Preface x A Tribute to James Stewart xxii About the Authors xxiii Technology in the Ninth Edition xxiv To the Student xxv Diagnostic Tests xxvi A Preview of Calculus 1. Functions and Models 1.1 Inverse Functions and Logarithms 8 2. Limits and Derivatives 2.1 The Limit of a Function 32 2.2 Calculating Limits Using the Limit Laws 44 2.3 The Precise Definition of a Limit 54 2.4 Continuity 64 2.5 Limits at Infinity; Horizontal Asymptotes 76 2.6 Derivatives and Rates of Change 89 2.7 The Derivative as a Function 102 3. Differentiation Rules 3.1 Derivatives of Polynomials and Exponential Functions 124 3.2 The Product and Quotient Rules 135 3.3 Derivatives of Trigonometric Functions 141 3.4 The Chain Rule 149 3.5 Implicit Differentiation 159 3.6 Derivatives of Logarithmic and Inverse Trigonometric Functions 167 3.7 Linear Approximations and Differentials 175 3.8 Hyperbolic Functions 183 4. Applications of Differentiation 4.1 Maximum and Minimum Values 202 4.2 The Mean Value Theorem 212 4.3 What Derivatives Tell Us about the Shape of a Graph 218 4.4 Indeterminate Forms and I’Hospital‘s Rule 231 4.5 Optimization Problems 242 4.6 Antiderivatives 257 5. Integrals 5.1 The Area and Distance Problems 274 5.2 The Definite Integral 286 5.3 The Fundamental Theorem of Calculus 301 5.4 Indefinite Integrals and the Net Change Theorem 3 11 5.5 The Substitution Rule 321 6. Applications of Integration 6.1 Areas Between Curves 338 6.2 Volumes 348 6.3 Volumes by Cylindrical Shells 362 6.4 Average Value of a Function 369 7. Techniques of Integration 7.1 Integration by Parts 382 7.2 Trigonometric Integrals 389 7.3 Trigonometric Substitution 396 7.4 Integration of Rational Functions by Partial Fractions 403 7.5 Improper Integrals 413 8. Further Applications of Integration 8.1 Arc Length 430 8.2 Area of a Surface of Revolution 437 8.3 Applications to Physics and Engineering 446 8.4 Applications to Economics and Biology 457 9. Parametric Equations and Polar Coordinates 9.1 Curves Defined by Parametric Equations 468 9.2 Calculus with Parametric Curves 479 9.3 Polar Coordinates 490 9.4 Calculus in Polar Coordinates 500 9.5 Conic Sections 508 10. Sequences, Series, and Power Series 10.1 Sequences 524 10.2 Series 538 10.3 The Integral Test and Estimates of Sums 551 10.4 The Comparison Tests 560 10.5 Alternating Series and Absolute Convergence 565 10.6 The Ratio and Root Tests 576 10.7 Power Series 579 10.8 Representations of Functions as Power Series 584 10.9 Taylor and Maclaurin Series 593 10.10 Applications of Taylor Polynomials 610 11. Vectors and the Geometry of Space 11.1 Three-Dimensional Coordinate Systems 628 11.2 Vectors 634 11.3 The Dot Product 645 11.4 The Cross Product 653 11.5 Equations of Lines and Planes 662 11.6 Cylinders and Quadric Surfaces 673 12. Vector Functions 12.1 Vector Functions and Space Curves 688 12.2 Derivatives and Integrals of Vector Functions 696 12.3 Arc Length and Curvature 702 12.4 Motion in Space: Velocity and Acceleration 714 13. Partial Derivatives 13.1 Functions of Several Variables 732 13.2 Limits and Continuity 749 13.3 Partial Derivatives 759 13.4 Tangent Planes and Linear Approximations 772 13.5 The Chain Rule 783 13.6 Directional Derivatives and the Gradient Vector 792 13.7 Maximum and Minimum Values 806 13.8 Lagrange Multipliers 818 14. Multiple Integrals 14.1 Double Integrals over Rectangles 836 14.2 Double Integrals over General Regions 849 14.3 Double Integrals in Polar Coordinates 860 14.4 Applications of Double Integrals 867 14.5 Surface Area 877 14.6 Triple Integrals 880 14.7 Triple Integrals in Cylindrical Coordinates 893 14.8 Triple Integrals in Spherical Coordinates 900 14.9 Change of Variables in Multiple Integrals 907 15. Vector Calculus 15.1 Vector Fields 922 15.2 Line Integrals 929 15.3 The Fundamental Theorem for Line Integrals 942 15.4 Green’s Theorem 952 15.5 Curl and Divergence 959 15.6 Parametric Surfaces and Their Areas 968 15.7 Surface Integrals 980 15.8 Stokes' Theorem 993 15.9 The Divergence Theorem 999 Appendixes A Trigonometry A2 B Proofs of Theorems A14 C Answers to Odd-Numbered Exercises A26 Index A87