9TH EDITION. George B. Thomas, Jr. Massachusetts Institute of Technology. Ross L. Finney. With the collaboration of Maurice D.

Transcription

1 9TH EDITION Calculus and Analytic Geometry George B. Thomas, Jr. Massachusetts Institute of Technology Ross L. Finney With the collaboration of Maurice D. Weir Naval Postgraduate School ^ Addison-Wesley Publishing Company Reading, Massachusetts Menlo Park, California New York Don Mills, Ontario Wokingham, England Amsterdam Bonn Sydney Singapore Tokyo Madrid San Juan Milan Paris

2 Contents To the Instructor To the Student viii xvii Real Numbers and the Real Line 1 Coordinates, Lines, and Increments Functions 17 Shifting Graphs 27 Trigonometric Functions 35 QUESTIONS TO GUIDE YOUR REVIEW 47 PRACTICE EXERCISES 48 ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS 49 DLimits and Continuity 1.1 Rates of Change and Limits Rules for Finding Limits Target Values and Formal Definitions of Limits 1.4 Extensions of the Limit Concept Continuity Tangent Lines 97 QUESTIONS TO GUIDE YOUR REVIEW 103 PRACTICE EXERCISES ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS DDerivatives 2.1 The Derivative of a Function Differentiation Rules Rates of Change Derivatives of Trigonometric Functions The Chain Rule Implicit Differentiation and Rational Exponents 2.7 Related Rates of Change 172 QUESTIONS TO GUIDE YOUR REVIEW 180 PRACTICE EXERCISES ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS BApplications of Derivatives 3.1 Extreme Values of Functions The Mean Value Theorem The First Derivative Test for Local Extreme Values 205 III

3 iv Contents 3.4 Graphing with / and y" Limits as x > ±», Asymptotes, and Dominant Terms Optimization Linearization and Differentials Newton's Method 260 QUESTIONS TO GUIDE YOUR REVIEW 268 PRACTICE EXERCISES 269 ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS 272 Integration 4.1 Indefinite Integrals Differential Equations, Initial Value Problems, and Mathematical Modeling 4.3 Integration by Substitution Running the Chain Rule Backward Estimating with Finite Sums Riemann Sums and Definite Integrals Properties, Area, and the Mean Value Theorem The Fundamental Theorem Substitution in Definite Integrals Numerical Integration 346 QUESTIONS TO GUIDE YOUR REVIEW 356 PRACTICE EXERCISES 357 ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS Applications of Integrals 5.1 Areas Between Curves Finding Volumes by Slicing Volumes of Solids of Revolution Disks and Washers Cylindrical Shells Lengths of Plane Curves Areas of Surfaces of Revolution Moments and Centers of Mass Work Fluid Pressures and Forces The Basic Pattern and Other Modeling Applications 434 QUESTIONS TO GUIDE YOUR REVIEW 444 PRACTICE EXERCISES ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS Transcendental Functions Inverse Functions and Their Derivatives 449 Natural Logarithms 458 The Exponential Function 467 a x and\og a x 474 Growth and Decay 482 L'Hopital's Rule 491 Relative Rates of Growth 498 Inverse Trigonometric Functions 504 Derivatives of Inverse Trigonometric Functions; Integrals Hyperbolic Functions 520 First Order Differential Equations 529 Euler's Numerical Method; Slope Fields 541 QUESTIONS TO GUIDE YOUR REVIEW 547 PRACTICE EXERCISES ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS Techniques of Integration 7.1 Basic Integration Formulas 7.2 Integration by Parts

4 Contents v 7.3 Partial Fractions Trigonometric Substitutions Integral Tables and CAS Improper Integrals 594 QUESTIONS TO GUIDE YOUR REVIEW 606 PRACTICE EXERCISES ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS Infinite Series Limits of Sequences of Numbers 613 Theorems for Calculating Limits of Sequences 622 Infinite Series 630 The Integral Test for Series of Nonnegative Terms 640 Comparison Tests for Series of Nonnegative Terms 644 The Ratio and Root Tests for Series of Nonnegative Terms Alternating Series, Absolute and Conditional Convergence Power Series 663 Taylor and Maclaurin Series 672 Convergence of Taylor Series; Error Estimates 678 Applications of Power Series QUESTIONS TO GUIDE YOUR REVIEW 699 PRACTICE EXERCISES ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS Conic Sections, Parametrized Curves, and Polar Coordinates Conic Sections and Quadratic Equations Classifying Conic Sections by Eccentricity Quadratic Equations and Rotations 728 Parametrizations of Plane Curves 734 Calculus with Parametrized Curves 744 Polar Coordinates 751 Graphing in Polar Coordinates 756 Polar Equations for Conic Sections 764 Integration in Polar Coordinates 770 QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES 778 ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS 783 Vectors and Wm Analytic A Geometry in Space Vectors in the Plane 787 Cartesian (Rectangular) Coordinates and Vectors in Space Dot Products 806 Cross Products 815 Lines and Planes in Space 822 Cylinders and Quadric Surfaces 829 Cylindrical and Spherical Coordinates 841 QUESTIONS TO GUIDE YOUR REVIEW 847 PRACTICE EXERCISES ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS Vector-Valued Functions and Motion in Space 11.1 Vector-Valued Functions and Space Curves Modeling Projectile Motion Arc Length and the Unit Tangent Vector T Curvature, Torsion, and the TNB Frame Planetary Motion and Satellites 893 QUESTIONS TO GUIDE YOUR REVIEW 902 PRACTICE EXERCISES ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS

5 vi Contents Multivariable Functions and Partial Derivatives 12.1 Functions of Several Variables Limits and Continuity Partial Derivatives Differentiability, Linearization, and Differentials The Chain Rule Partial Derivatives with Constrained Variables Directional Derivatives, Gradient Vectors, and Tangent Planes Extreme Values and Saddle Points Lagrange Multipliers Taylor's Formula 989 QUESTIONS TO GUIDE YOUR REVIEW 993 PRACTICE EXERCISES ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS Multiple Integrals 13.1 Double Integrals Areas, Moments, and Centers of Mass Double Integrals in Polar Form Triple Integrals in Rectangular Coordinates Masses and Moments in Three Dimensions Triple Integrals in Cylindrical and Spherical Coordinates 13.7 Substitutions in Multiple Integrals QUESTIONS TO GUIDE YOUR REVIEW 1055 PRACTICE EXERCISES ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS y[ Integration in Vector Fields 14.1 Line Integrals Vector Fields, Work, Circulation, and Flux Path Independence, Potential Functions, and Conservative Fields 14.4 Green's Theorem in the Plane Surface Area and Surface Integrals Parametrized Surfaces Stokes's Theorem The Divergence Theorem and a Unified Theory QUESTIONS TO GUIDE YOUR REVIEW 1134 PRACTICE EXERCISES 1134 ADDITIONAL EXERCISES THEORY, EXAMPLES, APPLICATIONS 1137 Appendices A.1 Mathematical Induction A-l A.2 Proofs of Limit Theorems in Section 1.2 A-4 A.3 Complex Numbers A-7 A.4 Simpson's One-Third Rule A-17 A.5 Cauchy's Mean Value Theorem and the Stronger Form of l'hopital's Rule A.6 Limits That Arise Frequently A-20 A.7 The Distributive Law for Vector Cross Products A-21 A.8 Determinants and Cramer's Rule A-22 A.9 Euler's Theorem and the Increment Theorem A-29 A-18 Answers A-35 Index 1-1 A Brief Table of Integrals T-1