Calculus and Analytic Geometry.

Calculus and Analytic Geometry 1. An Overview of Calculus. 1. 1 The Derivative 1. 2 The Integral 1. 3 Survey of the Text 2. Functions, Limits, and Continuity. 2. 1 Functions 2. 2 Composite Functions 2. 3 The Limit of a Function 2. 4 Computations of Limits 2. 5 Some Tools for Graphing 2. 6 A Review of Trigonometry 2. 7 The Limit of (sin A) /A as A Approaches 0 2. 8 Continuous Functions 2. 9 Precise Definitions of lim (x->infinity) f (x) =infinity and lim (x->infinity) f (x) =L 2. 10 Precise Definition of lim (x->a) f (x) =L 2. S Summary 3. The Derivative. 3. 1 Four Problems with One Theme 3. 2 The Derivative 3. 3 The Derivative and Continuity 3. 4 The Derivative of the Sum, Difference, Product, and Quotient 3. 5 The Derivatives of the Trigonometric Functions 3. 6 The Derivative of a Composite Function 3. S Summary 4. Applications of the Derivative. 4. 1 Three Theorems about the Derivative 4. 2 The First Derivative and Graphing 4. 3 Motion and the Second Derivative 4. 4 Related Rates 4. 5 The Second Derivative and Graphing 4. 6 Newton's Method for Solving an Equation 4. 7 Applied Maximum and Minimum Problems 4. 9 The Differential and Linearization 4. 10 The Second Derivative and Growth of a Function 4. S Summary 5. The Definite Integral. 5. 1 Estimates in Four Problems 5. 2 Summation Notation and Approximating Sums 5. 3 The Definite Integral 5. 4 Estimating the Definite Integral 5. 5 Properties of the Antiderivative and the Definite Integral 5. 6 Background for the Fundamental Theorems of Calculus 5. 7 The Fundamental Theorems of Calculus 5. S Summary 6. Topics in Differential Calculus. 6. 1 Logarithms 6. 2 The Number e 6. 3 The Derivative of a Logarithmic Function 6. 4 One-to-One Functions and Their Inverses 6. 5 The Derivative of bˣ 6. 6 The Derivatives of the Inverse Trigonometric Functions 6. 7 The Differential Equation of Natural Growth and Decay 6. 8 l'Hopital's Rule 6. 9 The Hyperbolic Functions and Their Inverses 6. S Summary 7. Computing Antiderivatives. 7. 1 Shortcuts, Integral Tables, and Machines 7. 2 The Substitution Method 7. 3 Integration by Parts 7. 4 How to Integrate Certain Rational Functions 7. 5 Integration of Rational Functions by Partial Fractions 7. 6 Special Techniques 7. 7 What to Do in the Face of an Integral 7. S Summary 8. Applications of the Definite Integral. 8. 1 Computing Area by Parallel Cross Sections 8. 2 Some Pointers on Drawing 8. 3 Setting Up a Definite Integral 8. 4 Computing Volumes 8. 5 The Shell Method 8. 6 The Centroid of a Plane Region 8. 7 Work 8. 8 Improper Integrals 8. S Summary 9. Plane Curves and Polar Coordinates. 9. 1 Polar Coordinates 9. 2 Area in Polar Coordinates 9. 3 Parametric Equations 9. 4 Arc Length and Speed on a Curve 9. 5 The Area of a Surface of Revolution 9. 6 Curvature 9. 7 The Reflection Properties of the Conic Sections 9. S Summary 10. Series. 10. 1 An Informal Introduction to Series 10. 2 Sequences 10. 3 Series 10. 4 The Integral Test 10. 5 Comparison Tests 10. 6 Ratio Tests 10. 7 Tests for Series with Both Positive and Negative Terms 10. S Summary 11. Power Series and Complex Numbers. 11. 1 Taylor Series 11. 2 The Error in Taylor Series 11. 3 Why the Error in Taylor Series Is Controlled by a Derivative 11. 4 Power Series and Radius of Convergence 11. 5 Manipulating Power Series 11. 6 Complex Numbers 11. 7 The Relation between the Exponential and the Trigonometric Functions 11. S Summary 12. Vectors. 12. 1 The Algebra of Vectors 12. 2 Projections 12. 3 The Dot Product of Two Vectors 12. 4 Lines and Planes 12. 5 Determinants 12. 6 The Cross Product of Two Vectors 12. 7 More on Lines and Planes 12. S Summary 13. The Derivative of a Vector Function. 13. 1 The Derivative of a Vector Function 13. 2 Properties of the Derivative of a Vector Function 13. 3 The Acceleration Vector 13. 4 The Components of Acceleration 13. 5 Newton's Law Implies Kepler's Laws 13. S Summary 14. Partial Derivatives. 14. 1 Graphs 14. 2 Quadratic Surfaces 14. 3 Functions and Their Level Curves 14. 4 Limits and Continuity 14. 5 Partial Derivatives 14. 6 The Chain Rule 14. 7 Directional Derivatives and the Gradient 14. 8 Normals and the Tangent Plane 14. 9 Critical Points and Extrema 14. 10 Lagrange Multipliers 14. 11 The Chain Rule Revisited 14. S Summary 15. Definite Integrals over Plane and Solid Regions. 15. 1 The Definite Integral of a Function over a Region in the Plane 15. 2 Computing |R f (P) dA Using Rectangular Coordinates 15. 3 Moments and Centers of Mass 15. 4 Computing |R f (P) dA Using Polar Coordinates 15. 5 The Definite Integral of a Function over a Region in Space 15. 6 Computing |R f (P) dV Using Cylindrical Coordinates 15. 7 Computing |R f (P) dV Using Spherical Coordinates 15. S Summary 16. Green's Theorem. 16. 1 Vector and Scalar Fields 16. 2 Line Integrals 16. 3 Four Applications of Line Integrals 16. 4 Green's Theorem 16. 5 Applications of Green's Theorem 16. 6 Conservative Vector Fields 16. S Summary 17. The Divergence Theorem and Stokes' Theorem. 17. 1 Surface Integrals 17. 2 The Divergence Theorem 17. 3 Stokes' Theorem 17. 4 Applications of Stokes' Theorem 17. S Summary Appendices: A. Real Numbers. B. Graphs and Lines. C. Topics in Algebra. D. Exponents. E. Mathematical Induction. F. The Converse of a Statement. G. Conic Sections. H. Logarithms and Exponentials Defined through Calculus. I. The Taylor Series for f (x, y). J. Theory of Limits. K. The Interchange of Limits. L. The Jacobian. M. Linear Differential Equations with Constant Coefficients. Answers to Selected Odd-Numbered Problems and to Guide Quizzes List of Symbols Index