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 b^x 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