Calculus

Preface To the Student To the Instructor Acknowledgments What Is Calculus? Preliminaries P.1 Real Numbers and the Real Line P.2 Cartesian Coordinates in the Plane P.3 Graphs of Quadratic Equations P.4 Functions and Their Graphs P.5 Combining Functions to Make New Functions P.6 Polynomials and Rational Functions P.7 The Trigonometric Functions 1. Limits and Continuity 1.1 Examples of Velocity, Growth Rate, and Area 1.2 Limits of Functions 1.3 Limits at Infinity and Infinite Limits 1.4 Continuity 1.5 The Formal Definition of Limit Chapter Review 2. Differentiation 2.1 Tangent Lines and Their Slopes 2.2 The Derivative 2.3 Differentiation Rules 2.4 The Chain Rule 2.5 Derivatives of Trigonometric Functions 2.6 The Mean-Value Theorem 2.7 Using Derivatives 2.8 Higher-Order Derivatives 2.9 Implicit Differentiation 2.10 Antiderivatives and Initial-Value Problems 2.11 Velocity and Acceleration Chapter Review 3. Transcendental Functions 3.1 Inverse Functions 3.2 Exponential and Logarithmic Functions 3.3 The Natural Logarithm and Exponential 3.4 Growth and Decay 3.5 The Inverse Trigonometric Functions 3.6 Hyperbolic Functions 3.7 Second-Order Linear DEs with Constant Coefficients Chapter Review 4. Some Applications of Derivatives 4.1 Related Rates 4.2 Extreme Values 4.3 Concavity and Inflections 4.4 Sketching the Graph of a Function 4.5 Extreme-Value Problems 4.6 Finding Roots of Equations 4.7 Linear Approximations 4.8 Taylor Polynomials 4.9 Indeterminate Forms Chapter Review 5. Integration 5.1 Sums and Sigma Notation 5.2 Areas as Limits of Sums 5.3 The Definite Integral 5.4 Properties of the Definite Integral 5.5 The Fundamental Theorem of Calculus 5.6 The Method of Substitution 5.7 Areas of Plane Regions Chapter Review 6. Techniques of Integration 6.1 Integration by Parts 6.2 Inverse Substitutions 6.3 Integrals of Rational Functions 6.4 Integration Using Computer Algebra or Tables 6.5 Improper Integrals 6.6 The Trapezoid and Midpoint Rules 6.7 Simpson's Rule 6.8 Other Aspects of Approximate Integration Chapter Review 7. Applications of Integration 7.1 Volumes by Slicing Solids of Revolution 7.2 More Volumes by Slicing 7.3 Arc Length and Surface Area 7.4 Mass, Moments, and Centres of Mass 7.5 Centroids 7.6 Other Physical Applications 7.7 Applications in Business, Finance, and Ecology 7.8 Probability 7.9 First-Order Differential Equations Chapter Review 8. Conics, Parametric Curves, and Polar Curves 8.1 Conics 8.2 Parametric Curves 8.3 Smooth Parametric Curves and Their Slopes 8.4 Arc Lengths and Areas for Parametric Curves 8.5 Polar Coordinates and Polar Curves 8.6 Slopes, Areas, and Arc Lengths for Polar Curves Chapter Review 9. Sequences, Series, and Power Series 9.1 Sequences and Convergence 9.2 Infinite Series 9.3 Convergence Tests for Positive Series 9.4 Absolute and Conditional Convergence 9.5 Power Series 9.6 Taylor and Maclaurin Series 9.7 Applications of Taylor and Maclaurin Series 9.8 The Binomial Theorem and Binomial Series 9.9 Fourier Series Chapter Review 10. Vectors and Coordinate Geometry in 3-Space 10.1 Analytic Geometry in Three Dimensions 10.2 Vectors 10.3 The Cross Product in 3-Space 10.4 Planes and Lines 10.5 Quadric Surfaces 10.6 A Little Linear Algebra 10.7 Using Maple for Vector and Matrix Calculations Chapter Review 11. Vector Functions and Curves 11.1 Vector Functions of One Variable 11.2 Some Applications of Vector Differentiation 11.3 Curves and Parametrizations 11.4 Curvature, Torsion, and the Frenet Frame 11.5 Curvature and Torsion for General Parametrizations 11.6 Kepler's Laws of Planetary Motion Chapter Review 12. Partial Differentiation 12.1 Functions of Several Variables 12.2 Limits and Continuity 12.3 Partial Derivatives 12.4 Higher-Order Derivatives 12.5 The Chain Rule 12.6 Linear Approximations, Differentiability, and Differentials 12.7 Gradients and Directional Derivatives 12.8 Implicit Functions 12.9 Taylor Series and Approximations Chapter Review 13. Applications of Partial Derivatives 13.1 Extreme Values 13.2 Extreme Values of Functions Defined on Restricted Domains 13.3 Lagrange Multipliers 13.4 The Method of Least Squares 13.5 Parametric Problems 13.6 Newton's Method 13.7 Calculations with Maple Chapter Review 14. Multiple Integration 14.1 Double Integrals 14.2 Iteration of Double Integrals in Cartesian Coordinates 14.3 Improper Integrals and a Mean-Value Theorem 14.4 Double Integrals in Polar Coordinates 14.5 Triple Integrals 14.6 Change of Variables in Triple Integrals 14.7 Applications of Multiple Integrals Chapter Review 15. Vector Fields 15.1 Vector and Scalar Fields 15.2 Conservative Fields 15.3 Line Integrals 15.4 Line Integrals of Vector Fields 15.5 Surfaces and Surface Integrals 15.6 Oriented Surfaces and Flux Integrals Chapter Review 16. Vector Calculus 16.1 Gradient, Divergence, and Curl 16.2 Some Identities Involving Grad, Div, and Curl 16.3 Green's Theorem in the Plane 16.4 The Divergence Theorem in 3-Space 16.5 Stokes's Theorem 16.6 Some Physical Applications of Vector Calculus 16.7 Orthogonal Curvilinear Coordinates Chapter Review 17. Ordinary Differential Equations 17.1 Classifying Differential Equations 17.2 Solving First-Order Equations 17.3 Existence, Uniqueness, and Numerical Methods 17.4 Differential Equations of Second Order 17.5 Linear Differential Equations with Constant Coefficients 17.6 Nonhomogeneous Linear Equations 17.7 Series Solutions of Differential Equations Chapter Review Appendix I Complex Numbers Appendix II Complex Functions Appendix III Continuous Functions Appendix IV The Riemann Appendix V Doing Calculus with Maple Answers to Odd-Numbered Exercises Index