Calculus

Note: Each chapter includes Review Exercises and P.S. Problem Solving. 1. Preparation for Calculus 1.1 Graphs and Models 1.2 Linear Models and Rates of Change 1.3 Functions and Their Graphs 1.4 Fitting Models to Data 1.5 Inverse Functions 1.6 Exponential and Logarithmic Functions 2. Limits and Their Properties 2.1 A Preview of Calculus 2.2 Finding Limits Graphically and Numerically 2.3 Evaluating Limits Analytically 2.4 Continuity and One-Sided Limits 2.5 Infinite Limits Section Project: Graphs and Limits of Trigonometric Functions 3. Differentiation 3.1 The Derivative and the Tangent Line Problem 3.2 Basic Differentiation Rules and Rates of Change 3.3 Product and Quotient Rules and Higher-Order Derivatives 3.4 The Chain Rule 3.5 Implicit Differentiation Section Project: Optical Illusions 3.6 Derivatives of Inverse Functions 3.7 Related Rates 3.8 Newton's Method 4. Applications of Differentiation 4.1 Extrema on an Interval 4.2 Rolle's Theorem and the Mean Value Theorem 4.3 Increasing and Decreasing Functions and the First Derivative Test Section Project: Rainbows 4.4 Concavity and the Second Derivative Test 4.5 Limits at Infinity 4.6 A Summary of Curve Sketching 4.7 Optimization Problems Section Project: Connecticut River 4.8 Differentials 5. Integration 5.1 Antiderivatives and Indefinite Integration 5.2 Area 5.3 Riemann Sums and Definite Integrals 5.4 The Fundamental Theorem of Calculus Section Project: Demonstrating the Fundamental Theorem 5.5 Integration by Substitution 5.6 Numerical Integration 5.7 The Natural Logarithmic Function: Integration 5.8 Inverse Trigonometric Functions: Integration 5.9 Hyperbolic Functions Section Project: St. Louis Arch 6. Differential Equations 6.1 Slope Fields and Euler's Method 6.4 Differential Equations: Growth and Decay 6.5 Differential Equations: Separation of Variables 6.4 The Logistic Equation 6.5 First-Order Linear Differential Equations Section Project: Weight Loss 6.6 Predator-Prey Differential Equations 7. Applications of Integration 7.1 Area of a Region Between Two Curves 7.2 Volume: The Disk Method 7.3 Volume: The Shell Method Section Project: Saturn 7.4 Arc Length and Surfaces of Revolution 7.5 Work Section Project: Tidal Energy 7.6 Moments, Centers of Mass, and Centroids 7.7 Fluid Pressure and Fluid Force 8. Integration Techniques, L'Hopital's Rule, and Improper Integrals 8.1 Basic Integration Rules 8.2 Integration by Parts 8.3 Trigonometric Integrals Section Project: Power Lines 8.4 Trigonometric Substitution 8.5 Partial Fractions 8.6 Integration by Tables and Other Integration Techniques 8.7 Indeterminate Forms and L'Hopital's Rule 8.8 Improper Integrals 9. Infinite Series 9.1 Sequences 9.2 Series and Convergence Section Project: Cantor's Disappearing Table 9.3 The Integral Test and p-Series Section Project: The Harmonic Series 9.4 Comparisons of Series Section Project: Solera Method 9.5 Alternating Series 9.6 The Ratio and Root Tests 9.7 Taylor Polynomials and Approximations 9.8 Power Series 9.9 Representation of Functions by Power Series 9.10 Taylor and Maclaurin Series 10. Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 10.2 Plane Curves and Parametric Equations Section Projects: Cycloids 10.3 Parametric Equations and Calculus 10.4 Polar Coordinates and Polar Graphs Section Project: Anamorphic Art 10.5 Area and Arc Length in Polar Coordinates 10.6 Polar Equations of Conics and Kepler's Laws 11. Vectors and the Geometry of Space 11.1 Vectors in the Plane 11.2 Space Coordinates and Vectors in Space 11.3 The Dot Product of Two Vectors 11.4 The Cross Product of Two Vectors in Space 11.5 Lines and Planes in Space Section Project: Distances in Space 11.6 Surfaces in Space 11.7 Cylindrical and Spherical Coordinates 12. Vector-Valued Functions 12.1 Vector-Valued Functions Section Project: Witch of Agnesi 12.2 Differentiation and Integration of Vector-Valued Functions 12.3 Velocity and Acceleration 12.4 Tangent Vectors and Normal Vectors 12.5 Arc Length and Curvature 13. Functions of Several Variables 13.1 Introduction to Functions of Several Variables 13.2 Limits and Continuity 13.3 Partial Derivatives Section Project: Moire Fringes 13.4 Differentials 13.5 Chain Rules for Functions of Several Variables 13.6 Directional Derivatives and Gradients 13.7 Tangent Planes and Normal Lines Section Project: Wildflowers 13.8 Extrema of Functions of Two Variables 13.9 Applications of Extrema of Functions of Two Variables Section Project: Building a Pipeline 13.10 Lagrange Multipliers 14. Multiple Integration 14.1 Iterated Integrals and Area in the Plane 14.2 Double Integrals and Volume 14.3 Change of Variables: Polar Coordinates 14.4 Center of Mass and Moments of Inertia Section Project: Center of Pressure on a Sail 14.5 Surface Area Section Project: Capillary Action 14.6 Triple Integrals and Applications 14.7 Triple Integrals in Cylindrical and Spherical Coordinates Section Project: Wrinkled and Bumpy Spheres 14.8 Change of Variables: Jacobians 15. Vector Analysis 15.1 Vector Fields 15.2 Line Integrals 15.3 Conservative Vector Fields and Independence of Path 15.4 Green's Theorem Section Project: Hyperbolic and Trigonometric Functions 15.5 Parametric Surfaces 15.6 Surface Integrals Section Project: Hyperboloid of One Sheet 15.7 Divergence Theorem 15.8 Stoke's Theorem Section Project: The Planimeter Appendices Appendix A Proofs of Selected Theorems Appendix B Integration Tables Appendix C Business and Economic Applications Additional Appendices The following appendices are available at the textbook website, on the HM mathSpace Student CD-ROM, and the HM ClassPrep with HM Testing CD-ROM: Appendix D Precalculus Review Appendix E Rotation and General Second-Degree Equation Appendix F Complex Numbers