Calculus with Analytic Geometry, Alternate
Note: Each chapter concludes with Review Exercises. The chapter organization of this supplement is different from that of Calculus: Early Transcendental Functions; users should work by topic rather than by chapter. Correlation guidelines are available. 1. The Cartesian Plane and Functions 1.1 Real Numbers and the Real Line; 1.2 The Cartesian Plane; 1.3 Graphs of Equations; 1.4 Lines in the Plane; 1.5 Functions 2. Limits and Their Properties 2.1 An Introduction to Limits; 2.2 Techniques for Evaluating Limits; 2.3 Continuity; 2.4 Infinite Limits; 2.5 e-d Definition of Limits 3. Differentiation 3.1 The Derivative and the Tangent Line Problem; 3.2 Velocity, Acceleration, and Other Rates of Change; 3.3 Differentiation Rules for Powers, Constant Multiples, and Sums; 3.4 Differentiation Rules for Products and Quotients; 3.5 The Chain Rule; 3.6 Implicit Differentiation; 3.7 Related Rates 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; 4.4 Concavity and the Second Derivative Test; 4.5 Limits at Infinity; 4.6 A Summary of Curve Sketching; 4.7 Optimization Problems; 4.8 Newton's Method; 4.9 Differentials; 4.10 Business and Economics Applications 5. Integration 5.1 Antiderivatives and Indefinite Integration; 5.2 Area; 5.3 Riemann Sums and the Definite Integral; 5.4 The Fundamental Theorem of Calculus; 5.5 Integration by Substitution; 5.6 Numerical Integration 6. Applications of Integration 6.1 Area of a Region Between Two Curves; 6.2 Volume: The Disc Method; 6.3 Volume: The Shell Method; 6.4 Arc Length and Surfaces of Revolution; 6.5 Work; 6.6 Fluid Pressure and Fluid Force; 6.7 Moments, Centers of Mass, and Centroids 7. Exponential and Logarithmic Functions 7.1 Exponential Functions; 7.2 Differentiation and Integration of Exponential Functions; 7.3 Inverse Functions; 7.4 Logarithmic Functions; 7.5 Logarithmic Functions and Differentiation; 7.6 Logarithmic Functions and Integration; 7.7 Growth and Decay; 7.8 Indeterminate Forms and L'Hopital's Rule 8. Trigonometric Functions and Inverse Trigonometric Functions 8.1 Review of Trigonometric Functions; 8.2 Graphs and Limits of Trigonometric Functions; 8.3 Derivatives of Trigonometric Functions; 8.4 Integrals of Trigonometric Functions; 8.5 Inverse Trigonometric Functions and Differentiation; 8.6 Inverse Trigonometric Functions: Integration and Completing the Square; 8.7 Hyperbolic Functions 9. Integration Techniques and Improper Integrals 9.1 Basic Integration Formulas; 9.2 Integration by Parts; 9.3 Trigonometric Integrals; 9.4 Trigonometric Substitution; 9.5 Partial Fractions; 9.6 Integration by Tables and Other Integration Techniques; 9.7 Improper Integrals 10. Infinite Series 10.1 Sequences; 10.2 Series and Convergence; 10.3 The Integral Test and p-Series; 10.4 Comparisons of Series; 10.5 Alternating Series; 10.6 The Ratio and Root Tests; 10.7 Taylor Polynomials and Approximations; 10.8 Power Series; 10.9 Representation of Functions by Power Series; 10.10 Taylor and Maclaurin Series 11. Conic Sections 11.1 Parabolas; 11.2 Ellipses; 11.3 Hyperbolas; 11.4 Rotation and the General Second-Degree Equation 12. Plane Curves, Parametric Equations, and Polar Coordinates 12.1 Plane Curves and Parametric Equations; 12.2 Parametric Equations and Calculus; 12.3 Polar Coordinates and Polar Graphs; 12.4 Tangent Lines and Curve Sketching in Polar Coordinates; 12.5 Area and Arc Length in Polar Coordinates; 12.6 Polar Equations for Conics and Kepler's Laws 13. Vectors and Curves in the Plane 13.1 Vectors in the Plane; 13.2 The Dot Product of Two Vectors; 13.3 Vector-Valued Functions; 13.4 Velocity and Acceleration; 13.5 Tangent Vectors and Normal Vectors; 13.6 Arc Length and Curvature 14. Solid Analytic Geometry and Vectors in Space 14.1 Space Coordinates and Vectors in Space; 14.2 The Cross Product of Two Vectors in Space; 14.3 Lines and Planes in Space; 14.4 Surfaces in Space; 14.5 Curves and Vector-Valued Functions in Space; 14.6 Tangent Vectors, Normal Vectors, and Curvature in Space 15. Functions of Several Variables 15.1 Introduction to Functions of Several Variables; 15.2 Limits and Continuity; 15.3 Partial Derivatives; 15.4 Differentials; 15.5 Chain Rules for Functions of Several Variables; 15.6 Directional Derivatives and Gradients; 15.7 Tangent Planes and Normal Lines; 15.8 Extrema of Functions of Two Variables; 15.9 Applications of Extrema of Functions of Two Variables; 15.10 Lagrange Multipliers 16. Multiple Integration 16.1 Iterated Integrals and Area in the Plane; 16.2 Double Integrals and Volume; 16.3 Change of Variables: Polar Coordinates; 16.4 Center of Mass and Moments of Inertia; 16.5 Surface Area; 16.6 Triple Integrals and Applications; 16.7 Cylindrical and Spherical Coordinates; 16.8 Triple Integrals in Cylindrical and Spherical Coordinates; 16.9 Change of Variables: Jacobians 17. Vector Analysis 17.1 Vector Fields; 17.2 Line Integrals; 17.3 Conservative Vector Fields and Independence of Path; 17.4 Green's Theorem; 17.5 Parametric Surfaces; 17.6 Surface Integrals; 17.7 Divergence Theorem; 17.8 Stokes's Theorem 18. Differential Equations 18.1 Definitions and Basic Concepts; 18.2 Separation of Variables in First-Order Equations; 18.3 Exact First-Order Equations; 18.4 First-Order Linear Differential Equations; 18.5 Second-Order Homogeneous Linear Equations; 18.6 Second-Order Nonhomogeneous Linear Equations; 18.7 Series Solutions of Differential Equations Appendixes: A. Proofs of Selected Theorems; B. Basic Differentiation Rules for Elementary Functions; C. Integration Tables Answers to Odd-Numbered Exercises