Table of Contents:

P. PRELIMINARIES, p.3 -- p.1 Real Numbers and the Real Line p.3 -- p.2 Cartesian Coordinates in the Plane, p.11 -- p.3 Graphs of Quadratic Equations, p.17 -- p.4 Functions and Their Graphs, p.23 -- p.5 Combining Functions to make New Functions, p.33 -- p.6 Polynomials and Rational Functions, p.39 -- p.7 The Trigonometric Functions, p.46 -- Chapter 1. LIMITS AND CONTINUITY, p.56 -- 1.1 Examples of Velocity, Growth Rate, and Area, p.59 -- 1.2 Limits of Functions, p.64 -- 1.3 Limits at Infinity and Infinite Limits, p.73 -- 1.4 Continuity, p.79 -- 1.5 The Formal Definition of Limit, p.88 -- Chapter 2. DIFFERENTIATION, p.95 -- 2.1 Tangent Lines and Their Slopes, p.95 -- 2.2 The Derivative, p.100 -- 2.3 Differentiation Rules, p.108 -- 2.4 The Chain Rule, p.116 -- 2.5 Derivatives of Trigonometric Functions, p.121 -- 2.6 Higher Order Derivatives, p.127 -- 2.7 Using Differentials and Derivatives, p.130 -- 2.8 The Mean-Value Theorem, p.137 -- 2.9 Implicit Differentiation, p.144 -- 2.10 Antiderivatives and Initial-Value Problems, 148 -- 2.11 Velocity and Acceleration, p.154 -- Chapter 3. TRANSCENDENTAL FUNCTIONS, p.163 -- 3.1 Inverse Functions, p.164 -- 3.2 Exponential Logarithmic Functions, p.170 -- 3.3 The Natural Logarithm and Exponential, p.174 -- 3.4 Growth and Decay, p.183 -- 3.5 The Inverse Trigonometric Functions, p.191 -- 3.6 Hyperbolic Functions, p.199 -- 3.7 Second-Order Linear DEs with Constant Coefficients, p.204 -- Chapter 4. MORE APPLICATIONS OF DIFFERENTIATION, p.214 -- 4.1 Related Rates, p.214 -- 4.2 Finding Roots of Equations, p.220 -- 4.3 Indeterminate Forms, p.228 -- 4.4 Extreme Values, p.233 -- 4.5 Concavity and Inflections, p.240 -- 4.6 Sketching the Graph of a Function, 245 -- 4.7 Graphing with Computers, p.253 -- 4.8 Extreme-Value Problems, p.259 -- 4.9 Linear Approximations, p.267 -- 4.10 Taylor Polynomials, p.272 -- 4.11 Roundoff Error, Truncation Error, and Computers, p.281.

Chapter 5. INTERGRATION, p.289 -- 5.1 Sums and Sigma Notation, p.289-- 5.2 Areas as Limits of Sums, p.294 -- 5.3 The Definite Integral, p.300 -- 5.4 Properties of the Definite Integral, p.305 -- 5.5 The Fundamental Theorem of Calculus, p.311 -- 5.6 The Method of Substitution, p.317 -- 5.7 Areas of Plane Regions, p.325 -- Chapter 6. TECHNIQUES OF INTEGRATION, p.332 -- 6.1 Integration by Parts, p.332 -- 6.2 Integrals of Rational Functions, p.338 -- 6.3 Inverse Substitutions, p.347 -- 6.4 Other Methods for Evaluating Integrals, p.354 -- 6.5 Improper Integrals, p.360 -- 6.6 The Trapezoid and Midpoint Rules, p.369 -- 6.7 Simpson's Rule, p.376 -- 6.8 Other Aspects of Approximate Integration, p.380 -- Chapter 7. APPLICATIONS OF INTEGRATION, p.391 -- 7.1 Volumes by Slicing-Solids of Revolution, p.391 -- 7.2 More Volumes by Slicing, p.400 -- 7.3 Arc Length and Surface Area, p.404 -- 7.4 Mass, Moments, and Centre of Mass, p.411 -- 7.5 Centroids, p.418 -- 7.6 Other Physical Applications, p.423 -- 7.7 Applications in Business, Finance, and Ecology, p.430 -- 7.8 Probability, p.434 -- 7.9 First-Order Differential Equations, p.446 -- Chapter 8. CONICS, PARAMETRIC CURVES, AND POLAR CURVES, p458 -- 8.1 Conics, p.458 -- 8.2 Parametric Curves, p.469 -- 8.3 Smooth Parametric Curves and Their Slopes, p.476 -- 8.4 Arc Lengths and Areas for Parametric Curves, p.479 -- 8.5 Polar Coordinates and Polar Curves, p.483 -- 8.6 Slopes, Areas, and Arc Lengths for Polar Curves, p.490

Chapter 9. SEQUENCES, SERIES, AND POWER SERIES, p.496 -- 9.1 Sequence and Convergence, p.469 -- 9.2 Infinite Series, p.504 -- 9.3 Convergence Tests for Positive Series, p.510 -- 9.4 Absolute and Conditional Convergence, p.521 -- 9.5 Power Series, p.527 -- 9.6 Taylor and Maclaurin Series, p.537 -- 9.7 Applications of Taylor and Maclaurin Series, p.546 -- 9.8 The Binomial Theorem and Binomial Series, p.550 -- 9.9 Fourier Series, p.554 -- Chapter 10. VECTORS AND COORDINATE GEOMETRY IN 3-SPACE, p.564 -- 10.1 Analytic Geometry in Three Dimensions, p.565 -- 10.2 Vectors, p.570 -- 10.3 The Cross Product in3-Space, p.580 -- 10.4 Planes and Lines, p.587 -- 10.5 Quadric Surfaces, p.595 -- 10.6 Cylindrical and Spherical Coordinates, p.598 -- 10.7 A Little Linear Algebra, p.602 -- 10.8 Using Maple for VEctor and Matrix Calculators, p.612 -- Chapter 11. VECTOR FUNCTIONS AND CURVES, p.623 -- 11.1 Vector Functions of One Variable, p.623 -- 11.2 Some Applications of Vector Differentiation, p.630 -- 11.3 Curves and Parametrizations, p.637 -- 11.4 Curvature, Torion, and the Frenet Frame, p.644 -- 11.5 Curvature and Torsion for General Parametrizations, p.651 -- 11.6 Kepler's Laws of Planetary Motion, p.659 -- Chapter 12. PARTIAL DIFFERENTIATION, p.671 -- 12.1. Functions of Several Variables, p.671 -- 12.2. Limits and Continuity, p.679 -- 12.3. Partial Derivatives, p.683 -- 12.4. Higher-Order Derivatives, p.690 -- 12.5. The Chain Rule, p.695 -- 12.6. Linear Approximations, Differentiability, and Differentials, p.705 -- 12.7. Gradients and Directional Derivatives, p.716 -- 12.8. Implicit Functions, p.727 -- 12.9. Taylor's Formula, Taylor Series, and Approximations, p.737 -- Chapter 13. APPLICATIONS OF PARTIAL DERIVATIVES, p.745 -- 13.1. Extreme Values, P.745 -- 13.2. Extreme Values of Functions Defined on Restricted Domains, p.753 -- 13.3. Lagrange Multipliers, p.766 -- 13.4. Lagrange Multipliers in n-Space, p.766 -- 13.5. The Method of Least Squares, p.775 -- 13.6. Parametric Problems, p.782 -- 13.7. Newton's Method, p.791 -- 13.8. Calculations with Maple, p.794.

13.9. Entropy in Statistical Mechanics and Information Theory, p.799 -- Chapter 14. MULTIPLE INTEGRATION, p.807 -- 14.1. Double Integrals, p.807 -- 14.2. Iteration of Double Integrals in Cartesian Coordinates, p.813 -- 14.3. Improper Integrals and a Mean-Value Theorem, p.820 -- 14.4. Double Integrals in Polar Coordinates, p.825 -- 14.5. Triple Integrals, p.835 -- 14.6. Change of Variables in Triple Integrals, p.841 -- 14.7. Applications of Multiple Integrals, p.847 -- Chapter 15. VECTOR FIELDS, p.859 -- 15.1. Vector and Scalar Fields, p.859 -- 15.2. Conservative Fields, p.866 -- 15.3. Line Integrals, p.875 -- 15.4. Line Integrals of Vector Fields, p.879 -- 15.5. Surfaces and Surface Integrals, p.887 -- 15.6. Oriented Surfaces and Flux Integrals, p.898 -- Chapter 16. VECTOR CALCULUS, p.906 -- 16.1. Gradient. Divergence, and Curl, p.906 -- 16.2. Some Identities Involving Grad, Div, and Curl, p.914 -- 16.3. Green's Theorem in the Plane, p.921 -- 16.4. The Divergence Theorem in 3-Space, p.925 -- 16.5. Stokes's Theorem, p.931 -- 16.6. Some Physical Applications of Vector Calculus, p.935 -- 16.7. Orthogonal Curvilinear Coordinates, p.943 -- Chapter 17. DIFFERENTIAL FORMS AND EXTERIOR CALCULUS, p.955 -- 17.1. k-Forms, p.956 -- 17.2. Differential Forms and the Exterior Derivative, p.962 -- 17.3. Integration on Manifolds, p.969 -- 17.4. Orientations, Boundaries, and Integration of Forms, p.975 -- 17.5. The Generalized Stokes Theorem, p.982 -- Chapter 18. ORDINARY DIFFERENTIAL EQUATIONS, p.990 -- 18.1. Classifying Differential Equations, p.991 -- 18.2. Solving First-Order Equations, p.994 -- 18.3. Existence, Uniqueness, and Numerical Methods, p.999 -- 18.4. Differential Equations of Second Order, p.1007 -- 18.5. Linear Differential Equations with Constant Coefficients, p.1010 -- 18.6, Nonhomogeneous Linear Equations, p.1014 -- 18.7. Series Solutions of Differential Equations, p.1021.