EBOOK: Calculus: Early Transcendental Functions

Chapter 0: Preliminaries 0.1 Polynomials and Rational Functions 0.2 Graphing Calculators and Computer Algebra Systems 0.3 Inverse Functions 0.4 Trigonometric and Inverse Trigonometric Functions 0.5 Exponential and Logarithmic Functions 0.6 Transformations of Functions Chapter 1: Limits and Continuity 1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve 1.2 The Concept of Limit 1.3 Computation of Limits 1.4 Continuity and Its Consequences 1.5 Limits Involving Infinity; Asymptotes 1.6 Formal Definition of the Limit 1.7 Limits and Loss-of-Significance Errors Chapter 2: Differentiation 2.1 Tangent Lines and Velocity 2.2 The Derivative 2.3 Computation of Derivatives: The Power Rule 2.4 The Product and Quotient Rules 2.5 The Chain Rule 2.6 Derivatives of Trigonometric Functions 2.7 Derivatives of Exponential and Logarithmic Functions 2.8 Implicit Differentiation and Inverse Trigonometric Functions 2.9 The Hyperbolic Functions 2.10 The Mean Value Theorem Chapter 3: Applications of the Derivative 3.1 Linear Approximations and Newton's Method 3.2 Indeterminate Forms and L'Hôpital's Rule 3.3 Maximum and Minimum Values 3.4 Increasing and Decreasing Functions 3.5 Concavity and the Second Derivative Test 3.6 Overview of Curve Sketching 3.7 Optimization 3.8 Related Rates 3.9 Rates of Change in Economics and the Sciences Chapter 4: Integration 4.1 Antiderivatives 4.2 Sums and Sigma Notation 4.3 Area 4.4 The Definite Integral 4.5 The Fundamental Theorem of Calculus 4.6 Integration by Substitution 4.7 Numerical Integration 4.8 The Natural Logarithm as an Integral Chapter 5: Applications of the Definite Integral 5.1 Area Between Curves 5.2 Volume: Slicing, Disks and Washers 5.3 Volumes by Cylindrical Shells 5.4 Arc Length and Surface Area 5.5 Projectile Motion 5.6 Applications of Integration to Physics and Engineering 5.7 Probability Chapter 6: Integration Techniques 6.1 Review of Formulas and Techniques 6.2 Integration by Parts 6.3 Trigonometric Techniques of Integration 6.4 Integration of Rational Functions Using Partial Fractions 6.5 Integration Tables and Computer Algebra Systems 6.6 Improper Integrals Chapter 7: First-Order Differential Equations 7.1 Modeling with Differential Equations 7.2 Separable Differential Equations 7.3 Direction Fields and Euler's Method 7.4 Systems of First-Order Differential Equations Chapter 8: Infinite Series 8.1 Sequences of Real Numbers 8.2 Infinite Series 8.3 The Integral and Comparison Tests 8.4 Alternating Series 8.5 Absolute Convergence and the Ratio Test 8.6 Power Series 8.7 Taylor Series 8.8 Applications of Taylor Series 8.9 Fourier Series Chapter 9: Parametric Equations and Polar Coordinates 9.1 Plane Curves and Parametric Equations 9.2 Calculus and Parametric Equations 9.3 Arc Length and Surface Area in Parametric Equations 9.4 Polar Coordinates 9.5 Calculus and Polar Coordinates 9.6 Conic Sections 9.7 Conic Sections in Polar Coordinates Chapter 10: Vectors and the Geometry of Space 10.1 Vectors in the Plane 10.2 Vectors in Space 10.3 The Dot Product 10.4 The Cross Product 10.5 Lines and Planes in Space 10.6 Surfaces in Space Chapter 11: Vector-Valued Functions 11.1 Vector-Valued Functions 11.2 The Calculus of Vector-Valued Functions 11.3 Motion in Space 11.4 Curvature 11.5 Tangent and Normal Vectors 11.6 Parametric Surfaces Chapter 12: Functions of Several Variables and Partial Differentiation 12.1 Functions of Several Variables 12.2 Limits and Continuity 12.3 Partial Derivatives 12.4 Tangent Planes and Linear Approximations 12.5 The Chain Rule 12.6 The Gradient and Directional Derivatives 12.7 Extrema of Functions of Several Variables 12.8 Constrained Optimization and and Lagrange Multipliers Chapter 13: Multiple Integrals 13.1 Double Integrals 13.2 Area, Volume and Center of Mass 13.3 Double Integrals in Polar Coordinates 13.4 Surface Area 13.5 Triple Integrals 13.6 Cylindrical Coordinates 13.7 Spherical Coordinates 13.8 Change of Variables in Multiple Integrals Chapter 14: Vector Calculus 14.1 Vector Fields 14.2 Line Integrals 14.3 Independence of Path and Conservative Vector Fields 14.4 Green's Theorem 14.5 Curl and Divergence 14.6 Surface Integrals 14.7 The Divergence Theorem 14.8 Stokes' Theorem 14.9 Applications of Vector Calculus Chapter 15: Second Order Differential Equations 15.1 Second-Order Equations With Constant Coefficients 15.2 Nonhomogeneous Equations: Undetermined Coefficients 15.3 Applications of Second-Order Equations 15.4 Power Series Solutions of Differential Equations
Appendix A: Proofs of Selected Theorems Appendix B: Answers to Odd-Numbered Exercises