Calculus, Single Variable

Chapter 0: Preliminaries 0.1 The Real Numbers and the Cartesian Plane 0.2 Lines and Functions 0.3 Graphing Calculators and Computer Algebra Systems 0.4 Trigonometric Functions 0.5 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 The Method of Bisections 1.5 Limits Involving Infinity Asymptotes 1.6 The Formal Definition of the Limit 1.7 Limits and Loss-of-Significance Errors Computer Representation or Real Numbers Chapter 2: Differentiation 2.1 Tangent Lines and Velocity 2.2 The Derivative Numerical Differentiation 2.3 Computation of Derivatives: The Power Rule Higher Order Derivatives Acceleration 2.4 The Product and Quotient Rules 2.5 The Chain Rule 2.6 Derivatives of the Trigonometric Functions 2.7 Implicit Differentiation 2.8 The Mean Value Theorem Chapter 3: Applications of Differentiation 3.1 Linear Approximations and Newton's Method 3.2 Maximum and Minimum Values 3.3 Increasing and Decreasing Functions 3.4 Concavity and the Second Derivative Test 3.5 Overview of Curve Sketching 3.6 Optimization 3.8 Related Rates 3.8 Rates of Change in Economics and the Sciences Chapter 4: Integration 4.1 Antiderivatives 4.2 Sums and Sigma Notation Principle of Mathematical Induction 4.3 Area under a Curve 4.4 The Definite Integral Average Value of a Function 4.5 The Fundamental Theorem of Calculus 4.6 Integration by Substitution 4.7 Numerical Integration Error bounds for Numerical Integration 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 Chapter 6: Exponentials, Logarithms and other Transcendental Functions 6.1 The Natural Logarithm 6.2 Inverse Functions 6.3 Exponentials 6.4 The Inverse Trigonometric Functions 6.5 The Calculus of the Inverse Trigonometric Functions 6.6 The Hyperbolic Function Chapter 7: Integration Techniques 7.1 Overview of Formulas and Techniques 7.2 Integration by Parts 7.3 Trigonometric Techniques of Integration Trigonometric Substitution 7.4 Integration of Rational Functions using Partial Fractions General Strategies for Integration techniques 7.5 Integration Tables and Computer Algebra Systems 7.6 Indeterminate Forms and L'Hopital's Rule 7.7 Improper Integrals A Comparison Test 7.8 Probability Chapter 8: First-Order Differential Equations 8.1 modeling with Differential Equations Growth and Decay Problems Compound Interest 8.2 Separable Differential Equations Logistic Growth 8.3 Direction Fields and Euler's Method 8.4 Systems of First Order Equations Predator-Prey Systems Chapter 9: Infinite Series 9.1 Sequences of Real Numbers 9.2 Infinite Series 9.3 The Integral Test and Comparison Tests 9.4 Alternating Series Estimating the Sum of an Alternating Series 9.5 Absolute Convergence and the Ratio Test The Root Test Summary of Convergence Test 9.6 Power Series 9.7 Taylor Series Representations of Functions as Series Proof of Taylor's Theorem 9.8 Applications of Taylor Series The Binomial Series 9.9 Fourier Series Chapter 10: Parametric Equations and Polar Coordinates 10.1 Plane Curves and Parametric Equations 10.2 Calculus and Parametric Equations 10.3 Arc Length and Surface Area in Parametric Equations 10.4 Polar Coordinates 10.5 Calculus and Polar Coordinates 10.6 Conic Sections 10.7 Conic Sections in Polar Coordinates Appendix A: Proofs of Selected Theorems Appendix B: Answers to Odd-Numbered Exercises