Calculus

1 Functions and Graphs 1.1 What Is Calculus? 1.2 Preliminaries 1.3 Lines in the Plane; Parametric Equations 1.4 Functions and Graphs 1.5 Inverse Functions; Inverse Trigonometric Functions Chapter 1 Review Book Report: Ethnomathematics by Marcia Ascher Chapter 1 Group Research Project 2 Limits and Continuity 2.1 The Limit of a Function 2.2 Algebraic Computation of Limits 2.3 Continuity 2.4 Exponential and Logarithmic Functions Chapter 2 Review Chapter 2 Group Research Project 3 Differentiation 3.1 An Introduction to the Derivative: Tangents 3.2 Techniques of Differentiation 3.3 Derivatives of Trigonometric, Exponential, and Logarithmic Functions 3.4 Rates of Change: Modeling Rectilinear Motion 3.5 The Chain Rule 3.6 Implicit Differentiation 3.7 Related Rates and Applications 3.8 Linear Approximation and Differentials Chapter 3 Review Book Report: Fermat's Enigma by Simon Singh Chapter 3 Group Research Project 4 Additional Applications of the Derivative 4.1 Extreme Values of a Continuous Function 4.2 The Mean Value Theorem 4.3 Using Derivatives to Sketch the Graph of a Function 4.4 Curve Sketching with Asymptotes: Limits Involving Infinity 4.5 Hopital's Rule 4.6 Optimization in the Physical Sciences and Engineering 4.7 Optimization in Business, Economics, and the Life Sciences Chapter 4 Review Chapter 4 Group Research Project 5 Integration 5.1 Antidifferentiation 5.2 Area as the Limit of a Sum 5.3 Riemann Sums and the Definite Integral 5.4 The Fundamental Theorems of Calculus 5.5 Integration by Substitution 5.6 Introduction to Differential Equations 5.7 The Mean Value Theorem for Integrals; Average Value 5.8 Numerical Integration: The Trapezoidal Rule and Simpson's Rule 5.9 An Alternative Approach: The Logarithm as an Integral Chapter 5 Review Chapter 5 Group Research Project Cumulative Review Problems - Chapters 1–5 6 Additional Applications of the Integral 6.1 Area Between Two Curves 6.2 Volume 6.3 Polar Forms and Area 6.4 Arc Length and Surface Area 6.5 Physical Applications: Work, Liquid Force, and Centroids 6.6 Applications to Business, Economics, and Life Sciences Chapter 6 Review Book Report: To Infinity and Beyond, A Cultural History of the Infinite, by Eli Maor Chapter 6 Group Research Project 7 Methods of Integration 7.1 Review of Substitution and Integration by Table 7.2 Integration By Parts 7.3 Trigonometric Methods 7.4 Method of Partial Fractions 7.5 Summary of Integration Techniques 7.6 First-Order Differential Equations 7.7 Improper Integrals 7.8 Hyperbolic and Inverse Hyperbolic Functions Chapter 7 Review Chapter 7 Group Research Project 8 Infinite Series 8.1 Sequences and Their Limits 8.2 Introduction to Infinite Series; Geometric Series 8.3 The Integral Test; p-series 8.4 Comparison Tests 8.5 The Ratio Test and the Root Test 8.6 Alternating Series; Absolute and Conditional Convergence 8.7 Power Series 8.8 Taylor and Maclaurin Series Chapter 8 Review Chapter 8 Group Research Project Cumulative Review Problems - Chapters 6–8 9 Vectors in the Plane and in Space 9.1 Vectors in R2 9.2 Coordinates and Vectors in R3 9.3 The Dot Product 9.4 The Cross Product 9.5 Lines in R3 9.6 Planes in R3 9.7 Quadric Surfaces Chapter 9 Review Chapter 9 Group Research Project 10 Vector-Valued Functions 10.1 Introduction to Vector Functions 10.2 Differentiation and Integration of Vector Functions 10.3 Modeling Ballistics and Planetary Motion 10.4 Unit Tangent and Principal Unit Normal Vectors; Curvature 10.5 Tangential and Normal Components of Acceleration Chapter 10 Review Chapter 10 Group Research Project Cumulative Review Problems - Chapters 1–10 11 Partial Differentiation 11.1 Functions of Several Variables 11.2 Limits and Continuity 11.3 Partial Derivatives 11.4 Tangent Planes, Approximations, and Differentiability 11.5 Chain Rules 11.6 Directional Derivatives and the Gradient 11.7 Extrema of Functions of Two Variables 11.8 Lagrange Multipliers Chapter 11 Review Book Report: Hypatia's Heritage by Margaret Alic Chapter 11 Group Research Project 12 Multiple Integration 12.1 Double Integration over Rectangular Regions 12.2 Double Integration over Nonrectangular Regions 12.3 Double Integrals in Polar Coordinates 12.4 Surface Area 12.5 Triple Integrals 12.6 Mass, Moments, and Probability Density Functions 12.7 Cylindrical and Spherical Coordinates 12.8 Jacobians: Change of Variables Chapter 12 Review Chapter 12 Group Research Project 13 Vector Analysis 13.1 Properties of a Vector Field: Divergence and Curl 13.2 Line Integrals 13.3 The Fundamental Theorem and Path Independence 13.4 Green's Theorem 13.5 Surface Integrals 13.6 Stokes' Theorem and Applications 13.7 Divergence Theorem and Applications Chapter 13 Review Chapter 13 Group Research Project Cumulative Review Problems - Chapters 11–13 14 Introduction to Differential Equations 14.1 First-Order Differential Equations 14.2 Second-Order Homogeneous Linear Differential Equations 14.3 Second-Order Nonhomogeneous Linear Differential Equations Chapter 14 Review Book Report: Mathematical Experience by Philip J. Davis and Reuben Hersh Chapter 14 Group Research Project Appendices A: Introduction to the Theory of Limits B: Selected Proofs C: Significant Digits D: Short Table of Integrals E: Trigonometry F: Parabolas G: Ellipses H: Hyperbolas I: Determinants J: Answers to Selected Problems