| |
| |
| |
| Before Calculus | |
| |
| |
| |
| Functions | |
| |
| |
| |
| New Functions from Old | |
| |
| |
| |
| Families of Functions | |
| |
| |
| |
| Inverse Functions | |
| |
| |
| |
| Limits and Continuity | |
| |
| |
| |
| Limits | |
| |
| |
| |
| |
| Computing Limits | |
| |
| |
| |
| Limits at Infinity; End Behavior of a Function | |
| |
| |
| |
| Limits | |
| |
| |
| |
| |
| Continuity | |
| |
| |
| |
| Continuity of Trigonometric Functions | |
| |
| |
| |
| The Derivative | |
| |
| |
| |
| Tangent Lines and Rates of Change | |
| |
| |
| |
| The Derivative Function | |
| |
| |
| |
| Introduction to Techniques of Differentiation | |
| |
| |
| |
| The Product and Quotient Rules | |
| |
| |
| |
| Derivatives of Trigonometric Functions | |
| |
| |
| |
| The Chain Rule | |
| |
| |
| |
| Implicit Differentiation | |
| |
| |
| |
| Related Rates | |
| |
| |
| |
| Local Linear Approximation; Differentials | |
| |
| |
| |
| The Derivative in Graphing and Applications | |
| |
| |
| |
| Analysis of Functions I: Increase, Decrease, and Concavity | |
| |
| |
| |
| Analysis of Functions II: Relative Extrema; Graphing Polynomials | |
| |
| |
| |
| Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents | |
| |
| |
| |
| Absolute Maxima and Minima | |
| |
| |
| |
| Applied Maximum and Minimum Problems | |
| |
| |
| |
| Rectilinear Motion | |
| |
| |
| |
| Newton's Method | |
| |
| |
| |
| Rolle's Theorem; Mean-Value Theorem | |
| |
| |
| |
| Integration | |
| |
| |
| |
| An Overview of the Area Problem | |
| |
| |
| |
| The Indefinite Integral | |
| |
| |
| |
| Integration by Substitution | |
| |
| |
| |
| The Definition of Area as a Limit; Sigma Notation | |
| |
| |
| |
| The Definite Integral | |
| |
| |
| |
| The Fundamental Theorem of Calculus | |
| |
| |
| |
| Rectilinear Motion Revisited: Using Integration | |
| |
| |
| |
| Average Value of a Function and Its Applications | |
| |
| |
| |
| Evaluating Definite Integrals by Substitution | |
| |
| |
| |
| Applications of the Definite Integral in Geometry, Science and Engineering | |
| |
| |
| |
| Area Between Two Curves | |
| |
| |
| |
| Volumes by Slicing; Disks and Washers | |
| |
| |
| |
| Volumes by Cylindrical Shells | |
| |
| |
| |
| Length of a Plane Curve | |
| |
| |
| |
| Area of a Surface Revolution | |
| |
| |
| |
| Work | |
| |
| |
| |
| Moments, Centers of Gravity, and Centroids | |
| |
| |
| |
| Fluid Pressure and Force | |
| |
| |
| |
| Exponential, Logarithmic, and Inverse Trigonometric Functions | |
| |
| |
| |
| Exponential and Logarithmic Functions | |
| |
| |
| |
| Derivatives and Integrals Involving Logarithmic Functions | |
| |
| |
| |
| Derivatives of Inverse Functions; Derivatives and Integrals Involving Exponential Functions | |
| |
| |
| |
| Graphs and Applications Involving Logarithmic and Exponential Functions | |
| |
| |
| |
| L'H�opital's Rule; Indeterminate Forms | |
| |
| |
| |
| Logarithmic and Other Functions Defined by Integrals | |
| |
| |
| |
| Derivatives and Integrals Involving Inverse Trigonometric Functions | |
| |
| |
| |
| Hyperbolic Functions and Hanging Cubes | |
| |
| |
| |
| Principles of Integral Evaluation | |
| |
| |
| |
| An Overview of Integration Methods | |
| |
| |
| |
| Integration by Parts | |
| |
| |
| |
| Integrating Trigonometric Functions | |
| |
| |
| |
| Trigonometric Substitutions | |
| |
| |
| |
| Integrating Rational Functions by Partial Fractions | |
| |
| |
| |
| Using Computer Algebra Systems and Tables of Integrals | |
| |
| |
| |
| Numerical Integration; Simpson's Rule | |
| |
| |
| |
| Improper Integrals | |
| |
| |
| |
| Mathematical Modeling with Differential Equations | |
| |
| |
| |
| Modeling with Differential Equations | |
| |
| |
| |
| Separation of Variables | |
| |
| |
| |
| Slope Fields; Euler's Method | |
| |
| |
| |
| First-Order Differential Equations and Applications | |
| |
| |
| |
| Infinite Series | |
| |
| |
| |
| Sequences | |
| |
| |
| |
| Monotone Sequences | |
| |
| |
| |
| Infinite Series | |
| |
| |
| |
| Convergence Tests | |
| |
| |
| |
| The Comparison, Ratio, and Root Tests | |
| |
| |
| |
| Alternating Series; Absolute and Conditional Convergence | |
| |
| |
| |
| Maclaurin and Taylor Polynomials | |
| |
| |
| |
| Maclaurin and Taylor Series; Power Series | |
| |
| |
| |
| Convergence of Taylor Series | |
| |
| |
| |
| Differentiating and Integrating Power Series; Modeling with Taylor Series | |
| |
| |
| |
| Parametric and Polar Curves; Conic Sections | |
| |
| |
| |
| Parametric Equations; Tangent Lines and Arc Length for Parametric Curves | |
| |
| |
| |
| Polar Coordinates | |
| |
| |
| |
| Tangent Lines, Arc Length, and Area for Polar Curves | |
| |
| |
| |
| Conic Sections | |
| |
| |
| |
| Rotation of Axes; Second-Degree Equations | |
| |
| |
| |
| Conic Sections in Polar Coordinates | |
| |
| |
| |
| Three-Dimensional Space; Vectors | |
| |
| |
| |
| Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces | |
| |
| |
| |
| Vectors | |
| |
| |
| |
| Dot Product; Projections | |
| |
| |
| |
| Cross Product | |
| |
| |
| |
| Parametric Equations of Lines | |
| |
| |
| |
| Planes in 3-Space | |
| |
| |
| |
| Quadric Surfaces | |
| |
| |
| |
| Cylindrical and Spherical Coordinates | |
| |
| |
| |
| Vector-Valued Functions | |
| |
| |
| |
| Introduction to Vector-Valued Functions | |
| |
| |
| |
| Calculus of Vector-Valued Functions | |
| |
| |
| |
| Change of Parameter; Arc Length | |
| |
| |
| |
| Unit Tangent, Normal, and Binormal Vectors | |
| |
| |
| |
| Curvature | |
| |
| |
| |
| Motion Along a Curve | |
| |
| |
| |
| Kepler's Laws of Planetary Motion | |
| |
| |
| |
| Partial Derivatives | |
| |
| |
| |
| Functions of Two or More Variables | |
| |
| |
| |
| Limits and Continuity | |
| |
| |
| |
| Partial Derivatives | |
| |
| |
| |
| Differentiability, Differentials, and Local Linearity | |
| |
| |
| |
| The Chain Rule | |
| |
| |
| |
| Directional Derivatives and Gradients | |
| |
| |
| |
| Tangent Planes and Normal Vectors | |
| |
| |
| |
| Maxima and Minima of Functions of Two Variables | |
| |
| |
| |
| Lagrange Multipliers | |
| |
| |
| |
| Multiple Integrals | |
| |
| |
| |
| Double Integrals | |
| |
| |
| |
| Double Integrals over Nonrectangular Regions | |
| |
| |
| |
| Double Integrals in Polar Coordinates | |
| |
| |
| |
| Surface Area; Parametric Surfaces} | |
| |
| |
| |
| Triple Integrals | |
| |
| |
| |
| Triple Integrals in Cylindrical and Spherical Coordinates | |
| |
| |
| |
| Change of Variable in Multiple Integrals; Jacobians | |
| |
| |
| |
| Centers of Gravity Using Multiple Integrals | |
| |
| |
| |
| Topics in Vector Calculus | |
| |
| |
| |
| Vector Fields | |
| |
| |
| |
| Line Integrals | |
| |
| |
| |
| Independence of Path; Conservative Vector Fields | |
| |
| |
| |
| Green's Theorem | |
| |
| |
| |
| Surface Integrals | |
| |
| |
| |
| Applications of Surface Integrals; Flux | |
| |
| |
| |
| The Divergence Theorem | |
| |
| |
| |
| Stokes' Theorem | |
| |
| |
| Appendix [order of sections TBD] | |
| |
| |
| |
| Graphing Functions Using Calculators and Computer Algebra Systems | |
| |
| |
| |
| Trigonometry Review | |
| |
| |
| |
| Solving Polynomial Equations | |
| |
| |
| |
| Mathematical Models | |
| |
| |
| |
| Selected Proofs | |
| |
| |
| Web Appendices | |
| |
| |
| |
| Real Numbers, Intervals, and Inequalities | |
| |
| |
| |
| Absolute Value | |
| |
| |
| |
| Coordinate Planes, Lines, and Linear Functions | |
| |
| |
| |
| Distance, Circles, and Quadratic Functions | |
| |
| |
| |
| Second-Order Linear Homogeneous Differential Equations; The Vibrating String | |
| |
| |
| |
| The Discriminant | |
| |
| |
| Answers | |
| |
| |
| Photocredits | |
| |
| |
| Index | |