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Preface | |

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Preliminaries | |

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Real Numbers, Estimation, and Logic | |

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Inequalities and Absolute Values | |

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The Rectangular Coordinate System | |

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Graphs of Equations | |

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Functions and Their Graphs | |

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Operations on Functions | |

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Trigonometric Functions | |

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Chapter Review | |

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Review and Preview Problems | |

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Limits | |

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Introduction to Limits | |

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Rigorous Study of Limits | |

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Limit Theorems | |

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Limits Involving Trigonometric Functions | |

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Limits at Infinity; Infinite Limits | |

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Continuity of Functions | |

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Chapter Review | |

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Review and Preview Problems | |

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The Derivative | |

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Two Problems with One Theme | |

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The Derivative | |

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Rules for Finding Derivatives | |

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Derivatives of Trigonometric Functions | |

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The Chain Rule | |

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Higher-Order Derivatives | |

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Implicit Differentiation | |

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Related Rates | |

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Differentials and Approximations | |

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Chapter Review | |

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Review and Preview Problems | |

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Applications of the Derivative | |

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Maxima and Minima | |

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Monotonicity and Concavity | |

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Local Extrema and Extrema on Open Intervals | |

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Practical Problems | |

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Graphing Functions Using Calculus | |

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The Mean Value Theorem for Derivatives | |

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Solving Equations Numerically | |

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Antiderivatives | |

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Introduction to Differential Equations | |

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Chapter Review | |

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Review and Preview Problems | |

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The Definite Integral | |

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Introduction to Area | |

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The Definite Integral | |

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The First Fundamental Theorem of Calculus | |

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The Second Fundamental Theorem of Calculus and the Method of Substitution | |

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The Mean Value Theorem for Integrals and the Use of Symmetry | |

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Numerical Integration | |

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Chapter Review | |

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Review and Preview Problems | |

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Applications of the Integral | |

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The Area of a Plane Region | |

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Volumes of Solids: Slabs, Disks, Washers | |

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Volumes of Solids of Revolution: Shells | |

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Length of a Plane Curve | |

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Work and Fluid Force | |

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Moments and Center of Mass | |

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Probability and Random Variables | |

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Chapter Review | |

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Review and Preview Problems | |

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Transcendental Functions | |

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The Natural Logarithm Function | |

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Inverse Functions and Their Derivatives | |

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The Natural Exponential Function | |

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General Exponential and Logarithmic Functions | |

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Exponential Growth and Decay | |

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First-Order Linear Differential Equations | |

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Approximations for Differential Equations | |

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The Inverse Trigonometric Functions and Their Derivatives | |

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The Hyperbolic Functions and Their Inverses | |

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Chapter Review | |

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Review and Preview Problems | |

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Techniques of Integration | |

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Basic Integration Rules | |

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Integration by Parts | |

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Some Trigonometric Integrals | |

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Rationalizing Substitutions | |

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Integration of Rational Functions Using Partial Fractions | |

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Strategies for Integration | |

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Chapter Review | |

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Review and Preview Problems | |

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Indeterminate Forms and Improper Integrals | |

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Indeterminate Forms of Type 0/0 | |

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Other Indeterminate Forms | |

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Improper Integrals: Infinite Limits of Integration | |

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Improper Integrals: Infinite Integrands | |

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Chapter Review | |

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Review and Preview Problems | |

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Infinite Series | |

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Infinite Sequences | |

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Infinite Series | |

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Positive Series: The Integral Test | |

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Positive Series: Other Tests | |

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Alternating Series, Absolute Convergence, and Conditional Convergence | |

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Power Series | |

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Operations on Power Series | |

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Taylor and Maclaurin Series | |

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The Taylor Approximation to a Function | |

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Chapter Review | |

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Review and Preview Problems | |

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Conics and Polar Coordinates | |

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The Parabola | |

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Ellipses and Hyperbolas | |

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Translation and Rotation of Axes | |

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Parametric Representation of Curves in the Plane | |

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The Polar Coordinate System | |

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Graphs of Polar Equations | |

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Calculus in Polar Coordinates | |

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Chapter Review | |

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Review and Preview Problems | |

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Geometry in Space and Vectors | |

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Cartesian Coordinates in Three-Space | |

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Vectors | |

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The Dot Product | |

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The Cross Product | |

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Vector-Valued Functions and Curvilinear Motion | |

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Lines and Tangent Lines in Three-Space | |

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Curvature and Components of Acceleration | |

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Surfaces in Three-Space | |

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Cylindrical and Spherical Coordinates | |

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Chapter Review | |

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Review and Preview Problems | |

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Derivatives for Functions of Two or More Variables | |

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Functions of Two or More Variables | |

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Partial Derivatives | |

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Limits and Continuity | |

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Differentiability | |

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Directional Derivatives and Gradients | |

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The Chain Rule | |

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Tangent Planes and Approximations | |

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Maxima and Minima | |

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The Method of Lagrange Multipliers | |

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Chapter Review | |

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Review and Preview Problems | |

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Multiple Integrals | |

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Double Integrals over Rectangles | |

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Iterated Integrals | |

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Double Integrals over Nonrectangular Regions | |

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Double Integrals in Polar Coordinates | |

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Applications of Double Integrals | |

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Surface Area | |

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Triple Integrals in Cartesian Coordinates | |

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Triple Integrals in Cylindrical and Spherical Coordinates | |

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Change of Variables in Multiple Integrals | |

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Chapter Review | |

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Review and Preview Problems | |

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Vector Calculus | |

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Vector Fields | |

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Line Integrals | |

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Independence of Path | |

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Green's Theorem in the Plane | |

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Surface Integrals | |

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Gauss's Divergence Theorem | |

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Stokes's Theorem | |

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Chapter Review | |

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Differential Equations | |

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Linear Homogeneous Equations | |

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Nonhomogeneous Equations | |

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Applications of Second-Order Equations | |

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Chapter Review | |

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Appendix | |

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Mathematical Induction | |

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Proofs of Several Theorems | |

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Answers to Odd-Numbered Problems | |

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Index | |

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Photo Credits | |