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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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| Exponential and Logarithmic Functions | |
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| The Trigonometric Functions | |
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| The Inverse 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 at Infinity; Infinite Limits | |
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| Limits Involving Trigonometric Functions | |
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| Natural Exponential, Natural Log, and Hyperbolic Functions | |
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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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| Derivatives of Exponential and Logarithmic Functions | |
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| Derivatives of Hyperbolic and Inverse Trigonometric Functions | |
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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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| Exponential Growth and Decay | |
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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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| Techniques of Integration and Differential Equations | |
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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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| First-Order Linear Differential Equations | |
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| Approximations for Differential Equations | |
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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 | |