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| Preliminaries | |
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| Real Numbers and the Real Line | |
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| Lines, Circles, and Parabolas | |
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| Functions and Their Graphs | |
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| Identifying Functions; Mathematical Models | |
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| Combining Functions; Shifting and Scaling Graphs | |
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| Trigonometric Functions | |
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| Graphing with Calculators and Computers | |
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| Limits and Derivatives | |
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| Rates of Change and Limits | |
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| Calculating Limits Using the Limit Laws | |
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| Precise Definition of a Limit | |
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| One-Sided Limits and Limits at Infinity | |
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| Infinite Limits and Vertical Asymptotes | |
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| Continuity | |
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| Tangents and Derivatives | |
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| Differentiation | |
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| The Derivative as a Function | |
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| Differentiation Rules | |
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| The Derivative as a Rate of Change | |
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| Derivatives of Trigonometric Functions | |
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| The Chain Rule and Parametric Equations | |
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| Implicit Differentiation | |
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| Related Rates | |
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| Linearization and Differentials | |
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| Applications of Derivatives | |
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| Extreme Values of Functions | |
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| The Mean Value Theorem | |
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| Monotonic Functions and the First Derivative Test | |
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| Concavity and Curve Sketching | |
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| Applied Optimization Problems | |
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| Indeterminate Forms and L'Hopital's Rule | |
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| Newton's Method | |
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| Antiderivatives | |
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| Integration | |
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| Estimating with Finite Sums | |
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| Sigma Notation and Limits of Finite Sums | |
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| The Definite Integral | |
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| The Fundamental Theorem of Calculus | |
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| Indefinite Integrals and the Substitution Rule | |
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| Substitution and Area Between Curves | |
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| Applications of Definite Integrals | |
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| Volumes by Slicing and Rotation About an Axis | |
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| Volumes by Cylindrical Shells | |
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| Lengths of Plane Curves | |
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| Moments and Centers of Mass | |
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| Areas of Surfaces of Revolution and The Theorems of Pappus | |
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| Work | |
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| Fluid Pressures and Forces | |
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| Transcendental Functions | |
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| Inverse Functions and their Derivatives | |
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| Natural Logarithms | |
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| The Exponential Function | |
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| ax and loga x | |
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| Exponential Growth and Decay | |
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| Relative Rates of Growth | |
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| Inverse Trigonometric Functions | |
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| Hyperbolic Functions | |
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| Techniques Of Integration | |
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| Basic Integration Formulas | |
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| Integration by Parts | |
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| Integration of Rational Functions by Partial Fractions | |
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| Trigonometric Integrals | |
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| Trigonometric Substitutions | |
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| Integral Tables and Computer Algebra Systems | |
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| Numerical Integration | |
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| Improper Integrals | |
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| Further Applications of Integration | |
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| Slope Fields and Separable Differential Equations | |
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| First-Order Linear Differential Equations | |
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| Euler's Method | |
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| Graphical Solutions of Autonomous Equations | |
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| Applications of First-Order Differential Equations | |
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| Conic Sections and Polar Coordinates | |
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| Conic Sections and Quadratic Equations | |
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| Classifying Conic Sections by Eccentricity | |
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| Quadratic Equations and Rotations | |
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| Conics and Parametric Equations; The Cycloid | |
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| Polar Coordinates | |
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| Graphing in Polar Coordinates | |
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| Area and Lengths in Polar Coordinates | |
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| Conic Sections in Polar Coordinates | |
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| Infinite Sequences and Series | |
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| Sequences | |
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| Infinite Series | |
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| The Integral Test | |
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| Comparison Tests | |
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| The Ratio and Root Tests | |
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| Alternating Series, Absolute and Conditional Convergence | |
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| Power Series | |
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| Taylor and Maclaurin Series | |
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| Convergence of Taylor Series; Error Estimates | |
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| Applications of Power Series | |
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| Fourier Series | |
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| Vectors and the Geometry of Space | |
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| Three-Dimensional Coordinate Systems | |
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| Vectors | |
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| The Dot Product | |
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| The Cross Product | |
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| Lines and Planes in Space | |
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| Cylinders and Quadric Surfaces | |
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| Vector Valued Functions and Motion in Space | |
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| Vector Functions | |
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| Modeling Projectile Motion | |
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| Arc Length and the Unit Tangent Vector T | |
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| Curvature and the Unit Normal Vector N | |
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| Torsion and the Unit Binormal Vector B | |
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| Planetary Motion and Satellites | |
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| Partial Derivatives | |
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| Functions of Several Variables | |
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| Limits and Continuity in Higher Dimensions | |
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| Partial Derivatives | |
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| The Chain Rule | |
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| Directional Derivatives and Gradient Vectors | |
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| Tangent Planes and Differentials | |
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| Extreme Values and Saddle Points | |
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| Lagrange Multipliers | |
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| Partial Derivatives with Constrained Variables | |
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| Taylor's Formula for Two Variables | |
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| Multiple Integrals | |
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| Double Integrals | |
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| Areas, Moments and Centers of Mass | |
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| Double Integrals in Polar Form | |
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| Triple Integrals in Rectangular Coordinates | |
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| Masses and Moments in Three Dimensions | |
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| Triple Integrals in Cylindrical and Spherical Coordinates | |
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| Substitutions in Multiple Integrals | |
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| Integration in Vector Fields | |
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| Line Integrals | |
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| Vector Fields, Work, Circulation, and Flux | |
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| Path Independence, Potential Functions, and Conservative Fields | |
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| Green's Theorem in the Plane | |
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| Surface Area and Surface Integrals | |
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| Parametrized Surfaces | |
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| Stokes' Theorem | |
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| The Divergence Theorem and a Unified Theory | |
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| Appendices | |
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| Mathematical Induction | |
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| Proofs of Limit Theorems | |
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| Commonly Occurring Limits | |
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| Theory of the Real Numbers | |
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| Complex Numbers | |
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| The Distributive Law for Vector Cross Products | |
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| Determinants and Cramer's Rule | |
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| The Mixed Derivative Theorem and the Increment Theorem | |
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| The Area of a Parallelogram's Projection on a Plane | |