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P. Preliminaries | |
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Lines | |
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Functions and Graphs | |
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Exponential Functions | |
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Inverse Functions and Logarithms | |
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Trigonometric Functions and Their Inverses | |
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Parametric Equations | |
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Modeling Change | |
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Limits and Continuity | |
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Rates of Change and Limits | |
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Finding Limits and One-Sided Limits | |
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Limits Involving Infinity | |
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Continuity | |
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Tangent Lines | |
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Derivatives | |
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The Derivative as a Function | |
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The Derivative as a Rate of Change | |
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Derivatives of Products, Quotients, and Negative Powers | |
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Derivatives of Trigonometric Functions | |
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The Chain Rule | |
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Implicit Differentiation | |
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Related Rates | |
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Derivatives of Inverse Trigonometric Functions | |
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Derivatives of Exponential and Logarithmic Functions | |
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Applications of Derivatives | |
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Extreme Values of Functions | |
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The Mean Value Theorem and Differential Equations | |
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The Shape of a Graph | |
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Graphical Solutions of Autonomous Differential Equations | |
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Modeling and Optimization | |
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Linearization and Differentials | |
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Newton's Method | |
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Integration | |
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Indefinite Integrals, Differential Equations, and Modeling | |
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Integral Rules | |
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Integration by Substitution | |
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Estimating with Finite Sums | |
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Riemann Sums and Definite Integrals | |
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The Mean Value and Fundamental Theorems | |
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Substitution in Definite Integrals | |
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Numerical Integration | |
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Applications of Integrals | |
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Volumes by Slicing and Rotation About an Axis | |
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Modeling Volume Using Cylindrical Shells | |
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Lengths of Plane Curves | |
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First Order Separable Differential Equations | |
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Springs, Pumping and Lifting | |
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Fluid Forces | |
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Moments and Centers of Mass | |
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Transcendental Functions and Differential Equations | |
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Logarithms | |
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Exponential Functions | |
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Linear First-Order Differential Equations | |
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Euler's Method | |
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Population Models | |
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Hyperbolic Functions | |
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Integration Techniques, L'Hôpital's Rule, and Improper Integrals | |
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Basic Integration Formulas | |
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Integration by Parts | |
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Partial Fractions | |
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Trigonometric Substitutions | |
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Integral Tables, Computer Algebra Systems, and Monte Carlo Integration | |
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L'Hôpital's Rule | |
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Improper Integrals | |
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Infinite Series | |
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Limits of Sequences of Numbers | |
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Subsequences, Bounded Sequences, and Picard's Method | |
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Infinite Series | |
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Series of Nonnegative Terms | |
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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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Applications of Power Series | |
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Fourier Series | |
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Fourier Cosine and Sine Series | |
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Vectors in the Plane and Polar Functions | |
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Vectors in the Plane | |
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Dot Products | |
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Vector-Valued Functions | |
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Modeling Projectile Motion | |
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Polar Coordinates and Graphs | |
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Calculus of Polar Curves | |
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Vectors and Motion in Space | |
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Cartesian (Rectangular) Coordinates and Vectors in Space | |
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Dot and Cross Products | |
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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 Space Curves | |
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Arc Length and the Unit Tangent Vector T | |
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The TNB Frame | |
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Tangential and Normal Components of Acceleration | |
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Planetary Motion and Satellites | |
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Multivariable Functions and Their 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, Gradient Vectors, and Tangent Planes | |
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Linearization 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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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 in Section | |
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Proof of the Chain Rule | |
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Complex Numbers | |
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Simpson's One-Third Rule | |
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Cauchy's Mean Mean Value Theorem and the Stronger Form of L'Hôpital's Rule | |
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Limits That Arise Frequently | |
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Proof of Taylor's Theorem | |
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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 | |