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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 | |