Functions and Limits | |

Functions and Their Graphs | |

Combining Functions | |

Shifting and Scaling Graphs | |

Rates of Change and Tangents to Curves | |

Limit of a Function and Limit Laws | |

Precise Definition of a Limit | |

One-Sided Limits | |

Continuity | |

Limits Involving Infinity | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Differentiation | |

Tangents and Derivatives at a Point | |

The Derivative as a Function | |

Differentiation Rules | |

The Derivative as a Rate of Change | |

Derivatives of Trigonometric Functions | |

Exponential Functions | |

The Chain Rule | |

Implicit Differentiation | |

Inverse Functions and Their Derivatives | |

Logarithmic Functions | |

Inverse Trigonometric Functions | |

Related Rates | |

Linearization and Differentials | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Applications of Derivatives | |

Extreme Values of Functions | |

The Mean Value Theorem | |

Monotonic Functions and the First Derivative Test | |

Concavity and Curve Sketching | |

Parametrizations of Plane Curves | |

Applied Optimization | |

Indeterminate Forms and L'Hopital's Rule | |

Newton's Method | |

Hyperbolic Functions | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Integration | |

Antiderivatives | |

Estimating with Finite Sums | |

Sigma Notation and Limits of Finite Sums | |

The Definite Integral | |

The Fundamental Theorem of Calculus | |

Indefinite Integrals and the Substitution Rule | |

Substitution and Area Between Curves | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Techniques of Integration | |

Integration by Parts | |

Trigonometric Integrals | |

Trigonometric Substitutions | |

Integration of Rational Functions by Partial Fractions | |

Integral Tables and Computer Algebra Systems | |

Numerical Integration | |

Improper Integrals | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Applications of Definite Integrals | |

Volumes by Slicing and Rotation About an Axis | |

Volumes by Cylindrical Shells | |

Lengths of Plane Curves | |

Exponential Change and Separable Differential Equations | |

Work and Fluid Forces | |

Moments and Centers of Mass | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Infinite Sequences and Series | |

Sequences | |

Infinite Series | |

The Integral Test | |

Comparison Tests | |

The Ratio and Root Tests | |

Alternating Series, Absolute and Conditional Convergence | |

Power Series | |

Taylor and Maclaurin Series | |

Convergence of Taylor Series | |

The Binomial Series | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Polar Coordinates and Conics | |

Polar Coordinates | |

Graphing in Polar Coordinates | |

Areas and Lengths in Polar Coordinates | |

Conics in Polar Coordinates | |

Conics and Parametric Equations | |

The Cycloid | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Vectors and the Geometry of Space | |

Three-Dimensional Coordinate Systems | |

Vectors | |

The Dot Product | |

The Cross Product | |

Lines and Planes in Space | |

Cylinders and Quadric Surfaces | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Vector-Valued Functions and Motion in Space | |

Vector Functions and Their Derivatives | |

Integrals of Vector Functions | |

Arc Length and the Unit Tangent Vector T | |

Curvature and the Unit Normal Vector N | |

Torsion and the Unit Binormal Vector B | |

Planetary Motion | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Partial Derivatives | |

Functions of Several Variables | |

Limits and Continuity in Higher Dimensions | |

Partial Derivatives | |

The Chain Rule | |

Directional Derivatives and Gradient Vectors | |

Tangent Planes and Differentials | |

Extreme Values and Saddle Points | |

Lagrange Multipliers | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Multiple Integrals | |

Double and Iterated Integrals over Rectangles | |

Double Integrals over General Regions | |

Area by Double Integration | |

Double Integrals in Polar Form | |

Triple Integrals in Rectangular Coordinates | |

Moments and Centers of Mass | |

Triple Integrals in Cylindrical and Spherical Coordinates | |

Substitutions in Multiple Integrals | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Integration in Vector Fields | |

Line Integrals | |

Vector Fields, Work, Circulation, and Flux | |

Path Independence, Potential Functions, and Conservative Fields | |

Green's Theorem in the Plane | |

Surface Area and Surface Integrals | |

Parametrized Surfaces | |

Stokes' | |

The Divergence Theorem and a Unified Theory | |

Questions to Guide Your Review | |

Practice and Additional Exercises | |

Appendices | |

Real Numbers and the Real Line | |

Mathematical Induction | |

Lines, Circles, and Parabolas | |

Trigonometric Functions | |

Basic Algebra and Geometry Formulas | |

Proofs of Limit Theorems and L'Hopital's Rule | |

Commonly Occurring Limits | |

Theory of the Real Numbers | |

Convergence of Power Series and Taylor's Theorem | |

The Distributive Law for Vector Cross Products | |

The Mixed Derivative Theorem and the Increment Theorem | |

Taylor's Formula for Two Variables | |

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