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Functions and Limits | |
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Functions and Their Graphs | |
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Combining Functions | |
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Shifting and Scaling Graphs | |
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Rates of Change and Tangents to Curves | |
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Limit of a Function and Limit Laws | |
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Precise Definition of a Limit | |
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One-Sided Limits | |
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Continuity | |
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Limits Involving Infinity | |
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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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Differentiation | |
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Tangents and Derivatives at a Point | |
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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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Exponential Functions | |
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The Chain Rule | |
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Implicit Differentiation | |
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Inverse Functions and Their Derivatives | |
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Logarithmic Functions | |
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Inverse Trigonometric Functions | |
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Related Rates | |
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Linearization and Differentials | |
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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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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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Parametrizations of Plane Curves | |
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Applied Optimization | |
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Indeterminate Forms and L'Hopital's Rule | |
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Newton's Method | |
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Hyperbolic Functions | |
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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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Integration | |
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Antiderivatives | |
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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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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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Techniques of Integration | |
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Integration by Parts | |
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Trigonometric Integrals | |
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Trigonometric Substitutions | |
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Integration of Rational Functions by Partial Fractions | |
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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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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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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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Exponential Change and Separable Differential Equations | |
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Work and Fluid Forces | |
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Moments and Centers of Mass | |
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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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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 | |
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The Binomial Series | |
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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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Polar Coordinates and Conics | |
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Polar Coordinates | |
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Graphing in Polar Coordinates | |
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Areas and Lengths in Polar Coordinates | |
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Conics in Polar Coordinates | |
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Conics and Parametric Equations | |
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The Cycloid | |
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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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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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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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Vector-Valued Functions and Motion in Space | |
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Vector Functions and Their Derivatives | |
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Integrals of Vector Functions | |
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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 | |
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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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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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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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Multiple Integrals | |
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Double and Iterated Integrals over Rectangles | |
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Double Integrals over General Regions | |
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Area by Double Integration | |
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Double Integrals in Polar Form | |
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Triple Integrals in Rectangular Coordinates | |
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Moments and Centers of Mass | |
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Triple Integrals in Cylindrical and Spherical Coordinates | |
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Substitutions in Multiple Integrals | |
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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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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' | |
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The Divergence Theorem and a Unified Theory | |
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Questions to Guide Your Review | |
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Practice and Additional Exercises | |
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Appendices | |
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Real Numbers and the Real Line | |
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Mathematical Induction | |
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Lines, Circles, and Parabolas | |
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Trigonometric Functions | |
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Basic Algebra and Geometry Formulas | |
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Proofs of Limit Theorems and L'Hopital's Rule | |
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Commonly Occurring Limits | |
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Theory of the Real Numbers | |
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Convergence of Power Series and Taylor's Theorem | |
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The Distributive Law for Vector Cross Products | |
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The Mixed Derivative Theorem and the Increment Theorem | |
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Taylor's Formula for Two Variables | |