Real Numbers and the Real Line Lines, Circles, and Parabolas Functions and Their Graphs Identifying Functions; Mathematical Models Combining Functions; Shifting and Scaling Graphs Trigonometric Functions Graphing with Calculators and Computers
2. Limits and Derivatives
Rates of Change and Limits Calculating Limits Using the Limit Laws Precise Definition of a Limit One-Sided Limits and Limits at Infinity Infinite Limits and Vertical Asymptotes Continuity Tangents and Derivatives
The Derivative as a Function Differentiation Rules The Derivative as a Rate of Change Derivatives of Trigonometric Functions The Chain Rule and Parametric Equations Implicit Differentiation Related Rates Linearization and Differentials
4. Applications of Derivatives
Extreme Values of Functions The Mean Value Theorem Monotonic Functions and the First Derivative Test Concavity and Curve Sketching Applied Optimization Problems Indeterminate Forms and L'Hopital's Rule Newton's Method 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
6. Applications of Definite Integrals
Volumes by Slicing and Rotation About an Axis Volumes by Cylindrical Shells Lengths of Plane Curves Moments and Centers of Mass Areas of Surfaces of Revolution and The Theorems of Pappus Work Fluid Pressures and Forces
7. Transcendental Functions
Inverse Functions and their Derivatives Natural Logarithms The Exponential Function ax and loga x Exponential Growth and Decay Relative Rates of Growth Inverse Trigonometric Functions Hyperbolic Functions
8. Techniques of Integration
Basic Integration Formulas Integration by Parts Integration of Rational Functions by Partial Fractions Trigonometric Integrals Trigonometric Substitutions Integral Tables and Computer Algebra Systems Numerical Integration Improper Integrals
9. Further Applications of Integration
Slope Fields and Separable Differential Equations First-Order Linear Differential Equations Euler's Method Graphical Solutions of Autonomous Equations Applications of First-Order Differential Equations
10. Conic Sections and Polar Coordinates
Conic Sections and Quadratic Equations Classifying Conic Sections by Eccentricity Quadratic Equations and Rotations Conics and Parametric Equations; The Cycloid Polar Coordinates Graphing in Polar Coordinates Area and Lengths in Polar Coordinates Conic Sections in Polar Coordinates
11. 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; Error Estimates Applications of Power Series Fourier Series
12. 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
13. Vector-Valued Functions and Motion in Space
Vector Functions Modeling Projectile Motion Arc Length and the Unit Tangent Vector T Curvature and the Unit Normal Vector N Torsion and the Unit Binormal Vector B Planetary Motion and Satellites
14. 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 Partial Derivatives with Constrained Variables Taylor's Formula for Two Variables
15. Multiple Integrals
Double Integrals Areas, Moments and Centers of Mass Double Integrals in Polar Form Triple Integrals in Rectangular Coordinates Masses and Moments in Three Dimensions Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals
16. 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' Theorem The Divergence Theorem and a Unified Theory Appendices Mathematical Induction Proofs of Limit Theorems Commonly Occurring Limits Theory of the Real Numbers Complex Numbers The Distributive Law for Vector Cross Products Determinants and Cramer's Rule The Mixed Derivative Theorem and the Increment Theorem The Area of a Parallelogram's Projection on a Plane
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