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Functions, Graphs, and Models | |
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Functions and Mathematical Modeling | |
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Graphs of Equations and Functions | |
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Polynomials and Algebraic Functions | |
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Transcendental Functions | |
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Preview: What Is Calculus? | |
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Prelude to Calculus | |
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Tangent Lines and Slope Predictors | |
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The Limit Concept | |
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More about Limits | |
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The Concept of Continuity | |
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The Derivative | |
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The Derivative and Rates of Change | |
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Basic Differentiation Rules | |
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The Chain Rule | |
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Derivatives of Algebraic Functions | |
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Maxima and Minima of Functions on Closed Intervals | |
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Applied Optimization Problems | |
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Derivatives of Trigonometric Functions | |
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Successive Approximations and Newton's Method | |
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Additional Applications of the Derivative | |
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Implicit Functions and Related Rates | |
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Increments, Differentials, and Linear Approximation | |
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Increasing and Decreasing Functions and the Mean Value Theorem | |
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The First Derivative Test and Applications | |
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Simple Curve Sketching | |
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Higher Derivatives and Concavity | |
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Curve Sketching and Asymptotes | |
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The Integral | |
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Introduction | |
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Antiderivatives and Initial Value Problems | |
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Elementary Area Computations | |
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Riemann Sums and the Integral | |
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Evaluation of Integrals | |
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The Fundamental Theorem of Calculus | |
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Integration by Substitution | |
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Areas of Plane Regions | |
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Numerical Integration | |
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Applications of the Integral | |
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Riemann Sum Approximations | |
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Volumes by the Method of Cross Sections | |
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Volumes by the Method of Cylindrical Shells | |
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Arc Length and Surface Area of Revolution | |
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Force and Work | |
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Centroids of Plane Regions and Curves | |
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Calculus of Transcendental Functions | |
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Exponential and Logarithmic Functions | |
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Indeterminate Forms and L'Hop?tal's Rule | |
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More Indeterminate Forms | |
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The Logarithm as an Integral | |
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Inverse Trigonometric Functions | |
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Hyperbolic Functions | |
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Techniques of Integration | |
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Introduction | |
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Integral Tables and Simple Substitutions | |
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Integration by Parts | |
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Trigonometric Integrals | |
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Rational Functions and Partial Fractions | |
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Trigonometric Substitutions | |
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Integrals Involving Quadratic Polynomials | |
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Improper Integrals | |
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Differential Equations | |
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Simple Equations and Models | |
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Slope Fields and Euler's Method | |
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Separable Equations and Applications | |
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Linear Equations and Applications | |
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Population Models | |
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Linear Second-Order Equations | |
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Mechanical Vibrations | |
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Polar Coordinates and Parametric Curves | |
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Analytic Geometry and the Conic Sections | |
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Polar Coordinates | |
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Area Computations in Polar Coordinates | |
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Parametric Curves | |
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Integral Computations with Parametric Curves | |
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Conic Sections and Applications | |
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Infinite Series | |
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Introduction | |
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Infinite Sequences | |
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Infinite Series and Convergence | |
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Taylor Series and Taylor Polynomials | |
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The Integral Test | |
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Comparison Tests for Positive-Term Series | |
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Alternating Series and Absolute Convergence | |
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Power Series | |
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Power Series Computations | |
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Series Solutions of Differential Equations | |
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Vectors, Curves, and Surfaces in Space | |
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Vectors in the Plane | |
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Three-Dimensional Vectors | |
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The Cross Product of Vectors | |
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Lines and Planes in Space | |
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Curves and Motions in Space | |
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Curvature and Acceleration | |
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Cylinders and Quadric Surfaces | |
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Cylindrical and Spherical Coordinates | |
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Partial Differentiation | |
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Introduction | |
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Functions of Several Variables | |
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Limits and Continuity | |
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Partial Derivatives | |
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Multivariable Optimization Problems | |
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Increments and Linear Approximation | |
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The Multivariable Chain Rule | |
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Directional Derivatives and the Gradient Vector | |
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Lagrange Multipliers and Constrained Optimization | |
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Critical Points of Functions of Two Variables | |
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Multiple Integrals | |
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Double Integrals | |
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Double Integrals over More General Regions | |
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Area and Volume by Double Integration | |