Preliminaries | |
Polynomials and Rational Functions | |
Graphing Calculators and Computer Algebra Systems | |
Inverse Functions | |
Trigonometric and Inverse Trigonometric Functions | |
Exponential and Logarithmic Functions Hyperbolic Functions Fitting a Curve to Data | |
Transformations of Functions | |
Limits and Continuity | |
A Brief Preview of Calculus: Tangent Lines and the Length of a Curve | |
The Concept of Limit | |
Computation of Limits | |
Continuity and its Consequences The Method of Bisections | |
Limits Involving Infinity Asymptotes | |
Formal Definition of the Limit Exploring the Definition of Limit Graphically | |
Limits and Loss-of-Significance Errors Computer Representation of Real Numbers | |
Differentiation | |
Tangent Lines and Velocity | |
The Derivative Numerical Differentiation | |
Computation of Derivatives: The Power Rule Higher Order Derivatives Acceleration | |
The Product and Quotient Rules | |
The Chain Rule | |
Derivatives of the Trigonometric Functions | |
Derivatives of the Exponential and Logarithmic Functions | |
Implicit Differentiation and Inverse Trigonometric Functions | |
The Mean Value Theorem | |
Applications of Differentiation | |
Linear Approximations and Newton’s Method | |
Indeterminate Forms and L’Hopital’s Rule | |
Maximum and Minimum Values | |
Increasing and Decreasing Functions | |
Concavity and the Second Derivative Test | |
Overview of Curve Sketching | |
Optimization | |
Related Rates | |
Rates of Change in Economics and the Sciences | |
Integration | |
Antiderivatives | |
Sums and Sigma Notation Principle of Mathematical Induction | |
Area | |
The Definite Integral Average Value of a Function | |
The Fundamental Theorem of Calculus | |
Integration by Substitution | |
Numerical Integration Error Bounds for Numerical Integration | |
The Natural Logarithm as an Integral The Exponential Function as the Inverse of the Natural Logarithm | |
Applications of the Definite Integral | |
Area Between Curves | |
Volume: Slicing, Disks, and Washers | |
Volumes by Cylindrical Shells | |
Arc Length and Surface Area | |
Projectile Motion | |
Applications of Integration to Physics and Engineering | |
Probability | |
Integration Techniques | |
Review of Formulas and Techniques | |
Integration by Parts | |
Trigonometric Techniques of Integration Integrals Involving Powers of Trigonometric Functions Trigonometric Substitution | |
Integration of Rational Functions Using Partial Fractions Brief Summary of Integration Techniques | |
Integration Tables and Computer Algebra Systems | |
Improper Integrals A Comparison Test | |
First-Order Differential Equations | |
Growth and Decay Problems Compound Interest Modeling with Differential Equations | |
Separable Differential Equations Logistic Growth | |
Direction Fields and Euler's Method | |
Systems of First-Order Differential Equations Predator-Prey Systems | |
Infinite Series | |
Sequences of Real Numbers | |
Infinite Series | |
The Integral Test and Comparison Tests | |
Alternating Series Estimating the Sum of an Alternating Series | |
Absolute Convergence and the Ratio Test The Root Test Summary of Convergence Tests | |
Power Series | |
Taylor Series Representations of Functions as Series Proof of Taylor’s Theorem | |
Applications of Taylor Series The Binomial Series | |
Fourier Series Parametric Equations and Polar Coordinates | |
Plane Curves and Parametric Equations | |
Calculus and Parametric Equations | |
Arc Length and Surface Area in Parametric Equations | |
Polar Coordinates | |
Calculus and Polar Coordinates | |
Conic Sections | |
Conic Sections in Polar Coordinates | |
Vectors and the Geometry of Space | |
Vectors in the Plane | |
Vectors in Space | |
The Dot Product Components and Projections | |
The Cross Product | |
Lines and Planes in Space | |
Surfaces in Space | |
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