| |
| |
| Preliminaries | |
| |
| |
| Polynomial and Rational Functions | |
| |
| |
| Graphing Calculators and Computer Algebra | |
| |
| |
| Inverse Functions | |
| |
| |
| Trigonometric and Inverse Trigonometric Functions | |
| |
| |
| Exponential and Logarithmic Functions | |
| |
| |
| Transformations of Functions | |
| |
| |
| Parametric Equations and Polar Coordinates | |
| |
| |
| Limits and Continuity | |
| |
| |
| A Brief Preview of | |
| |
| |
| The Concept of Limit | |
| |
| |
| Computation of Limits | |
| |
| |
| Continuity and its Consequences | |
| |
| |
| Limits Involving Infinity | |
| |
| |
| Limits and Loss-of-Significance Errors | |
| |
| |
| Differentiation | |
| |
| |
| Tangent Lines and Velocity | |
| |
| |
| The Derivative | |
| |
| |
| Computation of Derivatives: The Power Rule | |
| |
| |
| The Product and Quotient Rules | |
| |
| |
| The Chain Rule | |
| |
| |
| Derivatives of Trigonometric and Inverse Trigonometric Functions | |
| |
| |
| Derivatives of Exponential and Logarithmic Functions | |
| |
| |
| Implicit Differentiation | |
| |
| |
| 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 Overview of Curve Sketching | |
| |
| |
| Optimization | |
| |
| |
| Rates of Change in Economics and the Sciences | |
| |
| |
| Related Rates and Parametric Equations | |
| |
| |
| Integration | |
| |
| |
| Area Under a Curve | |
| |
| |
| The Definite Integral | |
| |
| |
| Antiderivatives | |
| |
| |
| The Fundamental Theorem of Calculus | |
| |
| |
| Integration by Substitution | |
| |
| |
| Integration by Parts | |
| |
| |
| Other Techniques of Integration | |
| |
| |
| Integration Tables and Computer Algebra Systems | |
| |
| |
| Numerical Integration | |
| |
| |
| Improper Integrals | |
| |
| |
| Applications of the Definite Integral | |
| |
| |
| Area Between Curves | |
| |
| |
| Volume | |
| |
| |
| Arc Length and Surface Area | |
| |
| |
| Projectile Motion | |
| |
| |
| Applications of Integration to Physics and Engineering | |
| |
| |
| Probability | |
| |
| |
| Differential Equations | |
| |
| |
| Growth and Decay Problems | |
| |
| |
| Separable Differential Equations | |
| |
| |
| Euler's | |
| |
| |
| Second Order Equations with Constant Coefficients | |
| |
| |
| Nonhomogeneous Equations: Undetermined Coefficients | |
| |
| |
| Applications of Differential Equations | |
| |
| |
| Infinite Series | |
| |
| |
| Sequences of Real | |
| |
| |
| Infinite Series | |
| |
| |
| The Integral Test and Comparison Tests | |
| |
| |
| Alternating Series | |
| |
| |
| Absolute Convergence and the Ratio Test | |
| |
| |
| Power Series | |
| |
| |
| Taylor Series | |
| |
| |
| Applications of Taylor Series | |
| |
| |
| Fourier Series | |
| |
| |
| Power Series Solutions of Differential Equations | |
| |
| |
| Vectors and the Geometry of Space | |
| |
| |
| Vectors in the Plane | |
| |
| |
| Vectors in Space | |
| |
| |
| The Dot | |
| |
| |
| The Cross Product | |
| |
| |
| Lines and Planes in Space | |
| |
| |
| Surfaces in Space | |
| |
| |
| Vector-Valued Functions | |
| |
| |
| Vector-Valued | |
| |
| |
| Parametric Surfaces | |
| |
| |
| The Calculus of Vector-Valued Functions | |
| |
| |
| Motion in Space | |
| |
| |
| Curvature | |
| |
| |
| Tangent and Normal Vectors | |
| |
| |
| Functions of Several Variables and Differentiation | |
| |
| |
| Functions of Several Variables | |
| |
| |
| Limits and Continuity | |
| |
| |
| Partial Derivatives | |
| |
| |
| Tangent Planes and Linear Approximations | |
| |
| |
| The Chain Rule | |
| |
| |
| The Gradient and Directional Derivatives | |
| |
| |
| Extrema of Functions of Several Variables | |
| |
| |
| Constrained Optimization and Lagrange Multipliers | |
| |
| |
| Multiple Integrals | |
| |
| |
| Double Integrals | |
| |
| |
| Area, Volume and Center of Mass | |
| |
| |
| Double Integrals in Polar Coordinates | |
| |
| |
| Surface Area | |
| |
| |
| Triple Integrals | |
| |
| |
| Cylindrical Coordinates | |
| |
| |
| Spherical Coordinates | |
| |
| |
| Change of Variables in Multiple Integrals | |
| |
| |
| Vector Calculus | |
| |
| |
| Vector Fields | |
| |
| |
| Curl and Divergence | |
| |
| |
| 12.3 | |
| |
| |
| Independence of Path and Conservative Vector Fields | |
| |
| |
| Green's Theorem | |
| |
| |
| Surface Integrals | |
| |
| |
| The Divergence Theorem | |
| |
| |
| Stokes' Theorem | |
| |
| |
| Applications of Vector Calculus | |
| |
| |
| Graphs of Additional Polar Equations | |
| |
| |
| F | |