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Functions And Limits | |
Functions and Their Representations | |
A Catalog of Essential Functions | |
The Limit of a Function | |
Calculating Limits | |
Continuity | |
Limits Involving Infinity | |
Review | |
Derivatives | |
Derivatives and Rates of Change | |
The Derivative as a Function | |
Basic Differentiation Formulas | |
The Product and Quotient Rules | |
The Chain Rule | |
Implicit Differentiation | |
Related Rates | |
Linear Approximations and Differentials | |
Review | |
Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions | |
Exponential Functions | |
Inverse Functions and Logarithms | |
Derivatives of Logarithmic and Exponential Functions | |
Exponential Growth and Decay | |
Inverse Trigonometric Functions | |
Hyperbolic Functions | |
Indeterminate Forms and l'Hospital's Rule | |
Review | |
Applications of Differentiation | |
Maximum and Minimum Values | |
The Mean Value Theorem | |
Derivatives and the Shapes of Graphs | |
Curve Sketching | |
Optimization Problems | |
Newton's Method | |
Antiderivatives | |
Review | |
Integrals | |
Areas and Distances | |
The Definite Integral | |
Evaluating Definite Integrals | |
The Fundamental Theorem of Calculus | |
The Substitution Rule | |
Review | |
Techniques of Integration | |
Integration by Parts | |
Trigonometric Integrals and Substitutions | |
Partial Fractions | |
Integration with Tables and Computer Algebra Systems | |
Approximate Integration | |
Improper Integrals | |
Review | |
Applications of Integration | |
Areas between Curves | |
Volumes | |
Volumes by Cylindrical Shells | |
Arc Length | |
Applications to Physics and Engineering | |
Differential Equations | |
Review | |
Series | |
Sequences | |
Series | |
The Integral and Comparison Tests | |
Other Convergence Tests | |
Power Series | |
Representing Functions as Power Series | |
Taylor and Maclaurin Series | |
Applications of Taylor Polynomials | |
Review | |
Parametric Equations and Polar Coordinates | |
Parametric Curves | |
Calculus with Parametric Curves | |
Polar Coordinates | |
Areas and Lengths in Polar Coordinates | |
Conic Sections in Polar Coordinates | |
Review | |
Vectors and the Geometry of Space | |
Three-Dimensional Coordinate Systems | |
Vectors | |
The Dot Product | |
The Cross Product | |
Equations of Lines and Planes | |
Cylinders and Quadric Surfaces | |
Vector Functions and Space Curves | |
Arc Length and Curvature | |
Motion in Space: Velocity and Acceleration | |
Review | |
Partial Derivatives | |
Functions of Several Variables | |
Limits and Continuity | |
Partial Derivatives | |
Tangent Planes and Linear Approximations | |
The Chain Rule | |
Directional Derivatives and the Gradient Vector | |
Maximum and Minimum Values | |
Lagrange Multipliers | |
Review | |
Multiple Integrals | |
Double Integrals over Rectangles | |
Double Integrals over General Regions | |
Double Integrals in Polar Coordinates | |
Applications of Double Integrals | |
Triple Integrals | |
Triple Integrals in Cylindrical Coordinates | |
Triple Integrals in Spherical Coordinates | |
Change of Variables in Multiple Integrals | |
Review | |
Vector Calculus | |
Vector Fields | |
Line Integrals | |
The Fundamental Theorem for Line Integrals | |
Green's Theorem | |
Curl and Divergence | |
Parametric Surfaces and Their Areas | |
Surface Integrals | |
Stokes' Theorem | |
The Divergence Theorem | |
Review | |
Trigonometry | |
Proofs | |
Sigma Notation | |
The Logarithm Defined as an Integral. | |
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