Calculus: Single and Multivariable

Chapter 1: Foundation For Calculus: Functions and Limits

• • 1.1 Functions and Change
• • 1.2 Exponential Functions
• • 1.3 New Functions From Old
• • 1.4 Logarithmic Functions
• • 1.5 Trigonometric Functions
• • 1.6 Powers, Polynomials, and Rational Functions
• • 1.7 Introduction To Limits and Continuity
• • 1.8 Extending The Idea of A Limit
• • 1.9 Further Limit Calculations Using Algebra
• • 1.10 Preview of The Formal Definition of A Limit Online

Chapter 2: Key Concept: The Derivative

• • 2.1 How Do We Measure Speed?
• • 2.2 The Derivative At A Point
• • 2.3 The Derivative Function
• • 2.4 Interpretations of The Derivative
• • 2.5 The Second Derivative
• • 2.6 Differentiability

Chapter 3: Short-Cuts To Differentiation

• • 3.1 Powers and Polynomials
• • 3.2 The Exponential Function
• • 3.3 The Product and Quotient Rules
• • 3.4 The Chain Rule
• • 3.5 The Trigonometric Functions
• • 3.6 The Chain Rule and Inverse Functions
• • 3.7 Implicit Functions
• • 3.8 Hyperbolic Functions
• • 3.9 Linear Approximation and The Derivative
• • 3.10 Theorems About Differentiable Functions

Chapter 4: Using The Derivative

• • 4.1 Using First and Second Derivatives
• • 4.2 Optimization
• • 4.3 Optimization and Modeling
• • 4.4 Families of Functions and Modeling
• • 4.5 Applications To Marginality
• • 4.6 Rates and Related Rates
• • 4.7 L’hopital’s Rule, Growth, and Dominance
• • 4.8 Parametric Equations

Chapter 5: Key Concept: The Definite Integral

• • 5.1 How Do We Measure Distance Traveled?
• • 5.2 The Definite Integral
• • 5.3 The Fundamental Theorem and Interpretations
• • 5.4 Theorems About Definite Integrals

Chapter 6: Constructing Antiderivatives

• • 6.1 Antiderivatives Graphically and Numerically
• • 6.2 Constructing Antiderivatives Analytically
• • 6.3 Differential Equations and Motion
• • 6.4 Second Fundamental Theorem of Calculus

Chapter 7: Integration

• • 7.1 Integration By Substitution
• • 7.2 Integration By Parts
• • 7.3 Tables of Integrals
• • 7.4 Algebraic Identities and Trigonometric Substitutions
• • 7.5 Numerical Methods For Definite Integrals
• • 7.6 Improper Integrals
• • 7.7 Comparison of Improper Integrals

Chapter 8: Using The Definite Integral

• • 8.1 Areas and Volumes
• • 8.2 Applications To Geometry
• • 8.3 Area and Arc Length In Polar Coordinates
• • 8.4 Density and Center of Mass
• • 8.5 Applications To Physics
• • 8.6 Applications To Economics
• • 8.7 Distribution Functions
• • 8.8 Probability, Mean, and Median

Chapter 9: Sequences and Series

• • 9.1 Sequences
• • 9.2 Geometric Series
• • 9.3 Convergence of Series
• • 9.4 Tests For Convergence
• • 9.5 Power Series and Interval of Convergence

Chapter 10: Approximating Functions Using Series

• • 10.1 Taylor Polynomials
• • 10.2 Taylor Series
• • 10.3 Finding and Using Taylor Series
• • 10.4 The Error In Taylor Polynomial Approximations
• • 10.5 Fourier Series

Chapter 11: Differential Equations

• • 11.1 What is a Differential Equation?
• • 11.2 Slope Fields
• • 11.3 Euler’s Method
• • 11.4 Separation of Variables
• • 11.5 Growth and Decay
• • 11.6 Applications and Modeling
• • 11.7 The Logistic Model
• • 11.8 Systems of Differential Equations
• • 11.9 Analyzing The Phase Plane
• • 11.10 Second-Order Differential Equations: Oscillations
• • 11.11 Linear Second-Order Differential Equations

Chapter 12: Functions of Several Variables

• • 12.1 Functions of Two Variables
• • 12.2 Graphs and Surfaces
• • 12.3 Contour Diagrams
• • 12.4 Linear Functions
• • 12.5 Functions of Three Variables
• • 12.6 Limits and Continuity

Chapter 13: A Fundamental Tool: Vectors

• • 13.1 Displacement Vectors
• • 13.2 Vectors In General
• • 13.3 The Dot Product
• • 13.4 The Cross Product

Chapter 14: Differentiating Functions of Several Variables

• • 14.1 The Partial Derivative
• • 14.2 Computing Partial Derivatives Algebraically
• • 14.3 Local Linearity and The Differential
• • 14.4 Gradients and Directional Derivatives In The Plane
• • 14.5 Gradients and Directional Derivatives In Space
• • 14.6 The Chain Rule
• • 14.7 Second-Order Partial Derivatives
• • 14.8 Differentiability

Chapter 15: Optimization: Local and Global Extrema

• • 15.1 Critical Points: Local Extrema and Saddle Points
• • 15.2 Optimization
• • 15.3 Constrained Optimization: Lagrange Multipliers

Chapter 16: Integrating Functions of Several Variables

• • 16.1 The Definite Integral of A Function of Two Variables
• • 16.2 Iterated Integrals
• • 16.3 Triple Integrals
• • 16.4 Double Integrals In Polar Coordinates
• • 16.5 Integrals In Cylindrical and Spherical Coordinates
• • 16.6 Applications of Integration To Probability

Chapter 17: Parameterization and Vector Fields

• • 17.1 Parameterized Curves
• • 17.2 Motion, Velocity, and Acceleration
• • 17.3 Vector Fields
• • 17.4 The Flow of A Vector Field

Chapter 18: Line Integrals

• • 18.1 The Idea of A Line Integral
• • 18.2 Computing Line Integrals Over Parameterized Curves
• • 18.3 Gradient Fields and Path-Independent Fields
• • 18.4 Path-Dependent Vector Fields and Green’s Theorem

Chapter 19: Flux Integrals and Divergence

• • 19.1 The Idea of A Flux Integral
• • 19.2 Flux Integrals For Graphs, Cylinders, and Spheres
• • 19.3 The Divergence of A Vector Field
• • 19.4 The Divergence Theorem

Chapter 20: The Curl and Stokes’ Theorem

• • 20.1 The Curl of A Vector Field
• • 20.2 Stokes’ Theorem
• • 20.3 The Three Fundamental Theorems

Chapter 21: Parameters, Coordinates, and Integrals

• • 21.1 Coordinates and Parameterized Surfaces
• • 21.2 Change of Coordinates In A Multiple Integral
• • 21.3 Flux Integrals Over Parameterized Surfaces