# Alignment: Calculus • Single and Multivariable • Seventh Edition

• 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 Optional Preview of the Formal Definition of a Limit
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 13 A Fundamental Tool: Vectors
• 13.1 Displacement Vectors
• 13.2 Vectors in General
• 13.3 The Dot Product
• 13.4 The Cross Product
• 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
• 15 Optimization: Local and Global Extrema
• 15.1 Critical Points: Local Extrema and Saddle Points
• 15.2 Optimization
• 15.3 Constrained Optimization: Lagrange Multipliers
• 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
• 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
• 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
• 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
• 20 The Curl and Stokes’ Theorem
• 20.1 The Curl of a Vector Field
• 20.2 Stokes’ Theorem
• 20.3 The Three Fundamental Theorems
• 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

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