# Alignment: Calculus • Single and Multivariable • Seventh Edition

Use Nagwa in conjunction with your preferred textbook. The recommended lessons from Nagwa for each section of this textbook are provided below. This alignment is not affiliated with, sponsored by, or endorsed by the publisher of the referenced textbook. Nagwa is a registered trademark of Nagwa Limited. All other trademarks and registered trademarks are the property of their respective owners.

• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 13 A Fundamental Tool: Vectors
• 13.1 Displacement Vectors
• 13.2 Vectors in General
• 13.3 The Dot Product
• 13.4 The Cross Product
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 15 Optimization: Local and Global Extrema
• 15.1 Critical Points: Local Extrema and Saddle Points
• 15.2 Optimization
• 15.3 Constrained Optimization: Lagrange Multipliers
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• 20 The Curl and Stokes’ Theorem
• 20.1 The Curl of a Vector Field
• 20.2 Stokes’ Theorem
• 20.3 The Three Fundamental Theorems
• Review Problems
• Projects
• 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
• Review Problems
• Projects
• Appendices
• A Roots, Accuracy, and Bounds
• B Complex Numbers
• C Newton’s Method
• D Vectors in the Plane
• E Determinants