Contents

• APEX Calculus
• Contributing Authors
• Preface
• Calculus I
  • 1 Limits
    • 1.0 Chapter Prerequisites
    • Chapter Introduction
    • 1.1 An Introduction To Limits
    • 1.2 Epsilon-Delta Definition of a Limit
    • 1.3 Finding Limits Analytically
    • 1.4 One Sided Limits
    • 1.5 Limits Involving Infinity
    • 1.6 Continuity
    • Chapter Summary
  • 2 Derivatives
    • 2.0 Chapter Prerequisites
    • Chapter Introduction
    • 2.1 Instantaneous Rates of Change: The Derivative
    • 2.2 Interpretations of the Derivative
    • 2.3 Basic Differentiation Rules
    • 2.4 The Product and Quotient Rules
    • 2.5 The Chain Rule
    • 2.6 Implicit Differentiation
  • 3 The Graphical Behavior of Functions
    • Chapter Introduction
    • 3.1 Extreme Values
    • 3.2 The Mean Value Theorem
    • 3.3 Increasing and Decreasing Functions
    • 3.4 Concavity and the Second Derivative
    • 3.5 Curve Sketching
  • 4 Applications of the Derivative
    • Chapter Introduction
    • 4.1 Related Rates
    • 4.2 Optimization
    • 4.3 Differentials
    • 4.4 Newton’s Method
  • 5 Integration
    • Chapter Introduction
    • 5.1 Antiderivatives and Indefinite Integration
    • 5.2 The Definite Integral
    • 5.3 Riemann Sums
    • 5.4 The Fundamental Theorem of Calculus
    • 5.5 Substitution
  • 6 Applications of Integration
    • Chapter Introduction
    • 6.1 Area Between Curves
    • 6.2 Volume by Cross-Sectional Area; Disk and Washer Methods
    • 6.3 The Shell Method
    • 6.4 Work
    • 6.5 Fluid Forces
• Calculus II
  • 7 Inverse Functions and L’Hôpital’s Rule
    • Chapter Introduction
    • 7.1 Inverse Functions
    • 7.2 Derivatives of Inverse Functions
    • 7.3 Exponential and Logarithmic Functions
    • 7.4 Hyperbolic Functions
    • 7.5 L’Hôpital’s Rule
  • 8 Techniques of Integration
    • Chapter Introduction
    • 8.1 Integration by Parts
    • 8.2 Trigonometric Integrals
    • 8.3 Trigonometric Substitution
    • 8.4 Partial Fraction Decomposition
    • 8.5 Integration Strategies
    • 8.6 Improper Integration
    • 8.7 Numerical Integration
  • 9 Sequences and Series
    • Chapter Introduction
    • 9.1 Sequences
    • 9.2 Infinite Series
    • 9.3 The Integral Test
    • 9.4 Comparison Tests
    • 9.5 Alternating Series and Absolute Convergence
    • 9.6 Ratio and Root Tests
    • 9.7 Strategy for Testing Series
    • 9.8 Power Series
    • 9.9 Taylor Polynomials
    • 9.10 Taylor Series
  • 10 Curves in the Plane
    • 10.0 Chapter Prerequisites — Conic Sections
    • Chapter Introduction
    • 10.1 Arc Length and Surface Area
    • 10.2 Parametric Equations
    • 10.3 Calculus and Parametric Equations
    • 10.4 Introduction to Polar Coordinates
    • 10.5 Calculus and Polar Functions
• Calculus III
  • 11 Vectors
    • Chapter Introduction
    • 11.1 Introduction to Cartesian Coordinates in Space
    • 11.2 An Introduction to Vectors
    • 11.3 The Dot Product
    • 11.4 The Cross Product
    • 11.5 Lines
    • 11.6 Planes
  • 12 Vector Valued Functions
    • Chapter Introduction
    • 12.1 Vector-Valued Functions
    • 12.2 Calculus and Vector-Valued Functions
    • 12.3 The Calculus of Motion
    • 12.4 Unit Tangent and Normal Vectors
    • 12.5 The Arc Length Parameter and Curvature
  • 13 Functions of Several Variables
    • Chapter Introduction
    • 13.1 Introduction to Multivariable Functions
    • 13.2 Limits and Continuity of Multivariable Functions
    • 13.3 Partial Derivatives
    • 13.4 Differentiability and the Total Differential
    • 13.5 The Multivariable Chain Rule
    • 13.6 Directional Derivatives
    • 13.7 Tangent Lines, Normal Lines, and Tangent Planes
    • 13.8 Extreme Values
    • 13.9 Lagrange Multipliers
  • 14 Multiple Integration
    • Chapter Introduction
    • 14.1 Iterated Integrals and Area
    • 14.2 Double Integration and Volume
    • 14.3 Double Integration with Polar Coordinates
    • 14.4 Center of Mass
    • 14.5 Surface Area
    • 14.6 Volume Between Surfaces and Triple Integration
    • 14.7 Triple Integration with Cylindrical and Spherical Coordinates
  • 15 Vector Analysis
    • Chapter Introduction
    • 15.1 Introduction to Line Integrals
    • 15.2 Vector Fields
    • 15.3 Line Integrals over Vector Fields
    • 15.4 Flow, Flux, Green’s Theorem and the Divergence Theorem
    • 15.5 Parameterized Surfaces and Surface Area
    • 15.6 Surface Integrals
    • 15.7 The Divergence Theorem and Stokes’ Theorem
• Appendices
  • Solutions To Selected Problems
  • Index
  • Important Formulas