Content of APEX Calculus, Version 3.0

APEX Calculus comprises 13 chapters; each chapter and section is listed below. This text was written to have the same basic organization as most traditional calculus textbooks. 

The whole text is 850+ pages long. For multiple reasons (lower cost per semester, lower weight per semester, increased likelihood of students bringing their text to class, etc.), APEX Calculus has been broken up into 3 overlapping volumes, roughly along traditional "Calc 1, 2 & 3" lines, as follows:

  • Calculus 1: Chapters 1 through 6.1
    • Limits through Integration by Substitution
  • Calculus 2: Chapters 5 through 8
    • Integration through Sequences and Series
  • Calculus 3: Chapters 9 through 13
    • Curves in the Plane through Multiple Integration

If this division does not match up well with your department's division of Calculus material:

  • Tell me so I know what people are most interested in,
  • Consider editing the source files at GitHub to create your own version, and/or
  • Consider "just making it work as is"; while not ideal, the book is still a great text at a great price.
  1. Limits
    1. An introduction to Limits
    2. Epsilon-Delta Definition of a Limit
    3. Finding Limits Analytically
    4. One-Sided Limits
    5. Continuity
    6. Limits Involving Infinity
  2. Derivatives
    1. Instantaneous Rates of Change: The Derivative
    2. Interpretations of the Derivative
    3. Basic Differentiation Rules
    4. The Product and Quotient Rules
    5. The Chain Rule
    6. Implicit Differentiation
    7. Derivatives of Inverse Functions
  3. The Graphical Behavior of Functions
    1. Extreme Values
    2. The Mean Value Theorem
    3. Increasing and Decreasing Functions
    4. Concavity and the Second Derivative
    5. Curve Sketching
  4. Applications of the Derivative
    1. Newton's Method
    2. Related Rates
    3. Optimization
    4. Differentials
  5. Integration
    1. Antiderivatives and Indefinite Integration
    2. The Definite Integral
    3. Riemann Sums
    4. The Fundamental Theorem of Calculus
    5. Numerical Integration
  6. Techniques of Integration
    1. Substitution
    2. Integration by Parts
    3. Trigonometric Integrals
    4. Trigonometric Substitution
    5. Partial Fraction Decomposition
    6. Hyperbolic Functions
    7. L'Hopital's Rule
    8. Improper Integration
  7. Applications of Integration
    1. Area Between Curves
    2. Volume by Cross-Sectional Area: Disk and Washer Methods
    3. The Shell Method
    4. Arc Length and Surface Area
    5. Work
    6. Fluid Forces
  8. Sequences and Series
    1. Sequences
    2. Infinite Series
    3. Integral and Comparison Tests
    4. Ratio and Root Tests
    5. Alternating Series and Absolute Convergence
    6. Power Series
    7. Taylor Polynomials
    8. Taylor Series
  9. Curves in the Plane
    1. Conic Sections
    2. Parametric Equation
    3. Calculus and Parametric Equations
    4. Introduction to Polar Coordinates
    5. Calculus and Polar Functions
  10. Vectors
    1. Introduction to Cartesian Coordinates in Space
    2. An Introduction to Vectors
    3. The Dot Product
    4. The Cross Product
    5. Lines
    6. Planes
  11. Vector-Valued Functions
    1. Vector-Valued Functions
    2. Calculus and Vector-Valued Functions
    3. The Calculus of Motion
    4. Unit Tangent and Normal Vectors
    5. The Arc Length Parameter and Curvature
  12. Functions of Several Variables
    1. An Introduction to Multivariable Functions
    2. Limits and Continuity of Multivariable Functions
    3. Partial Derivatives
    4. Differentiability and the Total Differential
    5. The Multivariable Chain Rule
    6. Directional Derivatives
    7. Tangent Lines, Normal Lines, and Tangent Planes
    8. Extreme Values
  13. Multiple Integration
    1. Iterated Integration and Area
    2. Double Integration and Volume
    3. Double Integration with Polar Coordinates
    4. Center of Mass
    5. Surface Area
    6. Volume Between Surfaces and Triple Integration
  • Appendix:
    • Solutions to Odd Numbered Exercises
    • Index
    • Useful formulas