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Chapters

This textbook is organized into 23 chapters covering 380 concepts from the AP Calculus AB and BC curriculum.

Chapter Overview

  1. Foundations of Calculus - Prerequisite concepts from precalculus including functions, trigonometry, and coordinate systems.
  2. Understanding Limits - Introduction to limits, notation, one-sided and two-sided limits, and limit laws.
  3. Evaluating Limits - Techniques for evaluating limits including algebraic manipulation and the Squeeze Theorem.
  4. Continuity - Continuity definitions, types of discontinuities, and the Intermediate Value Theorem.
  5. Asymptotes and End Behavior - Vertical and horizontal asymptotes, limits at infinity, and growth rate comparisons.
  6. The Derivative Concept - Rate of change, tangent lines, and the limit definition of the derivative.
  7. Differentiability - Differentiability conditions, non-differentiable points, and estimating derivatives.
  8. Basic Derivative Rules - Power rule, sum/difference rules, and polynomial derivatives.
  9. Product, Quotient, and Transcendental Derivatives - Product and quotient rules plus derivatives of trigonometric, exponential, and logarithmic functions.
  10. The Chain Rule - Chain rule for composite functions and its applications.
  11. Implicit Differentiation - Differentiating implicit equations and the inverse function theorem.
  12. Inverse Function Derivatives - Derivatives of inverse trigonometric functions and logarithmic differentiation.
  13. Higher-Order Derivatives and Motion - Second and higher derivatives with applications to position, velocity, and acceleration.
  14. Related Rates and Linear Approximation - Related rates problems and tangent line approximations.
  15. L'Hospital's Rule and Contextual Applications - L'Hospital's Rule for indeterminate forms and interpreting derivatives in context.
  16. Mean Value Theorem and Extrema - MVT, Rolle's Theorem, EVT, and finding critical points.
  17. Derivative Tests and Concavity - First and second derivative tests, concavity, and inflection points.
  18. Curve Sketching - Complete curve analysis using derivative information.
  19. Optimization - Setting up and solving optimization problems.
  20. Basic Antiderivatives - Antiderivatives, indefinite integrals, and basic integration rules.
  21. Transcendental Integrals - Integrals of trigonometric, exponential, and inverse trigonometric functions.
  22. Riemann Sums and the Fundamental Theorem - Riemann sums, definite integrals, and both parts of the FTC.
  23. Integral Properties and Techniques - Properties of definite integrals, average value, and u-substitution.

How to Use This Textbook

This textbook is designed with concept dependencies in mind. Each chapter builds on concepts from previous chapters, so it's recommended to progress through the chapters in order. Foundation concepts in early chapters are prerequisites for understanding more advanced topics later.

The learning graph ensures that you won't encounter a concept before mastering its prerequisites. If you're reviewing specific topics, check the prerequisites section of each chapter to ensure you have the necessary background.

Note: Each chapter includes a list of concepts covered. Make sure to complete prerequisites before moving to advanced chapters.