The AP® Calculus AB textbook

8 units in framework order, then the skills the exam tests. Each topic is a notes page you can read on its own, free.

Units

Unit 1: Limits and Continuity

  1. Can Change Happen at an Instant? 1.1
  2. What a Limit Is and How to Write It 1.2
  3. Reading Limits from a Graph 1.3
  4. Estimating Limits from Tables 1.4
  5. Limit Laws 1.5
  6. Limits by Algebra: Factor, Rationalize, Simplify 1.6
  7. Choosing a Way to Find a Limit 1.7
  8. The Squeeze Theorem 1.8
  9. Limits Across Graphs, Tables and Formulas 1.9
  10. Types of Discontinuity 1.10
  11. Continuity at a Point 1.11
  12. Continuity on an Interval 1.12
  13. Removing a Discontinuity 1.13
  14. Infinite Limits and Vertical Asymptotes 1.14
  15. Limits at Infinity and Horizontal Asymptotes 1.15
  16. The Intermediate Value Theorem 1.16

Unit 2: Differentiation: Definition and Fundamental Properties

  1. Average and Instantaneous Rates of Change 2.1
  2. The Derivative and Its Notation 2.2
  3. Estimating a Derivative at a Point 2.3
  4. Differentiability and Continuity 2.4
  5. The Power Rule 2.5
  6. Constant, Sum, Difference and Constant Multiple Rules 2.6
  7. Derivatives of sin x, cos x, e to the x and ln x 2.7
  8. The Product Rule 2.8
  9. The Quotient Rule 2.9
  10. Derivatives of tan, cot, sec and csc 2.10

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

  1. The Chain Rule 3.1
  2. Implicit Differentiation 3.2
  3. Derivatives of Inverse Functions 3.3
  4. Derivatives of Inverse Trigonometric Functions 3.4
  5. Choosing a Differentiation Rule 3.5
  6. Second and Higher Derivatives 3.6

Unit 4: Contextual Applications of Differentiation

  1. The Derivative in Context 4.1
  2. Straight-Line Motion: Position, Velocity, Acceleration 4.2
  3. Rates of Change in Other Contexts 4.3
  4. Introduction to Related Rates 4.4
  5. Solving Related Rates Problems 4.5
  6. Local Linearity and Linearization 4.6
  7. L'Hospital's Rule 4.7

Unit 5: Analytical Applications of Differentiation

  1. The Mean Value Theorem 5.1
  2. Extreme Value Theorem, Global and Local Extrema, Critical Points 5.2
  3. Where a Function Increases or Decreases 5.3
  4. The First Derivative Test 5.4
  5. The Candidates Test 5.5
  6. Concavity and Points of Inflection 5.6
  7. The Second Derivative Test 5.7
  8. Sketching f, f' and f'' 5.8
  9. Connecting f, f' and f'' 5.9
  10. Introduction to Optimization 5.10
  11. Solving Optimization Problems 5.11
  12. Behavior of Implicit Relations 5.12

Unit 6: Integration and Accumulation of Change

  1. Accumulating Change 6.1
  2. Estimating Area with Riemann Sums 6.2
  3. Riemann Sums, Sigma Notation and the Definite Integral 6.3
  4. The Fundamental Theorem and Accumulation Functions 6.4
  5. Behavior of Accumulation Functions 6.5
  6. Properties of Definite Integrals 6.6
  7. The Fundamental Theorem and Definite Integrals 6.7
  8. Antiderivatives and Indefinite Integrals 6.8
  9. Integrating by Substitution 6.9
  10. Integrating by Long Division and Completing the Square 6.10
  11. Choosing an Antidifferentiation Technique 6.14

Unit 7: Differential Equations

  1. Writing Differential Equations 7.1
  2. Verifying Solutions to Differential Equations 7.2
  3. Sketching Slope Fields 7.3
  4. Reasoning with Slope Fields 7.4
  5. General Solutions by Separation of Variables 7.6
  6. Particular Solutions from Initial Conditions 7.7
  7. Exponential Models 7.8

Unit 8: Applications of Integration

  1. Average Value of a Function 8.1
  2. Motion with Integrals 8.2
  3. Accumulation in Context 8.3
  4. Area Between Curves (Functions of x) 8.4
  5. Area Between Curves (Functions of y) 8.5
  6. Area Between Curves That Cross More Than Twice 8.6
  7. Volumes with Square and Rectangle Cross Sections 8.7
  8. Volumes with Triangle and Semicircle Cross Sections 8.8
  9. Disc Method About the Axes 8.9
  10. Disc Method About Other Lines 8.10
  11. Washer Method About the Axes 8.11
  12. Washer Method About Other Lines 8.12

Skills

Math you need

  1. Functions and Their Graphs
  2. Exponent Rules and Rewriting Powers
  3. Algebra for Limits: Factoring and Rationalizing
  4. Trigonometry and the Unit Circle
  5. Exponentials and Logarithms
  6. Inverse Functions
  7. Geometry for Calculus