calculus for the ap course table of contents

2 min read 11-01-2025

calculus for the ap course table of contents

This table of contents outlines the key topics covered in a typical Advanced Placement (AP) Calculus course, whether AB or BC. The depth of coverage for each topic will vary depending on the specific curriculum and instructor. This guide serves as a roadmap for students preparing for the AP exam.

Part 1: Limits and Continuity (Foundation for Calculus)

Chapter 1: Introduction to Limits
- Informal Definition of a Limit
- One-Sided Limits
- Limit Laws
- Properties of Limits
- Limits at Infinity
- Infinite Limits
- Determining Limits Graphically and Numerically
- Squeeze Theorem (Sandwich Theorem)
Chapter 2: Continuity
- Definition of Continuity
- Types of Discontinuities (Removable, Jump, Infinite)
- Intermediate Value Theorem (IVT)
- Properties of Continuous Functions
- Applications of Continuity

Part 2: Derivatives (The Language of Change)

Chapter 3: Introduction to Derivatives
- Definition of the Derivative (using limits)
- Interpretations of the Derivative (slope, rate of change)
- Differentiability vs. Continuity
- Power Rule
- Constant Multiple Rule
- Sum and Difference Rules
- Product Rule
- Quotient Rule
- Chain Rule
Chapter 4: Applications of Derivatives
- Related Rates Problems
- Optimization Problems (Finding Maximum and Minimum Values)
- Curve Sketching (using first and second derivatives)
- Mean Value Theorem (MVT)
- Rolle's Theorem
- Increasing and Decreasing Functions
- Concavity and Inflection Points
- L'Hôpital's Rule (for indeterminate forms)
Chapter 5: Implicit Differentiation and Derivatives of Inverse Functions
- Implicit Differentiation
- Derivatives of Inverse Trigonometric Functions
- Logarithmic Differentiation
- Derivatives of Exponential Functions

Part 3: Integrals (Accumulation and Area)

Chapter 6: Introduction to Integrals
- Definite Integrals (Riemann Sums)
- Fundamental Theorem of Calculus (Part 1 and Part 2)
- Indefinite Integrals (Antiderivatives)
- Properties of Definite Integrals
- Substitution Rule (u-substitution)
Chapter 7: Applications of Integrals
- Area Between Curves
- Volumes of Solids of Revolution (Disk/Washer Method, Shell Method)
- Average Value of a Function
- Accumulation Functions
- Integration by Parts
- Partial Fraction Decomposition (for certain integrands)

Part 4: Differential Equations (Modeling Change)

Chapter 8: Introduction to Differential Equations
- Differential Equations and Slope Fields
- Separable Differential Equations
- Exponential Growth and Decay
- Logistic Growth

Part 5: Sequences and Series (Infinite Sums) (AP Calculus BC Only)

Chapter 9: Sequences
- Sequences and Their Limits
- Monotonic Sequences
- Bounded Sequences
Chapter 10: Infinite Series
- Convergence and Divergence of Series
- Tests for Convergence (Integral Test, Comparison Test, Ratio Test, etc.)
- Power Series
- Taylor and Maclaurin Series
- Radius and Interval of Convergence

Part 6: Polar, Parametric, and Vector Functions (AP Calculus BC Only)

Chapter 11: Parametric Equations and Polar Coordinates
- Parametric Equations and Curves
- Polar Coordinates and Graphs
- Calculus with Parametric Equations (Derivatives and Integrals)
- Area in Polar Coordinates
Chapter 12: Vectors and Vector-Valued Functions
- Vectors in Two and Three Dimensions
- Vector-Valued Functions and Their Derivatives
- Motion in Space (Velocity and Acceleration)

This table of contents provides a comprehensive overview. The specific order and depth of coverage may vary. Remember to consult your textbook and course syllabus for the most accurate and detailed information.