Concept Overview

1. Limits and Continuity

• Understanding the concept of a limit.

• Limits at infinity.

• Continuity and the Intermediate Value Theorem.

• Limits of indeterminate forms (L'Hopital’s Rule).

2. Differentiation

• Definition of the derivative.

• Basic differentiation rules (power, product, quotient, chain rule).

• Implicit differentiation.

• Derivatives of trigonometric, exponential, logarithmic, and inverse functions.

• Applications of the derivative:

  • Related rates.

  • Optimization problems.

  • Linear approximation and differentials.

3. Integration

• Definition of the integral (antiderivative).

• Fundamental Theorem of Calculus.

• Techniques of integration:

  • Substitution.

  • Integration by parts.

  • Trigonometric integrals.

  • Partial fractions.

• Improper integrals.

• Numerical methods for integration (e.g., trapezoidal rule, Simpson's rule).

4. Applications of Integrals

• Area under curves.

• Volume of solids of revolution (disk and washer methods).

• Work, fluid pressure, and other physical applications.

• Average value of a function.

• Arc length and surface area.

5. Series

• Sequences and series.

• Convergence and divergence.

• Power series.

• Taylor and Maclaurin series, including their radius and interval of convergence.

• Error bounds for approximating functions using series.

6. Polar Coordinates and Parametric Equations

• Parametric equations and their derivatives.

• Polar coordinates and graphs.

• Calculus with parametric and polar equations (area, length, etc.).

7. Differential Equations

• Solving separable differential equations.

• Slope fields.

• Euler’s method.

8. Applications of Derivatives and Integrals

• Analyzing functions using the first and second derivatives (increasing/decreasing, concavity, and points of inflection).

• Optimization and motion problems.

• Work and related physical applications.

These topics make up the majority of what is assessed on the AP Calculus BC exam. The test consists of multiple-choice questions and free-response questions, with a focus on problem-solving, conceptual understanding, and mathematical reasoning.