Concept Overview

1. Limits and Continuity

• Understanding the concept of a limit and applying limit properties.

• Evaluating limits analytically (including one-sided limits and limits at infinity).

• Continuity of functions and the Intermediate Value Theorem.

2. Differentiation

• Definition of the derivative and interpreting it as a rate of change.

• Techniques of differentiation, including the power, product, quotient, and chain rules.

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

• Implicit differentiation.

• Higher-order derivatives.

• Applications of the derivative:

  • Critical points and extrema (local and global).

  • Mean Value Theorem and its application.

  • Concavity, inflection points, and the second derivative test.

  • Optimization problems and related rates.

3. Integration

• Antiderivatives and the Fundamental Theorem of Calculus.

• Techniques of integration, including substitution, integration by parts, and partial fractions.

• Definite integrals and properties of definite integrals.

• Applications of integrals:

  • Area under curves.

  • Average value of a function.

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

  • Work, area between curves, and other applied problems.

4. Applications of Differentiation and Integration

• Slope fields and differential equations.

• Mathematical modeling using calculus.

• Numerical methods (such as the Trapezoidal Rule and Simpson’s Rule).

5. Differential Equations

• Solving basic differential equations, including separable equations and growth/decay models.

• Slope fields and their relationship to differential equations.
  

The AP Calculus AB exam also emphasizes:

• Conceptual Understanding: Understanding the principles behind the mathematics.

• Problem Solving: Applying mathematical concepts to solve real-world problems.

• Communication: Clearly explaining reasoning and work in the context of calculus.