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Comprehensive AP Calculus AB Review

May 10, 2025

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AP Calculus AB Review

Limits and Continuity

  • Limits from Left and Right: A limit exists if it approaches the same value from the left and the right.
  • Evaluating Limits:
    • Plug in the value first; if you get a number, you're done.
    • If 0/0 or ∞/∞, use L'Hôpital's Rule.
    • If a number divided by 0, consider the sign of the small number (positive or negative).
  • Horizontal Asymptotes: Limits at infinity; may differ for positive and negative infinity.
  • Continuity: No breaks or gaps; a function is continuous if the limit approaches the function value at that point.

Derivatives

  • Limit Definition of Derivative: Know the concept even if not heavily tested.
  • Basic Derivative Rules: Product rule, quotient rule, and chain rule are crucial.
  • Differentiability: A function is not differentiable if:
    • It's not continuous.
    • The slopes from the left and right don't match (corner).
    • The tangent line is vertical (cusp).
  • Implicit Differentiation: Derivatives of equations with both x and y; solve for dy/dx.
  • Logarithmic Differentiation: Useful for functions raised to functions; involves taking the natural log.

Applications of Derivatives

  • Tangent Lines: Need a slope and a point.
  • Curve Behavior:
    • Increasing if derivative > 0, decreasing if < 0.
    • Concave up if second derivative > 0, concave down if < 0.
    • Points of Inflection: Where concavity changes.
  • Optimization and Related Rates: Not heavily tested; briefly know the basics.

Integration

  • Riemann Sums: Know how to calculate using a table.
    • Right/left sums as over/underestimates depending on function behavior.
    • Trapezoidal rule related to concavity.
  • Fundamental Theorem of Calculus:
    • Integrate by taking the antiderivative, plug endpoints, subtract.
    • Derivatives of integrals involve evaluating at bounds and applying chain rule.
  • Applications of Integration:
    • Area between curves.
    • Volumes using disk and washer methods.
    • Net change and average value.

Motion and Differential Equations

  • Velocity and Acceleration: Derivatives of position and velocity, respectively.
  • Solving Differential Equations: Know separation of variables, slope fields, and general vs. particular solutions.

Theorems

  • Three Major Theorems:
    • Extreme Value Theorem: Continuous functions have absolute max and min.
    • Intermediate Value Theorem: If a function passes values, it must pass through all intermediate values.
    • Mean Value Theorem: There exists a tangent line that matches the slope of the secant line.

Free Response Questions (FRQ) Types

  • Rate In vs. Rate Out: Calculate net rates.
  • Data Table and Riemann Sums: Estimate and interpret integrals and derivatives.
  • Graph Analysis: Relate graphs of derivatives to original functions.

These notes provide a comprehensive review of the key topics in AP Calculus AB, focusing on limits, derivatives, integrals, and common applications that are likely to appear on the exam.

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