AP Calculus AB Review

Limits and Continuity

Limits: Limit exists if it approaches the same value from left and right.
- Plug in value first; if you get a number, you're done.
- Use L'Hôpital's Rule for 0/0 or ∞/∞.
- If number/0, result is infinity; determine sign (positive or negative).
Horizontal Asymptotes: Limits at infinity; check both positive and negative infinity.
Continuity: A function is continuous if the limit equals the function’s value at a point.
- Continuous means no breaks or gaps.

Limit Definition: Understand the limit definitions, though not heavily tested.
Basic Rules: Know differentiation rules: product, quotient, chain rules.
- Symbolic differentiation from tables or given functions.
Differentiability
- Not differentiable if not continuous, if slopes mismatch (corners), vertical tangent (cusp), or undefined.
Implicit Differentiation: Differentiate both sides of an equation; isolate dy/dx.
Logarithmic Differentiation: Use when dealing with functions raised to functions.

Tangent Lines: Need slope and point; slope from derivative, point may be given or calculated.
Curve Behavior: Know increasing (f' > 0), decreasing (f' < 0), concave up (f'' > 0), concave down (f'' < 0).
- Relative extrema by 1st or 2nd derivative test; inflection points where concavity changes.
Optimization & Related Rates: Not heavily tested but know basic setup and solving.

Riemann Sums: Know how to perform and interpret left, right, trapezoid sums.
- Understand overestimates and underestimates based on function's behavior.
Fundamental Theorem of Calculus: Integral as antiderivative evaluation, understand derivatives of integrals.
Applications of Integration
- Area Between Curves
- Volumes: Know disk and washer methods (shell method not covered in AP).
- Net Change: Integrate net rate (inflow minus outflow) for amount changes.
- Motion: Velocity, acceleration, displacement from integrals and derivatives.

Separation of Variables: Solve for general and particular solutions.
- Slope Fields: Draw and interpret slope fields.

Extreme Value Theorem: Absolute max and min exist for continuous functions.
Intermediate Value Theorem: Continuous function passes through all values between known points.
Mean Value Theorem: Slope of tangent equals slope of secant at some point.

Rate In/Out Problems
Data Table Problems: Estimate derivatives, integrals; interpret results with units.
Graph Analysis: Relate derivative graphs to original function.
Differential Equations
Areas and Volumes of Revolution

For students preparing for BC Calculus, additional topics will be covered in a separate session.