AP Calculus ABAP
Limits, derivatives, integrals, and applications.
Overview
AP Calculus AB covers differential and integral calculus including limits, derivatives, integrals, and applications.
Why it matters
It builds a foundation for STEM majors and develops rigorous problem-solving and modeling skills valued by colleges.
Skills you’ll build
- Modeling with functions
- Differentiation & applications
- Integration & accumulation
- Analyzing graphs & motion
Topic Breakdown (Units)
Unit 1: Limits and Continuity
- Limit notation and properties
- One-sided limits and continuity
- Infinite limits and asymptotes
- The Intermediate Value Theorem
Unit 2: Differentiation — Definition and Basic Rules
- Derivative as a limit
- Power, product, quotient rules
- Basic derivative formulas
Unit 3: Differentiation — Composite, Implicit, and Inverse
- Chain rule
- Implicit differentiation
- Derivatives of inverse functions (incl. ln, exp, trig)
Unit 4: Contextual Applications of Differentiation
- Related rates
- Motion analysis
- Optimization
Unit 5: Analytical Applications of Differentiation
- Mean Value Theorem
- Increasing/decreasing & concavity
- Curve sketching
Unit 6: Integration and Accumulation of Change
- Antiderivatives
- Riemann sums
- Definite integrals
Unit 7: Differential Equations
- Slope fields
- Separable differential equations
- Exponential growth/decay
Unit 8: Applications of Integration
- Area between curves
- Volumes of revolution
- Accumulation models
Lessons & Notes
Unit 1: Limits and Continuity
Understand how functions behave near a point and what it means to be continuous. Limits underpin derivative definitions.
- lim f(x)
- continuity
- IVT
- removable vs. jump discontinuities
Unit 2: Differentiation — Definition and Basic Rules
Define the derivative using limits and apply rules to compute derivatives efficiently for polynomials and rational functions.
- f' (x)
- instantaneous rate of change
- tangent line
- derivative rules
Unit 3: Differentiation — Composite, Implicit, and Inverse
Differentiate composite functions and relationships not easily solved for y. Connect derivatives of inverses via reciprocals.
- chain rule
- implicit
- inverse derivative
Unit 4: Contextual Applications of Differentiation
Apply derivatives to model motion, rates, and maximize/minimize quantities in real contexts.
- position/velocity/acceleration
- optimization
- critical points
Unit 5: Analytical Applications of Differentiation
Use derivatives to analyze function behavior, justify conclusions, and draw accurate graphs.
- MVT
- first derivative test
- concavity
- inflection points
Unit 6: Integration and Accumulation of Change
Introduce integration as accumulation and area. Approximate with Riemann sums and compute exact values using FTC.
- ∫ f(x) dx
- Riemann sums
- FTC Part 1 & 2
Unit 7: Differential Equations
Model dynamic systems with differential equations. Use slope fields and separation to solve initial value problems.
- dy/dx
- slope field
- separable DE
- IVP
Unit 8: Applications of Integration
Apply integrals to compute areas, volumes, and accumulated change in physical and statistical contexts.
- washer/shell methods
- area between curves