The Complete AP Calculus AB Curriculum Roadmap.
Structured strictly according to the College Board Course & Exam Description (CED). Master every single theorem, derivation, and problem type with Socratic AI guidance.
Course Syllabus
All 8 AP Calculus Units in Detail
Every unit is fully seeded in Athena with adaptive daily practice sets, step-by-step whiteboard derivations, and targeted misconception probes.
Unit 1: Limits & Continuity
College Board Course & Exam Description (CED)
Foundations of calculus: intuitive definition of limits, algebraic manipulation, squeeze theorem, and the Intermediate Value Theorem (IVT).
Core Learning Objectives
- Estimating limits from graphs & tables
- Algebraic properties of limits & indeterminate forms
- Continuity at a point & over an interval
- Intermediate Value Theorem (IVT)
Continuity conditions (limit exists, value exists, and they equal) are frequently tested in FRQ justification questions.
Unit 2: Differentiation: Definition & Properties
College Board Course & Exam Description (CED)
Average vs. instantaneous rates of change, derivative as a limit of difference quotients, and basic differentiation rules.
Core Learning Objectives
- Limit definition of the derivative
- Differentiability implies continuity
- Power, sum, product, and quotient rules
- Derivatives of trigonometric & exponential functions
Always recognize difference quotient limit structures as derivative representations without computing the limit manually.
Unit 3: Composite, Implicit, & Inverse Functions
College Board Course & Exam Description (CED)
The Chain Rule, implicit differentiation for curves in the plane, and derivatives of inverse trigonometric functions.
Core Learning Objectives
- Chain rule for composite functions
- Implicit differentiation on curves
- Derivatives of inverse functions
- Logarithmic differentiation
Remember to apply the product rule when differentiating mixed terms like x²y implicitly with respect to x.
Unit 4: Contextual Applications of Differentiation
College Board Course & Exam Description (CED)
Connecting derivatives to physical scenarios: rectilinear motion (position, velocity, acceleration), related rates, and linear approximations.
Core Learning Objectives
- Straight-line particle motion & total distance
- Related rates geometric scenarios
- Local linearity & tangent line approximations
- L'Hôpital's Rule for limits of 0/0 and ∞/∞
Distinguish between displacement (net change) and total distance traveled (integral of speed) in motion problems.
Unit 5: Analytical Applications of Differentiation
College Board Course & Exam Description (CED)
Mean Value Theorem, extreme value theorem, first and second derivative tests, concavity, points of inflection, and optimization.
Core Learning Objectives
- Mean Value Theorem (MVT) & Extreme Value Theorem (EVT)
- Increasing/decreasing intervals & First Derivative Test
- Concavity, points of inflection & Second Derivative Test
- Applied optimization modeling
Always check endpoints when finding global extrema on a closed interval [a, b]. State continuity and differentiability hypotheses before citing MVT.
Unit 6: Integration & Accumulation of Change
College Board Course & Exam Description (CED)
Riemann sums, definite integrals as net signed area, the Fundamental Theorem of Calculus (FTC), and u-substitution techniques.
Core Learning Objectives
- Left, right, midpoint Riemann sums & trapezoidal approximations
- Fundamental Theorem of Calculus (Parts 1 & 2)
- Indefinite integrals & antiderivatives
- Integration by substitution (u-substitution)
FTC Part 1 questions with functions in integration bounds (e.g. d/dx ∫₀^{x²} f(t)dt) require applying the chain rule.
Unit 7: Differential Equations
College Board Course & Exam Description (CED)
Slope fields, verifying solutions, and solving separable differential equations with initial conditions for exponential growth and decay models.
Core Learning Objectives
- Interpreting & sketching slope fields
- Reasoning with differential equations
- Separation of variables
- Exponential growth & decay models
Separation of variables is an absolute staple of FRQ Section II — separating variables correctly earns the first crucial points.
Unit 8: Applications of Integration
College Board Course & Exam Description (CED)
Average value of a function, area between intersecting curves, and volumes of solids with known cross-sections and solids of revolution (disk/washer methods).
Core Learning Objectives
- Average value of a function on an interval
- Area between curves along x and y axes
- Volumes of solids with cross-sections (squares, semicircles)
- Volumes of solids of revolution (disk and washer methods)
Carefully determine whether to integrate with respect to x or y, and watch for inner vs. outer radii in washer method rotations.
AP Calculus AB Exam Format
Understanding the timing and structure of the 3-hour and 15-minute exam is half the battle.
Section I: Multiple Choice
45 Questions · 105 Minutes · 50% of Total Score
Tests algebraic, numerical, and analytical differentiation and integration techniques without electronic aids.
Requires graphing functions, solving equations numerically, finding numerical derivatives, and calculating definite integrals.
Section II: Free Response
6 Questions · 90 Minutes · 50% of Total Score
Real-world modeling, rate-in/rate-out differential scenarios, and particle motion requiring computational graphing.
The core analytical crucible: Riemann sums, FTC justifications, differential equations, and area/volume cross sections.
What an AP 4 or 5 is Actually Worth
Scoring a 4 or 5 on AP Calculus AB earns college credits that can save thousands in tuition and advance your STEM degree trajectory by a full semester.
Extremely Well Qualified
Calculus I (and often Calculus II placement) at 99% of US universities including MIT, Stanford, and Ivy League schools.
Well Qualified
Full Calculus I college credit (typically 4–5 semester credit hours) at almost all public and private universities.
Qualified
Accepted for general math or STEM-introductory requirements at the majority of state universities and colleges.
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