AP Calculus AB (College Board): complete guide to the units, the mathematical practices and the exam

College Board AP Calculus AB is designed to be the equivalent of a first-year, two-semester college calculus course. The course is built on a small set of big ideas (change, limits, and the analysis of functions) and a set of mathematical practices, and the content is organized into eight units. There is no coursework; computation, reasoning and justification are examined directly in both sections of the exam. This page is the index: below is a map of the units, the exam structure, and how to study each one.

This library now covers both AP Calculus AB and AP Calculus BC. AP Calculus BC contains everything in AB plus additional BC-only topics: extra integration techniques in Unit 6, Euler's method and logistic models in Unit 7, arc length in Unit 8, and two entire BC-only units, Unit 9 (Parametric Equations, Polar Coordinates, and Vector-Valued Functions) and Unit 10 (Infinite Sequences and Series). The AB pages below apply to BC students too, and the BC-only material is added under its units and in the new Unit 9 and Unit 10 sections.

The eight AP Calculus AB units

The College Board organizes the content into eight units. Each carries an exam weighting (the share of multiple-choice questions it tends to contribute).

Unit 1 Limits and Continuity (10 to 12%)

The meaning and notation of limits, estimating limits from graphs and tables, evaluating limits by algebraic properties and manipulation, the squeeze theorem, types of discontinuity, continuity at a point and over an interval, infinite limits and asymptotes, and the Intermediate Value Theorem.

Unit 2 Differentiation: Definition and Fundamental Properties (10 to 12%)

Average and instantaneous rates of change, the limit definition of the derivative and its notation, estimating derivatives, the connection between differentiability and continuity, and the fundamental rules: power, constant-multiple, sum, difference, product and quotient, plus the derivatives of sine, cosine, the natural exponential, the natural logarithm, and the remaining trigonometric functions.

Unit 3 Differentiation: Composite, Implicit, and Inverse Functions (9 to 13%)

The chain rule, implicit differentiation, and derivatives of inverse and inverse trigonometric functions.

Unit 4 Contextual Applications of Differentiation (10 to 15%)

Rates of change in motion and other contexts, related rates, linear approximation, and L'Hopital's rule for indeterminate forms.

Unit 5 Analytical Applications of Differentiation (15 to 18%)

The Mean Value Theorem, the Extreme Value Theorem, increasing and decreasing behavior, the first and second derivative tests, concavity, and optimization.

Unit 6 Integration and Accumulation of Change (17 to 20%)

Riemann sums, the definite integral, the Fundamental Theorem of Calculus, antiderivatives, and basic integration techniques. BC adds integration by parts, linear partial fractions, and improper integrals.

Unit 7 Differential Equations (6 to 12%)

Slope fields, separable differential equations, and exponential models. BC adds Euler's method and logistic models.

Unit 8 Applications of Integration (10 to 15%)

Average value, area between curves, volumes, and accumulation in context. BC adds arc length and distance traveled.

Unit 9 Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC only, 11 to 12%). Differentiating and integrating parametric and vector-valued functions, planar motion, arc length, and the calculus of polar curves including polar area.

Unit 10 Infinite Sequences and Series (BC only, 17 to 18%). Convergence and divergence of infinite series, the standard convergence tests, alternating series, power series and their intervals of convergence, and Taylor and Maclaurin series with error bounds.

Exam structure

The AP Calculus AB exam is 3 hours 15 minutes and has two equally weighted sections. A graphing calculator is required on the designated parts and forbidden on the others.

• Section I, multiple choice - 45 questions, 1 hour 45 minutes, 50%. A no-calculator part and a calculator part.
• Section II, free response - 6 questions, 1 hour 30 minutes, 50%. A calculator part and a no-calculator part.

The free-response questions are written from the mathematical practices, so they ask you to compute, connect graphical, numerical and analytical representations, justify conclusions with definitions and theorems (such as the IVT), and use correct notation throughout.

How to study AP Calculus AB

AP Calculus AB rewards fluent algebra, conceptual understanding of limits and rates, and clear justification.

1. Work from the Course and Exam Description. Each topic (for example 2.8 The Product Rule) maps to specific learning objectives and essential knowledge statements that exam questions are written from.
2. Learn the practices, not just the answers. Practice justifying with theorems, translating among representations, and writing correct notation, because the FRQs are scored on these skills.
3. Master the no-calculator algebra. Evaluating limits, differentiating from first principles, and applying the differentiation rules by hand all recur on the no-calculator parts.
4. Connect limits to derivatives. The derivative is a limit; seeing Unit 2 as the payoff of Unit 1 makes the rules feel inevitable rather than arbitrary.
5. Rehearse both calculator and no-calculator timing. Know which tools are allowed where, and practice presenting setups and justifications even when the calculator does the arithmetic.

The units, topic by topic

Each topic has a Course-and-Exam-Description-level answer page with worked exam questions and cross-links, plus an overview guide and quiz. Browse the set at /ap/calculus/syllabus. This library covers all eight units of AP Calculus AB in full, plus the BC-only topics (added under Units 6 to 8) and the two BC-only units, Unit 9 and Unit 10:

• Unit 1: can change occur at an instant, defining limits and limit notation, estimating limits from graphs, estimating limits from tables, limits using algebraic properties, limits using algebraic manipulation, selecting procedures for limits, the squeeze theorem, connecting representations of limits, types of discontinuities, continuity at a point, continuity over an interval, removing discontinuities, infinite limits and vertical asymptotes, limits at infinity and horizontal asymptotes, the Intermediate Value Theorem.
• Unit 2: average and instantaneous rates of change, defining the derivative, estimating derivatives at a point, differentiability and continuity, the power rule, constant, sum and difference rules, derivatives of sin, cos, e^x and ln x, the product rule, the quotient rule, derivatives of tan, cot, sec and csc.
• Unit 3: the chain rule, implicit differentiation, differentiating inverse functions, derivatives of inverse trig functions, selecting procedures for derivatives, higher-order derivatives.
• Unit 4: interpreting the derivative in context, straight-line motion, rates of change in applied contexts, introduction to related rates, solving related rates problems, linear approximation and linearization, L'Hospital's rule.
• Unit 5: using the Mean Value Theorem, Extreme Value Theorem and critical points, increasing and decreasing intervals, the first derivative test, the candidates test for absolute extrema, concavity and points of inflection, the second derivative test, sketching graphs from derivatives, connecting f, f-prime and f-double-prime, introduction to optimization, solving optimization problems, behavior of implicit relations.
• Unit 6: accumulations of change, approximating areas with Riemann sums, Riemann sums and the definite integral, accumulation functions and the FTC, behavior of accumulation functions, properties of definite integrals, evaluating definite integrals with the FTC, basic antiderivatives, integration by u-substitution, selecting antidifferentiation techniques. BC only: integration by parts, linear partial fractions, improper integrals.
• Unit 7: modeling with differential equations, verifying solutions, sketching slope fields, reasoning using slope fields, separation of variables, particular solutions and initial conditions, exponential growth and decay models. BC only: Euler's method, logistic models.
• Unit 8: average value of a function, motion with integrals, accumulation in applied contexts, area between curves (functions of x), area between curves (functions of y), area with multiple intersections, volumes by cross section (squares and rectangles), volumes by cross section (triangles and semicircles), disc method about a coordinate axis, disc method about other axes, washer method about a coordinate axis, washer method about other axes. BC only: arc length and distance traveled.
• Unit 9 (BC only): differentiating parametric equations, second derivatives of parametric curves, arc length of parametric curves, differentiating vector-valued functions, integrating vector-valued functions, planar motion problems, polar coordinates and differentiation, area of a polar region, area between two polar curves.
• Unit 10 (BC only): convergent and divergent series, geometric series, the nth term test, the integral test, harmonic series and p-series, comparison tests, the alternating series test, the ratio test, absolute and conditional convergence, alternating series error bound, Taylor polynomial approximations, the Lagrange error bound, radius and interval of convergence, Taylor and Maclaurin series, representing functions as power series.

For the official Course and Exam Description

The College Board publishes the full Course and Exam Description, released free-response questions, scoring guidelines and sample questions at apcentral.collegeboard.org. Always study from the current Course and Exam Description and the College Board's own released exams, because question style and the mathematical practices are board-specific.