AP Calculus AB

What the course covers

8 units, with exam weights

1. Unit 1

   Limits and Continuity

   10-15% of examFree
   What a limit is and how to find one from a graph, a table, or an expression; the three kinds of discontinuity; asymptotes as infinite limits and limits at infinity; and the Intermediate Value Theorem, the first theorem you must justify in writing.
   16 topics

   1. 1.1Introducing Calculus: Can Change Occur at an Instant?
   2. 1.2Defining Limits and Using Limit Notation
   3. 1.3Estimating Limit Values from Graphs
   4. 1.4Estimating Limit Values from Tables
   5. 1.5Determining Limits Using Algebraic Properties of Limits
   6. 1.6Determining Limits Using Algebraic Manipulation
   7. 1.7Selecting Procedures for Determining Limits
   8. 1.8Determining Limits Using the Squeeze Theorem
   9. 1.9Connecting Multiple Representations of Limits
   10. 1.10Exploring Types of Discontinuities
   11. 1.11Defining Continuity at a Point
   12. 1.12Confirming Continuity over an Interval
   13. 1.13Removing Discontinuities
   14. 1.14Connecting Infinite Limits and Vertical Asymptotes
   15. 1.15Connecting Limits at Infinity and Horizontal Asymptotes
   16. 1.16Working with the Intermediate Value Theorem (IVT)

2. Unit 2

   Differentiation: Definition and Fundamental Properties

   10-15% of exam
   The derivative as the limit of average rates of change, the tangent line, estimating derivatives from tables and graphs, the link between differentiability and continuity, and the basic rules: power, sum, product, quotient, and the derivatives of the core trig, exponential, and log functions.
   10 topics

   1. 2.1Defining Average and Instantaneous Rates of Change at a Point
   2. 2.2Defining the Derivative of a Function and Using Derivative Notation
   3. 2.3Estimating Derivatives of a Function at a Point
   4. 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
   5. 2.5Applying the Power Rule
   6. 2.6Derivative Rules: Constant, Sum, Difference, and Constant Multiple
   7. 2.7Derivatives of cos x, sin x, e^x, and ln x
   8. 2.8The Product Rule
   9. 2.9The Quotient Rule
   10. 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

3. Unit 3

   Differentiation: Composite, Implicit, and Inverse Functions

   5-10% of exam
   The chain rule and everything built on it: implicit differentiation, derivatives of inverse functions and inverse trig functions, choosing a strategy for any expression, and second and higher derivatives, including $\frac{d^2y}{dx^2}$ of an implicit relation.
   6 topics

   1. 3.1The Chain Rule
   2. 3.2Implicit Differentiation
   3. 3.3Differentiating Inverse Functions
   4. 3.4Differentiating Inverse Trigonometric Functions
   5. 3.5Selecting Procedures for Calculating Derivatives
   6. 3.6Calculating Higher-Order Derivatives

4. Unit 4

   Contextual Applications of Differentiation

   10-15% of exam
   Reading the derivative as a rate with units in context, straight-line motion, related rates, local linear approximation and whether it over- or underestimates, and L'Hospital's Rule for the forms 0/0 and $\infty/\infty$.
   7 topics

   1. 4.1Interpreting the Meaning of the Derivative in Context
   2. 4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration
   3. 4.3Rates of Change in Applied Contexts Other Than Motion
   4. 4.4Introduction to Related Rates
   5. 4.5Solving Related Rates Problems
   6. 4.6Approximating Values of a Function Using Local Linearity and Linearization
   7. 4.7Using L'Hospital's Rule for Determining Limits of Indeterminate Forms

5. Unit 5

   Analytical Applications of Differentiation

   15-20% of exam
   The Mean Value and Extreme Value Theorems, critical points, increasing and decreasing intervals, the first and second derivative tests, the candidates test, concavity and inflection points, connecting the graphs of $f$, $f'$, and $f''$, optimization, and analyzing implicitly defined curves. The heaviest unit on the exam, and the core of most justification points.
   12 topics

   1. 5.1Using the Mean Value Theorem
   2. 5.2Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
   3. 5.3Determining Intervals on Which a Function Is Increasing or Decreasing
   4. 5.4Using the First Derivative Test to Determine Relative (Local) Extrema
   5. 5.5Using the Candidates Test to Determine Absolute (Global) Extrema
   6. 5.6Determining Concavity of Functions over Their Domains
   7. 5.7Using the Second Derivative Test to Determine Extrema
   8. 5.8Sketching Graphs of Functions and Their Derivatives
   9. 5.9Connecting a Function, Its First Derivative, and Its Second Derivative
   10. 5.10Introduction to Optimization Problems
   11. 5.11Solving Optimization Problems
   12. 5.12Exploring Behaviors of Implicit Relations

6. Unit 6

   Integration and Accumulation of Change

   15-20% of exam
   Accumulation as area under a rate, Riemann and trapezoidal sums from tables, the definite integral as a limit, both parts of the Fundamental Theorem of Calculus, accumulation functions $g(x)=\int_a^x f(t)\,dt$, properties of integrals, and antiderivatives by basic rules, substitution, long division, and completing the square. (AB skips BC topics 6.11-6.13.)
   11 topics

   1. 6.1Exploring Accumulations of Change
   2. 6.2Approximating Areas with Riemann Sums
   3. 6.3Riemann Sums, Summation Notation, and Definite Integral Notation
   4. 6.4The Fundamental Theorem of Calculus and Accumulation Functions
   5. 6.5Interpreting the Behavior of Accumulation Functions Involving Area
   6. 6.6Applying Properties of Definite Integrals
   7. 6.7The Fundamental Theorem of Calculus and Definite Integrals
   8. 6.8Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
   9. 6.9Integrating Using Substitution
   10. 6.10Integrating Functions Using Long Division and Completing the Square
   11. 6.14Selecting Techniques for Antidifferentiation

7. Unit 7

   Differential Equations

   5-10% of exam
   Writing and verifying differential equations, drawing and reading slope fields, separation of variables with an initial condition (the five-point free-response part), and exponential growth and decay from $\frac{dy}{dt}=ky$. Light on the multiple-choice section but almost always a full free-response question. (AB skips BC topics 7.5 and 7.9.)
   7 topics

   1. 7.1Modeling Situations with Differential Equations
   2. 7.2Verifying Solutions for Differential Equations
   3. 7.3Sketching Slope Fields
   4. 7.4Reasoning Using Slope Fields
   5. 7.6Finding General Solutions Using Separation of Variables
   6. 7.7Finding Particular Solutions Using Initial Conditions and Separation of Variables
   7. 7.8Exponential Models with Differential Equations

8. Unit 8

   Applications of Integration

   10-15% of exam
   Average value, position and total distance from velocity, accumulation of amounts in context (initial value plus net change, rate in minus rate out), area between curves in $x$ or $y$, and volumes by cross sections, discs, and washers about the axes and other lines. (AB skips BC topic 8.13.)
   12 topics

   1. 8.1Finding the Average Value of a Function on an Interval
   2. 8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals
   3. 8.3Using Accumulation Functions and Definite Integrals in Applied Contexts
   4. 8.4Finding the Area Between Curves Expressed as Functions of x
   5. 8.5Finding the Area Between Curves Expressed as Functions of y
   6. 8.6Finding the Area Between Curves That Intersect at More Than Two Points
   7. 8.7Volumes with Cross Sections: Squares and Rectangles
   8. 8.8Volumes with Cross Sections: Triangles and Semicircles
   9. 8.9Volume with Disc Method: Revolving Around the x- or y-Axis
   10. 8.10Volume with Disc Method: Revolving Around Other Axes
   11. 8.11Volume with Washer Method: Revolving Around the x- or y-Axis
   12. 8.12Volume with Washer Method: Revolving Around Other Axes

The exam, part by part

4 parts, 3 h 10 min in all.

• Section I, Part A: Multiple Choice (no calculator)

  Questions

  29

  Time

  1 h 2 min

  Weight

  35%

  No calculator

  Format details

  Four answer choices (A-D), all stand-alone items (no shared-stimulus sets). Analytical, graphical, tabular, and verbal representations of algebraic, exponential, logarithmic, trigonometric, and general functions. Taken on screen in Bluebook. New count and timing for May 2027 (was 30 questions in 60 minutes).

• Section I, Part B: Multiple Choice (graphing calculator required)

  Questions

  13

  Time

  38 min

  Weight

  15%

  Calculator allowed

  Format details

  Four answer choices (A-D). A graphing calculator is required for some items (numerical derivatives and integrals, solving equations, graphing); others need none. The Bluebook built-in Desmos graphing calculator or an approved handheld may be used. Numeric choices are usually given to three decimal places. New count and timing for May 2027 (was 15 questions in 45 minutes).

• Section II, Part A: Free Response (graphing calculator required)

  Questions

  2

  Time

  30 min

  Weight

  16.7%

  Calculator allowed

  Format details

  Task types: Modeling, Area and Volume, Particle Motion

  Two 9-point questions, about 15 minutes each. Questions are viewed in Bluebook and answered by hand in a paper booklet. Show the setup (the integral, derivative, or equation) before any calculator result; calculator syntax is not accepted as work. 2023-2026 Part A pairs: Modeling + Particle Motion (2023, 2024), Modeling + Area and Volume (2025, 2026).

• Section II, Part B: Free Response (no calculator)

  Questions

  4

  Time

  1 h

  Weight

  33.3%

  No calculator

  Format details

  Task types: Graphical Analysis, Differential Equations, Analytic Functions, Particle Motion, Modeling, Area and Volume

  Four 9-point questions, about 15 minutes each. Almost every year includes a question built on the graph of f' (or of f with an accumulation function) and a differential equations question; the other two slots rotate among particle motion, implicit curves, table-of-values derivative questions, tabular modeling, and area/volume.

Skills the exam scores

• MP1Implementing Mathematical Processes

  Choose and carry out the right rule or procedure, with and without technology: pick a method from the form of an expression (chain rule for a composite) or from a relationship between ideas (rate and accumulation), apply it accurately, and explain how an approximation relates to the true value (over or under). CED skills 1.C-1.F. Weighted 50-70% of multiple choice and 35-60% of free response.

• MP2Connecting Representations

  Read mathematical information from graphs, tables, formulas, and words, re-express it in another form, and connect the features of $f$, $f'$, and $f''$ across representations (a zero of $f'$ on a graph is a candidate extremum of $f$). CED skills 2.A-2.E. Weighted 15-30% of multiple choice and 10-20% of free response.

• MP3Justification

  Pick the right definition, theorem, or test (IVT, MVT, EVT, first or second derivative test, candidates test, FTC), confirm its hypotheses, apply it, and state the conclusion; give reasons, interpret answers in context with units, and check that results are reasonable. CED skills 3.B-3.G. Weighted 10-20% of multiple choice and 35-60% of free response.

• MP4Communication and Notation

  Use precise mathematical language and notation ($f'(x)$, $\frac{dy}{dx}$, limits, definite integrals with $dx$), correct units, sound graphing, and three-decimal rounding. CED skills 4.A-4.E. Assessed only on free response, 10-25% of that section.