AP Calculus AB · Mathematics
AP Calculus AB: free lessons and course outline
A full year of college Calculus I (limits, derivatives, and integrals) taught unit by unit to the College Board course framework, one lesson for each required topic (closely related topics share a lesson). Every lesson quotes the required content, works examples with the numbers checked, and adds figures where a picture makes a problem clear. Work through the lessons at your own pace, and book a tutor the moment a topic doesn't click. Exam format from 2027: this is a hybrid digital exam: multiple choice in Bluebook, free response handwritten in a paper booklet. Work the practice problems on paper and check your steps against the worked answer, the way you will sit the free-response section in May.
AP Calculus AB follows the eight units of the College Board course framework, in 72 lessons. Each unit below shows its share of the multiple-choice section, so you can see where to spend your time: Units 5 and 6 together are roughly a third of the exam.
- U1Limits and Continuity10–15%
- U2Differentiation: Definition and Fundamental Properties10–15%
- U3Differentiation: Composite, Implicit, and Inverse Functions5–10%
- U4Contextual Applications of Differentiation10–15%
- U5Analytical Applications of Differentiation15–20%
- U6Integration and Accumulation of Change15–20%
- U7Differential Equations5–10%
- U8Applications of Integration10–15%
Writing the free-response questions? Read the Calculus FRQ guide.
Every lesson, unit by unit
U1 is free to read below. The other units open with a free month or a paid plan.
U1 Limits and Continuity 10–15% of the examFREE
- 1.1 Introducing calculus: can change occur at an instant?
- 1.2 Defining limits and using limit notation
- 1.3 Estimating limit values from graphs and tables
- 1.4 Determining limits using algebraic properties of limits
- 1.5 Determining limits using algebraic manipulation
- 1.6 Determining limits using the squeeze theorem
- 1.7 Selecting procedures and connecting representations
- 1.8 Types of discontinuities and continuity at a point
- 1.9 Confirming continuity over an interval
- 1.10 Removing discontinuities
- 1.11 Connecting infinite limits and vertical asymptotes
- 1.12 Connecting limits at infinity and horizontal asymptotes
- 1.13 Working with the Intermediate Value Theorem (IVT)
U2 Differentiation: Definition and Fundamental Properties 10–15% of the exam
- 2.1 Average and instantaneous rates of change at a point
- 2.2 The derivative as a function and derivative notation
- 2.3 Estimating derivatives of a function at a point
- 2.4 Connecting differentiability and continuity
- 2.5 The power rule and the sum and constant multiple rules
- 2.6 Derivatives of sin x, cos x, e^x, and ln x
- 2.7 The product rule
- 2.8 The quotient rule
- 2.9 Derivatives of tangent, cotangent, secant, and cosecant
U3 Differentiation: Composite, Implicit, and Inverse Functions 5–10% of the exam
- 3.1 The chain rule
- 3.2 Implicit differentiation
- 3.3 Differentiating inverse functions
- 3.4 Differentiating inverse trigonometric functions
- 3.5 Selecting procedures for calculating derivatives
- 3.6 Calculating higher-order derivatives
U4 Contextual Applications of Differentiation 10–15% of the exam
- 4.1 Interpreting the meaning of the derivative in context
- 4.2 Straight-line motion: position, velocity, and acceleration
- 4.3 Rates of change in applied contexts other than motion
- 4.4 Introduction to related rates
- 4.5 Solving related rates problems
- 4.6 Local linearization
- 4.7 Using L'Hospital's rule for determinate forms
U5 Analytical Applications of Differentiation 15–20% of the exam
- 5.1 Using the Mean Value Theorem
- 5.2 Extreme Value Theorem and critical points
- 5.3 Intervals of increase and decrease
- 5.4 First derivative test for relative extrema
- 5.5 Candidates test for absolute extrema
- 5.6 Determining concavity of functions over their domains
- 5.7 Using the second derivative test to determine extrema
- 5.8 Sketching graphs of functions and their derivatives
- 5.9 Connecting a function and its derivatives
- 5.10 Introduction to optimization problems
- 5.11 Solving optimization problems
- 5.12 Exploring behaviors of implicit relations
U6 Integration and Accumulation of Change 15–20% of the exam
- 6.1 Exploring accumulations of change
- 6.2 Approximating areas with Riemann sums
- 6.3 Riemann sums and definite integral notation
- 6.4 The Fundamental Theorem and accumulation functions
- 6.5 Behavior of accumulation functions
- 6.6 Applying properties of definite integrals
- 6.7 The Fundamental Theorem of Calculus and definite integrals
- 6.8 Antiderivatives and indefinite integrals
- 6.9 Integrating using substitution
- 6.10 Long division and completing the square
- 6.11 Selecting techniques for antidifferentiation
U7 Differential Equations 5–10% of the exam
- 7.1 Modeling situations with differential equations
- 7.2 Verifying solutions for differential equations
- 7.3 Sketching and reasoning with slope fields
- 7.4 Finding general solutions using separation of variables
- 7.5 Particular solutions with separation of variables
- 7.6 Exponential models with differential equations
U8 Applications of Integration 10–15% of the exam
- 8.1 Finding the average value of a function on an interval
- 8.2 Position, velocity and acceleration with integrals
- 8.3 Accumulation in applied contexts
- 8.4 Finding the area between curves expressed as functions of x
- 8.5 Area between curves in y and with several crossings
- 8.6 Volumes with known cross sections
- 8.7 Volume with the disc method: revolving around axes
- 8.8 Volume with the washer method: revolving around axes
Keep going in the full AP Calculus AB course
All of U1 (Limits and Continuity) is free to everyone, with no account. The course has 72 lessons, each with worked examples and practice problems that come with full solutions. A free account can also start a free month of the whole course, with no card.
Open this lesson in the courseSee the whole course