AP Calculus BC · Mathematics

AP Calculus BC: free lessons and course outline

Everything in Calculus AB plus the BC extensions (integration by parts and partial fractions, Euler's method and logistic growth, parametric, polar, and vector-valued functions, and infinite series) taught unit by unit to the College Board 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. The BC exam also reports an AB subscore. 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.

3H 10M EXAM 42 MCQ 6 FRQ 100 LESSONS 100 PRACTICE PROBLEMS PREREQ: CALCULUS AB

AP Calculus BC follows the ten units of the College Board course framework, in 100 lessons. Units 1–8 share their content with Calculus AB (with BC-only topics added in Units 6, 7 and 8); Units 9 and 10 are BC-only. Notice the weights: Units 6 and 10 alone are roughly a third of the exam, and the AB material is weighted far lighter than it is on the AB exam.

U1Limits and Continuity5–10%
U2Differentiation: Definition and Fundamental Properties5–10%
U3Differentiation: Composite, Implicit, and Inverse Functions5–10%
U4Contextual Applications of Differentiation5–10%
U5Analytical Applications of Differentiation10–15%
U6Integration and Accumulation of Change15–20%
U7Differential Equations5–10%
U8Applications of Integration5–10%
U9Parametric Equations, Polar Coordinates, and Vector-Valued Functions10–15%
U10Infinite Sequences and Series15–20%

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 5–10% of the examFREE

U2 Differentiation: Definition and Fundamental Properties 5–10% 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 5–10% 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 10–15% 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 Integrating using integration by parts
6.12 Integrating using linear partial fractions
6.13 Evaluating improper integrals
6.14 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 Approximating solutions using Euler's method
7.5 Finding general solutions using separation of variables
7.6 Particular solutions with separation of variables
7.7 Exponential models with differential equations
7.8 Logistic models with differential equations

U8 Applications of Integration 5–10% 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
8.9 Arc length of a smooth planar curve

U9 Parametric Equations, Polar Coordinates, and Vector-Valued Functions 10–15% of the exam

9.1 Parametric equations and their derivatives
9.2 Finding arc lengths of curves given by parametric equations
9.3 Defining and differentiating vector-valued functions
9.4 Integrating vector-valued functions
9.5 Motion problems with parametric and vector functions
9.6 Defining polar coordinates and differentiating in polar form
9.7 Area of a polar region bounded by one curve
9.8 Finding the area of the region bounded by two polar curves

U10 Infinite Sequences and Series 15–20% of the exam

10.1 Defining convergent and divergent infinite series
10.2 Working with geometric series
10.3 The nth term test for divergence
10.4 The integral test, harmonic series, and p-series
10.5 Comparison tests for convergence
10.6 The alternating series test for convergence
10.7 The ratio test for convergence
10.8 Determining absolute or conditional convergence
10.9 Alternating series error bound
10.10 Finding Taylor polynomial approximations of functions
10.11 Lagrange error bound
10.12 Radius and interval of convergence of power series
10.13 Finding Taylor or Maclaurin series for a function
10.14 Representing functions as power series

Keep going in the full AP Calculus BC course

All of U1 (Limits and Continuity) is free to everyone, with no account. The course has 100 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.

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