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Contemplanda course syllabus

AP® Calculus BC

A Contemplanda online course.

This course follows the College Board's published AP Calculus BC framework.

AP® Calculus AB and AP Calculus BC are two College Board courses built on a single framework. The framework asks students to grasp the central ideas of calculus and put its methods to use, to move between a function's graph, a table of its values, its formula, and a description in words, and to back each conclusion with an argument that rests on definitions and theorems.

The College Board describes AP Calculus AB as the equivalent of a first-semester college calculus course in differential and integral calculus, and AP Calculus BC as the equivalent of both the first and the second semester of college calculus.

AP Calculus BC includes everything in AP Calculus AB and goes further. Within Units 6 to 8 it adds integration by parts, linear partial fractions, improper integrals, Euler's method, logistic models, and the arc length of a planar curve, and its two further units take up parametric equations, polar coordinates, and vector-valued functions (Unit 9) and infinite sequences and series (Unit 10).

The College Board's framework has ten units. AP Calculus AB covers Units 1 to 8, and AP Calculus BC covers all ten: Units 9 and 10, and topics 6.11, 6.12, 6.13, 7.5, 7.9, and 8.13, are part of BC only. The College Board presents the framework as helping to prepare students for college credit or placement.

Course at a glance

Course title
AP Calculus
Publisher
Contemplanda
Framework
Follows the College Board's published AP Calculus BC framework
Structure
10 teaching units over two semesters (1, 2, 3, 4, and 5 in semester 1; 6, 7, 8, 9, and 10 in semester 2), plus 3 units holding a single unit test for the AB path (6A, 7A, and 8A) and 6 semester exams: 19 units in all
Lessons
131 lessons across the two paths: 112 teaching lessons, 13 unit tests, and 6 semester exams. A student on the BC path takes 125 lessons (111 teaching lessons, 10 unit tests, and 4 semester exams), and a student on the AB path takes 93 lessons (81 teaching lessons, 8 unit tests, and 4 semester exams)
Credit
Not awarded by Contemplanda
Enrollment
Purchasing is not open yet
Course page
www.contemplanda.com/courses/ap-calculus

Credit and enrollment

Contemplanda does not award high school credit. A student who wants credit for the course should ask their own school whether it will accept it.

Under College Board rules, a course may carry the AP designation on a student's transcript only if it is authorized through the College Board's AP Course Audit.

Purchasing is not open yet. Questions about enrolling can go to info@contemplanda.com.

Students who are already enrolled sign in at direct.contemplanda.com.

How the course works

  • Practice comes one question at a time and continues until the student answers four in a row correctly. After a wrong answer, the practice keeps drawing questions.
  • Many questions have hints that open one step at a time. After a wrong answer, the next hint is offered when the question has one. Opening a hint costs nothing and never lowers a mastery level, but the top level, mastered, also requires at least one graded question answered correctly without opening a hint.
  • Answers take many forms: algebraic expressions checked for mathematical equivalence and for the form a question asks for; points placed on a graph; graphs sketched; Riemann rectangles set up on a graph; the region an integral measures, shaded on a graph; tables filled in; along with multiple-choice and numeric answers.
  • Every figure has a written text description.

Recommended preparation

The College Board expects every calculus student to have completed the equivalent of four years of college-preparatory high school mathematics, including algebra, geometry, trigonometry, analytic geometry, and elementary functions, with a solid grounding in algebraic reasoning and algebraic structures.

For AP Calculus AB and BC, a working knowledge of

  • Linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions
  • The properties, composition, algebra, and graphs of functions
  • The language of functions: domain and range, odd and even, periodic, symmetry, zeros, intercepts, and increasing and decreasing
  • The sine and cosine functions as defined from the unit circle, and the values of the trigonometric functions at 0, π/6, π/4, π/3, and π/2 and at their multiples

For AP Calculus BC, also

  • Basic familiarity with sequences and series
  • Some exposure to parametric and polar equations

Units and lessons

Every published lesson, by semester and unit. In the titles below, (BC) marks a lesson, unit test, or exam for AP Calculus BC only, and (AB) marks one for AP Calculus AB only; every other lesson is for both. A student follows one path: the BC path leaves out every (AB) item, and the AB path leaves out every (BC) item, including Units 9 and 10. One lesson has a version for each path: Selecting techniques for antidifferentiation.

Semester 1

1
Limits and continuity
  1. Introducing calculus: can change occur at an instant?
  2. Defining limits and using limit notation
  3. Estimating limit values from graphs
  4. Estimating limit values from tables
  5. Determining limits using algebraic properties of limits
  6. Determining limits using algebraic manipulation
  7. Selecting procedures for determining limits
  8. Determining limits using the squeeze theorem
  9. Connecting multiple representations of limits
  10. Exploring types of discontinuities
  11. Defining continuity at a point
  12. Confirming continuity over an interval
  13. Removing discontinuities
  14. Connecting infinite limits and vertical asymptotes
  15. Connecting limits at infinity and horizontal asymptotes
  16. Working with the intermediate value theorem (IVT)
  17. Unit 1 testUnit test15 questions
2
Differentiation: definition and fundamental properties
  1. Defining average and instantaneous rates of change at a point
  2. Defining the derivative of a function and using derivative notation
  3. Estimating derivatives of a function at a point
  4. Determining when derivatives do and do not exist
  5. Applying the power rule
  6. Derivative rules: constant, sum, difference, and multiple
  7. Derivatives of cos(x), sin(x), e^x, and ln(x)
  8. The product rule
  9. The quotient rule
  10. Finding the derivatives of tan(x), cot(x), sec(x), and csc(x)
  11. Unit 2 testUnit test13 questions
3
Differentiation: composite, implicit, and inverse functions
  1. The chain rule
  2. Implicit differentiation
  3. Differentiating inverse functions
  4. Differentiating inverse trigonometric functions
  5. Procedures for calculating derivatives
  6. Calculating higher-order derivatives
  7. Unit 3 testUnit test12 questions
Exam

Semester 1 midterm

  1. Semester 1 midterm exam28 questions

Covers units 1 to 3.

4
Contextual applications of differentiation
  1. Interpreting the meaning of the derivative in context
  2. Straight-line motion: connecting position, velocity, and acceleration
  3. Rates of change in applied contexts other than motion
  4. Introduction to related rates
  5. Solving related rates problems
  6. Approximating values of a function using local linearity and linearization
  7. Using L'Hôpital's rule for determining limits of indeterminate forms
  8. Unit 4 testUnit test12 questions
5
Analytical applications of differentiation
  1. Using the mean value theorem
  2. Extreme value theorem, global versus local extrema, and critical points
  3. Determining intervals on which a function is increasing or decreasing
  4. Using the first derivative test to determine relative (local) extrema
  5. Using the candidates test to determine absolute (global) extrema
  6. Determining concavity of functions over their domains
  7. Using the second derivative test to determine extrema
  8. Sketching graphs of functions and their derivatives
  9. Connecting a function, its first derivative, and its second derivative
  10. Introduction to optimization problems
  11. Solving optimization problems
  12. Exploring behaviors of implicit relations
  13. Unit 5 testUnit test14 questions
Exam

Semester 1 final

  1. Semester 1 final exam32 questions

Covers all of semester 1, units 1 to 5.

Semester 2

6
Integration and accumulation of change
  1. Exploring accumulations of change
  2. Approximating areas with Riemann sums
  3. Riemann sums, summation notation, and definite integral notation
  4. The fundamental theorem of calculus and accumulation functions
  5. Interpreting the behavior of accumulation functions involving area
  6. Applying properties of definite integrals
  7. The fundamental theorem of calculus and definite integrals
  8. Finding antiderivatives and indefinite integrals: basic rules and notation
  9. Integrating using substitution
  10. Integrating functions using long division and completing the square
  11. Integrating using integration by parts (BC)
  12. Integrating using linear partial fractions (BC)
  13. Evaluating improper integrals (BC)
  14. Selecting techniques for antidifferentiation (BC)
  15. Selecting techniques for antidifferentiation (AB)
  16. Unit 6 test (BC)Unit test15 questions
6A
Unit 6 test (AB)
  1. Unit 6 test (AB)Unit test15 questions
7
Differential equations
  1. Modeling situations with differential equations
  2. Verifying solutions for differential equations
  3. Sketching slope fields
  4. Reasoning with slope fields
  5. Approximating solutions using Euler's method (BC)
  6. Finding general solutions using separation of variables
  7. Finding particular solutions using separation of variables
  8. Exponential models with differential equations
  9. Logistic models with differential equations (BC)
  10. Unit 7 test (BC)Unit test12 questions
7A
Unit 7 test (AB)
  1. Unit 7 test (AB)Unit test12 questions
Exam

Semester 2 midterm (AB)

  1. Semester 2 midterm exam (AB)24 questions

Covers units 6 and 7.

8
Applications of integration
  1. Finding the average value of a function on an interval
  2. Connecting position, velocity, and acceleration of functions using integrals
  3. Using accumulation functions and definite integrals in applied contexts
  4. Finding the area between curves expressed as functions of x
  5. Finding the area between curves expressed as functions of y
  6. Finding the area between curves intersecting at more than two points
  7. Volumes with cross sections: squares and rectangles
  8. Volumes with cross sections: triangles and semicircles
  9. Volume with disc method — revolving around the x- or y-axis
  10. Volume with disc method — revolving around other axes
  11. Volume with washer method — revolving around the x- or y-axis
  12. Volume with washer method — revolving around other axes
  13. The arc length of a smooth, planar curve and distance traveled (BC)
  14. Unit 8 test (BC)Unit test14 questions
8A
Unit 8 test (AB)
  1. Unit 8 test (AB)Unit test14 questions
Exam

Semester 2 midterm (BC)

  1. Semester 2 midterm exam (BC)26 questions

Covers units 6 to 8.

9
Parametric equations, polar coordinates, and vector-valued functions (BC)
  1. Defining and differentiating parametric equations (BC)
  2. Second derivatives of parametric equations (BC)
  3. Finding arc lengths of curves given by parametric equations (BC)
  4. Defining and differentiating vector-valued functions (BC)
  5. Integrating vector-valued functions (BC)
  6. Solving motion problems (BC)
  7. Defining polar coordinates and differentiating in polar form (BC)
  8. Finding the area of a polar region (BC)
  9. Finding the area of the region bounded by two polar curves (BC)
  10. Unit 9 test (BC)Unit test12 questions
10
Infinite sequences and series (BC)
  1. Defining convergent and divergent infinite series (BC)
  2. Working with geometric series (BC)
  3. The nth term test for divergence (BC)
  4. Integral test for convergence (BC)
  5. Harmonic series and p-series (BC)
  6. Comparison tests for convergence (BC)
  7. Alternating series test for convergence (BC)
  8. Ratio test for convergence (BC)
  9. Determining absolute or conditional convergence (BC)
  10. Alternating series error bound (BC)
  11. Finding Taylor polynomial approximations of functions (BC)
  12. Lagrange error bound (BC)
  13. Radius and interval of convergence of power series (BC)
  14. Finding Taylor or Maclaurin series for a function (BC)
  15. Representing functions as power series (BC)
  16. Unit 10 test (BC)Unit test15 questions
Exam

Semester 2 final (AB)

  1. Semester 2 final exam (AB)32 questions

Covers units 6 to 8.

Exam

Semester 2 final (BC)

  1. Semester 2 final exam (BC)32 questions

Covers all of semester 2, units 6 to 10.

Mathematical practices

The College Board organizes the skills of AP Calculus AB and BC into four mathematical practices, which it expects students to develop throughout the course. Every task on the AP Exams draws on these skills, and Practice 4 is assessed only in the free-response section.

1Implementing Mathematical Processes

Finding expressions and values by carrying out mathematical procedures and rules.

1.A
Recognize what question is being asked or what problem needs solving (not assessed).
1.B
Pick out the information that matters for answering the question or solving the problem (not assessed).
1.C
Choose a suitable rule or procedure from the form of the expression at hand, such as the chain rule for a composite function.
1.D
Choose a suitable rule or procedure from how concepts or processes are related, such as rate of change and accumulation, or differentiation and antidifferentiation.
1.E
Carry out the appropriate rules or procedures, with and without technology.
1.F
Explain how an approximation relates to the value it approximates.

2Connecting Representations

Moving mathematical information within one representation or between several.

2.A
Recognize the common structure beneath problems set in different contexts.
2.B
Read mathematical information from graphical, numerical, analytical, and verbal representations.
2.C
Recognize the same mathematical information when it is expressed in another form.
2.D
Recognize how the characteristics or properties of a function appear in different representations.
2.E
Describe how representations of a function and of its derivatives relate to one another.

3Justification

Supporting reasoning and solutions with justified arguments.

3.A
Use technology to form claims and conjectures (not assessed).
3.B
Choose the definition, theorem, or test that applies.
3.C
Check that the hypotheses or conditions of the chosen definition, theorem, or test are met.
3.D
Apply the chosen definition, theorem, or test.
3.E
Give reasons for solutions and conclusions.
3.F
Explain what a mathematical solution means in its context.
3.G
Check that solutions are accurate and appropriate.

4Communication and Notation

Communicating results and solutions with correct notation, language, and mathematical conventions.

4.A
Express ideas in precise mathematical language.
4.B
Attach the correct units of measure.
4.C
Write with correct mathematical symbols and notation, including the notations for a derivative.
4.D
Draw graphs with appropriate technique.
4.E
Round results appropriately.

Three big ideas run through the College Board's framework: change, limits, and analysis of functions. Derivatives describe rates of change and definite integrals describe net change, and the Fundamental Theorem of Calculus ties the two together.

Assessment

Each of the ten teaching units closes with a unit test. On the AB path, the unit tests in Units 6A, 7A, and 8A take the place of the (BC) tests of Units 6, 7, and 8. A student takes 10 unit tests on the BC path and 8 on the AB path.

Each semester has a midterm exam and a final exam. In semester 2, the midterm and the final each come in an (AB) version and a (BC) version, one for each path.

Unit tests and semester exams are graded automatically, and all graded work counts toward the student's mastery record, which shows each skill as attempted, familiar, proficient, or mastered.

Contemplanda publishes no grade weighting for this course. A school that decides to grant credit for it also decides how the work counts toward a grade.

Academic integrity

Work in the course must be the student's own.

Unit tests and semester exams are closed book: no notes, no lesson look-ups, and no outside help. Each question is worked on paper and the answer entered in the course.

Answer keys for graded questions are never sent to the student's device.

Graphing calculator

The unit tests and semester exams are written to be worked without a calculator.

The College Board treats a graphing calculator as an integral part of AP Calculus AB and BC, and one is required on two parts of each AP Exam. It also expects students to explore ideas with technology, a graphing calculator above all, across the whole course.

The College Board expects a calculator used on the exam to graph a function in any viewing window, find zeros of functions, and compute a derivative and a definite integral numerically. When a free-response answer uses one of those capabilities, the student must show the setup, such as the equation to be solved or the derivative or integral to be computed, together with the calculator's result; any other calculator feature needs the mathematical steps that produce the result.

A student preparing for the AP Exam should also practice with a graphing calculator, such as the Desmos graphing calculator, the kind built into the exam.

The AP Exam

The AP Calculus AB and AP Calculus BC Exams share one format. Each lasts 3 hours and 10 minutes and has two sections, with 42 multiple-choice questions and 6 free-response questions. The AB Exam assesses Units 1 to 8, and the BC Exam assesses all ten units.

Section I, Part A: multiple choice

Questions
29
Time
62 minutes
Share of score
35%
Calculator
Not permitted

Section I, Part B: multiple choice

Questions
13
Time
38 minutes
Share of score
15%
Calculator
Graphing calculator required

Section II, Part A: free response

Questions
2
Time
30 minutes
Share of score
16.7%
Calculator
Graphing calculator required

Section II, Part B: free response

Questions
4
Time
60 minutes
Share of score
33.3%
Calculator
Not permitted

On the exam, a Desmos graphing calculator is built into the testing app, available only in the parts that require a calculator, and students may also bring up to two handheld graphing calculators from the College Board's approved list.

Three of the six free-response questions are common to the AB and BC Exams and assess AB content, and each exam includes at least two questions set in a real-world context.

Students answer the multiple-choice questions and read the free-response questions in the College Board's digital testing app, and they handwrite their free-response answers in a paper booklet.

Scores are reported on a scale of 1 to 5. A student who takes the AP Calculus BC Exam also receives a Calculus AB subscore on the same scale, which reports how the student did on the AB material, about 60 percent of the exam.

This is the exam format in effect from the May 2027 exams.

The AP Exams are given in schools in May, and a student may not take both the AB and the BC Exam in the same year.

A student who takes the AP Exam at a school other than the one where they take the course, or who takes the exam without taking the course, gets a join code from the AP coordinator at the school where they will test and joins an exam-only section there.

Materials and technology

Where the course runs
In a web browser, through Contemplanda Direct
Graphing calculator
Not needed for the unit tests or semester exams; recommended for AP Exam practice outside this course
Textbook
None required. The course's own lessons are the text.
Course framework
apcentral.collegeboard.org/courses/ap-calculus-bc

Published at www.contemplanda.com/courses/ap-calculus/syllabus

Units, lessons, and counts in this syllabus are generated from the published course.

Questions: info@contemplanda.com

© 2013–2026 Ethan Alvarée. All rights reserved.

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